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= Nu =− (10) W ∂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).
0.5 TT− UdYθ Nu== Nuwe Nu ∫0 (13) Wm W W 0.5 TTwm− UdY ∫0 19 RESULTS AND DISCUSSION Br=-0.1 representing the cold wall or the wall cooling case (Tw
0.50 0.50 0.50
n=2.0 n=2.0 n=2.0 0.40 n=1.5 0.40 0.40 n=1.5 n=1.5 n=1.0 n=1.0 n=1.0 n=0.5 n=0.5 n=0.5 0.30 0.30 0.30 Y Y Y
0.20 0.20 0.20
0.10 0.10 0.10 Br=-0.1 Br=-0.1 Br=-0.1 0.00 0.00 0.00 0.00 0.20 0.40 0.60 0.80 0.00 0.20 0.40 0.60 0.80 0.00 0.20 0.40 0.60 0.80 θ θ θ
0.50 0.50 0.5
0.40 0.40 0.4
0.30 0.30 n=2.0 n=1.5 0.3 n=2.0 n=1.0 n=1.5 Y Y n=0.5 Y n=1.0 0.20 0.20 n=0.5 0.2
0.10 0.10 0.1 Br=0.0 n=0.5,1.0,1.5,2.0 Br=0.0 Br=0.0 0.00 0.00 0.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.01 0.02 0.03 0.04 θ θ θ
0.50 0.50 0.50
0.40 0.40 0.40 n=0.5 n=1.0 0.30 n=1.5 0.30 n=0.5 0.30 n=2.0 n=1.0 n=0.5 Y Y Y n=1.5 n=1.0 0.20 0.20 n=2.0 0.20 n=1.5 n=2.0 0.10 0.10 0.10 Br=0.1 Br=0.1 Br=0.1 0.00 0.00 0.00 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 -0.40 -0.30 -0.20 -0.10 0.00 0.10 θ θ θ Z=0.1 Z=0.3 Z=0.5
Figure 1. Dimensionless temperature distributions in terms of the power-law index for different Br at the constant wall temperature case. 20
0.50 0.50 0.50
0.40 0.40 0.40
n=0.5 n=0.5 0.30 0.30 n=0.5 0.30 n=1.0 n=1.0 n=1.0 n=1.5 n=1.5 n=1.5 Y Y Y n=2.0 n=2.0 n=2.0 0.20 0.20 0.20
0.10 0.10 0.10 Br =-0.1 Brq=-0.1 Brq=-0.1 q
0.00 0.00 0.00 -0.60 -0.40 -0.20 0.00 0.20 0.40 -1.20 -0.80 -0.40 0.00 0.40 0.80 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 θ θ θ
0.50 0.50 0.5 0.40 0.40 0.4 n=0.5,1.0,1.5,2.0 n=0.5,1.0,1.5,2.0 n=0.5,1.0,1.5,2.0 0.30 0.30 0.3
Y Y Y 0.20 0.20 0.2
0.10 0.10 0.1 Br =0.0 Br =0.0 Brq=0.0 q q 0.00 0.00 0.0 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 θ θ θ
0.50 0.50 0.50 n=0.5 0.40 n=1.0 0.40 n=1.5 0.40 n=2.0 n=0.5 n=0.5 n=1.0 0.30 0.30 n=1.0 n=1.5 0.30 n=1.5 n=2.0 n=2.0 Y Y Y 0.20 0.20 0.20
0.10 0.10 0.10 Brq=0.1 Br =0.1 Brq=0.1 q
0.00 0.00 0.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.50 1.00 1.50 2.00 2.50 3.00 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 θ θ θ
Z=0.1 Z=0.3 Z=0.5
Figure 2. Dimensionless temperature distributions in terms of the power-law index for different Brq at the constant wall heat flux case.
As seen, for the case without the viscous dissipation superior Nu values for the n>1-fluids than n<1, as effect, NuWm immediately settles to its fully developed opposed the common behavior. Further downstream, we values obtained in the Part 1 [3], which validates the finally reach the common behavior, an enhancement in numerical method used here. At Br=0, the case without the heat transfer for the n<1-fluids with the decreases viscous dissipation, Nu decreases with increasing n for for the n>1-fluids. Interestingly, it should be noted that n>1 (the dilatant behavior), while an enhancement in in the presence of the viscous dissipation (Br≠0), the heat transfer is observed (i.e. Nu increases) with steady-state value of the Nusselt number does not decreasing n for n<1 (the pseudoplastic behavior). For change for any values of the Brinkman number at a the wall heating case, at Br=0.01, we observe some constant value of the power-law index. For n=0.5, 1, 1.5 singularities. A decrease at n shifts the singularity point and 2, Nu receives the following corresponding values, ahead in the downstream. For Br=0.1, these singular respectively: 10.7800, 8.7413, 8.0543 and 7.7107 for points are attained earlier. These singularities can be Br≠0 while they are 3.9697, 3.7706, 3.6857 and 3.6391 explained in terms the competition between the heat for Br=0. flux supplied by the wall and the viscous dissipation generated heat. As seen initially, NuWm decreases up to Figure 4 illustrates the behavior of NuWm in the the singular point as a result of decreasing temperature downstream for different values of the power-law index difference between the wall temperature and the bulk at different values of Brq for the constant heat flux case fluid temperature resulting from the viscous dissipation at the wall. For Brq=0, Nu immediately receives its heat. For the cold or cooling wall situation, the viscous steady state value for each n given in Part 1 [3], see Fig. dissipation leads to higher temperature differences 4a. Including the viscous dissipation decreases Nu for between the wall and the bulk fluid with the increasing each n as result of decreasing temperature differences Br. Interestingly, in the downstream, initially, we obtain between the wall and the bulk fluid (Fig. 4b,c). And, an
21
12 40 (a) (b) n=0.5 n=1.0 10 n n=1.5 0.5 20 n=2.0 1.0 8 1.5 2.0 Wm Wm 0
Nu Nu 6
-20 4
Br=0.01 Br=0.0 2 -40 0.0 0.2 0.4 0.6 0.8 0.00.20.40.60.81.01.21.4 Z Z
40 20 (c) n=0.5 n=0.5 n=1.0 (d) n=1 n=1.5 n=1.5 20 15 n=2.0 n=2.0
Wm 0 Wm 10 Nu Nu
-20 5
Br=0.1 Br=-0.01 -40 0 0.0 0.2 0.4 0.6 0.8 0.00.20.40.60.81.01.21.4 Z Z
35 (e) n=0.5 30 n=1.0 n=1.5 n=2.0 25
20
Wm
Nu 15
10
5 Br=-0.1 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Z Figure 3. Variation of NuWm with n for different Br at the constant wall temperature case. enhancement in the heat transfer is obtained for the use a different Brinkman number definition, which is pseudoplastic fluids (n<1), while the opposite is true for based on the duct entrance temperature, than that in Part the dilatant fluids (n>1). For the cooling wall case, we 1. can easily observe increases at Nu values resulting from decreased temperature differences due to viscous CONCLUSION heating inside the fluid. At Brq=-0.1, interestingly, the pseuplastic fluid (n=0.5) gives lower Nu than the We have studied hydrodynamically developed, but Newtonian fluid (n=1), as opposed to the common thermally developing forced convection flow of power- behavior (see Fig. 4e). For the dilatant fluids, at n=2, law fluids in a plane duct (the Graetz problem) by just right after the entrance, a singularity at Nu is taking the effect of viscous dissipation into account. The observed (Fig. 4e). axial conduction in the fluid is neglected. Two types of wall thermal boundary condition have been considered, Finally, it should be noted that comparisons are possible namely: constant heat flux (H1-type) and constant wall for steady-state values of the pseudoplastic and dilatant temperature (T-type). Either wall heating or wall fluids in the presence of the viscous dissipation since we cooling case is examined. 22 10 14 (a) (b) 12 n n 0.5 8 0.5 10 1.0 1.0 1.5 1.5 2.0 2.0 8 Wm
6 Wm Nu Nu 6
4 4
2 Brq=0.0 Brq=0.01 2 0 0.0 0.2 0.4 0.6 0.8 0.00.20.40.60.8 Z Z
20 7.0 (c) (d) 6.5 n 15 2.0 0.5 Brq=0.1 6.0 1.5 1.0 Brq=-0.01 Wm 10 Wm 5.5 Nu Nu n=0.5 n=1.0 5.0 n=1.5 5 n=2.0 4.5
0 4.0 0.0 0.2 0.4 0.6 0.8 0.00.20.40.60.8 Z Z
40 (e) n=1.5 n=1.0 20 n=0.5
Wm 0 Nu n=2.0
-20
Brq=0.1 -40 0.00.20.40.60.8 Z
Figure 4. Variation of NuWm with n for different Brq at the constant wall heat flux case.
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23 Irvine, T. F. and Kami, J., Non-Newtonian Fluid Flow Oosthuizen, P. H. and Naylor, D., Introduction to and Heat Transfer, In Handbook of Single-Phase Convective Heat Transfer Analysis, McGraw-Hill, New Convective Heat Transfer (Edited by S. Kakaç, R.K. York, 1999, Chap. 4. Shah, W. Aung), Wiley, New York, 1987, Chap. 20. Wei, D. and Luo, H., Finite element solutions of heat Lawal, A. and Mujumdar, A. S., Forced-convection- transfer in molten polymer flow in tubes with viscous heat-transfer to a power-law fluid in arbitrary cross- dissipation, Int. J. Heat Mass Transfer 46, 3097-3108, section ducts, Can. J. Chemical Eng. 62, 326-333, 1984. 2003.
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