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Engineering e-Transaction (ISSN 1823-6379) Vol. 7, No.2, December 2012, pp 54-61 Online at http://ejum.fsktm.um.edu.my Received 9 July, 2012; Accepted 19 Sep, 2012 MATHEMATICAL MODELING FOR HEAT TRANSFER ENHANCEMENT IN A VENTILATED ENCLOSURE WITH A CIRCULAR HEAT-GENERATING BLOCK M.M. Billaha, M.J.H. Munshib, A.R. Khanc, A. Rashidc and M.M. Rahmanb a Department of Arts and Sciences Ahsanullah University of Engineering and Technology (AUST), Dhaka-1208, Bangladesh b Department of Mathematics Hamdard University Bangladesh, Gozaria, Munshiganj, Bangladesh c Department of Mathematics Jahangirnagar University, Savar, Dhaka-1342, Bangladesh d Department of Mathematics, Bangladesh University of Engineering & Technology (BUET), Dhaka-1000, Bangladesh Email: [email protected] convective heat transfer from periodic open cavities in a channel with oscillatory through flow. Khanafer et al. (2002) performed a numerical investigation on mixed convection heat transfer in open-ended enclosures for three different flow angles. They found that thermal insulation of cavity can be achieved through the use of high horizontal velocity flow. A numerical analysis of laminar mixed convection in a channel with an open cavity and a heated wall bounded by a horizontally insulated plate was presented by Manca et al. (2003), where the authors considered three heating modes: assisting flow, opposing flow and heating from below. Later, a similar problem for the case of assisting forced flow configuration was tested experimentally by Manca et al. (2006). Leong et al. (2005) performed a numerical study on the mixed convection from an open cavity in a horizontal channel. Authors found that the heat transfer rate was reduced, and the flow became unstable in the mixed convection regime. Aminossadati and Ghasemib (2009) performed a numerical study on the mixed convection in a horizontal channel with a discrete heat source in an open cavity. They considered three different heating modes and found noticeable differences among the indicated three heating modes. Wong and Saeid (2009) numerically investigated the opposing mixed convection arises from jet impingement cooling of a heated bottom surface of an open cavity in a horizontal channel filled with porous medium. ABSTRACT In this article, mixed convection flow and heat transfer characteristics inside a ventilated enclosure have been investigated. A heat-generating circular block positioned at the centre of the enclosure. Discretization of governing equations is achieved using a finite element technique based on Galerkin weighted residuals. The computation is carried out for a wide range of pertinent parameters such as Aspect ratio and Richardson number. Numerical results are reported for the effect of above mentioned parameters on the contours of streamline and isotherm. In addition, the heat transfer rate in terms of the average Nusselt number and average temperature of the fluid in the cavity as well as drag force are presented for the mentioned parametric values. The obtained results show that the flow and thermal field are strongly influenced by the aforesaid parameters. Keywords: Finite Element Method; circular block; ventilated cavity and mixed convection. 1.0 INTRODUCTION Mixed convection is that type of heat transfer in which there is a noteworthy interaction between free and forced convection. Mixed convective heat transfer in open/ventilated cavities has long been considered and has received increases attention owing to its application of practical interest, such as nuclear reactors, solar receiver, thermal storage and open cavity packaging of semiconductors. Pavlovic and Penot (1991) made an experimental investigation of the mixed convection heat transfer in an open isothermal cubic cavity. The authors found that the convective heat loss for the central solar receiver. Fusegi (1997) carried out a numerical study on A literature review on the subject shows that a sizeable number of authors had considered mixed convection in enclosures with heated body. Hsu and How (1999) studied mixed convection flow in an enclosure with a finite-size heat source on the sidewall and a heat conducting body. They observed that both the heat 54 transfer coefficient and the dimensionless temperature in the body center strongly depend on the configurations of the system. Omri and Nasrallah (1999) performed the mixed convection effect in an air-cooled cavity with differentially heated vertical isothermal sidewalls having inlet and exit ports by a control volume finite element method. They investigated two different placement configurations of the inlet and exit ports on the sidewalls. Best configuration was selected analyzing the cooling effectiveness of the cavity, which suggested that injecting air through the cold wall was more effective in the heat removal and placing inlet near the bottom and exit near the top produce effective cooling. A numerical model of laminar mixed convection in an open cavity with a heated wall bounded by a horizontally insulated plate was offered by Manca et al. (2003), where three heating modes were considered: assisting flow, opposing flow and heating from below. The effect of the ratio of channel height to the cavity height was found to play a significant role on streamline and isotherm patterns for different heating configurations. Shokouhmand and Sayehvand (2004) carried out a study of flow and heat transfer in a square driven cavity numerically. House et al. (1990) considered natural convection in a vertical square cavity with heat conducting body, placed on center in order to understand the effect of the heat conducting body on the heat transfer process in the cavity. They concluded that the heat transfer across the enclosure enhanced by a body with the thermal conductivity ratio less than unity. Gau and Sharif (2004) conducted mixed convection in rectangular cavities at various aspect ratios with moving isothermal side walls and constant flux heat source on the bottom wall. Bhoite et al. (2005) presented a numerical model for the problem of mixed convection flow and heat transfer in a shallow enclosure with a series of block-like heat generating component for a range of Reynolds and Grashof numbers and block-to-fluid thermal conductivity ratios. Rahman et al. (2009) have done an analysis on mixed convection in a rectangular ventilated cavity with a heat conducting circular cylinder. The authors highlighted the influence of the mixed convection parameter and the cavity aspect ratio on the flow structure and temperature distribution. Gau et al. (2000) performed experiments on mixed convection in a horizontal rectangular channel with side heating. Billah et al. (2011) performed a numerical investigation on heat transfer and flow characteristics for MHD mixed convection in a lid driven cavity with heat generating obstacle. The authors showed that both fluid flow and thermal fields is strongly affected by magnetic field. The literature review presented above indicates that not much attention has been paid to the problem of mixed convection in a ventilated cavity having a circular heat generating block. The present work focuses on conducting a comprehensive study on the effect of various flow and thermal configurations on mixed convection for a wide range of pertinent controlling parameters in a ventilated cavity. These parameters include Aspect ratio AR, and Richardson number Ri. 2.0 MODEL SPECIFICATION The geometry of the problem herein investigated is depicted in Figure 1. A Cartesian co-ordinate system is used with origin at the lower left corner of the computational domain. The system consists of a rectangular cavity with height H and length L, within which a heat generating solid circular block with diameter d. The circular block has a thermal conductivity of k and generates uniform heat q per unit volume. The side walls of the cavity are considered to be adiabatic. It is assumed that the incoming flow is at a uniform velocity, ui and at the ambient temperature, T i. The inlet opening is located on the bottom of the left vertical wall, whereas the outlet opening is at the top of the opposite side wall and the size of the inlet port is w = 0.1L which is the same as the exit port. y w g outle t d lx inlet w ly ui, Ti x L of the studied Fig. 1 Schematic configuration 3.0 MATHEMATICAL FORMULATION A two-dimensional, steady, laminar, incompressible, mixed convection flow is considered inside the enclosure. The fluid properties are assumed to be constant except for the density which is considered to very linearly with temperature according to the Boussinesq approximation. The working fluid is assumed to be air (Pr = 0.071). Taking into account the 55 above mentioned assumptions, the governing equations may be written in the non-dimensional form as follows: = U V 0 X Y (1) U U P 1 2U 2U U V X Y X Re X 2 Y 2 conductivity. The average Nusselt number at the L force D 0 (3) 0 U d and the average temperature of the N 3.1 Computational Procedure The momentum and energy balance equations are the combination of a mixed elliptic-parabolic system of partial differential equations that have been solved by using the Galerkin weighted residual finite element technique by Rahman et al. (2009). The continuity equation has been used as a constraint due to mass conservation. The basic unknowns for the above differential equations are the velocity components (U, g T L , Pr and Ri are V0 2 α Reynolds number, Prandtl number and Richardson V0 L generating parameter in ( T Tb Tc and k C p are N d , Drag cavity volume. (5) Q L (4) Q 0 number, respectively and 1 L fluid is defined as av dV / V , where V is the For solid block the energy equation is Where, Re is the dimensionless ratio of the thermal (2) 1 2 2 V X Y Re Pr X 2 Y 2 K 2 s 2 s RePr X 2 Y 2 k s / k heated surface is calculated as Nu V V P 1 2V 2V Ri U V X Y Y Re X 2 Y 2 U where N is the non-dimensional distances either along X or Y direction acting normal to the surface and K q L2 k s T is the heat V), the temperature and the pressure P. The six node triangular element is used in this work for the development of the finite element equations. All six nodes are associated with velocities as well as temperature; only the corner nodes are associated with pressure. This means that a lower order polynomial is chosen for pressure and, which is satisfied through continuity equation. The velocity component and the temperature distributions and linear interpolation for the pressure distribution according to their highest derivative orders in the differential Equations (2) - (5) the solid block the temperature difference and thermal diffusivity respectively). The above equations are non-dimensionalized by employing the following dimensionless quantities x y u v p d ,Y ,U ,V , P ,D , L L ui ui L ui 2 ly T Ti , Ts Ti l Lx x , L y , L L Th Ti s Th Ti X as U X , Y N U , V X , Y N V , X , Y N , Where X and Y are the coordinates varying along horizontal and vertical directions respectively, U and V are the velocity components in the X and Y directions respectively, θ is the dimensionless temperature and P is the dimensionless pressure. s X , Y N s , P X , Y H P , where β = 1, 2, … … , 6; λ = 1, 2, 3. Substituting the element velocity component distributions, the temperature distribution and the pressure distribution from Equations (2) - (5) the finite element equations can be written in the form The boundary conditions for the present problem are specified as follows: At the inlet: U = 1, V = 0, = 0 U 0,V 0, 0 X X At the solid block boundaries: U 0,V 0, b at the cavity walls: U V 0 N s U U K V U M x P x y At the outlet: K 1 S xx S yy U Q u Re (6) U V K V V M y P x y K K At the fluid-solid interface: N fluid N solid 1 S xx S yy V Ri K Q v Re 56 (7) 1 U K V S S x y RePr xx yy K Table 1 Comparison of average Nusselt numbers for Re = 1000, Gr = 105, C(x = 0.5, y = 0.5) the present data with those of Oztop et al. (2009) (8) Q S xx S yy Q s Oztop et al. (2009) (9) Where the coefficients in element matrices are in the form of the integrals over the element area and along the element edges S0 and Sw as A N N , xdA K y N N , y dA A K N N N dA x A , x K N N N dA y A , y K N N dA A S xx N , x N , x dA A S yy N , y N , y dA A M x H H , x dA A M y H H , y dA A Q u N S x dS0 S0 K x (11) (12) (13) (14) (15) (16) (17) (18) Hot wall Cold wall Hot wall 7.21 9.13 7.03 9.15 7.14 10.37 7.45 10.59 6.78 11.10 7.87 11.65 (19) S0 N S ydS0 (20) Q Sw N q1wdSw (21) Sw N q2wdSw Cold wall right side while the rest of the boundaries of the cavity were insulated. Moreover, the inner body is isothermal. Firstly, the comparison of the average Nusselt number (at the hot wall) between the outcome of the present code and the results found in the literature Oztop et al. (2009) for various diameter of the cylinder as shown in Table 1. The comparisons reveal an excellent agreement with the reported studies. Finally, the streamlines and isotherms for four different values diameter are presented in the Figures 3 and 4 respectively. It can be seen from the figures that the present results and those reported in Oztop et al. (2009). This validation boosts the confidence in the numerical outcome of the present work. (10) Q v Q s Present work (22) The set of non-linear algebraic Equations (6) - (9) are solved using reduced integration technique and Newton-Raphson method [More details in Reddy (1993), Zeinkiewicz et al. (1971), Roy and Basak (2005)]. The convergence of solutions is assumed when the relative error for each variable between consecutive iterations is recorded below the Fig. 2 Grid independency study for different grid elements, while Re = 100, Ri = 1.0, D = 0.2, K = 5.0, and Q = 1.0. convergence criterion ε such that n 1 n 104 , n is 4.0 RESULTS AND DISCUSSION number of iteration and is a function of U, V, θ and θs. A numerical study on mixed convection inside a vented cavity having a heat-generating circular block is governed by different controlling parameters. These parameters are heat generation Q, solid fluid thermal conductivity ratio K, Reynolds number Re, Prandtl number Pr Aspect Ratio AR and Richardson number Ri. Analysis of the results is made for two parameters AR and Ri, which affect flow fields and temperature distribution inside the cavity. Cavity aspect ratio varies 3.2 Code Validation The present numerical code was validated against the problem of mixed convection in a lid-driven enclosure having a circular body studied by Oztop et al. (2009). The cavity was heated at the left wall and cooled at the 57 at the three different regimes of flow, viz., pure forced convection, mixed convection and dominating natural convection with Ri = 0, 1 and 5 respectively. The variations of the average Nusselt number at the heated surface, average fluid temperature in the cavity are plotted for the different values of the aforesaid parameters. a from 0.5 to 2 while other parameters Q = 1, K= 5, Re= 100 and Pr = 7.1 are kept fixed. The results are presented in terms of streamline and isotherm patterns c b Figure 5 shows the effect of cavity aspect ratio on the streamlines (on the left) and isotherms (on the right) at Ri = 0. Four different values of cavity aspect ratios, AR = 0.5, 1, 1.5, and 2, have been considered for assessing the effect. At Ri = 0 and AR = 2, it is observed that a uni-cellular clockwise vortex is developed at the inlet port which we call primary vortices generated due to the heat generating block. The size of the primary cell becomes small with the decreasing AR. It also be seen that the shape of the eddy changes while AR changes. The corresponding effect of cavity aspect ratio for thermal field in the cavity is shown in the right column of Figure 5. Thermal boundary layer thickness decreases and the isothermal lines become closer at the adjacent area of heat source at Ri = 0 for all values AR. Besides thermal plume formed on the heat source changes its shape from inclined to vertical with decreasing AR. AR = 2 AR = 1.5 AR = 1 AR = 0.5 Fig. 3 Comparison of the streamlines between the present work (right) and that of Oztop et al. (2009) (left) at different diameter (a) D = 0.3L, (b) D = 0.4L and (c) D = 0.5L. Fig. 4 Comparison of the isotherms between the present work (right) and that of Oztop et al. (2009) (left) at different diameter (a) D = 0.3L, (b) D = 0.4L and (c) D = 0.5L. Streamlines Isotherms Fig. 5 Streamlines and Isotherms for different values of cavity aspect ratio at Ri = 0. of cavity aspect ratio at Ri = 0. 58 It is noticed that flow field for the lower value of AR (= 0.5) is almost identical with all values of Ri. The corresponding effect of cavity aspect ratio for thermal field in the cavity is presented in the right column of Figure 7. It is noticed that the thermal plume formed on the heat source does not change its shape with changing AR. AR = 1 AR = 0.5 Figure 6 illustrates the effect of cavity aspect ratio on the streamlines (on the left) and isotherms (on the right) at Ri = 1. It is observed that the primary vortices remain unchanged for the lower values of AR (= 0.5) while it compares with those at Ri = 0. The flow field remains almost identical for higher values of AR (= 2, and 1.5) except increasing the size of the primary cell. When AR = 1, a two-cellular secondary vortices developed near the left vertical wall due to the mixed convection effect. The corresponding effect of cavity aspect ratio for thermal field in the cavity is depicted in the right column of Figure 6. Making a comparison of the isothermal lines for different values of AR at Ri = 0 and Ri = 1, a slight change is found. The open isotherm lines increases at the higher values of AR (= 2.0, 1.5). AR = 2 AR = 1.5 Figure 7 presents the influence of cavity aspect ratio on the streamlines (on the left) and isotherms (on the right) at Ri = 10. It is found that the size of the primary vortex increases rapidly. Further, at Ri = 10 and the higher values of AR (= 1.0, 1.5 and 2.0), it is seen that the primary vortex spreads and covers most of the part of the cavity, indicating a sign of supremacy of natural convection in the cavity. AR = 0.5 Streamlines Isotherms Fig. 7 Streamlines and Isotherms for different values of cavity aspect ratio at Ri = 10. AR = 2 AR = 1.5 AR = 1 The effect of cavity aspect ratio on average Nusselt number (Nu), Drag force (Δ) and average fluid temperature (θav) in the cavity at three different values of Ri, is revealed in Figure 8. From the Figure 8, it is clearly observed that the distribution of average Nusselt number (Nu) goes up sharply in the forced convection dominated region (Ri ≤ 1) and beyond this region it goes down gradually in the cavity for increasing values of Ri for AR=1.5 and 2. On the other hand, average Nusselt number increases monotonically for the lowest value of AR (= 0.5). Beside this, average Nusselt number decreases for the value of AR (= 1). Drag force (Δ) decreases slightly in the forced convection dominated region (Ri ≤ 1) and beyond this region it goes up gradually in the cavity for increasing values of Ri for higher values of AR. In contrast, the average Nusselt number increases steadily for the lowest value of AR (= 0.5). On the other hand, for higher values of AR, the average fluid temperature (θav) increases Streamlines Isotherms Fig. 6 Streamlines and Isotherms for different values of cavity aspect ratio at Ri = 1. 59 NOMENCLATURE (i) (ii) d dimensional diameter of the cylinder (m) D dimensionless diameter of the cylinder g gravitational acceleration (ms-2) Gr Grashof number h convective heat transfer coefficient (Wm–2K–1) k thermal conductivity of fluid (Wm-1K-1) ks thermal conductivity of the solid cylinder (Wm1 -1 K ) K Solid fluid thermal conductivity ratio L length of the cavity (m) lx dimensional distance between y-axis and the cylinder center (m) ly dimensional distance between x-axis and the cylinder center (m) Lx dimensionless distance between y-axis and the cylinder center Ly dimensionless distance between x-axis and the cylinder center Nu Nusselt number N non-dimensional distances either X or Y (iii) direction acting normal to the surface Fig. 8 Effect of Aspect Ratio AR on (i) average Nusselt number, (ii) Drag force and (ii) average fluid temperature in the cavity. p dimensional pressure (Nm-2) P dimensionless pressure Pr Prandtl number Ra Rayleigh number dramatically in the forced convection dominated region (Ri ≤ 1) and beyond this region it changes gradually in the cavity for increasing values of Ri. It is also seen that the average fluid temperature decreases smoothly at the lowest value of AR (= 0.5). Re Reynolds number Ri Richardson number T dimensional temperature (K) u, v dimensional velocity components (ms-1) 5.0 CONCLUSION V cavity volume (m3) A computational study is performed to investigate the mixed convection flow in a ventilated enclosure with a heat-generating circular block. The nature of flow and thermal fields as well as characteristics of heat transfer process particularly its augmentation due to the introduction of heat generating circular block has been evaluated in this study. On the basis of the analysis the following conclusions have been drawn. w height of the opening (m) U, V dimensionless velocity components x, y Cartesian coordinates (m) X, Y dimensionless Cartesian coordinates Greek symbols The influence of cavity aspect ratio on fluid flow and temperature field is found to be pronounced. 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