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Experiments in Transport Phenomena A Manual for Use in Chemical Engineering 324, Transport Phenomena Laboratory By E.J. Crosby Revised by Thomas W. Chapman Updated by Rafael Chavez, 2002 Chemical Engineering Department University of Wisconsin-Madison Madison, Wisconsin 53706 Copyright © 1999 by T.W. Chapman

ChE 324 Lab Manual

Preface Page I-1

Chemical Engineering Department University of Wisconsin-Madison

ChE 324 Transport Phenomena Laboratory

Experiments in

Transport Phenomena by E.J. Crosby revised by T.W. Chapman updated by Rafael Chavez

Preface Chemical Engineering 324, Transport Phenomena Laboratory, is an important course in the chemical engineering curriculum. It is intended to accomplish three objectives: a. to demonstrate experimentally the major principles of the subject,

Transport Phenomena, which are presented in the lecture course, ChE 320; b. to develop skills in engineering experimentation and data analysis; and,

c. to provide instruction and practice in methods of technical communication.

The textbook Transport Phenomena by Bird, Stewart, and Lightfoot (2002) is the main source for the theoretical aspects of most of the topics treated in the laboratory. Generally the notation used in this manual will be the same as that used in that book.

In this revision of the manual, the references were updated to the Second Edition of the Transport Phenomena book. Some content was also added or modified to make the manual more self-contained and easier to use. The book by William Pfeiffer (2001) provides guidance regarding technical-communication skills. Also useful are the books by Beer and McMurrey (1997), M. Alley (1996), and the web pages of the technical-communication courses taught in Engineering

ChE 324 Lab Manual

Preface Page I-2

Professional Development (http://www.engr.wisc.edu/epd/tc/). The elegant little book by Strunk and White (1959) is an extremely valuable source of advice for writers.

During the semester the students work in small groups, performing weekly experiments. Individual reports are prepared and submitted at the subsequent class session. Each week students will be asked to prepare either a formal technical report or a shorter technical memo. Each student will also make one oral presentation. This lab manual provides general guidelines regarding the operation of the course as well as descriptions of each of the laboratory experiments. Students are expected to review the subject of each week's laboratory prior to the class in order to understand better the significance of the lab exercises. Also, a plan for data collection and analysis should be prepared ahead of time. Planning prepares the students to complete many of the necessary calculations during the lab period. Short quizzes may be given at the beginning of the lab sessions to confirm such preparation. The course will be much less time consuming for students who can complete most of the data analysis during the lab session. Each week the assigned experiment is put into context by a hypothetical memo written by a fictitious industrial supervisor to his engineering staff. These memos, included in Appendix 15, are intended to give the students a practical motivation for conducting the assigned study. With a concrete context, the students should find it easier to write a realistic and relevant report rather than simply commenting on whether their data agreed with “theory”, that is, what they perceive as the “right” answer because it comes from a textbook. Thus, student reports should be written in response to these assignment memos. References Alley, M., The Craft of Scientific Writing, 3rd edition, Springer-Verlag, New York (1996) Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, 2nd.Edition, John Wiley & Sons, New York (2002) Beer, D., and D. McMurrey, A Guide to Writing as an Engineer, John Wiley & Sons, New York (1997) Pfeiffer, William S., Pocket Guide to Technical Writing, 2nd Edition, Prentice Hall, Upper Saddle River, New Jersey (2001) Strunk, W., Jr., and E.B. White, The Elements of Style, 3rd edition, Macmillan, New York (1972)

ChE 324 Lab Manual

Table of Contents Page I-3

Chemical Engineering Department University of Wisconsin-Madison

ChE 324 Transport Phenomena Laboratory

Experiments in Transport Phenomena by E.J. Crosby revised by T.W. Chapman updated by R. Chavez Contents

Preface

I-1

Introduction

I-5

I. The Need for Experimentation in Chemical Engineering I-6

II. Course Guidelines for ChE 324 I-8

III. Use of Computers in ChE 324 I-11

IV. Safety in the Laboratory I-13

V. Technical Communication I-14

A. Format of Formal Reports I-14

B. Content of Memos I-19

C. Oral Reports I-20

VI. Experimental Design and Statistical Analysis of Data I-23

Part A. Measurement of Transport Properties A.1 Viscosity of Newtonian Liquids A.1-1A.2 Thermal Conductivity of Solids A.2-1

ChE 324 Lab Manual

Table of Contents Page I-4

Part. B. Measurement of Profiles of Velocity, Temperature, and Concentration B.1 Velocity Profiles in Steady Turbulent Flow B.1-1B.2 Temperature Profiles in Solid Rods B.2-1B.3 Concentration Profiles in a Stagnant Film B.3-1 Part C. Measurement of Transport Coefficients C.1 Friction Factors for Flow in Circular Tubes C.1-1C.2. Heat-transfer Coefficients in Circular Tubes C.2-1 Part D. Analysis of Macroscopic Systems D.2 Efflux Time for a Tank with Exit Pipe D.2-1D.3 Heating Liquids in Tank Storage D.3-1 Appendices Appendix 1. Sample Laboratory Report Ap.1-1Appendix 2. (Suppressed) Appendix 3. (Suppressed) Appendix 4. Density and Viscosity of Aqueous Sucrose and Glycerol Solutions

Ap.4-1

Appendix 5. Density and Viscosity of Water Ap.5-1Appendix 6. Eigenvalues for Experiment A.2 Ap.6-1Appendix 7. (Suppressed) Appendix 8. MathCAD Program to Calculate Temperature Profiles in Rods for Experiment B.2

Ap.8-1

Appendix 9. MathCAD Program to Calculate Unsteady-state Concentration Profiles for Experiment B.3

Ap.9-1

Appendix 10. Vapor Pressure of Acetone Ap.10-1Appendix 11. Excel Spreadsheet for Preliminary Data Analysis of Friction Factors. Experiment C.1

Ap.11-1

Appendix 12. Excel Spreadsheet for Analysis of Data in Experiment C.2, Heat Transfer Coefficients

Ap.12-1

Appendix 13. Statistical Analysis of Experimental Data Ap.13-1Appendix 14. Experiment D.3 Benchmark Problem Ap.14-1

Introduction

ChE 324 Lab Manual

I. The need for experimentation in Chemical Engineering Page I-6

I. The Need for Experimentation in Chemical Engineering

One of the general objectives of chemical engineers is to develop quantitative

models of chemical processes that are useful for process design, simulation, and control. Many chemical engineering course, particularly Transport Phenomena, present fundamental principles that lead to deterministic models of chemical and physical processes. Combination of the basic concepts of conservation, of mass, momentum, energy, and chemical species, with relevant rate laws leads to either macroscopic or differential balances that, in principle, can predict the behavior of chemical processes. Thus, it is demonstrated that such processes can be analyzed on a rational basis.

The mathematical models derived by this approach contain physical properties on

the material that is involved. Properties such as density, viscosity, heat capacity, thermal conductivity, diffusion coefficient, etc. necessarily appear in the theoretical equations. Thus, quantitative application of the models requires numerical values of such properties. For pure materials, property values may or may not be available in the literature. For mixtures or materials at extreme conditions of temperature and pressure, experimental values are quite rare.

In principle, thermodynamic and transport properties might be calculated

theoretically from molecular properties, but at this time, only the properties of simple molecules in the low-density gas state can be estimated theoretically with reasonable accuracy. Therefore, one is frequently faced with the need to measure material properties experimentally.

On the macroscopic scale, the methods of transport phenomena allow one to

compute transport rates and detailed profiles of velocity, temperature, and concentration but only in very simple geometries and for streamline or laminar fluid flow. When geometries or boundary conditions become complicated, the mathematics of solving the differential equations of change may become intractable or at least require numerical solution on a computer. A more serious obstacle arises when fluid flow is turbulent. Turbulent flow is inherently random and chaotic; no rigorous theoretical method is yet available for predicting velocity, temperature, and concentration profiles.

With complex geometries or with turbulent fluid flow, the macroscopic balances

of transport phenomena are still relevant, but empirical models must be used to characterize interfacial rates. Thus, one defines fluid-film transport coefficients such as the friction factor, the heat-transfer coefficient, and the mass-transfer coefficient. Although dimensional analysis of the differential conservation equations can identify what independent groups of variables should appear in the functional dependence of rate quantities on operating conditions, the actual relationships must be determined

ChE 324 Lab Manual

I. The need for experimentation in Chemical Engineering Page I-7

experimentally. Fortunately, results for a given geometry can be generalized as dimensionless empirical correlations that apply to a wide range of materials.

The purpose of this course is that students recognize that the physical quantities

discussed in their transport phenomena course can indeed be measured. The experiments are grouped into four categories: measurement of transport properties, observation of profiles, measurements of transport coefficients, and analyses of macroscopic systems. In each section there are experiments that deal with fluid flow, with heat transfer, and with mass transfer.

Too often undergraduate students approach laboratory courses with the idea that

their objective is to prove basic theories or to obtain results that agree with published information. Laboratory reports then focus on whether the experimental results agree with the "right" answer and on explanations of why the agreement is not perfect. Although there may be a correct value for an intrinsic property value of a material, such as density or thermal conductivity, the same can not be said of transport characteristics. Transport coefficients and similar efficiency factors of chemical-process systems depend on many variables. Thus, no generalized correlation given in the literature can be expected to predict the behavior of a particular experimental system perfectly.

Experimental data are precious. Quantitative (and qualitative) observations of the

performance of a particular system are costly in time, materials, instrumentation, and effort. But, provided that the experiments are well designed and the measurements done carefully, the results possess the quality of uniqueness. The results tell one how this particular system behaves under the conditions tested. Discrepancies between direct measurements and published results for similar systems should be analyzed to ensure that no critical features have been overlooked in the measurements or data analysis. But, almost always, the direct results should be taken as the best indication of that system's operating characteristics.

In order to combat the tendency of students to view their experiments as simply

an exercise in replicating known results, each experiment is introduced to the students with a memo that provides a hypothetical context for experiment. The students are encouraged to imagine themselves working in the chemical industry, and their supervisor gives them an assignment to conduct an experiment for some particular purpose in their company. These memos are intended to present a realistic and practical motivation for doing each experiment. Accepting this context and motivation should make it easier for students to write interesting lab reports that address the basic questions of how the experiment system actually behaves.

ChE 324 Lab Manual

Course Guidelines Page I-8

II. Course Guidelines 1. WEEKLY ACTIVITIES

The course meets weekly for one four-hour period. The first hour is normally used for instruction, discussion, oral presentations, and occasional quizzes. The remaining three hours are devoted to measurements and calculations in the laboratory.

Students conduct the experiments in groups, as assigned by the instructor.

However, except when specifically indicated, each individual must prepare and submit a separate report. Reports are due by class time in the following week. To minimize the time required for this course, students should come to class prepared for the experiment of the week. That is, they should review the experimental procedure and the underlying theory before coming to class. Also, they should do as much data analysis as possible during the lab session.

2. GRADING PROCEDURE The grade for this course will be based approximately upon the following

distribution of credit:

Category Weight Class Quizzes and Exercises 10 % Oral reports 5 % Laboratory Reports 80 % Professionalism 5 %

Reports are due at the next class following the experimental session. Late reports

will be penalized 10%/day. Weekends count as two days.

3. DAMAGE TO EQUIPMENT Fee cards with charges for costs will be issued to anyone who, in the opinion of

the instructional staff, damages or destroys equipment because of carelessness or negligence.

4. HOUSEKEEPING Cleanliness in the laboratory is mandatory. Because of the large number of

students using the laboratory, it is difficult to keep the laboratory clean and orderly unless each student cleans up his or her work area at the close of each instructional period. Each

ChE 324 Lab Manual


group of students is responsible for its work area. All utilities are to be turned off, and all spills are to be cleaned up before any member of the group leaves the laboratory. 5. SAFETY

All personnel are required to wear safety glasses and proper clothing when in the

laboratory in accordance with the specified safety procedures. Each individual student must acquire his or her own safety glasses and wear them at all times in the laboratory. Substantial footware, other than sandals, and proper clothing that provides protection from accidental spills and burns should be worn.

6. SMOKING, EATING, AND DRINKING

As indicated in the laboratory safety procedures, no smoking, eating, or drinking

in any form is allowed in the laboratory or adjoining rooms. 7. COMPUTATIONAL AIDS

As calculations can be made while experimental measurements are in progress,

students should bring to the laboratory hand calculators, handbooks, graph paper, etc., for the analysis of data. Computers connected to the college network and file servers are available adjacent to the lab and should be used as needed. These computers are available to all ChE students during open hours, but ChE 324 students have priority during the lab period.

8. LABORATORY REPORTS

Reports are expected to be submitted in typewritten form except that sample

calculations in the appendix may be hand written. Only the the body of the main report need be submitted on high-quality paper. Appendices may be printed on draft-quality paper,or on paper with printing on the reverse side, in order to minimize paper costs. Nevertheless, students who print at CAE should expect to purchase some additional paper beyond the initial semester allotment.

The laboratory instructor will present examples of proper formats for reports and

memos and will indicate how the reports should be bound.

9. WRITING SKILLS

A primary objective of this course is to develop the students’ writing skills. Dictionaries and other references should be used to avoid errors in spelling, punctuation, grammar, and word usage. In addition to the books recommended with the syllabus, there are other references available in the library. In particular, 15 relevant videotapes produced for EPD 201, Basic Technical Writing, can be viewed at Wendt Library. Detailed guidelines for writing technical reports are available from EPD on the web at

ChE 324 Lab Manual


http://fbox.vt.edu:10021/eng/mech/writing/. Students who have difficulty with basic writing skills should consult the campus Writing Lab in the Helen C. White Building. Use a style similar to that used in chemical engineering publications to present equations, figures, tables and citations.

10. ORAL PRESENTATIONS

Ability in oral communciation is just as important to an engineer as effective

written communication. Therefore, among the class exercises each student will give a short technical presentation. Individual students will be asked to present his or her results from one of the lab experiments in the course. Alternatively, one may propose another technical topic, such as a project from a summer job or co-op. Specific individual assignments will be given by the instructors. Presentations should be kept short and to the point. Effective visual aids, i.e., overhead transparencies, should be used to enhance the presentation. Guidance on presentation methods will be given by the lab instructor. Students should also consult the textbooks on technical communication.

ChE 324 Lab Manual

Use of Computers in ChE324 Page I-11

III. Use of Computers in ChE 324

As engineers, students in this course are expected to be computer literate. Also, they are expected to develop the ability to produce reports of a high professional standard. This means that data should be presented and analyzed using the most modern tools, and reports should be presented in a polished form, typed and containing neat graphs and tables. Therefore, it is necessary for students to use commonly accepted computer tools in preparing their reports.

Software The majority of students in ChE 324 use Microsoft Word for their reports and Microsoft Excel or Mathcad for the data analysis and preparation of graphs. Use of these particular programs is not required, but they are readily available through the Computer-Aided Engineering (CAE) facilities in the College. Also, in general, the lab instructors are familiar with these common programs and thus are prepared to offer assistance to students performing specific tasks with their data or reports. It will be assumed that the students already have a working knowledge of computers and programs of this type so detailed instruction will not be presented. Those people who need help should see the teaching assistants during their office hours and make use of the handouts and manuals and of the consultants at CAE. Computer Resources Students are encouraged to use the many CAE computers provided in the College to work on their reports. During the laboratory, a lab-group member or two should be able to do much of the data analysis while others are collecting more data. There is a satellite CAE computer room adjacent to the transport lab that now contains Windows-based machines. These machines are connected to the CAE network so that students can save their files directly. Of course, one can also save documents on Diskettes. MSDS (Material Safety Data Sheet) information is available at a number of websites, e.g., http://msds.pdc.cornell.edu. Information is provided for the safe handlling of most chemical compounds, including toxicity, flash point, volatility, etc. It is always wise and prudent to check on potential hazards with any chemical that one may be using. Course Homepages Most ChE324 instructors take advantage of internet resources to distribute materials and to communicate with students. Materials related to ChE 324 are posted on a course homepage at http://courses.engr.wisc.edu/ecow/get/che/324/. Students are expected to

ChE 324 Lab Manual

Use of Computers in ChE324 Page I-12

make use of this method to obtain course materials. The Appendix of this manual contains several Excel and Mathcad programs that have been set up to facilitate the data analysis in particular experiments. Those files are also available on the course homepage. The materials can be accessed from off-campus computers as in the CAE labs.

ChE 324 Lab Manual

Safety in the Laboratory Page I-13

IV. Safety in the Laboratory

The Department of Chemical Engineering and the University have provided a safe laboratory, safe equipment, and safe experimental procedures for this course. However, safety in any laboratory ultimately rests with each individual working there. A few specific safety procedures will be emphasized in ChE 324 to reduce the chance of an accident or injury and to help develop proper habits of laboratory safety. Issues of concern include

1. Safety Glasses: All persons in the laboratory will wear safety glasses whenever they are whithin the restricted area of the laboratory.

2. Shoes: Footware must provide protection. Sandals and open-toed shoes will not be permitted.

3. Eating, Drinking: Eating, drinking in the laboratory is not permitted.

4. Chemical Hazards: Review the hazards, toxicity, and proper disposal of any chemicals used in the lab. Such information is available on the Cornell web site mentioned in Section III.

5. Broken Glass: Be especially careful handling glass equipment or materials. There is a special receptacle for disposal of broken glassware. Do not discard in any waste baskets.

6. Contact Lenses: Wearing contact lenses in a chemical laboratory is hazardous. Is is recommended that persons avoid wearing them to the extent possible.

7. Clothing: The wearing of shorts and sleeveless shirts in the laboratory will not be permitted. Trousers and skirts must extend below the knees. The use of a laboratory coat is recommended.

8. Medical Insurance: The University does not carry blanket accident insurance to cover medical costs in case of accidents involving students. Students should obtain adequate medical or accident insurance for themselves.

9. Housekeeping: Housekeeping and safety are closely related. Sloppy housekeeping and poor safety practices in the laboratory and workroom/classroom should be reported to the instructor. We do not intend to allow any unsafe practices to develop. This will help insure the safety of all.

10. Paying attention: Most accidents can be avoided by diligence, awareness, and caution. No horseplay or practical jokes are allowed.

ChE 324 Lab Manual

Technical Communication Formal Reports

Page I-14

V. Technical Communication A major objective of this course is to provide instruction and practice in technical communication. Students will report on the laboratory experiments with complete, formal technical reports, which are appropriate for archival purposes, with shorter memoranda, which are commonly used to convey progress reports in industry, and with oral summaries, which are also used quite often for providing overviews to colleagues and supervisors. This section summarizes guidelines and suggestions to help in preparing each type of report. A. Formal Reports

1. DEADLINE FOR SUBMISSION

Reports are due at the beginning of the period following that period during which

the experiment was performed. Late reports will be penalized 10% of the grade for each day that submission is delayed.

2. MECHANICS OF PRESENTATION

Reports are to be word-processed (or typed) with double spacing on one side only

of standard 8 1/2" by 11" white paper. A 12-point font size is required and the Time New Roman font is suggested. The left-hand margin should be 1-1/4". All pages, including those with accompanying graphs and figures as well as appendices, are to be assembled in proper order, numbered, and stapled. Figures and tables cited in the body of the report should appear immediately following the citation, either at the end of the paragraph in the case of tables or on the next page following. (The body of the report should be printed on good-quality paper. If reports are printed at CAE, students may find it necessary to purchase an additional paper allowance. Appendices may be printed, written, or drawn on lower-quality or used paper, provided that the face side is clean.)

3. CONTENTS

There are few if any absolute rules governing the style or format of technical

reports, other than the basic requirements of clarity and neatness. The writer must adopt a style that is appropriate for the purposes of the report and recognizes the interests and background of the likely audience. Different formats may be appropriate for different reports, but often a standardized format is imposed on the writer by an organization or publication editor. For the sake of consistency in ChE 324, reports will comprise the following designated sections in the order given. Some reports will be submitted in a

ChE 324 Lab Manual


Page I-15

memo format. In that case, some for the following sections may be condensed, combined, or even eliminated. Standards for memos are summarized in Section V.B.

3.1 TITLE PAGE This page is to be written in the following form:

(top of page, centered)

Department of Chemical Engineering University of Wisconsin

Madison, Wisconsin

(middle of page, centered)

ChE324. Transport Phenomena Laboratory Experiment number

Experiment title

(lower, right-hand corner) Student's name Partners' names Date experiment was

completed Date report is submitted 3.2 ABSTRACT The abstract is a simple, clear, and concise paragraph or two covering (i) what

was done, (ii) how it was done, and (iii) what was accomplished. This differs from a Summary of Results in that it is written as an advertisement, to interest the passing reader in the rest of the report, or as a very general summary of the nature and scope of the report. It should be self-contained, i.e., no references to figures, tables, or equations should be included. The abstract should contain text only.

3.3 INTRODUCTION Reports normally begin with an introduction that sets the context for the report.

(The format at some companies and laboratories might include an executive summary before the introduction, either as an alternative to the abstract or in addition to it.) In general, the introduction specifies the topic of the report and states the motivation for the reported study, giving the objectives or purpose. Some background is given, such as references to earlier related work or to the theoretical basis for subsequent analysis. For a

ChE 324 Lab Manual


Page I-16

longer report, it is helpful to the reader to give a paragraph or two outlining the structure and content of the report and even the nature of the results.

3.4 THEORY (usually not required in ChE 324) For a research study, there may be a separate section devoted to the development

of the relevant theoretical analysis, which provides the basis for the subsequent data analysis and interpretation of results. In ChE 324, the theory is available in common textbooks and references so no separate theory section is required. Reference to relevant theory and its source should be given in the introduction or in a later section when it is used. Equations that will be used in presenting the results might be given in the Introduction in order to define terms. Use the textbook Transport Phenomena as a guide to the proper format for presenting equations within the text. Equations should appear as part of a sentence.

3.5 EXPERIMENTAL APPARATUS AND PROCEDURE (usually not required

in ChE 324) In an original experimental study there is normally a section describing apparatus,

procedure, and methods used to obtain the results. If these are not a central point of the presentation, or if they are fairly routine, they might appear in an appendix. In ChE 324 the apparatus and procedures are described in this lab manual so they need not be included in the student reports; reference to the descriptions given in the manual is sufficient. Incidental modifications in the standard procedures or any difficulties with the equipment might be mentioned in the discussion section.

3.6 RESULTS AND DISCUSSION This part of the report should include a summary form the specific results of the

experiment, presented in neat, clear-cut tabular and graphical form, and a thoughtful analysis of their significance, implications, or possible applications. Pertinent discussion of the experimental procedure (especially deviations from that suggested or possible sources of error), theoretical analysis of the system studied as needed to interpret the results, analysis of the data (including estimates of error), and conclusions are all given in this section. While the earlier sections should be quite objective, this is the place where the author can present his or her observations and interpretations. As a basis for the analysis or to support specific conclusions, one should refer freely to relevant citations in the literature.

In writing both the Introduction and the Discussion, the author should keep in

mind both the presumed motivation for the study and the likely interests of the reader. For each experiment this manual presents a hypothetical context for the experiment, other than an academic exercise. Thus, the report should be written in response to the presumed motivation. If specific questions are posed in the manual or by the instructors, they should be used as suggestions for topics to be discussed in the overall context of the report.

ChE 324 Lab Manual


Page I-17

3.7 CONCLUSIONS AND RECOMMENDATIONS (optional in ChE 324) Most reports lead to specific conclusions based on the reported results and the

analysis given. Recommendations for future work or applications and other implications might be stated in a final section. In shorter reports this section may be omitted with the relevant conclusions and recommendations given at the end of the discussion section. In ChE 324 reports a separate conclusions section is usually not warranted.

3.8 APPENDICES Appendices contain supplemental information that is not an integral part of the

main report but is often included for reference by the excessively curious reader (or grader). All materials covered in the following subsections are to be located in the appendix of the report. These particular sections are not usually included in industrial reports or academic papers.

3.8.1 SAMPLE CALCULATIONS

This section should include one example of each calculation made in connection

with the analysis of the experimental data. Each calculation is to be accompanied by the formula involved, written in terms of the clearly defined variables, with the item being calculated appearing on the left-hand side of the equation. The numerical values and units of each quantity appearing on the right-hand side of the equation are to be given when substituted into same. The calculations must be presented in neat, logical order with the answers either underlined or set off by blocks. Extensive arithmetic manipulations need not be shown. Identify the part of the experiment to which each calculation applies. Simply presenting a spreadsheet with the calculated values is not sufficient.

3.8.2 MATHEMATICAL DERIVATIONS

Any derivations performed in connection with the data analysis and discussion are

to be demonstrated in this section. This includes analysis of errors based on equations appearing elsewhere.

3.8.3 ORIGINAL DATA SHEET(S)

A neat, orderly data sheet, for recording the original data, is to be prepared prior

to beginning the experiment and preferably before coming to the lab. All data taken in the laboratory are to be included in their original form. Lab partners may make photocopies for inclusion in individual reports. For the case of computer-acquired data files, hard copies of the files are usually not required in the report.

ChE 324 Lab Manual


Page I-18

4. GRAPHS AND TABLES Key graphs and tables should be presented in the body of the reports, either in the

Summary of Results or in the Discussion. Less important ones, or those used in intermediate calculations, should be placed in the Appendix. Graphs and tables in the body of the report should be placed on or immediately following the page where they are first mentioned, as is done in textbooks. Graphs, as well as drawings and diagrams, are numbered sequentially and called “Figure x.” Tables are also numbered in a separate series. Graphs require complete, self-explanatory captions, placed below the figure but within the margins. Tables should include a descriptive heading at the top of the table. The main text should say what is presented in each figure and describe the significance of each graph and table.

Graphs are to be presented on standard 8-1/2" by 11" paper with generous

margins. Theoretical or computed functions should be shown as smooth curves without points. Plotted experimental points, used to define empirical curves, must be clearly indicated by some appropriate symbol. The variables plotted on the ordinate and abscissa as well as any parameters should be indicated with the units designated. The graphs are to be inserted in the assembly of the report pages with either the bottom or the right-hand edge as the base of the figures as drawn. Graphs are to be lettered to read from the "front" and "right side." Any hand-drawn curves are to be drawn with the aid of ship's curves or French curves. If more than one set of data are shown on a graph, different symbols should be used and defined either in a legend or in the caption. Computer-generated graphs are preferred. Column headings in tables should include the units of the entries.

5. GRADING

Reports will be graded on the basis of neatness, grammar, spelling, and clarity as

well as technical validity. Strive to be clear and concise.

ChE 324 Lab Manual

Technical Communication Memorandum Reports

Page I-19

B. Memorandum Reports Much communication within a technical or industrial organization is accomplished with memos. Memos are short documents that can be read quickly and easily. They are designed to convey their message clearly and concisely. Memos may be used to give a notice, to make a request, or to provide a report. In Chemical Engineering 324 memos are written for the latter purpose. The memos used here resemble those that an engineer in industry would use to provide a progress report or summary of a project to supervisors and colleagues. Typical formats for memos and guidelines for their content are provided in technical writing textbooks and at the EPD Technical Communication web site. The difference between a memo and a complete formal report is the greater amount of detail contained in the latter. A report is intended for the purposes of communicating with a wide, varied audience and of establishing a relatively permanent record of what was done in a project. A memo is generally written for more immediate needs, such as conveying recent results, e.g. a weekly progress report, to a knowledgeable reader such as a supervisor. A memo is also designed to be read and understood quickly. Thus, a memo does not need to present all the background material that goes into a formal report. The format is a bit different, and it is obviously shorter. The emphasis is on the results and your interpretation. Recommendations for additional work are usually appropriate. You may abbreviate the formal title page of a report into a memo heading such as that used in each weekly memo to students from the ChE 324 instructor, and one may omit an abstract. Separate sections headings may or may not be required, depending on the length and complexity of the memo. Headings are probably appropriate and helpful if a memo is longer than a page or two. Whether there are headings or not, several sections appearing in a report may be combined logically within a memo. For example, the introduction, theory, apparatus, and procedure may all be combined into several coherent paragraphs, and the discussion and conclusions might flow together. It is a good idea to give a summary of the most important results and conclusions in the first paragraph, which then serves as a type of abstract or summary. This is the information that a busy reader is most eager to obtain. Depending on the complexity of the material in the memo, you may or may not need an appendix. In any case, you should make liberal use of figures and tables, but make sure that the headings and captions are thorough and descriptive. Generally, you should include a list of references. For the purposes of this course, sample calculations are always required as an appendix with memos. It is conventional to provide copies to all parties with an interest in the content of a memo, and the names of those receiving copies are listed with a copy (cc:) notation at the bottom of the document. In ChE 324 the laboratory partners should be identified by including them in the copy list.

ChE 324 Lab Manual

Technical Communication Oral Reports

Page I-20

C. Oral Reports

Engineers are frequently called upon to present oral reports. These may be brief summaries to a team of colleagues on a project in process, a proposal to senior management for a major investment, a tutorial to other engineers on a specialized subject, or a paper at a technical conference reporting on a completed project. Oral reports may be presented to a small group around a conference table or delivered in an auditorium to an audience of hundreds. The duration may be only five or ten minutes or as long as an hour or even more. In all cases, the purpose of an oral report is to convey information to the audience rapidly and efficiently, preferably with a sense of the speaker's attitudes and personality that is no so readily conveyed in a written report. Although an oral report may not be able to cover a topic in as much detail as a written document can, it allows the speaker an opportunity to emphasize and communicate his more important points. Another advantage of an oral report is that the audience is often able to ask questions to clarify the speaker's intent.

Probably everyone who delivers an oral report feels some nervousness about standing up and talking before an audience. Such nervousness should not be a cause for concern but a source of energy for the presentation. Nervousness diminishes with experience, but for inexperienced speakers as well as old hands, preparation is the key to avoiding any feared awkwardness or embarrassment during the oral presentation.

Beer and McMurrey (1997) present a very sensible discussion about giving oral reports. Some of their primary points are summarized below. Pfeiffer (2001) presents a similar discussion.

Preparation

The key to a successful presentation is preparation. As with writing the speaker should first analyze his or her audience. Why are you giving the talk, why is an audience coming to hear it, what do they already know about your subject, and what do they want to learn from you? Those are the questions that the speaker must answer and keep in mind while preparing and delivering a talk. Of course, there are situations where the audience is rather diverse, and there is not a single set of answers. Then the speaker faces the challenge of balancing the talk to offer something of interest to everyone, without baffling anyone completely nor boring others excessively.

Analyzing the audience also involves identifying the primary purpose of giving the talk. After that purpose has been clearly stated, the speaker has a basis for selecting and organizing the content of the talk. Another key factor, however, is the time frame available for the presentation. If one has only 10 minutes available for the presentation, he must select his material judiciously to be both complete and concise and to communicate his key points. It is an unforgivable sin of technical presentations to exceed the time allotted. The audience is busy, and each member has his own agenda for the day. When the stated termination time comes, the audience stops listening so the entire point of the talk may be lost. Usually, time should be left at the end of a talk for questions and discussion. And no one will object if a technical talk ends a few minutes early.

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After a speaker has identified her primary purpose and the key points to be conveyed, she next must select a structure for the talk. That is, a logical sequence must be selected as the path by which she will lead the audience through the subject at hand. Every talk must have a beginning, a central part, and an end. The beginning is an introduction and a preview that prepares the audience and sets the stage for what follows. The end is the summary of what has been covered, with conclusions and perhaps recommendations. The end should reiterate key points just as the beginning might suggest what key issues are to be covered.

The central part of a talk is the technical development of the specific subject. This part of the talk, just like a written report, should be organized to make the trip from the original objective and premises to the conclusions as effortless as possible for the audience. Designing such a path requires selecting a logical structure. As indicated by Beer and McMurray, there are a number of alternative strategies that may be selected, depending on the topic and the audience. One may proceed chronologically or spatially; one may go from simple to complex or vice versa, one may organize the points in order of decreasing or increasing importance, familiarity, difficulty, etc. Regardless of what logical sequence is selected, the speaker should be consistent so that the audience does not get confused. Also, it is imperative that the degree of detail presented be adjusted to fit the time allotment for the talk and the technical level of the audience.

After the overall structure of the talk has been designed and the content selected, the speaker should design visual aids and graphics to enhance the clarity and efficiency of his presentation. Slides or overhead transparencies should be used to reinforce what the speaker is saying, helping to convey the overall logic of the presentation. As a picture is worth many words, the same is true of well-designed graphs and diagrams. Each graphic should have a descriptive heading, summarizing the significance of the illustration. On all sheets one should use large letters that are easy to read and avoid cluttering it up with too much information. Each page to be displayed should be kept quite simple and contain lots of blank space so that the observer does not get overloaded and can focus on the key point.

In preparation for the presentation, the speaker should give special thought to what will be said in the introduction and in the conclusion. These portions of a talk should appear to be ad lib, but they should be quite polished to make a good impression on the audience. A speaker may want to make some notes as an aid in the presentation, but for most of the talk, the visuals themselves should be sufficient reminders of what needs to be said and in what order.

Finally, in preparation for an oral presentation, a speaker should practice the talk. If possible, some friends or colleagues should be asked to listen to trial runs. Such practice is needed, first of all to ensure that the talk will not be too long but also to check the quality of the visual aids, to practice speaking on one's feet, and to test the planned wording of the introduction and the conclusion.

Presentation Giving a speech in front of an audience is always stressful, even for the most

accomplished speakers. You can reduce the stress by following the guidelines given by Pfeiffer (2001).

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When the time comes for the actual presentation, there are a few other issues to keep in mind. Think about the many bad talks (or lectures) that you have attended and think of all of the mistakes that the speaker made. These are mistakes that you wish to avoid.

With respect to delivery, remember to speak at a sufficiently audible level that those in the back of the room can easily hear what you say. The graphics should be designed so that the same people can easily read them. Look at your own projected graphics from the same distance to see how they work. Be careful to speak at a comfortable pace, neither too rapidly nor too slowly, and inject some dynamics into your delivery. Maintain eye contact with your audience to sense whether they are following you. If you see a puzzled face, you might ask whether there is a question.

When displaying projected graphics, use a pointer to help keep the audience with you. Use the hand closer to the screen to avoid blocking the view or turning your back to the audience. Leave the graphics up on the screen long enough for the audience to absorb the content. Although it is not desirable to read one's entire talk from the screen, some reading is helpful for the audience. That is, one should not be expecting the audience to be reading an outline or a statement on the screen and at the same time listening to the speaker make a separate point.

Speak naturally, not too stiffly, but avoid also being too informal. That is, use proper English and avoid slang and clichés. Also, try to eliminate nervous gestures and hemming and hawing that will distract or annoy the audience.

Finally, try to make the planned logical structure of the talk transparent to the audience. Orally and with visual aids, emphasize clear transitions as you step through the presentation. Also, it is very helpful to the audience when the speaker repeats the key points of the talk. One old recommendation, “The Preacher’s Maxim” is:

First tell them what you are going to tell them, then tell them, and finally tell them what you told them.

People are generally not very good listeners. They remember only a portion of what they hear and a bit more of what they read. They do remember the most when they both hear and see the information. It is the speaker’s obligation to help the audience to absorb and to remember the most important information from a talk. This can be done through planning, preparation, and practice of the presentation.

References Beer, David, and David McMurrey, A Guide to Writing as an Engineer, John Wiley & Sons, Inc., New York, 1997, Chapter 8. Pfeiffer, William S., Pocket Guide to Technical Writing, 2nd Edition, Prentice Hall, Upper Saddle River, New Jersey (2001), Chapter 3.

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Experimental Design and Statistical Analysis of Data Page I-23

VI. Experimental Design and Statistical Analysis of Data As discussed in Section I, the successful application of chemical engineering methods to practical problems requires experimentation. Although we have a sound theoretical framework for structuring our treatment of processes or chemical materials, the underlying theories almost always involve specific parameters that must be determined with experimental tests.

The textbook Transport Phenomena (Bird et al. 2002) develops a systematic approach for analyzing a wide variety of processes and systems, but applications require values of thermodynamic and transport properties. For example, flow problems require information about fluid viscosity as well as fluid densities. Heat transfer processes involve thermal conductivity as well as enthalpy and heat capacity of the materials involved. Separation processes and chemical reactors cannot be treated quantitatively unless one knows material properties such as vapor pressures, activity coefficients, solubility, equilibrium constants, reaction-rate constants, etc. All of these quantities must be determined experimentally for any given material as functions of the state variables such as temperature, pressure, and chemical composition.

Furthermore, although conservation laws yield the differential equations of change

for predicting profiles of velocity, pressure, temperature, and concentrations within a fluid, as well as the associated fluxes of momentum, energy, and mass, either mathematical complications associated with complex geometries and boundary conditions or the inherently random nature of turbulent flow make rigorous computations and a priori predictions impractical in most cases. For that reason, one uses phenomenological relations to define transport coefficients such as the friction factor, drag coefficients, heat-transfer coefficients, and mass-transfer coefficients, which are useful quantities, but they must be determined experimentally for a given situation.

There are generally two types of quantities that must be measured. There are the

material properties, and there are macroscopic characteristics of a certain type of system or process. The unknown quantity may be a single constant, or it may be an unknown function that varies with changes in local conditions. In the case of basic materials properties, thermodynamics usually reveals a set of independent variables upon which a quantity should depend. In the more general case, dimensional analysis often helps one to identify an appropriate set of independent variables to be considered. Based on theoretical analysis, one may know the functional form that the unknown quantity should follow. When a theoretical form is known, the task of the experimenter is to find the specific parameter values that enable the function to fit the behavior of the particular material or process of interest. When there is no theoretical guidance or experience that provides a functional form for the expected dependence of the measured quantity on its independent variables, the experiment has to seek an empirical functional form that represents the

ChE 324 Lab Manual


system behavior. One must also determine the associated parameter values that provide a quantitative description of the phenomenon.

Thus, in the conduct of practical chemical engineering there are two related

activities that are crucial to the effectiveness of an experimental program. First, one must consider the matter of experimental design. The other issue is statistical analysis of the data.

When one has available a deterministic model for a system that is based on a

rigorous theoretical analysis, the purpose of experimentation might be simply to confirm the validity of the theory. In this case one might use statistical analysis, combined with replicated measures, to discrimiate experimental error from shortcomings in the theoretical model. More often, the chemical engineer is working with a deterministic process model that contains some unknown parameters. These unknown parameters may be thermodynamic or transport properties of the material or they may be parameters such as transport coefficients that depend on the detailed geometry or flow conditions in the equipment. In this case, the objective is to estimate the value of the unknown parameter or parameters from the experimental tests.

Estimation of model parameters from data is often referred to as "curve fitting" or

regression analysis. A brief summary of curve fitting and parameter estimation is given in Appendix 13 of this manual. The statistical approach to parameter estimation not only provides a quantitative result from the experiment but also indicates the chance that the estimated value is accurate. Statistical analysis considers different kinds of experimental errors and provides a criterion for omitting data points that are clearly inconsistent with the bulk of the data set. (Of course, one should be alert to possible messages from the data that there is some effect occurring in the system that is not accounted for in the basic model.)

In the case where one does not know initially what functional form should

represent the magnitude and variation of a quantity of interest, it is appropriate to adopt a strategy of experimental design. That is, one has to decide which independent variables might have an effect on the outcome of an experiment. Then one must choose values of those variables to use in setting up the experiment. A number of experiments must be conducted at different settings of the suspected independent variables to see what effect each actually has. Because experiments are usually costly and time-consuming, one hopes to answer this question with a minimum number of tests.

For example, suppose one were interested in maximizing the yield of a particular

reaction in a certain type of reactor. The independent variables that might be relevant could include temperature, pressure, reactant concentrations, residence time in the reactor, as well as mixing characteristics. Another variable might be the concentration of a possible catalyst. To find out under what conditions the amount of product produced from the reactants is maximized, one could do many experiments at different settings of the various variables, but an exhaustive study might be prohibitively expensive.

ChE 324 Lab Manual


To make the experimental study of a problem like this most efficient, statisticians have developed techniques known as factorial design. The first objective of statistical design is to determine which variables have a large effect and which have little or no effect on the outcome of the experimental system. More advanced analysis considers whether the effects of variables are independent or whether there are interactions among the variables. Then there is the question of how the dependence on the variables can be represented by quantitative formula.

Box, Hunter, and Hunter (1978) provide a good treatment of experimental design

and factorial analysis. Many of their examples are taken from the field of chemistry. The fourth and last part of this book deals with model building, that is, identification of quantitative functions that can successfully describe observed experimental behavior. Although models based on statistics and empiricism, rather than a rigorous underlying theory, are limited in their predictive capacity to the actual range of variables studied, they are nevertheless quite useful for practical purposes.

Reference Box, G.E.P., W.G. Hunter, and J.S. Hunter, Statistics for Experimenters, An Introduction to Design, Data Analysis, and Model Building, John Wiley & Sons, New York (1978

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Syllabus Page I-26

ChE 324 Lab Manual

Part A Measurement of Transport Properties

ChE 324 Lab Manual

Experiment A.1 Viscosity of Newtonian Fluids

Page A.1-1

Experiment A.1

VISCOSITY OF NEWTONIAN LIQUIDS

Viscosity is a fundamental property of fluids that indicates a material's capacity to transport momentum by molecular mechanisms. The magnitude of liquid viscosity, which depends on temperature and chemical composition, determines not only the quantitative but also the qualitative nature of fluid flow (Bird et al., 2002). Numerous experimental methods have been devised for the measurement of the viscosity of a Newtonian fluid. An accurate determination of absolute viscosity, which is denoted by µ, demands a careful analysis of the experimental technique that is used. As an example of some of the problems involved in practical viscometry, this experiment employs a simple capillary-tube viscometer to measure the viscosities of several Newtonian liquids.

Theory Application of Newton’s law of viscosity and conservation of momentum to the steady flow of a constant-density fluid through a straight tube of uniform circular cross section of length L leads to the Hagen-Poiseuille relationship (Bird et al., 2002)

Q( )R4

8 L=

−π

µ

∆℘ (A.1-1)

where (-∆℘) is the net driving force for the flow, Q is the volumetric flow rate of fluid, and R is the tube radius. The quantity ℘ is defined as (p+ρgh) where p is static, or thermodynamic, pressure, ρ is fluid density, g is the acceleration of gravity, and h is vertical elevation above a datum plane. Thus, ℘ represents the combined effects of pressure and gravity in causing the fluid motion. (The notation in this manual follows that used in Transport Phenomena by Bird et al., 2002, which presents a summary table on pp. 757-764. For example, ∆x ≡ x2-x1 where the subscript 1 indicates the value of a quantity x at the fluid entrance and 2 the value at the exit.)

There are a number of assumptions involved in the development of the Hagen-Poiseuille law. Among other conditions, the flow must be laminar and free from end effects. If the construction and operation of an experimental apparatus can conform accurately to the key assumptions, it is possible to use Equation A.1.1 to measure the viscosity of Newtonian fluids.

A simple experimental arrangement which could yield a viscosity determination based on Equation A.1.1 is the steady flow Q of a fluid in a long, straight tube that is maintained at a constant temperature and is equipped with a device to measure the pressure gradient ∆p/L at some distance from the ends of the tube; a capillary manometer, for example. In most instances the control of the operating conditions over the entire length of the tube, the cleaning difficulties, and the need for a large sample of liquid to fill the length of the tube prohibit or make very difficult the use of such a device. Other more convenient and compact types of viscometers to which the Hagen-Poiseuille

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Page A.1-2

equation may be applied have been developed. The Cannon-Fenske viscometer and other modifications of the Ostwald pipette are examples (ASTM, 1955).

When the total change in the driving force ℘ associated with a flow rate Q through a tube is due to hydrostatic head alone, Equation A.1.1 may be written as

νπ

=−g h R

8Q L

( )∆4

(A.1-2)

where (-∆h) is upstream elevation minus downstream elevation, called the hydrostatic head difference, and the quantity ν is defined as

ν = µ / ρ (A.1-3)

and is called the kinematic viscosity. The kinematic viscosity is often expressed in units of cm2/sec, which is called stoke.

Consider steady fluid flow through a straight capillary tube of fixed length L for which the hydrostatic head differential ∆h is constant. If one measures the time for a fixed volume of fluid V to pass through a particular tube, the kinematic viscosity should be related to the observed efflux time te as follows:

ν = C te (A.1-4)

where C is called the viscometer constant. If C is evaluated by observing te with a liquid of known viscosity, C may be calculated for the apparatus. Then measurements of te for the same V in the same cell with other fluids allows the kinematic viscosities of the latter to be calculated from Equation A.1.4. Dynamic viscosity value is then obtained by multiplication with the density of the liquid.

Equation A.1.4 is derived by substituting the relation

Q = V/te (A.1-5)

into Equation A.1.2 and combining all constant factors into one term. The viscometer constant C is thus identified to be

C g h RVL

=−π ( )∆ 4

8 (A.1-6)

Although the preceding equations are derived for constant ∆h and constant Q, they may be applied with reasonable success to a pipette-type viscometer in which a liquid drains under a slowly varying hydrostatic head. In that case, one may use average values of Q and ∆h in Equations A.1.2 and 6, and the constant C should still be a property of only the viscometer geometry and not depend on the properties of the fluid.

Apparatus

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The capillary-tube viscometer apparatus consists of the following items:

1. A commercial Number 200 Cannon-Fenske pipette type viscometer, which is designed for a 20-to-80 centistoke range in kinematic viscosity

2. A stopwatch

3. Several constant temperature baths (plain, cylindrical Pyrex jar; 12-inch outside-diameter by 12-inch height; 4.5 gallon capacity), set at various temperatures

4. Automatic temperature controls

a. Temperature sensing element (mercury-contact, wide-range, quick setting, 0.01°C thermoregulator)

b. Electric heating-elements (200 watt, hairpin, immersion heaters)

5. A 10 ml. graduated pipette

6. Cleaning solution

7. Acetone

8. Distilled water

9. Source of dry filtered air

10. A mounting device for holding the viscometer in the water baths

11. A stock of 60-weight-percent aqueous sucrose solution

12. A stock of approximately 85-weight-percent aqueous glycerol solution

13. A glass pycnometer (25 ml)

14. An analytical balance

The design and construction details of the Cannon-Fenske pipette-type viscometer, including the dimensions for size No. 200, are shown in Figure A.1-1.

The viscometer may be filled with liquid such that there is an initial elevation difference, or static head (-∆h), between the liquid surface in the tube on the right side and that in the spherical bulb at the bottom of the cell. Both surfaces are at atmospheric pressure. Liquid is allowed to drain through the capillary tube, and the efflux time te is measured as the time for the liquid level on the right side to drop through the lower bulb. The volume V is the fixed volume contained between the two marks above and below that small reservoir. The elevation difference in Equation A.1.2 or 6 is taken to be the average difference between the liquid level in the larger bulb on the left side, which changes only slightly, and that in the lower bulb on the right as the liquid level drops from the upper mark to the lower one.

The cell constant C is determined from Equation A.1.4 by measuring the efflux time for a liquid of known kinematic viscosity. Then Equation A.1.4 may be used to calculate the kinematic viscosity of an unknown liquid from its efflux time in the same apparatus.

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Page A.1-4

Working capillary

Figure A.1-1. The Cannon-Fenske Pipette-Type Viscometer For Transparent

Liquids. Dimensions are given for size Number 200.

Procedure 1. Ensure that the thermostat baths have attained the predetermined temperatures at

which the viscosity measurements are to be made, 30, 45, and 60oC.

2. Clean the viscometer thoroughly before using. In the case where aqueous solutions of organic materials are involved, clean with cleaning solution, rinse with distilled water followed by acetone, and dry with filtered air. NOTE: In order for the viscometer to operate properly, it must be absolutely clean.

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Page A.1-5

3. Calibrate the viscometer using the 60-weight-percent aqueous sucrose solution that is provided. Measurements are to be taken at three temperatures, 30, 45, and 60oC.

4. With the viscometer in a vertical position, use the 10-ml graduated pipette to introduce exactly 6.5 ml of the sucrose solution into the wider leg of the viscometer. NOTE: All liquids are to be introduced into the viscometer at room temperature.

5. Place the viscometer in a constant temperature bath. It should be submerged such that the bath-water is at least one centimeter above the upper of the two small reservoirs. Allow at least ten minutes for the viscometer and its contents to reach thermal equilibrium with the bath, particularly at the higher temperatures. The filled viscometer will be moved from bath to bath to obtain data at various temperatures.

6. Before measuring any efflux times, align the viscometer vertically in the constant temperature bath, in the orientation shown in Figure A.1-1.

7. Apply suction to the narrow leg of the viscometer until the liquid level is about 0.5 cm above the etched mark between the two small reservoirs.

8. Place a thumb over open end of narrow leg to maintain the liquid level. At this point an unbroken column of liquid should extend from the large bulb at the bottom to a level near the bottom of the upper small reservoir.

9. Remove thumb and measure with the stopwatch the time required for the liquid meniscus to pass from the upper etched mark to the lower etched mark.

10. Repeat Steps 7. through 9 to obtain replicate data points. The runs go faster at the higher temperatures so it is more convenient to take replicates in the warmer baths.

11. Clean the viscometer thoroughly and dry it, as described in Step 2, both when a new liquid is to be introduced into the cell and when no further measurements are to be made.

12. Repeat Steps 4 through 11, replacing the sucrose solution first with the 85-weight-percent aqueous glycerol solution, then with distilled water. For glycerol use the same temperatures as in the calibration process. With pure water, it is sufficient to make a measurement only at 30oC; this measurement will be used to test the applicability of the method to less viscous fluids.

13. Collect the following data needed for the determination of the density of the sucrose and glycerol solutions, using water as a standard:

a) Weight of the empty, dry pycnometer b) Weight of the pycnometer plus distilled water c) Weight of pycnometer plus sucrose solution d) Weight of pycnometer plus glycerol solution e) Temperatures of all solutions weighed

These measurements may be done at room temperature or in the 30oC bath. Calculate densities in order to determine the actual solution concentrations from the density tables given in Appendix 4.

14. Note also approximate values of the quantities appearing in Equation A.1.6. These can

be used to estimate the expected value of C.

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15. Always pour the solutions slowly. Otherwise, they will entrain air bubbles that are very slow to escape and can affect the experimental results.

16. Be especially careful while cleaning and drying your viscometer. Return test solutions to their containers, and wipe up any spills. Rinse out the glassware as thoroughly as possible with distilled water and with Alconox if necessary. Rinse with a minimal amount of acetone, disposing of the waste acetone in the waste-solvent container provided, and dry very gingerly with compressed air. The greatest risk of breaking the glass occurs during the drying.

Data Analysis 1. For the 60-weight-percent aqueous sucrose solution plot:

a. density vs. temperature

b. absolute viscosity vs. temperature

c. kinematic viscosity vs. temperature

These properties are given in Appendix 4 and 4a.

2. Use these sucrose-solution data with measured efflux times to determine the viscometer constant C. Consider whether the data indicate any dependence of C on temperature.

3. Determine the experimental kinematic viscosity of the glycerol solution as a function of temperature. Plot the results, and for comparison include in the plot a literature value for the kinematic viscosity of an 85-weight-percent aqueous glycerol solution at 20°C. Properties of glycerol solutions are given in Appendix 4.

4. Compare the experimentally determined viscosity of water with published values. Properties of water are given in Appendix 5.

5. As a check on the validity of the viscometer calibration, estimate the geometrical parameters of the cell, and calculate the expected value of C from Equation A.1.6.

6. For laminar-flow conditions, the entrance length Le , i.e., the distance in the tube required for the flow patterns to become fully developed, has been found to be a function of the Reynolds number:

Le ≅ (0.05)(2R)(Re) (A.1-7)

where Re ≡ (2R<vz>ρ)/µ is the Reynolds number , and <vz> is the average velocity in the tube. Estimate the volume V and the mean velocity <vz> in order to estimate Re and the entrance length for both the glycerol solution and the water at 45oC. Compare the estimated entrance-length values with the actual capillary length to check the validity of neglecting end effects in the data analysis.

References Bird, R.B., W.E. Stewart and E.N. Lightfoot, Transport Phenomena, 2nd Edition, John Wiley and Sons, Inc., New York (2002).

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American Society for Testing Materials, Book of ASTM Standards, Part 5, Fuels, Petroleum, Aromatic Hydrocarbons, Engines Antifreezes. Philadelphia (1955). Tentative Method of Test for Kinematic Viscosity, ASTM Designation. D 445-53T, pp. 197, 200-224. Viscometers, Bulletin 19, Cannon Instrument Company, Box 812, State College, Pennsylvania. Prandtl, L. and O.G. Tietjens, Applied Hydro- and Aeromechanics, Dover Publications, Inc., New York (1934), pp. 26-27. Cannon, M.R., R.E. Manning, and J.D. Bell, Anal. Chem., 32, 355-358 (1960). Cannon, M.R., and M.R. Fenske, Ind. Eng. Chem. (Anal. Ed.), 10, 297-301 (1938).

Experiment A.2

THERMAL CONDUCTIVITY OF SOLIDS

Thermal conductivity, like viscosity, is an important transport property of matter. The rates of heat flow, particularly in solids, are determined by the magnitude of thermal conductivity, which reflects the capacity of the material to transmit energy by molecular mechanisms. Like density or heat capacity, thermal conductivity is a state property, and its value is generally a function of local temperature, pressure, and chemical composition of a material.

Thermal conductivity, denoted by k, is defined by Fourier's law as the proportionality factor between the heat flux q and a temperature gradient, which is the driving force for heat flow:

q k T= − ∇ (A.2-1)

The minus sign in Fourier's law indicates that heat always flows from regions of high temperature to regions of lower temperature.

The thermal conductivity of solids can exhibit values that range over many orders of magnitude. Good conductors such as metals have high conductivity, while good insulators, like wool, have much smaller values. It is necessary to measure the conductivity of a material experimentally in order to ascertain the correct value of k to use in quantitative heat-transfer calculations. This experiment demonstrates one method for measuring the thermal conductivity of solids.

This experiment is based on measurement of transient temperature changes in a sample of an initially cool solid material after it is immersed in a hot fluid bath. The experiment is modeled by use of Fourier's law, combined with the principle of conservation of energy, in order to obtain a theoretical relation for temperature as a function of time in the unsteady-state heating process. Comparison between experimental data for temperature as a function of time and the theoretical prediction allows calculation of the thermal conductivity.

Unfortunately, there is some uncertainty concerning the effect of the fluid mixing in the bath on the rate of heating in the solid. Therefore the apparatus must be calibrated with solids of known conductivities in order to determine the efficiency of heat transfer from the stirred fluid to the outer surface of the solid samples.

Theory

A microscopic energy balance in a homogeneous solid, where the physical properties are assumed to be constant yields

ChE 324 Lab Manual

Experiment A.2 Thermal Conductivity of Solids

Page A.2-2

ρ∂∂

~CTt

k Tp = ∇2 (A.2-2)

In solids of simple geometry, Equation A.2-2 can be used along with appropriate boundary conditions to solve for the temperature T within the solid body as a function of time t as well as position. Spatial derivatives ∇ and ∇2 are given for various coordinate systems in Bird et al. (2002, §A.7).

As an example consider a thin, wide slab of solid material with thickness 2b that is initially at a uniform temperature To. At time t=0 the slab is exposed on its surfaces to a fluid held at a different temperature T∞. The temperature profiles in the solid can be calculated from Equation 2 if the heat conduction from the edges of the slab is neglected and the temperature profile is taken to be a function only of time and distance y. The position coordinate y is measured from the center plane of the slab, and Equation A.2-2 becomes,

2

2~yTk

tTC p ∂

∂=

∂∂ρ (A.2-3)

It is convenient to express Equation A.2-3 in dimensionless form. Let a

dimensionless temperature be defined as

Θ=−−

∞

∞

T T y tT To

( , ), (A.2-4)

where T∞ is the surrounding fluid temperature and To is the solid’s initial temperature. A dimensionless position η is defined as

η= yb

, (A.2-5)

where b is the distance from the center to the surface of the slab. The dimensionless time, τ, is defined as

τα

=t

b2 , (A.2-6)

where the thermal diffusivity, α, is

αρ

=kCp~ . (A.2-7)

Observe that the magnitude of the thermal diffusivity is proportional to the value of the thermal conductivity, k. Equation A.2-3 becomes

2

2

η∂τ∂

∂Θ∂

=Θ (A.2-8)

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Page A.2-3

Two boundary conditions and an initial conditions are needed to solve the problem. At t=0, T=To, which in dimensionless from becomes,

1=Θ at τ = 0 (A.2-9)

In general, heat transfer from a stirred fluid to a solid surface is not perfectly efficient. In that case T1 is not equal to T∞ at all times, and the boundary condition for solving Equation 2 must be established accordingly. Although the flow patterns and convective heat transfer in the fluid phase surrounding the solid may be quite complex, it is common to represent the heat-transfer efficiency by use of Newton's "law" of cooling: q h T T1 1= − ∞( ) (A.2-10) where q1 is the flux of heat crossing the solid-fluid interface into the fluid. Equation A.2-10 is not a fundamental law; it is merely a convenient approximation used to describe the efficiency of the fluid-side heat-transfer process. It defines the proportionality factor h, the fluid-film heat transfer coefficient, the value of which depends on the flow conditions and geometry as well as the properties of the fluid. Better mixing and more efficient heat transfer give larger values of h. Assuming that in a given situation, one can estimate the value of h, Equation A.2-10 can be used as a more realistic boundary condition on the solid surface instead of the condition of a constant temperature. That is, the appropriate boundary condition for solving Equation A.2-2 for a slab becomes

q kTy

h T T1 = − = − ∞

∂∂

( ) at y = ±b (A.2-11)

where the first relation expresses Fourier's law for the heat flux on the solid side of the interface and the second gives the flux on the fluid side.

In dimensionless form, Equation A.2-11 becomes

∂∂ ηΘ

Θ+ =Bi 0 at η=±1 (A.2-12)

where the dimensionless parameter Bi is defined as

Bibhk

= (A.2-13)

and called the Biot number. The magnitude of the Biot number indicates the resistance to heat flow of the solid body relative to that in the surrounding fluid.

This problem has been solved and is given in many textbooks on heat conduction (Carslaw and Jaeger, 1959, Jacob, 1949). The result is an infinite series solution of the form

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Page A.2-4

∑∞

=

−

++=Θ

12

)(

)1()cos()sec(2

2

n n

nn

BiBieBi n

βηββ τβ

(A.2-14)

where the βn quantities are called eigenvalues and are identified as the positive roots of the relation β βtan( ) − =Bi 0 (A.2-15) The solution given in Equation A.2-14 converges slowly at short times. On the other hand, at longer times, as τ gets large, the exponential factors in each term get smaller, particularly those of higher order with large values of βn. At sufficiently long times the first term, with the smallest βn value, dominates, and the approach to the equilibrium temperature is everywhere in the solid a pure exponential decay. For example, at times sufficiently long that Θ has fallen below 0.8 everywhere in the solid, the dimensionless temperature at the center of the slab may be approximated accurately by ln ( , ; ) lnΘ τ β τ0 1

21Bi A≈ − + (A.2-16)

where A1 is a combination of constants appearing in the first term in Equation A.2-14. At these longer times a semilog plot of Θ versus τ should become a straight line with a slope of 2

1β− and an intercept of lnA1. Similar relations can be derived for unsteady conduction in a cylinder or in a sphere by solving Equation A.2-2 in cylindrical or spherical coordinates. The results for a cylinder are

[ ]Θ=+

−

=

∞

∑22

2 21

Bi J eBi J

o n

n o nn

n( )( )

β ηβ β

β τ

(A.2-17)

where Jo(x) is a Bessel function of the first kind and zero order. The eigenvalues in this case are determined as the roots of β β βJ Bi Jo1 0( ) ( )− = (A.2-18) where J1(x) is the Bessel function of the first kind and first order. The temperature profile in a sphere is given as

[ ]Θ =

− +

−

=

∞

∑22

2 21

Bi eBi Bi

n

n nn

nsin( )sin( )

β ηη β β

β τ

(A.2-19)

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Page A.2-5

with the eigenvalues being given as the roots of β βcot( ) + − =Bi 1 0 (A.2.20)

For the cylinder and sphere the characteristic distance b used to define the dimensionless groups in Equations A.2-5, 6, and 13 is the radius of the body, and the dimensionless distance η is the fractional distance from the center of the body to the surface. For these two cases as well as the slab, the behavior of the temperature at the center of the body at longer times takes on the form of Equation A.2-16.

Values of the first (smallest) eigenvalue, β1, calculated from Equations A.2-15, 18,

and 20, are given for the three geometrical cases as functions of Bi in Figure A.2-1. Tabulated values are given in Appendix 6. These values can be used in Equation A.2-16, which is valid for all three cases.

0

2

4

6

8

10

12

14

16

18

20

0 0.5 1 1.5 2 2.5 3 3.5β 1

1/B

i=k/

hb

Slab

Cylinder

Sphere

Figure A.2-1. The first eigenvalue β1 for heat conduction in a slab, cylinder, or sphere, given in relation to the Biot number. At a given value of Bi, the magnitude of the first eigenvalue is largest for the sphere and smallest for the slab.

For very large values of Bi, it is also possible to derive approximate asymptotic

forms for the β1 factor that appears in Equation A.2-16. The following approximations, which are accurate within about 1% for the range indicated, may be more convenient that tabulated values for use in data analysis:

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Page A.2-6

[ ]β π ε ε ε ε12 3

21 01775 0 3= − + − + <. ... .for (A.2-21)

for a slab,

β ε ε ε1

2

2 405 12

0 2= − +

+ <. ..., .for (A.2-22)

for a cylinder, and

[ ]β π ε ε ε131 329 0 2= − + + <. ..., .for (A.2-23)

for a sphere, where ε = 1/ Bi.

Plots of the complete forms of Equations A.2-14, 17, and 19 for a slab, cylinder, and sphere can be found in Perry's handbook or in one of a number of heat-transfer textbooks. (Kreith, 1958).

Apparatus The apparatus for this experiment consists of (1) a relatively large constant-temperature bath with automatic temperature control; (2) a circulation chamber for contacting a solid specimen with the bath fluid under controlled flow conditions;(3) a pump to circulate the bath fluid from the thermostat through the circulation chamber; (4) a mercury thermometer; (5) copper-constantan thermocouples connected to a digital thermometer; (6) a stop watch; and (7) solid test specimens of various shapes and materials, each with a copper-constantan thermocouple inserted at its center and mounting brackets attached for suspending it in the circulation chamber. The test specimens are shown schematically in Figure A.2-2. Physical properties and dimensions of the materials used are given in Table A.2-1. A diagram of the apparatus is given in Figure A.2-3.

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Figure A.2-2. Geometry of the test specimens. Table A.2-1. Physical properties and dimensions of the specimens used in Experiment A.2

Material of Construction

Density, ρ (lbm/in3)

Heat Capacity, CP (Btu/°F·lbm)

Thermal Conductivity, k (Btu/hr·ft·°F)

Specimen Shape and size

Aluminum Bronze 0.274 0.170 41 Sphere: D = 3.0 in.

Chrome Steel 0.283 0.113 26 Sphere: D = 3.0 in.

Carbon Steel 0.256 0.112 31 Cylinder: D = 2.0 in Slab: 2b = 2.0 in.

Brass 0.307 0.136 58 Cylinder: D = 2.0 in. Slab: 2b = 2.0 in.

Nylon 0.040 0.4 -- Cylinder D = 2.0 in. Slab: 2b = 2.0 in. Sphere: D = 3.0 in.

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Page A.2-8

Figure A.2-3. Diagram of the experimental apparatus.

Procedure 1. Turn on the thermostat, if it has not been done, and set the bath temperature at 60o C. 2. Turn on the circulation pump, and check that there is flow through the test chamber. 3. Ensure that the bath has reached the pre-determined temperature and that it is holding constant. 4. Inspect the digital thermometers and make sure that they are all reading room temperature properly. Note any offset observed with a particular thermocouple. 5. Choose a particular geometry for your experiment, slab, cylinder, or sphere. Check the dimensions, and note the materials of the solid test specimens that will be used. See Appendix 7. 6. Just prior to placing a test specimen in the circulation chamber, measure the bath temperature with the mercury thermometer, and record the temperature at the center of the test specimen. 7. Place the test specimen in the circulation chamber and record the temperature history at the center of the specimen by recording both time and temperature as the solid heats up.

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Page A.2-9

This may be done most conveniently by choosing a temperature, using the stopwatch to record when that temperature is reached, and then selecting the next temperature to be recorded. Be careful not to turn off the stopwatch when interval readings are taken. Take as many readings as possible at the beginning of a run while the temperature is changing most rapidly. Later the data may be taken in a more leisurely manner. Take data until the solid centerline temperature reaches at least 95% of its ultimate change, that is, until Θ ≤ 0.05 at η=0. 8. Just after the measurement of the temperature history at the center of the test specimen has been completed, once again measure the bath temperature with the mercury thermometer. 9. Repeat this procedure with two known materials (iron or steel and copper or brass) and with the unknown (nylon), all of the same shape and size. Note: Initial filling and any make-up of water lost from the bath should be done with hot tap water to minimize bath-temperature recovery time. Data Analysis The logic of this experiment is that the thermal conductivity of nylon, presumably unknown, can be determined in an apparatus in which the rate of heating of a nylon object can be observed. We have a theoretical model for the rate of heating of simple solid shapes that relates the changing temperature of the solid body to the thermal conductivity of the solid. There is a complication, however, in that the rate of heating also depends on the efficiency of the fluid in transferring heat to the surface of the solid. This efficiency depends on the properties of the bath fluid, the intensity of fluid mixing, and the geometry. These factors are characterized by the parameter h. In order to calibrate the apparatus, that is, to determine the value of h for the bath being used here, one observes heating rates with one or two materials of the same shape and size but with known thermal properties. Calculations:

1. For each of the known materials tested, plot the temperature-time data in dimensionless form on a semi-log graph, according to Equation A.2-16. Obtain the value of β1 from the slope of the linear region and calculate the Biot number, hL/k, from the appropriate equation (A.2-15,18, or 20) depending on the shape. Calculate the value of h. The fluid-film heat-transfer coefficient, h, is characteristic of the water bath, its flow rate, and of the shape of the solid body, but it is independent of the thermal properties of the solid. Therefore, the solid specimens should yield similar values of h.

2. After you have fit the semi-log plots of your temperature data for the two known materials and estimated the corresponding values of β1 (and Bi) from the slopes of the linear regions, estimate the values of the intercept ln A1 in Equation A.2-16 calculated from β1 and the truncated theoretical model. This theoretical intercept may be compared

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Page A.2-10

with the intercepts of your linear fit of the data for each known material in order to check for consistency. The theoretical values of A1 are the following:

ABi

Bi Bi11

12

21

=+ +sec

( )ββ

(A.2-24)

for the slab,

[ ] ( )

ABi

Bi J112 2

0 1

2=

+β β (A.2-25)

for a cylinder, and

( )

ABi

Bi Bi11

12 2

1

2=

+ −β

β β( ) sin (A.2-26)

for a sphere. Values of the Bessel function Jo(β1) are tabulated in Appendix 6.

3. From the temperature measurements on the nylon object, make an initial estimate of k by comparing the temperature-versus-time data with the theoretical form. Again, use only the data falling in the linear region of the semi-log plot. Because we can guess that the thermal conductivity of this polymer is relatively small, we can get an initial estimate of k by assuming that the Biot number is very large, that is, k/hb ≅ 0, and use the corresponding theoretical value of β1. (If this approximation turns out to be a good assumption, we would not need to determine the actual value of h from measurements with known materials.) 4. Refine your estimate of k of nylon by accounting for the effect of a finite Bi on β1, that is, the effect of the finite resistance to heat transfer in the water. This could be done by iterative calculations, starting with the initial estimate of k obtained in the previous step and successively revising the values of β1, α, and k until a good fit of the data is obtained. More conveniently, you may solve for k directly by noting that you have two independent expressions that may be solved simultaneously for β1 and k with nylon. The procedure is the following.

First, you have in Appendix 6 or Figure A.2-1 a relation between β1 and Bi into which you can substitute the definition of Bi from Equation A.2-13 and the known value of h to obtain β1 as a function of k. Also, you can plot the dimensionless temperature calculated from the experimental data with nylon versus real time. According to Equation A.2-16 and the definition of τ in Equation A.2-6, the slope of the linear region of such a plot, called m, will be

m b= − β α1

2 2( / ) (A.2-27)

Substituting the definition of α into Equation A.2-27 yields a second relation between β1 and k. Simultaneous solution of the two relations gives the values of k and β1 for the nylon.

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Page A.2-11

The two relations may be solved graphically by plotting them both as curves of β1 versus k and determining the location of their intersection. References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, Inc., New York (2002). Carslaw, H.S., and J.C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press, London (1959), pp. 121-124 Jacob, M., Heat Transfer, Volume 1, John Wiley and Sons, Inc., New York (1949), pp. 270-287 Kreith, F., Principles of Heat Transfer, International Textbook Company, Scranton (1958), pp. 137-145

ChE 324 Lab Manual

Part B Measurement of Profiles of Velocity, Temperature, and Concentration

ChE 324 Lab Manual

Experiment B.1 Velocity Profiles in Steady Turbulent Flow

Page B.1-1

Experiment B.1

VELOCITY PROFILES IN STEADY TURBULENT FLOW

For fluids in steady laminar flow in simple geometries, it is often possible to predict theoretically the exact velocity distribution of the fluid. This is the case in straight tubes with cross sections that are either circular, concentric annular, rectangular, or elliptical (Bird et al., 2002, pp. 48-56). In order to calculate such velocity distributions from the differential equations of motion that are based on conservation of mass and momentum in the fluid, one must express the flux of momentum by molecular mechanisms, called τ, in terms of the velocity gradient ∇v. Such expressions are given by Newton’s law of viscosity, for Newtonian fluids, and by a number of empirical models for non-Newtonian fluids (Bird et al., 2002, pp. 231-257).

In the case of steady turbulent flow, it is not possible to predict exactly the time-smoothed velocity distribution of a fluid because of the complex nature of the “turbulent momentum flux”, that is, the average rate of momentum transport by random turbulent eddies. A number of empirical and semi-empirical relationships have been used to describe this additional momentum flux and the resulting time-smoothed velocity distributions in certain types of turbulent flows. In this experiment the nature of turbulent flow is examined by measuring velocity distributions for water flowing through a smooth tube, under turbulent conditions, at various flow rates. The results will be analyzed in terms of standard models of the structure of turbulent boundary layers and of the turbulent transport of momentum.

Theory

When the velocity is fast relative to the fluid viscosity, fluid flow becomes unstable with respect to various disturbances, and the streamlines experience instantaneous fluctuations in both magnitude and direction called turbulent eddies. For pipe flow this usually occurs when the Reynolds number, Re = D<vz>ρ/µ, exceeds 2100. The fluctuations in velocity provide an additional mechanism for momentum transport across the time-averaged streamlines, with the result that the rate of momentum transport to a solid wall is increased. Also, the enhancement of the momentum flux by turbulent eddies modifies the time-smoothed velocity profile of the fluid; in regions of effective turbulent transport steep velocity gradients are not needed to drive momentum transport by viscous forces. Thus, the time-smoothed velocity profile can be flatter than the corresponding laminar flow field. On the other hand, near a solid wall, where the velocity fluctuations are blocked or damped out, viscous forces must carry the momentum flux into the wall. A laminar sublayer exists near the wall in which the velocity gradient becomes steep according to Newton's law of viscosity.

The principle of conservation of momentum and the associated equations of motion, are still valid in the case of turbulent flow. When the equations are averaged

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Page B.1-2

over a time period that is long compared with the frequency of the turbulent fluctuations one obtains equations for the time-smoothed velocity profiles. As shown by Bird et al. (2002, §5.2),

Time-smoothed equation of continuity

0)( =•∇ v (B.1-1)

Time-smoothed equation of motion gττv ρρ +•∇−•∇−−∇= ][][ )()( tvp

DtD

(B.1.2)

These equations contain, in addition to the usual viscous transport and inertial terms, extra terms that arise from the mixing effects of the eddies. The over bars indicate time-smoothed quantities. The extra terms are identified as the turbulent momentum flux,

)(tτ , and called the Reynolds stresses. If one could identify a general relation between the turbulent momentum flux and the time-smoothed velocity gradient for turbulent flows, in a form analogous to Newton's law of viscosity, then one could solve the time-smoothed equations of motion to obtain the averaged velocity profile and the shear stress on the walls.

A number of empirical relationships have been proposed to describe the turbulent momentum flux; Bird et al. (2002, §5.4) summarizes some of them. The empirical relations of Prandtl and Diessler were combined with the time-smoothed equation of motion and experimental data to yield the so-called Universal Velocity Profile, which agrees closely with experimental data for the time-smoothed velocity distributions in pipes at Reynolds numbers greater than 20,000. The profile has three identifiable regions: the laminar sublayer, the buffer layer, and the turbulent core. The three following semi-empirical expressions and ranges given by McCabe et al.(2001) describe the profile very closely:

Laminar sublayer: + = yv+

for 50 ≤≤ +y (B.1-3)

Buffer layer: 05.3ln .05 −= ++ yv for 305 ≤≤ +y (B.1-4)

Turbulent core: ( ) 5.5 + ln 2.5 = ++ yv for 30≥+y

(B.1-5)

The dimensionless velocity v+ is defined as

*v

vv z=+ (B.1-6)

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Page B.1-3

where ρτ o=*v and 0τ is the normal shear stress (or momentum flux) at the wall. Also,

µρ*vy

y =+ (B.1-7)

where y is the distance from the wall. The Universal velocity profile is plotted, along with numerous experimental data for turbulent flow in pipes, in Figure 5.5-3 of Bird et al. (2002). Although these equations and the corresponding plot fit data on the turbulent velocity profile in pipes at high Reynolds numbers, an awkward aspect of this approach is that Equation B.1-5 (and the corresponding graph) do not recognize the existence of the centerline of the pipe, where the velocity profile should be flat. The pipe radius R does not appear in these correlations because they focus on the effect of the wall, namely the shear stress τ0, on the structure of the turbulent boundary layer. The average shear stress at the wall can be determined from a macroscopic force balance on the pipe. For steady flow in a horizontal pipe the shear stress on the wall balances the net pressure force acting axially on the fluid. That is, ( )2 0

20π τ πRL R p pL= − (B.1-8)

Therefore, the wall shear stress is given by

( )

τ 00

2=

−p p RL

L (B.1-9)

As an alternative to fitting the turbulent momentum flux in order to derive the time-smoothed velocity profile in turbulent flow, one may simply correlate experimental data on the velocity profile in a particular geometry. For pipe flow at Reynolds numbers between 104 and 105, Prengle and Rothfus (1955) reported that

71

max,

1v

v

−=

Rr

z

z (B.1-10)

Schlichting (1951) has broadened the applicability of Equation B.1-10 by letting the exponent be an empirical function of Reynolds number. That is, he proposed the following empirical equation to describe the velocity distribution for steady flow in round tubes:

n/1

maxz, Rr 1v = v

−z (B.1-11)

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Page B.1-4

where the constant n reported by Schlichting depends on the Reynolds number as summarized in Table B.1-1. Although extremely simple and in certain respects unsatisfactory, Equation B.1-11 is convenient. For example, it allows one to relate the maximum velocity in a pipe to the average velocity (See Problem 5B.1 in Bird et al., 2002), but it cannot be used to calculate shear stress at the wall nor pressure drop. Table B.1-1. The Constant n of Equation B.1-11 as a Function of Reynolds Numbers

Re 4 x 103 7.3 x 104 1.1 x 105 1.1 x 106 2.0 x 106 3.2 x 106 n 6.0 6.6 7.0 8.8 10 10

One of the simplest methods of measuring point velocities within a flowing fluid is with an impact tube, also called a pitot tube, which is described by McCabe et al. (2001). By conversion of kinetic energy head to static pressure head at the mouth of a tubular probe, the undisturbed velocity in an impinging streamline can be related to the rise in pressure within the impact tube above the static pressure in the fluid at the point of impact. When this pressure difference is measured by a manometer, the local velocity of the fluid impacting the mouth of the tube vn is related to the manometer reading by the relation

( )

OH

c

OH

ban

pg

22

2sinh g 2 = v

2/1

ρρρρ

θ∆

=

−∆ (B.1-12)

where vn is fluid velocity normal to the mouth of the tube, ∆h is the differential length reading on the manometer scale, θ is the angle of the manometer relative to the horizon, and the subscripts in the density-difference term refer to the heavier manometer fluid (a) and to the lighter fluid (b) above it. The second expression can be used when the pressure drop is measured directly using an electronic transducer.

When the manometer is damped, as it is in this experiment, and the impact tube is aligned with the pipe axis, Equation B.1.12 may be used to relate the time-averaged value of the differential reading of the manometer, ∆ h , to the time-averaged axial point velocity, )(v rz . Be careful that the calculated velocities are dimensionally correct.

Apparatus

The apparatus for this experiment is illustrated in Figures B1-1 and 2. The equipment consists of 1. A test section of cylindrical pipe that is equipped with two piezometer rings for

measuring the local static pressure and a traversing impact tube with a static pressure tap. (Inside diameter of test section = 1-1/16 inches; distance between piezometer rings = 3 ft; length of test section before impact tube = 5 ft; length of test section

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Page B.1-5

after impact tube = 1 ft.) The configuration of the impact tube is shown in Figure B.1-1. A scale on the probe-positioning mechanism is graduated in tenths of an inch.

2. Two manometers (24-inch air-over-water, and 15-inch water-over-mercury). The manometer board can be oriented at several angles relative to the horizontal in order to amplify the ∆h reading.

3. A source of clean water and a 55-gallon galvanized steel drum supply reservoir. 4. A scale, collection container, and stopwatch for measuring the mass flow rate of

water. 5. A thermometer 6. Auxiliary piping (1.5 inch, Schedule 40, galvanized iron pipe) as shown in Figure

B.1-2. Valve V-10 controls the flow rate through the pipe. 7. A centrifugal pump driven by a 1.5 horsepower, 60 cycle, 220 volt, 3-phase electric

motor at 1800 rpm. The pump is rated to deliver 45 gal/min under 25 ft of liquid head at 1750 rpm.

Figure B.1-1. Diagram of the test section and traversing impact tube.

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Page B.1-6

Figure B.1-2. Diagram of the piping configuration for Experiment B.1.

Procedure Measure the time-smoothed velocity profile in the pipe at three Reynolds numbers according to the following procedure. 1. Locate all the valves and become familiar with the operation of the equipment,

particularly that concerning functions of the manometers and the pressure taps on the test section. Check and record dimensions of the apparatus, and measure the water temperature.

2. Purge the manometer lines of air as follows:

a) Start with all valves closed except V-10, which is to be fully opened. Let water flow through the test section. b) Turn valves V-1 and V-2 to the “piezometer” position. c) Open valves V-3, V-4, and V-5. When no more air bubbles are visible in the manometer, close V-3, V-4, and V-5.

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Page B.1-7

d) Open V-8 and V-9. When no more air bubbles are visible in the transparent plastic tubing, close V-8 and V-9. e) Turn valves V-1 and V-2 to the “impact tube” position. f) Open valves V-3, V-4, and V-5. When no more air bubbles are visible in the manometer, close V-3, V-4, and V-5. g) Open V-8 and V-9. When no more air bubbles are visible in the transparent tubing, close V-8 and V-9.

3. Adjust the height of the water columns in the air-over-water manometer as follows:

a) Start with all valves closed. There is to be no water flowing through the test section, but the test section is to be full of water.

b) Open valves V-8 and V-9. c) If the water columns are too high on the manometer scale, open V-6 and turn

V-1 and V-2 to the “piezometer” position. Inject air into the top of the manometer by means of the pressurizing bulb. When the heights of the water columns have been adjusted to the desired level, close valves V-1, V-2, and V-6.

d) If the water columns are too low on the manometer scale, turn on the pump, open V-7 and turn V-1 and V-2 to the “piezometer” position. Slowly open V-10. When the heights of the water columns have been adjusted to the desired level, close valves V-1, V-2, and V-7.

e) Close valves V-8 and V-9.

4. Turn on the pump, and adjust the flow rate through the test section to the approximate flow rate desired. Determine the actual mass flow rate by collecting and weighing the discharge from the test section. It is suggested that you make runs with valve V-10 fully open and with it set approximately at two-thirds and at one-half of fully open. The by-pass valve on the pump should remain closed. Also, the water level in the reservoir should be kept fairly constant and above the entrance of the outlet line.

5. Determine the rates of momentum transfer in the test section by measuring the

pressure drop between the piezometer rings. Valves V-1 and V-2 should be set to the “piezometer” position. Record no manometer readings when the manometer differential is less than two inches. In order to get reasonable accuracy with a significant ∆h value, either adjust the angle of the manometer or switch from the water-over-mercury to the air-over-water manometer by closing valves V-3 and V-5 and opening V-8 and V-9. On the other hand, be careful not to blow mercury from the manometer with a large pressure difference.

6. Determine the fluid velocity at a radial position in the test section by measuring the

pressure differential between the impact tube and the static pressure tap for various positions of the probe. (Turn valves V-1 and V-2 to the “impact tube” position). Again, take no manometer readings when the differential is less than two inches. Also, be certain to allow ample time for the flow and manometer reading to become stable. Record the position of the probe for each pressure measurement.

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Page B.1-8

7. Perform Step 6 for several radial distances from the center of the pipe. Make

measurements on both sides of the centerline of the pipe to obtain a complete velocity profile. In particular, take readings at the extreme positions of the probe, close to the walls, in order to identify the relationship between the scale on the probe-positioning device and the coordinates within the pipe. Do not, however, use excessive force in positioning the probe against the pipe wall.

8. Perform Step 5 once for each flow rate. 9. Perform Steps 4 through 8 for at least three different flow rates. (It is suggested that

the first measurement be made at the highest attainable water rate and that subsequent measurements be made at successive reductions of the flow rate.)

10. Immediately mop up any spilled water from the floor. Data Analysis

1. For each volumetric flow rate:

o Calculate τo from the pressure drop measurement using Equation B1-9.

o Calculate the mass flow rate and average water velocity from the timed collection and/or the electronic flow meter data.

o Calculate the velocity profile and plot it as max,v)(v zz r versus r/R. Plot the profiles for all flow rates on the same graph.

o Integrate the velocity profile to compute the flow rate of water. Calculate the corresponding average velocity >< zv and the Reynolds number. Tabulate the mass flow rate obtained from weighing and from this integration and discuss the differences.

o Fit the experimental local velocity data to Equation B.1-11 and determine the apparent value of n; compare the results with values reported in Table B.1-1. Calculate the average velocity in the pipe for each flow rate from the resulting equations, and compare with those determined by weighing.

2. For the largest flow rate only, plot the velocity profile in the form ++ yversusv using the calculated value of τo. In the same graph, plot the

universal velocity profile using Equations B.1-3, 4, and/or 5, as appropriate, or Figure 5.5-3 of Bird et al. (2002). Discuss the results.

3. Check whether the shear stress and pressure drop can be calculated from Equation B.1-11.

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Page B.1-9

References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, Inc., New York (2002). McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering, 6th Edition, McGraw-Hill, Inc., New York (2001), pp. 105-106, 232-234 Prengle, R.S., and R.R. Rothfus, Ind. Eng. Chem., 47, 379-386 (1955). Schlichting, H., Boundary Layer Theory, Fourth Edition, McGraw-Hill Book Co., Inc., New York (1960), pp. 504-506.

ChE 324 Lab Manual

Experiment B.2 Temperature profiles in solid rods

Page B.2-1

Experiment B.2

TEMPERATURE PROFILES IN SOLID RODS

When heat flows through a solid body by conduction, Fourier's law gives the local rate of heat flow. As shown Experiment A.2, it is often possible to predict temperature profiles within the body for both steady and unsteady-state conditions. Solution of the differential equation that expresses conservation of energy within the solid to obtain the temperature profiles depends, however, on the thermal boundary conditions imposed upon the body. At a surface, possible boundary conditions include specification of the temperature, specification of the heat flux, or knowledge of some fixed relation between temperature and the flux, such as Newton's law of cooling.

In Experiment A.2, the flow of heat was essentially one dimensional, and Newton's law of cooling was used to represent the resistance of heat flow in the fluid phase, which lay in series with the resistance of the solid itself. By calibrating the apparatus to obtain the magnitude of the heat-transfer coefficient in the fluid adjacent to the solid surface, one was able to calculate the thermal conductivity of a solid material from the observed temperature changes. Experiment B.2 is similar in nature except that the thermal conductivities of the solid materials used are all known, and the heat transfer to the fluid is by free--not forced--convection. Although the flow of heat here is actually two-dimensional, a one-dimensional approximation is used to describe the temperature profiles.

In this experiment, long, solid metal rods of different materials and diameters are heated by steam at one end. Heat flows down the rod and it is transferred to the surrounding air by free convection. Initially, the heat flow is in an unsteady state, while the rods are heating up. Eventually, when the rate of input of heat from the steam is balanced by the loss of heat to the air, the temperature profiles in the rods reach a steady state. A simplified theoretical model of this process is presented. Measurements of the axial temperature profiles allow comparison with the theoretical model and estimation of the fluid-film heat-transfer coefficient for the air surrounding the horizontal cylinders.

The situation studied in this experiment is representative of heat-exchanger fins, that is, masses of metal attached to heat-exchanger surfaces to provide increased surface area for transfer of heat from a solid to a fluid. Such structures enhance the overall rate of heat transfer if the resistance to heat flow in the solid is less than that in the fluid film.

Theory

A long, solid rod of uniform circular cross section with radius R, and initially at a uniform temperature equal to that of the surrounding air, is suddenly heated at one end by a constant-temperature heat source. Assume that the following approximations are acceptable: The physical properties of the solid are independent of temperature, the rod is very long, a temperature gradient ∇T exists only in the axial direction, and the fluid-film heat-transfer coefficient, h, of the air on the cylindrical surface is independent of position.

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Page B.2-2

The key simplification in this analysis is to neglect the radial variations of temperature within the rod and to solve for the temperature as a function only of axial position z and time t. The justification for this assumption is that the rod diameter is small compared to its length and that, because the metal has a high thermal conductivity, the resistance to radial heat flow lies primarily in the air phase. Thus, at any axial position, the temperature across the rod is essentially uniform, and the surface temperature is equal to the average across the cross section.

Based on this assumption, one can consider a thin slice of the rod, of thickness ∆z, apply an energy balance, and take the limit as ∆z goes to zero to derive a differential equation for temperature T. The result of this shell-balance procedure is

( )∂∂

α ∂∂

βTt

Tz

T Ta= − −2

2 (B.2-1)

where Ta is the bulk temperature of the ambient air. In Equation B.2-1 the two parameters are the thermal diffusivity of the solid

αρ

=kCp

(B.2-2)

and a factor containing the heat-transfer coefficient h

βρ

=2h

R Cp

(B.2-3)

The β factor has dimensions of (time)-1 and should no be confused with the eigenvalues used in Experiment A.2.

Note that treating the problem as a one-dimensional conduction problem has put the heat-transfer coefficient, which normally appears in a boundary condition, into the differential equation. Thus, the last term in Equation B.2-1 accounts for the local loss of energy from the metal to the surrounding air.

The appropriate boundary conditions for this problem are the following:

T T for t and all za= ≤ 0 (B.2-4)

T T for z and ts= = >0 0 (B.2-5)

T T for za= → ∞ (B.2-6)

where Ts is the temperature of the energy source at the base of the rod. The last condition applies if the rod is long enough that its temperature far from the source remains unaffected and in equilibrium with the air.

One may verify that the solution to this problem, given by Carslaw and Jaeger (1959), is

T t z T

T Te erfc z

tt e erfc z

tta

s a

z z( , ) / /−−

= +

+ −

−12 4 4

β α β α

αβ

αβ (B.2-7)

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Page B.2-3

where erfc is the complementary error-function, defined as

( ) ( )erfc y erf y e dnny

= − = − −∫1 1 2 2

0π (B.2-8)

Figure B.2-1 shows a plot of the error function and the complementary error function.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

x

erf(x

), er

fc(x

)

Figure B.2-1. Error function and complementary error function. The error function starts at 0 and asymptotically approaches 1, whereas the complementary error function starts at 1 and asymptotically approaches 0.

At steady state, Equation B.2-7 becomes simply

T z TT T

ea

s a

z( ) /−−

= − β α (B.2-9)

The instantaneous rate of total heat discharge by the rod to the air Q may be computed either by applying Fourier's law at the base of the rod,

Q R kTz

z

= −=

π∂∂

2

0

(B.2-10)

or by using Newton's law of cooling for the cylindrical surface

( )Q R h T T dza= −∞

∫20

π (B.2-11)

The solution for T(t,z) from Equation 7 may be used in either of the latter two equations. In general, the result is

( )Q t R k T T et

erf ts a

t

( ) ( )= − +

−

ππ α

βα

ββ

2 (B.2-12)

erf

erfc

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Page B.2-4

and at steady state it becomes

( )Q R k T Tss s a= −π βα

2 (B.2-13)

Apparatus

Figure B.2-2 illustrates the apparatus for this experiment. The equipment consists

of:

1. A steam chest that serves as a constant-temperature heat source. The steam chest is connected to the low-pressure steam line by a quick-release flexible hose. The steam pressure should remain constant at about 8-10 psig with the main valve completely open and the steam trap operating properly.

2. Three metal rods mounted horizontally with one end inserted into the steam chest. Two of the rods are made of aluminum (Alloy 2011-T3), and one is made of steel (Type 304 SS). The aluminum rods have diameters of 0.5 and 1.0 inch; the steel rod has a 1.0-inch diameter.

3. Copper-constantan thermocouples installed on the centerlines of the rods at specific axial distances from the steam chest. The locations of the thermocouples measured from the first thermocouple are 0.1, 0.3, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, and 3.0 feet.

4. Either a portable precision potentiometer with manual switches to record thermocouple readings or a computer system interfaced with the equipment to log temperature measurements from the array of thermocouples as functions of time.

5. A copper-constantan thermocouple probe designed to measure surface temperatures.

6. A stopwatch.

The properties of the metal rods are given in Table B.2-1.

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Page B.2-5

Table B.2-1. Physical Properties of the Metal Rods

Material Thermal

Conductivity, k (cal/cm-s-oC)

Density, ρ (g/cm3)

Heat Capacity, Cp (cal/g-oC)

Stainless steel (304) 0.039 8.0 0.12 Aluminum (2011-T3) 0.53 2.7 0.23

Figure B.2-2. Diagram of the Apparatus for Measuring Temperature Profiles in Solid Rods Procedure Measure the transient and steady-state temperature profiles in all three rods according to the following procedure. 1. Record the air temperature in the room.

2. Drain any condensate from the steam lines and steam chest before beginning the test.

3. The time-position-temperature data will be logged on a computer using LabVIEW software. The control program is located on the following path:

Labview/324 Thermocouples.llb/324 Expt. B.2 - Temp Profiles

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Page B.2-6

Only two switches on the virtual instrument panel are used to record the data:

a) The "Run" arrow ( ⇒ ) on the left side of the tool bar starts data acquisition.

b) The "Enable/Off" green switch stops acquisition and saves the data to a file.

The default sampling-time interval is 30 seconds, which should be adequate for this experiment.

To become familiar with the data acquisition system, check the rod temperatures (ambient) sampled by the computer before the steam is turned on. Start acquisition by clicking on the Run arrow, and stop acquisition after a few minutes by clicking on the green Enable/Off switch. DO NOT CLICK ON THE STOP ICON AS THIS WILL PREVENT THE DATA FILE FROM CLOSING PROPERLY AND ALL YOUR DATA WILL BE LOST. A prompt for a filename will appear within 2 cycles of the sampling interval (default is 2 x 30 sec). Save the initial thermocouple readings to a file on a diskette. Temporary data storage in the Labview directory on the C: drive is permissible, but data will be erased weekly.

After the ambient temperature profile has been recorded in a file, it is safest to close and re-open the LabVIEW program file (“324 Expt. B.2 - Temp Profiles”) so that all settings are re-initialized.

4. Measure and record the lengths of the rods and the distance of the closest thermocouple position from the outside of the steam chest. Measure the positions of the other thermocouples to check the values given above.

5. Open the steam valve and monitor the pressure in the steam chest. Start the computer program by clicking on the Run arrow just after the steam is introduced. Wait until steady state has been reached in all of the rods before stopping the data acquisition. (This may take ~ 2 hours for the stainless steel rod.)

During the data-acquisition period it is very important that the computer program is not interrupted or stopped because the time scale will subsequently be incorrect. When steady state has been reached, the data may be saved to a spreadsheet-compatible file by clicking on the Enable/Off switch. It may take up to 2 cycles of the sampling time for a prompt to appear. DO NOT "CANCEL" THE FILENAME PROMPT WHILE WAITING FOR THE FILE TO BE SAVED, OR ALL DATA WILL BE LOST.

During data acquisition it is possible to change the scaling on the graphical display or to scan through the data appearing in spreadsheet format on the screen. Be careful not to accidentally interrupt the acquisition process.

6. Record the steam pressure in the steam chest, and look up the corresponding saturation temperature for steam. This is the value to be used as Ts.

7. With the surface-temperature probe, attempt to measure the surface temperatures on the rods at several locations to see if they differ significantly from the interior temperatures.

8. Observe and record once again the air temperature in the room and the steam pressure to confirm that the boundary-condition temperatures have not changed significantly

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Page B.2-7

during the experiment. If either Ta or Ts change, it will be necessary to use average values in the data analysis.

9. If there is time at the end of the lab, use a fan to blow air across the rods, and observe the new steady-state temperature profiles.

Be careful not to touch any hot surfaces, and do not disconnect the steam line nor open the steam chest while there is live, pressurized steam in the apparatus.

Data Analysis

From the steady-state temperature profiles, one can estimate average values for

the air-film heat-transfer coefficient on each rod. Plot the steady-state profiles on a semi-log graph, according to Equation B.2-9, to estimate the values of h. If the measured rod temperatures do not seem to approach the steam temperature Ts at the edge of the steam chest, adjust the nominal location of the origin of z by choosing z=0 to be the position where the steady-state profile extrapolates to Ts. This adjustment may be a bit different for each rod because of different conduction effects within the steam chest itself.

Estimate the steady-state flow rate of heat Qss from the steam chest to each rod by

estimating from the data the temperature derivative dTdz z=0

and using Equation B.2-10.

Compare these values with the results calculated from Equation B.2-11 and the estimated values of h.

Using the temperatures of the rod and the air at locations 2, 5, and 8, calculate the predicted local heat transfer coefficient. Equation B.2-14 shows the correlation between the Nusselt,Grashof, and Prandtl numbers for free convection (Bird et al., 2002, §14.6).

[ ]4/1

9/416/9Pr)(

Pr)/492.0(1671.0772.0 GrNu

+= (B.2.14)

Equations B.2-15,16, and 17 show the definitions of the dimensionless numbers

fkhDNu = (B.2-15)

f

f

kCp

Prµ

= (B.2-16)

2

23

f

ff TgDGr

µβρ ∆

= (B.2-14)

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Page B.2-8

Note that βf in Equation B.2-14 is the coefficient of thermal expansion for air (1/Tf if ideal gas is assumed), not the convection factor defined in Equation B.2-3. The subscript f indicates that the properties are evaluated at the “film” temperature, i.e., the arithmetic average of the metal surface and air bulk temperatures.

Test the consistency of your data and the validity of the theoretical model by comparing the transient temperature data for each of the three rods with values predicted by Equation B.2-7 and the values of h obtained from the steady-state temperature profiles. Specifically, plot the experimental dimensionless temperatures at the location of the third thermocouple, counting from the steam chest, as a function of β t , and show the theoretically predicted curve on the same plot.

The equations developed in the Theory section have been programmed in a Mathcad sheet, and it is presented as Appendix 8. Students may copy this Mathcad program from the Appendix on the course web page and use it as a template to calculate the theoretical temperature profiles and heat flows for various values of h.

As an optional exercise, one might estimate the local heat flux from the rod surfaces by radiation. Especially at higher temperatures radiation can be a significant mechanism for heat transfer that acts in parallel with conduction and convection. Approximate calculations can indicate whether this mechanism is significant a particular situation and should be considered in the process model; see for example Bird et al. (2002, §16.5).

References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, Inc., New York (2002) Carslaw, H.S., and J.C. Jaeger, Conduction of Heat in Solids, Second Edition, Oxford University Press, London (1959), pp.133-135.

Revised 1/2002 by R.Chavez

ChE 324 Lab Manual

Experiment B.3 Concentration profiles in a stagnant gas

Page B.3-1

Experiment B.3

CONCENTRATION PROFILES IN A STAGNANT GAS

Because of the similarity between the mathematical functions that describe diffusion of chemical species and those for conduction of heat, it is often possible to identify analogies between these two phenomena. For example, for one-dimensional transport across a stagnant layer of material at steady state, one finds that the profiles of either temperature or concentration should be linear. Also, the flux of heat or mass is linearly proportional to the relevant driving force, i.e., the difference in temperature or concentration across the layer, and is inversely proportional to the thickness of the layer. The proportionality factor in each case is the respective transport property, thermal conductivity or molecular diffusivity. This linearity provides the basis for the film model in defining heat and mass-transfer coefficients and leads directly to useful analogies between the two processes.

With diffusion, however, an intrinsic complication can cause the analogy with heat conduction to break down. The diffusion process itself can constitute a net mass flux across an otherwise stagnant layer. Thus, convection occurs as a result of diffusion, and the concentration profile of a solute is distorted from its otherwise linear form. Furthermore, the driving force for the net flux of the solute in this case is no longer its concentration difference across the film. Rather, the driving force for the molar flux becomes a more complex function of concentration. The impact of these complicating effects depends on the nature of the transport process itself, that is, whether components other than the solute are also moving across the film, and on the level of concentration of the solute.

In this experiment acetone is allowed to evaporate into nitrogen. The steady-state concentration profiles of acetone vapor in an otherwise stagnant layer of nitrogen are measured to assess the degree to which they deviate from the straight-line form predicted by a simple analogy with heat conduction. Also, an unsteady-state concentration profile is measured and used to estimate the diffusivity of acetone vapor in nitrogen. The theoretical analysis of the diffusivity determination takes into account the net convective velocity arising from the diffusion process.

Theory

Consider the physical situation illustrated by Figure B.3-1. A reservoir of volatile liquid A is placed at the bottom of a column of length L containing stagnant, insoluble gas B. At the bottom of the column, z=0, the gas is saturated with vapor at a mole fraction xA0. At the top of the column at z=L there is a cross flow of pure gas B that

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Page B.3-2

maintains a concentration xA0=0. The column of gas is thus viewed as a stagnant film of B across which A diffuses.

Figure B.3-1. Diffusion of a vapor A through a stagnant fluid B

If one assumes a pseudo-steady state in the gas column, application of a steady-state, one-dimensional shell balance, for each gas indicates that the flux Niz of each component must be constant across the film. Specifically, the flux of vapor is everywhere equal to its value at the interface:

N NAz A= 0 (B.3-1)

As Bird et al. (2002, p. 543) show, Equation B.3-2 gives Fick’s law for a binary

mixture

( )N cdxdz

x N NAz ABA

A Az Bz=− + +D (B.3-2)

where c is the molar density of the gas mixture.

Because component B is insoluble, its flux at the gas-liquid interface must be zero, and since the flux is a constant (steady state), it must therefore be zero everywhere:

N Bz =0 (B.3-3)

Therefore, Equation B.3-2 becomes

( )

Nc

xdxdzA

AB

A

A0 1= −

− D

(B.3-4)

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Page B.3-3

Equation B.3-4 may be integrated, to obtain the concentration profile

11

110 0

−−

=

−

xx x

A

A A

zL

(B.3-5)

which becomes linear as xA0 gets small, thus

−=

→ Lzxzx AA

xA

1)(lim 00

(B.3-6)

Equation B.3-6 is analogous to the temperature profile in a similar situation.

From combination of Equations B.3-4 and 5 the flux of vapor is given by

( )Nc

L xc

L xxA

AB

A

AB

BA0

00

11

=−

=

D Dln

( ) ln

(B.3-7)

where (xB)ln, the log mean of the mole fraction of B across the film, is defined as

( )xx x

xx

BBL B

BL

B

lnln

=−

0

0

(B.3-8)

In the limit of very small liquid volatility, (xB)ln→1.0, and the flux becomes

000lim A

ABAx

xL

cNA

D=

→ (B.3-9)

which is analogous to the comparable heat-conduction result because the convection term in Equation B.3-2 becomes negligible as the solute concentration becomes very small.

The unsteady-state diffusion in this geometry, for the case where the concentration of vapor A in the gas is initially zero and the time is short enough that the concentration xA at z=L is not yet significantly different from zero, is analyzed in Bird et al. (2002, §20.1). As a simplifying mathematical approximation, the top of the column is taken to be at z=∞. The result for the concentration profile is given in Equation B.3-10:

( )x z t

xerf Z

erfA

A

( , )( )0

11

=− −+

ϕϕ

(B.3-10)

where erf(y) is the error function and

Z ztAB

=4D

(B.3-11)

The quantity ϕ is a dimensionless interfacial velocity that is a function of xA0 given implicitly by

( )[ ] 12exp11

1−

++=

ϕϕϕπ erfxAo (B.3-12)

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Page B.3-4

A table of ϕ(xA0) is presented in Appendix 9.

The unsteady-state concentration profile given by Equation B.3-10 for a specified concentration is included in Appendix 9. It is also plotted in Figure 20.1-1 of Bird et al. (2002) with xA0 as a parameter The interfacial flux of vapor corresponding to this solution is

N ctAAB

0 =ϕD

(B.3-12)

where ϕ is the function of xA0 indicated above, and c is the molar density of the gas-vapor mixture, which is assumed to be uniform across the film and can be estimated from the ideal-gas law. In the limit of small xA0, ϕ becomes equal to x A0 / π . In this case the flux given by Equation B.3-12 becomes directly analogous to the unsteady-state heat flux given by Equation 12.1-10 in Bird et al. (2002).

Observe that the steady-state concentration profile given by Equation B.3-5 depends on xA0 but is independent of the diffusivity DAB. The diffusivity of the vapor may be determined by measuring the flux NA0 in either the steady or unsteady state and then by applying either Equation B.3-7 or B.3-11. Alternatively, one can determine the diffusivity by observing the unsteady-state concentration profile and comparing it with Equation B.3-9.

Apparatus The apparatus for this experiment consists of the following: 1. A diffusion cell of uniform cross-sectional area that is divided into seven

compartments, as shown in Figure B.3-2. By rotation of the cell body, the compartments may be lined up to form a vertical column, or the compartments may be displaced and isolated from one another. Each compartment is fitted with small ports covered with rubber septa through which gas-sampling syringes may be inserted. The bottom of the cell is shaped to receive a cylindrical liquid reservoir that establishes a gas-liquid interface. At the top of the cell are a screen and a gas manifold through which one can pass a gas stream. The diameter of the cell chambers is 2.0 inches. The height of each section is 1.25 inches. The distance from the top of the upper cell section to the screen separating the diffusion column from the gas manifold is 1.0 inch.

2. A liquid reservoir in the form of a glass cup fitted with a vertical side arm to permit observation of the liquid level in the cup.

3. A gas-storage cylinder providing pure nitrogen gas to be used as the inert gas phase and a rotameter for monitoring the gas flow rate;

4. Pure liquid acetone for use as the evaporating material;

5. A gas chromatograph with an integrating chart recorder and gas-sampling syringes for measuring the gas-phase concentration of acetone vapor;

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Page B.3-5

6. Thermostatted polyethylene bottles containing acetone and air to provide calibration standards of saturated vapor at several temperatures

7. A barometer;

8. A thermometer;

9. A stop watch.

Figure B.3-1. Diagram of the Gas Diffusion Cell used in Experiment B.3

Procedure Diffusion of acetone vapor in nitrogen gas is observed by means of the following procedure: 1. Locate all components of the equipment and become familiar with their operation,

including the gas chromatograph.

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Page B.3-6

2. Calibrate the gas chromatograph for acetone-concentration determination using samples of (i) pure nitrogen or air, (ii) air saturated with acetone at room temperature, and (iii) air containing one or two intermediate concentrations of acetone vapor. Obtain the latter by sampling acetone-saturated gas from bottles held at 10oC and at 15oC in the thermostats. Construct a calibration curve of chromatogram peak-area ratio (acetone:air) versus mole fraction acetone. Calculate mole fractions assuming gas-liquid equilibrium at the thermostat temperatures, and use the ideal-gas law. Note that vapor-pressure data and Antoine-equation parameters for acetone are given in Appendix 10. Use about 3.0 ml gas samples. Do not change any settings on the chromatograph. Be careful not to introduce any liquid into the chromatograph and avoid contaminating the syringes with any liquid. (Note that air and nitrogen generate the same response in these particular chromatographs so that it is possible to use air in the calibration procedure.)

3. Check the gas-chromatograph calibration by running replicate samples. Each chromatograph gives a slightly different response, according to its column temperature, detector temperature, helium flow rate, etc. Therefore, each group should work on only one chromatograph. It should turn out, however, that the area-ratio calibration curves for the several available instruments are quite similar.

4. Align the diffusion cell to form a single vertical column, and close off all the sampling ports with the rubber caps provided.

5. While the chromatograph is being calibrated, start the diffusion process as follows:

a) Place the empty liquid reservoir in position at the bottom of the diffusion cell. Note that the locking screws holding the reservoir in place are spring-loaded; do not tighten them to the maximum extent.

b) Introduce the gas stream flow through the gas manifold at the top of the diffusion cell at a rate of 500-600 ml/min.

c) Block off the outlet of the gas manifold so that the nitrogen flushes the diffusion cell free of all unwanted gases. The gas flow is directed downward through the cell and out the side arm of the liquid reservoir.

d) Close the sampling-port valves.

e) Allow the gas flow to pass through its manifold-exit port and, as rapidly as possible, introduce the liquid acetone into the side arm of the liquid reservoir until the liquid level is nearly at the top of the reservoir.

f) Begin to time the diffusion process from the moment that the reservoir is filled with liquid.

g) Measure and record the liquid level in the reservoir relative to the bottom of the cell.

6. Allow the diffusion process to proceed for about 40-50 minutes, by which time the concentration profile should attain its steady state. Be careful not to disturb the flow of the outlet as this would destroy the profile.

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Page B.3-7

7. Stop the diffusion process by rotating, smoothly and slowly but not too slowly, the movable section of the cell to displace and isolate the alternate sections of the gas column. Be careful not to obstruct the outlet stream.

8. After the cell compartments have been isolated, determine the average composition of acetone vapor in each section of the cell in the following manner:

a) Open the valve on the gas-sampling port.

b) Insert a syringe through the septum and mix the gas in the chamber by a gentle pumping action with the syringe.

c) Slowly draw a gas sample, of about 3.0 ml, and run it in the chromatograph.

d) Run duplicate samples from several cell compartments to check the accuracy of the GC data.

9. Measure at least one unsteady-state concentration profile. Flush the cell and set it up for diffusion by following the same procedure as above, but close the cell and stop the diffusion process about five minutes after introducing the liquid into the reservoir.

Data Analysis

From the chromatography data calculate the concentration profiles in the cell for each run. Tabulate the average mole fraction of acetone in each compartment as a function of position (and time in the unsteady-state case) by using the distance from the gas-liquid interface to the mid-point of the cell chamber. Note that xA0 will not correspond exactly to that based on the vapor pressure of acetone at room temperature because the liquid surface will be cooled a bit by the evaporation process. Also, although xAL should be close to zero because of the flushing flow of pure nitrogen through the gas manifold, it is possible that the actual value may be small but finite.

. Extrapolate the concentration profile to the position of the interface, z=0, to estimate xA0, and extrapolate to the screen at z=L to check xAL, by comparing the experimental steady-state concentration profile with Equations B.3-5 and 6. This may be done by plotting experimental values of xA versus z on rectangular coordinates along with theoretical curves computed for various values of xA0 (with xAL = 0 as assumed in the Theory section). Identify the value of xA0 that agrees best with your data. Calculate the interfacial temperature that would give such a mole fraction at equilibrium.

Calculate (xB)ln to assess the magnitude of its influence in Equation B.3-7. Plot the ratio of the acetone flux given by Equation B.3-7 to that given by Equation B.3-9 as a function of xA0. This plot shows the relative deviation from the simple analogy with heat conduction across a stagnant film.

With the unsteady-state data, use the concentration measured in each compartment and Equation B.3-10 to estimate the binary diffusivity of acetone vapor in nitrogen DAB at the prevailing temperature and pressure. This calculation is set up as a Mathcad program given in Appendix 9. Average the values of DAB calculated, and compare the result with experimental values found in the literature and with a theoretical estimate calculated from, Equation B.3-13 (Bird et al., 2002, §17.3).

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Page B.3-8

ABDAB

BAAB

MMT

cD,

25

11

102646.2Ω

+

×= −

σ (B.3-13)

In Equation B.3-13, c is the molar concentration in mol cm-3, DAB the diffusivity in cm2 sec-1, T is the temperature in Kelvin, MA and MB are the molecular weights of the two species, and σAB and ΩD,AB are the binary Lennard-Jones parameters given by Equations B.3-14 and 15.

)(21

BAAB σσσ += (B.3-14)

κε

κε

κε BAAB = (B.3-15)

And the collision integral, ΩD,AB is a function of κT/εAB . The tabulated collision function and the Chapman-Enskog calculation are given in Appendix 9.

For the calculation of the parameters in Equations B.3-14 and 15 one may use the molecular parameters given in Table B.3-1 (McCabe et al.,2001).

Table B.3-1. Lennard-Jones parameters for diffusivity calculation.

σ, Å ε/κ, K

Nitrogen 3.798 71.4

Acetone 4.600 560.2

References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, Inc., New York, 2002. McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering, McGraw-Hill Book Co., New York, 2001, p. 1090.

ChE 324 Lab Manual

Part C Measurement of Transport Coefficients

ChE 324 Lab Manual

Experiment C.1 Friction factors for flow in circular tubes

Page C.1-1

Experiment C.1

FRICTION FACTORS FOR FLOW IN CIRCULAR TUBES

When a fluid flows through a pipe, the fluid exerts a drag force on the pipe wall. This force is the manifestation of axial momentum being transmitted from the fluid to the wall. One may view it as fluid friction because the equal and opposite force of the wall on the fluid tends to retard the flow. The tangential drag force per unit area of fluid-solid interface is called the shear stress, τ0. At steady state, a macroscopic force balance on a section of straight, horizontal, circular pipe of length L and radius R indicates that the fluid friction causes a pressure drop in the direction of flow:

( )p p R RLL02

0 2− =π τ π , (C.1-1)

where p0 is the fluid pressure at the entrance of the length of pipe, and pL is the pressure at the exit. Equation C.1-1 means that the force due to the upstream pressure, acting on the fluid in the direction of the flow, is balanced by the tangential drag force and by the downstream pressure acting in the opposite direction.

For practical purposes, it is desirable to relate τ0 to the flow rate of the fluid so that one may use Equation C.1-1 to calculate the relationship between flow rate and pressure drop in pipes. For Newtonian fluids, Newton's law of viscosity gives the shear stress at the wall:

τ µ∂∂0 = −

=

vrz

r R

(C.1-2)

where µ is the fluid viscosity, vz is the fluid velocity in the axial direction, and r is the radial coordinate in the pipe. For laminar flow, one can solve for the steady-state velocity profile vz(r) as shown in Section 2.3 of Bird et al. (2002). With that profile, one can then calculate the total volumetric flow rate of fluid Q as well as the derivative in Equation C.1-2. These two results are combined to obtain the Hagen-Poiseuille law for a horizontal pipe,

( )

Qp p R

LL=

−πµ

04

8 , (C.1-3)

which may be used to calculate either Q from (p0-pL) or vice versa.1

With turbulent flow, however, the chaotic nature of the flow and the radial transport of momentum by random eddies make it impossible to calculate the velocity profile from first principles. In Experiment B.1 it was seen that it is even difficult to measure the velocity profile near the pipe wall very accurately. In this case Equation

1 For a non-horizontal pipe, one must substitute the pressure p with the expression p-ρgz to include the effect of the static head. This quantity is designated by a script P in Bird et al.,2002.

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Page C.1-2

C.1-2 is of limited utility, and one must resort to empiricism in order to relate pressure drop to fluid flow rate and to the properties of the system. Based on the physical principles that govern fluid flow, even when the related differential equations cannot be solved, one can define a useful quantity for correlating experimental data on fluid flow behavior in laminar or turbulent flow and in more complex geometries. This quantity, called the friction factor, is calculated from measured values of pressure drops and flow rates for various geometries and fluids. According to dimensional analysis, friction factor values may be presented in fairly general dimensionless correlations that can be used for practical predictions of flow behavior. In this experiment, one measures pressure drop as a function of flow rate for water passing through horizontal pipes of several sizes. From the data a general correlation of the friction factor is constructed.

Theory

Consider the physical situation illustrated by Figure C.1-1.

Figure C.1-1. Flow inside a pipe.

For fully developed, steady-state flow the local, time-smoothed shear stress at the wall τ0, given by Equation C.1-2, is independent of position on the wall because the time-smoothed velocity profile is independent of the axial distance and of the angular position. A dimensionless friction factor f for this case may be defined as the proportionality factor in the relation

τ ρ021

2= < >

v fz (C.1-4)

where <vz> is the average velocity in the pipe, defined by

2vRQ

z π=>< (C.1-5)

This f is called the Fanning friction factor for pipe flow.

Combination of Equation C.1-4 with Equation C.1-1 yields the relation

f RQ

p pL

L=−

πρ

2 5

20 (C.1-6)

Pressure po Pressure

R D

z= z=L

z

r

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which allows calculation of f from experimental data on pressure and flow rate. At the same time, combination of Equation C.1-4 with Equation C.1-2 gives another expression for f:

2v

2

><

−

= =

z

Rr

z

rv

fρ

∂∂

µ (C.1-7)

which provides a theoretical form for computing f if the detailed velocity profile is known.

Even if the velocity profile cannot be computed from basic principles, as in turbulent flow for example, the underlying differential equations may be subjected to dimensional analysis as shown in Section 3.7 of Bird et al. (2002). The conclusion of such a dimensional analysis for steady, fully developed Newtonian flow in a straight pipe is that form and that the solution of the basic equations must be a general function of the form

Re);(vv

),,,(v *

RrRr

zz

z =><µρ (C.1-8)

Where the left hand side of equation C.1-8 is the dimensionless velocity profile. The Reynolds number is defined as

µ

ρ><= zD vRe (C.1-9)

where D is the pipe diameter, D=2R.

Substituting Equation C.1-8 into Equation C.1-7 leads to the conclusion that the friction factor function for incompressible, Newtonian fluids in a straight, smooth pipes should be simply

f f= (Re) (C.1-10)

More generally, the friction factor can depend on dimensionless geometrical ratios. For example, for shorter pipes in which the velocity profile is developing and τ0 depends on the distance downstream from an entrance, the expected relationship for the average friction factor is of the form

f f LD

= (Re, ) (C.1-11)

Or if the pipe wall is rough with a scale of roughness of average magnitude k, the correlation for f in fully developed flow is expected to be

f f kD

= (Re, ) (C.1-12)

One can calculate values of f from flow data on selected fluids in straight pipes using Equation C.1-6. Those values may then be fitted into an empirical dimensionless correlation to obtain the dimensionless function indicated in Equation C.1-10 (or C.1-11

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for short pipes). According to the principle of dimensional analysis, the resulting dimensionless function should be valid for all Newtonian fluids for pipes of all sizes. Thus, when the flow rate is specified for a different fluid in a pipe of any size, one may calculate the Reynolds number and estimate f from the general correlation. Finally, the expected pressure drop in the pipe may be predicted with Equation C.1-6. A similar approach is applied in the definition and use of heat or mass transfer coefficients, as explained in Chapters 14 and 22 of Bird et al. (2002).

A well-known general correlation of f values has been constructed from a large body of data and is shown in Figure C.1-2. A larger diagram is presented in Figure 5.10 of McCabe et al. (2001) and in Figure 6.2-2 of Bird et al. (2002).

Figure C.1-2. Friction factor as a function of Reynolds numbers and pipe roughness. The lowest curve represents smooth pipes and f increases with roughness. The straight line at the left is for laminar flow.

On this plot is included the theoretical relationship obtained from Equation C.1-7 for fully developed laminar flow, for which the velocity profile is given by

−

−=

22

14

)(Rr

LRpp

v Loz µ

(C.1-13)

Also shown are empirical curves for turbulent flow in pipes of various degrees of roughness, characterized by the scale dimension k. Although there are two possible solutions for f at any value of Re, corresponding to both laminar and turbulent flow, the former is stable only at Reynolds numbers lower than 2100. The turbulent-flow friction factor is always greater than that for laminar flow because turbulent mixing enhances momentum transport to the pipe wall. Similarly, finite values of L/D or k/D tend to increase the value of f.

In many cases of pipe flow the flow rate is not known a priori so the Reynolds number and friction factor cannot be calculated directly. For example, in the case where the pressure drop is imposed by external conditions, the flow rate and hence the Reynolds

100 1000 10000 100000 1000000

0.001

0.010

0.100

Re

f

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number are dependent variables to be determined from the f correlations. In this case one may rearrange the relation given by Equation C.1-10 to solve for the quantity

Re( )

f D p p DL

L=−ρ

µ ρ0

2 (C.1-14)

which is independent of <vz> in order to obtain a new correlation

( )f f f= Re (C.1-15)

For example, Figure C.1-2 may be replotted in the form of Equation C.1-15. The quantity fRe can be calculated from a known pressure difference by means of Equation C.1-14 and used to estimate the value of f. Then, Equation C.1-6 may be used to predict the volumetric flow rate Q.

In the present experiment, one measures pressure drops and flow rates for water passing through horizontal pipes of three different diameters. From the data one may calculate values of both the Reynolds number and the friction factor f, from which a correlation corresponding to Figure C.1-2 may be constructed. It is instructive to compare regions of the empirical friction-factor correlation with relations predicted from various forms of the velocity profile according to Equation C.1-7. For example, the parabolic laminar velocity profile yields the theoretical relation, which can be confirmed experimentally for Re<2100,

f = 16Re

(C.1-16)

If the time-smoothed velocity profile in fully developed turbulent flow is fit by the power-law form given in Equation B.1-10 the corresponding result from Equation C.1-7 is

f = 0 07911 4

.Re / (C.1-17)

On the other hand, if one fits the velocity profile in turbulent flow with Equation B.1-5, a specific form of Equation C.1-15 is obtained:

[ ]1 4 0 0 4010ff= −. log Re . (C.1-18)

Equation C.1-17 is generally valid up to Re = 105, while Equation C.1-18 fits smooth-pipe turbulent-flow data up to Re=4x106 (Bird et al, 2002, §6.2).

Apparatus

The apparatus for this experiment is a rig mounted with three straight horizontal pipes of two different diameters. The dimensions of the pipes are given in Table C.1-1. The test rig is illustrated in Figure C.1-3. Each pipe has affixed a pair of piezometer rings for measuring pressure drops between the locations of the two rings. Tubing connects the

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piezometer rings to a pair of 60-inch manometers, one containing air over water and the other containing water over mercury.

Clean water is supplied to the inlet ends of the three pipes by a manifold fed by a roller-type, positive-displacement pump, and there are valves on each line to allow flow to pass, to connect with the manometers, and to control the flow rate, as indicated in Figure C.1-2. The following materials and equipment should be available: containers to collect water leaving the system outlet, a platform scale for weighing the amount of water collected, a stop watch for measuring collection time, and a thermometer for measuring the temperature of the water.

Table C.1-1. Dimensions of the test-section pipes

Tube

Location on test rig

Inside pipe diameter (in)

Length of test section between pressure taps (ft)

A top 0.250 5.0

B middle 0.141 5.0

C Bottom 0.141 2.5

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Figure C.1-3. Diagram of the rig with three test pipes and manometers for measuring flow rates and pressure drops in the pipes. The inlet flow comes from a positive-displacement pump fed from a water reservoir. The outlet flow is returned to the reservoir through a flexible rubber hose, which can also be directed into the weighing bucket. Procedure The relation between pressure drop and flow rate in the pipes is observed in the following manner. 1. Locate all valves and the pump controls. Become familiar with the operation of the

equipment, particularly the manometers but also the platform scales and stop watch. Check the dimensions of the system, particularly the distance between the manometer taps. Weigh the empty water-collection bucket, and record room temperature.

2. Purge the manometer lines of air by the following procedure:

a. Open valves V-2, V-3, V-4, and V-7, and let water flow through Tube A.

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b. Open valves V-18, V-19, and V-20 and then open valves V-12 and V-15. When no more air bubbles are visible in the transparent sections of the tubing, close valves V-12 and V-15.

c. Direct the flow through Tube B by opening valves V-5 and V-8 and closing V-4 and V-7.

d. Open valves V-11 and V-14. When no more air bubbles are visible, close valves V-11 and V-14.

e. Repeat with Tube C by opening valves V-6 and V-9 and closing valves V-5 and V-8.

f. Open valves V-10 and V-13. When no more air bubbles can be detected in the tubing to the manometers, close valves V-10 and V-13 as well as V-20.

3. Adjust the height of the water columns in the air-over-water manometer as follows:

a. Start with all valves closed except V-3 and the valves associated with the pump.

b. Open valve V-9 on Tube C.

c. If the water levels are too high on the manometer scale,

i. Open valves V-16, V-17, V-10, and V-13.

ii. Open valve V-22 and V-6, and let the water columns fall to a desired level near the mid-point on the scale.

iii. Close valves V-22, V-16, V-17, V-6, and V-9.

d. If the water levels are too low on the manometer scale,

i. Start the pump with the inlet-line and by-pass valves open.

ii. Open valves V-16, V-17, V-10, and V-13.

iii. Crack valve V-6 open only slightly.

iv. Open valve V-22, and let the water columns rise in both columns to a level somewhat higher than that desired.

v. Close valve V-22.

vi. Close valve V-6.

vii. Stop the pump.

viii. Adjust the levels following the procedure given above in Step 3.c.

4. Set the approximate water flow rate desired in the following manner:

a. Start with all valves closed except the inlet and by-pass valves associated with the pump.

b. With valve V-3 closed, open completely the valves that will allow water to flow through one of the pipes, except for the respective control valve, V-4, V-5, or V-6.

c. Open the valves needed to connect the appropriate manometer to the test pipe.

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d. Slowly open valve V-4, V-5, or V-6 and adjust it in order to set the pressure drop in the test pipe to the level desired. Table C.1-2 contains suggested manometer readings for which data might be taken, in order to cover a wide range of flow rates. These values are given only as a guide; they need not be set exactly. Set a steady flow rate in the test pipe that yields a manometer reading corresponding approximately to a value suggested in the table. Record the manometer’s actual, steady state reading.

5. Determine the flow rate by collecting a mass of water from the system outlet in the weighing receptacle during a timed period. Hold the outlet hose in a fixed position, and direct some flow into the floor drain. Then place the collection bucket under the hose outlet without changing its position or elevation. Weigh the mass collected, and record its weight and the corresponding collection time. Record also the water temperature for each run.

6. Adjust the control valve (V-4, V-5, or V-6) and, if necessary, the valves in the manometer-connection lines, to attain flow rates corresponding to the six or seven levels indicated in Table C.1-2.

7. Close valve V-3 after the flow rate and pressure drop values have been recorded. Close all other valves that were opened in Step 4.

8. Repeat Steps 4 through 7 for the other two pipes.

9. Turn off the pump and close the valve on the pump inlet.

Table C.1-2. Suggested values of manometer readings at which data should be taken. Entries are manometer differential in inches.

Tube Air-over-Water Manometer Water-over-Mercury Manometer A 1.0 2.0 9.0 --- 8.0 19.0 34.0 B 1.0 2.6 5.5 18.0 5.5 18.0 30.0 C 0.9 2.0 3.0 15.0 4.5 12.0 33.5

An Excel spreadsheet program for data entry and calculations of flow rates and friction factors is presented in Appendix 11. Data should be entered during the laboratory period and a preliminary log-log plot of friction factor vs. Reynolds number should be prepared. If there are any obvious gaps in the graph, make additional runs to complete the correlation of f with Re. Because the pump is a positive-displacement type, with only moderate slippage, there must be an open flow path downstream whenever it is turned on, or the pump by-pass valve must be open, to accommodate the nearly constant flow rate generated by the pump. Also, all valves upstream of the pump must be fully open at all times. Data Analysis

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Much of the data analysis is done automatically by the spreadsheet given in Appendix 11, which not only converts the manometer readings directly to friction factors but also calculates mass flow rates and Reynolds numbers. Calculate also the actual pressure gradients, (p0-pL)/L, in the test sections and report them in an appendix. Plot these values on log-log graph to demonstrate the varying dependence of pressure drop on flow rate as it moves from laminar flow to increasingly turbulent conditions. In addition to presenting a log-log plot of friction factor versus Reynolds number, prepare a plot in the form suggested by Equation C.1-15. Show your experimental results as data points and show the accepted relations given in the literature (Bird et al., 2002, McCabe et al., 2001, Perry and Green, 1984) as solid curves. Fit the turbulent-flow results for various regions according to Equations C.1-17 and C.1-18, and determine whether the experimental data fit better with modified values of the constants given in those equations. References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, Inc., New York, 2002. McCabe, W.L., J.M. Smith, and P. Harriott, Unit Operations of Chemical Engineering, 6th ed., McGraw-Hill Book Co., New York, 2001. Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed., McGraw-Hill Book Co., New York, 1984.

ChE 324 Lab Manual

Experiment C.2 Heat-Transfer Coefficients in Circular Tubes

Page C.2-1

Experiment C.2

HEAT-TRANSFER COEFFICIENTS IN CIRCULAR TUBES

When a fluid flows through a pipe whose wall temperature is different from that of the fluid, heat flows between the pipe wall and the flowing fluid, raising or lowering the fluid's temperature. This is the situation in most practical heat exchangers. The transport of heat in the flowing fluid occurs by both conduction and convection. If the flow is turbulent, the dominant mechanism of transport in the bulk fluid is turbulent mixing, that is, the convection provided by turbulent eddies. Near the pipe wall the turbulence is damped out, and the flow of heat is ultimately carried across the solid-fluid interface by molecular conduction. In laminar flow conduction is the mechanism of radial transport throughout the pipe's cross section, but axial convection also affects the temperature profiles.

In principle, theoretical methods based on Fourier’s Law and Newton’s viscosity law could be used to calculate temperature profiles and heat fluxes within the flowing fluid. In practice, however, especially with turbulent flow, the mathematical problem becomes intractable, and an alternative approach must be used to calculate the performance of heat-transfer equipment.

From a practical point of view, one usually wants to know how much the temperature of a flowing fluid will change as it passes through a heat-transfer pipe. That problem can be formulated in terms of a macroscopic energy balance. In many cases, specifically at steady state when kinetic energy, potential energy, and work effects are negligible, the macroscopic energy balance for a flowing fluid is

w H Q∆ = (C.2-1)

where w is the mass flow rate of fluid, H is its enthalpy per unit mass, and Q is the rate of heat transfer from the wall into the fluid. If thermal effects are dominant in the enthalpy change, if the fluid’s heat capacity, Cp, is assumed constant, and if no latent heat effects occur, Equation C.2-1 may be approximated by

wC T Qp ∆ = (C.2-2)

where T is the average temperature of the fluid. In both of these equations the symbol ∆ indicates the change in a quantity from the entrance to the exit of the system. If the value of Q in Equation C.2-2 is known or can be estimated, one can calculate the change in the fluid temperature as it passes through a heat-exchanger tube.

The heat transfer rate Q is defined as the integral of the heat flux over the surface area of the solid-fluid interface, and the flux is determined by the shape of the temperature profile at the wall according to Fourier's law (Equation A.2-1). The difficulty in calculating the detailed temperature profile in most flow situations, however, precludes the systematic calculation of Q from first principles. As an alternative one can

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implement an empirical approach based on a combination of theoretical analysis and experimentation, analogous to the use of the friction-factor approach used for pressure-drop calculations.

Instead of using Fourier's law to represent the heat flux at a solid-fluid interface, we may define a factor h, called the fluid-film heat transfer coefficient, by the empirical relation

q h T Tw w f= −( ) (C.2-3)

where qw is the normal heat flux at the interface, and the subscripts w and f refer to the wall surface and the bulk fluid, respectively. Equation C.2-3 is called Newton's law of cooling. It is based on the general observation that in most convective transport situations the interfacial heat flux is found to be proportional to the temperature difference across the heat-transfer boundary layer. The coefficient h is not a fundamental property of the fluid; it is generally a complex function of the fluid properties, the system geometry, and the flow conditions. Its utility lies in the fact that it can measured experimentally as an indication of the magnitude of the resistance to heat transfer in various situations. With sufficient experimental experience, combined with theoretical guidelines, it is possible to develop correlations for h from which typical values can be predicted.

In this experiment one observes the temperature changes in water flowing through tubes that are heated by steam. From the flow rates and temperature measurements, one can calculate the heat-transfer coefficient for various pipe diameters and flow rates in order to examine the typical behavior of h in pipe flow. Because the tube wall and the steam may also introduce some resistance to heat flow, it is necessary to consider the concept of the overall heat transfer coefficient in such a system.

Theory

Consider the physical situation illustrated by Figure C.2-1. A cool fluid is flowing through a pipe and being heated by a warmer fluid flowing on the other side of a separating wall.

Figure C.2-1. Heat transfer between two streams

It is necessary to distinguish among different definitions of heat-transfer coefficients for two reasons. First, because in heat-exchanger calculations one deals

Hot stream in Th1

Hot stream out Th2

Cold stream out Tc2

Cold stream in Tc1

d

d

z

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directly with the total heat rate, Q, rather than the local rate, qw, there is necessarily an averaging that takes place:

( ) ( )Q q dA h T T dA h T T Aw w f ave w f ave= = − = −∫∫∫∫ (C.2-4)

where A is the total interfacial area for heat transfer. The identity of have and its relation to the local value defined in Equation C.2-3 depend on what convention one chooses in calculating the average driving force (Tw-Tf)ave. If the local value of h is constant over the entire surface area A, a common approximation, it may be taken out of the integral in Equation C.2-4, and the appropriate choice for the average driving force is the logarithmic mean (Tw-Tf)ln, defined as

( ) ( ) ( )( )( )

T TT T T T

T T

T T

w fw f in w f out

w f in

w f out

− =− − −

−

−

ln

ln

(C.2-5)

The other complication that arises in the definition and use of heat-transfer coefficients is that the wall temperature is often not known. In most cases, heat transfer occurs from one fluid to another through a solid wall. Only the bulk fluid temperatures are readily measured. In this case it is appropriate to use an overall heat-transfer coefficient U. Locally, at a position at some distance z downstream in the pipe the flux across the inner wall may be written as

( )qr

dQdz

U T Tww

w h c= = −1

2π (C.2-6)

where rw is the inner radius of the pipe, and the subscripts h and c refer to the hot and cold fluids.

As shown in Bird et al. (2002, Example 10.6-1), the overall coefficient, which represents the total resistance to heat flow from the hot fluid to the cold fluid, is given by

( )

U rr h

r r

k r hw ww i

out j in j

j o oj

= + +

−

−

∑1

1

1 1ln /, , (C.2-7)

where hi is the local film heat-transfer coefficient of the fluid inside the pipe, ho is that for the fluid on the outside with its interface at radial position ro, and the summation accounts for solid cylindrical layers between the fluids, each with thermal conductivity kj. The subscript w on U indicates that it is based on the cylindrical surface area at rw. Other choices for the reference radius are possible. Because of the cylindrical geometry, the local radial flux and hence U vary according to the radial position where they are evaluated. The value of total heat flow from one side of the cylindrical wall to the other side is, however, fixed and invariant with this choice.

As with the film coefficient, there is a question of how U should be averaged when Equation C.2-6 is integrated to get Q. Normally, one assumes that U is constant with z. In that case, there is no distinction between local and average values of U if the integrated rate expression is written as

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Page C.2-4

( )( )Q U r L T Tw w h c= −2π ln (C.2-8)

that is, if one uses the log-mean temperature difference as the average driving force.

Note in Equation C.2-7 that the quantity 1/U may be viewed as a sum of resistances in series. In each fluid phase, 1/h is the contribution to the resistance. The addition of these terms is weighted by the radial positions of the interfaces because each fluid is exposed to a different area for heat transfer. The terms representing the solid layers may also be expressed in a simpler form. For example, if there is a layer of scale or deposit that has formed on a solid surface, and its exact thickness or conductivity are not known, the following substitution may be used:

( )r r rk h

or Rw o i

ff

ln /=

1 (C.2-9)

where either term on the right may be called a fouling factor (McCabe et al., 2001, pp 330-331). Which form is reported in a given study must be inferred from the units in which the fouling factor values are presented. Obviously, Rf is a resistance, whereas hf is its reciprocal.

For heating of a fluid in a double-pipe heat exchanger, one may combine the energy balance for each side, Equation C.2-1, with the integrated rate equation, Equation C.2-8, to derive a model of the process. If the value of U can be estimated, then the model allows calculation of the outlet temperatures and Q from the inlet temperatures, geometry, fluid properties, and flow rates. Conversely, if the inlet and outlet temperatures and flow rates are measured, one may calculate Q from Equation C.2-2 and U from Equation C.2-8.

In the special case where all of the resistance to heat transfer resides in one fluid phase, such measurements allow direct determination of h for that fluid. For example, if the resistances to heat transfer in the outside fluid and in the solid walls are negligible in Equation C.2-7,

U hw i≈ (C.2-10)

and one can calculate hi as a function of flow rate, fluid properties and geometry from measurements of temperatures and flow rates. Experiments conducted under such circumstances have been used to develop correlations for h values that can now be used for estimates in design calculations.

The correlation of film heat-transfer coefficients is guided by dimensional analysis of the equations of change (Bird et al. §11.5). For steady flow in a straight pipe, the conclusion of dimensional analysis is that the average heat-transfer coefficient may be represented by the functional relationship (Bird et al. §14.3)

Nu Nu LD

=

Re,Pr, (C.2-11)

where Nu is the Nusselt number, defined as

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Page C.2-5

Nu Dhk

= (C.2-12)

Re is the Reynolds number

Re= Dvρµ

(C.2-13)

and Pr is the Prandtl number

Pr =C

kpµ (C.2-14)

where D is the pipe diameter, v is the average fluid velocity, k is the thermal conductivity of the fluid, ρ is its density, µ the viscosity, and L is the length of the pipe. For long pipes, the effect of L/D usually becomes unimportant. On the other hand, the fact that the viscosity may be a strong function of temperature leads one to add another dimensionless group, the ratio of the viscosity at the pipe wall to that of the bulk fluid, into some correlations.

For laminar flow exact solutions for the temperature profiles and heat flux have been derived for some relatively simple situations. The results can be used to obtain a theoretical form for h, and they may be put into dimensionless form. For example, for a short heat-transfer section in a fully developed laminar pipe flow with constant wall temperature and constant fluid properties, the specific form of Equation C.2-11 is found to be

( )[ ]Nu D Lln/

. Re Pr /=1621 3

(C.2-15)

where the subscript on Nu indicates that the log-mean temperature difference is used to define have. Other theoretical forms have been obtained for other boundary conditions and limiting cases.

For turbulent flow correlations for h must be empirical and based on experimental data. It is often assumed that the general form of Equation C.2-11 for relatively long pipes can be represented by a function

( )Nu a b cw

d= Re Pr /µ µ (C.2-16)

where a, b, c, and d are constants, and µw is the viscosity at the wall temperature. One important result for pipe flow with Re>20,000 is

( )Nu w=0 026 0 8 1 3 0 14. Re Pr /. / .µ µ (C.2-17)

Bird et al. (2002, p. 436) present a plot of this equation and other correlations covering a range of Reynolds numbers from 103 to 105 for smooth pipes.

Thus, for geometries where individual film heat-transfer coefficients have been measured and correlated in generalized form, predicted values may be used to estimate the overall coefficient U from Equation C.2-7, and Equation C.2-8 may then be used for heat-exchanger design or simulation. Conversely, for unknown situations, experiments may be run to measure U. When estimates can be made for most of the components of U,

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Page C.2-6

either from empirical correlations or theoretical considerations, then one can calculate one unknown term, such as the h of one fluid or a fouling factor, from Equation C.2-7.

Experiments using steam simplify these calculations a bit. First of all, during condensation at a fixed pressure, the temperature of steam remains constant. Its enthalpy balance, Equation C.2-1, becomes

w Qs sλ = (C.2-18)

where ws is the rate of steam consumption and λs is its latent heat of condensation. Also, as long as the condensate drains from the heat-transfer surface readily, the film heat-transfer coefficient for steam is quite large, and its contribution in Equation C.2-7 usually becomes negligible.

Apparatus

The apparatus for this experiment is a rig mounted with three straight horizontal double-pipe heat exchangers. On each rig, all of the heated copper tubes have a length of 41 inches, but each has a different diameter. The dimensions of the tubes are given in Table C.2-1. The outside pipes, which are insulated and into which steam is introduced, all have a two-inch inside diameter, and they are fitted with a steam-pressure gauge.

Table C.2-1. Dimensions of the Heat-exchanger Tubes. All have length 41 inches.

Tube

Location on Test Rig

Inside Pipe Diameter (in)

Outside Pipe Diameter (in)

Unit No. 1

A top 0.1425 0.190

B middle 0.314 0.374

C bottom 0.788 0.874

Unit No.2

A top 0.1862 0.250

B middle 0.4325 0.501

C bottom 1.025 1.125

Unit No. 3

A top 0.250 0.3125

B middle 0.666 0.750

C bottom 0.913 1.050

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The test rigs are illustrated in Figure C.2-1. Each heat exchanger has affixed

copper-constantan thermocouples for measuring inlet and outlet temperatures of steam and water as well as a temperature on the outside wall of the copper tubing.

Also available are digital thermometers, collection buckets, stopwatches, and scales for determining water flow rate, a supply of low-pressure steam, and an auxiliary system for supplying water.

The construction of the heat exchangers is shown in Figure C.2-2.

Figure C.2-1. Diagram of the Rig Showing Three Double-pipe Heat Exchangers with Different Size Tubes for Heating Water with Steam.

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Figure C.2-2. Construction Details of the Double-pipe Heat Exchangers

Procedure Measure overall heat-transfer coefficients in the three heat exchangers by means of the following procedure: 1. Locate all valves and the pump controls. Become familiar with the operation of the

equipment. Confirm the dimensions of the system. Weigh the empty water-collection bucket.

2. Purge air from the water lines and from the tubes of the heat exchangers by the following procedure:

a. Open valves V-2A, V-2B, and V-2C, turn on the pump, and let water flow through all three tubes. Control water flow rate by means of these three valves. Keep the pump inlet valve wide open to avoid cavitation. The centrifugal pumps are robust and can tolerate any backpressure, but the piping manifold may leak when exposed to the maximum pump head. Keep the pump head below 100 psi by adjusting the pump by-pass valve. At the same time do not open the by-pass valve too wide, or the pump motor may overheat and trip the circuit breaker.

b. When there is no air remaining in the water lines, close the three water valves.

4. Open valve V-4 to clear the steam lines and jackets of condensate.

5. Test one heat exchanger at a time, in the order C, B, A, doing two runs with each. Start up each of the heat exchangers, A, B, or C, in the following manner:

a. Open valve V-2 A, B, or C, and allow water to run through one of the tubes at a maximum flow rate. The water flow is returned to the water storage reservoir.

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Page C.2-9

Check the flow rate by collecting and weighing a portion of water leaving the tube. Calculate the value of the Reynolds number. (See Appendix 12 for a spreadsheet that facilitates this calculation.) Cover a range of Re values, greater than 10,000, by making two steady-state runs with each tube. That is, in the second run with each exchanger reduce the water flow rate by 25 to 50% to obtain a lower value of Re. A lower-flow-rate run may be omitted in the smallest tube if two different values of Re greater than 10,000 can not be attained.

b. Open completely the steam-condensate drain valve, V-1 A, B, or C, for the heat exchanger to be tested. Then open the corresponding steam-supply valve, V-3 A, B, or C, to allow steam to enter. Control the steam pressure, if necessary, with the latter valve. Make sure that the purge valve V-4 is closed.

5. Determine the steady-state heat-transfer rate Q in each run with the following steps:

a. Adjust the water flow rate through the tube in order to obtain a Reynolds number of the desired magnitude.

b. Adjust the steam pressure in the shell of the exchanger to the lowest level that gives only a slight (2-4 oF) temperature difference between the steam inlet and the condensate outlet. Check the measured steam pressure and temperatures against the steam tables to confirm that all instruments are operating properly.

c. Adjust the water flow rate or the steam pressure such that the rise in the water temperature across the exchanger is between 10 and 40oF. This range allows reasonable accuracy in calculating the energy balance but does not introduce great variations in fluid properties. Take care especially to avoid boiling of water within the exchanger. That is, keep the outlet water temperature well below 212oF.

d. When all temperatures attain steady state, record their values.

e. Measure the water flow rate again at the final steady-state conditions.

6. Avoid cavitation in the pump by keeping the inlet water temperature below 160oF as well as keeping the valve on the water-intake line completely open. The water in the feed tanks heats up gradually as the experiments proceed. It may be cooled by city water in a double-pipe heat exchanger that is mounted on the back of the experimental rig. Crack open the cold-water inlet valve to keep the recycled water temperature in the storage tank from rising too much.

7. Shut down the operation of an exchanger by first closing the steam supply valve, V-3 A, B, or C, and then closing the condensate drain valve. Never allow steam to pass into an exchanger unless there is water flowing through the tube. Keep all valves closed except those for the exchanger being tested. Do not operate two heat exchangers at once.

Data Analysis Calculate the heat load Q and the overall coefficient Uw using Equations C.2-2 and C.2-8 with the measured temperatures and water flow rates for each steady-state run. Calculate also the Reynolds number attained by the water in each run. An Excel

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Page C.2-10

spreadsheet is presented in Appendix 12, and available on the web page, to facilitate the calculations.

Use Equation C.2-17, or another appropriate correlation, to estimate the water-side coefficient hi. Neglecting the resistance to heat transfer on the steam side, use Equation C.2-7 to estimate the fouling factor hf (or Rf) in each tube. Check published correlations for steam-condensation heat-transfer coefficients to examine whether neglecting the steam-side resistance is a reasonable approximation.

For order-of-magnitude calculations, one may neglect the curvature in the cylindrical geometry and use a planar approximation in Equation C.2-7. That is, the ratio of ro/rw may be taken to be unity. In this case the resistance of the scale or fouling may also be viewed as a planar layer, and Equation C.2-7 becomes (Bird et al., 2002, §10.6):

1

1 11−

−

+

∆+= ∑

j oojiwww hrk

xhr

rU (C.2-19)

where x∆ is the thickness of a solid layer.

From your experimental results for the fouling factor, estimate the thickness of the fouling layer, assuming that it consists of a layer of lime, i.e., calcium carbonate. You may find lime’s properties in Perry and Green (1984) or other sources.

References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, New York, 2002. McCabe, W.L., J.M. Smith, and P. Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill Book Co., New York, 2001. Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed., McGraw-Hill Book Co., New York, 1984.

ChE 324 Lab Manual

ChE 324 Lab Manual

Part D Analysis of Macroscopic Systems

ChE 324 Lab Manual

Experiment D.1 (Suppressed)

ChE 324 Lab Manual

Experiment D.2 Efflux time for a tank with an exit pipe

Page D.2-1

Experiment D.2

EFFLUX TIME FOR A TANK WITH AN EXIT PIPE

Although there exists a theoretical framework for describing fluid flow behavior by calculating complete profiles of velocity and pressure (Bird et al., 2002), many practical problems cannot be solved with such a fundamental approach. One difficulty that arises is that the geometry and boundary conditions are too complex to allow a precise mathematical formulation and solution. The other obstacle is that the flow may become unstable and turbulent, introducing a random component in local fluctuations that cannot be predicted in any precise way. In such situations the fundamental laws of physics still apply, but an alternative approach is needed to obtain descriptions with engineering utility.

For complex geometries or flow situations it is common to adopt a macroscopic approach rather than trying to solve the differential equations of change. Required equations include the mass balance, the momentum balance, and the mechanical energy balance, all applied to a general macroscopic system. A macroscopic system is defined by the boundaries that contain a phase and, for a flow system, by control planes at the entrance and exit points. Conservation principles are then applied to the entire volume of the system rather than to a differential element.

The macroscopic balances are written as algebraic equations for steady-state systems or as ordinary differential equation when an unsteady-state process is considered. The terms that appear in the macroscopic balances are of three types: First, the input and output terms that are evaluated at the entrance and exit planes of the system; second, terms that express the total-transport rate across the boundaries of the system; and finally, terms that involve a summation, i.e. an integral, of some quantity over the entire volume of the system.

Because the purpose of the macroscopic balances is to avoid calculating in detail the profiles of material properties, such as velocity and pressure, the calculation of the summation terms must be done approximately. Such terms may be analyzed by dimensional analysis, and the results combined with experimental data to develop generalized correlations that can be used for estimates in practical engineering calculations.

In this experiment one observes the rate of drainage of liquids from a tank through exit tubes of various sizes. This system is analyzed with a macroscopic mass balance combined with a macroscopic mechanical energy balance. In the later equation it is necessary to use dimensionless correlations to estimate the loss of mechanical energy caused by viscous dissipation within the fluid.

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Theory

Consider a vertical, cylindrical tank such as that shown in Figure D.2-1 that contains a Newtonian liquid up to the level h above the bottom of the tank. The top of the tank as well as the bottom of the drain pipe are exposed to the atmosphere. Thus, the liquid drains from the tank under the influence of gravity. A model of the draining process is sought in order to predict the rate at which the tank empties.

Figure D.2-1. A liquid draining from a cylindrical tank through a pipe

First, one may write a macroscopic mass balance for the liquid in the system, which is defined as the entire liquid volume from the top surface to the outlet of the pipe. The total mass of liquid in the system is

m R L R htot = +ρπ ( )02 2 (D.2-1)

where ρ is the density of the liquid, R is the radius of the tank, R0 is the pipe, and L is its length. A mass balance for the liquid contained in the system yields

ddt

m Rtot = − ρπ 02 v (D.2-2)

where v is the average velocity out of the bottom of the pipe. There is no velocity term at the top because there is no flow into the top of the tank.

Taking the density to be constant and substituting Equation D.2-1 into D.2-2, one obtains

dhdt

RR

=−

02

v (D.2-3)

This equation may be integrated from the initial liquid level h1 to a subsequent level h2 to obtain the corresponding efflux time te. That is,

t dt RR

dhe

t

h

he= =−

∫ ∫0

0

2

1

2

v (D.2-4)

R

R

h

L

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Page D.2-3

In order to perform the desired integration, one must know the velocity v as a function of liquid level h. That relationship comes from the macroscopic mechanical energy balance.

If one neglects the rate of change of the total mechanical energy in the system, by the so-called pseudo-steady-state approximation, then one can use the macroscopic mechanical energy balance, also called the Bernoulli equation (Bird et al., 2002, p.204).

vcm EEWwpLhgvv

−−=

+++

><><

∆ρ

)(21 3

(D.2-5)

. In addition to the pseudo-steady-state approximation, one can assume that there is no work term, that there is no overall pressure differential, that the fluid is incompressible, and that the velocity profile in the tube is essentially flat. With those assumptions Equation D.2-5 becomes

( )12

2vw

v− + =−

g L hE

(D.2-6)

where g is the acceleration of gravity, w is the mass flow rate through the pipe, and Ev is the rate of viscous dissipation in the liquid.

From the derivation of the Bernoulli equation, one can identify the source of the viscous dissipation term. For a Newtonian fluid it is possible to apply dimensional analysis to arrive at the conclusion that

E

evv 2

wv

=

12

(D.2-7)

where ev is a dimensionless function of Reynolds number and geometrical ratios and is called the friction loss factor (Bird et al., 2002, p.206).

The contributions to Ev or ev from various regions of the system are additive. For regions where most of the viscous dissipation arises from friction and steep velocity gradients, such as flow through valves or around elbows, one can identify a value of ev that is relatively constant. Some representative values for obstacles, valves, and fittings are tabulated in Bird et al. (2002, p 207). A more complete listing is given in Perry's Handbook (Perry and Green, 1984).

For the geometry considered in this experiment, the only ev value of interest is that associated with the entrance into the pipe. According to Bird et al.(2002), this value should be )1(45.0 β−=ve , where β is the ratio of the area of the pipe to that of the tank. There is no exit loss because the liquid leaves the tube in a streamline manner, without any extra turbulence generated by its contact with air. For the orifice, the value of ev is

( )( )2 7 1 1 2 2. /− −β β β .

For regions of straight pipe, one can replace the friction loss factor with the Fanning friction factor f:

e f LDv =4 (D.2-8)

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Page D.2-4

where D is the diameter of the pipe, L is its length, and f is a function of Reynolds number, Re (Bird et al., 2002, p.206)

For the case under consideration, the Bernoulli equation may now be written as

( )12

12

22

0

v2v,i− + = +

∑g L h v f L

Re

i

(D.2-9)

where the summation of friction loss factors includes all obstacles and disturbances that the fluid encounters. All terms in Equation D.2-9 may be viewed as energy per unit mass of fluid flowing. The first term on the left-hand side represents the kinetic energy carried out of the pipe by the fluid. Rearrangement yields an explicit expression for the instantaneous velocity in the pipe

( )

v=2g L + h

1+ 2f LR0

v,ii

+

∑e

12

(D.2-10)

Before Equation D.2-10 is substituted into Equation D.2-4, it is necessary to assess the relative magnitudes of the terms in the denominator. In particular, if the term containing the friction factor is significant and f varies with Re, is it necessary to deal with the velocity dependence of that term.

First, suppose that the liquid is in laminar flow, then

fR

=16

2 0

µρv

(D.2-11)

as given in Equation C.1-16. If one assumes in addition that the kinetic energy and other friction terms are negligible, then

( )

v=gR0

2ρµL hL+

8 (D.2-12)

which is the result of the Hagen-Poiseuille analysis. If this expression for v is substituted into Equation D.2-4, one obtains the efflux time to be

t LRgR

L hL he =++

8 2

04

1

2

µρ

ln (D.2-13)

At the other extreme, if the flow is sufficiently turbulent or the pipe is sufficiently rough that f is a constant (see Figure 6.2-2 in Bird et al., 2002), then the result of the integration is

( ) ( )t RR

f LR

e

gL h L he

v ii=

+ +

+ − +

∑2

1 2

20

2

0

12

1

12

2

12

,

(D.2-14)

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Page D.2-5

where any of the three resistance terms might be dominant.

Finally, in the intermediate range of turbulent flow, for 2100<Re<105, in smooth pipes, the friction factor may be expressed by the Blasius formula (Equation C.1-17),

f = 0 07911

4

.

Re (D.2-15)

If only the term containing f in Equation D.2-10 is significant, then the velocity is given by

( )( )

vL h R g

L=

+

2

0 0791

405 4

4 4

17ρ

µ. (D.2-16)

In that case, integration of Equation D.2-4 yields

( )( )

( ) ( )t RR

L

g RL h L he =

+ − +

73

0 0791

2

2

02

14

05

14

47

1

37

2

37

. µ

ρ (D.2-17)

It is possible that, as a liquid drains from a tank and the level h drops, the Reynolds number within the pipe changes from one regime to another. In that case, the integral of Equation D.2-4 must be evaluated in several parts, using the appropriate expression for the velocity in each region of h.

Apparatus

The apparatus for this experiment is a cylindrical tank fitted with a level gauge, as shown in Figure D.2-2. The tank has a 6-inch diameter, and its total depth is 10 inches. Various interchangeable exit pipes can be screwed into the bottom of the tank. The available exit pipes are listed in Table D.2-1. Also available are a blank plug and a simple plug containing an orifice with diameter of 0.188 inches.

Additional equipment includes weighing containers, platform scales, a stopwatch, a thermometer, and a graduated cylinder. In addition to water, a 60-weight-per cent aqueous sucrose solution is available to be used as a working fluid. Properties of aqueous sucrose solutions are given in Appendices 4 and 4a.

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Page D.2-6

Table D.2-1. Dimensions of the available exit pipes for the efflux time experiment.

Length, inches Inside Diameter, inches

24

24

24

12

6

3

0.125

0.188

0.314

0.188

0.188

0.188

Figure D.2-2. Sketch of the draining-tank apparatus with an exit pipe in place

Procedure

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Page D.2-7

1. Calibrate the depth gauge on the tank as follows:

a. Insert the blank plug in the bottom of the tank.

b. Fill the tank by adding successively known volumes or weights of water and noting the change in level on the gauge as each portion is added.

2. Determine efflux times for draining the tank and instantaneous flow rates as follows:

a. Attach one of the exit pipes or the orifice to the bottom of the empty tank, and fill the tank and pipe with water or sucrose solution. During the filling, seal off the end of the tube with a fingertip or a stopper.

b. Starting with a full tank, allow the liquid to start flowing from the pipe.

c. Record the times at which the liquid level in the depth gauge passes predetermined locations.

d. Measure the total time that it takes for the liquid level to travel from the top to the bottom of the depth gauge.

3. Make the above measurements with all pipes and with the simple orifice with water. Record the temperature of the liquid in each case. For ease of comparison, it is suggested that one begin each test with the same initial level of liquid in the tank.

4. Repeat the measurements for several pipes with the sucrose solution as time permits. Start with the shorter tubes, and then test the longer ones if there is time. Again, begin each test with the same initial liquid level.

5. During the laboratory session perform preliminary analysis of the data. In particular, estimate initial and final Reynolds numbers for each run. Try to schedule experiments so that data are taken for both laminar and turbulent flow.

6. When changing tubes, take care not to bend the tubes out of vertical alignment. Also, the O-ring seals need not be tightened very hard.

7. Return sucrose solution to its storage reservoir after use. It contains preservative to prevent spoilage. Do not taste.

8. Avoid entraining air bubbles in the sucrose solution by pouring it down the sides of the tank and of the storage reservoir.

9. Wash all equipment well after use with the sucrose solution, especially the depth gauge and the drain valve just below the gauge. Wash the tubes well before placing them in the soaking trays after use.

10. Try to avoid vortex formation in the tank during draining. Swirling flow can be prevented by placing a circle of polyethylene sheet or another floating material on the surface of the liquid.

11. Check and record the dimensions of the apparatus.

Data Analysis

Calculate the experimental exit velocity v as a function of h for each run. Use these values to calculate the Reynolds number at the beginning and end of each run. Use

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Page D.2-8

the appropriate equation from the Theory section to calculate te, and compare these values with the experimental values.

Plot efflux time te values from all water runs versus (L/R0) on a single graph as individual data points. Show on the same graph, as lines, the two theoretical relationships given by Equations D.2-13 and D.2-17.

In a table compare the experimental efflux times for the sucrose solution compared with the theoretical values calculated with the appropriate model, depending on the Reynolds number. In the table show also L/R0 and the average value of Re for each run.

By reference to the theoretical models, investigate the expected accuracy of the te predictions and compare the uncertainty in those values with the deviation from experimental values. As part of this exercise calculate the relative magnitudes of the contributions to the viscous dissipation term for the various experimental conditions studied.

References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, New York (2002). Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed., McGraw-Hill Book Co., New York, 1984.

ChE 324 Lab Manual

Experiment D.3. Heating Liquids in Tank Storage

Page D.3-1

Experiment D.3

HEATING LIQUIDS IN TANK STORAGE

Most practical process calculations in chemical engineering are based on macroscopic balances of mass, momentum, energy, and chemical species. The general equations that express the relevant conservation principles for macroscopic flow systems are given in Chapters 7, 15, and 23 of Bird et al.(2002). The general balances provide a framework for cataloging all of the various phenomena that influence the behavior a system. Use of appropriate expressions for rate processes and judicious neglect of unimportant terms result in relatively simple process models that describe system behavior for purposes of design, simulation, and quantitative characterization.

The macroscopic energy balance for nonisothermal systems involving pure fluids is (Bird et al., 2002, p. 456)

WQwHdt

dEtot −+

Φ++∆−= ˆv

21ˆ 2 (D.3-1)

where Etot is the total energy content of the macroscopic system, w is the mass flow rate of the fluid, H is enthalpy per unit mass of fluid, v is the average fluid velocity such that v2/2 is the kinetic energy per unit mass, Φ is the potential energy per unit mass, Q is the rate of heat addition to the system across the boundaries, and W is the rate of work done by the fluid on the surroundings. The notation ∆(x) indicates taking the difference between the value of a property x at the exit of the fluid from the system and its value at the entrance point. Equation D.3-1 may be viewed as a generalization of the first law of thermodynamics for flow systems.

Proper application of the macroscopic balances requires careful definition of the boundaries of the system to be considered. The boundaries across which Q flows are taken to be phase boundaries or the walls containing the flowing fluid. The entrance and exit planes must be chosen such that the fluid flow is perpendicular to the control plane, and the fluid properties, v, p, T, etc., are essentially constant over the plane. A subscript 1 denotes the quantities at the entrance, and a subscript 2 those at the exit. Then ∆(x)=x2-x1 can be defined unambiguously.

The utility of the macroscopic balances also depends on neglecting relatively unimportant terms in any particular application. Neglecting terms a priori may require a certain amount of judgment in order to obtain correct answers, but assumptions of relative magnitudes can generally be tested, either experimentally or mathematically.

In thermal processes, that is, processes involving primarily heat transfer and temperature changes, one can usually neglect the changes in kinetic and potential energy content of the flowing fluid. Also, the work term is zero unless a mechanical device that transmits power to or from the fluid is present. Finally, the total energy of the system, Etot, may be approximated by the total internal energy of the fluid Utot. In this situation a good approximation to Equation 1 is

ChE 324 Lab Manual


Page D.3-2

QHwHwdt

dU tot +−= 2211ˆˆ (D.3-2)

If the individual terms in Equation D.3-2 can be expressed in terms of temperatures, fluid properties, system geometry, and flow conditions, this approximate macroscopic balance provides a useful mathematical model of the thermal process. In this experiment, application of Equation D.3-2 is demonstrated by observing the heating of water in a storage tank with several modes of heating. The rate of temperature rise of the water is measured and compared with models derived from Equation D.3-2.

Theory

Consider a stirred storage tank that either contains a mass of cool liquid or is being filled with a flow of liquid. The liquid in the tank may be heated with steam either by a steam coil mounted in the tank or by circulating a stream of liquid through an external heat exchanger. The general case, showing both possibilities, is represented by Figure D.3-1.

Figure D.3-1. Diagram of a storage tank equipped for either internal or external heating.

It is desired to derive a mathematical model that describes the temperature rise in the liquid in the tank by appropriate treatment of the terms appearing in Equation D.3-2 for the various heating scenarios. Regardless of the mode of heating, a number of further simplifying assumptions are appropriate with this type of system. First, we shall assume that the fluid properties are constant. Second, assume that the liquid flow rates, into the

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Page D.3-3

tank or through the external heat exchanger, as well as the initial and inlet-stream temperatures and the steam temperature, are constant in each case. It is assumed that the heat losses to the surroundings are negligible. Finally, it is assumed that perfect mixing occurs in the tank, that is, the liquid temperature is uniform, if the agitator is employed, but that plug flow occurs in the heat exchanger and, if the agitator is not used, in the tank as well.

For the heat-transfer rate Q, appearing in Equation D.3-2, we shall use an overall heat transfer coefficient, as defined in Equation C.2-7. The overall coefficient is assumed to be constant over the heat-transfer surface area A. Thus, for the steam coil in the well-mixed tank the heat transfer rate is expressed as

( )Q UA T Ts= − (D.3-3)

where Ts is the steam temperature and T is the temperature of the liquid in the tank. For an external tubular heat exchanger the total rate of heat transfer is given by

( )Q UA T Ts= − ln (D.3-4)

where (Ts-T)ln is the log-mean temperature difference defined in Equation C.2-5. Be careful not to confuse the overall heat transfer coefficient with the internal energy, since the letter U denotes both.

The total internal energy of a liquid with constant heat capacity may be calculated as

( )U C T T Vtot p ave ref= −ρ (D.3-5)

where V is the volume of the liquid. The average temperature Tave is defined as

TV

TdVave = ∫∫∫1 (D.3-6)

and the reference temperature Tref is chosen as an arbitrary datum at which Utot=0.

The enthalpy per unit mass of a constant-density, constant-heat capacity liquid is similarly based on the chosen reference state. It may be written as

( ) ( )refrefp ppTTCH −+−=ρ1ˆˆ (D.3-7)

Usually the reference pressure pref is 1.0 atm. In many situations at low to moderate pressures and in systems where the pressure does not vary greatly, as in the present case, the pressure term in Equation D.3-7 is negligible.

For tank-heating processes such as those considered here, Equations D.3-4 to 7 may be substituted into Equation D.3-2 to obtain a model to predict the temperature variation in the tank. There are a number of specific possible cases to be considered. First, the liquid may be heated either by an internal steam coil, or it may be heated in an external tubular heat exchanger. With each of those possibilities, there are various cases depending on whether the tank is initially full or empty, whether there is flow into or out of the tank, and whether the tank is agitated. Some of the possible cases are treated

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Page D.3-4

below. It is assumed in the transient problems that the heat capacity of the tank walls and of the external heat-exchanger loop are zero so that only the volume of liquid in the tank is included in the accumulation term. The instructor will select the cases that a particular group should examine during the laboratory session.

Case I. The liquid in the tank is heated by an internal steam coil. Case I.a. The tank is initially full at room temperature. There is no flow. The tank is stirred. In this case one normally assumes that the temperature in the tank is uniform; that is, the local liquid temperature is everywhere equal to Tave. Equation D.3-2 becomes in this case

)(ˆ TTUAdtdTCV sp −=ρ (D.3-8)

because there are no flow terms. The initial condition is

T T= 1 at t = 0 (D.3-9)

where t is time measured from the instant that steam is introduced into the coil, and T1 is the initial temperature of the liquid. Equation D.3-8 may be integrated to obtain

lnT TT T

UAC V

ts

s p

−−

= −

1 ρ (D.3-10)

Case I.b. The tank is initially empty, but liquid flows in at a constant rate. The mass flow rate of liquid into the tank is w. The temperature of the inlet stream is T1. There is no flow of liquid out of the tank, so this case is applicable only up to the time when the tank is full. With agitation, one may again assume that T is uniform throughout the liquid in the tank.

For convenience one may choose the reference temperature to be the same as the inlet temperature. Then the enthalpy of the liquid feed is defined as zero and does not contribute to the energy balance. Thus, Equation D.3-2 becomes

( )[ ] ( )TTUAdt

TTVdC sp −=

− 0ˆρ (D.3-11)

In this case it is necessary to recognize that both V and A vary with time as the liquid level in the tank rises. The volume of liquid in the tank varies linearly with time and is proportional to the mass flow rate w. The variation of the heat-transfer-surface area depends on the geometry of the tank and the coil. As an example, one might assume that the submerged area of the coil is linearly proportional to the liquid volume. This case is examined by Bird et al. (2002, pp. 466-468). The result is

( )

( )pff

pff

s CVtUA

CVtUATTTT

ˆ/

ˆ/exp11

1

1

ρ

ρ−−−=

−− (D.3-12)

ChE 324 Lab Manual


Page D.3-5

where Af and Vf are the respective values of area and volume when the tank is full. This function starts at zero and approaches unity at infinite time. The temperature, Tf, attained when the tank is full, that is, when time reaches the value tf, identified as

tVwf

f=ρ

(D.3-13)

is

( )

( )pf

pf

s

f

CwUA

CwUATTTT

ˆ/

ˆ/exp11

1

1 −−−=

−

− (D.3-14)

Case I.c. The tank is initially full. Flow rates in and out are constant and equal. The tank is stirred Consider the case where the initial temperature in the tank and the temperature of the inlet stream are equal at T1. It is convenient to choose T1 as the reference temperature for calculating enthalpy and internal energy. The flow rate in and out of the tank is w. Equation D.3-2 then becomes

( ) ( )TTUATTCwdtdTCV spp −+−−= 1

ˆˆρ (D.3-15)

with the initial condition

T T= 1 at t = 0 (D.3-16)

This case is posed in Problem 15.B.8 of Bird et al .(2002)

The solution to this equation is

+−=

−−

∞

∞ tVw

CwUA

TTTT

p ρˆexp1

(D.3-17)

where T∞ is the steady-state temperature in the tank, given by

+

+

=∞

p

sp

CwUA

TCw

UAT

T

ˆ1

ˆ1

(D.3-18)

which is obtained by solving Equation D.3-16 with the time derivative set equal to zero.

Case II. The liquid is heated by pumping a side stream through an external tubular heat exchanger.

The mass flow rate through the heat exchanger is constant and equal to wh. Although the tank plus the heat-exchanger loop could be selected as the complete macroscopic system, it is more convenient to divide the process in two subsystems: the tank and the heat-exchanger loop. Such a separation allows one to calculate both the temperature of the

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Page D.3-6

stream returning from the heat exchanger Th and the average temperature in the tank T. It is assumed that the tank itself is insulated, so the heat-transfer rate Q for the tank is zero. Equation D.3-4 yields the rate, Qh, in the heat exchanger.

Case II.a. The tank is initially full when the heating is started, and the tank is perfectly mixed. Once again, the assumption of perfect mixing implies that the temperature in the tank is always uniform at T. Thus, the temperature of the stream entering the heat exchanger is T. On the other hand, the temperature of the stream returning from the heat exchanger is Th, thus, the log-mean temperature-difference driving force in the heat exchanger is

( ) ( )T T

T T T TT TT T

ss s h

s

s h

− =− − −

−−

ln

( )

ln (D.3-19)

Although both T and Th will be changing with time during the process, the total volume of liquid in the heat-exchanger loop may be assumed to be small compared with the volume in the tank. Under this condition, one can neglect the accumulation term in Equation D.3-2 for the heat exchanger and apply what is called the "pseudo-steady-state" approximation. That is, one uses a steady-state energy balance on the heat exchanger to obtain an equation relating Th to T at any given instant:

( ) ( )lnˆ TTUATTCw shph −=− (D.3-20)

where the log-mean driving force is given by Equation D.3-19.

The unsteady-state energy balance for the tank itself is simply

( )TTCwdtdTCV hphp −= ˆˆρ (D.3-21)

Combination of Equations D.3-19, 20, and 21 and integration yields the temperature in the tank T(t) as

lnT TT T

wV

KK

ts

s

h−−

=

−

1 1ρ

(D.3-22)

where K is an abbreviation for

=

phCwUAK ˆexp (D.3-23)

In this case the product UA refers to the overall heat-transfer coefficient and the heat-transfer surface area inside the tubular heat exchanger.

ChE 324 Lab Manual


Page D.3-7

Case II.b. The tank is initially only partly full. There is a constant flow rate into the tank, but no flow out. The tank is stirred. For this case, Equations D.3-19 and 20 still describe the performance of the heat exchanger, which is taken to be at pseudo-steady state. If the initial temperature in the tank and that of the feed are equal at T1, and the reference temperature for enthalpy and internal energy calculation is T1, the macroscopic energy balance for the tank is

( )[ ] ( )TTCwdt

TTVdC hphp −=

− ˆˆ 1ρ (D.3-24)

There is an inlet stream with flow rate w, but it does not appear in the energy balance because its enthalpy is by definition zero at T1. The right-hand side of Equation D.3-24 represents the net enthalpy input from the external heat exchanger. Equation D.3-20 can be used to obtain the temperature difference. Because the steam temperature Ts is constant, a simpler form of Equation D.3-20, relating Th to T, is

KCw

UATTTT

phhs

s ≡

=

−−

ˆexp (D.3-25)

Kern (1950) gives the solution to this problem as

( )

( ) ( )ln ln

ww

KK

T T

ww

KK

T T T T

ww

KK

V w tV

hs

hs

h

−

−

−

− − −

=

−

+

⋅

+

1

11 1

1

1

1

1

ρρ

(D.3-26)

where V1 is the initial volume of liquid in the tank. This solution is valid, of course, only up until the time when the tank is full, tf, which is given by Equation D.3-13.

Case II.c. The tank is initially full with no inflow or outflow, but there is no mixing in the tank. If there is no agitation of the liquid in the tank, there is likely to be some stratification with warm liquid remaining at the top and colder fluid settling to the bottom. Thus, one cannot take the temperature in the tank to be uniform, and the temperature of the fluid entering the heat-exchanger loop is not equal to the average temperature in the tank. If there is no mixing taking place, a more realistic model of the heating process is to take the movement of liquid in the tank to be plug flow. That is, one might imagine a sharp boundary between the cold fluid in the bottom of the tank and warmer fluid at the top. This boundary moves down the tank as the liquid is pumped around the loop through the external heat exchanger.

Initially, the liquid in the tank is at temperature T1, and this is the inlet temperature to the heat exchanger. The temperature in the top of the tank is that of the fluid returning from the heat exchanger, Th, given by Equation D.3-25 with T=T1. The boundary between the warm liquid and the cold liquid moves down the tank according to the rate of pumping wh and the volume of the tank. In particular, the time for the interface to reach the bottom of the tank is

ChE 324 Lab Manual


Page D.3-8

t Vwc

h

=ρ (D.3-27)

If the inlet and outlet pipes are placed essentially at the bottom and top of the tank, tc is the period of one cycle, that is, the time for the inlet to the heat exchanger to experience a step change in temperature from T1 to Th. Thus, for the initial cycle, Equation D.3-25 predicts that the temperature in the upper region of the tank Tu will be

( )T TK

T Tu s s( )1 11= − − for 0 < ≤t t c (D.3-28)

Subsequently, the inlet temperature to the heat exchanger will be Tu(1) until t=2tc. By repeated application of Equation D.3-25, taking into account the step changes in the inlet and outlet temperatures in the heat exchanger loop, leads to the result

T T N

T TKs u

s

N−−

= −( )

1

for ( )N tt

Nc

− < ≤1 (D.3-29)

where N is an integer that counts the number of cycles that the liquid at the top of the tank has passed through the heat exchanger. Thus, if the liquid in the tank is truly stratified with no vertical mixing, one should observe successive step changes in Tu with time.

For all of these cases, the performance characteristic of the equipment that expresses the efficiency of the heating process is the product UA. The area A can be measured directly from the geometry of the system. Thus, measurements of the temperature response for any of the modes of heating considered allow experimental determination of the overall heat transfer coefficient U. Conversely, if U can be estimated from its basic definition and from correlations for the contributing film heat-transfer coefficients, such as those given in Bird et al.(2002, §§14.3, 14.7) , one could use the models given here to predict the process performance. Note that the precise definition of U depends on specifying which area is used for expressing A.

Apparatus

The apparatus for this experiment is a cylindrical tank fitted with an variable-speed agitator and a steam coil, as shown in Figure D.3-1. The geometry of the coil is designed to make the submerged area available for heat transfer directly proportional to the volume of liquid in the tank. Mounted on the side of the tank is a loop of piping with a pump and a tubular heat exchanger. A water line is available to provide the test liquid. Steam may be introduced into either the coil or the heat exchanger, and the steam pressure may be regulated with a manual control valve.

The tank has depth of 33.5 inches and inside diameter of 22.25 inches. The steam

coil is fabricated from ¾-inch galvanized iron pipe, 10.5 feet long, with a core diameter of 14.5 inches and 2.5 turns. The S-40 pipe has inside diameter of 0.824 inch and outside diameter of 1.05 inch. The same type of pipe is used for the external heating loop.

ChE 324 Lab Manual


Page D.3-9

The heat exchanger contains 10 copper tubes, ¼ inch in inside diameter and 18 inches long.

Also available are thermocouples with a digital thermometer, a stopwatch, a

tachometer for measuring agitator speed, and scales for weighing samples of water. The flow rate through the heat-exchanger loop may be controlled with the valve installed in the return line downstream from the heat exchanger. Pressure taps mounted across the heat exchanger and a water-over-mercury manometer may be used as a crude form of flow meter, while the actual flow rate may be determined by collection of water at the point of return to the tank. Procedure

The instructor will select the cases that a particular group should examine during the laboratory session. For each mode of heating and operation, the performance of the heating process is determined by measuring the temperature of water in the tank as a function of time. Characterization of the equipment is done by calculating the overall heat-transfer coefficient U from the temperature data and the appropriate macroscopic process model. Collection of the necessary data is accomplished by the following procedure:

1. Locate all valves, switches, and controls, and become familiar with the operation of the equipment. Check and record all key dimensions.

2. For tests involving water flow, determine the flow rates at specific valve settings by collecting and weighing samples of water before starting the experiment.

3. Similarly, for tests involving agitation of the tank, determine the rotational speed of the agitator with the tachometer. Measurement can be made by observing the end of the stirrer shaft that protrudes from the top of the motor.

4. Measure the temperature history within the tank for various modes of operation in the following manner:

a. Clear the steam line of condensate before passing steam into the equipment.

b. Start the steam flowing into the equipment, either the coil or the heat exchanger, by opening the steam valve. Maintain a constant steam pressure, at about 4 psig, by adjusting the steam valve, as needed.

c. As soon as all residual condensate is removed from the steam lines, as indicated by escape of live steam from the steam trap, start the flow(s) of water and start the stopwatch. At regular time intervals record the bulk temperature of the water in the tank. (Note the location of the thermocouple in the tank.) For the heat exchanger tests, periodically measure the temperatures of the water entering and leaving the exchanger. Record also the steam temperature at the entrance and exit of the heater.

5. Examine the validity of the assumptions made in the analysis, i.e., the heat loss from the tank wall is negligible, the liquid is stratified when the tank is not stirred, the temperature is uniform when the tank is stirred, and the effect of the stirring action on the temperature is negligible.

ChE 324 Lab Manual


Page D.3-10

Data Analysis

From the dimensions of the equipment, calculate the nominal heat-transfer area A upon which reported U values will be based. Use the appropriate model from the theory section to analyze the time-temperature data for the various heating processes. Generally, the data should be fit to determine an average value of the overall heat-transfer coefficient U. For the coil heating, report U as a function of agitator speed. For the heat exchanger, report U as a function of water flow rate.

For the case of the external heat exchanger, check the accuracy of the data and the validity of the model used by calculating the heat-input rates from both the temperature in the tank and the temperature change of the stream passing through the exchanger.

Compare the experimental values of U for the two heaters with one another, with representative values in the literature, and with general correlations. Consider both the absolute magnitudes and the variations with operating conditions, such as agitation rate or flow rate.

Discuss the validity of the assumptions made in the analysis.

References Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, Second Edition, John Wiley and Sons, New York, 2002. Kern, D.Q., Process Heat Transfer, McGraw-Hill Book Co., Inc., New York, 1950. Perry, R.H., and D. Green (eds.), Perry's Chemical Engineers' Handbook, 6th ed., McGraw-Hill Book Co., New York, 1984.

ChE 324 Lab Manual

Appendices

Appendix 1. A Sample Report

The following report was submitted by a ChE 324 student in the fall semester of 1996. It is generally well written so it is offered as a model to 324 students, particularly as an example of proper report format. It has been rewritten a bit, and editorial comments have been inserted to indicate where improvements are needed. Editorial comments are highlighted in bold type.

Physical Measurements Laboratory Department of Chemical Engineering

University of Wisconsin Madison

B.K. Badger Partners: Xxxx Xxxx Experiment Completed: 11/6/96 Report Submitted: 11/13/96

ChE 324 - Transport Phenomena Laboratory Experiment 3

Friction Factors for Flow in Circular Tubes

ChE 324 Lab Manual

ChE 324 Lab Manual Appendix 1. A Sample Laboratory Report Page 2

ABSTRACT

Steady-state pressure-drop measurements reveal a correlation between friction

factor and Reynolds number for Newtonian flow in smooth, cylindrical pipes. The

Reynolds number was determined by measurement of mass flow rates through pipes of

known dimensions. The friction factor was calculated by measuring the pressure drop

along a finite length of pipe, using mercury/water and air/water manometers connected

by piezometer rings to the pipe. Results show two different correlations for friction

factor versus Reynolds number, one for laminar flow and one for turbulent flow. The

transition region from laminar to turbulent flow was found to occur at a Reynolds number

between 2000 and 4900 . New parameters for both the Blasius formula and the Prandtl

resistance-law are presented for prediction of friction factors in the turbulent-flow

regime.

INTRODUCTION

In the design of industrial flow processes, one factor that must be accounted for is

the pressure drop associated with fluid flow through a pipe. The pressure drop is the

result of the shear stress between the flowing fluid and the stationary solid at the wall of

the pipe. The shear stress is the result of frictional drag and is proportional to the inside

surface area of the pipe and the kinetic energy of the fluid flowing through it. This

pressure drop must be accounted for in the design process in order to properly size pumps

and compressors and to minimize the energy requirements of the given system.

The design of flow processes requires a relationship between flow rate through a

pipe and the corresponding pressure drop along the length of the pipe. This relationship

can be determined for many systems if the velocity and pressure profiles in the fluid are

ChE 324 Lab Manual


known. However, measurement of the velocity and pressure profiles, especially in a

large process system, is either impracticable or impossible. Therefore, various empirical

correlations have been developed to represent the flow characteristics for geometrically

similar systems.

One such correlation for flow of fluids through a conduit is known as the friction

factor correlation. As a fluid flows through a pipe, it exerts a tangential force, F, on the

solid surface of the pipe. This force is proportional to the interfacial area and to a

characteristic flow kinetic energy with the proportionality factor being identified as the

friction factor, f. Such a friction factor is defined by Bird et al.(1960) as

=

><

−=

DLf

LDPP

f L Re,v

214

12

0

ρ (1)

where D is the diameter of the pipe, L is the length of the pipe, (Po-PL) is the pressure

drop along the length of the pipe, ρ is the density of the fluid, and >< v is the average

velocity of the fluid. For fully developed flow and long, smooth pipes, the friction factor,

f, is independent of L/D and is therefore a function only of the Reynolds number. The

Reynolds number Re is defined as

µ

ρ><=

vRe D (2)

where µ is the viscosity of the fluid flowing through the pipe. A graph of the friction

factor, f, versus the Reynolds number, Re, provides a convenient method to estimate

pressure drops across a length of pipe given the physical characteristics of the pipe and

the flow rate.

ChE 324 Lab Manual


For circular pipes where the Reynolds number is less than 2100, the flow is

known to be laminar, meaning that all of the velocity streamlines are smooth, steady, and

parallel in the axial direction. At a Reynolds number above 2100, the flow becomes

unstable and enters the turbulent region. In the turbulent region, the velocities throughout

the tube fluctuate randomly, with the average bulk velocity being in the direction of fluid

flow. Because of the qualitative difference between turbulent and laminar flow, it is

expected that there will be two regions of the friction factor correlation, one for laminar

flow and one for turbulent flow.

Although the velocity distribution and drag forces in laminar flow can be

calculated theoretically, the chaotic nature of turbulent flow requires that the macroscopic

mechanical behavior in this region be characterized empirically on the basis of

experimental data. Bird et al. (1960) report that the friction factor, f, can be represented

in the turbulent regime by one of several empirical forms, including the Blasius formula

and the Prandtl resistance law. Each of these equations applies in only a limited range of

Reynolds-number values.

The Blasius expression is written as

b

afRe

= for 4000 < Re < 100000 (3)

where the parameters are reported to be a = 0.0791 and b = 0.25 (Bird et al., 1960). On

the other hand, the Prandl formula is

baf

−= Relog1 for 4000 < Re < 50000 (4)

ChE 324 Lab Manual


with the accepted parameter values appearing here being a = 4.0 and b = 0.40 (Bird et

al., 1960).

The goal of this study was to test the reported relations for f and to develop a

correlation of friction factors with Reynolds number over a range of flow rates for

smooth, cylindrical pipes. Experimental results for pressure drop as a function of flow

rate were used to compute friction-factor values. Experimental f values were compared

with predictions from the standard correlations commonly used in designing flow

processes, namely the Blasius formula and the Prandtl resistance law. This comparison

showed the agreement of the accepted correlations with the experimental data. Revised

values of the a and b parameters were calculated to obtain a better representation of the

flow-resistance properties of the test system.

[Note: Some more information should be given about the test system, particularly its dimensions. This could be done by making reference to Experiment C.1 the lab manual. Also, it would be helpful to give page references in BSL.] SUMMARY OF RESULTS

Figure 1 shows the relationship observed between the friction factor, f, and

Reynolds number, Re. [Note by editor: The writer should have made clear that the

figure presents values calculated from experimental data. There should be a brief

description of the equipment and methods given in the Introduction or in an

additional Apparatus and Procedure section. Then, there should be a table

summarizing all of the experimental conditions and the corresponding results. This

information is needed to make the presentation of results clear and to lay the

groundwork for any subsequent discussion of experimental problems or potential

ChE 324 Lab Manual


improvements in the experimental methods. The missing table might take the

following form.

Table 1. A summary of experimental conditions, measured flow rates and

pressure drops, and calculated values of the friction factor.

Run Pipe Length L, ft

Pipe diameter

D, in

Flow rate

Q, ft3/sec

Pressure drop

P0-PL, lbf/in2

Re

f

There should be a sentence saying what is in Table 1. Then the earlier

sentence introducing Figure 1 should appear below the table.]

]

0.001

0.01

0.1

11000 10000 100000

Reynolds Number (dimensionless)

Fric

tion

Fact

or (d

imen

sion

less

)

Figure 1. Friction factor as a function of Reynolds number.

[Note: The caption of this figure should be more detailed, indicating the conditions under which the values were obtained. The graph should include lines through the points, either best-fit lines or lines based on Equations 3 and 4 as well as the theoretical line for laminar flow. Probably the Introduction should include the theoretical equation for f in the laminar region. The caption should comment on the transition to turbulent flow from laminar flow. The grid lines should be omitted

ChE 324 Lab Manual


from this graph; they just make it look cluttered. The word “dimensionless” could be omitted from the axes. Finally, the numbers on the x axis should appear at the bottom of the graph.] The experimental values for the friction factor and Reynolds number shown in

Figure 1 [and in the missing table summarizing the results] were used to estimate

values of the a and b parameters for both the Blasius formula and the Prandtl resistance

law. [Comment on how the curve fitting was done. Give some estimate of accuracy,

data scatter, or confidence limits.] The results are presented in Table 1 along with

values previously reported in the literature for these parameters.

Table 1 - Parameters for the Blasius formula and Prandtl resistance law which correlate friction factor with Reynolds number for turbulent flow in pipes. The literature values were taken from Bird et al. (1960). [Omit the boxes in tables. A horizontal line between the headings and the entries would be sufficient.]

Parameter

Blasius formula (literature

value)

Blasius formula (experimental

values)

Prandtl resistance law

(literature)

Prandtl resistance law (experimental)

a 0.0791 0.112 2.0 3.08

b 0.25 0.26 0.40 2.37

Figure 2 compares the experimentally determined values of friction factor as a

function of Reynolds number with predictions of the Blasius formula and of the Prandtl

resistance law based on the parameters previously reported by Bird et al.(1960).

ChE 324 Lab Manual


0

0.005

0.01

0.015

0.02

0.025

0 10000 20000 30000 40000

Re (dimensionless)

f(di

men

sion

less

)

f (experimental)f (Blasius formula)f (Prandtl Resistance-law)

Figure 2 - Experimental values and model predictions of the friction factor as a function of Reynolds number. The curves calculated from the models are based on the literature parameters given in Table1.

[There should be no points given for the model predictions; they are smooth curves. Also, there should be no box around the figure. It might be helpful to indicate in some way which data points came from which tubes.]

DISCUSSION

As shown in Figure 1, there are clearly two different relations between the friction

factor f and Reynolds number Re for the laminar and the turbulent flow regions. Figure 1

also shows that the friction factor generally decreases as the Reynolds number increases.

This decrease is less pronounced, however, in the turbulent region. The theoretical form

that can be developed for laminar flow [which should have been given in the

Introduction] predicts that f varies inversely with the second power of the Reynolds

number. According to the form of Equation 1, this means that the pressure drop varies

linearly with the flow rate through a pipe. The turbulent-flow behavior shown in Figure 1

indicates that pressure drop at the higher Reynolds numbers increases more rapidly with

flow rate.

Figure 1 also confirms that the transition from laminar to turbulent flow occurs

somewhere between a Reynolds number of 2000 and 4900. A lack of data in this region

prevents a more exact identification of the transition point from laminar to turbulent flow.

Also, the apparatus did not allow very accurate measurement of flow rate for slow flows.

ChE 324 Lab Manual


The exit hose may need to be removed to allow more accurately measure laminar flow

rates. Due to the extremely low flow rate in the laminar region and elevation changes of

the exit hose, pooling of the exit stream occurred within the hose. [Note that reference

to a hose and suggestions for changing the apparatus make no sense here because

the apparatus and procedure have not been described. There should have been at

least a reference to the lab manual to define the system that was used in the lab.]

Figure 2 shows that the Blasius formula and Prandtl resistance law do not agree

exactly, but each provides a fair description of the variation of the friction factor with

increasing Reynolds number for the range of flow rates studied. The Prandtl resistance

law, however, overestimates the measured friction factor by 30-50%, while the Blasius

formula underestimates the measured friction factor by 9-40%. Therefore, the parameters

in each of these formulas should be replaced with those listed in Table 1 to provide a

more accurate representation of our experimental system.

The discrepancies between the predicted and experimental values of friction

factor could be due to two sources of systematic error in our measurements. First, both

equations assume that the pipes are hydraulically smooth. Slight roughness on the inside

surfaces of the pipes would yield higher values for the measured friction factors.

Second, air bubbles in the manometer connector lines were a persistent problem that

introduced inaccuracy into the pressure-drop measurement, which would give artificially

low values for the measured friction factor. [Why low values? This is not obvious.]

This problem was most likely due to leaks within the test-system valves.

In this experiment, a correlation was obtained between the friction factor and the

Reynolds number for both turbulent and laminar pipe flow. The relationship obtained,

ChE 324 Lab Manual


which is presented in Figure 2, can be used in the design of flow processes in order to

estimate the pressure drop associated with a given flow of fluid through a circular pipe.

The form of the correlation is significantly different for laminar and turbulent flow.

Further measurements should be taken to clarify at what exact Reynolds number the

transition from laminar to turbulent flow occurs. The experimentally measured

correlation was compared with the Blasius formula and Prandtl resistance law to illustrate

the validity of these models for predicting friction factors in turbulent flow. New

parameters for both the Blasius and Prandtl formulas were determined to reflect more

accurately the flow behavior in the test system.

REFERENCES

1. Bird, R.B., Stewart, W.E. and Lightfoot, E.N., Transport Phenomena, John Wiley & Sons, New York, (1960), Chapters 5-6.

2. Crosby, E.J., Experiments in Transport Phenomena, Department of Chemical Engineering, University of Wisconsin-Madison, August 1961, pp. 1-14, B10-B17.

[No specific reference was made to the Crosby manual in the text of the report. Such reference should have been given in order to document the equipment and procedure used. Otherwise, the citation should not appear in the list of references. A bibliography, on the other hand, lists references that an author has consulted in preparing an article but does not necessary cite explicitly. Bibliographies are more common with articles expressing opinion rather than with scientific reports.] APPENDICES Appendix I. Sample Calculations Appendix II. Graphical Analyses Appendix III. Experimental Data App01-Sample Report, Revised 9/17/98; TWC

ChE 324 Lab Manual

ChE 324 Lab Manual Ap.2-1 Appendices 2 and 3

1

Appendix 2 (Suppressed).

Appendix 3 (Suppressed).

Den

sity

of

Aqu

eous

Gly

cero

l Sol

utio

ns

Tem

pera

ture

(C

)0

1020

3040

5060

7080

9010

0W

eigh

t Per

cent

Gly

cero

lD

ensi

ty (

g/cc

)5

1.01

015

1.00

735

101.

0251

1.02

411.

0221

01.

0190

51.

0155

1.01

131.

0062

1.00

060.

9946

0.98

830.

9818

151.

0345

01.

0313

020

1.05

661.

0495

1.04

690

1.04

350

1.03

951.

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1.02

961.

0239

1.01

751.

0110

1.00

3925

1.05

980

1.05

605

301.

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1.07

621.

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01.

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51.

0637

1.05

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1.04

731.

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1.03

431.

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351.

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01.

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540

1.10

681.

1035

1.09

930

1.09

475

1.08

931.

0839

1.07

811.

0719

1.06

571.

0591

1.05

2345

1.11

280

1.10

795

501.

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1.13

111.

1263

01.

1211

01.

1154

1.10

951.

1035

1.09

731.

0909

1.08

431.

0774

551.

1400

51.

1347

060

1.16

351.

1591

1.15

380

1.14

830

1.14

251.

1366

1.13

021.

1239

1.11

751.

1110

1.10

3965

1.16

750

1.16

195

701.

1922

1.18

721.

1812

51.

1756

51.

1694

1.63

601.

1571

1.15

061.

1441

1.13

751.

1304

751.

1948

51.

1890

080

1.21

971.

2144

1.20

850

1.20

240

1.19

611.

1900

1.18

361.

1768

1.16

991.

6320

1.15

6185

1.22

180

1.21

565

901.

2470

1.14

241.

2351

01.

2289

01.

2223

1.21

591.

2093

1.20

281.

1966

1.18

971.

1828

951.

2482

51.

2419

010

01.

2725

1.26

681.

2610

81.

2549

51.

2487

1.24

261.

2360

1.22

940

1.22

241.

2153

1.20

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The

se d

ensi

ties

are

rel

ativ

e to

that

of

wat

er a

t 4C

whe

re th

e m

ass

of 1

cc

is ta

ken

as 1

gm

. The

se d

ensi

ties

hav

e be

en c

alcu

late

d fr

om m

easu

red

dens

itie

s at

20

and

30C

and

the

data

for

ther

mal

exp

ansi

on d

eter

min

ed b

y G

erla

ch, G

, Th.

, and

rep

orte

d by

Fre

sniu

s, W

., Z

. ana

l. C

hem

., 24

, 99-

119

(188

5);

see

Law

rie,

J.W

.,

"G

lyce

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nd th

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lyco

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A.C

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onog

raph

Ser

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No.

44,

The

Che

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al C

atal

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ny, I

nc.,

New

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k (1

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, p. 1

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Dat

a fo

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and

30

C:

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, and

A.O

. Sno

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Ind.

Eng

. Che

m.,

20, 1

377-

1379

(19

28).

Ap.4-1

Appendix 4. Properties of glycerol and sucrose solutions

Vis

cosi

ty o

f A

queo

us G

lyce

rol S

olut

ions

1

Tem

pera

ture

(C

)0

1020

3040

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7080

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eigh

t Per

cent

Gly

cero

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isco

sity

(cp

)10

2.44

1.74

1.31

1.03

0.82

60.

680

0.57

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500

203.

442.

411.

761.

351.

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879

0.73

10.

635

305.

143.

492.

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571.

51.

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956

0.81

60.

690

408.

255.

373.

722.

722.

11.

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090.

918

0.76

30.

668

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213.

12.

371.

861.

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251.

050.

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2.85

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3.78

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.521

.213

.69.

256.

615.

013.

803.

002.

4380

255

116

60.1

33.9

20.8

13.6

9.42

6.94

5.13

4.03

3.18

8554

022

310

958

.033

.521

.214

.210

.07.

285.

524.

2490

1310

498

219

109

60.0

35.5

22.5

15.5

11.0

7.93

6.00

9536

9012

7052

323

712

167

.039

.926

.417

.512

.49.

0896

4600

1585

624

281

143

77.8

45.4

29.7

19.6

13.6

10.1

9757

7019

5076

534

016

688

.951

.933

.621

.915

.110

.998

7370

2460

939

409

196

104

59.8

38.5

24.8

17.0

12.2

9994

2030

9011

5050

023

512

269

.143

.627

.819

.013

.210

012

070

3900

1412

612

284

142

81.3

50.6

31.9

21.3

14.8

1

Seg

ur, J

.B.,

and

H.E

. Obe

rsta

r, In

d. E

ng. C

hem

., 43

, 211

7-21

20 (

1951

).

Ap.4-2

D

ensi

ty o

f A

queo

us S

ucro

se S

olut

ions

1

Wei

ght P

erce

nt S

ucro

se10

2030

4050

6070

75

Tem

pera

ture

(C

)

Den

sity

(g/

cc)

01.

0413

1.08

561.

1328

1.18

341.

2377

1.29

571.

3572

1.38

915

1.04

081.

0847

1.13

511.

1818

1.23

591.

3035

1.35

481.

3863

101.

0401

1.08

361.

1315

1.18

011.

2338

1.29

131.

3523

1.38

4015

1.03

921.

0824

1.12

870

1.17

831.

2318

1.28

891.

3497

1.38

1520

1.03

8556

1.08

1403

1.12

6984

1.17

6960

1.22

7567

1.28

6456

1.34

7174

1.37

8971

251.

0368

1.07

951.

1252

1.17

431.

2274

1.28

411.

3445

1.37

6430

1.03

531.

0778

1.12

331.

1722

1.22

491.

2815

1.34

181.

3737

351.

0336

1.07

581.

1212

1.16

981.

2225

1.27

881.

3390

1.37

1040

1.03

161.

0737

1.11

891.

1675

1.21

991.

2762

1.33

621.

3683

451.

0295

1.07

151.

1165

1.16

501.

2174

1.27

351.

3334

1.36

5450

1.02

921.

0690

1.11

391.

1624

1.21

461.

2706

1.33

051.

3625

551.

0247

1.06

641.

1113

1.15

971.

2118

1.26

771.

3275

1.35

9560

1.02

201.

0637

1.10

851.

1568

1.20

891.

2647

1.32

451.

3564

651.

0191

1.06

071.

1053

1.15

331.

2056

1.26

151.

3222

1.35

3370

1.01

621.

0579

1.10

241.

1503

1.20

251.

2584

1.31

881.

3501

751.

0134

1.05

481.

0994

1.14

731.

1993

1.25

511.

3155

1.34

6980

1.01

031.

0517

1.09

621.

1440

1.19

611.

2518

1.31

201.

3436

851.

0070

1.04

831.

0929

1.14

081.

1927

1.24

841.

3087

1.34

0290

1.00

361.

0447

1.08

941.

1374

1.18

911.

2449

1.30

511.

3367

951.

0001

1.04

121.

0858

1.13

391.

1856

1.24

141.

3015

1.33

3210

00.

9964

1.03

771.

0823

1.13

021.

1819

1.23

781.

3421

1.32

980

The

se d

ensi

ties

are

rel

ativ

e to

that

of

wat

er a

t 4C

whe

re th

e m

ass

of 1

cc

is ta

ken

as 1

gm

. The

se d

ensi

ties

hav

e be

en c

alcu

late

d fr

om m

easu

red

dens

itie

s at

20

and

30C

and

the

data

for

ther

mal

exp

ansi

on g

iven

by

Bat

es, F

.J.,

et a

l., "

Pola

rim

etry

, Sac

char

imet

ry a

nd th

e Su

gars

," C

ircu

lar

of th

e N

atio

nal B

urea

u of

Sta

ndar

ds C

440,

U.S

. Dep

t. of

Com

mer

ce, U

.S. G

over

nmen

t Pri

ntin

g O

ffic

e. W

ashi

ngto

n D

.C. (

1942

), T

able

119

, pp.

648

-649

Dat

a fo

r 20

C: B

ates

, F.J

. et a

l., o

p . c

it.,

Tab

le 1

13, p

p. 6

26-6

31.

Ap.4-3

Vis

cosi

ty o

f A

queo

us S

ucro

se S

olut

ions

1

Wei

ght P

erce

nt S

ucro

se10

2030

4050

6070

7172

7374

75

Tem

pera

ture

(C

)V

isco

sity

(cp

)0

3.78

26.

692

14.5

544

.79

235.

136

2052

5377

9211

876

1863

930

207

53.

137

5.43

511

.44

33.1

815

9.1

2038

2871

4136

6103

9245

1428

010

2.64

24.

487

9.19

625

.21

110.

912

0616

5823

2933

4049

0674

0215

2.25

43.

757

7.47

319

.53

79.4

974

6.9

1006

1379

1929

2759

4039

201.

340

1.94

53.

187

6.16

715

.43

58.4

948

1.6

636.

385

4.9

1170

1631

2328

251.

695

2.73

55.

164

12.4

040

.03

321.

641

7.8

551.

073

8.9

1009

1405

301.

493

2.37

34.

375

10.1

133

.82

221.

628

3.4

367.

648

4.3

648.

588

4.8

351.

325

2.07

83.

747

8.35

926

.46

157.

019

8.0

253.

032

7.9

431.

657

7.4

401.

184

1.83

33.

241

6.99

121

.04

114.

014

2.0

178.

922

8.5

296.

038

9.5

451.

071.

642.

845.

9417

.084

.910

513

116

520

927

150

0.97

1.47

2.49

5.07

14.0

64.4

78.4

96.5

121

152

193

550.

881.

322.

214.

3811

.649

.759

.973

.089

.711

214

160

0.80

1.20

1.97

3.81

9.66

39.0

46.6

56.1

68.4

84.1

105

650.

741.

091.

763.

348.

1731

.036

.743

.952

.964

.679

.670

0.68

1.00

1.60

2.94

6.98

25.0

29.4

34.9

41.7

50.3

61.4

750.

630.

921.

452.

616.

0020

.423

.828

.033

.339

.948

.280

0.59

0.85

1.32

2.34

5.20

16.8

19.5

22.8

26.9

32.0

38.3

850.

550

1.22

14.

7390

1.12

34.

1595

1.03

73.

7210

00.

960

3.34

1 S

win

dells

, J.F

., Sn

yder

, C.F

., H

ardy

, R.C

., an

d G

olde

n P.

E.,

"Vis

cosi

ties

of

Sucr

ose

Solu

tion

s at

Var

ious

Tem

pera

ture

s: T

able

s of

Rec

alcu

late

d V

alue

s,"

Sup

plem

ent t

o C

ircu

lar

of th

e N

atio

nal B

urea

u of

Sta

ndar

ds C

440,

U.S

. Dep

t. of

Com

mer

ce, U

.S G

over

nmen

t Pri

ntin

g O

ffic

e, W

ashi

ngto

n, D

.C.,

July

31,

195

8, T

able

s 13

2 an

d 13

3.

Dat

a fo

r 10

% s

ucro

se a

t 20

C: I

bid .

, Tab

le 1

31.

Dat

a fo

r 85

-100

C: L

ange

, N.A

., ed

itor

, "H

andb

ook

of C

hem

istr

y,"

9th e

diti

on, H

andb

ook

Publ

ishe

rs In

c., S

andu

sky,

Ohi

o (1

956)

, p. 1

667.

Ap.4-4

0.01

0.1

1

10

40 45 50 55 60 65 70

Sucrose concentration (%)

Kin

emat

ic v

isco

sity

(cm

2 /s)

65ºC

60ºC

55ºC

50ºC

45ºC

40ºC

30ºC

20ºC

35ºC

25ºC

15ºC

Figure Ap.4-1. Kinematic viscosity of aqueous sucrose solutions Ap.4-5

Temperature Absolute Density 1 Absolute Density 2 Viscosity 3,4

(°C) (g/cm3) (g/cm3) (cp)

0 0.9998681 0.999841 1.79215 0.9999919 0.999965 1.518810 0.9997282 0.9997 1.307715 0.9991266 0.999099 1.140420 0.9982343 0.998203 1.0000

25 0.997077 0.997044 0.893730 0.995678 0.995646 0.800735 0.994061 0.722540 0.9922497 0.656045 0.99025 0.5988

50 0.98807 0.549455 0.98573 0.506460 0.98324 0.468865 0.98059 0.435570 0.97781 0.4061

75 0.97489 0.379980 0.97183 0.356585 0.96865 0.335590 0.96534 0.316595 0.96192 0.2994100 0.95838 0.2838

Ap.5-1

4 The viscosity of water is 1.0000 cp at .20°C; Hodgman, C.D. op. cit., p. 2257.

Appendix 5. Density and Viscosity of Water

1 These densities are relative to that at 4°C where the mass of 1 cm3 is taken as 1 g; Forsythe, W.E.,

“Smithsonian Physical Tables,”9th edition, The Smithsonian Institution, Washington, D.C. (1954), Tables 287 and 290, pp. 296 and 298.2 Hodgman, C.D., editor, Handbook of Chemistry and Physics , 44th edition, The Chemical Rubber Publishing Company, Cleveland, Ohio (1962), p. 2197.3 Forsyth, W.E., op. cit., Table 311, p. 319.

To be used for data analysis in ChE 324, Experiment A.2, Thermal Conductivity of Solids.T.W. Chapman, 10/7/96Reviewed by R. Chavez 2002

The cell formulas for 1/Bi are:Slab 1/(Beta1*TAN(Beta1))Cylinder BESSELJ(Beta1,0)/(Beta1*BESSELJ(Beta1,1))Sphere TAN(Beta1)/(TAN(Beta1)-Beta1)

Beta1 Jo(Beta1) J1(Beta1) Tan(Beta1)Slab Cylinder Sphere

0.25 15.6653 31.7493 47.7996 0.984436 0.124026 0.2553420.3 10.7758 21.9713 33.1328 0.977626 0.148319 0.309336

0.35 7.8272 16.0752 24.2891 0.969609 0.172334 0.3650280.4 5.9131 12.2483 18.5491 0.960398 0.196027 0.422793

0.45 4.6003 9.6244 14.6136 0.950012 0.219353 0.4830550.5 3.6610 7.7474 11.7986 0.93847 0.242268 0.5463020.6 2.4362 5.3017 8.1312 0.912005 0.286701 0.6841370.7 1.6961 3.8264 5.9196 0.881201 0.328996 0.8422880.8 1.2140 2.8681 4.4837 0.846287 0.368842 1.0296390.9 0.8817 2.2102 3.4989 0.807524 0.40595 1.260158

1 0.6421 1.7389 2.7940 0.765198 0.440051 1.5574081.1 0.4627 1.3893 2.2720 0.719622 0.470902 1.964761.2 0.3240 1.1224 1.8745 0.671133 0.498289 2.5721521.3 0.2136 0.9137 1.5647 0.620086 0.522023 3.6021021.4 0.1232 0.7471 1.3183 0.566855 0.541948 5.7978841.5 0.0473 0.6116 1.1190 0.511828 0.557937 14.10142

π/2 1.570796 0.0000 0.5301 1.0000 0.472001 0.566824 1.63E+161.6 0.4994 0.9553 0.455402 0.569896 -34.232531.7 0.4052 0.8191 0.397985 0.577765 -7.6966021.8 0.3248 0.7043 0.339986 0.581517 -4.2862621.9 0.2552 0.6064 0.281819 0.581157 -2.927098

2 0.1941 0.5221 0.223891 0.576725 -2.185042.1 0.1396 0.4488 0.166607 0.568292 -1.7098472.2 0.0902 0.3844 0.110362 0.555963 -1.3738232.3 0.0447 0.3273 0.05554 0.539873 -1.1192142.4 0.0020 0.2762 0.002508 0.520185 -0.916014

2.404826 0.0000 0.2739 1.31E-15 0.519147 -0.9071792.5 0.2301 -0.048384 0.497094 -0.7470222.6 0.1879 -0.096805 0.470818 -0.6015972.7 0.1490 -0.142449 0.441601 -0.4727282.8 0.1127 -0.185036 0.409709 -0.355532.9 0.0783 -0.224312 0.375427 -0.246405

3 0.0454 -0.260052 0.339059 -0.1425473.1 0.0132 -0.292064 0.300921 -0.041617

π 3.141593 0.0000 -0.304242 0.284615 -1.23E-16

Ap.6-1

1/Bi

Appendix 6. - Calculated values of the first (smallest) eigenvalues for unsteady heat conduction in a slab, cylinder, or sphere with a Newton's law of cooling boundary condition.

Relations between the first eigenvalue, Beta1, and the Biot number Bi=hR/k for unsteady heat conduction into a slab, a cylinder, and a sphere. These tables may be used in the data analysis of Experiment A.2 of ChE 324

Ap.6-2

Figure Ap.6-1. Relationship of the Biot number with the first eigenvalue for heat conduction in solids of simple geometry and convective surface

0

2

4

6

8

10

12

14

16

18

20

0 0.5 1 1.5 2 2.5 3 3.5β1

1/B

i=k/

hb

Slab

Cylinder

Sphere

β 0.4 hr1−=α 0.157

ft2

hr=

Cp 0.12BTU

lb R⋅=ρ 499.424

lb

ft3

=k 9.434BTU

ft hr⋅ R⋅=

Conversion. MathCAD will convert to default English units. You may edit any of the units to make a conversion to a different one.

Calculations

Qss α β,( ) D

2

βα

⋅:=

Dimensionless steady-state heat flow =Q/kπR(Ts-Ta) (From Equation B.2-13)

Q t α, β,( ) exp β− t⋅( )π α⋅ t⋅

erf β t⋅( ) βα

⋅+

D

2⋅:=

Dimensionless heat flow =Q/kπR(Ts-Ta) at base of rod (Eq. B.2-12):

θss z α, β,( ) expβα

− z⋅

:=

Steady-state temperature profile (Eq. B.2-9):

θ z t, α, β,( ) 1

2exp

βα

z⋅

1 erfz

4 α⋅ t⋅β t⋅+

−

⋅

expβα

z−( )⋅

1 erfz

4 α⋅ t⋅β t⋅ 1−( )⋅+

−

⋅+

...

⋅:=

Dimensionless Temperature (Equation B.2-7):

β2 h⋅

D

2ρ⋅ Cp⋅

:=Convection factor (Eq. B.2-3)αk

ρ Cp⋅:=Thermal diffusivity:

Definitions

h 0.5BTU

hr ft2⋅ R⋅

⋅:=Must choose a value of h. As an example choose:(The value of h may be changed here to modify results below.)

D 1 in⋅:=Cp 0.12cal

gm K⋅⋅:=ρ 8

gm

cm3

⋅:=k 0.039cal

cm sec⋅ K⋅⋅:=

Data for metalTable B.2-1

Appendix 8. Calculation of temperature profiles and heat flows into the rod. (Experiment B.2)

Default Unit system is set to U.S.(Example for the 1-inch steel rod. Students can copy this file into their folders and then modify it by changing property values, k, ρ, Cp, D, or h, in the equations below.)

Ap.8-1

h 0.5BTU

hr ft2⋅ R⋅

=β 0.4 hr1−=α 0.157

ft2

hr=

θss z α, β,( )1

0.727

0.528

0.384

0.279

0.203

0.147

0.107

0.078

0.057

0.041

=θ z t, α, β,( )1

0.657

0.404

0.23

0.119

0.056

0.024

9.133·10 -3

3.115·10 -3

9.471·10 -4

2.563·10 -4

=z

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ft

=

t 1 hr⋅:=z 0 ft⋅ .2 ft⋅, 2 ft⋅..:=

Temperature profiles at various values of position and time. One could specify z and t as range variables to calculate θ(z,t,α,β). In such case, the temperatures for all z values and for the first time value would be at the top of the array, followed by the values for subsequent times. Here we show a list for only t=1.0 hr. and z from 0 to 2, at 0.2 intervals.

Q t α, β,( ) 0.066=t 10 hr⋅:=

Q t α, β,( ) 0.069=t 3 hr⋅:=

For example:time in hours

0 1 20

0.13

0.25

Q t α, β,( )

Qss α β,( )

t

hr

Heat flow at the base as a function of time:

Qss α β,( ) 0.066=

θss z α, β,( ) 0.853=

Steady-state values:Q t α, β,( )0.195

0.143

0.121

0.108

0.1

0.094

0.09

0.086

0.084

0.082

0.08

0.078

0.077

0.076

0.075

0.074

0.073

0.073

0.072

0.072

=θ z t, α, β,( )0.565

0.676

0.725

0.754

0.773

0.787

0.797

0.805

0.812

0.817

0.821

0.825

0.828

0.83

0.833

0.835

0.836

0.838

0.839

0.84

=t

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

hr

=

t .1 hr⋅ .2 hr⋅, 2 hr⋅..:=

z .1 ft⋅:=Dimensionless temperature and heat flow at

Ap.8-2

θ

0 1 2 3 4 5

0

1

2

3

4

5

0 1 1 1 1 1

0 0.422 0.565 0.634 0.676 0.704

0 0.109 0.254 0.347 0.41 0.457

0 0.017 0.088 0.161 0.221 0.27

0 1.408·10 -3 0.023 0.063 0.105 0.144

0 6.625·10 -5 4.675·10 -3 0.02 0.044 0.07

=

The resulting matrix is obtained by typing "θ=" (shortcut for θ is q CTRL-G). The rest of the temperature values may be viewed by clicking on the box and scrolling across the window.

θi j, θ z i( ) t j( ), α, β,( ):=Generate the matrix

t j( ) jtmax

20:=z i( ) i

zmax

20⋅:=

h 0.5BTU

hr ft2⋅ R⋅

=β 0.4 hr1−=α 0.157

ft2

hr=tmax 1 hr⋅:=j 1 2, 20..:=

zmax 2 ft⋅:=i 0 1, 20..:=

The dimensionless temperature can also be displayed as a matrix.

0 0.5 1 1.5 20

0.5

1

Distance from base, ft.

Dim

ensi

onle

ss te

mpe

ratu

re θ z .01 hr⋅, α, β,( )

θ z .1 hr⋅, α, β,( )

θ z .5 hr⋅, α, β,( )

θ z 1 hr⋅, α, β,( )

θ z 2 hr⋅, α, β,( )

θss z α, β,( )

z

The following plot shows how we can plot spatial temperature profiles at various values of time, including the steady-state profile, without having to generate the lists for those times:

Ap.8-3

View the corresponding values of z and t by entering the row and column number, respectively, inside the parentheses in the expressions below. For example z(1)=0.1ft and t(1)=0.05hr:

z 1( ) 0.1 ft= t 1( ) 0.05 hr=

Plot individual rows to show temperature as a function of time with position as a parameter.

Plot individual columns to show temperature as a function of position with time as a parameter:

α 0.157ft

2

hr= β 0.4 hr

1−= h 0.5BTU

hr ft2⋅ R⋅

=

0.5 10

0.5

1

θ1 j,

θ2 j,

θ5 j,

θ10 j,

θ20 j,

t j( )

hr

0 1 20

0.5

1

Distance from base, ft

θi 1,

θi 5,

θi 10,

θi 20,

z i( )

Notes: - The time is shown divided by hr on the graph at left. This is because MathCAD plots the variables in the default units, in this case seconds. To change the plot to a different scale one must divide by the conversion factor; hr means 3600 sec/hr. The graph on the right is expressed in the default units (ft) so there is no need to scale.

- To change the limits of an axis: 1) Click on the plot; you will see four numbers showing the limits of each axis. Change the appropriate limit to modify the scale.

T.W.Chapman10/21/96

Revised by R.Chavez2/2002

Ap.8-4

Appendix 9. Concentration Profiles in a Stagnant Film and determination of diffusivity of acetone in nitrogen. (Experiment B.3)

(Written by T.W. Chapman, 3/18/97. Revision by R. Chavez, 11/2/2001)

This is a more complete form of Table 20.1-1 in BSL for use in the model of unsteady-state diffusion, Equation 20.1-22 in Transport Phenomena.The regions may be modified according to your experimental data. highlighted

Refer to Section 20.1 of BSL 2nd edition. First, we can calculate the relation between φ and xAo, as well as the factor ψ .

φ 0.1 0.2, 3.5..:=

xAo φ( ) 1

1 π 1 erf φ( )+( )⋅ φ⋅ exp φ2( )⋅

1−( )

+

:= Equation 20.1-18 in BSL.

ψ φ( ) φπ

xAo φ( )⋅:=You may click on the tables and scroll to see other values.

φ0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

= xAo φ( )0.166

0.311

0.436

0.543

0.634

0.71

0.773

0.824

0.866

0.899

0.925

0.945

0.96

0.972

0.98

0.986

= ψ φ( )1.067

1.14

1.22

1.306

1.398

1.498

1.606

1.721

1.843

1.972

2.108

2.251

2.4

2.554

2.712

2.875

=Plot of the phi and psi factors as functions ofxAo:

0 0.5 10

2

4

φ

ψ φ( )

xAo φ( )

Ap.9-1

(Eq. 19.1-16 BLS 2nd Ed)X φ Z,( ) 1 erf Z φ−( )−1 erf φ( )+

:=

X and Z are related by the equation

XxA z t,( )

xAo:=Z

z

4Dt:=

Z is a dimensionless variable that combines distance, time, and diffusivity, and X is the normalized mole fraction.

We can find the diffusivity, D, if we know the mole fraction of acetone at a specific time, t, and position, z, as follows:

φ 0.135=φ Find φ( ):=

xAo φ( ) xAovalue=

Given

If the given guess for φ doesn't work, use the plot or tables above to supply a better value.

φ 0.1:=Guess phi:

xAovalue 0.219=xAovaluepA

P:=

pA 10

7.117141210.595

T 229.664+−

:=

ºC T 17:=mmHg P 740:=

We calculate the theoretical concentration profile which can be compared with experimental values to estimate the diffusivity if xA0 is known. We estimate xAo (xAovalue) using Antoine's Equation and the ideal gas law. You must enter P and T from your experimental data.

ψ φ( )1.067

1.071

1.074

1.078

1.081

1.085

1.088

1.092

1.096

1.099

1.103

1.107

1.11

1.114

1.118

1.121

=xAo φ( )0.166

0.174

0.181

0.189

0.197

0.204

0.212

0.219

0.226

0.234

0.241

0.248

0.255

0.263

0.27

0.277

=φ0.1

0.105

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0.175

=

φ 0.1 .105, .2..:=

This table shows a narrower range of φ values

Ap.9-2

The following is the theoretical plot of Z vs. X for and

This is the same type of plot as that shown in Figure 20.1-1 of BSL.

This calculation may be repeated at each location and/or time for which data are taken, and the resulting values of D may be averaged.

cm2

sD 0.092=

D

z

Z

2

4 t⋅( ):=

we can solve for D to obtain its valueZz

4Dt:=Since

Z 0.544=Z Find Z( ):=

X φ Z,( ) Xvalue=

Given

If the given guess for Z doesn't work, use the plot above to supply a better value

Z 0.3:=Guess :

Xvalue 0.489=XvaluexA

xAo φ( ):=Thus

sect 300:=cmz 5.715:=xA 0.107:=

Now we are ready to obtain the diffusivity. Enter the following values of the measured xA, position z, in cm, and time in seconds obtained from your experiment.

X φ Z,( )1

0.952

0.903

0.854

0.805

0.756

0.708

0.661

0.615

0.57

0.526

0.484

0.444

0.405

0.368

0.334

=Z

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

=

0 0.5 10

0.5

1

X φ Z,( )

Z

Z 0 .05, 2..:=

φ 0.135=xAo φ( ) 0.219=

Ap.9-3

σB 3.798 A⋅:=εB

κ

71.4 K⋅:= MB 28g

mol⋅:= P 101.3 kPa⋅:=

Calculation:

εAB

κ

εA

κ

εB

κ

⋅:= σAB1

2σA σB+( )⋅:=

σAB 4.199 A=κT

ε

T

εAB

κ

:=εAB

κ

199.996 K=κT

ε

1.465=

ΩD ΩD_ABκT

ε

:= ΩD 1.21= cP

Rgas T⋅:= c 41.585 mol m

-3=

DAB 2.26 105−⋅

mol0.5

A2⋅ g

0.5⋅

cm s⋅ K0.5⋅

⋅

T1

MA

1

MB+

⋅

c σAB2

⋅ ΩD⋅⋅:=

DAB 0.1cm

2

s=

Calculation of molecular diffusivity using Chapman-Enskog Theory

The following scrollable table contains the collision integral values. The first column contains the

values for κT

ε , the second and third columns contain the collision integral values for the viscosity

and diffusivity, respectively.

Ω0 1 2

0

1

2

3

4

0.3 2.79 2.66

0.35 2.63 2.48

0.4 2.49 2.32

0.45 2.37 2.18

0.5 2.26 2.07

:=ΩD_AB x( ) linterp Ω 0⟨ ⟩ Ω 2⟨ ⟩, x,( ):=

Units definitions:

A 108−

cm⋅:= g 103−

kg⋅:=

kPa 103

Pa⋅:= kJ 103

J⋅:=

Rgas 8.314J

mol K⋅⋅:=

Enter here Lennard-Jones parameters and other data:

Acetone

σA 4.600 A⋅:=εA

κ

560.2 K⋅:= MA 58g

mol⋅:= T 293 K⋅:=

Nitrogen

Ap.9-4

Ap.10- 1

Appendix 10. Acetone data for Experiment B.3 Table Ap.10-1. Vapor pressure of acetone

Temperature (oC) Pvap (mm Hg) - 59.4 1 -40.5 5 -31.1 10 -20.8 20 -9.4 40

-2.0 60 7.7 100

22.7 200 39.5 400 56.5 760

Values in Table Ap.10-1 are plotted in Figure Ap.10-1 in the form log Pvap vs. 1/Tabs. The values can be approximated by the following Antoine equation:

where T is in oC and pvap is in mm Hg.

Figure Ap.10-1. Vapor Pressure of Acetone as a Function of Temperature

( )224

0.116102447.7log10 +

−=T

pvap

1

10

1 00

10 00

0 .0 03 0 .0 03 2 0 .0 03 4 0 .00 36 0 .0 03 8 0 .0 04 0 .0 04 2 0 .0 04 4 0 .00 46 0 .0 04 8 0 .0 05

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Appendix A.13. Statistical Analysis of Experimental Data

Ap.13- 1

Appendix 13. Statistical Analysis of Experimental Data Based on notes prepared for ChE 324 by Prof. D.J. Klingenberg, Fall 1997

Introduction

In this course, you will be measuring a variety of variables, then analyzing the data by calculating derived quantities, and often fitting the data to equations in order to estimate some parameters. Engineers must appreciate the fact that all measured data contain some degree of uncertainty; all reported results are only estimates. To make these estimates meaningful, we must also somehow report the uncertainty in the results, or how confident we are about the results.

The purpose of these notes is to provide some guidance for reporting uncertain results, especially for the cases where measured data are analyzed and fit to equations to extract useful engineering information.

Experimental Errors

Experimental errors are typically grouped into two categories: systematic errors and random errors. Systematic errors are usually relatively fixed, and they influence all the data in a series of measurement in essentially the same way. Examples of this type of error are pressure measurements with a gauge that is not properly calibrated, measurement of temperature rise in a system that is supposedly adiabatic but in fact is not, and incorrect thermometer readings due to thermal expansion of the thermometer glass. Such errors can often be eliminated by auxiliary experiments and calibrations or theoretical corrections to the experimental data.

Random errors are usually more difficult to detect and to deal with. Examples of these errors include mistakes by the experimenter when recording the data, random electronic fluctuations in measuring equipment, and incidental vibrations that affect a mechanical system. Even after systematic errors are eliminated from an experiment, random errors will still be present. It is the experimenter's job to take these persistent errors into account by assessing the uncertainty in the data, trying to minimize it, and indicating the degree of uncertainty in the reported results.

Some data points taken in an experiment will appear to be obviously in error because they are inconsistent with the majority of measurements. These are called "outliers". It is tempting just to throw away such data points and omit them from subsequent analysis and the reported results. However, this should not be done without employing some systematic statistical procedure to confirm that apparent outliers are


Ap.13- 2

indeed inconsistent rather than revealing an important characteristic of the system. Exceptions to this rule arise in instances where one can explain on physical grounds why a suspect data point does not reflect the system of interest. For example, if one realizes that part of the equipment was not operating properly during a test, for example, some controlled variable such as temperature or flowrate deviated from its set point , it is reasonable to throw away the data taken during such a disruption. Generally, however, one should investigate the relative validity of all data points in a set of measurements.

Measured data always contain some errors, or random fluctuations. The experimenter must assign some uncertainty to the measured values to indicate the level of confidence that may be placed in reported results. This is often a straightforward task. Manufacturers of measuring equipment will typically state output uncertainties in the owner's manual or user's guide. Multiple readings from a fluctuating display may be taken and an average deviation calculated. Uncertainty can also be estimated from calibration curves developed for an instrument, particularly if replicate measurements are made.

Experimental uncertainties are typically specified by stating the apparent, or average, value of a quantity, plus or minus some value of the typical error. For example, if one measured a pressure by reading the height of a manometer leg, say 47.2 mm H2O, and estimated the uncertainty resulting from the coarseness of the scale or from continuous fluctuations of the water level to be, say, 0.5 mm, one would write

P = 47.2 mm H2O ± 0.5 mm H2O or P = 47.2 ± 0.5 mm H2O. Here, the indicated uncertainty is based more on common sense rather than a systematic statistical analysis. Such uncertainties in measured values should be estimated conservatively. In this example, we would state an uncertainty of 0.5 mm H2O only if we were very confident that the true pressure lies somewhere between 46.7 and 47.7 mm H2O.

Propagation of Error

Very often the experimenter will measure certain types of data and then obtain derived quantities from the measured values. For example, one might measure the current and voltage across a resistor and then calculate the power loss by the formula P = VI. For purposes of a report and its conclusions, it may be more important to identify the uncertainty of the derived quantity P than that of the raw data, I and V.

A general approach for determining the uncertainty of derived quantities determined from measured data (for which we have estimated their individual uncertainties) is as follows. Let the derived quantity be R, which is a function of n variables that are measured directly, say xl, x2,...,xn. There will be a known functional relationship between the value of R and the other variables,


Ap.13- 3

R = R(xl, x2, • • •, xn)

from which the derived quantity is calculated from the measurements. In the previous example, the function is P = P(V, I) = VI. Now let wi be the uncertainty of variable xi ; that is, xi = <xi> ± wi where <xi> is the estimated value of xi. The uncertainty in the derived quantity, wR, may be estimated by the formula

w w w wR 1 2 n=

+

+ ⋅⋅⋅ +

∂∂

∂∂

∂∂

R

x

R

x

R

xn1

2

2

2 21

2

(A.13-1)

Equation 1 accounts for the propagation of the individual uncertainties into the calculated value of R. The partial derivatives of R with respect to each xi are taken while all other xj are held constant; these terms arise from a two-term Taylor-series analysis of the function R. The terms are squared before adding to avoid unjustified cancellation of errors. Thus, Equation 1 gives a conservative estimate of wR. The Engineering Equation Solver (EES) program can perform error propagation analysis for a system of equations. EES is available on CAE Windows stations (CAE Applications > Engineering > EES) or it can be downloaded from the CAE web site.

Statistical Analysis

Although one can often estimate the uncertainty in a measured variable on an ad hoc basis, such an approach is subjective and may be difficult to document. Statistical analysis provides a more objective and systematic method for identifying uncertainties and confidence limits, particularly if a large number of data points are available and if replicate measurements can be taken. Using well-defined statistical methods and quantities has the advantage of putting the estimate of errors on a standardized, accepted basis, which avoids ambiguities or possible confusion in the reporting of results.

Consider a set of m readings of the same variable, z, which is also called a

sample. Denote these readings by z1 , z2, ..., zm. These might be, for example, m replicated readings of the temperature or the pressure of a system. Here are some basic statistical definitions: • Sample Mean or Arithmetic Mean:

zm

zii

m

==∑1

1

(A.13-2)

• Deviation of a data point:


Ap.13- 4

d z zi i= − (A.13-3) This deviation is not very useful because the average of all readings is zero. • Standard Deviation of a Sample:

( )2

1

1

2

1

1

−−

= ∑=

m

ii zz

mσ (A.13-4)

The standard deviation σ is an estimate of the average magnitude of the errors that

arise in the measurements. If one assumes that systematic errors have been eliminated, the deviations are the random errors.

The definition of the sample standard deviation given in Equation 4 is used when

the underlying population is not known, that is, when we don't know what would happen if there were an infinite number of readings. If we compared a set of readings against a known distribution such that the "correct" value of z were known, for example, in the calibration of a pressure gauge with a known, reliable instrument such as a dead-weight tester, then the factor of 1/(m-1) in Equation 4 would be replaced with 1/m. The explanation of this difference is that, when the sample comes from an unknown distribution, one measurement is used in estimating the sample mean, and thus only (m-1) measurements remain to estimate the standard deviation. If the mean were known ahead of time, however, then all the measurements could be used to estimate the standard deviation.

When one reports the results of an experimental study, it is normal practice to report the value of a measured quantity as the mean, defined by Equation 2. Similarly, the uncertainty is usually given as the standard deviation, defined by Equation 4.

Linear Regression Engineers often need to analyze sets of data where each data point is represented

by a pair of numbers. For example, one may measure the vapor pressure of a liquid as a function of temperature. The data in this case will be a collection of paired values of temperature and pressure. In general, we are typically concerned with a set of n data points, each represented by a pair of numbers (xi, yi) where i goes from 1 to n. The objective of the data analysis is to infer a functional relationship between x and y.

Two different cases are common. In the first case, the engineer may simply want to consolidate a large number of data points into a simple equation describing the functional relationship between the x and y values, y = f(x). Here the objective is simply


Ap.13- 5

to present the results in a neat, concise form. The function f(x) may be chosen arbitrarily, and the result is called an empirical correlation.

The other case arises when a functional relationship between the x and y values is known or specified a priori, usually on theoretical grounds. Then the function f(x) is called a system model, and the objective of the experimental study is to estimate values of parameters appearing in the function. In the vapor-pressure example, thermodynamic analysis suggests that the relationship between pressure and absolute temperature should have the form ln P = A+B/T. In this case one might want to estimate from the measurements the parameter B because it would indicate the heat of vaporization of the material studied.

In both of these problems, one must somehow "fit" the experimental data to a specified functional form and to determine the parameter values that provide this best fit. Also, particularly in the former case, several different forms of the function f(x) might be tried to see which form is most successful in representing the variation of the data. In this course, we will concern ourselves with fitting data to straight lines, in which case the curve-fitting procedure is called linear regression. The method by which we fit the data to a straight line is called least-squares linear regression.

For linear regression, we wish to fit data to a line described by the equation

y = f(x) = a x + b

where the constants a and b are called the parameters of the function. Our job is to determine the values of a and b from the n sets of paired numbers (xi, yi). This procedure should always begin by creating plot of y vs. x to determine whether or not the data indeed follow something close to a straight-line relationship. Such a plot is depicted in Figure 1. In this figure, the data appear to follow a straight line approximately, but the points are scattered such that they will not fit a straight line perfectly. There is clearly an infinite number of straight lines that would go more or less through the data points. We could also come up with a large number of potential algorithms by which we determine the "best" line by minimizing some measure of the distance between the line and the data points. Without additional information, it is not clear that any such method would be better than any other.


Ap.13- 6

0

2

4

6

8

10

12

0 2 4 6 8 10

x

y

Figure A.13-1. An example of hypothetical data in which y is apparently a linear

function of x and in which there is scatter in the experimental y values.

It turns out that if we assume 1. that the uncertainty in the x values is much less than the uncertainty in the y values; 2. that the uncertainties in the y values are uncorrelated; and 3. that repeated measurements of y at any particular x value will distribute normally about the "true" value, that is, that the statistical distribution of deviations in y at a particular x-value is the so-called normal probability distribution; then the best fit of the data to a straight line can be determined by finding the values of a and b that minimize the sum of squared deviations of y values from those predicted by the straight line. Thus, we must find the values of a and b that minimize the sum-of-squares objective function

( )[ ]∑=

+−=n

iii baxybaS

1

2),( (A.13-5)

This function S, which is a measure of the total discrepancy between the function f(x) and the data, depends on a and b, and we want to find the values that make S the smallest. As long as all the xi values are distinct, minimizing S will give a unique answer. It also turns out that this procedure typically gives informative results even if the above assumptions are relaxed.


Ap.13- 7

By taking the partial derivatives of S with respect to a and b and setting both

derivatives to zero, one can show that the values of a and b that minimize S are

( )( )

( )a

xyn

x y

xn

x=

−

−

∑∑ ∑

∑ ∑

1

12 2 (A.13-6)

and

( )( ) ( )( )

( ) ( )b

y x xy x

n x xy ax=

−

−= −

∑ ∑ ∑ ∑∑ ∑

2

2 2 (A.13-7)

where the summations are taken over all n pairs of data points, and x yand are mean values of each of the measured variables. Remember that a and b are the slope and intercept of the straight line fit through the data. Note that it is possible to convert a more complicated functional relationship into the linear form by a transformation of variables. For example, in the case of the vapor-pressure relation, one might define x as 1/T and y as ln P. Then the plot of data would be ln p vs. 1/T. Then the fitting would give A=b, the intercept, and B=a, the slope of the straight line.

Estimating Uncertainty

The results given in Equations 6 and 7 are only estimates of the true values of a and b. Even if the assumed form of f(x) is indeed the actual relation between x and y, if systematic errors are eliminated, and if the conditions listed as conditions for the least-square method are satisfied, the inevitable occurrence of random errors in any real measurements means that a very large number of data points is required before the true values of a and b can be estimated with great confidence. Therefore, it is necessary for the experimenter to make some estimate of the reliability of his results.

One approach for estimating uncertainty in reported values of the parameters is to

calculate the coefficient of determination r, which is defined by

( )( )( ) ( )

rxy

nx y

xn

x yn

y

2

2

2 2 2 2

1

1 1=

−

−

−

∑ ∑∑

∑ ∑ ∑ ∑ (A.13-8)


Ap.13- 8

It can be shown that the absolute value of r is less than or equal to unity. A perfect fit of the data to the linear equation y = ax + b gives the limiting value of r2 = 1.0 . Completely random scatter of x and y with no correlation would yield r = 0. The coefficient of determination is not a particularly satisfactory measure of fit, however, because its specific value r2<1.0 in any real application does not provide any meaningful quantitative information. That is, one does not know how close r2 should be to 1.0 in order to indicate a satisfactory fit. One would like a criterion against which to gauge goodness of fit.

A much better method for describing the quality of the fit is to report the uncertainty in the estimated values of the parameters a and b. The estimated values of the standard error values for the two parameters, sa and sb are given by

( )

ss

x xa

e

i

22

2=

−∑ (A.13-9)

and

( )

s sx

n x xb e

i

i

2 22

2=

−∑

∑ (A.13-10)

where se, the estimate of the sample error variance, is given by

sn

S a be2 1

2=

−( , ) (A.13-11)

The Microsoft Excel spreadsheet program contains a function called LINEST that

performs linear regression on a set of data points. In addition to calculating the values of a and b, LINEST produces the statistical parameters sa and sb as well as the coefficient of determination of the fit, r.

Given these statistical quantities, we can now report values of the parameters a

and b along with their confidence limits as a ± t sa and b ± t sb .

The additional factor t in the estimated uncertainties is a parameter that arises in the so-called Student's t test and the t distribution. It is a function of the number of data points available as well as the confidence level that one wishes to put on the parameter estimates. Values of the factor t are given in Table 1 as a function of the number of degrees of freedom ν and a parameter α that expresses the confidence interval of the estimate. The number of degrees of freedom is the number of data points minus the number of


Ap.13- 9

parameters being fit to the data. For the straight-line equation with finite intercept b, ν = n - 2, where n is the number of data points, and α is given by

α = −10100

.X

(A.13-12)

where X is the width of the confidence interval expressed as a percentage. The confidence interval X is a value that the experimenter may choose for purposes of stating his or her results. The significance of X is that of a probability; if the experiment were repeated numerous times with n data points taken each time, X is the probability that the resulting values of a and b would fall within the stated ranges estimated from the original data set. That is, if one wishes to state confidence limits corresponding to a certain value X for results based on n data points, the appropriate value of t can be found using the TINV(α,ν) function in Excel and used along with the sa and sb values to calculate the appropriate limits. Table 1 shows a sample of t values. The higher the level of confidence (probablility) one wishes to place on an estimate, the wider the confidence limit would be. On the other hand, increasing the data points (degrees of freedom) narrows the confidence range for a given level of confidence.

Table A.13-1. Tabulated values of t as a function of the degrees of freedom (v) and tail probability α. ν α = 0.5 0.20 0.10 0.05 0.02 0.01 1

1.000

3.078

6.314

12.706

21.821

63.657

2 0.816 1.886 2.920 4.303 6.965 9.925 3 0.765 1.638 2.353 3.182 4.541 5.841 4 0.741 1.533 2.132 2.776 3.747 4.604 5 0.727 1.476 2.015 2.571 3.365 4.032 6 0.718 1.440 1.943 2.447 3.143 3.707 7 0.711 1.415 1.895 2.365 2.998 3.499 8 0.706 1.397 1.860 2.306 2.896 3.355 9 0.703 1.383 1.833 2.262 2.821 3.250 10 0.700 1.372 1.812 2.228 2.764 3.169 15 0.691 1.341 1.753 2.131 2.602 2.947 20 0.687 1.325 1.725 2.086 2.528 2.845 30 0.683 1.310 1.697 2.042 2.457 2.750 50 0.679 1.299 1.676 2.009 2.403 2.678 ∞ 0.674 1.282 1.645 1.960 2.326 2.576


Ap.13- 10

For engineering purposes, it is common to report results at a 95% confidence limit or higher. If X is chosen to be 95%, then α = 0.05 . Suppose that an experiment yielded 5 data points. Then ν = 3, and Table 1 indicates that the corresponding value of t is 3.182 . If twice as many data points were taken such that ν = 8, then t would be 2.306 . Taking more data thus tightens up the range of uncertainty in the parameter estimates. Also, decreasing the specified confidence probability X reduces the range of uncertainty. Conversely, using more data points and specifying a wider range of uncertainty in the parameter values increase the probability X that the true parameter values are within the stated uncertainty limits.

Application of least-squares analysis and linear regression to a set of data is straightforward but tedious. Most of these methods are available as routines in software packages such as Microsoft Excel. LINEST Command The LINEST command on Excel (there is an equivalent command in Sun Microsystem’s StarOffice) is very helpful in data analysis. It allows one to do a multivariate linear regression, that is, fitting of an algebraic equation to a set of data. The output gives the fitted parameters of a linear model of the form y=a0+a1*x1+a2*x2+…+an*xn and additional information telling the sum of squares of the predictions, the expected error in each parameter, and various other statistical values. Unfortunately, this is a rather tricky command to use. It is an array function, which means that one has to select the output area before entering the function. The output area of LINEST, that is, the area that is highlighted before entering this function, has to be of size 5 rows by, at least, n+1 columns, with n being equal to the number of independent variables. In a simple linear regression of the form y=mx+b, the output area must be at least 5 rows by 2 columns. After selecting a region for the output, one can activate the LINEST function either by entering its name directly on the input line or by choosing it from the function command (in the Insert menu). The help pages in Excel give information on the specifics of this command. There are four arguments in the LINEST function: the range of cells containing the dependent variable values, y, the range of cells containing the independent variables values, the "const" logic value, and the "stats" logic value. One usually wants the "const" value to be true; if const is false or zero, the fit equation is forced to pass through the origin, i.e., a0=0. The "stats" value needs to be "true" for LINEST to give detailed error information on the regression results. To fit a polynomial function using LINEST, one has to calculate explicitly the polynomial variables in order to force it to work. For example, if one were trying to fit a set of y(x) data to the polynomial function y=a0+a1*x+a2*x2+a3*x3, one would need to provide a column of y values, a column for the x values, and separate columns for the x2


Ap.13- 11

values and the x3 values. That is, x2 and x3 are treated like additional independent variables in order to calculate the coefficients a0, a1, a2, and a3. After one enters the LINEST function, execution requires hitting shift-control-enter on the Windows machines or shift-applekey-enter on the Macintosh machines. Then the statistical information and the values of the coefficients in the fitted function will appear in the output area.

Bibliography G.E.P. Box, W.G. Hunter, and J.S. Hunter, Statistics for Experimenters, John Wiley & Sons, New York, 1978 I. W. Burr, Applied Statistical Methods, Academic Press, New York, 1974 N. P. Cheremisinoff, Practical Statistics for Engineers and Scientists, Technomic, Lancaster, PA, 1987 J. P. Holman, Experimental Methods for Engineers, 3rd ed., McGraw-Hill, New York, 1978 S. J. Kline and F. A. McClintock, Mech. Eng., January 1953, p. 3 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes, the Art of Scientific Computing, Cambridge University Press, Cambridge, 1986 Appendix13 Revised 9/17/98; TWC Revised 1/20/2002; RCC

Expt D benchmark problem

Benchmark problem for Expt DChE

John J Kasab

December

The benchmark problem for Experiment D in Experiments in Transport Phenomena Crosby is as follows A tank similar to those in the pilot plant initially contains gal ofwater at F and then receives a feed stream of water at F at the rate of galminSteam heat is applied at a constant steam pressure of psig Initally there is no owout of the tank but once the tank is full an exit ow of galmin is started to maintaina constant level in the tank

We want to analyze the heating this system with a steam coil or with an external heat exchanger of the types used in the pilot plant This report contains the theoretical developmentnecessary to analyze this benchmark case

Steam coil heating while lling the tank

First we want to nd out how the water temperature in the tank changes as a function oftime when the steam coil is used The tank of total volume V starts with a volume V att The water in the tank starts at temperature T The ow rate in is F and the inlettemperature is T The outlet ow is FHt tf where tf VF is the time when thetank lls

Start with a mass balance on the tank system Assuming constant density we can solve thedierential mass balance for V t

V t Ft V

We can make the equations dimensionless using the following variables

T T

Ts T

Ft

V V

VV

VV

where Ts is the steam temperature Note that is dened dierently that in Crosby Equation becomes

For the steam coil assume that the heat transfer area is proportional to the volume in thetank Thus let AtA and


Now take an energy balance over the tank We start with

Cp

d

dtV T T UAtTs T

which includes the simplication that T T Making this equation dimensionless usingthe above denitions we obtain

VV V

d

d

UA

F Cp

If we let N UAF Cp and expand the dierential terms equation becomes

d

d

N

or

d

d N

We can recover equation from Bird et al if we dene

N

which turns equation into

d

d

with the initial condition at N The solution to this is equation fromBird et al Application of the initial conditions gives

h NeN

i

where N and are as dened above Note that equation reduces to equation Dfrom Crosby when

Heat exchanger heating while lling the tank

We start with the same mass balance but this time the energy balance on the tank is

Cp

d

dtV T T Fh CpTh T

where Th is the heat exchanger outlet temperature The energy balance on the heat exchangeris

whCpTh T UAh

Ts T Ts Th

lnTsTTsTh


where wh Fh If we let Nh UAhwhCp we get

Th Ts eNhTs T

This result assumes that the heat exchanger is at pseudosteady state compared to the tankheating The equations can be made dimensionless by using the same variables as denedabove and adding in

Fh

F

eNh

and

The heat exchanger energy balance becomes

h eNh

and the tank energy balance becomes

d

d

The initial condition for equation is when

Postulate a solution of the

form

b

r

The rst term comes from the powerseries solution of equation which does not satisfythe initial condition on its own When the initial condition is applied equation becomes

This equation should be equivalent to equation D in Crosby although thiscannot be checked by inspection

Steam coil heating lled tank

In this case the volume of water within the tank is held constant at V as the ow in equalsthe ow out The inlet temperature is T T and the tank starts at temperature Tf We want to know what the maximum temperature T is and how the tank temperatureapproaches this limit Here we reset the starting time to the moment when the tank is fulland the outow starts

Start with an energy balance on the tank

CpVdT

dt UATs T F CpT T


Again we can make the equations dimensionless this time using the following variables

T TTs T

Ft

V

N UA

F Cp

Equation becomesd

d N N

with the initial condition f at Thus at steady state

N

N

and the transient solution is

N

N

eN

fe

N

where f is calculated using the end result from the rst part of this development Thisequation is equivalent to equation D in Crosby when f

Heat exchanger heating lled tank

In this case the tank is full and the water starts at temperature Tf but the heating comesfrom the heat exchanger The value of Tf for this case will be dierent from that for thesteam coil case since the steam coil and heat exchanger heat at dierent rates while thetank is lling with water Again the inlet temperature T equals T

Again the volume is constant at V The energy balance on the tank is

CpVdT

dt Fh CpTh T F CpT T

and on the heat exchanger

Fh CpTh T UAh

Ts T Ts Th

lnTsTTsTh

These energy balances can be made dimensionless using the same denitions for and asfor the steam coil with a lled tank In addition we need to dene Nh UAhFh Cpas before with the heat exchanger


Assuming a pseudosteady state condition the dimensionless energy balance on the heatexchanger becomes

h eNh

For the tank we getd

d

where is as dened in equation The initial condition for equation is f at

The solution is therefore

e

fe

and

References

Bird R B Stewart W E and Lightfoot E N Transport Phenomena New YorkJohn Wiley Sons pp

Crosby E J Experiments in Transport Phenomena revised ed University of WisconsinMadison

experiments in transport phenomena · 2019-10-14 · che 324 lab manual table of contents page i-3...

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