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FORCED CONVECTION FLOW of VISCOUS DISSIPATIVE POWER-LAW FLUIDS in a PLANE DUCT Part 2

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FORCED CONVECTION FLOW of VISCOUS DISSIPATIVE POWER-LAW FLUIDS in a PLANE DUCT Part 2 Isı Bilimi ve Tekniği Dergisi, 28, 2, 17-24, 2008 J. of Thermal Science and Technology ©2008 TIBTD Printed in Turkey ISSN 1300-3615 FORCED CONVECTION FLOW OF VISCOUS DISSIPATIVE POWER-LAW FLUIDS IN A PLANE DUCT Part 2. Thermally Developing Flow Mete AVCI* and Orhan AYDIN** *Karadeniz Technical University, Department of Mechanical Engineering, 61080 Trabzon, [email protected] **Karadeniz Technical University, Department of Mechanical Engineering, 61080 Trabzon, [email protected] (Geliş Tarihi: 01. 02. 2008, Kabul Tarihi: 12. 05. 2008) Abstract: In this part of the study, we investigate the problem of hydrodynamically developed but thermally developing laminar forced convection of a power-law fluid in a plane duct when the viscous dissipation is included. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (H1 boundary condition) and the constant wall temperature (T boundary condition). The combined and interactive influences of the power-law index and the Brinkman number on the distributions for the developing temperature profile and local Nusselt number are determined in the entrance region for the wall heating and cooling cases at T and H1 boundary conditions. Keywords: Non-Newtonian fluid, Power-law fluid, Thermally developing, Viscous dissipation, Plane duct, Constant heat flux, Constant wall temperature. BİR DÜZLEMSEL KANAL İÇERİSİNDEKİ VİSKOZ YAYILIMLI POWER-LAW AKIŞKANLARIN ZORLANMIŞ TAŞINIM AKIŞI Kısım 2. Termal Gelişen Akış Özet: Çalışmanın bu bölümünde, düzlemsel bir kanalda, hidrodinamik olarak tam gelişmiş termal olarak gelişmekte olan power-law akışkanın laminer zorlanmış taşınımı, viskoz yayılım etkileri dahil edilerek incelenmiştir. Akışkan içerisindeki eksenel iletim ihmal edilmiştir. Sabit ısı akısı (H1 sınır koşulu) ve sabit yüzey sıcaklığı(T sınır koşulu) olmak üzere iki farklı termal sınır koşulu ele alınmıştır. Power-law indeksi ve Brinkman sayısının, giriş bölgesinde, gelişmekte olan sıcaklık dağılımı ve yerel Nusselt sayısı üzerindeki etkisi, T ve H1 sınır koşullarında, sıcak ve soğuk cidar durumları için belirlenmiştir. Anahtar kelimeler: Newtonyen olmayan akışkan, Power-law akışkan, Termal gelişen, Viskoz yayılım, Düzlemsel kanal, Sabit ısı akısı, Sabit yüzey sıcaklığı. NOMENCLATURE Br Brinkman number, Eq. (6) Greek symbols 2 Brq modified Brinkman number, Eq. (16) α thermal diffusivity [m /s] cp specific heat at constant pressure η consistency factor employed in equation (1) k thermal conductivity [W/mK] ρ density [kg/m3] n power-law index υ kinematic viscosity [m2/s] Nu Nusselt number θ dimensionless temperature, Eq. (5,14) 2 qw wall heat flux [W/m ] Re Reynolds number, Eq. (5) Subscripts Pr Prandtl number, Eq. (5) e fluids entering T temperature [K] m mean u velocity [m/s] w wall U dimensionless velocity W width of the duct (=2w) (m) Y dimensionless vertical coordinate z axial direction [m] Z dimensionless axial coordinate, Eq. (5) 17 INTRODUCTION ANALYSIS The flow and heat transfer of non-Newtonian fluids The flow is considered to be hydrodynamically fully through ducts have wide potential applications in many developed but thermally developing. This problem is engineering areas including the chemical, petroleum, traditionally termed as the “Graetz” problem. Steady, polymer, food processing, pharmaceutical and laminar flow having constant properties (i.e. The biochemical and biomedical engineering. The most non- thermal conductivity and the thermal diffusivity of the Newtonian fluids of practical interest are highly viscous fluid are considered to be independent of temperature) is and, therefore, are often processed in the laminar flow considered. The axial heat conduction in the fluid and in regime. Readers are referred to see the excellent reviews the wall is assumed to be negligible. The shear stress by Irvine and Karni (1987) and Hartnett and Choi and strain relationship for the power-law type Ostwald- (1998). de Waele fluid is given as n−1 In the first part of this study (Aydın and Avcı, 2008), it du du was shown that the viscous dissipation had a τηyz = (1) considerable effect on the hydrodynamically and dy dy thermally fully developed flow and heat transfer of a power-law fluid in a plane duct for the constant wall where n represents the power-law index and the case of temperature (T type) and the constant heat flux (H1 n<1, n=1 and n>1 correspond pseudoplastic, Newtonian type) thermal boundary conditions at the wall. Now, we and dilatant behaviors. The velocity profile for fully will focus our interest on the hydrodynamically developed plane duct flow is given as follows: developed, but thermally developing laminar flow (1)/nn+ problem (the so-called Graetz problem) for the same un⎛⎞⎛⎞21+ ⎡ y ⎤ geometry, fluid type and thermal conditions. =−⎜⎟⎜⎟⎢1 ⎥ (2) un+1 w m ⎝⎠⎝⎠⎣⎢ ⎦⎥ However, in the existing convective heat transfer literature on the non-Newtonian fluids, the effect of the Since a fully developed velocity profile is assumed for viscous dissipation has been generally disregarded. thermally developing flow, the energy equation There are only a few studies to be cited. Lawal and including viscous dissipation effect can be represented Mujumdar (1984,1992) studied viscous dissipation by effect on heat transfer for power law fluids in arbitrary cross-sectional ducts. Flores et al. (1991) studied the ∂∂2TuT1 du =−τ (3) Graetz problem for the case of a power-law fluid either ∂y2 α ∂zkyz dy in the cylindrical geometry or the plane geometry by considering a constant temperature at the duct wall in the presence of viscous dissipation. Dang (1983) where ρ, cp, k and µ are density, specific heat, thermal studied the effect of viscous dissipation in the thermal conductivity and viscosity. The second term in the right entrance region in a pipe using the uniform wall hand side is the viscous dissipation term. temperature. Barletta (1997) studied the asymptotic behavior of the temperature field for the laminar and Due to axisymmetry at the center, the thermal boundary hydrodynamically developed forced convection of a condition at y=0 can be written as: power-law fluid which flows in a circular duct taking ∂T the viscous dissipation into account. The asymptotic =0 (4) Nusselt number and the asymptotic temperature ∂ y y=0 distribution were evaluated analytically in the cases of either uniform wall temperature or convection with an Two kinds of thermal boundary condition at wall are external isothermal fluid. Wei and Luo (2003) considered in this study, namely: constant wall heat flux investigated a Graetz-Nusselt type problem of (H1 type) and constant wall temperature (T type). They incompressible non-Newtonian fluids with temperature are treated separately in the following: dependent power-law viscous dissipation by using a Galerkin method with linear axisymmetric triangular Constant Wall Temperature (T Type) finite elements. Introducing the following dimensionless variables In a recent study, Aydın and Avcı (2006) studied the thermally developing laminar forced convection flow of u (TT− ) y a Newtonian-fluid in plane duct considering the effect U = , θ = w , Y = , of the viscous dissipation. um ()TTwe− W ηc u n−1 ρuW2−nn This chapter is followed by the Analysis, Results and zW p ⎛⎞m m Z = , Pr = ⎜⎟, Re = (5) Discussion and Conclusions chapters. Re Pr kW⎝⎠ η 18 where um is the mean velocity in the plane duct and Re Constant Heat Flux (H1 Type) is the Reynolds number based on this mean velocity and the width of the duct, W, which is equal to 2 w. The Now, we will consider the constant heat flux case, dimensionless variable Z is termed the Graetz number. qcw = . In this case, the following dimensionless Then, Eq. 3 becomes temperature is used: 2 n+1 ∂∂θθ 21n + 2 ⎛⎞(1)/nnnn++ (1)/ (TT− e ) UBrY=−2 ⎜⎟2 (6) θ = (14) ∂∂ZnY ⎝⎠ ()qWw / k where Br is the Brinkman number, which is defined as: With this definition, the energy equation can be written in dimensionless form as: η u n+1 Br = m (7) n−1 2 n+1 (1)n+ 2 kW() Twe− T ∂∂θθ ⎛⎞21n + n (1)/nn+ UBrY=+2 q ⎜⎟2 (15) ∂∂ZnY ⎝⎠ For this T-type thermal boundary condition, the wall temperature of the wall is kept isothermal in the where Brq is the modified Brinkman number, which is entrance region, which is mathematically shown as: given as: For z > 0 : TT= w at yw= (8) n+1 ηum Brq = (16) qWn In dimensionless form, the thermal boundary conditions w that will be applied in the solution of the energy equations are given as: The entrance condition at the beginning of the thermally developing region in this case is defined: ∂θ Y =0 : =0 Z =0 : θ =0 (17) ∂Y Y = 0.5 : θ =0 (9) and the wall thermal boundary condition is The local Nusselt number is obtained from ∂T kq= (18) ∂ y w ∂θ yw= NuW =− (10) ∂Y Y =0.5 where q is positive when its direction is to the fluid w (wall heating), otherwise it is negative (wall cooling). It should be noted the Nusselt number being used here is In dimensionless form, it can be written as: based on the difference between the wall and the inlet temperatures, i.e., on Tw – Te, and not on the difference ∂θ between the wall and the mean temperatures. Now the =1 (19) mean temperature, i.e. the bulk mean temperature is ∂Y Y =1 given by Oosthuizen and Naylor (1999). For this case, the Nusselt number is given: ∫ ρ uT dA Tm = (11) 1 ρ udA Nu = (20) ∫ W θ w Rewriting this equation in terms of the dimensionless variables: and 0.5 1 UdYθ NuWm = (21) TTwm− 0 ∫ θwm−θ = 0.5 (12) TTwe− UdY ∫0 For each case, the energy equation has been solved numerically using the difference method. The details of The Nusselt number based on the difference between the solution procedure can be found in Oosthuizen and the wall and the mean temperature is then given by: Naylor (1999).
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