ME 152 - IT: Interactive Thermodynamics Computer Exercises

ME 259 - IHT: Interactive Heat Transfer Computer Exercises

1. Basic Equation Solving

Suppose you wish to solve the following set of equations, where q1, q2, q3, and T are unknowns:

q1 = 0.35(1000-T)

q2 = 5.67(10-8(T4-3004)

q3 = 1.4(T-300)4/3q1 = q2 + q3Note that two equations are nonlinear and the system cannot be solved analytically.

STEP 1 - Preprocessing

Simply type the equations as given onto the IHT workspace as shown below. You may insert comments or headings by preceding text lines with a double slash // or by bracketing long sections of text with /* . */.

// Nonlinear Equation System Example

q1 = 0.35*(1000-T)

q2 = 5.67E-08*(T^4-300^4)

q3 = 1.4*(T-300)^(4/3)

q1 = q2 + q3

STEP 2 - Solving

Click the Solve button to prepare the solver. If your equations have syntax errors, an error warning will appear - click OK and usually the solver will highlight the word or equation that contains the error. If you do not supply enough equations for the number of unknowns, another error warning will appear - click OK and a summary of equations and unknowns will appear to help you determine what is missing.

If there are no syntax or number of equations errors, an Initial Guesses table appears. Initial guesses are required for each unknown and the solver uses a value of 1 as a default guess. You may change these guesses, however, the default values are often satisfactory. You may also insert Minimum and Maximum solution values if you know the range of possible solution values. For example, we know that temperature in Kelvin can never be less than zero; so inserting a minimum value of 0 for a temperature unknown will accelerate convergence since the solver will then only search for positive values. In this example, set the minimum value for T to 300 (why?).

After initial guesses have been set, click OK and wait for the Data Browser table to appear. (Note: if a previous dataset exists, the solver will ask you whether you want to save it or discard it.) If the message "Equation set successfully solved" appears in the Data Browser, then the values for q1, q2, q3, and T represent a valid solution. In general, there may exist several valid solution sets for a system of nonlinear equations. The particular solution found by the solver depends upon the initial guesses and the search algorithm. In this example, only one valid solution set exists since the equations are based upon a physical problem:

T322.1

q1237.3

q2150.7

q386.59

STEP 3 - Postprocessing

Once a valid solution has been found, you may want to copy it to the clipboard for pasting. Highlight the solution by clicking the upper left rectangle in the table and then click Copy. Options are available for transposing rows/columns and including variable names. It is suggested that you paste your solution to the bottom of your equation worksheet before printing a hardcopy. This is done by locating the cursor at the bottom and then clicking Paste from the Edit menu. Also, be sure to type in appropriate units next to each solution value. You can now print your entire problem by clicking Print from the File menu. If you wish to have greater control of how your page looks in terms of font style/size, organization, bold/italic/underline, etc., copy and paste your entire worksheet to Notepad or Microsoft Word.

Performing parametric studies and plotting results are possible using the Explore and Graph functions. These features will be used in the next example.

2. Solving a Heat Transfer Problem: Insulated Steam Pipe Undergoing Convection and RadiationThe following worksheet example shows how to set up a typical heat transfer problem. Note that through the use of commenting, the problem can be presented in a structured format where the given information and knowns are segregated from the equations and unknowns. The solver recognizes any variable that is set to a numerical value as a known. You may use any alphanumeric name for a variable, however, please try to be conventional. There are the usual intrinsic functions such as pi, sqrt, exp, ln, sin, cos, tan, etc. Refer to the Help Index for a complete listing of the intrinsic functions. Also, IHT does not keep track of dimensional units you must make sure that consistent units are used.

// ME 259 Class Example// Given: Insulated steam pipe undergoing convection and radiation

ri = 0.05 // m

ro = 0.075 // m

L = 25

// m

Ti = 423.15 // K

To = 298.15 // K

hi = 100

// W/m2-K

ho = 10

// W/m2-K

eps = 0.8

sigma = 5.67E-08 // W/m2-K4// Find: a) Total heat loss from pipe (q)

// b) Outside surface temperature of insulation (Ts)

// c) Effect of varying ro from 0.06 to 0.25 m

// d) Effect of varying eps from 0.05 to 1.0

// Assumptions:

i) steady-state conditions

//

i) one-dimensional radial conduction through insulation

//

iii) constant properties

//

iv) large surroundings, Tsur = To

//

v) pipe wall has negligible thermal resistance

// Properties:

// 85% magnesia insulation at (360 K, from Table A.3, p.835

kins = 0.055 // W/m-K

// ANALYSIS:

// Thermal resistancesRtotal = Rconvi + Rins + Rconvo*Rrado/(Rconvo+Rrado)

Rconvi = 1/(hi*2*pi*ri*L)

Rins = ln(ro/ri)/(2*pi*kins*L)

Rconvo = 1/(ho*2*pi*ro*L)

Rrado = 1/(hr*2*pi*ro*L)

hr = eps*sigma*(Ts^2+To^2)*(Ts+To)

// Heat rate & temperature equations

q = (Ti-To)/Rtotal

q = (Ti-Ts)/(Rconvi+Rins)

Ts_C = Ts - 273.15

The solution to this problem is given below:

Rconvi

0.001273

K/W

Rconvo0.008488

K/W

Rins

0.04693

K/W

Rrado

0.01654

K/W

Rtotal

0.05381

K/W

Ts

311.2

K

Ts_C

38.03

C

hr

5.133

W/m2-K

q

2323

W

Now, try performing a parametric study where you vary the insulation thickness (i.e., the value of ro) over the range of 0.06 to 0.25 m in steps of 0.01 m To do this, click on Explore and choose the Variable to Sweep as ro and input the range parameters. Then click OK and wait for the Data Browser to appear. Here, you will see the results corresponding to each value of ro. Suppose you are interested in the effect of ro on the heat loss, q. This is best viewed by plotting q vs. ro. Click on Graph, then select ro as coordinate X and q as coordinate Y1. Click OK and a plot should appear. You can add titles, change scales, change lines, change data markers, and add a legend if you wish. The graph can be printed as a full page or copied, sized, and pasted to your worksheet. Next, try plotting Ts versus ro and then repeat for a parametric study with 0.05 < eps < 1.

There are many other features in IHT, including 1st-order differential equations, integrals, material property functions, user-defined functions, and pre-written heat transfer correlations, special equations, and models (see Help menu). Not bad for a $15 program !!