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Page 1: MIXED CONVECTION IN AN INCLINED CHANNEL WITH LOCALIZED HEAT SOURCES

This article was downloaded by: [Cornell University Library]On: 17 November 2014, At: 13:45Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Numerical Heat Transfer, Part A: Applications: AnInternational Journal of Computation and MethodologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/unht20

MIXED CONVECTION IN AN INCLINED CHANNEL WITHLOCALIZED HEAT SOURCESXiao Wang a & Luc Robillard aa Mechanical Engineering Department , Ecole Potytechnique, Uniuersity of Montreal , Box6079, City Center, Montreal, Quebec, Canada , H3C3A7Published online: 02 Mar 2007.

To cite this article: Xiao Wang & Luc Robillard (1995) MIXED CONVECTION IN AN INCLINED CHANNEL WITH LOCALIZED HEATSOURCES, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, 28:3,355-373, DOI: 10.1080/10407789508913750

To link to this article: http://dx.doi.org/10.1080/10407789508913750

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Page 2: MIXED CONVECTION IN AN INCLINED CHANNEL WITH LOCALIZED HEAT SOURCES

I MIXED CONVECTION IN AN INCLINED CHANNEL WITH LOCALIZED HEAT SOURCES

Xiao Wang and Luc Robillard Mechanical Engineering Depar~rnent, Ecole Polytechnique, University of Montreal, Box 6079, City Center, Montreal, Quebec, Canada, H3C 3A 7

Fully developed opposing mired conueawn is numerically studied in an inclined channel that har discrete M n g on the h t o m and is ins&d on the top. The numericnl approach is based on the hypo~hesis that the solution is periodic according to the imposed wavelength of the healing elements. Considering that the heal produced by the healing elements is totidly urm'ed downslream, the tempemhue increment fmm one M n g element to Ihe Orher is deJin.4 on the basis of an energy bnlnnce. To verify the accuracy of the compurntionnl code, an &tical shrdy of the ermme case with an entire& heated wall is investigated. Also, to ualidnre that the sohtion of the problem is periodic with a wauelength corresponding to the imposed perlurbalwn, a channel wilh entrance and erir sections comlzining four lo sir heaiing elements is simulated numeri&. In the present study, the relntive strength of the forced flmv and buoyancy effpcts is emmined for a broad range of Rayleigh numbers, Rey~kLr numbers, and inclination angles. Both o u e d and locnl recirculoting flows are obsewcd that are caused by buoyancy effects on the forced f low.

INTRODUCTION

Despite changing demands and the availability of new heat transfer technol- ogy, direct air cooling of electronic components continues to command substantial attention. A comprehensive overview of this subject may be found in the article by Hannemann I:1], which concludes that natural and forced convection are still very popular in the application of industry due to lower cost and convenient user environments.

Channels formed by parallel plates or fins are a frequently encountered configuration in natural and forced convection cooling in air of electronic equip- ment. Packaging constraints and electronic considerations lead to a wide variety of complex heat dissipation profiles along the channel walls. However, in many cases of interest, an isothermal or isoflux boundary representation, or use of an isother- mal/isoflux boundary, together with an insulated boundary condition along the adjoining plate, can yield acceptable accuracy in the prediction of the thermal performance of such configurations.

Received 22 April 1994; accepted 7 February 1995. Financial support by Natural Science and Engineering Research Council, Canada, and FCAR,

Province of Quebec, is acknowledged. Address correspondence to Dr. Luc Robillard, Department of Mechanical Engineering, Ecole

Polytechnique, University of Montreal, Box 6079, City Center, Montreal, Quebec H3C 3A7, Canada.

Numerical Heat Transfer, Part A, 28:355-373,1995 Copyright 0 1B5 Taylor & Francis

1040-7782/95 $10.00 + .00 355

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356 X. WANG AND L. ROBILLARD

NOMENCLATURE

a dimensionless length of heating elements b dimensionless periodicity in x direction of

heating elements c dimensionless increment of temperature

for a wavelength (= a /Re Pr) Fr Froude number (= u b / d m ) g gravitational acceleration, m/s2 h ' width of the channel, m k thermal conductivity, W/(m K) L dimensionless length of the whole channel Pr Prandtl number (- v/u) q ' imposed heat flux on heating elements,

W/(mZ S) q, flow reversal parameter (Eq. 15) Ra Rayleigh number (= g/3ATfh"/va) Re Reynolds number (= h'ub/v) r dimensionless time (= r 'ub/h') T dimensionless temperature [ - (T' -

T:)/AT'I T: reference temperature, K AT' temperature scale (= q'h'/k) u; imposed velocity in x direction averaged

over h', m/s u, u dimensionless velocity components in x

and y directions ( = u'/ub, u'/ub)

x, y dimensionless Cartesian coordinates (= x'/h', y'/h')

a thermal diffusivity, m2/s /3 thermal expansion coefficient, K-' v kinematic viscosity of fluid, m2/s p fluid density, kg/m3

I) dimensionless stream function (= $'/ h'ub)

ll dimensionless vorticity (= n'h'/ub)

Subscripts

b refers to bottom wall c refers to convective property t refers to top wall

Superscripts

' refers to dimensional variable - refers to value averaged over one wave. length refers to conductive property

Elenbasse 121 was the first to document a detailed studv of the thermal - - characteristics of one such configuration, and his experimental results for isother- mal plates in air were later confirmed numerically by Bodoia and Osterle [3] and shown to apply as well to the constant heat flux conditions 1:4]. Also, Aung et al. [5, 61 and Miyatake et al. [7, 81 extended the available results to include both asymmetric wall temperature and heat flux boundary conditions, including the single insulated wall. More recently, Hasnaoui et al. [9] investigated mixed wnvec- tion in a horizontal layer with heating elements regularly spaced on the lower boundary. Their results show that at low Reynolds numbers (Re), a steady state is possible for which the convective cells remain attached to the heating elements. Beyond a critical Reynolds number (Re,,), the cells are carried downstream, and reinforced and weakened periodically.

However, few studies have been reported thus far in the literature for channels with arbitrary inclination angles. Tomimura and Fuji [lo] have explored the aiding mixed convection flow and heat transfer between inclined parallel plates with localized heat flux over their length. Their results demonstrate that the temperature field is considerably affected by inclination angle and much affected by the discrete heating because' of the intermittent development of a thermal boundary layer. Lavine [11, 121 has derived an analytical model based on a parallel flow approach for mixed convection between parallel plates, both plates being heated with constant heat flux. According to the choice of the inclination angle, the

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CONVECTION IN A CHANNEL WITH HEAT SOURCES 357

buoyancy force can act in the flow direction (aiding mixed convection [Ill) or in the opposite direction (opposing mixed convection [12]). The experiments [I31 also demonstrated flow recirculation.

The purpose of the present article is to investigate numerically the idealized case of a two-dimensional laminar flow in a channel of infinite extent and of arbitrary inclination with one of the walls isolated while discrete heating elements with uniform heat flux are regularly spaced along the other wall that is elsewhere adiabatic. For zero forced flow, the heat produced by the heating elements cannot escape from the flow domain, and the asymptotic behavior in time consists of a temperature increasing at a steady rate at each point of the flow domain. In the present study, a forced flow of arbitrary intensity is imposed between the walls. The heat from each heating element is conveyed downstream, and a steady state is achieved for which the temperature field is characterized by a spatial increment that repeats at each heating element in the downstream direction. We assume here that the flow field and temperature gradients repeat periodically along the channel, according to the imposed periodicity of the heating elements. This assumption follows directly from the parallel flow approach used by Lavine [ll, 121 for entirely heated walls. As mentioned earlier, this approach assumes that the convective flow arising from buoyancy effects has a zero wavenumber. Consequently, when the lower wall is heated by individual heating elements, no spatial periodicity other than the spacing between heating elements is expected to affect the solution. Thus, the numerical flow domain used for most of the results may be restricted to one wavelength of the imposed periodicity, with periodic conditions applied at the two end boundaries. Some numerical results are also obtained from a whole-channel configuration with a few heating elements and specified entrance and outlet conditions. This last configuration requires a large mesh size and much more CPU time. Its main purpose is to validate the one-wavelength configuration by showing (1) that the flow field tends to repeat along the channel according to the imposed periodicity of the heating elements and (2) that the velocity and temperature profiles at a distance far enough from the entrance and outlet do agree with those of the one-wavelength configuration. Finally, an analytical solution is given in the appendix for the limiting case where the entire boundary is heated. The approach for this solution is based on a parallel flow hypothesis. The developments are similar to those by Bejan and Tien [14] and Vasseur et al. [IS] and follow closely the work by Lavine [ l l , 121.

MATHEMATICAL FORMULATION

The flow domain considered here is illustrated in Figure l a . It consists of a two-dimensional fluid layer of infinite extent in the x direction, bounded by two inclined parallel plates at an angle cp with the horizontal direction. The upper wall is insulated, the bottom wall is heated, and the uniform nondimensional heat flux - d T / d y is generated from heating elements of length a, which are regularly spaced at a distance b, while the remaining surface is adiabatic. Moreover, an external flow with downward mean velocity (opposing flow) u, is imposed. Assum- ing that the laminar flow is incompressible with constant properties, that the usual Boussinesq approximation holds, p = p,[l - P(T' - T:)], and that the heat gener-

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358 X. WANG AND L ROBILIARD

ated from the heating elements is totally carried downstream by the forced flow, we obtain the following set of dimensionless equations expressing the conservation of energy, momentum, and mass:

Equations (1)-(4) have been nondimensionalized by defining x , y, u, u, T , R, and

T' - T; T =

AT'

where the prime quantities are dimensional. The intensity of the natural convec- tion and the strength of the forced convection are characterized by the Rayleigh number, Ra = g p ~ ~ ' h r 3 / ( a v ) , and the Reynolds number, Re = u b h f / v , respec- tively. The other parameters a = a ' / h 8 , b = b1 /h ' , cp, and Pr = v / a are the dimensionless length of the heating elements, the dimensionless periodicity of the heating elements, the inclination angle, and the Prandtl number, respectively. The time derivative terms of Eqs. (1) and (2) serve uniquely to integrate numerically to steady state. There is no interest as such in the transient solution.

Two flow domains are considered for the solution of Eqs. (1)-(4): the whole-channel configuration with entrance and outlet sections, shown in Figure l a , and the one-wavelength configuration, shown in Figure l b . This last flow domain is based on the assumption that the solution is periodic in the x direction according to the periodic spacing of the heating elements. Consequently, it is aimed at reproducing the flow conditions in a channel of infinite extent. Both flow domains have the same thermal and dynamical boundaly conditions for the lower and upper walls. Those conditions are as follows.

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CONVECTION IN A CHANNEL WITH HEAT SOURCES

Figure 1. Numerical domain: ( a ) Whole-channel configuration, and ( b ) one-wavelength configuration.

Lower boundary ( y = 0)

for all x ,

for each element, and

elsewhere.

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360 X. WANG AND L. ROBILLARD

Upper boundary (y = 1)

for all x. The boundary conditions for the vertical boundaries are different for the two flow domains. For the whole-channel configuration, they are

where f stands for any physical variable. The conditions at the outlet correspond to those used by Tomimura and Fuiii [lo] and Yiicel et al. [16]. For the one- wavelength configuration, every physical quantity f , including the temperature gradient, satisfies the following periodic condition on the two vertical boundaries:

However, the temperature itself does not satisfy this condition. In fact, at each wavelength b, the temperature must be increased by a factor c such that

where c = a/(Re Pr) is defined on the basis of an energy balance. The increment c implies, of course, the existence of a conductive heat flux in the upstream direction that the convective heat flux must compensate.

The present problem contains six governing parameters, namely, Ra, Re, Pr, the spacing of the heating elements b, the individual length of the heating elements a, and the inclination angle cp. As shown in the appendix, for the limiting case of an entirely heated wall (a = b), the number of governing parameters is reduced to two, P, and P2, defined in Eq. (A9).

NUMERICAL METHOD

Finite difference techniques are used for both flow configurations. Owing to the moderate convective flow of the present problem, the convective and diffusive terms in Eqs. (1) and (2) are discretized by a central differences schema, while the time derivatives are approximated by forward differences. An alternating direction implicit (ADI) procedure is used to perform the time integration of the vorticity and energy equations while the stream function is solved at each time step by a line successive overrelaxation method (LSOR) for all interior grid points, and a point successive overrelaxation method (PSOR) for vertical boundary grid points. The periodic boundary conditions are treated in the AD1 approach with a matrix partition procedure similar to that utilized by Phillips [17].

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CONVECTION IN A CHANNEL WITH HEAT SOURCES 361

A uniform mesh size of 40 x 40 was used in the x and y directions of the computation domain for the periodic case of b = 2, which was found to be an acceptable compromise between accuracy and computer time, since several tests of 80 X 80 mesh size show that the differences between the two grid sizes are less than 3% in accuracy and more than a factor of 8 in CPU time. For the whole- channel configuration with five heating elements, a grid of 150 X 20 was used. Time steps were varied from At = for low Ra to At = for high Ra. The execution time on an IMB RISC 6000/540 was 5040 CPU seconds using 5000 iterations for the whole-channel configuration at Ra = 500, Re = 1, 4 = 75", while it took 180 CPU seconds and 800 iterations for the periodic case. Both configurations have a time step of At = 10-3-10-4.

The convergence criterion to steady state was twofold. On the one hand, the energy balance was checked for the computation domain, namely,

(heat flu~)in - (heat flu~)~,t <

(heat flux)in

The other criterion checks that the fractional change of a variable f between two time steps at any node is less than i.e.,

Results from the present numerical code were also compared with the analytical solution given in the appendk for the limiting case a = b (entirely heated bottom wall). Figure 2 shows velocity profiles obtained from both approaches for P, = 100 and P, = -70, 70, and 140.

RESULTS AND DISCUSSION

For all the cases investigated numerically in the present study, the length a of the heating elements and the spacing b were maintained equal to 1.0 and 2.0,

Figure 2. Velocity profiles of entirely heated wall.

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362 X. WANG AND L. ROBILLARD

respectively. Also, Pr was set equal to 0.7, which is the value of air at standard conditions. The ranges of Rayleigh and Reynolds numbers were 0 < Ra < 10000: and 0 < Re < 10, respectively. The inclination angle cp was varied between 0" and 90°, so that the forced flow was downward with the heating elements on the bottom wall.

In contrast with experiments on a similar problem published by Incropera et al. [ la ] , where turbulent flow was found to occur for the range 5000 < Re < 14000, the present study involves relatively low Re and Ra, for which laminar flow is expected.

The present approach assumes that the flow is two-dimensional. In a recent two-dimensional study on a similar problem, Zhang and Tangborn [19] found that spatially varying the temperature boundary conditions makes the two-dimensional free convection stable to three-dimensional disturbances. They mention, however, that this behavior of natural convection does not guarantee that the mixed convection flows can exist in two dimensions.

In Figures 3 and 4, a comparison is shown between numerical results obtained from the one-wavelength configuration and those from the whole-channel configuration for a typical case of Ra = 500, Re = 1, and cp = 75". Flow and temperature fields are shown by streamlines and isotherms in Figures 3a and 36, respectively. Heating elements correspond to the heavy line segments regularly spaced on the bottom wall. The whole-channel configuration includes five heating elements with entrance and outlet conditions given by Eq. (11). With those boundary conditions, a definite recurrence, or periodicity, of the flow field and temperature gradients is obsewed, which corresponds to the spacing 6 of the heating elements. This periodicity could be improved upon by increasing the length of the channel and the number of heating elements. However, this would require prohibitive CPU time. In Figures 4a and 46, one can notice the great similarity of flow and temperature fields for both configurations. Velocity and temperature

Figure 3. Comparison between the one- wavelength configuration and the whole- channel configuration (Ra = 500, Re = 1, q = 75'. and Pr = 0.7): (a) flow field, and (b ) temperature field.

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Figure 4. Velocity ( a ) and temperature (b) profiles at given locations A and B.

profiles taken at locations A and B are labeled in Figures 4a and 46. It is seen that profiles from both configurations 'are quite similar. Such a similarity was also found for the other values of the parameters Ra, Re, and (p (results are omitted for brevity), and it was concluded that results from the one-wavelength configuration could adequately represent the flow behavior in a long channel, away from the entrance and outlet regions.

Flow and temperature fields, as functions of the Ra, for a fixed Re (Re = 11, are shown in Figure 5, with streamlines (left) and isotherms (right). In this figure, as well as in Figure 6, A & is the constant increment that is added up from one streamline to the other, starting from the zero value of the bottom wall; ATi is the constant increment between adjacent isotherms. When the Rayleigh number is zero (Figure Sa), a parabolic velocity profile is produced, with the maximum velocity midway between the walls. The temperature field of that figure is the one obtained for pure forced convection, without any buoyancy effect. With Ra increasing from zero, the velocity is reduced near the top wall and increased near the bottom wall in order to maintain the mass conservation. This change in flow behavior corresponds to the occurrence of a convective roll of infinite extent superposed on the forced flow. Such a phenomenon also exists for the case of an entirely heated wall, for which an analytical solution is given in the appendix. The first term on the right-hand side of Eq. (A12) corresponds to that convective roll. This convective roll grows with Ra, and beyond a critical value, an upstream flow is produced near the top wall (Figure 5c). For such conditions, results obtained from the whole-channel configuration indicate that a clockwise recirculating flow occurs

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X. WANG AND L. ROBILLARD

Figure 5. Flow and temperature fields as function of Ra in a horizontal channel (Re = 1.0, Pr = 0.7, c = 1.4286): (a, b) A & = 0.077, ATi = 0.110; (c) A & = 0.111, ATi = 0.110; ( d ) A$: = 0.167, ATi = 0.130; ( e ) A & = 0.214, ATi = 0.143; ( f ) A & = 0.268, ATi = 0.159; and ( g ) A & =0.589, ATi = 0.238.

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Figure 6. Flow and temperature fields as function of inclination angle (Ra = 500, Re = 1, Pr = 0.7, c = 1.4286): (a, b ) A$, = 0.167, AT, = 0.130; ( c ) A& = 0.171, ATi = 0.130; (d) A& = 0.116, AT, = 0.119; (e) A& = 0.077, AT, = 0.110; (f) A+" = 0.143, ATi = 0.130; and (g) A+s = 0.250, AT, = 0.159.

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366 X. WANG AND L ROBILLARD

that extends over the whole channel length from entrance to outlet. Consequently, the upstream flow appearing in Figure 5c corresponds to a recirculating flow extending to infinity in both x directions (overall recirculating flow). This overall recirculating flow becomes more pronounced as Ra is increased beyond 250 (Figures 5c-5g). In Figure 5g, the upstream flow is almost equivalent to the downstream flow near the bottom wall, and the forced convection contributes in a negligible way to the velocity profile. The effect of localized heating is apparent in Figures 5c-5f, where two-dimensional features are caused by a recirculating flow that does not extend beyond one wavelength (local recirculating flow).

As can be expected, the above behavior in the flow field strongly influences the temperature gradients. For the whole set of Figures 5a-5g, the isotherms dways intersect the adiabatic boundaries at right angles, as they should, and the heating elements at an angle that ensures the constant heat flux boundary condition. However, it is noticed from the shape of the isotherms that, with increasing Ra;. the fluid near the top wall becomes hotter than the fluid near the bottom wall^.. a: effect of' the inclination angle is shown in Figure 6, with Figure 6a

corresponding, to, Figure 5d. It is noticed in Figures 6a-6d that the overall recirculating flow remains clockwise (flow reversal near the top wall) but decreases to zero) at cp - 70': At 70°, the overall recirculating flow is zero. However, there exists a, local' recirculating flow just above the heating element in the counterclock- wise disection~ (Figure 6e). For inclination angles above 70°, local and overall recirculating, nbws are in the counterclockwise direction. This reversal in the directibn~ of the recirculating flows modifies the temperature field, the fluid near the topwalll becomingcolder than the fluid near the bottom wall.

The overall. and' bcal recirculating flows may be defined from the following parameter. (flow reversal' parameter):

wliene: I&,,, d l $kin, are the maximum and minimum value of the stream function, resgectivell;, a t a: given' lbcation x along the wavelength. The flow reversal parame- te r i's:giveni in1 Figure 7 as a function of x for Ra = 500, Re = 1, and ' c p = O0, 20°, 607; 70:;. 80?:, andI'90::. Upstream flow near the top and near the bottom are idtntified~ by solid' and' dashed lines, respectively. The overall recirculating flow is given: by the minimum value of the flow reversal parameter along the wavelength Cq,~D);. whereas; the local recirculating flow is given by the difference qrm_ - q;m,n,.q;m,z being, the maximum value of the flow reversal parameter along the wavel'ength. The: flow reversal parameter is x independent for the entirely heated bottom, wall!, the: local recirculating flow being a feature of localized heating on the bottom: wall!

Figure 8, defines; the limit between flow reversal (either overall or local flow reversal# and: no, flow reversal. It is seen that for the case of an entirely heated bottom~wall\, ru = b;.'therc is no separate influence of Re, and this limit corresponds to, a) fiiedl miticall value of' - 50 for the coefficient ~ a / ( ~ e ' Pr) of Eq. (2). The coeffici'ent Ftaa./CFte2 Pr).= Fr-' (where Fr is Froude number) expresses the ratio of' buoyancy forces, to inertia forces. With discrete heating, for instance, with

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8.-3m 8.-l R-0 - w. "a .........--.. bollam "0. r...,d

., , , , , 1) Figure 7. Flow revers$ parameter as a function of x.

a = b/2, the critical value of the coefficient Ra/(Re2 Pr), as obtained numerically, decreases first, with Re increasing from zero. When Re increases further, it reaches an asymptotic value of - 62.

The variation of the total heat f l k (flux integrated over the width) along the channel is known a priori, as it must increase linearly over each heating element by the quantity a and remain constant elsewhere. This behavior can be used to check the local energy balance of the present numerical code. The 'local quantities Q*, Q,, and Q, defined as

are the conductive, convective, and total heat fluxes, respectively. Here we,compute Q* and Q, separately. When added up, these two quantities must reproduce the

Figure 8. Conditions for flow reversd.

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368 X. WANG AND L ROBILLARD

known function Q adequately. Also Q increases at each wavelength by the same increment Q(b) - Q(0) = a, so that only the difference AQ(x) = Q ( x ) - Q(0) is relevant within the present framework. Moreover, as the conductive heat flux based on the x derivative of T repeats integrally at each wavelength, the convec- tive heat flux is also an increasing function of x. Figure 9 provides the separate values AQ* = Q*(x) - Q*(0) and AQ, = Q,(x) - Qc(0) as obtained numerically, denoted by heavy and light symbols, respectively. In Figure 9a the effect of Ra is shown from three sets of data corresponding to Ra = 0, 500, and 1000, for the same Re = 1 and inclination angle cp = 0. In Figure 96 the effect of the inclination angle is shown with three sets of data corresponding to cp = 60°, 70°, and 80" for the same Re = 1 and Ra = 500. It may be verified in Figures 9a and 96 that each pair of curves AQ* and AQ, adds up to reproduce faithfully AQ.

Figure 10 shows the effect of Ra on the temperature distribution along the bottom wall for the case of a horizontal channel. In Figure 10a the difference between the bottom wall temperature at a given x and the bottom wall tempera- ture at x = 0 [AT, = T,(x) - T,(O)] is given as a function of x. It is observed that the mixed convection (Ra > 0 ) provides a more gradual increase of temperature along the wavelength. The local value (T , - TbUlk)-' , where T,,,, is the fluid bulk temperature, corresponds to the form of Nu used by Lavine 1201 and is given as a function of x in Figure 106. The general effect of mixed convection (Ra > 0 ) consists in increasing the difference between the bottom wall and the bulk temperatures. A comparable trend is found in Lavine's work [20] for Nu averaged over the two walls, when the channel is horizontal. This behavior is related to the

Figure 9. Local heat flux profiles: (a) effect of Ra; ( b ) effect of q.

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CONVECTION IN A CHANNEL WITH HEAT SOURCES

I., I

... ............ ~t:.!o?o .............................................................. Figure 10. Effect of the Rayleigh number o o 0 I J z on the temperature distribution at the

lower boundary (& = 0, Re = 1, Pr = 0.7): ( a ) AT, as a Function of x ; ( b ) (Tb -Tb,,,)-' as a function of x.

flow reversal and to the definition of the bulk temperature [T,,,, = Q,(x)/(Pr Re)], where Q,(x) is computed from Eq. (16h).

CONCLUSION

Fully developed mixed convection taking place within an inclined fluid layer with regularly spaced heating elements on its lower boundary and insulated on its top boundary has been studied numerically. Both a one-wavelength configuration and a whole-channel configuration are explored, from which we conclude that the results obtained from the one-wavelength configuration could represent adequately the flow behavior in a long channel, away from the entrance and outlet regions.

Numerical results reveal that buoyancy effects produce an overall recirculat- ing flow, i.e., a recirculating flow that extends over the whole channel length. This flow is comparable to the one obtained when the bottom wall is entirely heated. Its direction and intensity are related to the Rayleigh number and inclination angle of the channel. A particular feature of discrete heating is the occurrence of local recirculating flow. These local recirculating flows produce the two-dimensional features of the solution.

The heat generated from the heating elements must be carried downstream, and the mechanism involved in this process is forced convection. For this process to occur, the temperature must increase at each wavelength by the same incre-

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370 X. WANG AND L. ROBILLARD

ment. As a consequence, the conductive heat transfer is in the upstream direction and must be compensated by the forced convection.

APPENDIX: ANALYTICAL APPROACH

In the extreme case of a = b (entirely heated wall configuration), the flow in the channel is assumed to be parallel, such that u = u(y) and u = 0. Also, the temperature field can be divided into the sum of a linear x dependence and an unknown function in y. Thus, we take

and

where c is a constant temperature gradient in the x direction, which is determined on the basis of an energy balance, and T, is a reference temperature that can be set to zero without loss of generality. Similar approaches have been used by Bejan and Tien [14], Vasseur et al. [IS], and Lavine [ l l , 121. By these assumptions, the dimensionless governing Eqs. (1)-(4) then become

Ra u"' = - (c cos cp + 0' sin cp)

Pr Re

1 udy = 1

Integrating Eq. (A3) over the channel cross section and making use of boundary condition Eqs. ( A 9 and (A6), it may be readily deduced that c = l/(Pr Re). The solution of Eqs. (A3)-(A61 leads to the following two sets of velocity and tempera- ture profiles, each set corresponding to a specific range of the inclination angle.

Set 1

For 0 < cp < .n (downward forced flow),

u = C,[sinh(My) + sin(My)] + C,[cosh(My) - ws(My)l

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CONVECnON IN A CHANNEL W K H HEAT SOURCES

The governing parameters P, and P2 are

PI = Ra sin q/Re Pr

P2 = Ra cos q / ~ e ' Pr

with M = p,'I4. The constants C , and C, are specified according to boundary conditions, dl) = 0, O'(1) = 0:

Set 2

For - rr < cp < 0 (upward forced flow),

u =A,[cos(Ny)sinh(Ny) + sin(Ny)cosh(Ny)l +A,sin(Ny)sinh(Ny)

sin (Ny) cosh (Ny) (A10)

1 0 = - A [cosh (Ny) sin (Ny) - sinh (Ny) cos (Ny)]

2 N 2

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372 X. WANG AND L ROBILLARD

with N = (-P,/4)'I4. The constants A, and A, are specified according to boundaly conditions u(1) = 0 and O'(1) = 0.

In the particular case of a horizontal channel, cp = 0, PI = 0, P, =

Ra/ReZ PrZ, we obtained the following velocity and temperature distributions:

Equations (A12) and (A131 have the same form as the solution given by Bejan and Tien [14] in the analysis of laminar natural convection heat transfer in the core region of a slender. horizontal cavity with different end temperatures. These similarities indicate that a temperature increment a t each wavelength generated by the heating elements in channels plays an important role in governing the natural convection in the present mixed convection problem.

REFERENCES

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CONVECTION IN A CHANNEL WITH HEAT SOURCES 373

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