Numerical prediction of pressure loss coefficient and induced mass flux for laminal natural convective flow in a vertical channel

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Pergamon

NUMERICAL AND

Energy Vol.22, No. 4. pp. 413-423, 1997 © 1997ElsevierScienceLtd Printed in Great Britain. All rightsreserved 0360-5442/97 $17.00+0.00

PII:S0360-5442(95)00060-6 PREDICTION

INDUCED

CONVECTIVE

MASS

OF PRESSURE FLUX

FLOW

LOSS

FOR LAMINAL

IN A VERTICAL

COEFFICIENT NATURAL

CHANNEL

A. S. KHEIREDDINE,* M. H O U L A SANDA,* S. K. CHATURVEDI,** and T. O. MOHIELDIN ~ *Department of Mechanical Engineering and ~Engineering Technology,Old Dominion University, Norfolk, VA 23529, U.S.A. (Received 13 February 1996)

Abstract--The natural convection heat transfer and ventilation characteristics of heated and vented parallel wall channels have been studied numerically. The flow is assumed to be laminar and steady, and the governing two-dimensional Navier-Stokes equations are solved by a finite volume formulation to calculate the chimney effect that draws cooler ambient air from the lower opening. The fluid-dynamic and heat-transfer characteristics of vented vertical channels are investigated for both symmetric isothermal and constant heat-flux boundary conditions for the Rayleigh number ranging from 103 to l0 s and the aspect ratio in the 5-20 range. The non-dimensional entrance and exit pressure losses and the induced mass-flow are correlated with the Rayleigh number. The results indicate that although inlet- and exit-loss coefficients may vary significantly with Rayleigh number and aspect ratio, the total pressure-loss coefficient is a weak function of Rayleigh number. It is also shown that the total pressure loss coefficient and non-dimensional mass-flow rate results are better correlated with the modified Rayleigh number that is obtained by dividing the Rayleigh number by the aspect ratio. The elliptic flow results, obtained from the present procedure, are compared with fully developed flow results and with boundary-layer calculations of previous authors. Numerical results also yielded important information regarding the placement of the free pressure boundary. The results for the present geometry indicate that the effect of free-boundary location is negligible if it is placed at a distance of four times the channel width or greater. © 1997 Elsevier Science Ltd. All rights reserved

INTRODUCTION Buoyancy-induced flows abound in nature and engineering systems. These flows have significant potential for applications in material processing, electronic equipment design, building heat-transfer problems, thermo-syphon technology, and solar collectors such as the Trombe wall. The fluid motion in these systems is caused by differential heating of surfaces. In close cavities, this heating results in complex cellular or multi-cellular flow patterns depending on the Rayleigh and Prandtl numbers and on aspect ratio. In open channels, differential heating of vertical walls causes a chimney effect due to the buoyancy force in the channel. We analyze the inlet and exit pressure-loss characteristics of a differentially heated vertical channel, open at both the bottom and top ends. The laminar natural convective motion in this geometry has been considered by many previous authors, t-~° Application of this geometry to electronic cooling applications is discussed in Ref. 11. For example, in many electronic circuit boards, a parallel wall channel configuration is used for natural convection cooling. The aspect ratio and Rayleigh number for these cases are typically in the range of 5 - 1 0 and 250-106, respectively. A two-dimensional analysis has been used to determine the pressure-loss coefficient. This is supported by the work of Sparrow and Bahrami, ~2 which indicates that the three-dimensional edge effects are negligible for a modified Rayleigh number (Ra*) greater than 10. Since in many practical electronic board configurations Ra* is greater than 50, a two-dimensional analysis is justified. Most authors of studies dealing with the parallel-wall geometry make the boundary-layer approximation and solve the coupled governing equations for mass, momentum and energy transport analytically and/or numerically for both isothermal and constant-wall-

*Author for correspondence. 413

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A.S. Kheireddineet ai

heat-flux conditions. Although these studies have yielded valuable information about heat-transfer characteristics and mass flows induced through the system, they do not provide results pertaining to entrance- and exit-pressure losses. Since consideration of the flow regime outside the channel is excluded in the boundary-layer approximation, the earlier studies do not incorporate flow regimes required to predict the inlet- and exit-pressure losses. A linearized model by Tichy t3 considers the specified problem by invoking the Oseen approximation n4 for the non-linear convective terms in the governing equations. The effects of combined inlet- and exitpressure losses on the mass flowrate and heat-transfer characteristics of the channel are determined parametrically by assuming different values of combined inlet- and exit-pressure losses. However, Tichy's model does not predict the inlet and exit pressure losses as part of the solution. In fact, the paper citing ASHRAE practice pertaining to the air-ductwork coefficient suggests using a total pressureloss coefficient (Kt) in the range of 1.0-4.0. The present study actually predicts the total pressure loss coefficient as a function of Rayleigh number and channel-aspect ratio, t h e n et ai~5 have solved the same problem in the same manner as Tichy by employing the momentum integral approach but with assumed values of combined inlet- and exit-pressure losses. Analyses of the laminar convective channel-flow problem, employing Navier-Stokes equations, has been performed in Refs. 16 and 17. These analyses involve both flow inside and outside of the channel by including both inlet and exit pressure losses into a combined pressure value at the outset of numerical calculation. Although this step simplifies the numerical procedure, the results pertaining to inlet and exit losses cannot be extracted separately due to the formulation of the problem. Also, it is noted that the geometry considered in these studies is different from the geometry considered here. The present study predicts the inlet and exit pressure losses of a vertical channel in some detail. This is done by considering the region inside the channel, as well as regions upstream and downstream of the channel. The numerical code, based on the SIMPLER algorithm, 17 is used to solve the steady twodimensional Navier-Stokes equations for laminar flow. Heat transfer, mass flowrate and total pressure loss characteristics are predicted as a function of Rayleigh number and channel aspect ratio for both isothermal and constant-wall heat-flux cases. The present study makes a contribution since no direct pressure-loss measurements have been reported in the literature for the geometry studied here. Predicted results for heat transfer and mass flowrates are also compared with existing experimental and theoretical results.

PHYSICAL MODELAND SOLUTIONPROCEDURE Figure 1 shows a vertical channel whose inside wails are maintained at a constant temperature or a constant heat flux. The flow in the cavity is initiated and maintained by the body force due to the density difference between the heated channel air and cooler outside air. The ambient air enters the channel from the bottom opening, and the heated air is discharged from the top vent. Velocity and pressure values at the inlet and exit sections are not known a priori. In the numerical procedure used here, either the known value of velocity or pressure must be specified at the inlet and exit sections of the channel. To overcome this dilemma of two unknown quantities, the computational domain is extended well beyond the geometrical configuration. We take advantage of the fact that the differentially heated channel is attached to a constant pressure reservoir, namely the atmosphere. The free boundary ABCDA in Fig. 1 is the extended boundary, located far from heated surfaces where undisturbed atmospheric pressure condition can be applied. The computational domain has been extended laterally four times the channel width on both sides to boundaries AD and BC, and four times the channel width in both the upstream and downstream regions of the channel to boundaries AB and CD. The inside faces of the channel wails are maintained at a constant temperature or a constant wail heat flux while the outside faces are subjected to the adiabatic boundary condition. In order to accomplish this numerically, the walls are chosen with a small thickness equal to 0.065 times the channel width. Calculations with a larger wall thickness equal to 0.13 times the channel width showed no effect on the heat-transfer and pressure-drop characteristics in the channel. Non-dimensionai governing equations for the steady, two-dimensional laminar convective motion may be expressed as

Pressure loss coefficient and induced mass flux for laminal natural convective flow in a vertical channel

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(1)

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(2)

O(puv)/Ox + O(pvv)/Oy=-Op/Oy + Pr~-R-a(02v/ox2+ 02v/oy 2,

(3)

d(puO)/3x + 3(pvO)lOy = l / R ~ o z O / 3 x

(4)

z + 020/0y2).

Symbols u, v, 0, p, x and y are non-dimensional variables. The channel width I has been used to nondimensionalize x and y, and ~[3g(Th - T,o)l or ( x ~ q l 2 / k ) have been used as the characteristic velocities to non-dimensionalize velocity components. Symbols Ra and Pr represent Rayleigh and Prandtl numbers, respectively. The non-dimensional temperature 0 and Ra are defined as

0 = T - TJTh - T~ or ( T - T~)k/ql;

Ra = [3gATl3/va or [3gql4/vaK,

(5)

where q is the wall heat flux. Equations ( 1 ) - ( 4 ) are solved subject to the following boundary conditions:

u,v = 0 on all solid surfaces; 0 = 1 on both walls for the isothermal case; 30/3x = - 1 , y = O, O0/3x = +1, y = 1 (constant wall heat flux case); p = 0 on the extended boundary ABCDA. The non-dimensional induced mass flowrate m and m* are defined, respectively, as

m = fnll~,

m* = r h l l x ~ ,

(6)

where Ra* = Ra/Ar, and rh is the dimensional mass flowrate. It is noted that m is the conventionally defined non-dimensional mass flowrate. However, as our results will show, use of m* results in an aspect-ratio-independent correlation with Ra*.

416

A.S. Kheireddine et al

Equations ( 1 ) - ( 4 ) and the associated boundary conditions comprise the mathematical statement of the problem. Due to non-linearity and coupling of energy and momentum equations, the solution of the problem must be obtained numerically. In the present work, a finite volume method has been used to obtain the algebraic equations that govern the values of u, v, p, and 0 at grid points in a staggered numerical grid. t8 The mass, momentum and energy equations, Eqs. (1)-(4), are discretized on a numerical grid shown in Fig. 1. The resulting algebraic equation can be written in the following general form: [(puq~)e - (pu)edpt,] - [ ( p u q b ) w - (pU)wC~e]

(7)

+ [(pu~b)n - (pU)nq~p] - [(pu~b)~ - (pu)~$p] = S t r A y , where th represents u, v, p, and 0, respectively, and S, is the source term in Eqs. (1)-(4). The above algebraic equation can be solved provided that the unknowns u, v, p, and 0 are interpolated. In the present calculation, the second upwind scheme is used to perform this interpolation. The coupled mass, momentum and energy equations are solved by a semi-implicit method known in the literature as SIMPLE/8 Equation (7) for any variable ~b at grid point P may be rewritten as Ap~be = ~A~ndPlVn + S , ,

(8)

NB

where Ae and A s s are coefficients containing the convection and diffusion contribution in Eq. (7). The subscript N B refers to neighboring points (e.g. e, w, n and s). The solution of Eq. (8) is obtained by using a line-by-line solver in which the equations along a single grid line in the x or y directions are solved simultaneously by using the tridiagonal algorithm. More detail of this well documented procedure can be found in Patankar? 8 DISCUSSION OF RESULTS The numerical results in this study were obtained for 103 --< Ra <- 105 and for the channel aspect ratio (Ar) in the range 5 - A r <- 20. Prior to obtaining numerical results, a detailed investigation was conducted to assess effects of the grid size on various local and global flow parameters. Due to symmetric boundary conditions, only one half of the flow domain was considered for the numerical grid. We were initially interested in finding the optimal location of the free boundary ABCDA. Clearly, choosing the boundary as far away as possible from the channel walls is desirable to ensure the application of undisturbed ambient pressure boundary condition. However, from a numerical standpoint, this would mean excessive computational effort. Consequently, results were obtained for several configurations involving different locations of the free boundary ABCDA to establish its optimal location for the present numerical procedure. Although the grid independence of numerical results was demonstrated for several channel geometries (with different aspect ratios, Rayleigh numbers and boundary conditions), we present here results for the channel with aspect ratio of 10. The pressure losses at the entrance and exit are characterized by the vent-loss coefficient K which is defined by the following relation for the inlet vent: K~( p.u~ / 2 ) = [ P . + ( 1/ 2 )p~ou2.] - [ P , + ( 1/ 2 )pu2~ ] ,

(9)

where ui is the mean velocity at the entrance section, P® is the ambient pressure, and u. is taken as zero by definition. The variation of local inlet vent-loss coefficient (Ki) for the isothermal boundary condition and for Ra = l0 s is shown in Fig. 2 for three numerical grids (92 x 198, 70 x 140, and 47 x 100). It is noted that the variation of K~ across the channel width is essentially grid independent. The grid with 70 x 140 points yields results that are nearly the same as those obtained from the 92 x 198 grid, and it has been used for further calculations in this study. Results for the exit vent-loss coefficient (K,), calculated from a relationship similar to Eq. (6), are shown in Fig. 3. It is noted that the exit pressure loss results are also grid independent with the 70 x 140 grid. It should be indicated that all these results were obtained with the free pressure boundary placed at a distance of 4L from the walls in both x and y directions.

Pressure loss coefficient and induced mass flux for larninal natural convective flow in a vertical channel

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The effect of free pressure boundary location is shown in Figs. 4 and 5 where Ki and Ke are plotted for the case of Ra = 105 and Ar= 10. The free pressure boundary is placed at distance of 2l, 3l and 41 respectively from the solid walls for the three cases considered in Figs. 4 and 5. It can be seen that locating the pressure free boundary at a distance of 2l does not produce grid independent results for Ki but yields grid independent results for Ke. However, the difference between 3l and 41 locations is practically insignificant. In the present study, we have used the 4l free boundary location for all further calculations. Figure 6 shows the variation of average Nusselt number with Ra* (=- Ra/Ar) for the isothermal case. We note that numerical results for various aspect ratios are slightly lower than the results reported by Aihara. 6 This is primarily due to inclusion of entrance losses in the present study which were not included in the study by Aihara. ~ At low Ra*, the boundary layer results of Aihara deviate more significantly from the solutions of Navier-Stokes equations presented here. The variation of non-dimensional mass flowrate (m) induced through the channel, for the isothermal

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Fig. 4. Effect of pressure boundary location on inlet vent loss coefficient. 2.0

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Fig. 5. Effect of pressure boundary location on exit vent loss coefficient.

boundary condition case, is shown in Fig. 7 as a function of Ra and Ar. It is interesting to note that for aspect ratios of 10 and 20, the non-dimensional mass flowrate (m) shows a maximum around 8,000 and 10,000 respectively. This also corresponds, as shown in Fig. 8, with the minimum value of Kt for these aspect ratios. For the aspect ratio of 5.0, the mass flowrate shows only decreasing values as Ra increases. This is also consistent with the fact that in Fig. 8, a monotomically increasing value of K, for Ar = 5 is observed for increasing value of Ra. We have also plotted in Fig. 9 the mass flowrate m* = m/x/Ar as a function of Ra* for the isothermal case. It is noted that in this new parametric domain, all data points in Fig. 7 nearly collapse onto a single curve which has a broad maximum around Ra* = 10,000. For comparison, results obtained by Borgers and Akbari 9 are also shown in Fig. 9. We observe that numerical results for m* from the present study are slightly lower than those obtained from the boundary layer calculation of Ref. 9. It should be pointed out that Ref. 9 does not include entrance and exit losses in the calculations, and this explains the deviation between the two results. It is interesting to note that the differences between the r..i.-

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calculations of Ref. 3 and our results narrow at higher values of Ra*, where the boundary-layer assumption is more likely to be valid, but widens in the low-Ra* region where elliptic-flow effects dominate. Fully-developed flow results 5 are also shown in Fig. 9. At low Ra*, the m* results from the present study approach asymptotically the results for the fully-developed case. However, it is interesting to note that the present results in the low Ra* range somewhat underestimate m* values predicted from the fully-developed flow analysis. This is due to the fact that the entrance/exit losses incorporated in the present study are not included in the fully developed flow analysis. Figure 10 shows the variation of m* with Ar* for the constant heat flux case. It is again noted that the present results for m* approach an asymptotic limit, somewhat lower than the one predicted by the fully-developed flow analysis, primarily due to absence of entrance losses in the fully developed flow analysis. Streamline and temperature contours for a channel of aspect ratio of 10 are shown in Fig. 11. For

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low Rayleigh numbers, we note that the temperature field becomes quickly fully-developed after a short entrance region. In contrast, the Ra = 105 case shows a thermally developing region with a distinct thermal boundary layer regime. Since the boundary layer approximation becomes invalid at very low values of Rayleigh numbers, use of full Navier-Stokes equations becomes more appropriate to calculate heat transfer characteristics especially in the channel entrance region. In fact, this may also explain the fact that although present results are in better accord with results from the boundary calculations 6.9 at higher Rayleigh numbers, they show increased deviation at lower Rayleigh number. The streamline patterns for Ra = 105 also show a developing boundary layer type flow regime which draws in air strongly from the ambient. In contrast, the Ra = 10 2 case, due to lower buoyancy force, yields a weaker convective current through the channel. Figure 8 shows the variation of the total vent-loss coefficient (K,) as a function of Rayleigh number and aspect ratio for the isothermal case. The total loss coefficient is defined as the sum of the inlet and

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Fig. 1I. Numerical results for isotherms and streamlines. exit vent losses. Two trends are evident from the figure. As the aspect ratio increases, for a given Rayleigh number, the K, value increases. However, the increase is more pronounced in the lower Rayleigh number range, and is smaller in the higher Rayleigh number range. The second trend indicates that for aspect ratios of 10 and 20, the K, value shows a minimum at Rayleigh numbers approximately equal to 8,000 and 10,000 respectively. In contrast, the Ar = 5 case shows a monotonically increasing value of K, with increasing Ra value. Figure 12 shows the variation of K* =- K,/Ar °1° with Ra* for both isothermal and constant heat flux boundary condition cases. It is noted that the scatter in numerical data in Fig. 8 is substantially reduced by correlating K,/Ar °]° with the new variable, Ra*. In fact, all data points for both heat flux and temperature boundary conditions nearly collapse on to a curve K,/Ar ° t ° = 1.607. This is an interesting result since it points to the fact that K* is essentially independent of Ra* and has only weak dependence on the aspect ratio.

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CONCLUSIONS Results in the present study for laminar natural convective flow in a heated vertical channel show that flow patterns and heat transfer characteristics are governed by both Rayleigh number and aspect ratio. The total vent-loss coefficient has been determined for the range of parameters 103 -< Ra <- 105 and 5 <- A r <- 20. At high Rayleigh numbers, the total vent-loss coefficient is a weak function of aspect ratio. The A r = 5 case shows a monotonically increasing functional dependence of K, on Rayleigh number. In contrast, the cases for A r = 10 and 20 show a minimum in the value of K, over the range of Rayleigh numbers considered in this study. It is also observed that when K, I A r °'~° is correlated with Ra*, all data points for various aspect ratios, Ra and boundary conditions nearly collapse on a horizontal line, described by the relation K, I A r ° ~ ° = 1.607. The non-dimensional mass flowrate varies with both Ra and aspect ratio. For the cases involving A r = 10 and 20, the mass flowrate is maximized at values of Ra where K, exhibits minimum values. The case of A r = 5 does not show an extremum, and the non-dimensional mass flowrate decreases as Ra increases. It is also shown that non-dimensional mass flowrate m* correlates with only one variable, namely Ra* for both isothermal as well as wall heat flux boundary conditions. The present study also points to the need of using an outer free pressure boundary for problems involving natural convection phenomenon in open cavity configurations. Application of ambient pressure boundary condition at the inlet and exit sections can yield physically unrealistic (oscillatory) results, thus indicating that location of free boundary can affect the computation of actual phenomenon and produce spurious results if care is not exercised. For the geometry considered here, calculated results are independent of free boundary location when it is placed at a distance greater than four times the channel width. NOMENCLATURE Ar = Aspect ratio, H/l

Cp = Specific heat (kJ/kg K) g = Acceleration due to gravity H = Channel height (m) K = Vent-loss coefficient K* = Modified vent-loss coefficient, K* =

KI(Ar °.l°)

k = Thermal conductivity L = Dimensionless height, L = PrH/l( 1IRa)

l = Channel width (m) m = Dimensionless mass flow rate, m = ml O(RalPr ) jl2

m = Mass flow rate (kg/s m) m* = M o ~ e d mass flow rate, m* = _ _ ml ~Ar Nu = Average Nusselt number, Nu = qlK(Th - T~)I 1 Pr = Prandtl number

Pressure loss coefficient and induced mass flux for laminal natural convective flow in a vertical channel q - - A v e r a g e heat transfer rate from the wall Ra = Raleigh number, Ra = Cpp2g[JATl2/kix Ra* = Modified Rayleigh number, Ra* = Ra/Ar T = Te mperatu re (K) u = Dimensionless velocity along channel v = Dimensionless velocity across channel

423

v = Kinematic viscosity (m2/s) p = Density (kg/m 3) 0 = Dimensionless temperature, 0 = T - TJTh -- T~

Subscripts i = Inlet e = Exit or east h = Hot surface (wall) NB = Neighboring point

Greek letters

n= s= w= t= ~ =

/3 = Coefficient of thermal expansion (K-~) tx = Dynamic viscosity (N-s/m 2)

North South West Total Ambient condition

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Elenbaas, W., Physica, 1942, 9, 1. Bodia, J. and Osterle, J., J. Heat Transfer, 1962, 84, 40. Aihara T., Heat Transfer Jpn Res., 1986, 15, 69. Engel, R. and Muller, K., ASME Paper No. 67-HT-16,1967. Aung, W., Int. J. Heat Mass Transfer 15, 1977 (1972). Aihara, T. Trans. Jap. Soc. Mech. Engnr 29, 903 (1963). Aung, W., Fletcher, L. and Sernas, V., Int. J. Heat Mass Transfer, 1972, 15, 2293. Carpenter, J., Briggs, D. and Sernas, V., ASME J. Heat Transfer, 1976, 98, 95. Bogers, T. R. and Akbari, H., Sol. Energy, 1979, 22, 165. Ostrach, S. Laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperature, NACA TN 2863, Washington, D.C. 1952. Peterson, G. P. and Ortega, A., Adv. Heat Transfer, 1990, 20, 181. Sparrow, E. M. and Bahrami, P. A., J. Heat Transfer, 1980, 102, 221. Tichy, J. A., J. Sol. Energy Engng, 1983, 105, 187. Oseen, C. W., Neue Methoden und Ergebnisse in der Hydrodynamik. Akademischer Verlag, Leipzig, 1927. Chen, D., Chaturvedi, S. K. and Mohieldin, T. O., Energy--The International Journal, 1994, 19, 259. Kettleborough, C. F., Int. J. Heat and Mass Transfer 1972, 15, 883. Nakamura, H., Asako, Y. and Naitou, T., Num. Heat-Transfer 1982, 5, 95. Patankar, S. V., Nu. Heat-transfer and Fluid Flow. McGraw-Hill, New York, 1980.