NON-SIMILAR SOLUTIONS FOR NATURAL CONVECTION FROM A MOVING VERTICAL PLATE WITH A CONVECTIVE THERMAL BOUNDARY CONDITION

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1 NON-SIMILAR SOLUTIONS FOR NATURAL CONVECTION FROM A MOVING VERTICAL PLATE WITH A CONVECTIVE THERMAL BOUNDARY CONDITION by Asterios Pantokratoras School o Engineering, Democritus University o Thrace, Xanthi, Greece In a recent paper by Makinde (Thermal Science, 011, Vol. 15, Suppl. 1, pp. S137-S143.) the eect o thermal buoyancy along a moving vertical plate with internal heat generation was considered. The plate thermal boundary condition was a convective condition with a heat transer coeicient 1/ proportional to x. The luid thermal expansion coeicient was 1 proportional to x and the internal heat generation was assumed to decay 1 exponentially across the boundary layer and proportional to x in order that the problem accepts a similarity solution. In the present work, the same problem without heat generation is considered, with constant heat transer coeicient and constant thermal expansion coeicient which is more realistic and has much more practical applications. The present problem is non-similar and results are obtained with the direct numerical solution o the governing equations. The problem is governed by the Prandtl number, the non-dimensional distance along the plate and a convective Grasho number, which is introduced or the irst time. It is ound that the wall shear stress, the wall heat transer and the wall temperature, all increase with increasing distance and the wall temperature tends to 1. The inluence o the convective Grasho number is to increase the wall shear stress and the wall heat transer and to reduce the wall temperature. Key words: moving plate, boundary layer low, heat transer, convective parameter Introduction The above mentioned paper by Makinde [1]concerns the low along a vertical plate moving with constant velocity in a calm luid. The plate thermal boundary condition was a convective condition 1/ with a heat transer coeicient proportional to x and the luid thermal expansion coeicient being 1 proportional to x. In addition internal heat generation exists which changes both along and across 1 the plate (exponentially across the boundary layer and proportional to x in the vertical direction). All the above assumptions have been made in order that the problem accepts a similarity solution. Autor's apantokr@civil.duth.gr 1

2 However, the assumption o a heat transer coeicient varying along the plate as a unction o 1/ x is not realistic and very diicult to be obtained in practice. Concerning the variation o the luid thermal expansion coeicient there is a serious problem. The author mentions in the paper that the governing equations are based on the Boussinesq approximation. However, the Boussinesq approximation is based on the assumption that the luid thermal expansion coeicient is constant and equal to that o the ambient luid (see pages in Schlichting and Gersten [],and page 183 in Bejan [3]).Taking into account the above, the results o Makinde [1]have only theoretical value. In the present work we present results based on constant heat transer coeicient and constant luid thermal expansion coeicient which are compatible with real luids. Non-similar solution is obtained or this problem which depends on the distance along the plate. Problem deinition and solution procedure Consider the low along a vertical semi-ininite plate with u and v denoting respectively the velocity components in the x and y directions, where x is the coordinate along the plate and y is the coordinate perpendicular to x ( see Fig. 1 in Makinde [1]).For a steady, two-dimensional low, the boundary layer equations are u v continuity equation: 0 x (1) u u u momentum equation: u v g ( T T ) () x T T T energy equation: u v a x (3) subject to ollowing boundary conditions at the plate; T u uw, v 0, k h ( T T) on y 0, u 0,T T as y (4) where is the luid kinematic viscosity, β is the luid thermal expansion coeicient, a is the luid thermal diusivity, k is the luid thermal conductivity and T is the luid temperature. It is assumed that the plate is heated by convection rom a luid with constant temperature T with a heat transer coeicient h. Following Merkin and Pop [4] the ollowing dimensionless quantities have been introduced X h x (5) u k w h Y y (6) k

3 u U (7) u w kv V (8) h Pr (9) a T T T T (10) Y (11) 1/ X g ( T T ) k Grc (1) u h The quantity given by eq.(1) is a new non-dimensional parameter, which is introduced here and named as convective Grasho number. Using the above quantities the eq.(1)-(3) take the ollowing dimensionless orm w U X V Y 0 (13) U U U U V X Y Y Grc (14) U X V Y 1 Pr Y (15) The corresponding boundary conditions are: U ( X,0) 1, V( X,0) 0, U( X, ) 0 (16) ( 1 ) Y on Y=0, 0 as Y (17) The eq.(13)-(15) represent a two-dimensional parabolic problem. Such a low has a predominant velocity in the streamwise coordinate which in our case is the direction along the plate. In this type o low convection always dominates the diusion in the streamwise direction. Furthermore, no reverse low is acceptable in the predominant direction. The solution o this problem is obtained using a inite dierence algorithm as described by Patankar [5]. In order to obtain a complete orm o both the temperature and velocity proile at the same cross section we used a nonuniorm lateral grid. ΔY takes 3

4 small values near the surace (dense grid points near the surace) and increases along Y. A total o 500 lateral grid cells were used. It is known that the boundary layer thickness changes along X. For that reason the calculation domain must always be at least equal to or wider than the boundary layer thickness. In each case we tried to have a calculation domain wider than the real boundary layer thickness. This has been done by trial and error. I the calculation domain was thin the velocity and temperature proiles were truncated. In this case we used another wider calculation domain in order to capture the entire velocity and temperature proiles. The parabolic (space marching) solution procedure is described analytically in the textbook o Patankar [5] which remains to this day a model o simplicity and clarity and one o the most coherent explications o the inite volume technique ever written (Acharya and Murthy [6]). The above solution procedure is implicit and unconditionally stable (White [7], page 76), has been used extensively in the literature and has been included in luid mechanics and heat transer textbooks (see Anderson et al. [8]p. 364; White [7], p. 71; and Oosthuizen and Naylor [9], p. 14). The method has been used succeully in a series o papers by the present author (Pantokratoras, [10],Pantokratoras, [11], Pantokratoras, [1],Pantokratoras [13], Pantokratoras [14]). Results and discussion The problem is non-similar and iit s governed by three non-dimensional parameters. The Prandtl number, the convective Grasho number and the non-dimensional distance along the plate expressed by the quantity X {eq.(5)} The most important parameters or this problem are the non-dimensional wall temperature and the non-dimensional wall shear stress as well as the non-dimensional wall heat transer deined as U U'(0) 0 X 1/ U Y Y 0 (18) '(0) 0 X 1/ Y Y 0 (19) Beore applying the current solution procedure to the present problem it was applied to the case considered by Merkin and Pop [4] in order to check its accuracy. The problem considered by Merkin and Pop [4] concerns the low along a motionless plate inside a ree stream without buoyancy. The results are shown in table 1. 4

5 Table 1. Values o dimensionless wall temperature (0) or the non-similar case in orced convection low with constant heat transer coeicient without buoyancy or Pr=1 (validation test) X (0) Merkin And Pop (011) (0) present method Table 1 is a validation test o our numerical solution procedure. The values o Merkin and Pop [4] have been extracted rom their ig.4 due to lack o tabulated data in their work. Taking into account this act the agreement between the results o Merkin and Pop [4] and the results obtained by the present solution procedure is satisactory. Next results are presented or the non-similar case or mixed convection. Table. Values o wall shear stress U '(0), wall heat transer '(0), and wall temperature (0) or dierent parameter values or the non-similar case with constant heat transer coeicient and constant thermal expansion coeicient or Pr Grc 1 Grc 10 Grc 100 X U '(0) '(0) (0) U '(0) '(0) (0) U '(0) '(0) (0) From table, the ollowing conclusions can be drawn. For a ixed value o the distance X, an increase o the convective Grasho number causes a reduction o the non-dimensional wall temperature and an increase in the wall shear stress and wall heat transer. From Figs. 1 and, it is evident that as the convective Grasho number increases, the velocity and temperature boundary layer thicknesses 5

6 decrease. For a ixed value o the convective Grasho number an increase o the distance X causes an increase o the non-dimensional wall temperature which tends to 1. Figure 1. Velocity proiles or Pr=0.7, X=1 and dierent values o Grc. Figure. Temperature proiles or Pr=0.7, X=1 and dierent values o Grc. This means that the wall temperature tends to T, and the problem tends to become identical to the case o mixed convection along a vertical isothermal plate with plate temperature equal to T. In 6

7 addition, as X increases, the wall shear stress and the wall heat transer increase continuously. Figs. 3 and 4 show velocity and temperature proiles at dierent distances X or Grc 10. Figure 3. Velocity proiles or Pr=0.7, Grc=10 and dierent values o X. Figure 4. Temperature proiles or Pr=0.7, Grc=10 and dierent values o X. It can be seen that as X increases the velocity and temperature increases and the proiles become thinner. At very low values o X the velocity is practically identical with that o the Sakiadis low (orced convection) and the corresponding wall shear stress is near the value which is the wall 7

8 shear stress o the classical Sakiadis low ( Schlichting and Gersten [], page 177). The inluence o the Prandtl number is shown in Figs. 5 and 6 where it is seen that, as the Prandtl number increases, the temperature decreases, the thermal boundary layer thickness decreases and the velocity decreases. Figure 5. Velocity proiles or Grc=10, X=1 and dierent values o Prandtl number. Figure 6. Temperature proiles or Grc=10, X=1 and dierent values o Prandtl number. 8

9 Conclusions The mixed convection problem along a vertical plate with convective boundary condition with constant heat transer coeicient and constant thermal expansion coeicient has been investigated in this paper. The main conclusions can be summarized as ollows: 1. When the convective Grasho number increases or speciic value o X and Pr the wall shear stress and the wall heat transer increase, whereas the wall temperature and the velocity and thermal boundary layer thicknesses decrease. For Grc=0 the problem changes rom mixed convection to orced convection and the results o the momentum equation are identical with those o Sakiadis low.. When the distance X is increased, the wall shear stress, the wall heat transer and the wall temperature all increase, whereas the velocity and thermal boundary layer thicknesses decrease. The wall temperature tends to 1 as the distance X takes large values. 3. An increase o the Prandtl number causes a reduction in the temperature and velocity which is accompanied by a thermal boundary layer reduction and an increase in the velocity boundary layer thickness. At very high Prandtl numbers ( Pr ) the buoyancy orce near the plate tends to zero and the low approaches asymptotically the pure orced convection state (Sakiadis low [15]). Nomenclature Grc - convective Grasho number, [-] h - heat transer coeicient, [Wm - k -1 ] k - thermal conductivity, [Wm -1 K -1 ] Pr -Prandtl number, [-] T - luid temperature, [K] T -hot luid temperature, [K] u, v-velocity components, [ms -1 ] U,V-dimensionless velocity components, [-] x,y cartesian co-ordinates, [m] X,Y dimensionless co-ordinates, [-] Greek Letters a - thermal diusivity, [m s -1 ] β - thermal expansion coeicient, [K -1 ] - kinematic viscosity, [m s -1 ] - similarity variable, [-] - dimensionless temperature, [-] 9

10 Reerences [1] Makinde, O. D.,Similarity Solution or Natural Convection rom a Moving Vertical Plate with Internal Heat Generation and a Convective Boundary Condition, Thermal Science, 15 (011) pp. S137-S143. [] Schlichting, H.,Gersten, K., Boundary layer theory, 9th ed., Springer, Berlin, 003 [3] Bejan, A.,Convection Heat Transer, John Wiley & Sons, 3 rd edition, New Jersey, 004 [4] Merkin, J.,Pop, I.,The orced convection low o a uniorm stream over a lat surace with a convective surace boundary condition, Commun Nonlinear Sci Numer Simulat,16 (011),pp [5] Patankar, S.V.,Numerical Heat Transer and Fluid Flow, McGraw-Hill Book Company, New York, 1980 [6] Acharya, S.,Murthy, J., Foreword to the Special Issue on Computational Heat Transer, ASME Journal o Heat Transer,19 (007), pp [7] White, F.,Viscous Fluid Flow, 3 nd ed., McGraw-Hill, New York, 006 [8] Anderson, D., Tannehill, J., Pletcher, R., Computational Fluid Mechanics and Heat Transer, McGraw-Hill, New York,1984 [9] Oosthuizen, P., Naylor, D.,Introduction to Convective Heat Transer Analysis,Graw-Hill, New York,(1999) [10] Pantokratoras, A., Laminar ree convection o pure and saline water along a heated vertical plate, ASME Journal o Heat Transer, Vol. 11 (1999), pp [11] Pantokratoras, A., The classical plane Couette-Poiseuille low with variable luid properties, ASME Journal o Fluids Engineering, 18 (006), pp [1] Pantokratoras, A.,The nonsimilar laminar wall plume in a constant transverse magnetic ield, International Journal o Heat and Mass Transer, 5 (009a), pp [13] Pantokratoras, A.,The nonsimilar laminar wall jet with uniorm blowing or suction: New results, Mechanics Research Communications, 36 (009b), pp [14] Pantokratoras, A.,Nonsimilar aiding mixed convection along a moving cylinder in a ree stream, ZAMP, Vol. 61 (010), pp [15] Sakiadis, B.C.,Boundary layer behavior on continuous solid suraces: The boundary layer on a continuous lat surace, AIChE Journal, 7 (1961), pp