Applied Matheatics 13, 3(1): 7-37 DOI: 1.593/j.a.1331.4 erical Solution of the Extend Graetz Proble for a Bingha Plastic Fluid in Lainar Tube Flow M. E. Sayed-Ahe d 1,, H. K. Saleh, Ibrahi Hady 1 Engineering Matheatics and Physics Departent, Faculty of Engineering, Fayou University, Egypt Engineering Matheatics and Physics Departent Faculty of Engineering, Cairo University, Egypt Abstract The proble of therally lainar heat transfer based on the fully develod velocity for Bingha fluids in the entrance region of a circular duct is studied. The effect of the axial position, Peclet nuber, and Brinkan nuber are taken into account. Two cases of T (constant wall terature) and H (constant heat flux) therally boundary conditions are considered in this study. The energy equation is solved nuerically by finite difference ethods. The arching technique and the relaxation ethod are the ain factors of the coputer progra. The results of the present work including the variation of the sselt nuber and ean terature with the axial distance for the two cases are represented in Figures and copared with the available known results to support its accuracy. Keywords Graetz Proble, Bingha Fluid, Circular Duct, erical Analysis 1. Introduction Many of the fluids used for industrial purposes are non-newtonian such as pastes, polyer solutions and elts, plastics, pulps and eulsions in every day cheical engineering practice. Bingha odel is one of the siplest odels that describes aterials with yield stress and ost widely odel for the above aterials. Viscous dissipation changes the terature distributions by playing a role like an energy source, which affects heat transfer rates. The erit of the effect of the viscous dissipation dends on whether the pi is being cooled or heated. Many studies involving pi flows in the existing literature have neglected the effect of viscous dissipation. In fact, the shear stresses can induce a considerable power generation. However, in the existing convective heat transfer literature, this effect is usually regarded as iportant only in two cases of flow in capillary tubes and flow of very viscous fluids. The effects of viscous diss ipat ion in lainar flows have not yet been deeply investigated. For liquids with high viscosity and low theral conductivity, disregarding the viscous dissipation can cause appreciable errors. The first studies of heat transfer in a duct flow of Newtonian fluid were ade ore than a century ago by Greatz in1883-1885 and indendently by sselt in 191. Solutions to this proble and its various extensions continue to evoke any research efforts and a coprehens ive review of the works on heat transfer in Corresponding author: es@fayou.edu.eg (M. E. Sayed-Ahed) Published online at http://journal.sapub.org/a Copyright 13 Scientific & Acadeic Publishing. All Rights Reserved lainar duct flow was copiled by Shah and London[1]. Tyagi[] rfored a wide study on the effect of viscous dissipation on the fully develod lainar forced convection in cylindrical tubes with an arbitrary cross-section and unifor wall terature. Ou and Cheng[3] e ployed the separation of variables ethod to study the Graetz proble with finite viscous dissipation. They obtained the solution in the for of a series whose eigenvalues and eigenfunctions of which satisfy the Stur Liouville syste. The solution technique follows the sae approach as that applicable to the classical Graetz proble and therefore suffers fro the sae weakness of poor convergence behavior near the entrance. Lin et al.[4] showed that the effect of viscous dissipation was very relevant in the fully develod region if convective boundary conditions were considered. With these boundary conditions and if viscous dissipation was taken into account, the fully develod value of the sselt nuber was 48/5 for every value of the Biot nuber and of the other paraeters. On the other hand, it is well known that, if a forced convection odel with no viscous dissipation is eployed, the fully develod value of the sselt nuber for convective boundary conditions dends on the value of the Biot nuber. Basu and Roy[5] analyzed the Graetz proble for Newtonian fluid by taking the account of viscous dissipation but neglecting the effect of axial conduction. Zanchini[6] analyzed the asyptotic behavior of lainar forced convection in a circular tube, for a Newtonian fluid at constant prorties by taking into account the viscous dissipation effects. It was disclosed that particularly for the boundary conditions of unifor wall terature and of heat transfer by convection to an external fluid yielded the sae asyptotic behavior of the sselt nuber, naely
8 M. E. Sayed-Ahed et al.: erical Solution of the Extend Graetz Proble for a Bingha Plastic Fluid in Lainar Tube Flow 48/5. And therefore, He obviously stated that, for these boundary conditions, when the wall heat flux q w (x ) tended to zero, i.e., the local Brinkan nuber Br(x) tended to infinity, it is copletely wrong to neglect the effect of viscous dissipation on the asyptotic behavior of the forced convection proble. Aydin[7] investigated the effect of viscous dissipation on therally develod lainar forced convective flow in a pi for a Newtonian fluid with constant prorties. They showed that the effect of viscous dissipation could not be neglected when the wall terature was unifor. The effect of viscous dissipation on the theral entrance region in a pi for a power law non-newtonian flu id has been studied by Liou and Wang[8] using the unifor wall heat flux theral boundary condition, Berardi et al.[9] using the convection with an external isotheral flu id, Lawal and Mu ju dar[1] and Dang[11] using the unifor wall terature. Barletta[1] studied the asyptotic behavior of the terature flield for the lainar and hydrodynaically develod forced convection of a power-law fluid which flows in a circular duct taking the viscous dissipation into account. The asyptotic sselt nuber and the asyptotic terature distribution were evaluated analytically in the cases of either the unifor wall terature or convection with an external isotheral fluid. Also, Valko[13] studied the proble for power law fluid using the Laplace transfor Galerkin technique. Khatyr et al.[14] investigated the fully develod lainar forced convection in circular ducts for a Bingha plastic with viscous dissipation and negligible axial heat conduction in the fluid. Both the asyptotic sselt nuber and the asyptotic profile terature are obtained for any axial distributions of wall heat flux, which yield a therally develod region. Min et al.[15] studied therally fully develod flow of a Bingha plastic including axial conduction, to obtain the sselt nuber and terature profiles and to propose a correlation forula between the sselt nuber and the Peclet nuber and also studied therally developing flow (the Graetz proble) of a Bingha plastic including both axial conduction and viscous dissipation. The solution to this proble is obtained by using the ethod of separation of variables, where the resulting eigenvalue proble is solved approxiately by using the ethod of weighted residuals. In this par, we study the effects of the viscous dissipation and axial conduction on the theral entrance for a Bingha fluid in a tube. The flow is assued to be lainar, steady state and fully develod velocity. Two cases of T (constant wall terature) and H (constant heat flux) therally boundary conditions are considered. The energy equation is solved nuerically by finite difference ethod to obtain the terature distributions. Then the ean terature an sselt nuber are coputed for the two theral boundary conditions. Forulation of the Proble The circular tube configuration and boundary conditions are shown in Fig. 1. The flow is assued to be lainar steady state, and incopressible with constant physical prorties. The Bingha odel is chosen to characterize the Non-Newtonian behavior of the fluid. The fluid enters the duct at unifor terature T e and fully develod velocity u. The no-slip condition is applied at the circular tube walls and two cases of theral boundary conditions; unifor wall terature everywhere ( T ) and unifor heat flu x axially as well as ripherally ( H ) ) are assued. Figure 1. Circular tube configuration and boundary conditions The energy equation for a circular tube with viscous dissipation and axial condation effect ay be written as u T T 1 T T η( du / dr) (1) α x r x where η is the apparent viscosity for a Bingha odel and given by du dr µ τ y / () η The boundary conditions of the proble for the two cases are given by: Case (i) T- theral boundary T (, r) T e, T (x, r ) T w, T ( x,), T x as x (3) Case (ii) H -theral boundary T (, r) T e, T q ( x, r ) ", k T ( x,), T as x (4) x In addition to the non-diensional variables, we ay introduce the following: r u x r x r r, u, x / /, r u RePr Pe η η / µ and c τ y / τ w, ρc pudh, Pr k T Tw θ for Case (i) and T T e w µ c p τ y, c k τ w T Te θ ro q" for Case(ii) k
Applied Matheatics 13, 3(1): 7-37 9 Where r is the radius of the tube, τ w is the wall shear stress and D h is the hydraulic diaeter. The energy equation (1) is written in the following non-diensional for by M in et al.[14] as θ 1 θ 1 θ θ du Brη, (5) u x r r x dr where (6) θ θ ( x,) ( x,r ) as x (9) x The local sselt nubers N u, xt N, u according to xh, the two cases of the theral boundary conditions are given by θ r r 1 xt, (1) θ 1 θ θ ( x x, H w,1) (11) 1 u θr dr θ (1) 1 u r dr The boundary conditions of the proble for the two cases of theral boundary conditions are written in the following non- diensional fors: Case (i): θ (, r ) 1., θ ( x,1), θ ( x,), θ ( x,r ) x as x (8) Case (ii): θ (, r ), θ ( x,1). 5, The fin ite-difference for of the energy equation (5) is: u ( θ ( i 1, θ ( i 1, 1 θ ( i 1, x (7) 3. erical Solution Using central difference for the L.H.S of the energy equation (5) and the Crank-Nicolson finite difference schee for the R.H.S, the fin ite difference fors of the first and second derivatives are written as θ θ i 1, θ ( i 1, O( x x x θ θ i, j 1) θ O( ( ( θ θ ( i 1, 1 θ j 1) θ x r ( θ j 1) θ θ Brη φ( ) ( j (13) Where du φ dr Rearranging equation (13) in the for as: a5θ a1θ ( i 1, aθ ( i 1, a3θ j 1) Brη φ( ) a4θ ( j (14) For 1 i 1, j n 1 1 u ( 1 u ( where, a 1, a, x. 4 x x 4 x 1 1 a3, 1 1 a 4 and a 5 r r ( r( x ) )
3 M. E. Sayed-Ahed et al.: erical Solution of the Extend Graetz Proble for a Bingha Plastic Fluid in Lainar Tube Flow For j 1 ( r ) the ter In the equation (13) becoes; 1 θ r r u ( is undefined, therefore we apply the L Hopital s rule. Then θ ( i 1, θ ( i 1, 1 θ ( i 1, θ θ ( i 1, x x 1 r θ θ θ j 1) θ θ Brη ( φ( r (15) Rearranging equation (15) in the for as: a5θ a1θ ( i 1, aθ ( i 1, a3θ j 1) a4θ Brη φ( (16) where, 1 u ( 1 u ( j ) a1, a, a x. 4 x 3, x 4 x ( a and 4 a 5 x 4 The difference fors of the boundary conditions are: θ ) θ ) For case (i): θ ( 1, 1, θ n),, θ (, θ ( 1, x θ ) θ ) For case (ii): θ ( 1,,, θ ( 1, θ (, θ ( 1, x To copute sselt nuber and ean terature for two cases of terature boundary conditions we ay introduce finite-difference fors for the integrals and the first partial derivatives in equations (1) and (1).The Sipson rule for single integral is introduced by Jain et al.[16], is used to introduce a difference for for the integral equation (11) as n 1 n t 1) t n) 4 t l) t l) 3 l,4,6 l 3,5,7 θ ( i) n 1 n t,1), ) 4 1 ( i t1 ( i n t1 l) t1 l) 3 l,4,6 l 3,5,7 Where, t (, ), 1 i j r u r (j-1) Δ r, t( i, t1 θ 4. Method of Solution The governing equations are solved nuerically using the successive over relaxation ethod. For each axial position (i) (contain θ as unknown variable notation), in the first we get the velocity fro equation (7), and the energy equation (14) is then solved to obtainθ. The linear algebraic equations resulting fro equation (14) ay be written in the atrix for as A( i, j ) θ ( i, B( i, j ) (17) Where, A( i, j ) : Designate the coefficients of the unknowns in the algebraic equations θ ( i, : Designate the unknowns in the algebraic equations B( i, j ) : Designate the residual in the algebraic equations i 1,,3,,n and j1,,3,, The relaxation equation can be written in the for of : a i 1, jθ1( i 1, ai 1, jθ ( i 1, w e θ aij, 1 θ1 j 1) aij, 1θ j 1) (18) aij, Brηj φj ( 1 we) θ1 Where: θ 1 : Known values of the terature in the old iterations, θ : Unknown values of the terature in the new iterations, w : Relaxat ion factor, e a: the coefficient of θ. The resulting of a linear algebraic syste of equations are solved by using successive over relaxation ethod (S.O.R) with relaxation factor w e 1.1. The process reated up till the difference between the new and old terature value at 5 each point is less than 1.
Applied Matheatics 13, 3(1): 7-37 31 Figure. The variat ion of t he diensionless ean t erat ure with resct to axial dist ance for Pe 1 (for case(i)) Figure 3. The variation of the diensionless ean terature with resct to axial distance for Pe 1 (for case (i) )
3 M. E. Sayed-Ahed et al.: erical Solution of the Extend Graetz Proble for a Bingha Plastic Fluid in Lainar Tube Flow Figure 4. The variation of the diensionless ean terature with resct to axial distance for Pe 1 (for case (ii) ) Figure 5. The variation of the diensionless ean terature with resct to axial dist ance for Pe 1 (for case (ii) )
Applied Matheatics 13, 3(1): 7-37 33 Figure 6. The variation of the diensionless local sselt nuber with resct to axial distance for Pe 1 (for case (i) ) Figure 7. The variation of the diensionless local sselt nuber with resct to axial distance for Pe 1 (for case (i) )
34 M. E. Sayed-Ahed et al.: erical Solution of the Extend Graetz Proble for a Bingha Plastic Fluid in Lainar Tube Flow Figure 8. The variation of the diensionless local sselt nuber with resct to axial distance for Pe 1 (for case (ii) ) Figure 9. The variation of the diensionless local sselt nuber with resct to axial distance for Pe 1 (for case (ii) )
Applied Matheatics 13, 3(1): 7-37 35 Figure 1. Coparison of the local sselt nuber for case (i). for P e 1 5. Results and Discussion Figure 11. Coparison of the ean terature resct to axial distance for case (i) at c and Pe 1 erical results are presented in Figures ( to 11). The results are ade for Pe 1, 1, with Br,.1, 1, and c,.4,.6 over the range of.1 < x <.6 where the transverse esh spacing x.1 and.1, for the two cases of the theral boundary conditions. The variation of diensionless ean terature θ with axial coordinate x is shown for case(i) ( constant wall terature ) in Figures (a, b, c,d ) and 3(a, b, c,d ) for Peclet nubers, Pe 1,1, resctively and different Brinkan nuber Br,.1, 1,. The study of this Figures shows that the value of the diensionless ean terature θ decreases with increasing the values of c (ratio of the yield shear stress to the wall shear stress) for the values of Br,.1 and it increases with the increasing values of c for Br 1,. The value of the
36 M. E. Sayed-Ahed et al.: erical Solution of the Extend Graetz Proble for a Bingha Plastic Fluid in Lainar Tube Flow diensionless ean terature increasing the values of increases with increasing the values of θ decreases with x for values of Br,.1 and it x for Br 1,. The value of the diensionless ean terature decreases with increasing the values of Peclet nuber Pe at the sae value of Br and c. The variation of diensionless ean terature θ with axial coordinate x is shown for case(ii) ( constant wall heat flu x) in Figures 4(a, b, c,d ) and 5(a, b, c,d ) for Peclet nubers, Pe 1,1, resctively and different Brinkan nuber Br,.1, 1,. The study of this Figures shows that the value of the diensionless ean terature θ increases with increasing the values of c for the values of Br.1,1, and it apars the sae values with increasing values of c for Br. The value of the diensionless ean terature θ increases with increasing the values of x for all values of Br. The value of the diensionless ean terature θ decreases with increasing the values of Peclet nuber Pe at the sae value of Br and c. The value of the diensionless ean terature θ increases with increasing the values of Br for all values c and Pe. The variation of local sselt nuber θ with axial coordinate x is shown for case(i) in Figures 6(a, b, c,d ) and 7(a, b, c,d ) for Peclet nubers, Pe 1,1, resctively and different Brinkan nuber Br,.1, 1,. It has been found that the value of local sselt nuber, x T, increases with increasing the values of c for all values of Br. The value of local sselt nuber x, T decreases with increasing the values of x for all values of Br, 1, and it decreases with increasing the values of x then it increases with increasing the values of x for the value of Br.1 for the different value of Pe and c. It has been observed the local sselt nuber x, T decreases with increasing the values of Peclet nuber Pe at the sae value of Br and c. The value of local sselt nuber x, T increases with increasing the values of Br for all values c and Pe. The variation of local sselt nuber with axial coordinate x,h x is shown for case(ii) in Figures 8(a, b, c,d ) and 9(a, b, c,d ) for Peclet nubers, Pe 1,1, resctively and different Brinkan nuber Br,.1, 1,. The study of this Figures shows that the value of local sselt nuber x,h decreases with increasing the values of c for all values of Br.1, 1, but when Br it increases with increasing the values of c. Also it is observed that The value of local sselt nuber decreases with increasing the values of x for all values of Br. Also it is observed that the value of local sselt nuber x T x,h x,h decreases with increasing the values of Peclet nuber Pe at the sae value of Br and c. Also it is observed that The value of local sselt nuber ( x,h ) decreases with increasing the values of Br for all values c and Pe. Figure(1) shows the coparison of the local sselt nuber for case (i) with the results obtained by Min et al.[14]( c,.4,.6,br,.1,1, and Pe 1 ). It has been found that present results of this study are in good agreeent with the results of Min et al.[14]. Figure(11) shows the coparison of the ean terature θ for case (i) with the results obtained by Min et al.[15] for c ( Br,.1,1, and Pe 1 ). It has been found that present results of this study are in good agreeent with the results of Min et al.[15]. Tab les (1) and () show the coparison of the ean terature θ and the local sselt nuber, x T in the present work with previous work Blackwell[17] and exact solution for Newtonian fluid ( c, Br and Pe 1). It has been found that present results of this study are in good agreeent with the results of Blackwell[17] and exact solution. Ta ble 1. Coparison of the ean terature in this work (nuerical solution for c, Br and Pe 1) with Blackwell[17] and Exact solution x Exact solution Blackwell[17] erical solution (c ) 1.1.961731477656.9617496.96357588651.4.97363444847.973635.977576849741.1.836189374.836189.836385467466.4.6876547356.6876.688567893518.1.3959878135531.395988.39534918943781.4.439349846339.43935.439611167578.6.54583351369.5458.5445556553 Ta ble. Coparison of the local sslt nuber in this work (nuerical solution for c, Br and Pe 1) with Blackwell[14] and Exact solution x x, T Exact solut ion Blackwell[17] erical solution (c ) 1.1 1.8873537838 1.84 13.474831719748.4 8.361373463618 8.36 8.14435937.1 6.1515178348 6. 6.1545419345994.4 4.17433177399 4.17 4.1794679143.1 3.7998835848 3.71 3.796554965498.4 3.65679419479766 3.657 3.6564587937.6 3.6567934577639 3.657 3.656539475176 6. Conclusions A finite difference ethod has been used to nuerically solve the energy equations for steady lainar fluid flow and
Applied Matheatics 13, 3(1): 7-37 37 heat transfer in entrance region of a circular pi. A Bingha Non-Newtonian odel is used to characterize the fluid behavior. Two cases of theral boundary conditions (T and H ) are studied. Two-diensional storage and arching technique with a relaxation ethod line by line solution procedure for the difference equations are soe iportant features of the progra. the effects of the Brinkan nuber ( Br,.1,1, ), the theral axial position x,the Peclet nuber ( Pe 1, 1 ) and the ratio of the yield shear stress to the wall shear stress( c,.4,.6) on the terature θ, the ean terature θ and the local sselt nuber x, T and x,h are studied. It is observed that, 1- The value of the diensionless ean terature θ for case (i) decreases with increasing the values of x for values of Br,.1 and it increases with increasing the values of x for Br 1,. Also the value of the diensionless ean terature θ decreases with increasing the values of Peclet nuber Pe. - The value of the diensionless ean terature θ for case (ii) increases with increasing the values of x for all values of Br. Also the value of the diensionless ean terature( θ ) decreases with increasing the values of Peclet nuber Pe and it increases with increasing of the Br. 3- The value of local sselt nuber x, T for case(i) decreases with increasing the values of x for all values of Br.Also the value of local sselt nuber decreases with increasing the values of Peclet nuber Pe, while increases with increasing the values of Br. 4- The value of local sselt nuber for case (ii) decreases with increasing the values of x for all values of Br, and it decreases with increasing the values of Peclet nuber Pe. Also it is observed that The value of local sselt nuber x,h decreases with increasing the values of Br. REFERENCES x,h, [1] Shah, R. K. and London, A. L., Lainar flow forced convection in ducts. Advances in Heat Transfer (suppl.1). Acadeic Press, New York, 1978. [] Tyagi V.P., Lainar forced convection of a dissipative fluid in a channel. J Heat Transfer 1996;88:161 169. x T [3] Ou J.W. and Cheng K.C.,Viscous dissipation effects in the entrance region heat transfer in pis with unifor heat flux. Appl Sci Res 1973;8:89 31. [4] Lin T.F., Hawks K.H. and Leidenfrost W., Analysis of viscous dissipation effect on the theral entrance heat transfer in lainar pi flows with convective boundary conditions, War Stoffubertrag 1983;17: 97 15. [5] Basu T and Roy D.N., Lainar heat transfer in a tube with viscous dissipation, Int J Heat Mass Transfer 1985;8:699 71. [6] Zanchini E., Effect of viscous dissipation on the asyptotic behavior of lainar forced convection in circular tubes, Int J Heat Mass Transfer 1997;4:169 178. [7] Aydin O., Effects of viscous dissipation on the heat transfer in a forced Therally developing flow, Energy Conversion and Manageent 5;46: 391 31. [8] Liou C.T. and Wang F.S.,Solutions to the extended Graetz proble for a power-law fluid with viscous dissipation and different entrance boundary conditions, er Heat Transfer A 199;17:91 18. [9] Berardi P.G., Cuccurullo G., Acierno D. and Russo P.,Viscous dissipation in duct flow of olten polyers, In: Proceedings of Eurother Seinar 46, Pisa, Italy; July 1995. p. 39 43. [1] Lawal A.and Mujudar A.S., The effects of viscous dissipation on heat transfer to power law fluids in arbitrary cross sectional ducts, War Stoffubertrag 199;7:437 46. [11] Dang V.D., Heat transfer of a power-law fluid at low Peclet nuber flow, J Heat Transfer 1983;15:54 9. [1] Barletta A., Fully develod lainar forced convection in circular ducts for power-law fluids with viscous dissipation, Int J Heat Mass Transfer 1997;4(1):15 6. [13] Valko, P.P., Solution of the Graetz Brinkan proble with the Laplace transfor Galerkin ethod, International Journal of Heat and Mass Transfer 5;48: 1874 188. [14] Khatyr R., Ouldhadda D. and Il Idrissi A., Viscous dissipation effects on the asyptotic behavior of lainar forced convection for Bingha plastics in circular ducts, Int J Heat Mass Transfer 3;46:589 98. [15] [15] Min T., Yoo J.Y. and Choi H., Lainar convective heat transfer of a Bingha plastic in a circular pi I. Analytical approach therally fully develod flow and therally developing flow (the Graetz proble extended), Int J Heat Mass Transfer 1997;4:35 337. [16] Jain M.K., Lyenger S.R.K., and Jain R.K., erical Method for Scientific and Engineering Coputation, Wiely Eastern Liited New Delhi, second edition,1991. [17] Blackwell, B. F., erical solution of the Graetz proble for a Bingha plastic in lainar tube flow with constant wall terature, Journal of Heat Transfer, 1985; 17:466-468.