Thermal entrance lengths for laminar flow in various ducts for constant surface heat flux

Pergamon Press Printed in the United States

IN HEAT AND MASS TRANSFER Vol. 3, pp. 8 9 - 98, 1976

THERMAL ENTRANCE LENGTHS FOR LAMINAR FLOW IN VARIOUS DUCTS FOR CONSTANT SURFACE HEAT FLUX

Owen T. Hanna, Orville C. Sandall and Bernard A. Paruit Department of Chemical and Nuclear Engineering University of california, Santa Barbara California

(Communicated b y J . P .

93106

HartnettandW.J.

Minkowycz)

ABSTRACT A method for the determination of thermal or diffusion entrance lengths for transport in ducts with a uniform flux is illustrated. The method requires only a knowledge of the asymptotic downstream solution of the transport equation, which is far easier to obtain than the full solution. The results have a simple form and include effects of axial conduction. Geometries considered are parallel planes, concentric annuli and rectangular ducts of varying aspect ratio. Effects of asymmetric boundary conditions and flow development are considered.

Introduction In this paper an analysis is made of the length required for temperature profiles to reach a fully developed shape for the case of laminar flow heat transfer inside ducts having a uniform or specified surface heat flux.

The

method used to compute the thermal entrance length has been presented by Sandall and Hanna

[i] and is based on an overall energy balance over the

total entrance length.

To apply the method one needs to know only the down-

stream velocity and temperature profiles.

These solutions are relatively

easy to obtain for the case of uniform surface heat flux.

The technique is

generally applicable to situations in which the wall heat flux is specified.

89

90

O.T. Hanna, O.C. Sandall and B.A. Paruit

Vol. 3, No. 2

The approach given here should be useful primarily in problems where the solution of the energy equation in the entrance region is not available, such as for ducts of peculiar cross section, unsymmetrical boundary conditions, or for the flow of highly viscous, non-Newtonian fluids.

Such an approach might

also be useful for problems involving significant variations of fluid properties.

This method would also seem to have applications for the design of

compact heat exchangers. It must be emphasized that the entrance length technique discussed here is significant because of its broad and simple applicability to complicated cross sections and because of its inclusion of axial conduction.

The exam-

ples given, which correspond to well-known classical results of varying complexity, illustrate these features and indicate comparisons between the present method and classical results. The model used to compute the thermal entrance length may be illustrated with reference to FIG. i.

Fluid is flowing through a duct of arbitrary cross

section, A, having the perimeter, S.

The fluid enters the duct with a uni-

form temperature, T . An overall energy balance for the case of a uniform o wall heat flux is made including a section upstream from the start of heating, since due to axial conduction, the effect of heating propagates upstream. This balance is made over the length - L', the upstream point at which axial conduction effects are no longer felt, to L, the point at which the temperature profile first becomes fully developed.

This gives an expression for the

entrance length. 1 [u(T-T°)]LdA + ~

L = QsPCP ..... AI

AI

[Qa]LdA

(i)

~T The axial heat conduction flux, Qa' is given by Fourier's Law as Qa = - k ( ~ ) L For a uniform wall heat flux the axial temperature gradient becomes constant ~T Qs under fully developed (downstream) conditions and is given as ~ = uu~pCpA" Substitution of these relationships into Equation

(i) and dividing .

by

the equivalent diameter, D e gives L pCp D--= QSD e e

I A

1 [u(T-To)]L dA - --Pe

(2)

Thus it is seen that the effect of axial conduction on the entrance length is given by the term - i/Pe.

Vol. 3, NO. 2

THERMAL~TRANCE

I~%~THS I N ~

91

For the case of heating, the initial temperature must be less than the minimum temperature at the point where the temperature profile first becomes fully developed.

This inequality yields

pCp

_ _

L__ > De -- QSDe Equation

I A

u(T-T )dA m

1 Pe

(3)

(3) represents the basic model from which thermal entrance

lengths are estimated in this paper for laminar flow in various geometries. The integral in Equation

(3) may be expressed in terms of the fully developed

Stanton number and T -T by making the substitution, T-T = (T-Tw) + ( % - T m) w m m ' and noting the definition of the bulk temperature and bulk velocity. L__ > pcp

D

e

-- 4Q

(%-Tm) Ub

1 4St

1 Pe

(4)

Thus if the asymptotic Stanton number is known and if the temperature difference, Tw-T m, is known from a previous calculation, then the thermal entrance length may be simply determined using Equation

(4).

Parallel Planes The physical case considered here is laminar flow between parallel plates as shown in FIG. 2a.

The fluid enters the duct with a uniform temper-

ature, To, and with a fully developed velocity profile; each wall is heated with a uniform heat flux, Q.

The energy equation may be integrated, neglect-

ing viscous dissipation, to give the fully developed temperature distribution, T-T . Substitution of this result into Equation (3) and integrating gives an m expression for the thermal entrance length. L - - > 0.00871Pe D -e Equation

1 Pe

(5)

(5) predicts the interesting result that due to axial conduc-

tion for low Pe, as could occur in the case of liquid metals, or some polymer situations, the thermal entrance length vanishes. The predicted entrance length from Equation calculations of Cess and Schaffer

(5) may be compared to the

[2] which predict that the local Nusselt

number, neglecting axial conduction, has approached to within 5% of its fully developed value in an entrance length given by L/D e = 0.0115 Pe.

92

O.T. Hanna,

O.C.

Sandall

and B.A.

Concentric The case of a concentric and outer b o u n d a r i e s

The general

For the special

minimum this

algebraic

temperature

extremely reason,

grals were

evaluated

equation

to use because

thermal

the solution

of the m i n i m u m

tempera-

case of the outer wall insulated, at the insulated wall, analytically;

however,

r = R 2.

The results

entrance

length

of these

RI/R2,

of RI/R 2 over the complete

range

For is

For this

the inte-

integrations

L/D e is, within

the

the result

of the large number of terms.

they depend only on the p a r a m e t e r

less than 1%, not a function result may be e x p r e s s e d

of

(6)

for the location

common

numerically.

that the dimensionless

Application

heat fluxes on each wall requires

and most

and also because

flux on the inner

1 Pe

(6) may be e v a l u a t e d

inconvenient

heat

2b is now considered.

for heating occurs

case Equation

show

an error of in RI/R 2.

This

as

L__ > 0.0321 Pe - I__ D -Pe e

(7)

Equation

(7) may be compared

Lundberg

et al.

to the thermal

entrance

[3] for uniform wall h e a t i n g

RI/R 2 = 0.5 the exact calculations absence

of axial

2

Annuli

ur (T-T m) dr

case of different

of a n o n - l i n e a r ture.

in FIG.

3, No.

leads to

R 12 R1

L > pCp D--e -- (QIRI+O2R2) _ _ De

Vol.

annulus with a constant

as shown

the model to this geometry

Paruit

conduction,

in 5% of its fully d e v e l o p e d

calculations

in the concentric

of Lundberg

the local Nusselt value

region

et al. predict

annulus. that,

For

in the

number has a p p r o a c h e d

in an entrance

of

to with-

length given by L/D e =

0.039 Pe. The R e c t a n g u l a r Attention cross

section

right,

problem

is now focused on the case of a duct having as shown in FIG.

2c.

but the m e t h o d of solution

of arbitrary

Channel

cross

involving

Application

section.

is an interesting

problem

used here could also be applied

This point

unsymmetrical of Equation

This

a rectangular

is illustrated

in its own to a duct

by c o n s i d e r a t i o n

of a

heating.

(3) to this geometry

yields

b a 1 L > ,~ pcp S S u(T-T )dxdy ~ -eD e [ 2 a (QI+Q3)+2b(Q2+Q4 ) ] -b -a m -

(8)

Vol.

3, No. 2

THERMAL E N T R A N C E L E N G T H S

IN DUCI~

93

For a rectangular geometry it is possible to obtain analytical expressions for u(x,y)

and T(x,y), but in general,

differences might be used to determine ferences here.

for an arbitrary cross section, these profiles.

We employ finite dif-

The simplest finite difference approximations

second derivatives were used.

finite

to the various

The constant flux boundary conditions

in the

energy equation were implemented using a fictitious boundary approach in order to assure a uniformity of the difference approximations with respect to spatial increment. For the case of equal heat fluxes on all faces of the duct, the results of the entrance L D

--

e

>

--

length calculations

can be put into the form

1 Pe

8Pe

(9)

where 8 depends only on the aspect ratio b/a. b/a is shown in FIG.

3.

Numerical

The relationship between 8 and

experiments showed that an interior grid

of 256 points produced L/D e results having numerical errors of less than ten percent.

Since the model is itself approximate,

sufficient to demonstrate the approach.

this accuracy was felt to be

By using the calculated results for

a/b = 0.5, 0.2 and 0.i, the value of 8 for a/b = 0 (parallel planes) was estimated.

This estimate yielded 8(0) = 0.0074 versus 0.0087 from a direct

calculation.

When the finite difference and extrapolation

errors are consid-

ered, these values are not inconsistent. The previous straightforward, stream profile

calculations

for symmetrical heating were relatively

since the position of the minimum temperature

is known to be at the center of the duct.

arbitrary cross section or for unsymmetrical

heating,

cross section of the minimum of the temperature this more general state of affairs,

in the down-

In general,

the position in the

is unknown.

To illustrate

a rectangular duct case was considered

for which a/b = 0.5 and Q4 = 2QI" Q2 = Q3 = 0 with Q1 arbitrary. case the finite difference calculations located at approximately ter 8 is equal to 0.096

for an

In this

show that the minimum temperature

is

x = -b/2, y = +5a/16 and the entrance length parame(versus 0.083 for all heat fluxes being equal). Flow Development

It is of some interest to calculate

the effects of flow development

viscous dissipation on the thermal entrance length.

Results of this calcula-

tion for the circular tube geometry are presented here. flow development and heating begin simultaneously, temperature profiles that are radially uniform.

and

It is assumed that

with inlet velocity and

94

O.T. Hanna, O.C. Sandall and B.A. Paruit

Vol. 3, No. 2

It would be expected that the velocity profile would develop faster than the temperature profile for Prandtl numbers greater than about unity.

This

assumption is made; the calculation then produces results that are entirely consistent with this assumption. With the above assumptions, an overall energy balance is made between the tube entrance and the first point downstream where the developed temperature profile applies.

Evaluation of this energy balance requires a value for

the flow entrance length.

For simplicity, a formula given by Schlichting

[4]

is used for this value. Results of this calculation show that in general, if both thermal and mechanical

(p~)

(Q)

effects are important,

3 PUb 32 ) 4) + (Pr - .08 Pe - Pe QPeRe 3 32 P ~ 4+---Re Q

7 > (~ D--

When Q >> p ~

(i0)

(which is usually the case) we get the zero dissipation result

(Sandall and Hanna, 1973).

On the other hand, for p ~

>> Q, the entrance

length expression is seen to be virtually identical to the previous case, if Pr >> .08.

These results only apply for Pr greater than about unity, since

it has been assumed that the velocity develops faster than the temperature. It is interesting to note that the term 32/PeRe in Equation the interaction of dissipation and axial conduction.

(i0) represents

The effects of dissipa-

tion on the thermal entrance length may be important for heat transfer in polymers, if Prandtl numbers are large and Reynolds numbers small. Constant Wall Temperature Although the development given here applies to a uniform flux boundary condition, it is natural to inquire in general about the possibility of the approach for the case of constant wall temperature.

Again the circular tube

is interesting and simple so it will be considered for steady, developed laminar flow.

In this case, the downstream temperature solution is repre-

sented by the first eigenfunction of the Graetz solution

[5].

The essential difficulty in a specified surface temperature problem is that the unknown surface heat flux is required, and this flux varies with distance.

The simplest approach for estimation of the flux, without solving

the energy equation in the entrance region, is to use a Leveque-type solution

VOl. 3, NO. 2

[6].

THERMAL~fRANCE ~

IN[X~I~S

95

The problem with this procedure is that the Leveque sclution does not

apply accurately over the whole entrance region, and indeed its use also leaves in doubt the question of the preservation of the inequality, Equation (3).

If we proceed, in spite of these questions, we obtain L/D ~ 0.0207Pe.

This value compares with L/D = 0.036Pe corresponding to the position where the local Nusselt number is within five percent of the asymptotic value given by Sellars et al.

[ 7 ] . Although the two Nusselt numbers corresponding to the

quoted values of L/D are not far apart, it is clear that the constant wall temperature prediction is inferior to those given earlier for uniform flux as would be expected. Nomenclature A

duct cross-sectional area

a,b

rectangular duct half-widths

C

heat capacity per unit mass

P D

circular tube diameter

D

duct equivalent diameter, 4A/S

e

h

heat transfer coefficient, Q/(Tw-Tb)

k

thermal conductivity

-L I L

position upstream of heating where T=-T o entrance length; first axial position where downstream temperature profile applies

Q

heat flux per unit area

r

radial coordinate

R

tube radius

RI,R 2

inner and outer annulus radii

R

position of minimum temperature in annulus

m S

duct perimeter

St

Stanton number = h/pCpU b_

T

temperature

%

bulk temperature =

T

wall temperature

T T

w m

c

1 UbA

I A

uTdA

minimtun temperature in cross section centerline temperature for circular tube or parallel planes

u

axial velocity

%

1 mean velocity = ~

I

udA

A x,y,z

rectangular coordinates

g

half-width for parallel planes

(z axial)

96

O.T. Hanna, O.C. Sandall and B.A. Paruit

Vol. 3, No. 2

Greek Symbols thermal diffusivity aspect ratio function for rectangular ducts p

density

Subscripts 0

refers to entrance or upstream condition

1,2

denotes different wall fluxes in annulus or rectangular

3,4

duct geometries

Dimensionless Groups Pe

Peclet number, UbDe/~

Pr

Prandtl number, ~Cp/k

Re

Reynolds number, DUbP/~

St

Stanton number, h/(CpUbP) References

i.

O. C. Sandall and O. T. Hanna, AIChE Journal, 19, 867 (1973).

2.

R. D. Cess and E. C. Schaffer, App. Sci. Res., 8A, 339 (1959).

3.

R. E. Lundberg, W. C. Reynolds and W. M. Kays, NASA TN D-1972, Washington, D.C., Aug. (1963).

4.

H. Schlichting, Boundary Layer Theory, 6th ed., McGraw-Hill, New York (1968).

5.

M. Jakob, Heat Transfer, Vol. i, Wiley, New York

6.

J. Leveque, Ann. Mines, 13, 201, 305, 381 (1928); J. G. Knudsen and D. L. Katz, Fluid Dynamics and Heat Transfer, p. 363, McGraw-Hill, New York (1958).

7.

J. R. Sellars, M. Tribus and J. S. Klein, Trans. A.S.M.E. 78, 441

(1958).

(1956).

Vol. 3, No. 2

THERMAL~TRANCE~

INDUCTS

0.12

l

97

I

I

I

l

0.10 Q

s

+++,,.,. . . .

(~

+++++++I+,~I++++++~,

i)

x

o

+

--~

Y

L > riPe-~

0,08

.8

Cross-sectional

0.06

0.04 Q J',+

L'

-I

I_

L

"~

0.02

-I

bla~oo (ParalLel Planesj,C$- 0.00871 . . . .

I

I

J

I

I

2

4

6

8

10

0.00

ASPECT RATIO. b/a

FIG. 1

FIG. 3

Model for thermal entrance length estimation.

Thermal entrance lenqth for rectangular channels with uniform heat flux.

Center Li ne

+ol

(b) Concentric Annulus

(a) Parallel Plates

~+, oiz, yl

/ °'1

l°I

+ r ~ L--~' ,,'----I +------21b-+---~y-----~----I - CL~er

ceo,+,

Line

L---t- ---~+--L [

ioi

tot

(c) Rectangular Channel

(d) Circular Tube

FIG. 2 Geometries considered for thermal entrance length calculations.

12