Turbulent Heat Transfer to Non Newtonian Fluids in Circular Tubes

GBH Enterprises, Ltd.

Process Engineering Guide:
GBHE-PEG-HEA-518

Turbulent Heat Transfer to NonNewtonian Fluids in Circular Tubes
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Process Engineering Guide:

Turbulent Heat Transfer to NonNewtonian 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

THE CALCULATION OF HEAT TRANSFER
COEFFICIENTS FOR NON-NEWTONIAN AND DRAG
REDUCING FLUIDS IN TURBULENT PIPE FLOW

4

General
Drag Reducing Fibre Suspensions
Transition Delay

4
5
5

6

6.1
6.2
6.3


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

DOCUMENTS REFERRED TO IN THIS PROCESS
ENGINEERING GUIDE

9

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
(Subscript)
N
w

Meaning

Newtonian
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 nonNewtonian Fluids (referred to in 6.1 and 6.3).


Turbulent Heat Transfer to Non Newtonian Fluids in Circular Tubes

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