Analogy between Free Convection over a vertical flat plate and Forced Convection over a Wedge

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1 Analogy between Free Convection over a vertical flat plate and Forced Convection over a Wedge B. S. Mohammad and S. Abdallah Aerospace Engineering and Engineering Mechanics University of Cincinnati 71 Rhodes Hall, University of Cincinnati, Cincinnati, OH 451 United States of America abdelnbs@mail.uc.edu Abstract: - Forced and free convection heat transfer phenomena are similar in some aspects, but governed by different equations and correlations. While the forced convection over a wedge is governed by the Flakner-Skan equation derived for incompressible flow, the free convection is governed by the coupled compressible flow equations. In the current research, analogy between free convection over a vertical plate and forced convection over a 10 wedge is developed. Forced convection over a wedge is driven by an eternal pressure gradient while free convection is driven by buoyancy forces that generate a similar pressure gradient. Based on the similarity of pressure gradient, the Nusselt number for free convection over a wall was shown to be equal the Nusselt number for forced convection over a 10 wedge multiplied by a correction factor. The correction factor is a function of the Prandtl number. Numerical results for both modes confirm the developed analogy. Based on our results, the parameters of free convection (Rayleigh and Prandtl) could be used to solve a forced convection problem (that depends on Reynolds and Prandtl) and vice versa. Key-Words: - Analogy, Free Convection, Forced Convection, Similarity, Correlation, Numerical, classical 1 Introduction Convection heat transfer (free and forced) is the science that deals with the energy transport from a hot surface to a moving fluid. The main difference between free and forced convection is that the heat transfer in free convection is due solely to local buoyancy difference caused by the presence of the hot or cold body [1]. Free convection is a very important phenomenon for a number of engineering systems because it plays an important role in the design and performance of systems involving multimode heat transfer effects. Moreover, to minimize operating cost, free convection is often preferred to forced convection in power generating, electric devices and thermal manufacturing applications. Free convection is important in establishing temperature distribution in building and determining heat losses or heat loads for heating, ventilating, and air conditioning systems []. Free convection is in many cases chosen in design rather than forced convection based on the passive safety the system ehibits [3]. Unlike forced convection, free convection does not depend on the electric power to operate a fan or a pump to circulate the coolant to remove the heat. That is the main reason that the Gen IV reactors, epected to be in operation 08, depend on free convection to remove the heat from the core (for eample: Lead Cooled Fast Reactor). This is what they refer to in nuclear industry as passive safety reactors [4]. We would like to emphasize that the literature review in this research is not intended to be about the convective heat transfer which is a wide area of science. However, we are concerned with similarity solutions to convection problems. Similarity solutions are discussed in numerous references, for eample [1, 5-3]. We can divide the similarity solutions that were proposed to solve convective heat transfer problems into two main divisions, geometrical and physical similarity. First, the geometrical similarity which was first developed in 1931 by Falkner and Skan [17]. Basically, the method depends on the geometrical similarity between the velocity profiles along the convective surface. Using the transformation they developed, the governing equations were reduced from partial differential equations to third order non linear equations that could be solved using numerical analysis. Lin and Lin [18] introduced a similarity solution method to solve forced convection for any fluid over surfaces subjected to different boundary conditions (using Runge-Kutta scheme). Hsu et al. [19] used similarity transformation, series epansion method, Runge-Kutta integration and the shooting methods to solve the heat transfer problem over a wedge. Hsu and Hsiao [0] used series epansion, similarity transformation and finite difference to solve the heat transfer problem for a second grade fluid past a fin. B.L. Kuo [1] employed a differential transformation method to obtain series solutions for the Flakner-Skan boundary layer problem. ISSN: ISBN:

2 Second, the physical similarity of convective heat transfer. It is well known that the steady state convection and conduction could be solved similar to a simple resistor circuit. If we consider the Voltage drop ( V) to be equivalent to the temperature difference ( T) and the electrical resistance (R el ) to be equivalent to the thermal resistance (R th ) then simply the current (I) is equivalent to the heat transfer (q) as given by; V T Q= I= = R R el th The Analogy between transient convection and integrator circuit was proposed recently by B. S. Mohammad et al [] and S. Usman et al [3]. They showed that unsteady convection could be modeled using a simple RC electrical circuit. In the current paper we developed another form of physical analogy between free convection over an isothermal vertical flat plate and forced convection over an isothermal wedge. We would like to emphasize that the similarity we are proposing is between an incompressible and a compressible flow which are governed by totally different system of equations. The first is governed by uncoupled system of equations while the second is. As we will show later, the analogy is possible based on the similar pressure gradients even though they have different origins. In different heat transfer tetbooks (for eample F. M. White [10]), the convection problem over a vertical flat plate (forced or free) is usually solved using coordinate transformation. A. Bejan [13] introduced a new similarity variable than the one conventionally used in free convection problems. In this study, the heat transfer by free convection over a vertical flat plate is solved using A. Bejan [13] similarity parameter and the results are analyzed and used to obtain analogy with the forced convection over a wedge. A. Bejan [13] similarity parameter was selected by the authors because it reflects what really happens (growth of the boundary layer) as the Prandtl number increases (this is the other way round in the conventional solution). A complete eplanation can be found in A. Bejan [13] book sections 4.3 and 4.5). Numerical solutions are obtained for the similarity governing equations using double shooting scheme at Prandtl numbers of 0.01, 0.1, 0.7, 1,, 3, 5, 6, 6.7, 10, 30 and 100. To find the analogy between both modes of convection we also needed the solutions for the flow over a wedge with a 10 angle. Similarly, we solved the wedge equations numerically using a single shooting scheme for the same set of Prandtl numbers. This research is divided into four sections. Section one starts with the problem formulation and description for both free convection over an isothermal wall and forced convection over an isothermal wedge. The Numerical solution methodology for the equations is discussed in section two. Section three includes the solutions to the transformed governing equations for both modes of heat transfer. Consequently, the results are analyzed and the analogy is developed in section four. Finally the conclusions of the current research are given. Problem Formulation.1 Governing equations The heated vertical flat plate shown in Fig. 1 results in a flow and a temperature fields adjacent to the wall. The flow and heat transfer fields are governed by the boundary layer equations (Appendi A). The equations are well known. However, we remind the reader that the governing equations for the free convection case are coupled unlike the forced convection governing equations. That is due to the buoyancy force that drives the flow which is controlled by the temperature changes (density difference). Consequently, the problem is more comple than that of forced convection.. Similarity variables and transformed equations The transformation we are following here is well known and well established. However, we have to write the equations down in details because they will be needed to establish the similarity between both modes. First, we begin with the free convection equations. Assume that the similarity variable (η n ) is defined as shown in equation (1), the dimensionless temperature (θ n ) is defined as shown in equation () and the stream function (ψ n ) is defined as shown in equation (3) [13]. The velocities and their derivatives are then easily epressed in terms of F n (function of η and Pr). The velocity in the direction parallel to the wall (u) is given by equation (4) y η 1/4 n = Ra (1) T T θ n = () Tw T ψ 1/4 n =α Ra Fn (3) α u= Ra F n ' (4) Where: Ra is the rayleigh number defined as: gβ 3 ( T-T ) Ra = να ISSN: ISBN:

3 Second, we consider the forced convection over a wedge as shown in Fig. where the flow accelerates in the -direction. In this case the equation of the free stream velocity is given by equation (5) modes. That is the main idea of the problem we are studying and the details are given in section four. U c m = (5) Where φ / π m=, φ / π φ is the wedge angle, and c is a constant We transform the governing equations for forced convection over a wedge. Assume that the similarity variable (η f ) is defined as shown in equation (6), the dimensionless temperature (θ) is defined as shown in equation (7) and the stream function (ψ f ) is defined as shown in equation (8). The velocity in the direction parallel to the wall (u) is given by equation (9). U η f = y (6) ν Fig. 1 Free Convection from a vertical heated flat plate T T θ f = Tw T (7) ψ f = U ν Ff (8) u= U F f ' (9) The analogy we developed in the current research is given in section four. However, a simple eplanation of the main idea is given here. The free convection velocity profile in Fig.1 starts from zero reaches a maimum and then drops to zero again and this region is defined as the hydrodynamic boundary layer thickness. The temperature profile will always go from the wall temperature to the free stream temperature as shown in Fig. 1. On the other hand, the velocity profile for the case of forced convection over a wedge will go from zero and reach the free stream. However, the temperature behavior is the same manner as in the case of free convection. Now, we split the free convection hydrodynamic boundary layer into two parts as shown in Fig. 3. One observes that the shape of the inner portion of the hydrodynamic boundary layer (δ f ) is similar (see Fig. 3) to the shape of the hydrodynamic boundary layer for the case of forced convection over the wedge shown in Fig.. Consequently, one could take the epression of the maimum velocity over the wall in the case of free convection (equation (4)) and apply it as the free stream velocity for a wedge to study the analogy between both Fig. Forced convection over a wedge Fig. 3 Analogy between free and forced convection ISSN: ISBN:

4 3. Numerical Solution Methodology The method that was chosen for solution of the transformed equations is Runge Kutta method. It is well known and documented how forced and free convection are solved in such classical problems. However, the solution steps are summarized as follows (Details given in Appendi B): 1. The main equations were transformed to 1 st order coupled ODE s.. For the case of free convection, two values for F'' ( 0 ) and θ' ( 0) were assumed and a double shooting scheme (coupled equations) was developed to find the correct θ ' 0. For the initial guess we values of F'' ( 0 ) and ( ) made use from the values published in F. M. White [10], B. Gebhart et al. [9], S.Ostrach [5] and S. Kakac [11]. 3. The coupled ODE s were solved using the Runge Kutta. 4. Equations were solved for the set of mentioned Pr numbers. 5. Procedures were repeated for the case of forced convection over a wedge (m=0.5). The initial guesses θ' 0 were taken from W. Kays et al. for F'' ( 0 ) and ( ) [14] and a single shooting scheme (equations are not coupled)was programmed to find the correct values for both parameters. Now after implementing the code we have the values F '' 0, θ ' 0 and θ ' 0 (Table 1) at different Pr of ( ) ( ) ( ) n n f numbers which we will use to develop the analogy F '' 0 is constant and doesn t depend on Pr-equations ( ( ) f are not coupled for forced convection). Details are given in Appendi C. Pr θ ' n ( 0 ) F ( ) n '' 0 θ ' f ( 0 ) E E E E E E E E E E E E+00 Table 1 values of F ''( 0 ), '( 0) and '( 0) θ θ at different n n f Pr numbers. 4. Analogy between forced and free convection The velocity (u), starts from zero at the wall, reaches a maimum and drops to zero again away from the wall for the case of free convection. The plot of F n ' is basically the dimensionless velocity profile (see A. Bejan [13]). Equation (4) shows that u is maimum when F n ' becomes maimum at a specific. Then, the maimum velocity over the vertical plate in the case of free convection is written as shown in equation (10). In addition, for a specific Pr the value of F' n is ma constant and could be obtained easily from the numerical solution. Rearranging, this could be written as shown in equation (11). Going back to the case of forced convection, we note that the flow over a wedge has a free stream velocity that changes as shown by equation (5). Comparing equation (5) to (11), we notice that u ma is analogous to the free stream velocity in the case of forced convection over a wedge with m=0.5. That is because both have the same pressure gradients. Doing the math, we found out that this is a wedge with an angle φ=10 (that s the reason we solved the forced convection for that specific angle). α u ma = Ra F' n (10) ma 3 α gβ T uma = F' n να ma = K gβ T Where: K= αf' n ma να (11) From the solutions to the free convection we have obtained we constructed Table with the values of F' n and corresponding η ma n (Appendi C). From the definition of the Nu equations (1) and (13) could be written for the case of free convection over a wall and forced convection over a wedge respectively (A. Bejan [13]). ( )( ) 1/4 Nu = θ ' n n 0 Ra (1) ( ) Nu = θ ' f f 0 Re (13) For the case of forced convection over a wedge let s assume that the U takes the form of u ma given by equation (10). Substituting in equation (11) we obtain ISSN: ISBN:

5 equation (14). Again equation (14) could be written as equation (15) because the values of Pr, F' n and ma θ' f ( 0) are all constants for a specific problem. U uma Nu ' f f ( 0 = θ ) = θ' f ( 0) ν ν 3 F' α gβ T n Nu ( ) ma = θ' 0 f f να ν ( )( ) 1/4 1 Nu = θ ' 0 Ra F' f f n (14) Pr ma Nuf ( ) 1/4 = A Ra (15) Where: A= Constant = - θ ' ( 0) f F' n ma Pr Comparing equations (1) and (15) one finds out that they are similar. Now, we try to find the constants for a specific problem. Table 3 shows the constants of equations (1) and (15) for the different Prandtl numbers we solved for. Then, we conclude that the Nusselt number in the case of free convection is equal to that of forced convection over a wedge (10 ) multiplied by a correction factor as shown in equation (16). The correction factor is a function of Pr and is shown in Table 4 and plotted in Fig. 4. Equation (17) is a curve fit for the values in the range of Pr=0.1 to 100. Nu = C n r * Nu (16) f C r = Pr (0.01 Pr 100) (17) Now we take a close look at the developed analogy. The forced convection problem is governed with uncoupled momentum and energy equations. However, the free convection is governed with coupled momentum and energy equations. One would think that it is strange that both cases would reduce to be similar. In fact, that was true because the velocity epression used as the free stream for the forced convection is energy dependent (equations (10) and (4)). In other words, the coupling is maintained through the epression of the free stream velocity (equation 10). Table values of Pr F' n ma E F' n ma Pr ' ( 0) at different Pr numbers. θ A n Table 3 Comparison between Nu calculated based on free convection and that calculated based on analogy with forced convection over a wedge with angle =10. Pr Cr Table 4 Correction factor vs. Prandtl number ISSN: ISBN:

6 C r (Correction factor) Pr Fig. 4 Correction factor vs. Prandtl number. 5 Conclusion Using the similarity between the pressure gradient for free convection over a vertical isothermal plate and that for forced convection over a 10 angle isothermal wedge it was proved that the local Nusselt in the case of forced convection reduces to a constant multiplied by local Rayleigh 1/4. This is the same derived equation for the case of free convection over vertical isothermal plate. Therefore, the Nusselt number in the case of free convection can be epressed as that of forced convection multiplied by a correction factor which is a function of the Prandtl number (equation 16). The correction factor was calculated, plotted and fitted for the range of Pr numbers from 0.01 to 100. Hence, the parameters of free convection could be used to solve a forced convection problem and vice versa. References: [1] F. M. White, Heat Transfer, Addison Wesley, [] H.B. Awbi and A. Hatton, Natural convection from heated room surfaces, Energy and Buildings 30, 1999, pp [3] S. Khalil, Francesco D Auria and Mahmoud A. Salehi, Analysis of natural circulation phenomena in VVER-1000, Nuclear Engineering and Design 9, 004, pp [4] Gen IV International forum, available online at: systems/inde.htm [5] S.Ostrach, An analysis of laminar-free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force, NACA TN 635, 195. [6] H. Schlichting, Boundary Layer Theory, McGraw- Hill, Seventh Edition, [7] L. C. Burmeister, Convective Heat Transfer, Wiley, [8] V. S. Arpaci and Larsen, Convection Heat Transfer, Prentice-Hall, [9] B. Gebhart, Y. Jaluria, R. Mahajan, and B. Sammakia, Buoyancy-Induced Flows and Transport, Hemisphere, [10] F. M. White, Viscous Fluid Flow, McGraw Hill, nd edition, [11] S. Kakac and Y. Yener, Convective Heat Transfer, CRC Press, nd edition, [1] P. Oosthuizen and D. Naylor, Introduction to Convective Heat Transfer Analysis, McGraw-Hill, [13] A. Bejan, Convective Heat Transfer, Wiley & Sons, Inc., 3 rd edition, 004. [14] W. Kays, M. Crawford and B. Weigand, Convective Heat and Mass Transfer, McGraw-Hill, 4 th Edition, 005. [15] John H. Lienhard IV and John H. Lienhard V, A Heat Transfer Tetbook, Phlogiston Press, Cambridge Massachusetts, 3 rd edition, 001. [16] F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, Wiley, 5 th Edition, 00. [17] V.M. Falkner, S.W. Skan, Some approimate solutions of the boundary layer equations, Philos. Mag. 1 (80), 1931, pp [18] H.T. Lin, L.K. Lin, Similarity solutions for laminar forced convection heat transfer from wedges to fluids of any Prandtl number, Int. J. Heat Mass Transfer 30, 1987, pp [19] C.H. Hsu, C.S. Chen, J.T. Teng, Temperature and flow fields for the flow of a second grade fluid past a wedge, Int. J. Non-Linear Mech. 3 (5), 1997, pp [0] C.H. Hsu, K.L. Hsiao, Conjugate heat transfer of a plate fin in a second-grade fluid flow, Int. J. Heat Mass Transfer 41(8 9), 1998, pp [1] Bor-Lih Kuo, Heat transfer analysis for the Falkner Skan wedge flow by the differential transformation method, Int. J. Heat and Mass Transfer 48, 005, pp [] B. S. Mohammad, S. Usman and S. Abdallah, Transient response of a natural convection system, International Congress on Advances in Nuclear Power Plants (ICAPP), June 4-8, 006, Reno, NV. [3] S. Usman, S. Abdallah, M. Hawwari, M. Scarangella, L. Shoaib, Integrator Circuit as an Analogy for Convection, Nuclear Technology, 157, January 007, pp ISSN: ISBN:

7 Appendi A Governing equations The governing equations are written in Cartesian coordinates for free convection as follows: u v + = 0 y u u u u + v =ν + gβ T T y y ( ) (18) (19) u T + v T =α T (0) y y At y= 0, u= 0, v= 0, T= T w (1) and at y =, u= 0, T= T Since the continuity and energy equations are the same for free and forced convection then we only show the momentum equation for the case of forced convection over a wedge: u u u m u + v =ν + U y () y At y= 0, u= 0, v= 0, T= T w and at y =, u= U, T= T (3) For the case of free convection, the momentum and energy equations are transformed to equations 4 and 5 respectively (for free convection). The boundary conditions (in terms of the new variables) are given by equations A.9. 3 F nθ ' =θ '' (4) ( n ) 3 F ' Fn F n '' = F n ''' +θ Pr 4 At ηn =0, F n =0, F n'=0, θ=1 and At ηn =, F n'=0, θ=0 (5) (6) For the case of forced convection, the energy and momentum equations are transformed to equations 7 and 8 respectively. The boundary equations (in terms of the new variables) are given by equations 9. It should be mentioned that the transformed energy equation of the wedge is for an isothermal surface. m+ 1 θ '' + Pr F f θ ' = 0 (7) m+ 1 F f ''' + Ff F f '' + m 1 (F f ') = 0 (8) At ηf =0, F f =0, F f '=0, θ=1 and At ηf =, F f '=1, θ=0 (9) Appendi B Double Shooting Scheme The aim of this double shooting is to automate a method to find the proper values of F'' ( 0 ) and θ '( 0) which are not known prior to solving the coupled free convection transformed equations. θ are known (boundary The values of F' ( ) and ( ) conditions) and they are dependent on the assumed values of θ ' 0. This is epressed as follows: F'' ( 0) and ( ) F' ( ) = g1 F'' ( 0 ), θ' ( 0) θ( ) = g F'' ( 0 ), θ' ( 0) Using Taylor Series ( ) ( ) ( ) ( ) F'' ( 0) δ θ' ( 0 ) θ '( 0) θ =θ1 +γ F 1'' + ( ) ( ) ( ) β θ' ( 0 ) θ '( 0) F' = F 1 '' +α F'' 0 F 1'' + Where: 1 1 ( ) F '( ) ( ) ( ) F ' 1 α= F '' 0 F 1'' 0 ( ) ( ) ( ) ( ) θ θ1 γ= F '' 0 F 1 '' 0 ( ) F '( ) '( 0 ) '( 0) (30) (31) (3) (33) F 3 ' 1 β= (34) θ 3 θ 1 ( ) ( ) '( 0 ) '( 0) θ3 θ1 δ= (35) θ 3 θ 1 Such that: F 1 ''( 0 ), F ''( 0 ), 1 '( 0) and 3 '( 0 ) assumed any values. Also, F '( ), F '( ), ( ) θ θ are initially 1 F 3 ', θ1( ), θ( ) and θ3( ) are the corresponding solutions (at boundary) for three test runs (initially). In equation 30 and 31 we set the values of F'( ) and θ( ) equal to zero and we solve two equations in two unknowns to get the new values for the improved guesses. Doing so we get the following: ISSN: ISBN:

8 ( ) ( ) δf 1 ' βθ1 F'' ( 0) = F new 1 ''( 0) + αδ βγ ( ) F '( ) αθ1 γ 1 θ '( 0 ) =θ new 1 '( 0) + βγ αδ (36) (37) The steps of solution are then as follows: 1. Assume F 1 ''( 0 ), F ''( 0 ), θ 1 ' ( 0 ) and θ 3 '( 0) as discussed previously.. Obtain three equivalent solutions. 3. Obtain a new guess and its corresponding solution. 4. Check if the obtained values at boundary conditions are the required or not. 5. If yes then stop. If no we repeat again to obtain a new improved guess and keep going until convergence. Appendi C Dimensionless velocity and temperature profiles The dimensionless velocity profiles are shown for the case of free convection over an isothermal wall in Fig. 5. Only solutions for selected Prandtl numbers are displayed. Also, the dimensionless temperature profiles are given in Fig. 6. The solutions for Pr=0.01, 0.1, 1, 10, 100 and 100 are shown in A. Bejan [13]. The dimensionless velocity profile for the case of forced convection over an isothermal wedge with an angle of 10 is shown in Fig. 7. Only one curve is plotted because for the wedge the momentum and energy equations are not coupled and therefore the solution is not dependent on Prandtl number. Fig. 8 shows the dimensionless temperature profiles at different Prandtl numbers. θ n F f ' Pr=30 Pr=5 Pr= η n Pr=0.7 Fig 6 Dimensionless temperature profiles vs. similarity parameter (Free Convection) Fig. 7 Dimensionless velocity profile vs. similarity parameter (wedge forced convection (β=10 ). η f -F' n Pr=0.7 Pr=3 Pr=5 Pr=30 Pr=6 θ f Pr=100 Pr=30 Pr=10 Pr=5 Pr=3 Pr= Pr=1 Pr=0.7 Pr=0.1 Pr= Fig. 5 Dimensionless vertical velocity profiles vs. similarity parameter (free convection) η n Fig. 8 Dimensionless temperature profiles vs. similarity parameter (Forced convection over wedge (φ=10 ). η f ISSN: ISBN: