Fully developed laminar flow and heat transfer in a square duct with one moving wall

LEXTERS IN HEATANDMASS TRANSFER Vol. 3, pp. 355 - 364, 1976

Pergamon Press Printed in the United States

FULLY DEVELOPED LAMINAR FLOW AND HEAT TRANSFER IN A SQUARE DUCT WITH ONE MOVING WALL R. M. Abdel-Wahed,

S. V. Patankar, and E. M. Sparrow Department of Mechanical Engineering University of Minnesota, Minneapolis, Minnesota

(Comn~anicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT Numerical solutions for the velocity and temperature fields have been obtained to determine fully developed friction factors and Nusselt numbers. The friction factor increases markedly as a function of the Reynolds number Re based on the velocity of the moving wall w The effect of Re on the Nusselt number depends on the W Prandtl number. At higher Prandtl numbers, there is a significant increase in the Nusselt number with Re . For intermediate Prandtl numbers, the variation of Wthe Nusselt number with Re w is rather small. Introduction Forced convection heat transfer to a fluid flowing in a duct having a moving wall is encountered in many engineering applications such as clutch assemblies and screw extruders.

The

present study was initiated to provide information on the heat transfer and fluid friction characteristics a square duct having one moving wall. schematically in the inset of Fig.

for laminar flow in

The situation is depicted

I, which shows the cross sec-

tion of a square duct whose upper wall is moving transversely with a uniform velocity u w. to a recirculating

The movement of the wall gives rise

flow in the duct cross section which is super-

posed on the axial flow whose mean velocity is w.

The duct walls

are maintained at a common uniform temperature Tw. The solutions, which are obtained here via numerical techniques,

are aimed at providing fully developed Nusselt numbers 355

356

R.M. Abdel-Wahed, S.V. Patankar and E.M. Sparrow

and friction factors.

Vol. 3, No. 5

These results are parameterized by the

Reynolds number of the moving wall Re

(to which values were w

assigned between 0 and I000) and by the Prandtl number

(which

was given values of 0.7 and 5). To illustrate their calculation procedure for three-dimensional parabolic flows, Patankar and Spalding developing

(I) considered the

laminar flow and heat transfer in a duct similar

to that being studied here and presented a limited number of results.

The present results complement and extend-those of (i)

in that they provide information for the fully

developed regime,

encompass a much wider range of Reynolds numbers,

and pertain to

a different thermal boundary condition which may have broader practical applicability. Analysis The knowledge of the u, v, and w velocity components prerequisite fore,

for the solution of the energy equation.

is a

There-

the determination of the velocity field is the first task

to be undertaken.

For fully developed flow,

the fact that u, v,

and w are independent of the axial coordinate z brings about the decoupling of the recirculating field) from the axial flow. by the recirculation.

flow (i.e., the u, v velocity

However,

the axial flow is affected

These characteristics

suggest that the

velocity problem be attacked in successive stages, with the recirculating flow solved first and the axial flow solved thereafter. If dimensionless variables are introduced as follows X = x/L, U = uL/v,

Y = y/L V = vL/v,

(la) P = p/(p(~/L) 2]

the governing equations for the recirculating

(ib) flow take the form

~U/~X + ~V/~Y = 0

(2)

~U2/~X + ~(UV)/~Y = -~P/~X + ~2U/~X2 + ~2U/~y2

(3)

Vol. 3, No. 5

FIEW AND HFAT TRANSFER IN A SQUARE DUCT

3(UV)/@X + BV2/BY

=

357

-@P/~Y + B2V/@X 2 + ~2V/3y2

(4)

The velocity component V is zero on all the walls, whereas U is zero on all but the upper wall, where u

--

=

(s)

Re w

The numerical solution of equations (2) through (5) was accomplished using a finite-difference procedure for two-dimensional elliptic flows.

Such a procedure can be derived from the more

general procedure of (i) by dropping the terms pertaining to the axial coordinate. The axial flow may next be considered.

Here, it is assumed

that Bp/~z = dp/dz, where p is a cross-sectionally averaged pressure. To rid the z-momentum equation of parameters, it is convenient to define W -- w(u/L 2) / (-dp/dx)

(6)

with the result that @(UW)/@X + ~(VW)/BY = i + @2W/BX2 + @2W/By2

(7)

The boundary condition is that W = 0 on all the duct walls.

Al-

though equation (7) is free of parameters, the solution for W depends on Re w via the input velocity components U and V. finite-difference solution for W is straightforward.

The

Once a solution for W(X,Y) has been obtained, its mean value W can be evaluated

From the definition of the friction factor for a square duct (hydraulic diameter L) f = (-d~/dz)L/(%)p~2

(9)

it can readily be shown that fRe = 2/W

(I0)

358

R.M. Abdel-Wahed, S.V. Patankar and E.M. Sparrow

Vol. 3, No. 5

where Re is the Reynolds number of the axial flow

(ll)

Re = wL/~ Attention may now be turned to the energy equation.

The

condition for fully developed heat transfer is that the normalized temperature distribution e = (T - T w)/(r b - T w) is independent of z (2).

(12)

If this condition is introduced into the

energy equation and axial conduction is neglected,

there results

3(U0)/~X + ~(V0)/~Y + W0~ = (~20/3X2 + ~20/3Y2)/pr

(13)

where = (-dp/dz) (e4/p~ 2) (rb Since

-

Tw)

d(T b - Tw)/dz

(Tb - Tw) decays exponentially

is a constant.

The numerical

At the duct walls,

value

(14)

to zero as z increases, of ~ remains

to be determined.

0 = 0 since T = T W"

It is relevant to relate ~ to the Nusselt number. the fully developed heat transfer coefficient

First,

is defined as

(15)

h = (Q/A)/(T w - T b) Here, Q is the rate of heat transfer integrated around the circumference

of the duct and A = 4L, where both Q and A are

per unit length in the streamwise direction.

With this,

the

Nusselt number is Nu = hL/K

(16)

From a formal integration of equation section of the duct,

(13) over the cross

it can be shown that

Nu = -~WPr/4

(17)

Inasmuch as W is already available from the solution for the axial velocity field,

it is apparent

that the determination of

Vol. 3, NO. 5

~

AND HEAT TRANSFER IN A SQUARE [X~T

359

holds the key to the evaluation of the Nusselt number. The procedure for determining First a new variable

~ will now be described.

~ = e/~ is defined.

Then,

from the defini-

tion of the bulk temperature (Tb - Tw)ffwdxdy = ff(T - Tw)Wdxdy it follows

(18)

that

= ~WdXdY/~W~dXdY When e = ~ equation

(19)

is substituted

into equation

(13), along with ~ from

(19), there follows

(U~)/~X + ~(V~)/~Y + W ~ [ ~ W d X H Y / ~ W ~ d X d Y ) = (~2~/~X2 + The variable

(20)

~2~/~Y2)/pr

~ is also equal to zero on the duct walls.

The numerical

scheme for solving equation

tical to that for equation

(7) except for the nonlinearity

ing from the third term on the left of (20). a solution for (20), an iterative procedure guessed distribution

of ~, the nonlinear

the equation is then solved for ~.

is used.

With the resulting distribu-

tion for ~ is obtained.

The procedure

equation

From a

term is evaluated and

term is re-evaluated,

Finally,

aris-

In order to obtain

tion of ~, the nonlinear vergence.

(20) would be iden-

and a new solu-

is continued until con-

(19) is used to obtain ~, and e is

computed from e = ~ . The iterations mentioned above converge extremely rapidly because of the special nature of the nonlinear term,

~ appears

a result,

in both the numerator

term.

In this

and the denominator.

As

the value of the term is not dependent on the absolute

magnitude of the guessed ~ distribution, tion of ~ over the duct cross section. The finite-difference

but only on the varia-

solutions were carried out with a

28x28 grid, which corresponded

to the maximum number of grid

points consistent with the computer storage capacity. deployments

Three

of the grid points were employed with the view of

360

R.M. Abdel-Wahed, S.V. Patankar and E.M. Sparrow

examining

the accuracy

of the results.

points were equally

spaced,

grid was nonuniform,

with

walls

and the largest

Different latter

degrees

cases.

concluded percent

whereas

In one case,

in the other

the smallest

spacing

and the Nusselt

factors

were employed

numbers

are accurate

section.

in the

studies,

are accurate

the

the duct

of the cross

of these step-size

that the friction

the grid

two cases

spacing near

at the center

of grid nonuniformity

As a result

Vol. 3, No. 5

it was

to within

to within

one

two to

three percent. Results The friction product,

factor results,

are plotted

Re w in Fig.

i.

and discussion

as a function

expressed

in terms of the fRe

of the wall Reynolds

In the case of the Nusselt number

number

results,

I00

90

80

UW

yL

f Re

70

7

L

_.J.

X

60

501 0

I

I

I

I

2OO

400

600

8OO

Rew FIG. Friction

i

factor results

I000

to

Vol. 3, NO. 5

facilitate

~

AND HEAT TRANSFER IN A SQUARE [X,L~

the identification

of the enhancement

wall,

the ratio Nu/Nu ° has been plotted

Re w.

The quantity Nu o, which has a numerical

the fully developed Nusselt number walls

due to the moving 2 as a function

value of 2.98,

of

is

in a square duct in which all

are stationary. Examination

creases

of Fig.

i indicates

fact that fRe varies

is appreciable,

from about

Re w = 0 to Re w = 200.

factor

in-

Whereas

curve is somewhat

surprising.

respectively

the increase

by the

of the friction

There appear

increase

to be two distinct

by a rapid rise at small Re w for intermediate

and larger

The knee of the curve is between Re w = i00 and 200.

5

4 -

Pr=5

5

Nu Nu o 2

0

0

I

I

I

I

200

400

600

800

Re w FIG.

fac-

the shape of the fRe vs. Re w

characterized

and by a much more gradual

as witnessed

57 to 82 over the range from

tor with Re w is entirely expected,

regimes,

that the friction

as a result of the transverse motion of the duct wall.

The extent of the increase

Re w.

in Fig.

361

2

Nusselt number results

I000

362

R.M. Abdel-Wahed, S.V. Patankar and E.M. Sparrow

The Nusselt number changes

of Fig.

in the flow field and/or

transport

processes

Nusselt number tion factor situated

ferent behavior. diminishes creases.

i, although

lower values

to a shallow minimum, Over

in shape

to the fricis

an altogether

to a maximum,

dif-

then gradually

and subsequently

slowly

the range of Re w from 200 to I000,

for Pr = 0.7 does not differ

The

On the other hand,

curve for Pr = 0.7 displays It rises rapidly

in Re w.

the knee of the curve

of Rew.

of

of participating

an increase

for Pr = 5 is similar

curve of Fig.

at somewhat

2 are also suggestive

in the roles

which accompany

curve

the Nusselt number

number

results

Vol. 3, No. 5

in-

the Nusselt

significantly

from the value

of 1.5Nu o . Although appears

a precise

explanation

out of reach at present,

for the aforementioned

they are believed

to be related

to changes

in the shape and size of the recirculation

with Re w.

The motion of the wall

recirculation In addition,

zone which

duct.

fills most

to the lower

The sizes and shapes

change with Re

tends

recirculation corners

that the Nusselt numbers

This

to conductive

the scrubbing

duct walls provides that are used

section.

of all of the recirculation

of the zones

a high Prandtl

to convective

energy

transfers).

energy

to good advantage

number transfers

flow on the

for convective

by the higher

than

In the present

action of the recirculating

the opportunities

for Pr = 5

of the moving wall

is because

to be more responsive

(as contrasted problem,

left and lower right

affected by the presence

are those for Pr = 0.7. fluid

of the duct cross

W"

It may also be noted are much more

pattern

sets up a large clockwise

there may be small counterclockwise

zones adjacent

trends

transfer

Prandtl number

fluid. Acknowledgment This research was performed ENG-7518141.

under

the auspices

of NSF Grant

Vol. 3, 5]0. 5

FLOW AND HEAT TRANSFER IN A SQUARE DLL~

363

Nomenclature f

friction factor, equation (9)

h

heat transfer coefficient, equation (15)

k

thermal conductivity

L

side of square duct

Nu

Nusselt number, hL/k

Nu

O

Nusselt number for duct with stationary walls

P

pressure

P Pr

cross-sectionally averaged pressure Prandtl number

Re

Reynolds number of the axial flow, uL/~

Re w

Reynolds number based on the moving wall, UwL/~

T

temperature

Tb

bulk temperature

Tw

wall temperature

U,V

cross sectional velocity components

U

velocity of the moving wall

W

W

axial velocity component

W

mean velocity of axial flow

x,y

cross sectional coordinates

Z

axial coordinate

8

dimensionless temperature, (T - Tw)/(T b - Tw) energy equation parameter, equation (14) dynamic viscosity kinematic viscosity density

0

References i. .

S. V. Patankar and D. B. Spalding, Int. J. Heat Mass Transfer 15, 1787 (1972). W. M. Kays, Convective Heat and Mass Transfer, McGraw-Hill, New York (1966).