IOSR Journal of Matheatics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volue 10, Issue 4 Ver. IV (Jul-Aug. 014), PP 10-107 Effect of Darcy Dissipation on Melting Fro a Vertical Plate with Variable Teperature Ebedded In Porous Mediu Uaru Mohaed 1, Hakii Danladi 1 and Aliyu Ishaku Ma ali 1 Departent of Matheatics and Statistics, Federal University of Technology, Minna, Nigeria. Departent of Matheatics/coputer science Ibrahihi Badaasi Babangida University Lapai,Niger State, Nigeria Abstract: The effect of Darcy dissipation on elting fro a vertical plate with variable teperature ebedded in porous ediu is nuerically studied. The partial differential equations governing the proble under consideration have been transfored by a siilarity transforation into a syste of ordinary differential equation which is solved nuerically by Runge-Kutta-Gill ethods. Diensionless velocity, Teperature and concentration profiles are presented graphically for various values of physical paraeter. During the course of integration, it is found that Increasing the values of elting result into the decrease in local nusselt nuber. Keyword: Liquid phase;mied convection; Melting effect; and porous ediu I. Introduction A situation where both the forced and free convection effects are of coparable order is called ied or cobined convection. The analysis of ied convection boundary layer flow along a vertical plate ebedded in a fluid saturated porous edia has received considerable theoretical and practical interest. The phenoenon of ied convection occurs in any technical and industrial probles such as electronic devices cooled by fans, nuclear reactors cooled during an eergency shutdown, a heat echanger placed in a low-velocity environent, solar collectors and so on. Several authors have studied the proble of ied convection about different surface geoetries. The analysis of convective transport in a porous ediu with the inclusion of non-darcian effects has also been a atter of study in recent years. The inertia effect is epected to be iportant at a higher flow rate and it can be accounted for through the addition of a velocity squared ter in the oentu equation, which is known as the Forchheier s etension of the Darcy s law. A detailed review of convective heat transfer in Darcy and non-darcy porous ediu can be found in the book by Nield and Bejan [1]. The study of elting effect is considered by any previous authors. For eaple, Roberts [] firstly presented shielding effect to describe the elting phenoena of ice placed in a hot strea of air at a steady state. Later, fro the point of view of boundary layer theory, fil theory and penetration theory, Tien and Yen [3] studied the effect of elting on convective heat transfer between a elting body and surrounding fluid. Epstein and Cho [4] considered the lainar fil condensation on a vertical elting surface for 1-D and -D systes based on nusselt s ethod to discuss the elting rate. They pointed out that as long as the elting solid is large copared with the thickness of theral boundary layer, transient effects in the solid would be neglected. Sparrow et al. [5] studied the velocity and teperature fields, the heat transfer rate and the elting layer thickness by eans of a finite-difference schee in the elting region for natural convection. In probles dealing with porous edia, the effects of elting, radiation and heat generation or absorption becoe iportant (Bakier [6], Gorla et al. [7], Cheng and Lin, [8], Tashtoush [9],Chakha et al.[10] Kazierczak et al. [11], Kazierczak et al. [1] Chen et al. [13]). The proble of unsteady ied convection boundary layer flow near the stagnation point on a heated vertical plate ebedded in a fluid saturated porous ediu with theral radiation and variable viscosity was investigated by Hassanien and Al-arabi [14]. Murthy et al. [15] considered ied convection flow of an absorbing fluid up a unifor non Darcy porous ediu supported by a sei-infinite ideally transparent vertical flat plate due to solar radiation. Chakha et al. [16] presented a nuerical study of coupled heat and ass transfer by boundary-layer free convection over a vertical flat plate ebedded in a fluid-saturated porous ediu in the presence of therophoretic particle deposition and heat generation or absorption effects. Ali [17] discussed the effect of suction or injection on the free convection boundary layers induced by a heated vertical plate ebedded in a saturated porous ediu with an eponential decaying heat generation. The present study focuses on the effect of Darcy dissipation on elting fro a vertical plate with variable teperature ebedded in porous ediu. The governing syste of non-linear partial differential equations is transfored into siilarity non-linear ordinary differential equations which are solved nuerically using shooting technique together with Runge-Kutta sith-order integration schee. 10 Page
II. Matheatical Forulations Consider the ied convection heat transfer fro a heated vertical plate ebedded in a porous ediu saturated with a Newtonian fluid. It is assued that the plate constitutes the interface between the liquid phase and the solid phase during elting inside the porous atri. The co-ordinate syste and flow odel are shown in the Figure 1.The coordinate is taken along the plate, the y coordinate is easured noral to the plate, while the origin of the reference syste is taken at the leading edge of the plate. The fluid flow is oderate and the pereability of the ediu is assued to be low so that the Forchheier flow odel is applicable and the boundary-drag effect is neglected. The plate is at a variable teperature T at which the aterial of the porous atri elts. The liquid phase teperature is T (>T ) and the teperature of the solid far fro the interface is T (<T ). The flow is steady, lainar and two diensional. With the usual boundary layer and linear Boussinesq approiations, the governing equations, naely the equation of continuity, the oentu and the energy equations for the porous ediu ay be written as u v y u Kg y 0 T T u v y T y T y u c K p In the above equations, K,,, and g are the pereability, equivalent theral diffusivity, kineatic viscosity, theral epansion coefficient and acceleration due to gravity respectively. The above proble is solved subject to the following boundary conditions. dt y 0, T T T A, k 1 Cs T Ts, dy y, T T, u U (5) Where A>0, we will designate as aiding flow when the buoyancy force has a coponent in the T T T A direction of free strea velocity i.e.,the first boundary condition on the elting surface siply stated that the teperature of the interface equals the elting teperature of the aterials saturating the porous atri. The second boundary condition at y=0 is a direct result of heat balance. To seek siilarity solution to equation () and (3) with boundary conditions (4) and (5), we introduce the following diensionless variables u y 1 u 1 f, T T T T Substituting equation (6) into equation () and (3), we obtain the following transfored governing equations: f 0 1 Ec 1 f f f Da 0 (8) The boundary conditions are 0 M 0 0 0, 0, f, 1, f 1 (4) (1) () (3) (6) (7) (9) (10) 103 Page
Where C p M 1 C KgT T T s T T T T s is the elting paraeter, is the ashof nuber U is the ynolds nuber, is the Eckert nuber and is the Darcy nuber The ratio in equation (7) is a easure of the relative iportance of free and force convection and is the controlling paraeter for the present proble. The heat transfer rate along the surface of the plate q w T k y y0 qw can be coputed fro the Fourier heat conduction law. The heat transfer results can be represented by the local Nusselt nuber Nu, which is defined as h q Nu k k T w T Ec C Where h denotes the local heat transfer coefficient and k represent the liquid phase theral conductivity. Substituting equation (6) and (11) into equation (1) we obtain Nu 0 1. (13) Ra III. Method of solution The syste of non-linear ordinary differential equation (7) - (8) together with the boundary conditions (9)-(10) is solved nuerically using the Nuchtshei-Swigerstshooting iteration technique together with sith order Runge-Kutta integration schee. In a shooting ethod, the issing (unspecified) initial condition at the initial point of the interval is assued, and the differential equation is then integrated nuerically as an initial value proble to the terinal point. The accuracy of the assued issing initial condition is then checked by coparing the calculated value of the dependent variable at the terinal point with its given value there. If a difference eists, another value of the issing initial condition ust be assued and the process is repeated. This process is continued until the agreeent between the calculated and the given condition at the terinal point is within the specified degree of accuracy. For this type of iterative approach one naturally inquires whether or not there is a systeatic way of finding each succeeding. The present results are validated by direct coparison with those obtained by Bakier [6] as shown in Table 1. It seen fro this table that both results are in ecellent agreeent, which gives confidence in the nuerical results obtained for the present proble. IV. Discussion of sult In order to test the accuracy of our results, nuerical results are presented graphically in Figures.,3,4 and 5 and in Tables 1,,3 and 4 for several set of values of the pertinent paraeters such as elting paraeter and Eckert nuber. Figures and 3 present the influence of elting paraeter (M) on velocity and Teperature profiles respectively. It is obvious that increasing the elting paraeter causes higher acceleration to the fluid flow which in turn, increases its otion and causes decrease in teperature. This is established by respective increases in the boundary layer thickness of velocity and teperature. Figures 4, 5 show the effect of Eckert nuber (Ec) on velocity and teperature profiles in the ied convection flow. It is seen fro the figures that the velocity curve decreases with the increase of Eckert nuber. However, with the increase of Eckert nuber, there is significant increase in heat generation due to fluid otion and this translates to teperature increase as shown in figure 4. Table present the values of f 0 and 0 which are proportional respectively to the strea function and heat transfer rate fro surface of the plate for various value of elting paraeter and Eckert nuber. It is evident fro the table that f 0 and 0 decrease on increasing elting paraeter. On the other hand, the effect of Eckert nuber on 0 is opposite, that is, Ec has tendency to enhance 0. Table 3 shows the effect of buoyancy paraeter and Eckert nuber on 0 in the presence of T u 0 p T f 0 and Da (1) K (11) 104 Page
elting paraeter. It is observed fro the table that as increases, the strea function at wall decreases and the heat transfer rate increases. Also as the Eckert nuber increases, the strea function at wall decreases and heat transfer rate increases due to heating in the plate. In table 4 values of 0 0 are listed for soe values of λ and Ec. Fro table, it can be seen that as λ increases other hand f 0 decreases and 0 Table1: Values of f and f 0 increases and 0 increases with increase in Eckert nuber. 0, f 0 and f 0 decreases. On the for different values of buoyancy paraeter and elting paraeters, Ec 0 Bakair [4] Present work M / 0 f 0 f 0 0 f 0 f 0 0.4 0.0 0.4570-0.3656 1.0000 0.4570-0.3656 1.0000 1.4 0.678-0.50.4000 0.678-0.50.4000 0 1.6865-1.349 1.000 1.6866-1.3493 1.000.0 0.0 0.799-1.1197 1.0000 0.743-1.097 1.0000 1.4 0.383-1.591.4000 0.3808-1.53.4000 3.0 0.4754-1.9016 4.0000 0.4747-1.8988 4.0000 8.0 0.690 -.7611 9.0000 0.690 -.7607 9.0000 10.0 0.7594-3.0370 11.000 0.7593-3.0375 11.000 0.0 1.0383-4.1530 1.000 1.038-4.159 1.000 Table: Values of M Table3: Values of f 0 and 0 Ec=0.1 for different values of Eckert nuber and elting paraeters. Ec=0. f 0 0 f 0 Ec=0.3 0 f 0 0 0 0.3358 0 3.318 0 4.050 0. -0.8017.004-1.146.8114-1.3704 3.4760 0.4-1.451 1.7814-1.9914.4893 -.458 3.0330 0.8 -.3911 1,4944-3.3387.086-4.0684.548 1.0 -.7896 1.3948-3.8960 1.9480-4.7507.3753 Table4: Values of f 0 and 0 Ec=0.1 for different values of Eckert nuber and buoyacy paraeter. Ec=0. f 0 0 f 0 Ec=0.3 0 f 0 0 0-1.1683 1.1683-1.7549 1.7549 -.1999.1999 0. -1.539 1.539-1.7963 1.7963 -.1966.1966 0.4-1.3535 1.3535-1.9059 1.9059 -.300.300 0.6-1.4618 1.4618 -.0456.0456 -.4890.4890 0.8-1.5761 1.5761 -.011.011 -.6797.6797 f 0 and 0 Ec=0.1 for different values of Eckert nuber and constant paraeter. Ec=0. f 0 0 f 0 Ec=0.3 0 f 0 0 0.1-1.795 1.795 -.4809.4809-3.015 3.015 0. -1.7430 1.7430 -.415.415 -.945.945 0.3-1.6945 1.6945 -.3661.3661 -.883.883 0.4-1.6494 1.6494 -.314.314 -.834.834 0.5-1.6073 1.6073 -.656.656 -.7680.7680 V. Conclusion In this study, the effect of Darcy dissipation on elting fro a vertical plate with variable teperature ebedded in porous ediu are considered. The governing equations are derived using the boundary layer and boussineq approiation. These equations are transfored using a siilarity transforation and then solved by 105 Page
si-order runge-kutta ethod. aphical results for velocity and teperature as well as the nusselt nuber are presented and discuss for different paraetric conditions. It is noted that the velocity and teperature as well as the heat transfer coefficients are significantly affected by the elting and Eckert nuber in the ediu. The heat transfer coefficient is reducing with increasing value of the elting point. Copares with previously published work was perfored, and the results were found to be in good agreeent ference [1]. Nield D A, Bejan A (006) Convection in Porous Media, Springer-Verlag, New York. []. Roberts L (1958). On the elting of a sei-infinite body of ice placed in a hot strea of air, J. Fluid Mech. 4: 505 58. [3]. Tien C, Yen Y C (1965) The effect of elting on forced convection heat transfer, J. Appl. Meteorol. 53 57. [4]. Epstein M, Cho DH (1976) Lainar fil condensation on a vertical elting surface, ASMEJ. Heat Transfer 98: 108 113. [5]. Sparrow E M, Patankar S V, Raadhyani S (1977) Analysis of elting in the presence of natural convection in the elt region, ASME J. Heat Transfer 99: 50 56. [6]. Bakier AY. (1997) Aiding and opposing ied convection flow in elting fro a vertical flat plate ebedded in a porous ediu, Transport in Porous Media. 9:17 139. [7]. Gorla RSR, Mansour MA, Hassanien IA, Bakier AY (1999) Mied convection effect on elting fro a vertical plate, Transport in Porous [8]. Cheng WT, Lin CH (007), Melting effect on ied convection heat transfer with aiding and opposing flows fro the vertical plate in a liquid saturated porous ediu, Int. J. Heat Mass Transfer 50: 306 3034. [9]. Tashtoush B (005) Magnetic and buoyancy effects on elting fro a vertical plate ebedded in saturated porous edia, Energy Conversion and Manageent 46: 566 577. [10]. Chakha A J, Ahed S E, Aloraier A S(010) Melting and radiation effects on ied convection fro a vertical surface ebedded in a non-newtonian fluid saturated non-darcy porous ediu for aiding and opposing eternal flows International Journal of the Physical Sciences Vol. 5(7), pp. 11-14 [11]. Kazierczak M, Poulikakos D, Pop I (1986) Melting fro a flat plate ebedded in a porous ediu in the presence of steady natural convection, Nuer. Heat Transfer 10: 571 581. [1]. Kazierczak M, Poulikakos D, Sadowski D (1987) Melting of a vertical plate in porous ediu controlled by forced convection of a dissiilar fluid, Int. Co. Heat Mass Transfer 14: 507 517. [13]. Chen MM, Farhadieh R, Baker Jr L (1986) On free convection elting of a solid iersed in a hot dissiilar fluid, Int. J. Heat Mass Transfer 9: 1087 1093. [14]. Hassanien IA, Al-arabi TH (009) Non Darcy unsteady ied convection flow near the stagnation point on a heated vertical surface ebedded in a porous ediu with theral radiation and variable viscosity, Coun. Nonlinear. Sci. Nuer. Siulat. 14: 1366 1376. [15]. Murthy PVSN, Mukherjee S, Kharagpur D, Srinivasacharya PVS, Krishna SSR (004) Cobined radiation and ied convection fro a vertical wall with suction/injection in a non-darcy porous ediu, Acta Mech. 168: 145 156. [16]. Chakha AJ, Al-Mudhaf AF, Pop I (006) Effect of heat generation or absorption on therophoretic free convection boundary layer fro a vertical flat plate ebedded in a porous ediu, Int. Co. Heat Mass Transfer 33: 1096 110. [17]. Ali M E (007) The effect of lateral ass flu on the natural convection boundary layers induced by a heated vertical plate ebedded in a saturated porous ediu with internal heat generation, Int. J. Ther. Sci. 46: 157 163. Figures n Figure 1 : The investigated physical odel Figure. Velocity profile for different values of M, /=1.0, Ec=0.1,λ=0.3 and Da=0.1 106 Page
Figure 3. Teperature profile for different values of M /=1.0, Ec=0.1,λ=0.3 and Da=0.1 Figure 4.Velocity profile for different values of Ec, M=0.5, /=1.0, λ=0.3and Da=0.1 Figure 5.Teperature profile for different values of Ec, M=0.5, /=1.0, λ=0.3and Da=0.1 107 Page