and Technology, Dhaka, Bangladesh c Department of Mechanical Education, Technology Faculty, Fırat

This article was downloaded by: [University of Malaya] On: 04 September 2014, At: 23:36 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/unht20 Modeling of Unsteady Natural Convection for Double-Pipe in a Partially Cooled Enclosure M. M. Rahman a b, Hakan F. Öztop c, S. Mekhilef a, R. Saidur d, A. Ahsan e & Khaled Al-Salem f a Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia b Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh c Department of Mechanical Education, Technology Faculty, Fırat University, Elazig, Turkey d Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia e Department of Civil Engineering, and Institute of Advanced Technology, University Putra Malaysia, Selangor, Malaysia f Department of Mechanical Engineering, College of Engineering, King Saud University, Saudi Arabia Published online: 04 Jun 2014. To cite this article: M. M. Rahman, Hakan F. Öztop, S. Mekhilef, R. Saidur, A. Ahsan & Khaled Al-Salem (2014) Modeling of Unsteady Natural Convection for Double-Pipe in a Partially Cooled Enclosure, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, 66:5, 582-603, DOI: 10.1080/10407782.2014.885254 To link to this article: http://dx.doi.org/10.1080/10407782.2014.885254 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content

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Numerical Heat Transfer, Part A, 66: 582 603, 2014 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407782.2014.885254 MODELING OF UNSTEADY NATURAL CONVECTION FOR DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE M. M. Rahman 1,2, Hakan F. Öztop 3, S. Mekhilef 1, R. Saidur 4, A. Ahsan 5, and Khaled Al-Salem 6 1 Department of Electrical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia 2 Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh 3 Department of Mechanical Education, Technology Faculty, Fırat University, Elazig, Turkey 4 Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia 5 Department of Civil Engineering, and Institute of Advanced Technology, University Putra Malaysia, Selangor, Malaysia 6 Department of Mechanical Engineering, College of Engineering, King Saud University, Saudi Arabia In this article, a model is developed for unsteady natural convection heat transfer and fluid flow in a partially cooled enclosure with a hollow cylinder through it. The right vertical wall of the enclosure is cooled partially. The location of the partial cooling is set up in three different configurations; namely, bottom (P 1 ), middle (P 2 ), and top (P 3 ). A hollow cylinder is located at the middle of the enclosure to simulate a double-pipe heat exchanger. Three values of Grashof number are applied in this work, i.e., 10 4,10 5 and 10 6, and three lengths of the cooler, i.e., 0.2, 0.4 and 0.6. Finite element method was utilized to solve the unsteady dimensionless conservation equations of mass, momentum and energy. It is found that the length and location of cooler does not have a significant effect on the natural convection for the case of the low Grashof number. The maximum heat transfer rate is reached when the cooler is located at the middle of the vertical wall. 1. INTRODUCTION Natural convection heat transfer plays an important role in industrial applications and science, including nuclear system design, solar heating systems, heat exchangers [1], and cooling of electronic devices [2]. In other words, almost all technologies Received 14 June 2013; accepted 21 December 2013. Address correspondence to M. M. Rahman, Department of Mathamatics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh. E-mail: m71ramath@gmail.com or m71ramath@ um.edu.my Color versions of one or more of the figures in the article can be found online at www.tandfonline. com/unht. 582

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 583 NOMENCLATURE g gravitational acceleration, (m=s 2 ) k f fluid thermal conductivity, (W=mK) k s solid thermal conductivity, (W=mK) K solid fluid thermal conductivity ratio, k s =k f L length of the cavity, (m) Lc length of the cooler, (m) N any direction Nu Nusselt number p dimensional pressure, (N=m 2 ) P non-dimensional pressure, Pr Prandtl number, n=a P 1 location of cooler at bottom P 2 location of cooler at middle P 3 location of cooler at top Ra Rayleigh number T Temperature, (K) t dimensional time, s U ¼ (u, v) dimensional velocity components U ¼ (U, V) dimensionless velocity components. V cavity volume x ¼ (x, y) dimensional coordinates X ¼ (X, Y) dimensionless coordinates a thermal diffusivity, (m 2 =s) b thermal expansion coefficient, (1=K) r electrical conductivity m dynamic viscosity, (Pa.s) n kinematic viscosity, (m 2 =s) h non-dimensional temperature, q density, (kg=m 3 ) w stream function n heat function s dimensionless time r laplacian operator C general independent variable Subscripts av h c s average hot cold solid involving passive heat transfer as the main source of thermal dissipation rely on the natural convection mechanism [3]. These applications are also reviewed in earlier works [4 6]. Analysis of natural convection in hollow cylinders has been studied in the literature by many authors. Guj and Stella [7], Shu et al. [8], and Kuhen and Goldstein [9] were among the first to investigate heat and mass flow in annuli. Abu-Hijleh et al. [10] studied local entropy generation due to natural convection from a heated horizontal isothermal cylinder in oil. They observed that viscous dissipation plays a minor role in hollow cylinder configuration. Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids was studied by Abu-Nada et al. [11]. They found that the diameter ratio plays important role on heat transfer. Angeli et al. [12] worked on buoyancy-induced flow regimes for the basic case of a horizontal cylinder enclosed by a long co-axial square-sectioned cavity; a twodimensional time-dependent solution was found. They tested effects of the diameter of the heated cylinder and Rayleigh number on heat and fluid flow. Asan [13] studied the steady, laminar, two-dimensional natural convection in an annulus between two isothermal concentric square ducts. He solved the integral form of the streamfunctionvorticity formulation to find that the average Nusselt numbers on inner and outer squares are mainly dependent on the Rayleigh number and dimension ratio. Yui et al. [14] studied unsteady natural convection heat transfer off a heated horizontal circular cylinder into an air-filled coaxial triangular enclosure. Xu et al. [15] studied the steady, laminar natural convection around a concentric horizontal cylinder inside a triangular enclosure. The enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures. They concluded that at constant

584 M. M. RAHMAN ET AL. Figure 1. Schematic diagram for the problem with boundary conditions and coordinate system [25]. aspect ratio, both the inclination angle and cross-section geometry significantly affect the overall heat transfer rate. In another study [16], they solved the similar problem of annuli but, in this case, a triangular body was inserted inside the circular enclosure. Khanafer and Chamkha [17] analyzed the effects of heat generation rate on heat transfer in a horizontal annulus filled with porous medium. Mamun et al. [18] studied numerically the effect of a heated hollow cylinder on mixed convection in a ventilated cavity. They tested the effects of a hollow cylinder diameter. They observed that the cylinder diameter has a significant effect on both the flow and thermal fields and that the solid-fluid thermal conductivity ratio has significant Figure 2. Grid independency study with s ¼ 0.1, Lc ¼ 0.1 and Ra ¼ 10 5 for the case P 2.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 585 Figure 3. Comparison of results with the literature for average Nusselt number at the heated surface. effect only on the thermal field. Oztop et al. [19] developed a model to work on natural convection in empty tube located beneath a differentially heated enclosure. They also studied the case with a solid cylinder inside a lid-driven cavity to analyze the mixed convection in that configuration [20]. Varol et al. [21] solved the problem of two-dimensional natural convection in a porous-media-filled triangular enclosure with a square body inside. They applied different thermal boundary conditions for Figure 4. Effect of cooler position on streamlines for selected values of s with Lc ¼ 0.1 and Ra ¼ 10 4.

586 M. M. RAHMAN ET AL. Figure 5. Effect of cooler position on streamlines for selected values of s with Lc ¼ 0.1 and Ra ¼ 105. Figure 6. Effect of cooler position on streamlines for selected values of s with Lc ¼ 0.1 and Ra ¼ 106.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 587 the square body and found that temperature field and flow distributions are strongly dependent on the temperature boundary conditions. The number of studies on partially cooled enclosures is very limited, for example, Varol et al. [22, 23] and Oztop [24]. They observed in those studies that location and length of the cooler are the most effective parameters for controlling flow and temperature fields. Billah et al. [25] made some numerical work on fluid flow due to mixed convection in a lid driven cavity with a heated hollow cylinder. Rahman et al. [26, 27] investigated the effect of heat-generating solid body on mixed convection flow in a ventilated cavity. Rahman et al. [28, 29] analyzed magneto hydrodynamic mixed convection and joule heating in a lid-driven cavity with a square block. The present study focuses on modeling the unsteady natural convection heat transfer and flow field in a hollow cylinder located inside a square enclosure with partial cooling. 2. MODEL DEFINITION The problem under investigation here is that of a square-shaped enclosure with a warmer hollow cylinder in the middle of the enclosure [25]. The right wall of the enclosure has partial cooling while other walls are adiabatic. The length of the partial cooler is depicted a s Lc. Heated part of heat exchanger has constant temperature. Gravity acts in the downward vertical direction. The outer cavity is square with wall Figure 7. Effect of cooler position on isotherms for selected values of s with Lc ¼ 0.1 and Ra ¼ 10 4.

588 M. M. RAHMAN ET AL. length L and is filled with air. The diameter of the inner tube is set to one tenth of L and the outer tube two tenths. 3. GOVERNING EQUATIONS AND THEIR NUMERICAL SOLUTION The governing equations are based on the conservation laws of mass, momentum, and thermal energy in two-dimensional forms. Boussinesq approximation for buoyancy-driven flows is adopted in this study. The dimensionless variables are defined as follows. s ¼ ta L 2 ; X ¼ x L ; U ¼ ul a ; P ¼ ðp þ qgy qa 2 ÞL2 ; h ¼ T T c T h T c and h s ¼ T s T c T h T c The governing equations for this problem can be written in dimensionless forms as follows. r:u ¼ 0 qu qs þ U:rU ¼ rpþr2 U þ Ra qh Pr qx ð1þ ð2þ ð3þ Figure 8. Effect of cooler position on isotherms for selected values of s with Lc ¼ 0.1 and Ra ¼ 10 5.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 589 Figure 9. Effect of cooler position on isotherms for selected values of s with Lc ¼ 0.1 and Ra ¼ 10 6. Figure 10. Effect of cooler position on heat function for selected values of s with Lc ¼ 0.1 and Ra ¼ 10 5.

590 M. M. RAHMAN ET AL. qh qs þ U:rh ¼ 1 Pr r2 h ð4þ The energy balance for solids is written as follows. qh s qs þr2 h s ¼ 0 ð5þ As shown in Eqs. (3) (5), two parameters that preside over this problem are the Rayleigh number (Ra), and Prandtl number (Pr), which are shown below as follows. Ra ¼ gbðt h T c ÞL 3 na and Pr ¼ n a ð6þ The velocity at all solid boundaries is set to zero (Eq. (7)). All cavity walls, except for the cooler, are adiabatic (Eq. (8)), and the inner surface of the cylinder is isothermal (Eq. (9)). Equations (10) and (11) shows the boundary conditions for the cooler and the outer surface of the cylinder, respectively. U ¼ 0 qh qn ¼ 0 ð7þ ð8þ Figure 11. Effect of cooler position on heat function for selected values of s with Lc ¼ 0.1 and Ra ¼ 10 6.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 591 h ¼ 1 h ¼ 0 qh ¼ K qh s qn fluid qn solid ð9þ ð10þ ð11þ Where N symbolizes the nondimensional normal direction, and the nondimensional thermal conductivity value is given as K ¼ k s =k f. The Prandtl number is taken as 0.71. The average heat transfer rate in terms of average Nusselt number from heat exchanger is determined by the following. Nu av ¼ 1 p Z p 0 qh qn du and the average temperature of the fluid is evaluated as follows. ð12þ Figure 12. Average Nusselt number versus dimensionless time for cooler position at (a) Ra¼ 10 6, (b) Ra¼ 10 5, and (c) Ra¼ 10 4, while Lc ¼ 0.1.

592 M. M. RAHMAN ET AL. Z h av ¼ h dv=v ð13þ where V is the cavity volume. The nondimensional streamfunction and heatfunction can be defined as follows. U ¼ qw qy ; V ¼ qw qx ð14þ qn qx ¼ Vh 1 qh Pr qy ; qn qy ¼ Uh 1 qh ð15þ Pr qx The governing equations (3) (5) are solved using the finite element formulation with the Galerkin weighted residual technique [30]. Radiation heat transfer is neglected in this study. The coupled governing equations (3) (5) were transformed into sets of algebraic equations using finite element method. The second order triangular type elements were used and the dependent variables are approximated Figure 13. Average fluid temperature in the domain versus dimensionless time for cooler position at (a) Ra¼ 10 6,(b) Ra¼ 10 5, and (c) Ra¼ 10 4, while Lc ¼ 0.1.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 593 Figure 14. Effect of cooler length on streamlines for selected values of s and the case of P2 with Ra ¼ 104. Figure 15. Effect of cooler length on streamlines for selected values of s and the case of P2 with Ra ¼ 105.

594 M. M. RAHMAN ET AL. Figure 16. Effect of cooler length on streamlines for selected values of s and the case of P 2 with Ra ¼ 10 6. Figure 17. Effect of cooler length on isotherms for selected values of s and the case of P 2 with Ra ¼ 10 4.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 595 over each element using shape functions. The resulting algebraic equations were then solved using iterative method. The solution process was continued until the required convergent criterion was satisfied: jc n þ 1 C n j10 6, where n and C symbolize the number of iterations and the general dependent variable, respectively. Figure 2 shows a grid dependency test that lead to the choice of 5,936 elements for the calculations. The accuracy of results is verified with available data by Lee and Ha [31] and is depicted in Figure 3. As shown in the figure, the current results are in good agreement with the published data of Lee and Ha [31]. 4. RESULTS AND DISCUSSION A numerical analysis is performed using the finite element method to simulate unsteady heat and fluid flow in a hollow heated cylinder located inside a square enclosure. The problem was simulated to obtain temperature distribution for heat exchangers. Results were obtained for different cases based on location of cooler in top (P 3 ), middle (P 2 ), and bottom (P 1 ) of the enclosure wall. Streamlines are presented in Figure 4 to show the transient behavior of the solution for different locations of the cooler at Lc ¼ 0.1 and Ra ¼ 10 4. For all cases, two circulating cells are observed inside the cavity with different rotating directions. At the beginning, namely s ¼ 0.01, positions of cells and flow strength are independent from the changing of Figure 18. Effect of cooler length on isotherms for selected values of s and the case of P 2 with Ra ¼ 10 5.

596 M. M. RAHMAN ET AL. location of cooler due to moving hot fluid from the inserted object. As time progresses, flow strength is starting to be affected by the location of the cooler with maximum values of flow strength is obtained as 2.06. At s ¼ 1, the cell on the left is minimized and shifted to the top left corner while the main cell is off-center depending on the position of cooler due to impinging hot and cold fluid at the right side of the inserted body. As expected, flow strength is increased with the advancing of time and two minor cells are formed at the top and bottom left corners. Figure 5 shows the streamlines for Ra ¼ 10 5 and different instances of time. As seen in the figure, results are resembling the results in Figure 4 for s ¼ 0.01. Then, the right cell is getting smaller and the flow of it is twisting the small cell at the top left corner due to mixing of hot and cold fluids. The location of cooler becomes very effective in controlling the location of the circulation cell and flow strength. Thus, thermal boundary layer becomes strongly affected on cooler. Higher flow strength is obtained for P 3 position as 10.15. For the highest value of Rayleigh number, namely Ra ¼ 10 6, two circulating flow cells are impinged above the heat exchanger for s ¼ 0.01, as seen in Figure 6. The location of cooler becomes more significant as the solution evolves in time due to stronger kinetic energy of the fluid. The effect of cooler position on isotherms for selected values of dimensionless time at Lc ¼ 0.1 and Ra ¼ 10 4 is shown in Figure 7. A hollow like temperature distribution is observed around the double pipe for s ¼ 0.01 and cooler is not active in this case. The location of cooler becomes very active with increasing of time and temperature inside the cavity becomes constant depends on the location of the Figure 19. Effect of cooler length on isotherms for selected values of s and the case of P 2 with Ra ¼ 10 6.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 597 cooler. Figures 8 and 9 show the same effect for Ra ¼ 10 5 and Ra ¼ 10 6, respectively. In this case, a plumelike temperature distribution is observed on heated double pipe and direction of the plume change with location of cooler due to heat transport way. Figure 10 shows heat function for different dimensionless times for Lc ¼ 0.1 and Ra ¼ 10 5. Convection effect becomes higher with increasing of Rayleigh number, as shown in Figure 11 due to increasing of kinetic energy. Variation of average Nusselt number with time is presented in Figure 12 for different cooler position at different Rayleigh numbers with constant cooler length, Lc ¼ 0.1. Nusselt numbers are decreased with increasing time. As an expected result, higher heat transfer rate is obtained with higher Rayleigh numbers. Value of average Nusselt number is a function of cooler position. For all cases, the highest heat transfer occurs inside the cavity with the color at P 2 position. For all cases, Nusselt number values remain constant for s 0.5. Similarly, average temperature values are illustrated in Figure 13 with the same parameters of Figure 12. As seen from the figures, the average values of temperature are increased with advancing of time. For all values of Rayleigh numbers, the average temperature value becomes lowest with the cooler at P 2 and the highest value is obtained for the cooler at P 1.Temperature values are almost constant for s 0.5. Figure 14 shows the effect of changing the length of cooler with time for Ra ¼ 10 4. For all values of cooler length, two circulation cells are formed in two sides of the hollow cylinder with w exit ¼ 1.12 at s ¼ 0.01. At this instant of time, the length Figure 20. Effect of cooler length on heat function for selected values of s and the case of P 2 with Ra ¼ 10 5.

598 M. M. RAHMAN ET AL. of the cooler becomes insignificant. As time advances, the cell on the left moves to the top left corner and its strength is decreased with increasing of cooler length. At s ¼ 1, very small circulating cell is formed at the top left corner and a kidney shaped cell is formed near the right side of the heat exchanger and Its strength increases with increasing of cooler length. Effects of cooler length is tested for higher Rayleigh number, namely Ra ¼ 10 5, as shown in Figure 15. In this case, flow strength is increased with increasing of cooler length and Rayleigh number. The shape of the main cell is affected by the length of the cooler while the cell at the top left corner remains unchanged. With increasing Rayleigh number, multiple cells are formed, as seen in Figure 16 for Ra ¼ 10 6. In the figure, cooler length affects the shape of the cell and flow strength for the main flow and the thermal boundary layer becomes more effective as the cooler length increased. Figure 17 presents the isotherms for different values of cooler length and time. As seen from the figure, circular shaped temperature distribution is observed around the heat exchanger for s ¼ 0.01 and Ra ¼ 10 4. As expected, the thermal boundary layer becomes thinner around the heat exchanger. As time advances, isotherms move from the heat exchanger to the ceiling and heat transfer occurs between the heat exchanger and the cooler. Isotherms are diagonal between the heat exchanger and the cooler for Lc ¼ 0.2, and they start to move towards the vertical with the increase of cooler length. For Ra ¼ 10 5, isotherms show wavy variation around the perimeter of the heat exchanger, at the beginning, and they are almost parallel to the horizontal walls at the middle of the cavity for all Figure 21. Effect of cooler length on heat function for selected values of s and the case of P 2 with Ra ¼ 10 6.

DOUBLE-PIPE IN A PARTIALLY COOLED ENCLOSURE 599 lengths of the cooler, as seen in Figure 18. This distribution becomes more apparent for Ra ¼ 10 6 due to strong heat transfer between the cooler and the heat exchanger, as can be seen in Figure 19. This is similar to having a short circuit between the heat exchanger and the cooler where the majority of heat is transferred directly between the two. Figure 20 shows that the length of the cooler makes little effect on heat distribution in this case. The effect of cooler length on heat function at different instances of time for Ra ¼ 10 6 is showed in Figure 21. At the beginning, namely s ¼ 0.01, plume-like distribution is observed starting from the heat exchanger to the top of the cavity. As time advances, the plume moves towards the cooler due to flow motion inside the cavity. The average Nusselt number values are decreased with the increasing of time, as seen in Figures 22a 22c for different Rayleigh numbers. Heat transfer is decreased with the decreasing of the Rayleigh number. For all cases, heat transfer remains constant for s 0.5. The highest heat transfer rate is observed at the highest value of cooler length due to the increase of the heat transfer surface. Figure 23 shows the average fluid temperature in the domain against time for different cooler lengths and the cooler is located at the middle of the wall. Temperatures are increased with Figure 22. Average Nusselt number versus dimensionless time for cooler length of the case P 2 (a) Ra¼ 10 6,(b) Ra¼ 10 5, and (c) Ra¼ 10 4. at

600 M. M. RAHMAN ET AL. Figure 23. Average fluid temperature in the domain versus dimensionless time for cooler length of the case P 2 at (a) Ra¼ 10 6,(b) Ra¼ 10 5, and (c) Ra¼ 10 4. time and it becomes almost constant depending on the Rayleigh numbers. As an expected result, the average temperature is decreased with the increase of cooler length. 5. CONCLUSIONS Numerical work has been performed to simulate natural convection heat transfer from a double-pipe heat exchanger in a partially cooled enclosure. The finite element method was used to solve governing equations. Some important findings can be listed, as follows. 1. Location of the cooler plays an important role on heat transfer, temperature distribution, and flow field, along with the nondimensional time and Rayleigh number. The highest heat transfer rate is observed for the highest value of cooler length. 2. Flow strength decreases with the decreasing cooler length. The cooler length is not a significant parameter on flow field for lower values of Rayleigh numbers.

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