Journal of Mechanical Science and Technology 29 (7) (2015) 2995~3003 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online)
DOI 10.1007/s12206-015-0630-z
Effects of heater location and heater size on the natural convection heat transfer in a square cavity using finite element method† Ich-Long Ngo and Chan Byon* School of Mechanical Engineering, Yeungnam University, Gyeongsan, 712-749, Korea (Manuscript Received September 19, 2014; Revised February 17, 2015; Accepted March 2, 2015) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract Finite element method was used to investigate the effects of heater location and heater size on the natural convection heat transfer in a 2D square cavity heated partially or fully from below and cooled from above. Rayleigh number (5Í102 ≤ Ra ≤ 5Í105), heater size (0.1 ≤ D/L ≤ 1.0), and heater location (0.1 ≤ xh/L ≤ 0.5) were considered. Numerical results indicated that the average Nusselt number (Num) increases as the heater size decreases. In addition, when xh/L is less than 0.4, Num increases as xh/L increases, and Num decreases again for a larger value of xh/L. However, this trend changes when Ra is less than 104, suggesting that Num attains its maximum value at the region close to the bottom surface center. This study aims to gain insight into the behaviors of natural convection in order to potentially improve internal natural convection heat transfer. Keywords: Finite element method; Heat transfer; Natural convection; Square cavity ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction Heat transfer through natural convection flow has received considerable interest because of its broad applications in many engineering systems, such as cooling devices for electronic instruments, energy storage systems, compartment fires, doubleglazed windows, solar collectors, and gas-filled cavities around nuclear reactor cores [1-5]. For several years, researchers have investigated natural convection in a cavities, with various configurations for the heated position on the surrounding boundaries. Watson [6] was one of the pioneers in this topic. An experimental and analytical investigation of natural convection heat transfer in a rectangular cavity heated and cooled from the side walls was later developed by Seki et al. [7], who studied the effects of vessel geometry and the temperature difference between cold and hot walls on the average Nusselt number. Rudraiah et al. [8] also considered this cavity configuration but under the presence of a magnetic field. They showed that the magnetic field decreases the rate of convective heat transfer. By accounting for the effects of surface radiation, correlations of the Nusselt number with the Grashof number were proposed by Ramesh and Venkateshan [9] using a differential interferometer in experiments. Such a cavity configuration has been recently used for studies on heat transfer enhancement by adding nanoscale particles to the base fluid [10-14]. Natural
convection in a corner region formed by a vertical hot wall and a cold horizontal wall was studied by Kimura and Bejan [15] using scale analysis and numerical simulations. Other configurations of a cavity heated from below and cooled from one side were numerically investigated in other studies [1618]. November and Nansteel [19] investigated the natural convection in a cavity heated partially from a vertical or horizontal wall in regard to the effects of the unheated length ratio on the average Nusselt number. Oztop and Abu-Nada [20] numerically studied the natural convection in rectangular enclosures partially heated from the side wall using the finite volume method. Their work indicated that heater size and the location of the heating section affect the flow and temperature field. However, few researchers have addressed the significance of these factors when the heater is placed partially or fully at the bottom or top surface. Finite element method (FEM) was used to determine the effects of heater location and heater size on the natural convection heat transfer in a 2D square cavity heated partially or fully from below and cooled from above. COMSOL was used for the FEM study. The Rayleigh number (5Í102 ≤ Ra ≤ 5Í105), heater size (0.1 ≤ D/L ≤ 1.0), and location of the heating section (0.1 ≤ xh/L ≤ 0.5) were considered.
2. Methodology
*
Corresponding author. Tel.: +82 53 810 2452, Fax.: +82 53 810 4627 E-mail address:
[email protected] † Recommended by Associate Editor Ji Hwan Jeong © KSME & Springer 2015
2.1 Numerical model Fig. 1 shows the schematic diagram of the model and the
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æ ¶ 2T ¶ 2T ¶T ¶T ¶T +u +v = a çç 2 + 2 ¶t ¶x ¶y ¶y è ¶x
ö ÷÷ . ø
(4)
The last term on the right-hand side of Eq. (3) originated from Boussinesq approximation, in which density is assumed as a constant everywhere except in the body force term of the y-momentum equation [28]. The following non-dimensional variables are introduced to solve this problem in a nondimensional form: ta * x * y * uL * vL ;x = ; y = ; u = ;v = ; a a L L L2 T - Tc pL2 p* = ; T* = Th - Tc ra 2
t* =
Fig. 1. Schematic diagram of the model and boundary conditions used for natural convection of air flow in a square cavity.
boundary conditions considered for the analysis of the internal natural convection of air flow in a 2D cavity. The dimensions of the cavity and the heater perpendicular to the plane of the diagram are assumed infinite or long enough that the end effect can be disregarded, as indicated in previous studies [2125]. In addition, Sarris et al. [26] indicated that 2D and 3D solutions had no observable differences from each other when Rayleigh numbers range from 102 to 107. This range covers the range of Ra considered in the present study (from 5×102 to 5×105). The comparison of present results with others, specifically with the 3D simulation results obtained by Fusegi et al. [27], also verify the aforementioned argument, as shown in Fig. 2. Therefore, the problem can be considered to be 2D. All boundary conditions at the walls are imposed as impermeable and no-slip walls. The hot wall characterized by length D and its center position xh has a uniform temperature (Th), whereas the top surface is maintained at lower temperature Tc. The ratios D/L and xh/L denote the heater size and location of the heating section, respectively. The heated location is considered within half of the bottom surface because of the symmetry of this model. The remaining boundaries are specified as adiabatic walls. The air flow (Pr = 0.71) in this model is assumed to be an incompressible laminar flow. Therefore, the governing equations in dimensional form are given as follows: ¶u ¶v + =0 ¶x ¶y
(1)
æ ¶ 2u ¶ 2u ö æ ¶u ¶u ¶u ö ¶p +u +v ÷=+ m çç 2 + 2 ÷÷ ¶x ¶y ø ¶x ¶y ø è ¶t è ¶x
rç
2
¶u* ¶v* + =0 ¶x* ¶y*
(5)
æ ¶ 2u * ¶ 2u * ö ¶u* ¶u* ¶u* ¶p* + u* * + v* * = - * + Pr çç *2 + *2 ÷÷ * ¶t ¶x ¶y ¶x ¶y ø è ¶x
(6)
* * æ ¶ 2 v* ¶ 2 v* ö ¶v* ¶p* * ¶v * ¶v * + u + v = + Pr çç *2 + *2 ÷÷ + Ra Pr T ¶t * ¶x* ¶y* ¶y* ¶y ø è ¶x
(7) ¶T * ¶T * ¶T * ¶ 2T * ¶ 2T * + u* * + v* * = *2 + *2 * ¶t ¶x ¶y ¶x ¶y
(8)
where Ra and Pr are the Rayleigh and Prandtl numbers, respectively. They are defined as: Ra =
r g b (Th - Tc ) L3 hL m and Nu = . ; Pr = k ma ra
(9)
The initial conditions used are t* = 0; u* = v* = 0 and T* = (Th* + Tc*)/2 at 0 ≤ x* ≤ 1 and 0 ≤ y* ≤ 1. From an engineering perspective, the rate of heat transfer from the hot surface is the most important characteristic of the internal convection flow. It is expressed using the average (mean) Nusselt number as follows:
(2)
2
æ¶ v ¶ vö æ ¶v ¶v ¶v ö ¶p r ç + u + v ÷ = - + m çç 2 + 2 ÷÷ + r g b (T - Tc ) t x y y ¶ ¶ ¶ ¶ ¶y ø è ø è ¶x
(3) and
where L is the dimension of the cavity used as a characteristic length, as shown in Fig. 1. The non-dimensional form of the governing equations can be written as:
Nu m = -
1 D/L
ò
xh / L + D/(2L)
xh / L - D/(2L)
Nu x dx*
(10)
where the local Nusselt number ( Nu x ) on the hot wall may be obtained by a non-dimensional analysis of Fourier’s law, which is given as:
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Table 1. Grid independence test (Ra = 105, D/L = xh/L = 0.5 and Pr = 0.71). Grid No.
No. of elements
Num
1
400 (20 Í 20)
5.285483
8.932814
2
1600 (40 Í 40)
5.757625
2.606524
3
3600 (60 Í 60)
5.907699
1.368035 0.883647
4
6400 (80 Í 80)
5.988518
5
10000 (100 Í 100)
6.041436
Relative error (%)
Fig. 2. Comparison of the present work with previous results. Average Nusselt number as a function of Rayleigh number with Pr = 0.7.
Nu x = -
¶T * ¶y*
.
(11)
y* = 0
2.2 Model validation (a)
The numerical model shown in the previous section was implemented in COMSOL Multiphysics 4.3, in which FEM is used to discretize the governing equations into a set of solvable algebraic equations [29]. The computer code used in the present study has been thoroughly validated by comparing the obtained results with those of other research, as shown in Fig. 2. This figure indicates that our results are in very good agreement with others obtained under the same operating conditions. For Rayleigh numbers of 103, 104, 105, and 106, the average Nusselt numbers on the hot wall are 1.118, 2.244, 4.517, and 8.815, respectively, which were obtained using 80 Í 80 grid elements. The maximum relative differences are less than 0.13%, 0.33%, 1.93%, 0.18%, and 2.86% from the results obtained by Khanafer et al. [10], Barakos et al. [30], Markatos et al. [25], De Vahl Davis [31], and Fusegi et al. [27], respectively. The results were also qualitatively compared with those of Khanafer et al. [10], as illustrated in Fig. 3. In this figure, the streamlines and isotherms of the present study match well with those in Ref. [10]. The present work was also validated against the experimental results obtained by Krane and Jessee [32], as shown in Fig. 4. As shown in comparison, both solutions are in good agreement. Therefore, the present work can be easily applied for practical computations in cavities of different aspect ratios, fluids of different Prandtl numbers and even 3D enclosures. To test for grid independence, the average Nusselt numbers on the hot wall were determined for the case of Ra = 105 and Pr = 0.71 with different grid numbers, as listed in Table 1. This table indicates that the Nusselt number converges to a given value. In addition, the relative error is less than unity for grid No. 4. Therefore, to reduce computational cost, the grid
(b)
Fig. 3. Streamlines (a) and isothermal lines; (b) with various Rayleigh numbers for comparing the present work with that of Khanafer et al. [10] at Pr = 0.7.
(a)
(b) Fig. 4. Comparison of the present work with the experimental results by Krane and Jessee [32] (Ra = 1.89×105, Pr = 0.71).
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(a)
(b)
Fig. 6. Average Nusselt numbers evaluated at the hot wall as a function of heater size D/L for various Rayleigh numbers at xh/L = 0.5.
(c)
(d)
Fig. 5. Time history of average Nusselt number for various heater sizes: (a) D/L = 0.1; (b) D/L = 0.4; (c) D/L = 0.5; (d) D/L = 1.0.
model with 6400 (80 Í 80) elements was used in this study.
3. Results and discussions 3.1 Effects of heater size D/L The heat transfer enhancement of natural convection flow in a cavity can be achieved by adjusting the size of the hot or cold walls [19, 20, 22]. The characteristics and mechanism of natural convection flow in each configuration for the hot or cold walls change and depend significantly on the heater size in combination with Rayleigh numbers [23]. These conclusions were re-verified and extended for a wide range of heater size (0.1 ≤ D/L ≤ 1.0) while fixing the location of the heating section at the center of the bottom surface (xh/L = 0.5). To test for solution convergence, the variation of the average Nusselt number on the hot wall in terms of dimensionless time (t*) was first tested for various heater sizes, as indicated in Fig. 5. The numerical solutions of Num converge by around t* = 10 for all cases of D/L and Rayleigh numbers. The average Nusselt number is independent of time when t* > 5, as shown in Fig. 5 for all heater sizes. Therefore, the converged solutions are determined at t* = 10 for all cases. The steadyasymmetric regime obtained was also observed previously [33, 34]. On the basis of the converged solutions, the effects of heater size (D/L) on the average Nusselt number were determined at the various Rayleigh numbers, as shown in Fig. 6. The results indicate that the average Nusselt number increases as the heater size decreases. This finding is in good agreement with a previous study [22]. The average Nu approaches unity when
(a)
(b)
(c)
Fig. 7. (Top) Isotherms and (Bottom) streamlines for various heater sizes at Ra = 103: (a) D/L = 0.2; (b) D/L = 0.5; (c) D/L = 0.9.
D/L approaches unity for Ra ≤ 104. For a large value of D/L, all isothermal lines in the cavity seem to be in the horizontal direction, as illustrated in Fig. 7(c) (Top). This result indicates that the conductive heat transfer dominates in comparison with the convective heat transfer at large D/L and low Ra, thus explaining why the Nusselt number approaches unity. In general, the average Nu increases with increasing Rayleigh number [35, 36]. However, as indicated in Fig. 6, it is approximately constant regardless of increasing Ra in some cases, such as when Ra ≤ 103 for all D/L, when Ra ≤ 5Í103 for D/L = 0.1, and when Ra ≤ 104 for D/L = 1.0. The isothermal lines and streamlines appear symmetric with respect to the cavity’s vertical centerline with two circulations, as shown in Fig. 7 (Bottom). The centers of two circulations are close to the hot wall since the buoyancy-driven force is small at low Ra. The flow regime corresponding to this case is well known to be the steady-symmetric regime. Based on this regime, the thermal and flow characteristics obtained in this study are similar to the laminar symmetric regime proposed by Angeli et al. [24]. At intermediate Ra (Ra = 104), the thermal flow pattern in
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(a)
(b)
(c)
Fig. 8. (Top) Isotherms and (Bottom) streamlines for various heater sizes at Ra = 104: (a) D/L = 0.2; (b) D/L = 0.5; (c) D/L = 0.9.
Fig. 9. Variation of average dimensionless temperature over the vertical centerline of the cavity in terms of dimensionless time at Ra = 104 and D/L = xh/L = 0.5.
the cavity becomes asymmetric, and only one circulation appears in steady state [Figs. 8(a)-(c)]. The experiment of Pallares et al. [37, 38] also indicated that the initially formed S2 structure (with two circulations) changes to a stable S1 structure (with only one circulation) at Ra ≈ 104. These results were analyzed through both experimentation [37] and simulation [38]. To track the formation of this asymmetric phenomenon, the time variation of the average temperature over the vertical centerline of the cavity is presented in Fig. 9. The thermal energy is directly proportional to the temperature in a given system. As indicated in Fig. 9, two symmetric circulations occur at the beginning. Their centers are lifted to the middle of the cavity because of the higher buoyancydriven force, and the thermal energy level is defined at a given value during the initial non-steady heating process (symmetric pattern). When two circulations compete with each other, the one with the lower vortex strength then becomes smaller than the other corresponding to the transient process in the figure. The thermal energy in this process is reduced over time. Finally, only one circulation exists in steady state corresponding to an asymmetric pattern, and the thermal energy level is lower than in the previous state. The thermal energy in this system tends to minimize itself and approach a stable state when the symmetric pattern be-
(a)
(b)
(c)
Fig. 10. (Top) Isotherms and (Bottom) streamlines for various heater sizes at Ra = 5Í105: (a) D/L = 0.2; (b) D/L = 0.5; (c) D/L = 0.9.
comes asymmetric at a given Rayleigh number. This phenomenon is well documented as the symmetric-breaking phenomenon, and it is valid over the range of D/L at Ra = 104. The obtained results for several values of D/L are shown in Fig. 9. The symmetry-breaking phenomenon is not uncommon in confined buoyant flows, as reported by both Angeli et al. [24] and Desrayaud and Lauruat [39]. At high Ra (Ra ≥ 5Í104), the average Nu increases as Ra increases for all cases of D/L, as shown in Fig. 6. The heat transfer is dominated by convection rather than conduction at low Ra, as previously mentioned. This finding is consistent with results obtained by Sarris et al. [26]. In addition, in view of the high buoyancy-driven force, the two circulations appearing in the cavity are lifted up, and their centers are close to the top surface (or cold wall), as evident in Figs. 10(a) and (b) (bottom). However, an asymmetric pattern of thermal flow occurs with large heater size (D/L ≥ 0.7) and very high Rayleigh number (Ra > 105). The asymmetric pattern of the thermal and flow fields is shown in Fig. 10(c). The flow pattern is in good agreement with that of Hasnaoui et al. [40]. 3.2 Effects of heater location xh/L The rate of heat transfer, flow field, and temperature distribution in a cavity are affected by not only the heater size but also the heater location. The local Nusselt number is considerably higher when the hot wall is placed near the side walls [26]. Chu et al. [36] proposed a guideline for locating heaters on the vertical wall of a rectangular channel to obtain the maximum heat transfer. More recently, Türkoglu and Yücel [41] showed that the maximum Nusselt number is obtained when the heater is slightly above the bottom wall and the cooler is slightly below the top horizontal wall of a square cavity heated and cooled from the side walls. The effects of heater location are one of the key design parameters for heat transfer enhancement in numerous engineering applications. The effects of heater location on the heat transfer characteristics of natural convection flow in a square cavity were investigated for the location of the heating section ranging from 0.1
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(a)
(b)
(c)
Fig. 13. (Top) Isotherms and (Bottom) streamlines for various heater locations with Ra = 104: (a) xh/L = 0.1; (b) xh/L = 0.3; (c) xh/L = 0.5. Fig. 11. Average Nusselt number with respect to heater location for various Rayleigh numbers at D/L = 0.2.
(a) (a)
(b)
(c)
Fig. 12. (Top) Isotherms and (Bottom) streamlines for various heater locations with Ra = 103: (a) xh/L = 0.1; (b) xh/L = 0.3; (c) xh/L = 0.5.
to 0.5 for various Rayleigh numbers while keeping the heater size at 0.2. The numerical results in Fig. 11 show that the average Nusselt number increases with the increase in heater location (xh/L ≤ 0.4), but it decreases when the hot wall is located at xh/L = 0.5, except for the cases of low Ra (Ra ≤ 103) and intermediate Ra (Ra = 104). These results imply that the rate of heat transfer (Num) is higher when the heater is placed close to the center of the bottom surface. This finding is consistent with the results obtained by November and Nansteel [18]. The unexpected reduction of average Nu at xh/L = 0.5 and low Ra (Ra ≤ 103) is attributable to the variation of flow pattern with respect to different heater locations. When the heater location is less than 0.5, the steady-state flow pattern is always non-symmetric because of the geometrical asymmetry. This phenomenon is evident in Figs. 12-14. However, when the heater is located at the center of the bottom wall (xh/L = 0.5), the flow structure becomes symmetric at low [Fig. 11(c)] and high Rayleigh numbers [Fig. 14(c)]. As mentioned in the previous section, two circulations in this operating condition compete with each other, which may result in higher thermal resistance for the heat transfer process. The thermal resistance
(b)
(c)
Fig. 14. (Top) Isotherms and (Bottom) streamlines for various heater locations with Ra = 105: (a) xh/L = 0.1; (b) xh/L = 0.3; (c) xh/L = 0.5.
is not significant at low Ra (Ra ≤ 103) and Nusselt number, so it continues to increase in this case, whereas it becomes large enough to reduce the heat transfer rate throughout the entire system at high Ra. This implies that the competition between two circulations at high Rayleigh number confines the heat transfer ability from a hot place to another. Finally, at intermediate Rayleigh number (Ra = 104), only one circulation is formed in all cases of heater location (Fig. 13). The average Nusselt number increases for the entire range of xh/L. The average Nu in the case of Ra = 104 does not appear to decrease at xh/L = 0.5 and differs from those of low and high Ra, as shown in Fig. 11. The reason is that the asymmetric pattern of the thermal and flow fields at intermediate Ra satisfies the essential condition of minimizing the energy in the system, as mentioned in the previous section. This result provides good guidance for placing the heater on the bottom surface. The rate of heat transfer (Num) is most significant when the heater is placed near the center of the bottom surface.
4. Conclusions This study used FEM to present the effects of heater location and heater size on the natural convection heat transfer in a
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2D square cavity heated partially or fully from below and cooled from above. The Navier-Stokes and energy equations were solved computationally with the use of the Boussinesq approximation involved in the buoyancy-driven force term. The major results of this study are as follows: (a) The average Nusselt number increases as the heater size D/L decreases. It approaches unity when D/L approaches unity for Ra ≤ 104. In general, the average Nu increases with increasing Rayleigh number. However, it is approximately constant regardless of increasing Ra when Ra ≤ 103 for all D/L, when Ra ≤ 5Í103 for D/L = 0.1, and when Ra ≤ 104 for D/L = 1.0. (b) The flow pattern in steady state is significantly affected by the heater size in accordance with the Rayleigh number. At intermediate Ra (Ra = 104), the flow pattern becomes asymmetric for all cases of D/L. The thermal energy in this system tends to minimize itself, approaching a stable state when the symmetric pattern becomes asymmetric at a given Rayleigh number. (c) For xh/L ≤ 0.4, the average Nusselt number increases with the increase of heater location, but it decreases when the heater is located at xh/L = 0.5, except for the cases of low Ra (Ra ≤ 103) and intermediate Ra (Ra = 104). These results imply that the heat transfer rate (Num) attains its maximum value at the region close to the center of the bottom surface.
Acknowledgment This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Science, ICT, and Future Planning (2014R1A2A2A01007081), Republic of Korea.
Nomenclature-----------------------------------------------------------------------D g k L n Nu p Pr Ra t T u, v x, y
: Length of hot wall : Gravitational acceleration : Thermal conductivity : Cavity length : Unit normal vector : Nusselt number : Local pressure : Prandtl number : Rayleigh number : Time : Temperature : Velocity components : Horizontal and vertical coordinates
Greek symbols α β μ ρ
: Thermal diffusivity : Coefficient of thermal expansion : Dynamic viscosity : Fluid density
3001
Subscripts c h m x, y
: Cold : Hot : Mean (Average) Nusselt number : Horizontal and vertical direction
Superscripts * T
: Non-dimensional variables : Transposed operation in matrix
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Ich-Long Ngo received his B.S. degree in aeronautical engineering from Hanoi University of Technology, Hanoi, Vietnam, in 2009 and his M.S. degree in mechanical engineering from Changwon National University, Changwon, Korea, in 2013. He is a Ph.D. candidate at Yeungnam University, Gyeongsan, Korea.
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Chan Byon received his B.S., M.S., and Ph.D. degrees in mechanical engineering from KAIST, Daejeon, Korea. He was a visiting scholar at the University of California, Los Angeles, USA, and the Imperial College London, UK, in 2009 and 2012, respectively. He is currently working as a professor at Yeungnam University, Gyeongsan, Korea.