J. Basic. Appl. Sci. Res., 2(7)7270-7275, 2012 2012, TextRoad Publication ISSN 2090-4304 Journal o Basic and Applied Scientiic Research www.textroad.com Controlling the Heat Flux Distribution by Changing the Thickness o Heated Wall Mohammad Reza Hajmohammadi 1, Abouzar Pouzesh 2 and Sadegh Poozesh 1* 1 Department o Mechanical Engineering, Amirkabir University o Technology, Tehran, Iran. 2 Depatrtment o Mechanical Engineering, Gachsaran branch, Islamic Azad University, Gachsaran, Iran ABSTRACT In this paper the eect o the thickness o a trapezoidal wall placed between a heat source and a cold luid upon the hot spot temperature o the system (luid, sources, and the wall) was considered. It was shown that as the wall thickness increases, the axial conduction overweighs the increase in the thermal resistance and thereore, the hot spot temperature decreases. But, when the thickness exceeds a certain value, the increase in the thermal resistance overweighs the axial conduction and this makes the hot spot temperature increase. Obviously, there should be an optimal value or the wall thickness which will be extracted. Numerical simulation is perormed or a heat transer problem via ANSYS FLUENT 12.0.1 environment to optimize the thickness o a plate where the lower surace is heated with uniorm heat lux. At the other surace o the plate there is a laminar boundary layer low. It can be shown that by using a trapezoidal wall, the hot spot temperature o the wall reduces up to25.06%. Keywords: Constructal theory, Boundary layer, Optimization. INTROUCTION Problems discussing the eect o thickness o solid wall upon heat conduction and convective heat transer coeicient o the luid in contact with the solid wall are called Conjugated Heat Transer. These problems have been considered vastly due to their eects on thermal boundary layers and heat transer coeicient. The unknown temperature distribution at the luid-solid interace is the main concern in these problems. Luikov [1] and Payvar [2] studied a case where lower surace was at a constant temperature and the upper one was subjected to the boundary layer o the low. Both right and let sides o the wall were thermally isolated. Using two non-exact methods, Luikov [1] studied the problem. One method uses the dierential analysis or low-prandtl lows and the other is based on the integral solution by using sine-fourier transorm and considering a polynomial or the velocity and temperature proiles in the boundary layer. He developed an equation or the temperature distribution at the luid-solid interace. However, his equation was so complicated and included complex integrals and series, and thereore, it was very hard to compare the analytical and numerical results. Payvar [2] used the Lighthill approximation to develop an equation in integral orm or calculating the temperature distribution at the luid-solid interace and then, solved it numerically. The mentioned researches, neglect the heat conduction along the solid wall. Using iterative methods, Karvinen [3] studied the problem o orced convection over a lat plate with inite thickness and internal heat generation. Trevinoand linanh [4] studied an almost similar case but there was a uniorm, constant heat lux to the lower solid wall instead o internal heat generation in their case. They deined the parameter α as the wall capability to conduct heat to luid capability to transer heat rom the wall. Using Regular Perturbation or huge values o α and also Singular Perturbation or very small values o α, they extracted the integral-dierential orm o equations to calculate the temperature distribution at the interace and also solved it numerically. They considered a wall with negligible thickness and also, in the direction normal to the wall treated the temperature as a lumped variable. The numerical study reported by Campo and Schuler [5] used inite dierence to investigate the eects o thickness o the solid wall on the heat transer characteristics o the 1- and 2-dimentional low in a pipe. Vaporization over a solid wall o inite thickness was theoretically studied by Mori etal [6]. The region near to the wall has been considered 2-dimensional and then, temperature distribution and Nu number have been calculated when Prandtl and Schmidt numbers are equal to 1. Using the integral orm o the momentum and energy equations, Pozzy and Tognaccini [7] did research into a orced low over a semi-ininite lat plate and obtained the temperature at the luid-solid interace or a compressible, laminar low where Pr=1. Recently by using numerical methods, * Corresponding Author: Sadegh Poozesh, Department o Mechanical Engineering, Amirkabir University o Technology, Tehran, Iran. Email: Sadegh.poozesh@gmail.com and Phone: +98917741188 7270
Hajmohammadi et al., 2012 Mobedi [8] ound the heat transer coeicient in a square cavity with inite thickness horizontal walls where vertical walls were at a constant temperature and horizontal walls were adiabatic. Moreover, many did research into conjugated heat transer in order to optimize the thermal operation and obtain the optimum values or wall thickness to control the behavior o boundary layer. For example, Lim and Bejan [9] and Arici [10] investigated the optimal thickness o the solid walls. Lim and Bejan [9] investigated the optimal thickness o a wall, δ and k, with convection on one side where the lower surace was at constant temperature and the upper one was subjected to orced laminar low. They tried to determine the wall thickness such that the heat transer rom the wall was minimum. Using an analytical method and neglecting the vertical variation o wall temperature, they showed that the wall thickness had to decrease in the low direction. Besides, they reported that or Biot numbers less than 1, the decrease in the wall thickness had more eect on the heat transer. Then, they proved it numerically. Arici [10] did research on the optimal wall thickness o a pipe. They tried to reduce the axial conduction in the wall and increase the heat transer rom the low inside the pipe. They showed that to accomplish this goal, the wall thickness should reduce in the low direction. In this paper, we use the axial conduction along the wall by thickening it as a method to deal with this problem. The eect o axial heat conduction along the wall will be taken into account where the heat lux at the lower surace is assumed to be uniorm. The optimal value or dierent values o parameters aecting the problem will be obtained. The problem is simply simulated via ANSYS FLUENT (Release 12.0.1, 2009) environment but the results are worthwhile to be presented. Problem Description Contemplate Figure 1 in which conjugate heat transer takes place in a steady, two-dimensional regime. The lower surace o the plate is under uniorm and constant heat lux, q, and there is no heat generation inside the trapezoidal solid plate. The incompressible luid is lowing over the upper surace o the plate with the ree stream o velocity U and temperature T. The let and right sides o the trapezoidal plate are considered insulated whereas a constant heat lux is traveling through the bottom side toward the top o the plate. The upper side o the solid makes direct contact with the lowing luid and heat transer to the luid occurs via this surace. The eect o the plate thickness, b, and accordingly the heat conduction through the plate in both directions x and y on the hot spot temperature are considered since the 2D energy equations are solved in the both solid plate and luid domains in our analysis. It is noteworthy that in case o assuming constant heat lux over the lower surace o the plate, minimizing the hot spot temperature at the plate will be posed and in dimensionless orm we have: Tmax T max (1) '' q L / k s The global unction θ max can be determined via numerical simulation in ANSYS FLUENT environment, by solving or the temperature ield in each assumed coniguration while varying the dimension b, and then calculating θ max to see whether θ max can be minimized by arranging the coniguration in another style. Figure 1: Schematic o the problem. Follows to the objective o this paper, according to equation (1) the thickness has been assumed to vary linearly. b b 0 0 ( ) L b b (2) L L L L Where b 0 and b L are wall thickness at the beginning and at the end, respectively. 7271
J. Basic. Appl. Sci. Res., 2(7)7270-7275, 2012 Geometry The irst step in every simulation is generating the geometry. The domains are trapezoidal. The length o the domains, L, is 1m.The height o the solid domain is varied as the unknown o the study. By varying the thickness o the plate the boundary condition on the interace surace is varied between the uniorm-constant heat lux and nearly uniorm-constant temperature. As the thickness o the laminar boundary layer o the orced convection low at the end o the boundary layer or uniorm-constant temperature boundary condition on the wall, δ L,T, is thicker than that or the uniorm-constant heat lux boundary condition on the wall, δ L,H, the height o the luid domain is set to 10* δ L,T. This height is estimated as ollows using the classical result or the wall with uniorm temperature on the wall classical ormula [11], H 50L (3) Re L The height o the luid domain calculated by the above equation is varied by dierent number o Reynolds number, Re L. Not that in this study the thickness o the thermal boundary layer is thinner than the hydraulic one since Prandtl number, Pr, or the luid is larger than unity. Mesh The next step is to create Mesh. The mesh is generated in tetrahedrons (path independent) environment. The grid o luid domain is non-uniorm in both x and y directions, such that the grid adjacent to the walls and to the beginning o the boundary layer is iner compared to that in the other regions. The purpose o such ine mesh is to capture sharp gradients in the both hydraulic and thermal boundary layer. The appropriate mesh size was determined by successive reinements, increasing the number o elements our times rom the current mesh size to the next mesh j j 1 3 size, until the criterion j / 10 is satisied, such that the independency o the solutions with max max max j respect to the grid is achieved. Here, max represents the maximum temperature calculated using the current mesh j1 size, and max corresponds to the temperature using the next mesh, where the number o elements was increased by our times. The ollowing results are perormed by using between 10,000 and 20,000 rectangular elements or the luid domain and between 1000 and 5000 rectangular elements or solid domain. Determination o the properties o the domains and the corresponding boundary conditions is the next step to simulate the physics o the problem. The luid and the solid material are water liquid and Aluminium, respectively. Solution Our solution is perormed using FLUENT solver in ANSYS environment. The scheme o pressure-velocity coupling is set to SYMPLEC which is an appropriate option or uncomplicated problems, where convergence depends on pressure-velocity coupling. Like most commercial CFD packages, ANSYS Fluent uses a inite volume approach to convert the governing partial dierential equations into a system o discrete algebraic equations by discretizing the computational domain. 2ndorder discretization method is selected or both low and energy equations. Also the convergence criteria or all the equations is set to 10-8. Such high order discretization schemes and tight convergence criteria are desirable or accurate resolution o boundary thermal boundary layer. RESULTS AND DISCUSSION According to Re. [1], the non-dimensional temperature ield in the described problem is only the unction o one ree parameter, α, which represents the ratio o the ability o the plate to carry heat in the stream wise direction to the ability o the luid to carry heat rom the plate. This parameter is deined as in Re. [12], 2 / 3 2ksbPr (4) " 3 / 2 (0) ( u ) C L p Where, " (0) denotes the second derivative o the Blasius unction evaluated at the plate (0.4695). Also the nondimensional temperature in reerence [12] is given as, 7272
Hajmohammadi et al., 2012 ~ C T p ( u " ) (2L) q (0) Pr " 2/ 3 ( T T ) (5) Thus according to Eq. 3 and Eq. 4 we can yield that the unction θ max can be expressed as ollows, 1/ 2 1/ 3 max Re Pr ( m, b / L) (6) Where, 1/2 1/3 ks m Re Pr (7) k And φ is a particular unction. Consequently, the global unction θ max can be determined numerically by solving or the temperature ield in each assumed coniguration while varying the dimension b or dierent values o m. or simplicity, dierent values o m is provided using dierent values o Reynolds number in Eq. 6, and dierent values o Reynolds number is provided by applying dierent values o inlet velocity, U, in our simulation. It can be concluded rom equation (2) that b L and b L are two ree parameters o this case. In addition, L which is used to nondimensionalize Length and temperature, is the length o the inclined plane subjected to the heat lux so that the length o this plane and the rate o heat transerred rom the plane to luid remain constant. Thereore, i the length o the upper surace is a, the ollowing relation can be written. a b b L L L L 0 2 1 ( ) (8) As you notice, in all Figure 2, there is an optimal value or b /L or dierent values o b /L. In act in each case, at a constant b /L as b /L grows, temperature decreases irst and then, it increases because as b /L increases, the axial conduction becomes stronger at the region near to the end o the wall and the surace in contact with the cold luid, based on the relation (8) increases too. On the other hand, increase in the thermal resistance in the y-direction, like the previous case, prevent us rom increasing b /L. Figure 2: Eect o b /L on hot temperature or dierent values o b /L or α = 1.118. 7273
J. Basic. Appl. Sci. Res., 2(7)7270-7275, 2012 In Figure 3, the variation o optimal b /L and the corresponding hot spot temperature with respect to b /L at α = 1.118, have been shown. There is an optimal value or b /L in this igure, which means there is an optimal trapezoidal geometry or the wall so that the hot spot temperature gets as minimum as possible. Results show that atα = 2.5, by using a trapezoidal wall, the hot spot temperature reduces up to 25.06%. Conclusion Figure 3: the variation o optimal b /L and the corresponding hot spot temperature with respect to. In the present paper, changing the wall thickness can be used to minimize the hot spot temperature or maximize the heat removal in problems where the growth in the boundary layer results in high hot spot temperature. We tried, based on the uniorm distribution o imperections, to have a more uniorm temperature distribution. The eect o the thickness o a trapezoidal wall placed between a heat source and a cold luid (orced convection) upon the hot spot temperature o the system (luid, sources, and the wall) was considered. It was shown that as the wall thickness increases, the axial conduction overweighs the increase in the thermal resistance and thereore, the hot spot temperature decreases. But, when the thickness exceeds a certain value, the increase in the thermal resistance overweighs the axial conduction and this makes the hot spot temperature increase. Obviously, there should be an optimal value or the wall thickness which was extracted and also, it was calculated that as wall has its optimal thickness, the hot spot temperature decreases up to 25%. REFERENCES 1. Luikov, A. V, 1974. "Conjugated convective heat transer problems", Int. J. Heat Mass Transer, 17, pp: 257-265. 2. Payvar, P, 1977."Convective heat transer to laminar low over a plate o inite thickness", Int. J. Heat Mass Transer, 20, pp: 431-433. 3. Karvinen, R, 1978. "Note on conjugated heat transer in a lat plate", Int. J. Heat Mass Transer, 5, pp: 197-202. 4. Trevino, C., and linanh, a, 1984. "Forced convection over a lat plate with internal heat generation" Int. J. Heat Mass Transer, 27, pp: 1067-1073. 5. Campo, A., and Schüler, C. A, 1988. "Heat transer in laminar low through circular tubes accounting or twodimensional wall conduction", Int. J. Heat Mass Transer, 31, pp: 2251 2259. 6. Mori, S., Nakagawat, H., tanimoto, A. and Sakakibara, M. 1997. "Heat and mass transer with a boundary layer low past a lat plate o inite thickness", Int. J. Heat Mass Transer, 34, pp: 2899-2909. 7. Pozzi, A. and Tognaccini, R, 2000. "Coupling o conduction and convection past an impulsively started semiinnite lat plate", Int. J. Heat Mass Transer, 43 pp: 1121-1131. 8. Mobedi, M, 2008. "Conjugate natural convection in a square cavity with inite thickness horizontal walls", International Communications in Heat and Mass Transer, 35 pp: 503 513. 9. Lim, J. S., and Bejan, A, 1901. "The optimal thickness o a wall with convection on one side", International Communications in Heat and Mass Transer 35, pp: 161-167. 7274
Hajmohammadi et al., 2012 10. Arici, M. E., 2002. "Determination and Use o the Optimal Variation o Pipe Wall Thickness For Laminar Forced Convection", International Communications in Heat and Mass Transer, 29, pp: 663-672. 11. A. Bejan, 2004. "Convection Heat Transer". Hoboken, NJ: Wiley. 12. Da Silva, A.K., Lorente, S., and Bejan, A, 2005. Constructal multi-scale structures with asymmetric heat sources o inite thickness. Int J Heat Mass Transer, 48, pp: 2662 2672. List o Symbols b C p h k k s L plate thickness luid heat capacity at constant pressure Convective heat transer coeicient luid thermal conductivity solid plate thermal conductivity plate length m ree parameter deined in Eq. 6 Pr q Re T max prandtl number Heat lux Reynolds number based on plate length, Re= ρ U L/μ maximum temperature at the plate ~ T non-dimentional temperature deined in Eq. 4 T U Free stream temperature Free stream velocity Greek symbols θ max non-dimensional maximum temperature at the plate deined in Eq. 5 μ luid kinematic viscosity α ree parameter deined in Eq. 3 ρ luid density 7275