turbulent heat transfer to non newtonian fluids in circular tubes

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Process Engineering Guide: GBHE-PEG-HEA-518

Turbulent Heat Transfer to Non-Newtonian Fluids in Circular Tubes Information contained in this publication or as otherwise supplied to Users is believed to be accurate and correct at time of going to press, and is given in good faith, but it is for the User to satisfy itself of the suitability of the information for its own particular purpose. GBHE gives no warranty as to the fitness of this information for any particular purpose and any implied warranty or condition (statutory or otherwise) is excluded except to the extent that exclusion is prevented by law. GBHE accepts no liability resulting from reliance on this information. Freedom under Patent, Copyright and Designs cannot be assumed.



Process Engineering Guide: Turbulent Heat Transfer to Non-Newtonian Fluids in Circular Tubes

CONTENTS SECTION 0 INTRODUCTION/PURPOSE 2 1 SCOPE 2 2 FIELD OF APPLICATION 2 3 DEFINITIONS 2 4 THE INTEGRATION OF THE ENERGY EQUATION 2 5 THE EDDY VISCOSITY FOR NON-NEWTONIAN AND

DRAG REDUCING FLUIDS 3 6 THE CALCULATION OF HEAT TRANSFER

COEFFICIENTS FOR NON-NEWTONIAN AND DRAG REDUCING FLUIDS IN TURBULENT PIPE FLOW 4

6.1 General 4 6.2 Drag Reducing Fibre Suspensions 5 6.3 Transition Delay 5



7 NOMENCLATURE 6 8 BIBLIOGRAPHY 7 APPENDICES A CALCULATION PROCEDURE FOR THE COEFFICIENTS

y1+ and C1+ 8 FIGURES

1 PROCEDURE FOR THE CALCULATION OF THE CONSTANTS y1+ and C1+ IN THEVAN DRIEST EXPRESSION FOR THE EDDY VISCOSITY 9

DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE 24



0 INTRODUCTION/PURPOSE This Process Engineering Guide is one of a series of guides on non-Newtonian fluids. The most common method used to calculate heat transfer coefficients for the fully developed turbulent pipe flow of Newtonian fluids is via empirical correlations of the type:

Whilst this method is accurate enough for many engineering purposes, such empirical correlations do not have a physical basis and it is extremely difficult to extend them from Newtonian fluids to non-Newtonian and drag reducing fluids. The integration of the energy equation provides a much more satisfactory general approach. 1 SCOPE This guide presents a procedure for calculating heat transfer to non-Newtonian and drag reducing fluids under turbulent flow conditions in circular tubes. 2 FIELD OF APPLICATION This guide applies to the process engineering community in GBH Enterprises worldwide. 3 DEFINITIONS For the purposes of this guide, no specific definitions apply. 4 THE INTEGRATION OF THE ENERGY EQUATION Consider axisymmetric pipe flow of an incompressible fluid with a fully developed velocity profile. Neglecting axial heat conduction and viscous dissipation the equation of energy is:



Where q reff is the effective radial heat flux given by:

The eddy diffusivity ε hr in equation (3) can be replaced by the relationship:

where ε rz is the eddy viscosity. Pr tr is the turbulent Prandtl number and can be assumed equal to unity (see Ref. [1]). For fully developed flows equation (2) has been integrated for the case of constant wall flux by Lyon (see Ref. [2]) and for the case of constant wall temperature by Seban and Shimazaki (see Ref. [3]). The expressions developed by Lyon and by Seban and Shimazaki are very complex and require the solution of double integrals. However, simplified integrations of equation (2) have been presented by Edwards and Smith (see Ref. [1]). In this work it was recommended that the dimensionless heat transfer coefficient be calculated from:



Before solving equation (5) it is necessary to define the eddy viscosity. This is considered below. 5 THE EDDY VISCOSITY FOR NON-NEWTONIAN AND DRAG REDUCING FLUIDS Smith and Edwards (see Refs. [4] and [5]) have demonstrated that standard Newtonian eddy viscosity expressions can be successfully adapted to predict heat transfer to non-Newtonian and drag reducing fluids in turbulent pipe flow. The most accurate eddy viscosity expression in the wall region is that due to Van Driest (see Ref. [6]).

where:

A, y1+ and C1+ are constants which will be considered later.



Away from the wall it is recommended that the eddy viscosity expressions of Mizushina and Ogina (see Ref. [7]) be used i.e.:

The above expressions for the eddy viscosity can be applied to non-Newtonian fluids by replacing the Newtonian viscosity in the dimensionless variables vz,+ y+ and R+ by the apparent viscosity at the wall (see Ref. [8]). They can also be applied to drag reducing fluids by assuming: (a) That in the turbulent core of drag reducing pipe flow there is always a region where the velocity profile can be described by a logarithmic profile i.e:

(b) That the gradient of the logarithmic profile is constant i.e. the coefficient A in the above equations is constant and equal to its Newtonian value of 2.5 and the coefficient B varies with the degree of drag reduction. These assumptions are supported by the evidence of Elata, Lehrer and Kahanovitz (see Ref.[9]), Arunachalem, Hummel and Smith (see Ref. [10]) and Rollin and Seyer (see Ref.[11]).



6 THE CALCULATION OF HEAT TRANSFER COEFFICIENTS FOR

NONNEWTONIAN AND DRAG REDUCING FLUIDS IN TURBULENT PIPE FLOW

6.1 General The starting point for the calculation of heat transfer coefficients is the evaluation of the coefficient B. For Newtonian and purely viscous non-Newtonian flow the coefficient B has a constant value of 5.1. With drag reducing fluids it is necessary to determine B from experimental pressure drop data. The correlation of pressure drop data for drag reducing fluids has been discussed in GBHE-PEG-.FLO.304. Computer programs, CHEMCad can be used to calculate a heat transfer coefficient as the procedure, outlined below, is too complex to be carried out by hand.

(a) From pressure drop data calculate the coefficient B from the equation of Arunachalem, Hummel and Smith (see Ref. [10]) i.e.:

(b) Calculate R+ from the wall shear stress (i.e. pressure drop). (c) Calculate y2+ from equation (10). (d) Calculate y1+ and C1+ using the procedure outlined in Appendix A. (e) Calculate the heat transfer coefficient from equation (5). The integral should be divided into three zones 0 to y1+, y1+ to y2+ and y2+ to R+ and solved numerically.



6.2 Drag Reducing Fibre Suspensions Although it is recommended that the above design procedure be applied to all forms of drag reducing flow it should be noted that it has not been tested at all on fibre suspensions. When applied to drag reducing polymeric materials, the above procedure has been shown to be capable of producing heat transfer coefficients to within ± 20% of experimental data (see Refs. [4] and [5]). 6.3 Transition Delay The calculation procedures outlined so far do not apply to the transition delay behavior described in GBHE-PEG-.FLO.304. No design methods are currently available to take account of this kind of behavior. A conservative design can be obtained for transition delay by calculating heat transfer coefficients assuming the flow to be laminar.



7 NOMENCLATURE



Symbol Meaning (Subscript) N Newtonian w at the pipe wall 1 at the boundary between the viscous sublayer and the logarithmic region 2 at the boundary between the logarithmic region and the turbulent core 8 BIBLIOGRAPHY This Process Engineering Guide makes reference to the following: [1] M.F. Edwards and R. Smith

The Integration of the Energy Equation for Fully Developed Turbulent Pipe Flow. Trans. I. Chem. E. 58, 260-264 (1980).

[2] R.N. Lyon

Liquid Metal Heat-transfer Coefficients Chem. Eng. Prog. 47, 75-79 (1951).

[3] R.A. Seban and T.T. Shimazaki

Heat Transfer to a Fluid Flowing Turbulently in a Smooth Pipe with Walls at Constant Temperature. Trans. ASME73, 803-809 (1951).

[4] R. Smith and M.F. Edwards

Heat Transfer to non-Newtonian and Drag reducing Fluids in Turbulent Pipe Flow. Int. J. Heat Mass Transfer 24, 1059-1069 (1981).

[5] R. Smith, M.F.Edwards and H.Z. Wang

Pressure Drop and Mass Transfer in Dilute Polymer Solutions in Turbulent Drag reducing Pipe Flow Int. J. Heat Mass Transfer, accepted for publication.

[6] E.R. Van Driest

On Turbulent Flow near a Wall J. Aero, Sci 23, 1007-1011 (1956).



[7] T. Mizushina and F. Ogino Eddy Viscosity and Universal Velocity Profile in Turbulent Flow in a Straight Pipe. J. Chem. Engng Japan 3, 166-170 (1970).

[8] M.F. Edwards and R. Smith

The use of Eddy Viscosity Expressions for Predicting Velocity Profiles in Newtonian, non-Newtonian and Drag reducing Turbulent Pipe flow J. Non-Newtonian Fluid Mech 7, 153-169 (1980).

[9] C. Elata, J. Lehrer and A. Kahanovitz

Turbulent-shear Flow of Polymer Solutions Israel J Technol, 4, 87-95 (1966).

[10] V.T Arunachalem, R.L. Hummel and J.W. Smith

Flow Visualisation Studies of a Turbulent Drag reducing Solutions Can. J. Chem. Engng 50, 337-343 (1972).

[11] A. Rollin and F.A. Seyer

Velocity Measurements in Turbulent Flow of Viscoelastic Solutions Can. J. Chem. Engng 50, 714-718 (1972).



APPENDIX A CALCULATION PROCEDURE FOR THE COEFFICIENTS y1+ and C1+ The coefficients y1+ and C1+ in Van Driest expression for the eddy viscosity can be determined by matching the predicted mean velocity and viscosity gradient with those predicted by the logarithmic profile given in equation (11). If the eddy viscosity is known, it can be used to give the mean velocity profile by (see Ref. [8]):

Substituting the Van Driest expression into equation (13) and equating the velocity with that predicted by the logarithmic profile at y1+ gives:



Equating the velocity gradients gives an explicit expression for C1 +:

Given the values of the coefficients A and B then y1+ and C1+ can now be determined by an iterative search procedure (e.g. false position) solving the integral by a suitable numerical technique (e.g.Simpson's rule). Figure 1 shows a flow chart for the calculation of y1+ and C1+. This calculation can be carried out using previously mentioned computer programs. If the flow is Newtonian or purely viscous non-Newtonian then the coefficients A and B are fixed with the values of 2.5 and 5.1 respectively giving y1+ = 114.2 and C1+ = 25.33.



FIGURE 1 PROCEDURE FOR THE CALCULATION OF THE CONSTANTS y1+ and C1+ IN THE VAN DRIEST EXPRESSION FOR THE EDDY VISCOSITY



DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE This Process Engineering Guide makes reference to the following documents: PROCESS ENGINEERING GUIDES GBHE-PEG-.FLO.304 Pipeline Design for Isothermal, Turbulent Flow of non-Newtonian Fluids (referred to in 6.1 and 6.3).