Perganmn International Communications in Heat and Mass Transfer, Vol. 21, No. 4, pp. 469--475, 1994

Copyright O 1994 Elsevier Science Ltd Printed in the USA. All rights reserved

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A S Y M M E T R I C SOLUTION F O R V E L O C I T Y P R O F I L E

IN L A M I N A R F L O W T H R O U G H R E C T A N G U L A R DUCTS

M. Spiga and G.L. Morini Istituto di Fisica Tecnlca, Facolt~ di Ingegueria, Unlversit~ di Bologna

Viaie Risorgimento 2, 40136 Bologna, Italy

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT The present analysis is concerned with the velocity distribution in steady state, hydrodynamically fully developed, laminar flow for Newtonian fluids in rectangular ducts, The solutions describing velocity profiles and friction factors are derived by using the finite Fourier transform. The solution giving the two-dimensional velocity distribution is formally symmetric with respect to the rectangular Cartesian coordinates x and y, and provides more accurate numerical results, with shorter computational times, in comparison to the well known velocity distributions usually quoted!in literature.

Introduction

Although many papers were devoted in the past to laminar flow through rectangular ducts

and several monographs exist, a renewed interest can be noticed in the recent literature [1-2]

(mainly because of the wide application in compact heat exchangers) to provide an improved

accuracy for those.predictions that are susceptible to numerical errors.

The prediction, of the hydrodynamics of rectangular duct viscous flows requires a two-

dimensional analysis. The hydrodynamically fully developed velocity profile for ducts with

rectangular cross~ section was determined by series solution [3-4], using an analogy with the

Lagrange's stress function of the theory of elasticity [5], where the two independent Cartesian

coordinates x andy appear in trigonometric and hyperbolic functions, respectively. Later, approximate solutions were proposed as a series with less computational complexity [6]

and as binomial products [7-8], where again the two coordinates appear in non symmetric form,

being related to different numerical exponents.

A~.Q

470 M. Spiga and G.L. Morini Vol. 21, No. 4

The velocity profile given by the exact series solutions is in excellent agreement with the

experimental results of numerous authors [9+11], but the convergence of the series solution is

slow and requires an appreciable computational effort.

Aim of this communication is the presentation of a new analytical solution for the velocity

profile, characterized by a symmetric form with respect to the two Cartesian coordinates, and by

a fast numerical convergence.

Analysis

Consider a steady hydrodynamically developed laminar flow in a duct of rectangular cross

section. The origin of the Cartesian coordinate system is at the bottom left comer of the duct, as

shown in Fig. 1, with the flow direction along the ~ axis.

T!

O a

FIG. 1

Rectangular Duct and Coordinate System.

The following assumptions are introduced: the fluid is Newtonian with constant physical

properties, natural convection is negligible and the pressure gradient along the ~ axis is constant.

Under these conditions, the momentum Navier Stokes equation in dimensionless form becomes:

a2v e2v 0x 2 + - ~ - + 1 = 0 (1)

The dimensionless coordinates are x-~/a (0

Vol. 21, No. 4 LAMINAR FLOW THROUGH RECTANGULAR DUCTS 471

M e t h o d s

The Fourier sine transform constitutes a powerful tool to solve second order differential equations in finite domains, when the unknown function vanishes on the boundaries [12]. Considering a Fourier sine transform respect to x, the dimensionless velocity and its second

derivatives become:

IiV c~2V = nx[V(0,y)- (-1)nv(1,y)] - n2x2V(n,y) (4) "q(n,y) = (x,y) sin(n~x) dx and ~x 2

Resorting to the boundary conditions, multiplying each term by sin(nrcx) and integrating over [0,1 ], the momentum equation becomes:

-n212V4 c32V - 1 - ( - l ) n (5) c~ 2 n~

Considering a Fourier sine transform respect to y, multiplying each term of the equation (5) by sin(m~/13) and integrating over [0,[3], the twofold transformed momentum equation reads as:

= m2,n2 = -n2 ~2 V ( n , m ) - T V ( n , m ) + mx r" 0 m" ' n~-~-[1-(-1)n ] --~--[V(n, ) - ( -1 ) V(n,[3)] = [1-(-1) m]

whose solution, remembering the boundary conditions for the x-transformed velocity, is: .~ [33 [1- ( -1)n] [1- ( -1) m] V(n,m) - (6)

nmx 4 m 2 + [32 n 2

The inversion formula provides the velocity distribution:

V(x,y) = 16132 ~"~ ~ '~ sin(nx~) s in(mxb)

x 4 nm ([32 n 2 + m 2) ( 7 ) nodd modd

It is evident that the velocity profile is symmetric with respect to the coordinates ~ and 11, i.e., if ~a and rl/b are exchanged, the equation (7) remains unchanged. Of course the sinusoidal

functions in ~ and 11 guarantee the symmetry of the distribution with respect to the center of the rectangular cross section. A very fast convergence is assured by the third power ofn and m in the denominator of the equation (7). The uniform convergence is easily demonstrable by the Cauchy

criterion. Hence the average velocity over the whole domain can be easily obtained by integration and reads as:

64 P E 1 768 Wslab 1

W = - - ~ - n2m2( +-~- ] n odd modd nodd modd n 2 m2 - ~6 n2m2([32n2+m2) (8)

where Wslab is the average velocity in a slab with thickness b, and same pressure gradient and

duct inclination (Wslab = [32 P/12). It is easily verified that, as expected, lim W = Wslab. 13-~0

472 M. Spiga and G.L. Morini Vol. 21, No. 4

It must be pointed out that the usual velocity profile quoted in literature is given by a series

solution, where "q appears in a hyperbolic cosine function and ~, appears in a trigonometric cosine

function, where the reciprocity of the coordinates is not evident at all.

R e s u l t s

The peculiarities of the analytical solution, shown in the equation (7), are the perfect

symmetry (between ~/a and q/b), the uniform convergence of the series and the short

computational time. The numerical results presented in this section were obtained very rapidly

with a common 386 or 486 personal computer.

Table 1 shows the rigorous symmetry of the solution for V(~,q); the aspect ratio has been

put equal to unity and some values of ~/a and rl/b have been chosen. The classical solution of the

theory of elasticity [3+5] provides satisfactory results, but only three significant figures remain

unchanged when ~Ja and rl/b are changed, and the required computational time to reach

convergenge is higher. The approximate solutions by Savino [6] and Purday [7] put in evidence

an even greater lack of symmetry and accuracy.

TABLE 1 Velocit~ V(~,vl) in the Rectangular Duct at S ~ f i c Lo~fi9m, foJ

13=1

Ua=0.5 ~a = 0.75

~ a = 0.6

~/a = 0.75

~Ja = 0.85

~a = 0.75

U a = 0.9

~a = 0.75

13=1

q/b = 0.75

rl/b = 0.5

q/b = 0.75

rl/b = 0.6

q/b = 0.75

rl/b = 0.85

q/b = 0.75

Wqa = 0.9

[3-4-5]

0.05742985

0.05740838

0.05557022

0.05558934

0.03243877

0.03241640

0.02359903

0.02363724

[6] 0.05736592

0.05740392

0.05552668

0.05555864

0.03236313

0.03232557

0.02354681

0.02348536

[7]

0.05532648

0.05878895

0.05396095

0.05643739

0.03096753

0.02998236

0.02221032

0.02116402

present paper

0.05750080

0.05750080

0.05566930

0.05566930

0.03252332

0.03252332

0.02371376

0.02371376

The availability of the analytical solution and its fast convergence make the present

technique quite effective and unexpensive in terms of computer time.

The agreement between the two series solutions, equation (7) and [13], is satisfactory, but a

difference on the third figure appears for small value of the aspect ratio; the computational time is

shorter for the equation (7).

The velocity ratio profile V(x,y)Nma x is shown in Figs. 2 and 3, for different aspect ratios

(13 = 0.2 and 13 = 1). In the square duct the fluid velocity experiences a s l o w and gradual increase

from the walls to the center, in the rectangular duct there is a strong increase near the walls and

an extended central core with high velocity.

Vol. 21, No. 4 LAMINAR FLOW THROUGH RECTANGULAR DUCTS 473

1"

0 .8

0.6

0.4

0.2

C I

1

0.8

0.6

0.4

0.2 13

113 o y y (: v.,,

1 x 1 FIG. 2 FIG. 3

Velocity Profile in a Rectangular Duct with 13=0.2 Velocity Profile in the Square Duct.

The knowledge of the velocity profile allows to determine the shear stresses and all

parameters concerning the fluid flow (average and maximum velocity, momentum flux and

kinetic energy correction factors, friction factor). The surface integrals involved in the

determination of such parameters are easily solved, being related to first, second and third powers

of sinusoidal functions. Term by term derivation and integration are permitted by the uniform

convergence of the series in the equation (7). The momentum flux correction factor K d is defined

as the ratio between the momentum of the actual flow and the momentum of a slug flow, with the

same average velocity. Analogously for the kinetic energy correction factor K e.

The maximum-to-average velocity, the momentum flux and kinetic energy correction factors

are plotted in Fig. 4, versus [3, and compared with the results obtained by Lundgren et al.

(denoted by small circles) available in literature [ 13].

The friction factor-Reynolds number product, deduced by the equation (7) as a function of

the aspect ratio, is plotted in Fig. 5 and compared whir the classical results quoted in literature

[14-15]; the agreement is quite satisfactory, but we observe that in the range 0

474 M. Spiga and G.L. Morini Vol. 21, No. 4

At last a polynomial approximate representation, based on the velocity profile given by

equation (7), is presented for the maximum-to-average velocity ratio, the momentum flux, the

kinetic energy correction factors and the friction factor-Reynolds number product:

Vmaxw = 3(1 +0.54668813+ 1.552013132 -4.059427133 + 3.214927134 -0.857313135)

= 6 (1 + 1.029821113 +0.114993132 -0.691909133 +0.581376134 -0.153967135) KO (9)

54 (1+0.69363513+0.554608132 _ 1.887459133 +1.272092134 _0.236888135 ) K e = -~

fRe = 96 (1 - 1.20243813 +0.881188132 +0.888193133 - 1.698125134 +0. 723659135)

Agreement between the polynomials (9) and the exact series solution (7) is within 0.06 % all

over the rectangular cross section; the f-Re product presents the worst approximation, the results

are indicated in Fig.5. The results concerning the velocity ratio and the correction factors given

by the equation (9) are coincident, in Fig. 4, with the results obtained by the exact solution (7). On the basis of the velocity profile given by equation (7), future work will be focussed on

the prediction of the temperature profiles and the Nusselt numbers in the rectangular duct (see

the analogy between the equation (8) and the equation (348) in [13]), with constant temperature

and/or heat flux on the four different boundaries.

Acknowledzements

This research was financially supported by the Italian C.N.R. and MU.R.S.T.

a,b

D

f

Kd

Ke m,n

P

P Re

V(.) ~P(.)

Nomenclature

longer and shorter sides, respectively, of the rectangle (m),

hydraulic diameter of the duct 2ab/a+b (m),

friction factor,

G-component of the gravitational acceleration (m/s2),

momentum flux correction factor,

kinetic energy correction factor,

discrete transform variables in the sine Fourier transform,

velocity defined in equation (2), (m/s), pressure (Pa), Reynolds number pWD/tt, dimensionless velocity v/P,

Fourier sine transform of the dimensionless velocity V with respect to x,

two-fold Fourier transform of the velocity V,

Vol. 21, No. 4 LAMINAR FLOW THROUGH RECTANGULAR DUCTS 475

v(.) velocity (m/s), W average velocity (m/s),

x,y dimensionless rectangular Cartesian coordinates.

Greek letters

13 aspect ratio b/a,

B viscosity (Pa s),

p density (kg/m3),

~,rl,~ rectangular Cartesian coordinates (m).

Subscripts

max maximum value,

slab slab geometry.

References

1. S.X. Gao and J. P. Hartnett, Analytical Nusseit number predictions for slug flow in rectangular duct, Int. Comm. in HeatandMass Transfer 20, 751 (1993).

2. J. P. Hartnett and M. Kostic, Heat transfer to newtonian and non newtonian fluids in rectangular ducts, Adv. Heat Transfer 19, 247 (1989).

3. H. L. Dryden, F. D. Murnaghan and H. Bateman, Comm. Hydrodyn. Bull. 8__44, 197 (1932).

4. S. M. Marco and L. S. Han, A note on limiting laminar Nusselt number in ducts with constant temperature gradient by analogy to thin-plate theory, Trans. of ASME 77, 625 (1955).

5. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York (1970).

6. J. M. Savino and R. Siegel, Laminar forced convection in rectangular channels with unequal heat addition on adjacent sides, Int. J. Heat Mass Transfer 7, 733 (1964).

7. H. F. P. Purday, Streamline flow, Constable, London (1949).

8. N. M. Natarajan and S. M. Lakshmanan, IndianJ. Technol. 1__00, 435 (1972).

9. D. B. Holmes and J. R. Vermeulen, Velocity profiles in ducts with rectangular cross section, Chem. Eng. Sci. 23, 717 (1968).

10. E. M. Sparrow, C. W. Hixon and G. Shavit, Experiments on laminar flows development in a square duct using a Laser-Doppler flowmeter, J. Basic Eng. 89, 116 (1967).

11. G. F. Muchnik, S. D. Solomonov and A. R. Gordon, Hydrodynamic development of a laminar velocity field in rectangular channels, J. Eng. Phys. 25, 1268 (1973).

12. B. Brown, Bull. Amer. Math. Soc. 48, 522 (1942).

13. R, K. Shah and A. L. London, Laminar flow forced convection in duets, Adv. Heat Transfer 14 (suppl I), 196 (1978).

14. M. A. Tirunarayanan and A. Ramachandran, Correlation of isothermal pressure drop in rectangular ducts, Proc. 2nd Australas. Conf. Hydraul. Fluid Mech., A213 Auckland (1965).

15. F. S. Shih, Laminar flow in axisymmetric conduits by a rational approach, Ca. J. Chem. Eng. 4__55, 285 (1967).

Received February 22, 1994