272c_07p

December 2002 The Arabian Journal for Science and Engineering, Volume 27, Number 2C. 149

الخالصــة

سخان والحمل الحراري داخل وعاء مربع المقطع وساطة نتقال الحرارة باتعنى هذه الدراسة ب السطح المقابل عند درجة حرارة منخفضة يكون بينما ،وجهت على أحد األبـبعاد مثمحدود األ

لة التغير مع الزمن في حاعدثنائية الب الطاقة والحركة والكتلة حفظتم حل معادالت وقد .وثابتةن رقم نوسلت المحلي تنخفض قيمته كلما زادت المسافة من حافة أأوضحت النتائج و بالطرق العددية

.عتماد كل من خطوط الحرارة والتدفق على مكان السخاناالسخان عند ثبات رقم رالى ، وكذلك المدى يالدراسة تغطو. ن ستنباط عالقة تربط بين كل من رقم نوسلت ورالى ووضع السخااولقد تم

. ١٠٦ الى ١٠٣من لرقم رالى

LAMINAR NATURAL CONVECTION FROM AN ISOFLUX DISCRETE HEATER

IN A VERTICAL CAVITY

A.M. Al-Bahi Aeronautical Engineering Department

A.M. Radhwan and G.M. Zaki* Thermal Engineering Department

King Abdulaziz University Jeddah, Saudi Arabia

*Address for Correspondence: King Abdulaziz University P.O. Box 80204, Jeddah 21587 Saudi Arabia e-mail: [email protected]

A.M. Al-Bahi, A.M. Radhwan, and G.M. Zaki

150 The Arabian Journal for Science and Engineering, Volume 27, Number 2C. December 2002

ABSTRACT

Laminar natural convection heat transfer in an air filled vertical square cavity differentially heated with a single isoflux discrete heater on one wall with top and bottom adiabatic surfaces is numerically studied. The coupled unsteady two-dimensional conservation equations are solved by employing a forward time central space implicit finite difference scheme. Numerical results reveal that the local Nusselt number decreases along the length of the heater at constant value of the modified Rayleigh number (Ra*) with slight enhancement near the trailing edge. The heater location for the maximum heat dissipation rate is Rayleigh number dependent, which is in agreement with previous experimental and numerical results. A correlation for this location is obtained in addition to a relation for the dependence of the average Nusselt number on the modified Rayleigh number and the location parameter (s/L). Average Nusselt number variation for s/L = 0.25 to 0.75 and Ra* =103 to 106 is presented and compared to the available correlations for full contact and discretely heated enclosures.

Keywords: natural convection, square enclosure, discrete, laminar



LAMINAR NATURAL CONVECTION FROM AN ISOFLUX DISCRETE HEATER IN A VERTICAL CAVITY

1. INTRODUCTION

Natural convection in differentially heated enclosures plays an important role in many engineering applications, where it provides a means of heat transfer without the need for fans or pumps. Cooling of electronic chips [1], ventilation of buildings [2], thermal performance of solar collectors [3], and geophysics [4] are just some examples for engineering applications where basically a cavity is differentially heated. The recent advances in electronic circuit technology as well as the development of large integrated chips with increased heat dissipation rates made heat transfer a key factor in circuit board design. Cooling of these boards is fundamentally addressed as convective heat transfer in enclosures [5, 6] and has been extensively studied for the past two decades. These studies focus on the thermoconvection performance within a cavity filled with a fluid and asymmetrically heated for different geometrical parameters (aspect ratio), fluid properties (Prandtl number), heating mode (isoflux or isothermal), and a selected set of boundary conditions. The two-dimensional convection within a square cavity heated on one vertical side and cooled on the opposite side [7–9] is currently considered a reference or bench-mark solution for verifying other solution procedures. The problem has been extended to establish additional bench-mark solutions for fluids with temperature dependent viscosity [10], fluids with heat generation [11], and time dependent wall functions [12, 13].

Practically, Rayleigh number and geometric features characterize the heat transfer problem in cavities. Correlations in simple power-law form have been developed, but the constants showed dependency on the method and accuracy of solution. In Table 1 the correlation constants from Henkes and Hoogendoorn [14], Markatos and Pericleous [8], and Barakos et al. [9] are presented along with their validity ranges. The importance of the convection phenomena in cavities is revealed by the abundant studies reviewed by Ostrach [15] and Yang [16]. Studies of more relevance to circuit board cooling are characterized by heating over a segment of the wall and referred to as discrete heating. The work of Chu et al. [17] is one of the early contributions, where the temperature and velocity fields were computed for a range of different parameters (aspect ratio from 0.4 to 5, l/W = 0.2 to 1 and RaL ≤ 105). Though data was limited to isothermal heating, the complex dependence of Nusselt number on Rayleigh number and other parameters apparently rules out obtaining general correlations suitable for design. For example, the heater location for the maximum heat transfer rates shifts towards the bottom of the enclosure as Rayleigh number increases. This effect is attributed to the large space available for flow circulation when the heater is located close to the bottom. This circulation effect is more pronounced at high Rayleigh number, Ra = 105 [17], and is confirmed by the experimental results of Turner and Flack [18] and for an isothermal heater mounted on an isothermal heat sink, Radhwan and Zaki [19].

The effect of the discrete heater location has also been investigated for a single isoflux and isothermal heater, Refai and Yovanovich [20, 21], for two dual heaters Chadwick et al. [22] and for multiple heaters, Keyhani et al. [23]. The effect of the aspect ratio Ar on the heat transfer rate has also been studied. The results of Chu et al. [17], for Ar = 0.4–5, showed that Nusselt number generally decreases for wide enclosures at constant Rayleigh number. This trend was insignificant as reported by Sernas and Lee [24] for aspect ratios between 0.4 and 1. Although two-dimensional convection has been investigated numerically and experimentally, development of a correlation that combines the basic parameters (heater size, location, aspect ratio, and Rayleigh number) is a challenging problem because of the complex parametric interdependency and the arbitrary choice of the characteristic length in both Rayleigh and Nusselt numbers. The distance from the bottom of the enclosure [23], the width of the heater [25], the distance from the leading edge of the discrete heater [22], and (l/Ar) [20, 21] are just examples.

Therefore, it is uncertain to interpret or extrapolate data and/or correlations of different investigators. The correlations developed on basis of two-dimensional convection in cavities are summarized in Table 1 along with their validity range.

The preceding studies focus on two-dimensional convection flow in cavities with asymmetric discrete heating. Though three-dimensional convection received less attention a few correlations are available [26–28].



Table 1. Convection Correlations for Full Contact and Discretely Heated Enclosures.

Author [Ref] Study Configuration Heat transfer correlation Range of parameters

A1 Square enclosure, isothermal, full contact heated wall at Th and opposite wall at Tc

bL LNu a Ra= Ar = 1

Pr = 0.71

a b

Markatos and Pericleous [8]

0.143 0.299 103 ≤ RaL ≤ 106

Henkes & Hoogendooren [14]

0.304 0.25 103 ≤ RaL ≤ 106

Barakos et al. [9] 0.301 0.25 103 ≤ RaL ≤ 1011

Turner and Flack [18] E Rectangular enclosure, single and dual isothermal heaters

NuL = c1 GrLc2

l/L c1 c2

0.125 0.45 0.33

0.25 0.069 0.32

0.5 0.097 0.31

Ar = 1,

s/L = 0.5

5×106 ≤ Gr L ≤ 9×106

Pr = 0.71

Refai and Yovanovich [20, 21]

A Rectangular enclosure, isoflux or isothermal single discrete heater, results for L=W only

isoflux heater2

Nul′ = [ε1.511 (ε1.294)m

+ {0.21 ε–0.288 (Ra*)0.221

[1.261(0.7)n ε–0.012]n}8.5]0.118

m = 0 for heater at the center

m = 1 for top or bottom positions

ε = l/L

l′ = (lW/L)

0 ≤ Ra*l ′ < 106 ε4

n = 0 for 0.5 ≤ ε ≤ 1

n = 1 for ε = 0.25

Pr = 0.72

Chadwick et al. [22] A&E Rectangular enclosure, single and dual isoflux heaters

21

cy yNu c Gr′ ′=

s/L c1 c2

0.2 0.526 0.192

0.5 0.454 0.199

0.8 0.438 0.193

Ar = 5

l/L = 5

0 .133×104 < Gry′ ≤ 5×105

y′ measured from the leading edge of the heater

Keyhani et al. [23] E 11 discrete isoflux heaters mounted on an adiabatic wall.

Nuy = 1.009 Ray*0.1805 Ar = 16.5

Pr ~150

6 12*10 5 10yRa x< <

Ho and Chang [25] A&E Rectangular enclosure, four isoflux discrete heaters

wNu = i i* b ci w a Ra Ar

w = width of the enclosure

ai, bi, ci, see [25] Table 2

Ar = 1–10

Ra*w = 104 – 106

Pr = 0.71

1 A: Analytical or numerical study, E: Experimental 2 Another correlation is available for isothermal surfaces [21]



The objective of the present paper is to study the dependency of the heat transfer performance on the heater location for a reference case of a single isoflux discrete heater in a square enclosure and to correlate the results. The effect of location on the spatial Nusselt number variation along the vertical heat sink is also studied. The time-dependent two-dimensional mass, momentum and energy conservation equations are solved numerically. Second order implicit finite difference approximation with alternate direction scheme (ADI) is employed to solve the unsteady state equations.

2. MATHEMATICAL MODEL

The configuration under consideration is shown schematically in Figure 1. It is an enclosure of height L and width W (aspect ratio Ar = L/W ). The horizontal surfaces are assumed adiabatic while the vertical right wall is a heat sink at a constant temperature Tc .

Figure 1. Schematic of enclosure configuration and the solution grid.

The left vertical wall has a discrete flush-mounted heater of length l, at a distance s from its center to the bottom of the cavity. The heater represents a local isoflux heat source, while the remainder of the wall is adiabatic. The fluid inside the enclosure is assumed Newtonian and incompressible with constant properties except for the density in the buoyancy term of the Y momentum equation (Boussinesq approximation). For two dimensional laminar free convection flow, by eliminating the pressure between the X and Y momentum equations (by cross differentiation) the mass, momentum, and energy equations are presented in the dimensionless stream function, vorticity, and energy formulation as:

2 2

2 2X Y∂ ψ ∂ ψ

+ = −ω∂ ∂

(1)

2 2

2 2*

PrRaU V

X Y XX Y∂ω ∂ω ∂ω ∂ ω ∂ ω ∂θ

+ + = + +∂τ ∂ ∂ ∂∂ ∂

(2)

2 2

2 21Pr

U VX Y X Y

∂θ ∂θ ∂θ ∂ θ ∂ θ+ + = + ∂τ ∂ ∂ ∂ ∂

. (3)



The initial and boundary conditions are:

initial conditions:

At τ = 0, 0 ≤ X ≤ 1, 0 ≤ Y ≤ 1

U = V = 0, ψ = 0, θ = 0

adiabatic horizontal surfaces:

at 0 < X < 1, Y = 0 and Y = 1

U = V = 0, Y∂θ∂

= 0, ψ = 0

isothermal wall:

at 0 ≤ Y ≤ 1, X = W/L = 1/Ar

U = V = 0, θ = 0, ψ = 0

heat source wall:

at (s – l/2)/L ≤ Y ≤ (s + l/2)/L, X = 0

U = V = 0, X∂θ∂

= –1, ψ = 0

and at 0 ≤ Y < (s – l/2)/L, (s + l/2)/L < Y ≤ 1 and X = 0

U = V = 0, X∂θ∂

= 0, ψ = 0

The dimensionless variables are defined in the nomenclature. The modified Rayleigh and Nusselt numbers are based on the enclosure width W, for a square enclosure (Ar = 1)W = L.

The local heat transfer rate is obtained from the energy balance on the heater section and along the cold isothermal wall, from which Nusselt numbers are:

0h

X=

qL 1Nu = k T

=∆ θ

l1 ≤ Y ≤ l2 . (4)

Local Nusselt number at the isothermal wall is:

1c

X=Nu

X∂θ

= −∂

0 ≤ Y ≤ 1. (5)

The average Nusselt number along the discrete heater length is:

hqLNu

k T=

∆. (6)

A forward time central space implicit finite difference scheme is used to solve the governing equations. The numerical solution is obtained by employing a stabilizing correction splitting method to improve convergence. This splitting (ADI) scheme has been extensively used in its original form, Peaceman and Rachford [29] and in its generalized form [30–32]. The governing equations were discretized on a nonuniform 41×41 grid (Figure 1), in which the mesh is finer near the vertical walls to better resolve the hydrodynamic and thermal boundary layers there. The computation of the stream function, the temperature, and the vorticity were carried out using second order difference approximations. Central difference relations were used for the diffusion terms of the vorticity transport equation, while the convective term was



presented by an upwind difference scheme [33]. No slip boundary conditions were imposed to solve the stream function equation. A second order central difference approximation was used to obtain the second derivatives of the stream function at the boundaries, which are necessary to have an acceptable solution of the vorticity equation. The values of ψ outside the computational domain (image points) were calculated using Taylor second order expansion. In particular for equal nodes grid this technique leads to Koskova condition [34]. The steady state solution was obtained as the limit of the transient calculations. The stream function equation, which is highly coupled with the vorticity equation, was satisfied for every time step applying an iterative procedure. Five iterations were found to be sufficient for a dimensionless time step of 5×10–4. The criterion for convergence was examined for each variable through the normalized residual R, such that:

1, ,,

,,

Max

Max

n ni j i ji j

ni ji j

R

+

φ

φ − φ= ≤ ξ

φ, (7)

where φ is any variable (θ, ψ, or ω) and ξφ is the accuracy limit set to 2×10–8 for Ra* = 106 with a nonuniform 41 × 41 grid and n is the iteration level.

The developed computer code for the present problem has been verified by performing calculations for natural convection in a square enclosure and comparing the results to previous bench-mark solutions of de Vahl Davis [7], Markatos and Pericleous [8], and Radhwan and Zaki [19]. The streamlines and isotherm patterns obtained by the present solution procedure were found to be in good agreement with the published data [7, 8, and 19]. Details of this comparison for which the boundary conditions are of the first kind (θ = 1 at X = 0 and 0 at X = 1) are not presented for brevity. Instead, the values of the average Nusselt number are given in Table 2. The difference between the present results and the average of the three solutions is within 1%. Further validation was carried out for an air filled rectangular enclosure with an isoflux flush mounted single heater, Ar = 5, s/L=0.5, l/L = 0.133, and Gr = 5.16×105 [22]. The present isotherms and streamlines are in agreement with those obtained by the SIMPLER algorithm [22], as seen in Figure 2. Comparing local values of θ and ψ was not performed as they were not given in [22]. For this validity test a 61×61 nonuniform grid was used.

Present Chadwick [22]

Figure 2. Verification of the present solution as compared to that of Chadwick [22], L/W = 5, l/L = 0.13, and Gr = 5.16×10 5.



Table 2. Average Nusselt Number for an Air Filled Square Cavity with Isothermal Vertical Walls (Verification of Present Solution Algorithm).

Average Nusselt number, NuL RaL

de Vahl Davis [7] Markatos [8] Radhwan [19] Present

103 1.118 1.108 1.149 1.114

105 4.519 4.43 4.515 4.434

3. NUMERICAL RESULTS

The developed code is used to study the thermal convection in a vertical square enclosure (Ar = 1) filled with air (Pr = 0.71). The buoyancy assisted flow is created by a flush-mounted strip heater, l/L = 0.125 fixed at s/L=0.25, 0.5, and 0.75 (Figure 1). These parameters are selected to establish basic data for isoflux thin discrete heater in a square enclosure, a condition that has not been considered in previous studies (Table 1) and to focus on the effect of heater location on local and average heat transfer rates.

3.1. Isotherms and Streamlines

Figure 3 shows the effect of heater location on the development of isotherms and streamlines for Ra*=103 and 106. The computed isotherms for all locations are clustered near the hot surface indicating boundary layers build up. The parallel isothermal lines away from the heater at low Rayleigh number, Ra*=103, indicate conduction dominated mode of heat transfer, though convection is developed in the layer adjacent to the heating strip. The streamlines for all cases form a single cell filling the cavity and the effect of the heater location causes distortion of this cell. The form of distortion and the maximum value of ψ are location and Rayleigh number dependent. The magnitude of ψmax determines the extent of circulation, as seen in Figure 3, the highest value is obtained near the bottom (s/L=0.25) and the lowest value is at s/L= 0.75. This result is in agreement with the previous studies where maximum heat transfer is obtained for heaters placed close to the bottom of the enclosure.

As seen from the isotherms in Figure 3, at Ra*=103, weak convection is confined in the left side zone where the heat source is mounted. The parallel vertical isotherms for all heater positions reveal that the heat transfer across the enclosure is conduction dominated in particular within the region affected by the cold wall. Increase in Rayleigh number, up to Ra*=106, causes the convection heat transfer to be more confined along the heated vertical wall and the upper section of the cold wall where the isotherms cluster along these two sections irrespective of the heater location.

The temperature variation in the X direction gives an indication to the mode of heat transfer. The temperature distributions at the mid-height of the heater for the different locations are shown in Figure 4 for Ra*=103 and 106. The dimensionless surface temperature, θ (at X = 0), which is inversely proportional to Nusselt number (Equation (4)) decreases with increasing Rayleigh number. The figure depicts the change of the heat transfer mode along the horizontal plane, where at Ra*=103, the temperature gradient decreases to a constant linear distribution at X ≥ 0.3. At this distance the buoyancy effects diminish and conduction is dominating. On the other hand at Ra*=106 the gradient decreases to reach nearly zero near the center of the enclosure width. This trend is the same for all heater locations and has been observed in previous studies for isothermal full contact heaters [9] and isoflux discrete heater [20]. For the laminar convection, Ra*=106, θ at s/L =0.75 is the highest compared to other locations indicating less efficient heat transfer because of the limited space for circulation. The effect of location for low Ra* is different (Figure 4), where θ is the highest at s/L = 0.25 indicating poor local heat transfer along the mid-plane. The symmetrical isotherms and streamlines at s/L = 0.5 with ψmax = 0.14 indicates relatively uniform heat transfer across the enclosure. In general the horizontal mid-plane temperature variation at X=0 is an indication to the dominating heat transfer mechanism but the effectiveness of buoyancy driven cooling is determined by the magnitude of Nusselt number, which is discussed in the following section.



14.0max a 13.0max 12.0max

46.11max 9max 7max

Ra*

= 1

03

Ra

* =

10

6

s/L = 0.25 s/L = 0.5 s/L = 0.75

Figure 3. Isotherms and streamlines dependency on the heater location and Rayleigh number.

Figure 4. Horizontal temperature variation across the enclosure at the middle height of the heater for different location parameter, s/L and Rayleigh numbers.



3.2. Nusselt Number

The local Nusselt number variations along the heating element, Nuh, and the cold wall, Nuc (at X = 1), are shown in Figure 5. The variation of Nuh along the heater length shows that, at high modified Rayleigh numbers (Ra* = 106), the highest heat transfer rate takes place at the leading edge of the heating strip regardless of its position. This effect is less pronounced at low Ra*. Though the local Nusselt number decreases from a maximum at the leading edge towards the trailing edge, there is a slight enhancement in the local Nusselt number at the trailing edge. Results of Heindel et al. [26] showed a similar trend for isoflux discrete heaters for both two and three dimensional analysis. This may be explained as a result of disturbances introduced by mixing of hot air leaving the heater edge and the relatively cool air in the adiabatic zone beyond the trailing edge. The predicted Nuh variation along the heater is in agreement with the experimental data of Chadwick et al. [22] although this trailing edge effect does not appear in their experimental results.

Figure 5. Local Nusselt number variation along the isoflux discrete heater and the isothermal opposite cold wall.

The corresponding variation of Nuc along the vertical heat sink is shown in the same figure for Ra* = 103 and 106 for three heater locations. For low values of Ra*, Nuc is low and nearly uniform indicating a dominating conduction regime. Even though, Nuc is slightly affected by the heater location, where at s/L = 0.25 Nusselt number Nuc is slightly higher at the bottom of the enclosure (Nuc = 0.199 at Y ≅ 0 compared to the value of Nuc = 0.164 at Y = 1). This trend is reversed when the heater is located at Y = 0.75, for which Nuc = 0.134 at Y ≅ 0 and Nuc = 0.184 at Y = 1. Increase in Rayleigh number changes this distribution where the effective cooling zone with high Nuc is concentrated in the upper section of the cooling wall irrespective of the heater position. The value of Nuc along the cold vertical wall decreases towards the bottom (downstream direction). This trend is a direct effect of the boundary layer growth along the hot or cold walls where Nuh starts high at the heater leading edge and decreases monotonically towards the trailing edge. The same occurs during cooling, noting that the flow is downwards along the cooling section.

Figure 6 illustrates the dependency of the average Nusselt number, hNu on the heater location with Ra* as a parameter. It is seen that the position for maximum Nusselt number shifts towards the bottom as Rayleigh number



increases (s/L = 0.6 at Ra* = 104 and s/L = 0.39 at Ra* = 106). This result extends the previous findings of Chu et al. [17] and Turner and Flack [18] for isothermal heaters in a square enclosure and the data of Chadwick et al. [22] for an isoflux heater in a rectangular enclosure. The loci of the maxima of Nu–Ra* variation in Figure 6 presents the relation between s/L for maximum cooling location and Ra*. In an effort to find out this relation, the present data is reproduced for Ra* ≥ 104 in Figure 7 and fitted to

max.Nu

sL

= 1.396 Ra*– 0.093 104 < Ra* ≤ 106 (8)

which is valid for isoflux discrete heater in a square enclosure. The data of Chu et al. [17], for the isothermal discrete heater, is modified for Ra* = Ra Nu and plotted on the same figure, which shows the same trend as that of Equation (8).

The effect of Rayleigh number on the overall heat transfer rate from the discrete heater is presented in Figure 8. The computed average Nusselt number hNu is nearly constant up to Ra* ≤ 104 beyond this limit the average Nusselt number varies with Ra*n. The figure shows that the gradient of the present data is within the range of the previous correlations for full contact isothermal heat source [9, 14] and isoflux heater [22] (Table 1). By employing regression analysis the average Nusselt number has been correlated with Ra* for the heater to fit a power-law form as:

*nhNu a Ra= . (9)

The coefficients a and n showed dependency on the location parameter, Table 3.

The exponent n is slightly higher than the average found by Ho and Chang [25] (0.194–0.206) but the average of the three values is within that of [22] (n = 0.215). To include the effect of the location ratio on Nusselt number a relation is suggested so that the first derivative for maximum hNu leads to Equation (8). Fitting the data to the suggested form yields:

20.307 0.093* *0.973 0.358h

s sNu Ra RaL L

= −

. (10)

Figure 6. Effect of heater location and modified Rayleigh number on the average Nusselt number along the heater.



1E+4 1E+5 1E+6 Ra*

0.1

1.0

s/L

Equation 8

present Eq. 8

present isoflux heating

Chu et al [17] isothermal heating

Figure 7. Dependence of heater location for maximum rate of heat transfer on Rayleigh number.

1E+3 1E+4 1E+5 1E+6 1E+7

Ra*

1

10

Nu

h

s/L = 0.25

s/L = 0.5 present

s/L = 0.75

Refai [20,21] s/L = .5

Chu et al. [17] s/L = .5

full contact isothermal heater (Table.1)

Henkes and Hoogendoorn [14]

Barakos et al [9]

Figure 8. Average Nusselt number variation, compared to available correlations.

Table 3. Coefficients of Equation (9).

s/L a n

0.25 0.488 0.227

0.5 0.582 0.217

0.75 0.609 0.201



Equation (10) predicts the average Nusselt number for a single discrete isoflux heater including the shifting of the peak Nusselt number towards the bottom with Rayleigh number increase. The correlation covers the present range of parameters with maximum deviation of 19%.

In general discrete heating gives higher Nusselt number values compared to full contact heating, Figure 8. The results of Chu et al. [17] were used to obtain average Nusselt numbers for the present configuration l/L = 0.125 and s/L = 0.5. It is seen that the average Nusselt number for the discrete isothermal heater is on the average 38% higher than the corresponding value for full contact isothermal heater, at Ra* = 2×105. Furthermore the present results for isoflux discrete heater is 20% higher than those for isothermal heater under the same conditions (Ra* = 2×105). The only correlations available for single isoflux discrete heater that includes the effect of heater size l/L are those of Refai and Yovanovich [20, 21], Table 1. Unfortunately the correlation is limited to l/L > 0.25 and to a narrow range of Rayleigh number, Ra*L ≤ 106 (l/L)4. The correlation has been modified where the characteristic length L replaces (l/Ar), m = 1, and n = 1 to present the case for which the heater is located at the center, s/L = 0.5.

Figure 8 shows that Refai and Yovanovich [20, 21] correlation under predicts the present results by 10 to 19%. This difference may be attributed to extending the range of validity of the correlation and/or using a refined grid in the present study (stretched 41×41 grid) relative to the 20×20 uniform node distribution. However, Refai and Yovanovich correlation shows an interesting change in gradient at Ra* = 5×104 which marks the onset of convection. This transition occurs at Ra* = 104 for the present solution at s/L = 0.5. The correlation of Turner and Flack [18] (Table 1) is developed for a Grashof number range beyond the present laminar flow analysis, for which Ra* ≤ 106. For a heater size l/L = 0.125 and Ar = 1 the constants are c1 = 0.045 and c2 = 0.33. These give values of NuL between 7.3 and 8.8 for 2.6×107 ≤ Ra* ≤ 5.7×107. These values are less than those expected by the present study, therefore, extrapolation of Turner and Flack correlations [18] for isoflux heating is not recommended but the correlation of Refai [20] is the closest to the present conditions.

4. CONCLUSION

Natural convection in an air filled discretely heated square cavity is numerically studied. The heat source is a flush mounted isoflux strip heater (l/L = 0.125) at different locations (s/L = 0.25–0.75) while the heat sink is a vertical isothermal wall. The top and bottom surfaces are adiabatic. The time dependent two-dimensional conservation equations of mass, momentum, and energy are solved using finite difference scheme. After validation of the method, the streamlines and isotherms for different heater locations and Rayleigh numbers (103–106) were obtained. The results showed that for small heaters the flow is characterized by a single circulation cell, which prolongs to an elliptical shape at high Rayleigh numbers (Ra* = 106) with distortion towards the location of the heater. At low Ra* (103–104), heat transfer by conduction is dominating and Nusselt number is nearly constant.

The local Nusselt number decreases along the heater length from the leading edge with slight enhancement at the trailing edge. This is explained as a result of transition of moving fluid from a hot region at the heater’s edge to a relatively colder fluid at the adiabatic region next to the heater. In addition the heater location for maximum heat transfer was found to be Rayleigh number dependent and a new correlation is developed for this optimum location. Furthermore a correlation for the average Nusselt number in terms of the modified Rayleigh number and the heater location parameter has been obtained. The available correlations for full contact and discrete heating in enclosures are presented and compared to the present results.

NOMENCLATURE

Ar Aspect ratio = L/W cp specific heat at constant pressure, Ws/K kg g gravitational acceleration, m/s2

Gr Grashof number gβ∆TL3/ν2 k thermal conductivity, W/m K



l heater length, m l′ characteristic length (Table 3), lW/L l1, 2 dimensionless distance = [s ∓ l/2]/L L enclosure height, m Nu local Nusselt number, qW/k∆T Nu average Nusselt number, /qW k T∆ Pr Prandtl number, cp µ /k q heat flux, W/m2 R normalized residual Ra Rayleigh number, gβ∆TL3/(να) Ra* modified Rayleigh number gβqL4/(kνα) s distance to center of the heater, m t time, s ∆T temperature difference = T – Tc, K T Temperature, K u, v velocity components in the x and y directions U, V dimensionless velocity components U = uL/ν, V = vL/ν W width of enclosure, m x, y space coordinates in Cartesian system X, Y dimensionless Cartesian coordinates, x/L, y/L y′ distance measured from the heater leading edge, m.

Greek Symbols

α thermal diffusivity, k /ρc cp, m2/s β coefficient of volumetric thermal expansion, K–1 ε ratio between heater length to enclosure height, l/L θ dimensionless temperature, k(T – Tc)/qL µ dynamic viscosity, kg/m s ν kinematic viscosity, m2/s ρ variable density = ρc[1 – β(T – Tc)], kg m–3 ρc density at Tc, kg m–3 φ dummy variable ψ non-dimensional stream function = ψd /ν ω non-dimensional vorticity = ωd L2/ν τ non-dimensional time,= tν/L2

Subscripts

c cold surface d dimensional h hot surface L based on enclosure length

Superscripts

— average value



REFERENCES

[1] S. Sathe and B. Sammakia, “A Review of Recent Developments in Some Practical Aspects of Air-cooled Electronic Packages”, Journal of Heat Transfer, 120 (1998), pp 830–838.

[2] R.K. Caby, A.E. Holdo, and G.P. Hammond, “Natural Convection Heat Transfer Rates in Enclosures”, Energy and Buildings, 27(2) (1998), pp 137–146.

[3] H. Buchberg, I. Catton, and D.K. Edwards, “Natural Convection in Enclosed Spaces — A Review of Application to Solar Energy”, Journal of Heat Transfer, 98 (1976), pp. 182–188.

[4] R. Missner and U. Müller, “A Numerical Investigation of Three-Dimensional Magneto-Convection in Rectangular Cavities”, International Journal of Heat and Mass Transfer, 42 (1999), pp. 1111–1121.

[5] C.T. William and A. Hiroyuki, “Advances in Electronic Packaging”, Proceedings of the ASME/JSME Conference, New York, 1992.

[6] F.P. Incropera, “Convection Heat Transfer in Electronic Equipment Cooling, “Journal of Heat Transfer, 110 (1988), pp. 1097–1111.

[7] G. de Vahl Davis, “Natural Convection of Air in a Square Cavity: A Bench-Mark Numerical Solution”, International Journal of Numerical Methods in Fluids, 3 (1983), pp. 249–264.

[8] N.C. Markatos and K.A. Pericleous, “Laminar and Turbulent Natural Convection in an Enclosure Cavity”, International Journal of Heat and Mass Transfer, 27 (1984), pp. 755–772.

[9] G. Barakos, E Mitsoulis, and D. Assimacopoulos, “Natural Convection Flow in a Square Cavity, Revisited: Laminar and Turbulent Models with Wall Functions”, International Journal for Numerical Methods in Fluids, 18 (1994), pp. 695–719.

[10] R.J Cochran and R.E. Roy, “Comparison of Computational Results for Natural Convection in a Square Enclosure with Variable Viscosity, Bench-Mark Problem, “Proceedings of the 1995 ASME Cong. Part 1, 1995, pp. 1112–1117.

[11] M.M. El-Refaee, M.M. El-Sayed, N.M. Alnajem, and I.E. Megahid, “Steady State Solutions of Buoyancy-Assisted Internal Flows Using a Fast False Implicit Transient Scheme (FITS)”, J. Num. Meth. Heat Fluid Flow, 6 (1996), pp. 3–23.

[12] M. Kazmierczak, “Buoyancy Driven Flow in an Enclosure with Time Periodic Boundary Conditions”, Int. J. Heat Mass Transfer, 35 (1992), pp. 1507–1519.

[13] Q. Xia, K.T. Yang, and D. Mukutmoni, “Effect of Imposed Wall Temperature Oscillations on the Stability of Natural Convection in a Square Enclosure”, Journal of Heat Transfer, 117 (1995), pp. 113–120.

[14] W.M. Henkes and C.J. Hoogendoorn, “Scaling of the Laminar-Natural Convection Flow in a Heated Square Cavity”, International Journal of Heat and Mass Transfer, 11 (1993), pp. 913–2925.

[15] S. Ostrach, “Natural Convection in Enclosures”, Journal of Heat Transfer, 110 (1988), pp. 1175–1190. [16] K.T. Yang, “Natural Convection in Enclosures”, in: Handbook of Single-Phase Convective Heat Transfer, ch. 13. ed. S. Kakac,

R. Shah, and W. Aung. New York: Wiley, 1987. [17] H.H. Chu, S.W. Churchill, and C.V.P. Patterson, “The Effect of Heater Size, Location, Aspect Ratio and Boundary Conditions

on Two Dimensional Laminar Natural Convection in Rectangular Channels”, Journal of Heat Transfer, 98(2) (1976), pp. 194–201.

[18] B.L. Turner and R.D. Flack, “The Experimental Measurement of Natural Convective Heat Transfer in Rectangular Enclosure with Concentrated Energy Sources, Journal of Heat Transfer, 102 (1980), pp. 236–241.

[19] A. Radhwan and G.M. Zaki, “Laminar Convection in a Square Enclosure with Discrete Heating of Vertical Walls”, Journal of Engineering Science, King Abdulaziz University, Jeddah, Saudi Arabia, 12 (1999), pp. 70–77.

[20] G. Refai and M.M. Yovanovich, “Numerical Study of Natural Convection from Discrete Heat Sources in a Vertical Square Enclosure”, AIAA, 28th Aerospace Science Meeting, Reno, Nevada, 1990, paper No.90-0256.

[21] G.A. Refai and M.M. Yovanovich, “Influence of Discrete Heat Source Location on Natural Convection Heat Transfer in a Vertical Square Enclosure”, Journal of Electronic Packaging, 113 (1991), pp. 268–274.

[22] M.L. Chadwick, B.W. Webb, and H.S. Heaton, “Natural Convection from Two-dimensional Discrete Heat Sources in a Rectangular Enclosure”, International Journal of Heat and Mass Transfer, 34(7) (1991), pp. 1679–1693.

[23] M. Keyhani, V. Prasad, and R. Cox, “An Experimental Study of Natural Convection in a Vertical Cavity with Discrete Heat Sources”, Journal of Heat Transfer, 110 (1988), pp. 616–624.

[24] V. Sernas and E.I. Lee, “Heat Transfer in Air Enclosures of Aspect Ratio Less Than One”, ASME Winter Annual Meeting, San Francisco, California, 1978, ASME paper 78-WA/HT-7.

[25] C.J. Ho and J.Y. Chang, “A Study of Natural Convection Heat Transfer in a Vertical Rectangular Enclosure with Two-Dimensional Heating: Effect of Aspect Ratio”, International Journal of Heat and Mass Transfer, 37(6) (1994), pp. 917–925.



[26] T.J. Heindel, S. Ramadhyani, and F.P. Incropera, “Laminar Natural Convection in Discretely Heated Cavity: I. Assessment of Three-Dimensional Effects”, Journal of Heat Transfer, 117 (1995), pp. 902–909.

[27] T.J. Heindel, S. Ramadhyani, and F.P. Incropera, “Laminar Natural Convection in a Discretely Heated Cavity: II. Comparisons of Experimental and Theoretical Results”, Journal of Heat Transfer, 117 (1995), pp. 910–917.

[28] K.A. Park and A.E. Bergles, “Natural Convection Heat Transfer Characteristics of Simulated Microelectronic Chips”, Journal of Heat Transfer, 109 (1987), pp. 90–96.

[29] D.W. Peaceman and H.H. Rachford, “The Numerical Solution of Parabolic and Elliptic Equations”, Journal of the Society of Industrial and Applied Mathematics, 3 (1955), pp. 28–35.

[30] M.M. Abdelrahman and A.M. Al-Bahi, “Numerical Solution of the Unsteady Incompressible, Navier-Stokes Equations in Generalized Non-Orthogonal Curvilinear Coordinates”, Journal of Engineering and Applied Sci., 41(2) (1996), pp. 419–434.

[31] A.M. Al-Bahi and M.M. Abdelrahman, “On the Solution of Unsteady Viscous Incompressible Flow Problems”, 34th Aerospace Sciences Meeting, Reno, Nevada, 1996, AIAA-Paper 96-0826.

[32] J. Douglas and J.E. Gunn, “A General Formulation of Alternating Direction Method, Part I, Parabolic and Hyperbolic Problems”, Numerical Mathematics, 6 (1994), pp. 428–453.

[33] W. Shyy, S. Thakur, and J. Wright, “Second-Order Upwind and Central Difference Schemes for Recirculating Flow Computation”, Journal of the American Institute of Aeronautics and Astronautics, 30(4) (1992), pp. 923–932.

[34] E.F. Nogotov, Applications of Numerical Heat Transfer. New York: Mc-Graw Hill, 1976, pp. 100–109.

Paper Received 3 February 2002; Revised 25 June 2002; Accepted 23 October 2002.

Documents

272c_07p