Laminar Heat Transfer to Non-Newtonian Fluids in Circular Tubes

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Process Engineering Guide: GBHE-PEG-HEA-517

Laminar 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: Laminar 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 APPLICABILITY AND LIMITATIONS 2 4.1 Applicability 2 4.2 Limitations 3 5 THEORETICAL BACKGROUND 4 6 PRESENTATION OF RESULTS 5

7 PRESENTATION OF RESULTS 5



8 USE OF The VAULT 5 8.1 Limitations of The VAULT 6 9 NOMENCLATURE 6

10 BIBLIOGRAPHY 7 DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE 7 APPENDICES A



0 INTRODUCTION/PURPOSE This Process Engineering Guide is one of a series of guides on non-Newtonian fluids based on reports prepared for GBH Enterprises. Heat transfer to viscous fluids in laminar flow is a process which is frequently encountered. The fluids of commercial interest may be Newtonian or non-Newtonian. A notable feature of many such viscous fluids is that their rheological properties are very sensitive to temperature. For heat transfer the viscosity will vary markedly with radial position. This variation can have a large effect on both the radial temperature and velocity profiles and consequently on heat transfer rates and pressure drop. 1 SCOPE This guide introduces a mathematical model which is capable of accurately predicting temperature and velocity profiles, heat transfer rates and pressure drop for viscous Newtonian and non-Newtonian fluids during heat transfer in laminar flow in tubes. The model deals not only with the simple problem of laminar heat transfer to Newtonian fluids, but also with the following complications which can have a significant effect in some situations: (a) Fluid Rheology. (b) Boundary Conditions. (c) Viscous Shear Heating. (d) Expansion Cooling. (e) nternal Heat Generation. The complete mathematical treatment of this problem is complex and beyond the scope of this guide. 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 APPLICABILITY AND LIMITATIONS 4.1 Applicability

(a) Fluid Rheology The method will cater for a range of non-Newtonian fluid types, viz., power law fluids which can have a yield stress for which the rheological equation is of the form:

and also fluids whose rheological characteristics are more generally described in terms of shear stress and shear rate by:

where the viscosity is evaluated as a function directly from viscometric data. The Newtonian fluid is included merely as a special case. For more information on the interpretation of viscometric data for non-Newtonian fluids, see GBHE-PEG-FLO-302. (b) Boundary conditions

The inlet fluid temperature can be constant or vary with radial position. The wall temperature can be constant or an arbitrary function of distance. Uniform wall heat flux also can be dealt with.

(c) Viscous shear heating

Viscous shear heating is included but this will only be of significance with fluids of very high viscosity such as polymer melts which are subjected to high pressure drops (greater than 30 bar).



(d) Expansion cooling

This effect is included but except for very high pressure drop situations, such as in polymer melt flow, it will not be significant.

(e) Internal heat generation

Internal heat generation, such as due to heat of reaction, can be included provided this is uniform.

4.2 Limitations The analysis of heat transfer to non-Newtonian fluids requires that many simplifications be made to the equations of conservation of mass, momentum and energy in order to allow a solution to be obtained. The normal assumptions are: (a) Density, thermal conductivity and specific heat of the fluid are independent

of temperature (and these are usually realistic assumptions). (b) The shear stress at any point is a function of shear rate, temperature and

pressure only. (Time dependant rheological behavior is not considered) (c) The flow is steady, laminar and axisymmetric and radial and tangential

velocities are negligible. (d) Natural convection is negligible and all elastic forces are treated as

entrance losses.



5 THEORETICAL BACKGROUND The simplified conservation equations are as follows: Conservation of momentum:

Assuming

And

Conservation of mass:

Conservation of energy:

Simplifying the thermal expansion term gives:



Substituting this in equation (7) and rearranging gives:

These equations should be solved in conjunction with the definition of the rheological character of the fluid which is assumed to be viscous, non-Newtonian fluid. The shear stress may be expressed by Equation! (2), or the viscosity term in equation (2) may be expressed as a temperature dependent power law function from equation (1). Before solving, the equations and the corresponding boundary conditions are expressed in non-dimensional form. These equations are solved by the substitution of implicit finite difference approximations for the partial differentials. The resultant algebraic equations were solved using Thomas's method. The non-linear terms were rendered constant at each point by secondary calculations based upon their values at the previous position upstream, thus eliminating lengthy iterative calculations.



6 PRESENTATION OF RESULTS The results are computed in the form T(r,z), u(r,z) and P(z), i.e. the radial temperature and velocity profiles and the axial pressure profile. In some cases this is the result required, but for heat transfer calculations it is often useful to present the results in the form of a Nusselt number as a function of the Graetz number (a dimensionless axial distance). The appropriate definition of the Nusselt number depends on the wall boundary conditions as follows. 7 EXPERIMENTAL VERIFICATION OF PROCEDURE The procedures adopted in this guide have been substantiated by experiment for a range of fluids, including Newtonian oils and solutions of polymers (see Ref. [1]). 8 USE OF The VAULT As well as performing the calculations for laminar heat transfer, The VAULT will also calculate turbulent heat transfer and isothermal pressure drop for non-Newtonian fluid flow in circular pipes. The VAULT will handle laminar heat transfer for both Newtonian fluids and generalized Bingham plastics. The rheological properties of the generalized Bingham plastic are assumed to be described by the equation :

The consistency index K is considered to be a function of temperature, defined by :



The yield stress xy and the flow behavior index n are both assumed to be independent of temperature. (Equations (10) and (11) are equivalent to the equation (1).) Two alternative boundary conditions may be specified: a) Constant wall temperature. b) Constant wall heat flux An example of the output from The VAULT is given as Appendix A 8.1 Limitations of The VAULT General methods describe / generate temperatures and velocities as functions of both radial and longitudinal position, and pressures as functions of longitudinal position, the output from The VAULT only gives the total pressure drop and the radial variation of temperature and velocity at the exit of the pipe. It also gives the average Nusselt number and heat transfer coefficient. If the variation of these values along the pipe is required, it is necessary to perform several runs with different pipe lengths. Unfortunately, The VAULT does not accept data input from a file, so all the data has to be entered afresh for each different pipe length. The radial variation in inlet temperature described in Clause 4.1 (b) above cannot be used with VISFLO, nor can the variable wall temperature option. The latter may be partially simulated by dividing the pipe into a series of sections and performing repeated runs with differing inlet and wall temperatures, but this will not correctly model the radial variation of temperature from one section to the next.



9 NOMENCLATURE Symbol Meaning Unit CP Specific heat at constant pressure Jkg-1K-1 Cv Specific heat at constant volume Jkg-1K-1 f function K consistency index Ko reference consistency index at To k thermal conductivity Wm-1K-1 n power law index p pressure Nm-2 Q rate of internal heat generation W r radial distance m T temperature K u z axial velocity ms-1 z axial position m Symbol Meaning Unit (Greek) compressibility [m2/N] shear rate s-1 viscosity N.s m-2 density kgm-3 shear stress kgm-1s-2 w wall shear stress kgm-1s-2 coefficient of thermal expansion K-1

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Laminar Heat Transfer to Non-Newtonian Fluids in Circular Tubes