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Revisit on natural convection from vertical isothermal plate arrays--effects of extra plume buoyancy
International Journal of Thermal Sciences 120 (2017) 263e272
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Revisit on natural convection from vertical isothermal plate arrays–effects of extra plume buoyancy* Shwin-Chung Wong*, Shih-Han Chu Department of Power Mechanical Engineering, National Tsing Hua University, Hsin-Chu 300, Taiwan, ROC
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 January 2017 Received in revised form 13 May 2017 Accepted 18 June 2017
In this study, natural convection from vertical isothermal parallel-plate arrays is examined using 2-D steady-state numerical analysis. Extended computation domains encompassing the single- or multichannel plate arrays are adopted. Also investigated are the consequences of using computation domains without the inlet extension or/and the outlet extension. The plate height is fixed at 100 mm, the plate spacing is 7 mm, and the plate thickness is 1 mm. The Elenbaas Rayleigh number Ra’ is fixed at 62.8. The results show that setting the inlet boundary at the entrance of the channel would ignore the separation formed near the outer entrance corner and the associated flow resistance; while setting the outlet boundary at the exit of the channel would omit the extra buoyancy provided by the hot plume above the arrays. In multi-channel arrays, the average heat transfer coefficients in different individual channels (hs) are different from each other. The h is the highest in the central channel and the lowest in the edge channel. Except for the edge channels, the h values are higher than that of a single channel. The differences in h between the central and the edge channels increase with increasing number of channels. This phenomenon can be ascribed to (1) the stronger extra hot-plume buoyancy in the inner plume region and (2) the higher entrance-separation resistance in the outer channels. The overall convection heat transfer coefficients in the plate arrays increase with increasing number of plate channels. The conventional assumption that all the channels of a multi-plate array have similar heat transfer performance needs re-evaluation. © 2017 Elsevier Masson SAS. All rights reserved.
Keywords: Natural convection Vertical plates Heat sink Hot plume
1. Introduction Natural convection from a vertical parallel plate array, one of the cornerstone problems of natural convection heat transfer, has been widely studied. This heat transfer configuration can be found in many practical applications, such as heat exchanger, heat sinks for electronics or LED cooling, etc. Elenbaas [1] pioneered the measurement of average convection heat transfer coefficient (h) on the inner isothermal walls of two isolated square parallel plates. It was assumed that the flow characteristics in all the channels of a parallel-plate array are similar and can be represented by a single channel between two parallel plates. Examined parameters included the plate spacing (b), the plate height (H ¼ 5.95, 12, and 24 cm), the plate temperature (Tw), and the inclination angle. The 3-
* This manuscript has not been published in an archival journal, nor is presently submitted for publication in another journal. * Corresponding author. E-mail address: [email protected] (S.-C. Wong).
http://dx.doi.org/10.1016/j.ijthermalsci.2017.06.018 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.
D experimental data for square plates were transformed with approximations into a 2-D (for infinitely long plates) empirical correlation between Nusselt number (Nu) and the Elenbaas Rayleigh number (Ra’) over 0.2 < Ra’ < 1 105:
Nu ¼
i 1 0h 35 3=4 Ra 1 e Ra0 24
(1)
where Nu ≡ hb , Ra’ ≡ Hb Ra ¼ Hb b gbðTnaw T∞ Þ. k An empirical formula for the optimum plate spacing was further proposed based on the experimental data. The optimum spacing to yield maximum total heat transfer rate of the plate array was located at Ra’ ¼ 46. Since then, almost all numerical or theoretical studies on the natural convection form a vertical multi-plate array were simplified to 2-D analyses between two parallel plates [2e15]. Bar-Cohen and Rohsenow [2] utilized the circumstances that the flows between two vertical parallel plates are bounded by two extreme situations: a fully-developed flow within two very close plates and a single-plate boundary layer flow with an infinite plate distance. An integrated formula can be obtained by combining the 3
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m_ o
Nomenclature Aw b g
inner wall area (m2) plate spacing (mm) gravitational acceleration, 9.81 (m/s2) along negative y-direction
Gr
Grashof number, Gr ¼
h
average heat transfer coefficient for an individual channel, h ¼ Q/Aw(Tw -T∞) (W/m2K) average heat transfer coefficient in the nth channel of a multi-channel plate array (W/m2K) overall heat transfer coefficient for a multi-channel plate array (W/m2K) average heat transfer coefficient for a two-plate single channel (W/m2K) plate height (mm) thermal conductivity of air (W/mK) distance between plate's upper end and upper boundary of computational domain (mm) distance between side-end plate and side boundary of computational domain (mm) distance between plate's lower end and lower boundary of computational domain (mm) mass flow rate in an individual channel in multichannel plate array (kg/s)
hn ho hs H k Lb Ls Lt m_
b3 g br2 ðTw T∞ Þ
m2
two limiting asymptotic analytic solutions. This principle was applied for four different heating arrangements: symmetric isothermal, asymmetric isothermal, symmetric isoflux, and asymmetric isoflux. For the symmetric isothermal situation, the following formula was proposed:
Nu ¼
576 2:873 þ pffiffiffiffiffiffiffiffiffi Ra’ 2 Ra’
1=2
:
(2)
This relation matches well with the 2-D empirical Nu formula of Elenbaas [1]. With the assumption that all channels in a multichannel plate array are similar, the formulae for the optimal plate spacing were derived for these four heating arrangements. Among them, the optimum spacing for symmetric isothermal plate arrays occurs at Ra’ ¼ 54.3, slightly higher than Elenbaas' value. These formulae have been widely used for the optimum design of vertical fin arrays [16,17]. Bodoia and Osterle [3] and Aung et al. [4] performed 2-D boundary-layer analysis between two isothermal parallel plates. The computation domain was set between the entrance and the exit of the channel. A uniform flow velocity was imposed at the channel entrance. Their 2-D numerical Nu results agreed fairly well with 3-D experiments of Elenbaas [1], with about 10% overestimations for Ra’ > 400. The discrepancies were ascribed to the assumption of uniform velocity at the channel entrance which ignored the entrance flow resistance [4]. In addition, Bodoia and Osterle [3] found their Nu calculations significantly lower than the 3-D data of Elenbaas [1] at low Ra’ values. This discrepancy was attributed to the side leakage effects in the 3-D experiments when plate spacings were small. Anand et al. [5] also adopted the boundary-layer approximations and set the computation domain between the entrance and the exit of the channel. Wang and Pepper [6] formulated with the elliptic Navier-stokes equations and energy equation, but the computation domain was still set between the channel entrance and exit. To account for the entrance flow resistance, an extended inlet
m_ s n N Nu p qw Q
overall mass flow rate in multi-channel plate array (kg/ s) mass flow rate in two-plate single channel (kg/s) channel number, n ¼ 1 for the middle channel total number of channels in a plate array Nusselt number defined as Nu ¼ hb k pressure (Pa) heat flux on the channel wall per unit depth (W/m2) heat transfer rate in a channel (W)
Ra
Rayleigh number, Ra ¼
Ra’ t T Tw T∞ u, v x, y
Elenbaas Rayleigh number, Ra’ ¼ Hb b gbðTnaw T∞ Þ plate thickness (mm) temperature (K) inner plate wall temperature (K) ambient air temperature (K) velocities in x- and y-direction, respectively coordinate directions
b3 g bðTw T∞ Þ
na
3
Greek symbols a thermal diffusivity (m2/s) b thermal expansion coefficient (1/K) v kinematic viscosity (m2/s) r air density (kg/m3) r∞ air density under ambient temperature (kg/m3)
subdomain as well as elliptic governing equations was employed in Refs. [7e10]. But in these works, the outlet boundary was set at the channel exit. Martin et al. [9] focused on the low Ra’ regime for an isothermal parallel-plate channel, where streamwise diffusion and heat conduction to the upstream air and adjacent surfaces become influential. They found the low-Ra’ heat transfer dependent on the particular inlet and outlet configuration used. Naylor et al. [10] reported entrance separation as Ra’ increased to 291.7, owing to the edge effect with inlet flow turning. This separation was shown to have an adverse effect on the local heat transfer near the entrance. Knowing that setting the outlet boundary at the channel exit excludes the effect of the hot outflow region, Chang and Lin [11] and Andreozzi et al. [12] adopted I-shaped computation domains with extended inflow and outflow subdomains in their transient 2-D simulation for the channel flow between two symmetricallyheated parallel plates. Morrone et al. [13] also used I-shaped computation domains in their analysis aiming to obtain an empirical correlation for the optimum plate spacing. Shyy et al. [14] adopted a convergent inflow subdomain and a divergent outflow subdomain in the 2-D steady analysis. The inflow boundary conditions utilized the inviscid flow generated by a line source on the centerline at the channel inlet, coupled with free entrainment boundaries along the side. Similarly, the outflow boundary conditions correspond to a line source plume coupled with free entrainment boundaries along the side. Ramanathan and Kumar [15] sidestepped the problems of open boundary conditions at the inlet and the exit by considering a large virtual enclosure encompassing the flow channel. However, the channel flow might not be completely free from the natural recirculation associated with the enclosure. It is noted that the above studies [1e15] only treated a single vertical channel. To explore the thermal and flow characteristics in multi-channel plate arrays, Floryan and Novak [18] simulated isothermal plate arrays with two, three, or an infinite number of channels, adopting
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an extended inlet subdomain. Interaction between neighboring channels was shown to increase with increasing Grashof number. Under Gr ¼ 105, entrance separation manifested near the outer channel wall. Heat transfer was enhanced on the inner wall but reduced on the outer wall. For a two-channel case, heat transfer on the inner wall was 5% higher than that in a single isolated channel. On the contrary, heat transfer on the outer wall was 5% lower. However, the average heat transfer on both walls approached that in a single isolated channel. For the three-channel case, the above effect was more obvious in the edge channel. Heat transfer in the central channel was still affected by the flow condition in the edge channel. For the periodic configuration with an infinite number of channels, the edge effect was precluded and the heat transfer was 18% higher than that in a single channel. The results of Floryan and Novak [18] showed some special features in multi-channel plate arrays that could not be revealed by single-channel analysis. However, only the inflow effect was considered and the effect of the hot outflow was ignored in their study. In fact, the collective hot plume above a vertical parallel-plate array, as in heat exchangers and heat sinks, provides considerable extra buoyancy effect. This effect has been overlooked because previous studies which accounted for the hot outflow [11e15] treated only the single-channel conditions for which the hotplume effect is not strongly manifested. The collective hot-plume effect above vertical plate arrays cannot be explored without handling multiple channels. The present study will numerically analyze the natural convection from isothermal multi-channel parallel-plate arrays adopting a sufficiently extended open computation domain. In addition, the reliability of the conventional assumption that all the channels in a multi-plate array have similar heat transfer performance, which has been widely accepted since Elenbaas [1] and adopted as the basis for optimum plate spacing evaluation [1,2,5,13], will be examined. The hot-plume will be shown to enhance heat transfer markedly, with stronger effects for plate arrays with more channels. Furthermore, the combination of the hot-plume effect and the entrance resistance in the edge channels leads to significantly different heat transfer performances in different channels. 2. Numerical methods 2.1. Governing equations In this work, 2-D analysis (infinitely long plates) will be considered. The following assumptions are made: 1. The air flow field is incompressible, laminar and steady. 2. The plate surface temperature (Tw) is fixed at 350 K, and the ambient temperature (T∞) is 300 K. 3. The Boussinesq approximation is assumed with the coefficient of thermal expansion b ¼ 1/T∞ and the fluid density evaluated at T∞ [19]. With (Tw - T∞)/T∞ ¼ 0.167 in the present work, the errors for the heat transfer coefficients due to this approximation should be insignificant [20,21]. The other fluid properties are temperature-dependent by interpolating the data provided in Ref. [17]. 4. Radiation heat loss is ignored. The governing equations can be written as follows: Continuity equation:
vu vv þ ¼0 vx vy Momentum equations:
(3)
u
u
vu vu 1 vp þv ¼ þ V,ðnVuÞ vx vy r∞ vx
vv vv 1 vb p þv ¼ þ V,ðnVvÞ þ g bðT T∞ Þ vx vy r∞ vy
265
(4)
(5)
where b p ¼ p þ r∞ gy Energy equation:
u
vT vT þv ¼ V,ðaVTÞ vx vy
(6)
2.2. Computation domain and boundary conditions The computation domain and geometric parameters for singlechannel and multi-channel configurations are illustrated in Fig. 1. Channel numbering for multi-channel plate arrays is given in Fig. 1(b). The central channel is n ¼ 1 and the outermost channel is n ¼ (Nþ1)/2, where N is the number of channels in a plate array. As our main attention is on the hot-plume and entrance edge effects for multi-channel plate arrays, the following plate geometries are fixed: plate height H ¼ 100 mm, plate thickness t ¼ 1.0 mm, plate spacing b ¼ 7 mm. With a fixed (Tw-T∞) of 50 K, these parameters amount to Ra’ ¼ 62.8, approximating the optimum value (Ra’ ¼ 54.3) of Bar-Cohen and Rohsenow [2]. Open domains with sufficient extensions on four sides are adopted. The top extension length (Lt), the bottom extension length (Lb), and the side extension length (Ls) are respectively determined based on domain independence tests. For all the channel numbers (N ¼ 1, 5, 11, 21) investigated in this work, comparisons were made for Lt ¼ 100 mm, 150 mm, 200 mm, and 250 mm; Lb ¼ 50 mm, 100 mm, and 150 mm; and Ls ¼ 50 mm, 100 mm, and 150 mm. Table 1 displays the differences in h using various values of Lt, Lb, and Ls with the grid size of 0.25 mm for a single-channel case as an example. The differences in h for all individual channels of multi-channel arrays are also rather small. It is noted that when Lt 150 mm, slight unstable flow phenomenon may show near the upper boundary. But this has no effect on the magnitude of h. With differences in h within 0.47%, the selected Lt, Lb, and Ls are 100 mm, 50 mm, and 50 mm, respectively. Uniform square grids are adopted and tested with sizes of 0.1 mm, 0.25 mm, 0.5 mm, and 1 mm for different channel numbers at Lt ¼ 100 mm and Lb ¼ Ls ¼ 50 mm. Again, the differences in h for all individual channels are rather small. The test results of the average heat transfer coefficient h for the single-channel case are compared in Table 2 as an example. When the grid size is 0.25 mm, the differences in h compared with those for a 0.1 mm grid size are within 0.001%. Thus, the 0.25 mm grid size is selected. The grid numbers for the array with N ¼ 1, 5, 11, or 21 are 435,372, 558,383, 666,997, and 1,051,743, respectively. The boundary conditions of the computational domain are set as follows. Boundary sections AB, BC and DA are pressure-inlet boundaries with stagnation pressure of 1 atm and temperature of T∞. Section CD is a pressure-outlet boundary subjected to vu/ vy ¼ vv/vy ¼ vT/vy ¼ 0, and the static pressure is set at 1 atm. On the plate walls, no-slip boundary conditions are applied for the flow velocities (u ¼ v ¼ 0) and the temperature is fixed at Tw ¼ 350 K. The computational domain is meshed via commercial software ANSYS 15.0, and the velocity and temperature distributions are calculated by FLUENT [22]. Second-order upwind scheme is applied for momentum and energy equations. The SIMPLE algorithm [23] is utilized for pressure correction. The residuals for the continuity and momentum equations are under 1 104, while those for the energy equation are under 1 1010. The average convection heat transfer coefficient (h) in a certain
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Fig. 1. Computation domain and geometric parameters, (a) single channel, (b) multiple channels.
Table 1 Domain tests for a single channel at a grid size of 0.25 mm.
Test Test Test Test Test Test Test Test
1 2 3 4 5 6 7 8
Lt (mm)
Lb (mm)
Ls (mm)
h (W/m2K)
Difference in h (%)
100 150 200 250 100 100 100 150
50 50 50 50 100 100 50 100
50 50 50 50 50 100 100 100
5.749 5.748 5.748 5.748 5.742 5.766 5.776 5.757
-0.02 0.02 0.02 0.12 0.30 0.47 0.14
Table 2 Grid tests for a single channel at Lt ¼ 100 mm and Lb ¼ Ls ¼ 50 mm. Grid size
h (W/m2K)
Difference in h (%)
0.1 mm 0.25 mm 0.5 mm 1.0 mm
5.75 5.75 5.77 5.97
<0.001 e 0.42 3.77
Fig. 2. Comparison of Nu versus Ra’ between present calculations and those by the semi-empirical formula of Bar-Cohen and Rohsenow [2].
channel is the average of the values on both inner walls. The frontend and rear-end plate surfaces outside the channel are excluded in calculating h. The average convection heat transfer coefficient for a channel is determined by
1 h¼ 2H
y¼H Z
y¼0
qw;l þ qw;r dy; Tw T∞
smooth over Ra’ < 1000. This is because the widely different Ra’ values in the computations are formed by selecting different values of b, H, and (Tw-T∞). It will be carefully discussed in our future paper that variation of these three parameters would cause slightly different effects on h as the extra hot-plume buoyancy is taken into account. Nonetheless, the reliability of present computations can be verified by Fig. 2.
(7)
where H is the wall height, and qw,l and qw,r are the local heat fluxes per unit depth on the left and the right wall, respectively. To validate our computations, the Nu results are plotted versus Ra’ for single-channel cases along with the semi-empirical results (Eq. (2)) of Bar-Cohen and Rohsenow [2]. Good agreement is shown in Fig. 2. Except for the 7% difference at Ra’ ¼ 3000, the differences between two curves are within 5%. Since both the entrance resistance and hot-plume buoyancy are excluded in the derivation of Eq. (2) [2], this may be partially responsible for the differences. It will be shown in the following section that neglecting these effects in a single-channel analysis would over-estimate h significantly (about 13% for the present Ra’ ¼ 62.8). Note that our curve is not quite
3. Results and discussion The present study considers single- and multi-channel parallel vertical plate arrays. The plate height is fixed at 100 mm, the plate spacing is 7 mm, and the plate thickness is 1 mm. Under a fixed temperature difference of Tw-T∞ ¼ 50 K, the Elenbaas Rayleigh number Ra’ is fixed at 62.8. 3.1. Effects of computation domain arrangement First, the heat transfer performances for four different domain arrangements are examined using the single-channel configuration. The domain dimensions are shown in Table 3. Fig. 3 illustrates these domain arrangements and their corresponding hs. Case I is
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Table 3 Extension lengths in different arrangements of computation domain.
Case Case Case Case
I: Extended boundaries II: Outlet boundary at channel exit III: Inlet boundary at channel entrance IV: Outlet boundary at channel exit; Inlet boundary at channel entrance
Fig. 3. Average convection heat transfer coefficients for four different domain arrangements, H ¼ 100 mm, b ¼ 7 mm, t ¼ 1.0 mm, (Tw-T∞) ¼ 50 K, Ra’ ¼ 62.8.
Lt (mm)
Lb (mm)
Ls (mm)
100 0 100 0
50 50 0 0
50 50 50 0
the standard with open boundaries extended on all sides. It is consistent with the test arrangement of Elenbaas [1] except that his plate thickness was 6 mm for the 120 120 mm2 plates. The domain of Case II contains inlet extension but no outlet extension. Now, the upper boundary is set as the pressure-outlet boundary, i.e., vu/vy ¼ vv/vy ¼ vT/vy ¼ 0 with the static pressure equal to 1 atm. In contrast, the domain of Case III contains outlet extension but no inlet extension. The lower boundary is set as the pressureinlet boundary, with vu/vy ¼ vv/vy ¼ 0, T ¼ T∞, and the stagnation pressure equal to 1 atm. In Case IV, the domain contains neither the inlet nor the outlet extension. The pressure-outlet boundary and the pressure-inlet boundary are applied at the exit and the inlet of the channel, respectively. The temperature and velocity fields of Case I are shown in Fig. 4. In Fig. 4(a), a hot-plume zone is present above the plates. Fig. 4(b) shows that a convergent flow is drawn into the channel entrance by buoyancy. The h of this single-channel case is 5.75 W/m2K. Case II, with inlet extension but no outlet extension, exhibits a lowest h of 5.56 W/m2K. Case III,
Fig. 4. Temperature and velocity fields for the single channel with H ¼ 100 mm, b ¼ 7 mm, t ¼ 1.0 mm, (Tw-T∞) ¼ 50 K, Ra’ ¼ 62.8.
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Fig. 5. Ratios of hn/hs for multi-channel arrays with different Ns, hn: h in nth channel of the plate arrays; hs: h in a single isolated channel.
with outlet extension but no inlet extension, exhibits the highest h of 6.77 W/m2K, 18% higher than the Case-I value. The lowest Case II value results from two reasons. First, the entrance resistance is accounted for in Case II with the inlet extension. Second, with the outlet boundary set at the channel exit, the favorable effect of the downstream hot-plume buoyancy is excluded. On the contrary, in Case III the entrance resistance is excluded and the favorable hotplume effect is included. As both the entrance resistance and hotplume buoyancy are present in Case I, its h lies between those of Cases II and III. In Case IV, where neither of the two effects is included, h ¼ 6.50 W/m2K, 13% higher than the Case-I value. Comparing Case I with Case III, or Case II with Case IV, excluding entrance resistance can be found to raise h by about 17%. Similarly, comparing Case I with Case II, or Case III with Case IV, excluding hot-plume buoyancy decreases h by about 4%. Therefore, for the single-channel configuration, the entrance resistance appears more influential than the hot-plume buoyancy. Bodoia and Osterle [3] used the computation domain as Case IV,
Fig. 6. Temperature and velocity fields for plate arrays with N ¼ 5 (a, c) and N ¼ 21(b, d) at Ra’ ¼ 62.8.
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while the test arrangement of Elenbaas [1] was like Case I. The Nu results of Bodoia and Osterle [3] are higher than the experimental data of Elenbaas [1] by about 10% for Ra’ > 400. The discrepancies were ascribed to the omission of the entrance flow resistance [4]. Our present result of h for Case IV is higher than that of Case I by about 13%. 3.2. Heat transfer characteristics in multi-channel plate arrays To investigate the heat transfer characteristics of vertical multichannel plate arrays, configurations with N ¼ 5, 11, and 21 are analyzed. Fig. 5 presents the ratio of hn/hs for N ¼ 5, 11, and 21, where hn is the average heat transfer coefficient associated with the nth individual channel, and hs is the average heat transfer coefficient of the single-channel case. Except for the outermost channels for N ¼ 11 and 21, hn/hs > 1. For the same N, hn/hs is higher in a channel closer to the central channel. In addition, this trend is stronger for a larger N. In the central channel of the 21-channel plate array, hn/hs ¼ 1.329, reflecting a 32.9% enhancement of heat transfer compared to the single-channel configuration. Also shown in Fig. 5 is that the suppression of heat transfer in the outermost channels worsens for a larger N. Fig. 6 illustrates the temperature and velocity fields for N ¼ 5 and N ¼ 21. As shown in Fig. 6(a) and (b), the height of the hot-plume zone is maximum at the central channel and diminishes outwards, and the width of the hot-plume zone grows with increasing N. Consequently, the central channel is subjected to the strongest extra hot-plume buoyancy along with the part from the hot channel walls. This explains why hn/hs is higher in the inner channels and this trend appears stronger for a larger N. On the other hand, the entrance-flow characteristics can be seen in the velocity fields shown in Fig. 6 (c) and (d). With a large number of channels, abrupt flow turning around the edge entrance can influence a few neighboring channels. Fig. 7 presents closer illustrations of the flow fields in the outermost channels for N ¼ 5 and N ¼ 21. For N ¼ 5, only in the outermost channel appears a small entrance separation bubble. For N ¼ 21, a larger separation bubble forms in the outermost channel, and a very small one seems to exist in the next channel. That is, the more channels there are in a plate arrays, the stronger edge effect is induced in the outer channels. This is because when there are more channels, the
269
buoyancy grows not only with the number of channel walls, but also with the wider collective hot plume. Stronger buoyancy induces larger air flow rate and hence sharper flow turning around the edge entrance. Hence, the flow rates in the outermost channels are reduced by increasing entrance separation resistance. Also can be seen in Fig. 6 (c) and (d) is that, most of the inner channels are unaffected and the cool air enters these channels from below. In these channels, the air flow rates and heat transfer are enhanced due to the extra hot-plume buoyancy. For N ¼ 5, the h of the central channel is 11.7% higher than that of the edge channel, while the mass flow rate is 7.7% higher (not shown); for N ¼ 21, the h of the central channel is 40.3% higher than that of the edge channel, while the mass flow rate is 40.0% higher. Compared to the single-channel case, the h of the central channel for N ¼ 21 is 32.9% higher than hs and the mass flow rate is 32.2% higher. For N ¼ 5, the h of the central channel is 14.7% higher than hs and the mass flow rate is 8.7% higher. In the outermost channel of the N ¼ 21 array, the h is 5.3% lower than hs and the mass flow rate is 5.5% lower. Fig. 8 compares the overall heat transfer coefficients (ho) and the average mass flow rate per channel (m_ o =N) of plate arrays with different numbers of channels. From N ¼ 1 to N ¼ 21, ho increases from 5.75 W/m2K to 7.17 W/m2K, by 24.7%; m_ o =N increases from 1.44 103 kg/s to 1.76 103 kg/s, by 22.2%. These data cast doubt on the conventional assumption ever since Elenbaas [1] that all the channels, except the outermost ones, may be represented by a single channel. In fact, the h values in individual channels are significantly different due to the effects of hot-plume buoyancy and entrance separation. Furthermore, the overall heat transfer rate obtained by N times the heat transfer rate of a single channel would be considerably under-estimated, with a larger error for a larger N. To compare the relative contributions of the hot-plume buoyancy and the entrance resistance, domain arrangements without inlet or outlet extension were investigated. The temperature fields for N ¼ 21 are compared in Fig. 9. Without inlet extension (Fig. 9(a)), ho is 7.91 W/m2K, which is 11.3% higher than the value of 7.17 W/m2K with both inlet and outlet extensions. The increase in ho results from the omission of entrance resistance. When the outlet extension is excluded, ho decreases to 5.87 W/m2K by 17.4%. Hence, the favorable effect of hot-plume buoyancy exceeds the
Fig. 7. Velocity fields near the channel entrances for plate arrays with (a) N ¼ 5 and (b) N ¼ 21 at Ra’ ¼ 62.8.
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Adopting an inlet extension in the two-channel analysis, Floryan and Novak [18] have shown that entrance separation bubble appears at a smaller Ra’ for a narrower channel (smaller b/H). It is noted that the hot plume was omitted in Ref. [18]. Here in the present work at a fixed Ra’ of 62.8, separation is shown to intensify with increasing channel number. 3.3. Virtual chimney effect above vertical plate arrays
Fig. 8. Overall ho and m_ o =N for plate arrays with different Ns.
unfavorable entrance resistance for N ¼ 21. In Section 3.1, it has been shown for a single channel that the entrance resistance is more influential than the hot-plume buoyance. Therefore, the effect of the collective hot plume intensifies with increasing number of channels.
The phenomenon of extra hot-plume buoyancy may be perceived from a different view. It is essentially similar to the chimney effect enabled by attaching adiabatic extensions downstream of the heated channel or channels of a parallel plate array [24], a vertical parallel-plate heat sink [25], or two parallel plates [26e33]. The chimney effect has been intensively studied theoretically [24], experimentally [25e28], and numerically [26,29e33] for various heating conditions or chimney arrangements. The general finding is that the adiabatic extensions can result in significant heat transfer enhancement. However, for the situations without chimney shields, the role of the open plume has been ignored in the literature. For a heated plate array, the buoyancy comes from the hot air in two zones: in the plate channels and
Fig. 9. Temperature fields with (a) lower boundary set at the channel entrances, (b) upper boundary set at the channel exits, Ra’ ¼ 62.8.
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with N ¼ 11. With 100 mm-high chimney shields, the hot-air volume is greatly enlarged. Although the air temperature in the chimney is lower than that in the unshielded hot plume, the overall effect is still stronger. Without shields, ho ¼ 6.67 W/m2K and m_ o =N ¼ 1.61 103 kg/s for N ¼ 11. Compared to the single channel with h of 5.75 W/m2K and m_ o =N ¼ 1.44 103 kg/s, the virtual chimney effect leads to a 16% increase in ho and 11.8% increase in m_ o =N. With actual shield plates, ho is further increased to 8.84 W/ m2K (32.5% increase) and (m_ o =N) increased to 2.51 103 kg/s (55.9% increase). 4. Conclusions
Fig. 10. Computations on a periodic single-channel configuration, (a) computation domain, (b) results of h (Tw ¼ 330 K, T∞ ¼ 300 K).
above the plate array. To elucidate this point, a periodic singlechannel configuration, as shown in Fig. 10(a), is computed with symmetric boundaries along both vertical sides. The results for various Lts are presented in Fig. 10(b). Interestingly, h increases continuously with increasing Lt. This behavior results from the artificially enlarged hot-air zone above the plate array. The extended upper symmetric boundaries act as virtual chimney shields to raise heat transfer in the channel. Analogous with the Archimedes' Principle conventionally applied to an object in liquid, the buoyancy on the hot air is proportional to the product of density difference and the hot-air volume. With increasing Lt, the hot-air volume above the plates increases and thereby leads to higher buoyancy. In fact, the virtual chimney effect exists in the hot plume of a vertical plate array even without real chimney shields. However, without the chimney shields the hot plume narrows upwards with entrainment of surrounding air. Thus, the virtual chimney effect would be weaker than actual chimney effect. Fig. 11 illustrates the temperature fields of an open and a shielded plate array
This study numerically re-examined the classical problems of natural convection from vertical isothermal plate arrays using 2-D steady-state analysis. Extended computation domains encompassing the single- or multi-channel plate arrays are adopted. Also investigated are the consequences of using computation domains without the inlet extension or/and the outlet extension. The plate height is fixed at 100 mm, the plate spacing is 7 mm, the plate thickness is 1 mm, and the temperature difference between the plate wall and ambience is 50 K. These conditions correspond to an Elenbaas Rayleigh number of 62.8. The convection heat transfer _ in the individual channels coefficients (h) and mass flow rates (m) are calculated. The following conclusions have been reached: 1. Setting the inlet boundary at the entrance of the channels would over-estimate the values of h in the outer channels, as the flow resistance associated with the entrance separation therein is ignored. Setting the outlet boundary at the exit of the channel would significantly under-estimate the values of h in the inner channels, because the extra buoyancy provided by the hot plume above the plate arrays is omitted. 2. Multi-channel plate arrays exhibit higher values of h and m_ than a single channel except for the outermost channels. The values of h and m_ are highest at the central channel and lowest at the edge channel, with their differences increasing with increasing
Fig. 11. Temperature fields (a) without and (b) with adiabatic chimney shields, N ¼ 11, Ra’ ¼ 62.8.
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number of channels. For an array with 21 channels (N ¼ 21), h and m_ in the central channel are 32.9% and 32.2% higher than those in a single channel, respectively. In the outermost channels, h and m_ are respectively 5.3% and 5.5% lower than those in a single channel. This phenomenon can be attributed mainly to the stronger extra hot-plume buoyancy above the inner channels, and partly to the higher entrance-separation resistance in the outermost channels. 3. The overall convection heat transfer coefficients (ho) in the plate arrays increase with increasing number of channels, as a result of enhanced hot-plume buoyancy above the arrays. For N ¼ 21, ho is 24.7% higher than the single-channel value. Therefore, the overall heat transfer rates conventionally calculated by N times the heat transfer rate of a single channel would be considerably under-estimated, with a larger error for a larger N. 4. The accuracy of the assumption that all the channels of a multiplate array have similar heat transfer performance, which has been widely accepted since Elenbaas [1] and adopted as the basis for optimum plate spacing estimation, needs reevaluation. 5. The heat transfer enhancement by the extra hot-plume buoyancy for multi-channel plate arrays may be perceived as a virtual chimney effect. As the actual chimney yields enhanced heat transfer [24e33], the open hot plume also leads to heat transfer enhancement. However, without actual chimney shields, the virtual-chimney enhancement is not as large as the actual chimney effect. Future studies will investigate the heat transfer characteristics associated with different Ra's and 3-D array configurations. References [1] Elenbaas W. Heat dissipation of parallel plates by free convection. Physica 1942;9:1e28. [2] Bar-Cohen A, Rohsenow WM. Thermally optimum spacing of vertical, natural convection cooled, parallel plates. J Heat Transf 1984;106:116e23. [3] Bodoia JR, Osterle JF. The development of free convection between heated vertical plates. J Heat Transf 1962;84:40e3. [4] Aung W, Fletcher LS, Sernas V. Developing laminar free convection between vertical flat plates with asymmetric heating. Int J Heat Mass Transf 1972;15: 2293e308. [5] Anand NK, Kim SH, Fletcher LS. The effect of plate spacing on free convection between heated parallel plates. J Heat Transf 1992;114:515e8. [6] Wang X, Pepper DW. Numerical simulation for natural convection in vertical channels. Int J Heat Mass Transf 2009;52:4095e102. [7] Kettleborough CF. Transient laminar free convection between heated vertical plates including entrance effects. Int J Heat Mass Transf 1972;15:883e96. [8] Nakamura H, Asako Y, Naitou T. Heat transfer by free convection between two parallel flat plates. Numer Heat Transf 1982;5:95e106.
[9] Martin L, Raithby GD, Yovanovich MM. On the low Rayleigh number asymptote for natural convection through an isothermal, parallel plate channel. J Heat Transf 1991;113:899e905. [10] Naylor D, Floryan JM, Tarasuk JD. A numerical study of developing free convection between isothermal vertical plates. J Heat Transf 1991;113:620e6. [11] Chang TS, Kin TF. Transient buoyance-induced flow through a heated vertical channel of finite height. Numer Heat Transf Part A 1989;16:15e35. [12] Andreozzi A, Buonomo B, Manca O. Transient natural convection in vertical channels symmetrically heated at uniform heat flux. Numr Heat Transf Part A 2009;55:409e31. [13] Morrone B, Campo A, Manca O. Optimum plate separation in vertical parallel plate channels for natural convective flows: incorporation of large spaces at the channel extremes. Int J Heat Mass Transf 1997;40:993e1000. [14] Shyy W, Gingrich WK, Gebhart B. Adaptive grid solution for buoyancyinduced flow in vertical slots. Numer Heat Transf Part A 1992;22:51e70. [15] Ramanathan S, Kumar R. Correlations for natural convection between heated vertical plates. J Heat Transf 1991;113:97e107. [16] De Lieto Vollaro A, Grignaffini S, Gugliermetti F. Optimum design of vertical rectangular fin arrays. Int J Therm Sci 1999;38:525e9. [17] Incropera FP, DeWitt DP, Bergman TL, Lavine AS. Principle of heat and mass transfer. seventh ed. Wiley; 2013. [18] Floryan JM, Novak M. Free convection heat transfer in multiple vertical channels. Int J Heat Fluid Flow 1995;16:244e53. [19] Bejan A. Convection heat transfer. fourth ed. Wiley; 2013. [20] Chenoweth DR, Paolucci S. Natural convection in an enclosed vertical air layer with large horizontal temperature differences. J Fluid Mech 1986;169: 173e210. ndez J, Zamora B. Effects of variable properties and non-uniform heating [21] Herna on natural convection flows in vertical channels. Int J Heat Mass Transf 2005;48:793e807. [22] Handbook of fluent, ANSYS, Inc. [23] Patankar SV. Numerical heat transfer and fluid flow. Hemisphere Publishing Corporation; 1980. [24] Fisher TS, Torrance KE, Sikka KK. Analysis and optimization of a natural draft heat sink system. IEEE Trans Compon Pack Manuf Tech, Part A 1997;20: 111e9. [25] Fisher TS, Torrance KE. Experiments on chimney-enhanced free convection. J Heat Transf 1999;121:603e9. [26] Straatman AG, Tarasuk JD, Floryan JM. Heat transfer enhancement from a vertical, isothermal channel generated by the chimney effect. J Heat Transf 1993;115:395e402. [27] Auletta A, Manca O, Morrone B, Naso V. Heat transfer enhancement by the chimney effect in a vertical isoflux channel. Int J Heat Mass Transf 2001;44: 4345e57. [28] Auletta A, Manca O. Heat and fluid flow resulting from the chimney effect in a symmetrically heated vertical channel with adiabatic extensions. Int J Therm Sci 2002;41:1101e11. [29] Andreozzi A, Buonomo B, Manca O. Numerical study of natural convection in vertical channels with adiabatic extensions downstream. Numer Heat Transf 2005;47:741e62. [30] Andreozzi A, Buonomo B, Manca O. Thermal management of a symmetrically heated channel-chimney system. Int J Therm Sci 2009;48:475e87. [31] Andreozzi A, Buonomo B, Manca O. Thermal and fluid dynamic behaviors in symmetrical heated channel e chimney systems. Int J Num Meth Heat Fluid Flow 2010;20:811e33. [32] Andreozzi A, Buonomo B, Manca O. Numerical investigation of transient natural convection in a vertical channel-chimney system symmetrically heated at uniform heat flux. Int J Heat Mass Transf 2012;55:6077e89. [33] Nasri Z, Laatar AH, Balti J. Natural convection enhancement in an asymmetrically heated channel-chimney system. Int J Therm Sci 2015;90:122e34.
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Revisit on natural convection from vertical isothermal plate arrays–effects of extra plume buoyancy* Shwin-Chung Wong*, Shih-Han Chu Department of Power Mechanical Engineering, National Tsing Hua University, Hsin-Chu 300, Taiwan, ROC
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 January 2017 Received in revised form 13 May 2017 Accepted 18 June 2017
In this study, natural convection from vertical isothermal parallel-plate arrays is examined using 2-D steady-state numerical analysis. Extended computation domains encompassing the single- or multichannel plate arrays are adopted. Also investigated are the consequences of using computation domains without the inlet extension or/and the outlet extension. The plate height is fixed at 100 mm, the plate spacing is 7 mm, and the plate thickness is 1 mm. The Elenbaas Rayleigh number Ra’ is fixed at 62.8. The results show that setting the inlet boundary at the entrance of the channel would ignore the separation formed near the outer entrance corner and the associated flow resistance; while setting the outlet boundary at the exit of the channel would omit the extra buoyancy provided by the hot plume above the arrays. In multi-channel arrays, the average heat transfer coefficients in different individual channels (hs) are different from each other. The h is the highest in the central channel and the lowest in the edge channel. Except for the edge channels, the h values are higher than that of a single channel. The differences in h between the central and the edge channels increase with increasing number of channels. This phenomenon can be ascribed to (1) the stronger extra hot-plume buoyancy in the inner plume region and (2) the higher entrance-separation resistance in the outer channels. The overall convection heat transfer coefficients in the plate arrays increase with increasing number of plate channels. The conventional assumption that all the channels of a multi-plate array have similar heat transfer performance needs re-evaluation. © 2017 Elsevier Masson SAS. All rights reserved.
Keywords: Natural convection Vertical plates Heat sink Hot plume
1. Introduction Natural convection from a vertical parallel plate array, one of the cornerstone problems of natural convection heat transfer, has been widely studied. This heat transfer configuration can be found in many practical applications, such as heat exchanger, heat sinks for electronics or LED cooling, etc. Elenbaas [1] pioneered the measurement of average convection heat transfer coefficient (h) on the inner isothermal walls of two isolated square parallel plates. It was assumed that the flow characteristics in all the channels of a parallel-plate array are similar and can be represented by a single channel between two parallel plates. Examined parameters included the plate spacing (b), the plate height (H ¼ 5.95, 12, and 24 cm), the plate temperature (Tw), and the inclination angle. The 3-
* This manuscript has not been published in an archival journal, nor is presently submitted for publication in another journal. * Corresponding author. E-mail address: [email protected] (S.-C. Wong).
http://dx.doi.org/10.1016/j.ijthermalsci.2017.06.018 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.
D experimental data for square plates were transformed with approximations into a 2-D (for infinitely long plates) empirical correlation between Nusselt number (Nu) and the Elenbaas Rayleigh number (Ra’) over 0.2 < Ra’ < 1 105:
Nu ¼
i 1 0h 35 3=4 Ra 1 e Ra0 24
(1)
where Nu ≡ hb , Ra’ ≡ Hb Ra ¼ Hb b gbðTnaw T∞ Þ. k An empirical formula for the optimum plate spacing was further proposed based on the experimental data. The optimum spacing to yield maximum total heat transfer rate of the plate array was located at Ra’ ¼ 46. Since then, almost all numerical or theoretical studies on the natural convection form a vertical multi-plate array were simplified to 2-D analyses between two parallel plates [2e15]. Bar-Cohen and Rohsenow [2] utilized the circumstances that the flows between two vertical parallel plates are bounded by two extreme situations: a fully-developed flow within two very close plates and a single-plate boundary layer flow with an infinite plate distance. An integrated formula can be obtained by combining the 3
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m_ o
Nomenclature Aw b g
inner wall area (m2) plate spacing (mm) gravitational acceleration, 9.81 (m/s2) along negative y-direction
Gr
Grashof number, Gr ¼
h
average heat transfer coefficient for an individual channel, h ¼ Q/Aw(Tw -T∞) (W/m2K) average heat transfer coefficient in the nth channel of a multi-channel plate array (W/m2K) overall heat transfer coefficient for a multi-channel plate array (W/m2K) average heat transfer coefficient for a two-plate single channel (W/m2K) plate height (mm) thermal conductivity of air (W/mK) distance between plate's upper end and upper boundary of computational domain (mm) distance between side-end plate and side boundary of computational domain (mm) distance between plate's lower end and lower boundary of computational domain (mm) mass flow rate in an individual channel in multichannel plate array (kg/s)
hn ho hs H k Lb Ls Lt m_
b3 g br2 ðTw T∞ Þ
m2
two limiting asymptotic analytic solutions. This principle was applied for four different heating arrangements: symmetric isothermal, asymmetric isothermal, symmetric isoflux, and asymmetric isoflux. For the symmetric isothermal situation, the following formula was proposed:
Nu ¼
576 2:873 þ pffiffiffiffiffiffiffiffiffi Ra’ 2 Ra’
1=2
:
(2)
This relation matches well with the 2-D empirical Nu formula of Elenbaas [1]. With the assumption that all channels in a multichannel plate array are similar, the formulae for the optimal plate spacing were derived for these four heating arrangements. Among them, the optimum spacing for symmetric isothermal plate arrays occurs at Ra’ ¼ 54.3, slightly higher than Elenbaas' value. These formulae have been widely used for the optimum design of vertical fin arrays [16,17]. Bodoia and Osterle [3] and Aung et al. [4] performed 2-D boundary-layer analysis between two isothermal parallel plates. The computation domain was set between the entrance and the exit of the channel. A uniform flow velocity was imposed at the channel entrance. Their 2-D numerical Nu results agreed fairly well with 3-D experiments of Elenbaas [1], with about 10% overestimations for Ra’ > 400. The discrepancies were ascribed to the assumption of uniform velocity at the channel entrance which ignored the entrance flow resistance [4]. In addition, Bodoia and Osterle [3] found their Nu calculations significantly lower than the 3-D data of Elenbaas [1] at low Ra’ values. This discrepancy was attributed to the side leakage effects in the 3-D experiments when plate spacings were small. Anand et al. [5] also adopted the boundary-layer approximations and set the computation domain between the entrance and the exit of the channel. Wang and Pepper [6] formulated with the elliptic Navier-stokes equations and energy equation, but the computation domain was still set between the channel entrance and exit. To account for the entrance flow resistance, an extended inlet
m_ s n N Nu p qw Q
overall mass flow rate in multi-channel plate array (kg/ s) mass flow rate in two-plate single channel (kg/s) channel number, n ¼ 1 for the middle channel total number of channels in a plate array Nusselt number defined as Nu ¼ hb k pressure (Pa) heat flux on the channel wall per unit depth (W/m2) heat transfer rate in a channel (W)
Ra
Rayleigh number, Ra ¼
Ra’ t T Tw T∞ u, v x, y
Elenbaas Rayleigh number, Ra’ ¼ Hb b gbðTnaw T∞ Þ plate thickness (mm) temperature (K) inner plate wall temperature (K) ambient air temperature (K) velocities in x- and y-direction, respectively coordinate directions
b3 g bðTw T∞ Þ
na
3
Greek symbols a thermal diffusivity (m2/s) b thermal expansion coefficient (1/K) v kinematic viscosity (m2/s) r air density (kg/m3) r∞ air density under ambient temperature (kg/m3)
subdomain as well as elliptic governing equations was employed in Refs. [7e10]. But in these works, the outlet boundary was set at the channel exit. Martin et al. [9] focused on the low Ra’ regime for an isothermal parallel-plate channel, where streamwise diffusion and heat conduction to the upstream air and adjacent surfaces become influential. They found the low-Ra’ heat transfer dependent on the particular inlet and outlet configuration used. Naylor et al. [10] reported entrance separation as Ra’ increased to 291.7, owing to the edge effect with inlet flow turning. This separation was shown to have an adverse effect on the local heat transfer near the entrance. Knowing that setting the outlet boundary at the channel exit excludes the effect of the hot outflow region, Chang and Lin [11] and Andreozzi et al. [12] adopted I-shaped computation domains with extended inflow and outflow subdomains in their transient 2-D simulation for the channel flow between two symmetricallyheated parallel plates. Morrone et al. [13] also used I-shaped computation domains in their analysis aiming to obtain an empirical correlation for the optimum plate spacing. Shyy et al. [14] adopted a convergent inflow subdomain and a divergent outflow subdomain in the 2-D steady analysis. The inflow boundary conditions utilized the inviscid flow generated by a line source on the centerline at the channel inlet, coupled with free entrainment boundaries along the side. Similarly, the outflow boundary conditions correspond to a line source plume coupled with free entrainment boundaries along the side. Ramanathan and Kumar [15] sidestepped the problems of open boundary conditions at the inlet and the exit by considering a large virtual enclosure encompassing the flow channel. However, the channel flow might not be completely free from the natural recirculation associated with the enclosure. It is noted that the above studies [1e15] only treated a single vertical channel. To explore the thermal and flow characteristics in multi-channel plate arrays, Floryan and Novak [18] simulated isothermal plate arrays with two, three, or an infinite number of channels, adopting
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an extended inlet subdomain. Interaction between neighboring channels was shown to increase with increasing Grashof number. Under Gr ¼ 105, entrance separation manifested near the outer channel wall. Heat transfer was enhanced on the inner wall but reduced on the outer wall. For a two-channel case, heat transfer on the inner wall was 5% higher than that in a single isolated channel. On the contrary, heat transfer on the outer wall was 5% lower. However, the average heat transfer on both walls approached that in a single isolated channel. For the three-channel case, the above effect was more obvious in the edge channel. Heat transfer in the central channel was still affected by the flow condition in the edge channel. For the periodic configuration with an infinite number of channels, the edge effect was precluded and the heat transfer was 18% higher than that in a single channel. The results of Floryan and Novak [18] showed some special features in multi-channel plate arrays that could not be revealed by single-channel analysis. However, only the inflow effect was considered and the effect of the hot outflow was ignored in their study. In fact, the collective hot plume above a vertical parallel-plate array, as in heat exchangers and heat sinks, provides considerable extra buoyancy effect. This effect has been overlooked because previous studies which accounted for the hot outflow [11e15] treated only the single-channel conditions for which the hotplume effect is not strongly manifested. The collective hot-plume effect above vertical plate arrays cannot be explored without handling multiple channels. The present study will numerically analyze the natural convection from isothermal multi-channel parallel-plate arrays adopting a sufficiently extended open computation domain. In addition, the reliability of the conventional assumption that all the channels in a multi-plate array have similar heat transfer performance, which has been widely accepted since Elenbaas [1] and adopted as the basis for optimum plate spacing evaluation [1,2,5,13], will be examined. The hot-plume will be shown to enhance heat transfer markedly, with stronger effects for plate arrays with more channels. Furthermore, the combination of the hot-plume effect and the entrance resistance in the edge channels leads to significantly different heat transfer performances in different channels. 2. Numerical methods 2.1. Governing equations In this work, 2-D analysis (infinitely long plates) will be considered. The following assumptions are made: 1. The air flow field is incompressible, laminar and steady. 2. The plate surface temperature (Tw) is fixed at 350 K, and the ambient temperature (T∞) is 300 K. 3. The Boussinesq approximation is assumed with the coefficient of thermal expansion b ¼ 1/T∞ and the fluid density evaluated at T∞ [19]. With (Tw - T∞)/T∞ ¼ 0.167 in the present work, the errors for the heat transfer coefficients due to this approximation should be insignificant [20,21]. The other fluid properties are temperature-dependent by interpolating the data provided in Ref. [17]. 4. Radiation heat loss is ignored. The governing equations can be written as follows: Continuity equation:
vu vv þ ¼0 vx vy Momentum equations:
(3)
u
u
vu vu 1 vp þv ¼ þ V,ðnVuÞ vx vy r∞ vx
vv vv 1 vb p þv ¼ þ V,ðnVvÞ þ g bðT T∞ Þ vx vy r∞ vy
265
(4)
(5)
where b p ¼ p þ r∞ gy Energy equation:
u
vT vT þv ¼ V,ðaVTÞ vx vy
(6)
2.2. Computation domain and boundary conditions The computation domain and geometric parameters for singlechannel and multi-channel configurations are illustrated in Fig. 1. Channel numbering for multi-channel plate arrays is given in Fig. 1(b). The central channel is n ¼ 1 and the outermost channel is n ¼ (Nþ1)/2, where N is the number of channels in a plate array. As our main attention is on the hot-plume and entrance edge effects for multi-channel plate arrays, the following plate geometries are fixed: plate height H ¼ 100 mm, plate thickness t ¼ 1.0 mm, plate spacing b ¼ 7 mm. With a fixed (Tw-T∞) of 50 K, these parameters amount to Ra’ ¼ 62.8, approximating the optimum value (Ra’ ¼ 54.3) of Bar-Cohen and Rohsenow [2]. Open domains with sufficient extensions on four sides are adopted. The top extension length (Lt), the bottom extension length (Lb), and the side extension length (Ls) are respectively determined based on domain independence tests. For all the channel numbers (N ¼ 1, 5, 11, 21) investigated in this work, comparisons were made for Lt ¼ 100 mm, 150 mm, 200 mm, and 250 mm; Lb ¼ 50 mm, 100 mm, and 150 mm; and Ls ¼ 50 mm, 100 mm, and 150 mm. Table 1 displays the differences in h using various values of Lt, Lb, and Ls with the grid size of 0.25 mm for a single-channel case as an example. The differences in h for all individual channels of multi-channel arrays are also rather small. It is noted that when Lt 150 mm, slight unstable flow phenomenon may show near the upper boundary. But this has no effect on the magnitude of h. With differences in h within 0.47%, the selected Lt, Lb, and Ls are 100 mm, 50 mm, and 50 mm, respectively. Uniform square grids are adopted and tested with sizes of 0.1 mm, 0.25 mm, 0.5 mm, and 1 mm for different channel numbers at Lt ¼ 100 mm and Lb ¼ Ls ¼ 50 mm. Again, the differences in h for all individual channels are rather small. The test results of the average heat transfer coefficient h for the single-channel case are compared in Table 2 as an example. When the grid size is 0.25 mm, the differences in h compared with those for a 0.1 mm grid size are within 0.001%. Thus, the 0.25 mm grid size is selected. The grid numbers for the array with N ¼ 1, 5, 11, or 21 are 435,372, 558,383, 666,997, and 1,051,743, respectively. The boundary conditions of the computational domain are set as follows. Boundary sections AB, BC and DA are pressure-inlet boundaries with stagnation pressure of 1 atm and temperature of T∞. Section CD is a pressure-outlet boundary subjected to vu/ vy ¼ vv/vy ¼ vT/vy ¼ 0, and the static pressure is set at 1 atm. On the plate walls, no-slip boundary conditions are applied for the flow velocities (u ¼ v ¼ 0) and the temperature is fixed at Tw ¼ 350 K. The computational domain is meshed via commercial software ANSYS 15.0, and the velocity and temperature distributions are calculated by FLUENT [22]. Second-order upwind scheme is applied for momentum and energy equations. The SIMPLE algorithm [23] is utilized for pressure correction. The residuals for the continuity and momentum equations are under 1 104, while those for the energy equation are under 1 1010. The average convection heat transfer coefficient (h) in a certain
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Fig. 1. Computation domain and geometric parameters, (a) single channel, (b) multiple channels.
Table 1 Domain tests for a single channel at a grid size of 0.25 mm.
Test Test Test Test Test Test Test Test
1 2 3 4 5 6 7 8
Lt (mm)
Lb (mm)
Ls (mm)
h (W/m2K)
Difference in h (%)
100 150 200 250 100 100 100 150
50 50 50 50 100 100 50 100
50 50 50 50 50 100 100 100
5.749 5.748 5.748 5.748 5.742 5.766 5.776 5.757
-0.02 0.02 0.02 0.12 0.30 0.47 0.14
Table 2 Grid tests for a single channel at Lt ¼ 100 mm and Lb ¼ Ls ¼ 50 mm. Grid size
h (W/m2K)
Difference in h (%)
0.1 mm 0.25 mm 0.5 mm 1.0 mm
5.75 5.75 5.77 5.97
<0.001 e 0.42 3.77
Fig. 2. Comparison of Nu versus Ra’ between present calculations and those by the semi-empirical formula of Bar-Cohen and Rohsenow [2].
channel is the average of the values on both inner walls. The frontend and rear-end plate surfaces outside the channel are excluded in calculating h. The average convection heat transfer coefficient for a channel is determined by
1 h¼ 2H
y¼H Z
y¼0
qw;l þ qw;r dy; Tw T∞
smooth over Ra’ < 1000. This is because the widely different Ra’ values in the computations are formed by selecting different values of b, H, and (Tw-T∞). It will be carefully discussed in our future paper that variation of these three parameters would cause slightly different effects on h as the extra hot-plume buoyancy is taken into account. Nonetheless, the reliability of present computations can be verified by Fig. 2.
(7)
where H is the wall height, and qw,l and qw,r are the local heat fluxes per unit depth on the left and the right wall, respectively. To validate our computations, the Nu results are plotted versus Ra’ for single-channel cases along with the semi-empirical results (Eq. (2)) of Bar-Cohen and Rohsenow [2]. Good agreement is shown in Fig. 2. Except for the 7% difference at Ra’ ¼ 3000, the differences between two curves are within 5%. Since both the entrance resistance and hot-plume buoyancy are excluded in the derivation of Eq. (2) [2], this may be partially responsible for the differences. It will be shown in the following section that neglecting these effects in a single-channel analysis would over-estimate h significantly (about 13% for the present Ra’ ¼ 62.8). Note that our curve is not quite
3. Results and discussion The present study considers single- and multi-channel parallel vertical plate arrays. The plate height is fixed at 100 mm, the plate spacing is 7 mm, and the plate thickness is 1 mm. Under a fixed temperature difference of Tw-T∞ ¼ 50 K, the Elenbaas Rayleigh number Ra’ is fixed at 62.8. 3.1. Effects of computation domain arrangement First, the heat transfer performances for four different domain arrangements are examined using the single-channel configuration. The domain dimensions are shown in Table 3. Fig. 3 illustrates these domain arrangements and their corresponding hs. Case I is
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267
Table 3 Extension lengths in different arrangements of computation domain.
Case Case Case Case
I: Extended boundaries II: Outlet boundary at channel exit III: Inlet boundary at channel entrance IV: Outlet boundary at channel exit; Inlet boundary at channel entrance
Fig. 3. Average convection heat transfer coefficients for four different domain arrangements, H ¼ 100 mm, b ¼ 7 mm, t ¼ 1.0 mm, (Tw-T∞) ¼ 50 K, Ra’ ¼ 62.8.
Lt (mm)
Lb (mm)
Ls (mm)
100 0 100 0
50 50 0 0
50 50 50 0
the standard with open boundaries extended on all sides. It is consistent with the test arrangement of Elenbaas [1] except that his plate thickness was 6 mm for the 120 120 mm2 plates. The domain of Case II contains inlet extension but no outlet extension. Now, the upper boundary is set as the pressure-outlet boundary, i.e., vu/vy ¼ vv/vy ¼ vT/vy ¼ 0 with the static pressure equal to 1 atm. In contrast, the domain of Case III contains outlet extension but no inlet extension. The lower boundary is set as the pressureinlet boundary, with vu/vy ¼ vv/vy ¼ 0, T ¼ T∞, and the stagnation pressure equal to 1 atm. In Case IV, the domain contains neither the inlet nor the outlet extension. The pressure-outlet boundary and the pressure-inlet boundary are applied at the exit and the inlet of the channel, respectively. The temperature and velocity fields of Case I are shown in Fig. 4. In Fig. 4(a), a hot-plume zone is present above the plates. Fig. 4(b) shows that a convergent flow is drawn into the channel entrance by buoyancy. The h of this single-channel case is 5.75 W/m2K. Case II, with inlet extension but no outlet extension, exhibits a lowest h of 5.56 W/m2K. Case III,
Fig. 4. Temperature and velocity fields for the single channel with H ¼ 100 mm, b ¼ 7 mm, t ¼ 1.0 mm, (Tw-T∞) ¼ 50 K, Ra’ ¼ 62.8.
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Fig. 5. Ratios of hn/hs for multi-channel arrays with different Ns, hn: h in nth channel of the plate arrays; hs: h in a single isolated channel.
with outlet extension but no inlet extension, exhibits the highest h of 6.77 W/m2K, 18% higher than the Case-I value. The lowest Case II value results from two reasons. First, the entrance resistance is accounted for in Case II with the inlet extension. Second, with the outlet boundary set at the channel exit, the favorable effect of the downstream hot-plume buoyancy is excluded. On the contrary, in Case III the entrance resistance is excluded and the favorable hotplume effect is included. As both the entrance resistance and hotplume buoyancy are present in Case I, its h lies between those of Cases II and III. In Case IV, where neither of the two effects is included, h ¼ 6.50 W/m2K, 13% higher than the Case-I value. Comparing Case I with Case III, or Case II with Case IV, excluding entrance resistance can be found to raise h by about 17%. Similarly, comparing Case I with Case II, or Case III with Case IV, excluding hot-plume buoyancy decreases h by about 4%. Therefore, for the single-channel configuration, the entrance resistance appears more influential than the hot-plume buoyancy. Bodoia and Osterle [3] used the computation domain as Case IV,
Fig. 6. Temperature and velocity fields for plate arrays with N ¼ 5 (a, c) and N ¼ 21(b, d) at Ra’ ¼ 62.8.
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while the test arrangement of Elenbaas [1] was like Case I. The Nu results of Bodoia and Osterle [3] are higher than the experimental data of Elenbaas [1] by about 10% for Ra’ > 400. The discrepancies were ascribed to the omission of the entrance flow resistance [4]. Our present result of h for Case IV is higher than that of Case I by about 13%. 3.2. Heat transfer characteristics in multi-channel plate arrays To investigate the heat transfer characteristics of vertical multichannel plate arrays, configurations with N ¼ 5, 11, and 21 are analyzed. Fig. 5 presents the ratio of hn/hs for N ¼ 5, 11, and 21, where hn is the average heat transfer coefficient associated with the nth individual channel, and hs is the average heat transfer coefficient of the single-channel case. Except for the outermost channels for N ¼ 11 and 21, hn/hs > 1. For the same N, hn/hs is higher in a channel closer to the central channel. In addition, this trend is stronger for a larger N. In the central channel of the 21-channel plate array, hn/hs ¼ 1.329, reflecting a 32.9% enhancement of heat transfer compared to the single-channel configuration. Also shown in Fig. 5 is that the suppression of heat transfer in the outermost channels worsens for a larger N. Fig. 6 illustrates the temperature and velocity fields for N ¼ 5 and N ¼ 21. As shown in Fig. 6(a) and (b), the height of the hot-plume zone is maximum at the central channel and diminishes outwards, and the width of the hot-plume zone grows with increasing N. Consequently, the central channel is subjected to the strongest extra hot-plume buoyancy along with the part from the hot channel walls. This explains why hn/hs is higher in the inner channels and this trend appears stronger for a larger N. On the other hand, the entrance-flow characteristics can be seen in the velocity fields shown in Fig. 6 (c) and (d). With a large number of channels, abrupt flow turning around the edge entrance can influence a few neighboring channels. Fig. 7 presents closer illustrations of the flow fields in the outermost channels for N ¼ 5 and N ¼ 21. For N ¼ 5, only in the outermost channel appears a small entrance separation bubble. For N ¼ 21, a larger separation bubble forms in the outermost channel, and a very small one seems to exist in the next channel. That is, the more channels there are in a plate arrays, the stronger edge effect is induced in the outer channels. This is because when there are more channels, the
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buoyancy grows not only with the number of channel walls, but also with the wider collective hot plume. Stronger buoyancy induces larger air flow rate and hence sharper flow turning around the edge entrance. Hence, the flow rates in the outermost channels are reduced by increasing entrance separation resistance. Also can be seen in Fig. 6 (c) and (d) is that, most of the inner channels are unaffected and the cool air enters these channels from below. In these channels, the air flow rates and heat transfer are enhanced due to the extra hot-plume buoyancy. For N ¼ 5, the h of the central channel is 11.7% higher than that of the edge channel, while the mass flow rate is 7.7% higher (not shown); for N ¼ 21, the h of the central channel is 40.3% higher than that of the edge channel, while the mass flow rate is 40.0% higher. Compared to the single-channel case, the h of the central channel for N ¼ 21 is 32.9% higher than hs and the mass flow rate is 32.2% higher. For N ¼ 5, the h of the central channel is 14.7% higher than hs and the mass flow rate is 8.7% higher. In the outermost channel of the N ¼ 21 array, the h is 5.3% lower than hs and the mass flow rate is 5.5% lower. Fig. 8 compares the overall heat transfer coefficients (ho) and the average mass flow rate per channel (m_ o =N) of plate arrays with different numbers of channels. From N ¼ 1 to N ¼ 21, ho increases from 5.75 W/m2K to 7.17 W/m2K, by 24.7%; m_ o =N increases from 1.44 103 kg/s to 1.76 103 kg/s, by 22.2%. These data cast doubt on the conventional assumption ever since Elenbaas [1] that all the channels, except the outermost ones, may be represented by a single channel. In fact, the h values in individual channels are significantly different due to the effects of hot-plume buoyancy and entrance separation. Furthermore, the overall heat transfer rate obtained by N times the heat transfer rate of a single channel would be considerably under-estimated, with a larger error for a larger N. To compare the relative contributions of the hot-plume buoyancy and the entrance resistance, domain arrangements without inlet or outlet extension were investigated. The temperature fields for N ¼ 21 are compared in Fig. 9. Without inlet extension (Fig. 9(a)), ho is 7.91 W/m2K, which is 11.3% higher than the value of 7.17 W/m2K with both inlet and outlet extensions. The increase in ho results from the omission of entrance resistance. When the outlet extension is excluded, ho decreases to 5.87 W/m2K by 17.4%. Hence, the favorable effect of hot-plume buoyancy exceeds the
Fig. 7. Velocity fields near the channel entrances for plate arrays with (a) N ¼ 5 and (b) N ¼ 21 at Ra’ ¼ 62.8.
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Adopting an inlet extension in the two-channel analysis, Floryan and Novak [18] have shown that entrance separation bubble appears at a smaller Ra’ for a narrower channel (smaller b/H). It is noted that the hot plume was omitted in Ref. [18]. Here in the present work at a fixed Ra’ of 62.8, separation is shown to intensify with increasing channel number. 3.3. Virtual chimney effect above vertical plate arrays
Fig. 8. Overall ho and m_ o =N for plate arrays with different Ns.
unfavorable entrance resistance for N ¼ 21. In Section 3.1, it has been shown for a single channel that the entrance resistance is more influential than the hot-plume buoyance. Therefore, the effect of the collective hot plume intensifies with increasing number of channels.
The phenomenon of extra hot-plume buoyancy may be perceived from a different view. It is essentially similar to the chimney effect enabled by attaching adiabatic extensions downstream of the heated channel or channels of a parallel plate array [24], a vertical parallel-plate heat sink [25], or two parallel plates [26e33]. The chimney effect has been intensively studied theoretically [24], experimentally [25e28], and numerically [26,29e33] for various heating conditions or chimney arrangements. The general finding is that the adiabatic extensions can result in significant heat transfer enhancement. However, for the situations without chimney shields, the role of the open plume has been ignored in the literature. For a heated plate array, the buoyancy comes from the hot air in two zones: in the plate channels and
Fig. 9. Temperature fields with (a) lower boundary set at the channel entrances, (b) upper boundary set at the channel exits, Ra’ ¼ 62.8.
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with N ¼ 11. With 100 mm-high chimney shields, the hot-air volume is greatly enlarged. Although the air temperature in the chimney is lower than that in the unshielded hot plume, the overall effect is still stronger. Without shields, ho ¼ 6.67 W/m2K and m_ o =N ¼ 1.61 103 kg/s for N ¼ 11. Compared to the single channel with h of 5.75 W/m2K and m_ o =N ¼ 1.44 103 kg/s, the virtual chimney effect leads to a 16% increase in ho and 11.8% increase in m_ o =N. With actual shield plates, ho is further increased to 8.84 W/ m2K (32.5% increase) and (m_ o =N) increased to 2.51 103 kg/s (55.9% increase). 4. Conclusions
Fig. 10. Computations on a periodic single-channel configuration, (a) computation domain, (b) results of h (Tw ¼ 330 K, T∞ ¼ 300 K).
above the plate array. To elucidate this point, a periodic singlechannel configuration, as shown in Fig. 10(a), is computed with symmetric boundaries along both vertical sides. The results for various Lts are presented in Fig. 10(b). Interestingly, h increases continuously with increasing Lt. This behavior results from the artificially enlarged hot-air zone above the plate array. The extended upper symmetric boundaries act as virtual chimney shields to raise heat transfer in the channel. Analogous with the Archimedes' Principle conventionally applied to an object in liquid, the buoyancy on the hot air is proportional to the product of density difference and the hot-air volume. With increasing Lt, the hot-air volume above the plates increases and thereby leads to higher buoyancy. In fact, the virtual chimney effect exists in the hot plume of a vertical plate array even without real chimney shields. However, without the chimney shields the hot plume narrows upwards with entrainment of surrounding air. Thus, the virtual chimney effect would be weaker than actual chimney effect. Fig. 11 illustrates the temperature fields of an open and a shielded plate array
This study numerically re-examined the classical problems of natural convection from vertical isothermal plate arrays using 2-D steady-state analysis. Extended computation domains encompassing the single- or multi-channel plate arrays are adopted. Also investigated are the consequences of using computation domains without the inlet extension or/and the outlet extension. The plate height is fixed at 100 mm, the plate spacing is 7 mm, the plate thickness is 1 mm, and the temperature difference between the plate wall and ambience is 50 K. These conditions correspond to an Elenbaas Rayleigh number of 62.8. The convection heat transfer _ in the individual channels coefficients (h) and mass flow rates (m) are calculated. The following conclusions have been reached: 1. Setting the inlet boundary at the entrance of the channels would over-estimate the values of h in the outer channels, as the flow resistance associated with the entrance separation therein is ignored. Setting the outlet boundary at the exit of the channel would significantly under-estimate the values of h in the inner channels, because the extra buoyancy provided by the hot plume above the plate arrays is omitted. 2. Multi-channel plate arrays exhibit higher values of h and m_ than a single channel except for the outermost channels. The values of h and m_ are highest at the central channel and lowest at the edge channel, with their differences increasing with increasing
Fig. 11. Temperature fields (a) without and (b) with adiabatic chimney shields, N ¼ 11, Ra’ ¼ 62.8.
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number of channels. For an array with 21 channels (N ¼ 21), h and m_ in the central channel are 32.9% and 32.2% higher than those in a single channel, respectively. In the outermost channels, h and m_ are respectively 5.3% and 5.5% lower than those in a single channel. This phenomenon can be attributed mainly to the stronger extra hot-plume buoyancy above the inner channels, and partly to the higher entrance-separation resistance in the outermost channels. 3. The overall convection heat transfer coefficients (ho) in the plate arrays increase with increasing number of channels, as a result of enhanced hot-plume buoyancy above the arrays. For N ¼ 21, ho is 24.7% higher than the single-channel value. Therefore, the overall heat transfer rates conventionally calculated by N times the heat transfer rate of a single channel would be considerably under-estimated, with a larger error for a larger N. 4. The accuracy of the assumption that all the channels of a multiplate array have similar heat transfer performance, which has been widely accepted since Elenbaas [1] and adopted as the basis for optimum plate spacing estimation, needs reevaluation. 5. The heat transfer enhancement by the extra hot-plume buoyancy for multi-channel plate arrays may be perceived as a virtual chimney effect. As the actual chimney yields enhanced heat transfer [24e33], the open hot plume also leads to heat transfer enhancement. However, without actual chimney shields, the virtual-chimney enhancement is not as large as the actual chimney effect. Future studies will investigate the heat transfer characteristics associated with different Ra's and 3-D array configurations. References [1] Elenbaas W. Heat dissipation of parallel plates by free convection. Physica 1942;9:1e28. [2] Bar-Cohen A, Rohsenow WM. Thermally optimum spacing of vertical, natural convection cooled, parallel plates. J Heat Transf 1984;106:116e23. [3] Bodoia JR, Osterle JF. The development of free convection between heated vertical plates. J Heat Transf 1962;84:40e3. [4] Aung W, Fletcher LS, Sernas V. Developing laminar free convection between vertical flat plates with asymmetric heating. Int J Heat Mass Transf 1972;15: 2293e308. [5] Anand NK, Kim SH, Fletcher LS. The effect of plate spacing on free convection between heated parallel plates. J Heat Transf 1992;114:515e8. [6] Wang X, Pepper DW. Numerical simulation for natural convection in vertical channels. Int J Heat Mass Transf 2009;52:4095e102. [7] Kettleborough CF. Transient laminar free convection between heated vertical plates including entrance effects. Int J Heat Mass Transf 1972;15:883e96. [8] Nakamura H, Asako Y, Naitou T. Heat transfer by free convection between two parallel flat plates. Numer Heat Transf 1982;5:95e106.
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