Laminar Heat Transfer to Non Newtonian Fluids in Circular Tubes

GBH Enterprises, Ltd.

Process Engineering Guide:
GBHE-PEG-HEA-517

Laminar Heat Transfer to NonNewtonian Fluids in Circular Tubes

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Process Engineering Guide:

Laminar 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

APPLICABILITY AND LIMITATIONS

2

4.1
4.2

Applicability
Limitations

2
3

5

THEORETICAL BACKGROUND

4

6

PRESENTATION OF RESULTS

5

7


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

APPENDICES

7

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 nonNewtonian. 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


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 nonNewtonian 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
Cv
f
K
Ko
k
n
p
Q
r
T
uz
z

Specific heat at constant pressure
Specific heat at constant volume
function
consistency index
reference consistency index at To
thermal conductivity
power law index
pressure
rate of internal heat generation
radial distance
temperature
axial velocity
axial position

Jkg-1K-1
Jkg-1K-1

Symbol
(Greek)

Meaning

Unit

β
ẏ
µ
ρ
ṫ
ṫw
Ҝ

compressibility [m2/N]
shear rate
viscosity
density
shear stress
wall shear stress
coefficient of thermal expansion

Wm-1K-1
Nm-2
W
m
K
ms-1
m

s-1
N.s m-2
kgm-3
kgm-1s-2
kgm-1s-2
K-1


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