Solar Energy, 1972, Vol. 13, pp. 363-371. Pergamon Press. Printed in Great Britain

E F F E C T O F T U B E I N C L I N A T I O N O N L A M I N A R C O N V E C T I O N I N U N I F O R M L Y H E A T E D T U B E S

F O R F L A T - P L A T E S O L A R C O L L E C T O R S

K. C. CHENG* and S. W. HONG*

(Receiued 3 March 1971 ; in revised form 23 April 197 I)

Abstract- A numerical study using a combination of boundary vorticity method and line iterative relaxation method is carried out to determine the free convection effects on fully developed upward laminar forced flow in uniformly heated inclined tubes. The combined free and forced laminar convection for water with the inclined tube configuration in the low Reynolds number flow regime has practical application in flat-plate solar collectors for water heating. The tube inclination or gravitational force orientation effects on flow and heat transfer characteristics are clarified and show that in high Rayleigh number regime the tube orientation effect has considerable influence on the results, particularly in the neighborhood of horizontal direction. The numerical results show that the perturbation analysis in terms of power series of Rayleigh number is invalid for the present problem and reveal further that a maximum value for Nusselt number does not exist for any tube inclination angle with given values of the dimensionless parameters which is clearly contrary to the result from perturbation solution.

R ~ u n ~ - U n e Etude numErique est faite, en utilisant une combinaison de la mEthode de vorticit~ aux parois et d'une mEthode de relaxation it~rative, pour d~terminer les effets de convection libre sur un courant lamin- aire pleinement dEvelopl~ et dirigE vers le haut, dans des tubes inclines, chauffEs uniformEment. Lorsque la configuration du tube inclinE, dans lequel le regime d'Ecoulement a un nombre de Reynolds faible, le permet, la convection laminaire libre et forcEe de I'eau a une application pratique aux plaques collectrices solaires pour le chauffage de I'eau. Les effets de I'inclinaison du tube ou de I'orientation de la force de gravitation sur les caract~ristiques du courant et du transfert de chaleur sont expliqu~s; ainsi dans le cas d'un r~gime ayant un nombre de Rayleigh ElevE, I'orientation du tube a une influence tr~s grande sur les rEsultats, surtout Iorsqu'on s'approche de la direction horizontale.

Les ~sultats num~riques montrent que ranalyse de la perturbation par rapport aux series de puissance du nombre de Rayleigh ne peut pas s'appliquer au probl~me present; ils indiquent Egalement qu'il n'existe pas de valeur maximum du hombre de Nusselt pour aucun des angles d'inclinaison du tube avec les valeurs donnEes des param~tres sans dimension, ce qui est tout a fait contraire aux rEsultats de la solution de per- turbation.

Resumen - Se Ileva a cabo un estudio numErico, utilizando una comhinaci6n de m~todo de vorticidad limitrofe y m~todo de relajaci6n iterativa de linea, a fin de determinar los efectos de la convecci6n libre sobre un flujo forzado laminar ascendente plenamente desarrollado en tubos inclinados de calentamiento uniforme. La convecci6n laminar libre y forzada correspondiente al agua, combinada con la configuraci6n tubular in- clinada en regimen de flujo con nfimero Reynolds bajo, tiene una applicaci6n prfictica en los captadores de energia solar de placa plana para calentamiento del agua. Se aclaran los efectos ejercidos pot la inclinaci6n de los tubos o por la orientaci6n de las fuerzas gravitacionales sobre las caractisticas de flujo y transferencia de calor, demostr~ndose que. en r~gimen de ndmero Reynolds alto, el efecto de la orientaci6n de los tubos influye sensiblemente en los resultados, sobre todo en proximidad a la direcci6n horizontal.

Los resultados num~ricos indican que el anfilisis de la perturbaci6n, en tErminos de series de potencias de nfimero Rayleigh, es invfilido para el presente problema, y revelan ademfis que no existe valor mfiximo de ntimero Nusselt respecto de cualquier fingulo de inclinaci6n de los tubos, con valores indicados de los parfi- metros adimensionales, que sea claramente contrario al resultado de la soluci6n de perturbaci6n.

I N T R O D U C T I O N

THIS PAPER presents a numerical study of the free convection effects on fully devel- oped upward laminar forced flow in uniformly heated inclined tubes mounted in a flat-plate solar collector. The problem is concerned with combined free and forced

* Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada

363

364 K.C. CHENG and S. W. HONG

laminar convection for water with the inclined tube configuration in the relatively low Reynolds number flow regime for water heating. Since flat-plate solar collectors are usually used in an inclined position, the determination of an optimum inclination angle with the maximum heat transfer, rate will be one of the factors in the rational design of flat-plate solar-energy collectors. In addition, the clarification of flow and heat transfer characteristics in heated inclined tubes will shed some light on the proper arrangement of tubes or tubing in increasing the performance of solar water beaters Ill. Since the tube length is usually short for solar collectors, a thermal entrance region problem in inclined tubes is the problem one actually encounters in various types of solar water heaters using flat-plate solar collectors[2]. Nevertheless, the present study can be considered as a limiting case and also a starting point toward the numerical solution of the Graetz problem in inclined tubes, which is currently not available in the literature.

Recently Baker[2] reported film heat-transfer coefficients in solar collector tubes at low Reynolds numbers, based on the experimental work for forced laminar convection heat transfer in the thermal entrance region of a horizontal tube of the tube-in-strip type and discussed the free convection effects. The effects of the double helix secondary flow caused by density variation due to heating in a gravitational force field acting in a cross-section normal to the main flow in heated horizontal tubes are now well under- stood for the fully developed situation. In contrast, the flow and heat transfer charac- teristics in inclined tubes which are used widely as solar collector tubes for water heating are not well understood.

lqbal [3] presented a theoretical study of the effects of free convection superimposed on fully developed forced upward flow in uniformly heated inclined tubes using a per- turbation method[4]. However, recent studies[5, 6] for related problems show con- clusively that the perturbation method is only applicable for very low parameter region, and diverges quickly with the increase of the parameter. In view of the difficulty with the analytical solution using perturbation analysis, a numerical solution appears to be the only practical approach for the accurate solution of the combined free and forced laminar convection in inclined tubes.

The purpose of this paper is to present an accurate numerical solution using a boundary vorticity method [5, 7] for a steady, fully-developed, combined free and forced laminar convection in uniformly-heated inclined tubes valid up to a reasonably high value of the Rayleigh number. Besides presenting accurate flow and heat transfer results for water (Pr = 5), the significant discrepancy between the results from numeri- cal analysis and those from perturbation analysis will be pointed out.

MATHEMATICAL FORMULATION OF THE PROBLEM

The theoretical analysis is concerned with a steady hydrodynamically and thermally fully developed laminar upward flow of water in an inclined tube (see Fig. 1) subjected to uniform wall heat flux. The following assumptions, similar to those used in [3], are made in the present analysis: 1. Physical properties of water (Pr = 5) are constant except density variation with

temperature in buoyancy term (Boussinesq approximation). 2. Low-speed flow is considered. 3. Axial pressure gradient is constant.

Introducing the cylindrical coordinates (R, ~b, Z) as shown in Fig. 1 and the follow- ing dimensionless dependent variables, constants, parameters and a dimensionless

Laminar convection in uniformly heated tubes 365

\Fw-. ~--~d

Fig. 1. Coordinate system for inclined tube and numerical grid.

steam function ~, R = ar; U = (v /a)u; V = (v ia)v ; W = R e ( v / a ) w ; Tw - T = ReCaPrO; C = OT[OZ; A = - ( O P / O Z + p w g s in a); R e = Aa3/4pv2; Pr = v/r; R a = flgCa(/vK; u = O~blrOdp and v =--O0[Or, the Navier-Stokes equations and energy exluation[3] can be re~iuced to the following dimensionless forms after applying the foregoing assump- tions and eliminating the pressure terms between the momentum equations in R and directions by cross-differentiation:

Momentum equation for secondary flow

00 O~, v O ~ _ L R e R a ( ~ r S i n Or-trOd# ~ .100 d~) cos a. v'~ = u ~+7~cos

Vorticity equation

Axial momentum equation

Energy equation

Ow , v o w V~w = u _ - - - - t - - - - + RaO sin a - - 4 .

Or r O ~

= ~ [ aO, v ~0\ v

(1)

(2)

(3)

(4)

It is noted that the vorticity function ~ is introduced here to avoid using biharmonic function V4O in the momentum equation for secondary flow. Because of symmetry, only one half of the circular region need be considered as shown in Fig. I. The boundary conditions can be stated as follows:

~ = O r w = O = O

O0 a_E ~,=¢=~= a,=o

a t r = l

along ~ = 0 and ~r. (5)

366 K.C. CHENG and S. W. HONG

The governing Eqs. (1), (3) and (4) are quasi-linear, second-order partial differential 'equations of elliptic type, and the vorticity Eq. (2) is simply a Poisson's equation. The main departure of the present formulation from that reported in[3] lies in the introduc- tion of the vorticity function ~: for the purpose of employing the recently developed boundary vorticity method [5, 7]. Since the computational procedure for the numerical determination of the boundary vorticity and the numerical solution of a set of finite- difference equations by line iterative relaxation method are well discussed elsewhere [5, 6, 7], details of the numerical solution including finite difference approximations will not be given here for the sake of brevity. In the numerical computation, the prescribed error for all the dependent variables and the secondary velocity components is

, = 2 I.,lyl>l < lO-' (6)

where J~a is a dummy variable at a guide point (i,j). All the numerical results are obtained by using a mesh size of M, N = 28 and a relaxation factor of unity. The singu- larity at the origin of the cylindrical coordinates is avoided by using finite difference equation in Cartesian coordinates at the origin. The computing time required to obtain a complete solution for given values of Re, Ra, Pr and a is about 3 minutes on the IBM 360/67 system.

FLOW AND HEAT TRANSFER RESULTS

Flow and heat transfer results in the form of the product of friction factor and Reynolds number (fRe) and the Nusselt number (Nu) are of interest in design. The expressions forfRe and Nu can be obtained in two alternative forms [5] as

(fRe)l = [2~%/(p~2) ] (2aff" p)/l~ = 4[~'~/Orl,,/a,

(fRe)2 = (8/~) [1 -- (sin ot/4)RaO]

(Su), = h(2a)/k-- 2~,[~/Orj,d[~O[ (7)

(Nu)2 =

where the subscript 1 refers to the expression based on mean velocity or temperature gradient at wall and the subscript 2 refers to the expression based on an overall force or energy balance for the axial length dZ. The above two alternative ways of obtaining flow and heat transfer results enable one to check the convergence of the numerical results. Simpson's rule is employed in evaluating the average quantities indicated above. Since four independent parameters are involved in the present problem, a complete parametric study is impractical and hence only representative cases for water (Pr = 5) are given to illustrate the tube inclination angle or body-force orientation effects.

The effect of tube inclination angle a on flow result is demonstrated in Fig. 2 where the ratio fRe/(fRe)o is plotted against the Rayleigh number with a as a parameter for Pr = 5 and Re = 5. For the range of Rayleigh number under consideration, the effect of heating on flow result for horizontal tube (a = 0 °) is negligible. As Rayleigh number increases, the inclination angle effect on the productfRe is seen to become progressive- ly appreciable as a varies from 0 ° to 45 °.

From the viewpoint of solar water heating, the tube inclination angle effect on heat transfer result is of considerable interest. Heat transfer results are known to be similar

Laminar convection in uniformly heated tubes 367

/ 4 Pr =5 H

Re=5 / / ]

, ' ,=90 ° I I I

._,o 60",

3 45 °

~" 3 0 o

2

I I0 ~00 I(:300

Ro

Fig. 2. f R e / ( f R e )o vs. R a with a as a parameter for Pr = 5 and R e = 5.

to flow results and the effect of Rayleigh number on the Nusselt number ratio Nu/(Nu)o for several tube inclination angles is demonstrated in Fig. 3. It is seen that as Rayleigh number increases, the inclination angle effect on heat transfer becomes progressively significant, particularly in the range of a = 0 ° - 60 °.

The problem under consideration was solved by Iqbal [3] using perturbation analysis similar to that employed by Morton[4], and it is desirable to compare the results from this work with those of reference [3]. Figures 4 to 7 show clearly that there is a signifi- cant difference between the present numerical solution and the perturbation solution. It should be pointed out that for the horizontal tube case the perturbation solution is known to diverge quickly with the increase of the parameter, and the rather sharp increase in the Nusselt number with the increase of Ra shown in Fig. 4 for a = 0 is typical of the perturbation solution. It is noted that for horizontal tube, the Nusselt number is dependent on the parameter ReRa. The trend of heat transfer results from

2

1.5

Pr =5

Re =5 <z = 9 0 '=

60 °

4 5 "

tO iOO iOOO Ra

Fig. 3. N ul ( N u )o vs. R a with ~ as a parameter for Pr = 5 and R e = 5.

368 K . C . C H E N G and S. W. H O N G

1 5

pro5 o =0"

---Perturbotion / "~ i'~

- - T h , s work ,' ,' y// i i i i i /

Ro

Fig. 4. Nu/(Nu)o vs. Ra for Pr = 5 and a = 0 ° with compar ison made against perturbat ion solution.

perturbation analysis for the inclination angles a = 30 ° and 60 ° shown in Figs. 5 and 6, respectively, also shows the same blow-up trend similar to that for a = 0 °. The heat transfer results from finite-difference solution shown in Figs. 5 and 6 indicate that after reaching a Rayleigh number of approximately 103 the free convection effect becomes predominant over forced convection, and from thereon the heat transfer result is seen to be independent of Reynolds number. At ot = 60 ° the difference for heat transfer result between Re = 5 and 20 appears to be insignificant (see Fig. 6) which is opposite to that shown in Fig. 4 for a = 0 °. For the vertical case (o~ = 90 °) the Nusselt number depends on Ra only. The present numerical solution checks exactly with the known numerical results for the horizontal tube (a = 0)[4] and the vertical tube (a = 90°)[8]. Conse- quently, the convergence of the present numerical solution can be said to be confirmed.

The comparison shown in Fig. 7 leads one to conclude that the perturbation solution is invalid at Ra = 100, and the implication that for fixed values of the dimensionless parameters, the maximum heat transfer rate exists between a = 20 ° and 60°[3] is believed to be erroneous.

2 - Pr =5

a =~0 =

- - -Per turbot ion / / I / / / ~5 onolysis[~ I / i / / /

/ / /// / This work / / ~,'~ ~

/ / / / / •

I I l I I I J • I I I i I I I ] I I I I I l I I

I0 iOO IO00

Ro

Fig. 5. Nu/(Nu)o vs. Ra f o r Pr = 5 and a = 30 = wi th compar ison made against per turbat ion solution•

Laminar convection in uniformly heated tubes 369

o

a = 6 0 "

Re = 5 20 IO 15 2

_ - - P e r t u r b a t i o n ~ r~/ / / ~F" ~5 one lys is [3] / ,/'/,'/'

- - T h i s work , ~ / , " • " ' "

j / . ,

i i i i I i ' ' 1 t I I 1 I I I0 I00 I000

Ro

Fig, 6. Nul (Nu)o vs. Ra for Pr = 5 and a = 6 0 ° with comparison made against perturbation solution.

I - 5

. . ...... 8 Pr:5 ] ... ~ -,, Ro I 0 0

!'/ Re - 6 \",., '~ I . , I ~ ~ ~

- 4 . . . . : . ' . . .-_. '~-.. . . ,

,# / / / " / ~ @ ~ v _ __ per tur bation

, ,

I I I I i 0 15 30 45 60 75 90

a, degree

Fig. 7. Comparison of heat transfer results from this work with those from perturbation analysis.

CONCLUDING REMARKS

1. The most probable mode o f heating water in solar col lector tubes is considered to be mixed convection [2]. In view of the rather significant tube inclination angle effect with the increase of the Rayleigh number, it appears that the determination of the suitable film heat-transfer coefficients in solar collectors based on theoretical and ex- perimental data for horizontal tubes alone is inappropriate and unrealistic. The deter- ruination of the film heat-transfer coefficients in inclined solar collector tubes requires the solution o f Graetz problem in inclined tubes. The results o f the numerical solution presented in this paper for the limiting case of hydrodynamically and thermally fully developed situation will provide a guide for the determination of the film heat-transfer coefficient to be used in flat-plate solar collectors.

2. The thermal boundary condition of uniform wall heat flux around the tube is an idealized one neglecting the thermal resistance of the tube-plate bond in solar collec-

S.E. Vol. 13, No. 4 - B

370 K . C . C H E N G a n d S . W. H O N G

tots. The numerical technique[5.7] employed can be readily adapted to the case with non-uniform wall temperature around the tube [2].

3. The convergence of the present numerical solution is ascertained by comparison with the known numerical results for the two limiting cases of horizontal tube (a = 0) and vertical tube (~ = 90°). Besides presenting accurate numerical results, it is demon- strated clearly that the perturbation method[3, 4] as used in the literature diverges quickly with the increase of the characteristic parameter. It is believed to be significant to note that for any combination of Ra, Re and Pr, the maximum value of Nusselt number does not exist for any tube inclination angle which is evidently contrary to the result reported in [3] using perturbation method.

Acknowled~,ement-This work was supported by the National Research Council of Canada through Grant A i 655. The authors wish to thank Mrs. E. S. Buchanan for typing the manuscript.

A a

C f

h k

M N

Nu P

Pr R,6 ,Z

Ra Re

r T

U , V , W 14, D, ~ '

Ol

E

o K

iz p

p T

Subscript 0 w

N O M E N C L A T U R E Axial pressure gradient in fluid. - (OP/OZ + p,,.g sin a) Radius of tube Axial temperature gradient, aT/OZ Friction factor, 2 ~ / ( p W 2), or a dummy variable Gravitational acceleration Average heat transfer coefficient Thermal conductivity Number of divisions in R-direction N umber of divisions in d-direction Nusselt number, h (2a)/k Pressure Prandtl number Cylindrical coordinates Rayleigh number, [3gCa'/vK Reynolds number, Aa3/4p~ Dimensionless radial co-ordinate, R/a Local temperature Velocity components in R, ~b and Z directions Dimensionless velocity components in R, ~ and Z directions Angle of tube inclination, see Fig. 1 Coefficient of thermal expansion A prescribed error, see equation (6) Dimensionless temperature difference, ( Tw - T)/R eC aPr Thermal diffusivity Viscosity Kinematic viscosity,/~/p Vorticity function defined by Eq. (2) Density Shear stress Dimensionless stream function Dimensionless Laplacian operator, a~/c3r 2 + O/rOr + ~2/r20c~"

Condition for pure forced convection in horizontal tube Value at wall

Superscript n nth iteration - Average value.

R E F E R E N C E S [ll G. O. G. Lf f and D. J. Close, Solar water heaters. Low Temperature Engineering Application o f Solar

Energ, y, Am. Soc. Mech. Engrs. New York, Chapter VI (1967).

Laminar convection in uniformly heated tubes 371

[2] L. H. Baker, Film heat-transfer coefficients in solar collector tubes at low Reynolds number. Solar Energy 11, No. 2, 78 (1967).

[3] M. lqbal, Free-convection effects inside tubes of flat-plate solar collectors. Solar Energy 10, No. 4, 207 (1966).

[4] B. R. Morton, Laminar convection in uniformly heated horizontal pipes at low Rayleigh numbers. Quart. J. Mech. Appl. Math. 12, 4 ! 0 ( 1959l.

[5] G. J. Hwang and K. C. Cheng, Boundary vorticity method for convective heat transfer with secondary flow-appfication to the combined free and forced laminar convection in horizontal tubes. Heat Transfer 1970 (Proc. Fourth International Heat Transfer Conference), Vol. 4, NC 3.5, Elsevier, Amsterdam.

[6] M. Akiyama and K. C. Cheng, Boundary vorticity method for laminar forced convection heat transfer in curved pipes. Int. J. Heat Mass Transfer 14, i 659-1675 ( ! 97 i ).

[7] G. J. Hwans, Thermal instabifity and finite ampfitode convection with secondary flow. Ph.D. thesis, University of Alberta~ Edmonton, Alberta, Canada (1970).

[8] T. M. Hallman, Combined forced and free-laminar heat transfer in vertical tubes with uniform internal heat generation. Trans, ASME 78, 1831 ( 1956L