Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis

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1 Tamkang Journal of Science and Engineering, Vol. 12, No. 1, pp (2009) 99 Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis M. E. Sayed-Ahmed 1 *, A. Saif-Elyazal 1 and L. Iskander 2 1 Department of Engineering Mathematics and Physics, Faculty of Engineering, Fayoum University, Fayoum-63111, Egypt 2 Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza-12163, Egypt Abstract Laminar fully developed flow and heat transfer of Herschel-Bulkley fluids through rectangular duct is investigated numerically. The non-linear momentum and energy equations are solved numerically using finite-element approximations. We consider two cases of thermal boundary conditions H 1 and T thermal boundary conditions. The velocity, temperature profiles, product of friction factor-reynolds number and Nusselt number for H 1 and T thermal boundary conditions are computed for various values of the physical parameters of the Herschel-Bulkley fluids and aspect ratio of the duct. The present results have been compared with the known solution for Newtonian and power-law fluids and are found to be in good agreement. Key Words: Numerical Analysis, Non-Newtonian Fluids, Heat Transfer, Rectangular Duct 1. Introduction The fluid flow behavior of non-newtonian fluid has attracted special interest in recent years due to the wide application of these fluids in the chemical, pharmaceutical, petrochemical, food industries and electronic industries. A large number of fluids that are used extensively in industrial application are non-newtonian fluids exhibiting a yield stress y, stress that has to be exceeded before the fluid moves. As a result the fluid cannot sustain a velocity gradient unless the magnitude of the local shear stress is higher than this yield stress. Fluids that belong to this category include cement, drilling mud, sludge, granular suspensions, aqueores foams, slurries, paints, plastics, paper pulp and food products. Several studies have appeared, [1 6] studied the fully developed velocity profile and fraction factor- Reynolds number product and Nusselt number for Newtonian in a rectangular duct. In early studies for non- Newtonian fluid, Schechter [7] applied a variational *Corresponding author. method to obtain the velocity profile and the corresponding friction factor for a power-law fluid. Wheeler and Wissler [8] solved the same problem by finite difference method and furthermore proposed a simple friction factor-reynolds number correlation for the special case of a square duct. Kozicki et al. [9] presented fairly approximate relationship between friction factor and Reynolds number through rectangular ducts as well as for some other non-circular ducts. More recent studies in the field under discussion are mainly based on numerical approaches. Geo and Hartnett [10] used finite difference method to obtain friction factor-reynolds number product and velocity profile for power-law fluid through rectangular duct. On the other hand Seppo Serjala [11, 12] solved the same problem by using finite-element method. The flow of Herschel-Bulkley fluids has been considered for other geometries by Bettra and Eissa [13] in entrance region through parallel plate channels. Eissa [14] studied laminar heat transfer for thermally developing flow of Herschel-Bulkley fluid through special cases of square ducts by using finite difference method. Sayed-Ahmed and El-yazal [15,16] studied the flow and

2 100 M. E. Sayed-Ahmed et al. heat transfer of Robertson-Stiff fluids in rectangular duct using finite-difference method. In the present study, we investigate the problem of laminar fully developed flow of Herschel-Bulkley fluids through rectangular duct. The non-linear momentum equation is solved iteratively using a finite-element method to obtain the velocity profile and the value of the friction factor-reynolds number product. The energy equation is solved numerically to obtain the temperature profile and Nusselt number for two cases of thermal boundary conditions H 1 (axially uniform heat flux and peripherally uniform temperature) and T (axially and peripherally uniform temperature). Computations are given over wide range of duct aspect ratios and flow behavior index and yield stress of Herschel-Bulkley fluids. 2. Problem Formulation We consider a steady, fully developed, laminar, isothermal flow of in-compressible and purely viscous non-newtonian in a rectangular duct. The duct configuration and coordinate system are shown in Figure 1. For the previous assumptions, the momentum and energy equations reduced to (1) wall). Since the flow is hydro dynamically and thermally fully developed flow then the term T/ z in the energy equation (2) can be represented as [12] for case (1): H 1 thermal boundary condition, and as (4) (5) for case (2): T thermal boundary condition The fraction factor f and Reynolds number Re for Herschel-Bulkley fluids are given by The average velocity w av is given by (6) (7) Now it is convenient to write the above equations in the non-dimensional form. The relevant dimensionless quantities are defined by (2) In which w is the velocity component in z-direction and the apparent viscosity for Herschel-Bulkley fluids is described by (8) Substitution these quantities in equation (8) into (3) The boundary condition for the velocity is no slip boundary condition (i.e. the velocity at the walls of a rectangular duct equals zero). The boundary conditions for the temperature is T = T w (the temperature at the Figure 1. Configuration and coordinate system for a rectangular duct.

3 Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis 101 equations (1) and (3), and by making use of the relation (6) and (7). The dimensionless equation of motion is reduced to where dimensionless average velocity W av can be written as and the dimensionless viscosity is reduced to (9) The Nusselt number Nu is defined by (16) (17) (10) where h is the average heat transfer coefficient and given by The dimensionless energy equations are reduced to for case (1): H 1 thermal boundary condition and case (2): T thermal boundary condition (11) (12) (18) The Nusselt number for the two cases thermal boundary condition is given by (19) where m is the bulk mean temperature and evaluated by the form Noting that for symmetry reasons only a quarter of the flow domain has to be considered (0 X 0.5, 0 Y 0.5 ), the velocity boundary conditions can be written as 3. Numerical Solution (20) (13) and the temperature boundary conditions can be written as (14) The product of friction factor and Reynolds number reduce into the dimensionless form (15) The finite-element method is used to approximate the solution of partial differential equations (9), (11) and (12) with boundary conditions (13) and (14). We will divided the domain into finite rectangular elements, which is a suitable element for the domain as shown in Figure 2 with sides X and Y. Suppose that W and can be approximated by the expression (21) where j is a linear interpolation function of the rectangular element. Multiply equation (9), (11) and (12) by a test function v and integrate over the element domain e. The variational form becomes

4 102 M. E. Sayed-Ahmed et al. where m is the average viscosity m over the element and q is the average dimensionless temperature q over the element from the last iteration. The linear interpolation functions for rectangular duct with both sides DX and DY are given by [17] (28) Figure 2. The domain is divided into rectangular elements with four nodes. (22) Using linear interpolation functions in equations (28) into equations (25), (26) and (27) and evaluate the integrals then substituting the boundary conditions to get a system of linear equations on Wj and qj. For non linear equation (25) an iterative procedure is used to obtain the unknown Wj. The initial value of viscosity is assumed to be the same at each mesh node (m = 1.0 for Newtonian fluid) to obtain the velocity at each node of the elements. Making use the value of the velocity that are obtain to get the new viscosity by (23) (29) (24) Substituting equation (21) for W, q and yi for v into the variational form (22)-(24), we obtain (25) The process is repeated until the criteria of conver-5 old gence W(i,newj) - W(i,oldj) 10-5 and m new are (i, j) - m (i, j) 10 satisfied. The convergence has been achieved by taking elements. The average velocity Wm is evaluated by Simpson rule formula of the double integrals (26) (30) (27) For case (1): H1 thermal boundary condition, equation (26) is solved to obtain the temperature qj. For case (2): T thermal boundary conditions, non linear equation (27) an iterative procedure is used to obtain the unknown qj. We assume initial value of temperature the

5 Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis same at each mesh node (q = 1.0) and evaluating the bulk mean temperature to obtain the temperature at each node of the elements. Making use the new value of the temperature that are obtain to get the new value of the bulk mean temperature. The process is repeated un-5 old til the criteria of convergence q new are (i, j) - q (i, j) 10 satisfied. The convergence has been achieved by taking elements. 103 The velocity profile and friction factor fre are obtained for different values of (n = 1.5, 1.2, 1.0, 0.8, 0.5), (td = 0.0, 0.01, 0.03) and (a = 1.0, 0.5, 0.2). Figures 3a-3c, 4a-4c and 5a-5c show the distribution of velocity W along the centerline of the cross-section, which is parallel to the major side of the duct, for various values of n, td and for a = 1.0, 0.5, 0.2 respectively. The study of Figures (3-5) shows that the value of the velocity W increases from zero at the wall (no slip condition) to the maximum value at the mid point (0.5, 0.5a) for all values of a, n and td. The maximum value of the velocity increases as the n increases for all values of td and a as a result of decrease in the apparent viscosity of the non-newtonian fluid and, therefore, increasing the average velocity. Also the value of the velocity decreases with increasing in td for all values of n and a due to increasing the plug core formation, Figure 3. The distribution of velocity W along the centerline of the cross-section, which is parallel to the major side of the duct for a = 1.0. Figure 4. The distribution of velocity W along the centerline of the cross-section, which is parallel to the major side of the duct for a = Results and Discussions

6 104 M. E. Sayed-Ahmed et al. Figure 6. The variation of fre with flow behavior index n. Figure 5. The distribution of velocity W along the centerline of the cross-section, which is parallel to the major side of the duct for a = 0.2. which no flow occurs through the region, so the net flow region decreasing. The velocity gradient at the wall increases as n increases for all values of td and a. We can also observe that the velocity profile becomes flatter with a decrease in the value of the flow behavior index n (shear thinning n < 1.0) and a due to decreasing in apparent viscosity of the non-newtonian fluid. It is found also the velocity profile becomes increasingly flattened with increasing of the value of td this phenomenon is due to the formulation of the plug core of the yield non-newtonian fluids along the centerline of the duct. The variation of fre with flow behavior index n is shown in Figures (6a-6c) for a = 1.0, 0.5, 0.2 respec- tively. It has been found that the value of fre increases with increasing value of n due to the increasing of the viscosity of Herschel-Bulkley fluid with the flow index n for all values of td and a. It can found also the increasing of the value of td increases the value of fre for all values of n and a. This phenomenon is explained as follows an increasing in the value of the yield stress td increases the plug core formation, which no flow occurs through this region and net flow region decreasing. The decreasing in the value of a increases the value of fre for all values of n and td as a result of decreasing the cross section area. Nusselt number are obtained for different values of (n = 1.5, 1.2, 1.0, 0.8, 0.5), (td = 0.0, 0.01, 0.03) and (a = 1.0, 0.5, 0.2). Figures 7a-7c and 8a-8c show the variation of Nusselt number for case (1): H1 thermal boundary

7 Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis 105 Figure 7. The variation of Nusselt number Nu with flow behavior index n for case (1): H1 thermal boundary condition. Figure 8. The variation of Nusselt number Nu with flow behavior index n for case (2): T thermal boundary condition. conditions and case (2): T thermal boundary conditions for various values of n, td and for a = 1.0, 0.5, 0.2 respectively. Examination of Figures 7a-7c and 8a-8c shows that the value of Nu for case (1) and case (2) thermal boundary conditions increases as the flow behavior index n decreases for all values of td and a. The increase of td increases Nu for case (1) and case (2) thermal boundary conditions for all values of n and a. This phenomenon is explained as follows an increasing in the value of the yield stress td increases the plug core formation, which no flow occurs through this region and net flow region decreasing which decreases bulk mean temperature. The decreasing in the value of a increases the value of Nu for case (1) and case (2) thermal boundary conditions for all values of n and td as a result of decreasing the cross section area, which decreases bulk mean temperature. It also found that, the values of Nu for H1 thermal boundary conditions greater than the values of Nu for T thermal boundary conditions. Tables 1-3 show the comparison of the present results of fre and Nu for H1 and T thermal boundary condition with the previous work (Shah and London [5] and Syrjala [11,12]) for a = 1.0,0.5,0.2, n = 1.2, 1.0, 0.5, and td = 0.0 (power-law fluid). The present results are found to be in good agreement with the previous work. Nomenclature Dh hydraulic diameter (4 cross area/perimeter)

8 106 M. E. Sayed-Ahmed et al. Table 1. The comparison of the present results of fre with the previous work for D = 0.0 (power-law fluid) n Shah and London, [5] Syrjala, [11] Present result Table 2. The comparison of the present results of Nu for case (1): H1 thermal boundary condition with the previous work for D = 0.0 (power-law fluid) n Shah and London, [5] Syrjala, [12] Present result Table 3. The comparison of the present results of Nu for case (2): T thermal boundary condition with the previous work for D = 0.0 (power-law fluid) n Shah and London, [5] Syrjala, [12] Present result F fraction factor (e) f i element load force vector h average heat transfer coefficient k thermal conductivity of Herschel-Bulkley fluids (e) k ij element stiffness matrix L maximum length of the rectangular duct m consistency index of Herschel-Bulkley fluids n flow behaviour index of Herschel-Bulkley fluids Nu Nusselt number P Pressure q n secondary variable projection along unit vector Re Reynolds number T temperature T m bulk mean temperature T w wall temperature v test function w axial fluid velocity in duct w av average fluid velocity in duct W dimensionless axial velocity in duct Wav dimensionless average fluid velocity x, y, z rectangular Cartesian coordinates X, Y, Z dimensionless coordinate in x, y, z axes aspect ratio thermal diffusivity apparent viscosity of the model dimensionless viscosity of the model average dimensionless viscosity over the element dimensionless temperature m the bulk mean temperature is the average dimensionless temperature over the element y yield stress value of the model D dimensionless yield stress e The element domain e The boundary of the element j linear interpolation function X, Y sides length of the rectangular element References [1] Marco, S. M. and Han, L. S., A Note on Limiting Nusselt Number in Duct with Constant Temperature Gradient by Analogy to Thin Plate Theory, Trans. ASME, Vol. 77, pp (1955). [2] Holmes, D. B. and Vermeulen, J. R., Velocity Profile

9 Laminar Flow and Heat Transfer of Herschel-Bulkley Fluids in a Rectangular Duct; Finite-Element Analysis 107 in Ducts with Rectangular Duct Cross-Sections, Chem. Eng. Sci., Vol. 23, pp (1968). [3] Munchkin, G. F., Solomonov, S. D. and Gordon, A. R., Hydrodynamic Developed of a Laminar Velocity Field in Rectangular Channels, J. Eng. Phys. (USSR), Vol. 25, pp (1973). [4] Natarajan, N. M. and Lakshmanan, S. M., Laminar Flow in Rectangular Ducts: Prediction of the Velocity Profiles and Fraction Factors, Indian J. Yechnol., Vol. 10, pp (1972). [5] Shah, R. K. and London, A. L., Laminar Flow Forced Convection in Ducts, Advanced Heat Transfer (Suppl. 1) (1978). [6] Hartnett, J. P. and Kostic, M., Heat Transfer to Newtonian and Non-Newtonian Fluids in Rectangular Duct, Adv. Heat Transfer, Vol. 19, pp (1989). [7] Schechter, R. S., On the Steady Flow of a Non- Newtonian Fluid in Cylinder Ducts, AICHE J. Vol. 7, pp (1961). [8] Whreeler, J. A. and Wissler, E. H., The Friction Factor-Reynolds Number Relation for Steady Flow of Peseudoplastic Fluids through Rectangular Ducts, AICHE J. Vol. 11, pp (1965). [9] Kozicki, W., Chou, Ch and Tiu, C., Non-Newtonian Fluid Inducts of Arbitrary Cross-Sectional Shape, Chem. Eng. Sci, Vol. 21, pp (1971). [10] Gao, S. X. and Hartnett, J. P., Non-Newtonian Fluid Laminar Flow and Forced Convection Heat Transfer in Rectangular Ducts, Int. J. Heat Mass Transfer, Vol. 35, pp (1992). [11] Seppo Syrjala, Finite-Element Analysis of Fully Developed Laminar Flow of Power-Law Non-Newtonian Fluid in Rectangular Duct, Int. Comm. Heat Mass Transfer, Vol. 22, pp (1995). [12] Seppo Syrjala, Further Finite-Element Analysis of Fully Developed Laminar Flow of Power-Law Non- Newtonian Fluid in Rectangular Duct: Heat Transfer Prediction, Int. Comm. Heat Mass Transfer, Vol. 23, pp (1996). [13] Battra, R. L. and Eissa, M., Heat Transfer of a Herschel-Bulkley Fluid in a Thermal Entrance Region of Parallel-Plate Channels with Viscous Dissipation Effects, Proc. of the 19 th National conf. on FMEP, Bombay (I.T.T., Powai), India, pp. A4-1 A4-6 (1992). [14] Eissa, M., Laminar Heat Transfer for Thermally Developing Flow of a Herschel-Bulkley Fluid in a Square Duct, Int. Comm. Heat Mass Transfer, Vol. 27, pp (2000). [15] Sayed-Ahmed, M. E. and El-yazal, A. S., Laminar Fully Developed Flow and Heat Transfer of Robertson-Stiff Fluids in a Rectangular Duct with Temperature Dependent Viscosity, Int. Comm. Heat Mass Transfer, Vol. 30, pp (2003). [16] Sayed-Ahmed, M. E. and El-yazal, A. S., Laminar Fully Developed Flow and Heat Transfer of Robertson-Stiff Fluids in a Rectangular Duct, Can. J. Phys. Vol. 83, pp (2005). [17] Zienkiewicz, O. C. and Morgan, K., Finite Elements and Approximation, John Wiley & Sons, Inc. Canada, chapter (3) pp (1983). Manuscript Received: Nov. 29, 2006 Accepted: Sep. 5, 2007

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