PERIODIC FLOW OF NON-NEWTONIAN FLUIDS ACROSS AN ARRAY OF CYLINDERS

Abstract

The flow and thermal features of fluids across the periodic array of cylinders and/or over tube banks are considered as one of the classical problems in the fluid mechanics because of its widespread industrial applications. For instance, the flow of process streams in the shell side of tubular heat exchangers is often used to model the flow over the tube banks. Screens used to filter polymer melts are sometimes modeled as randomly oriented cylinders. Longitudinal flow over rod bundles is common in most fuel elements of nuclear power reactors where the main heat exchanging equipment is composed of a large number of parallel fuel rods arranged in a regular array. To allow sufficient and regularly distributed space for the coolant flow parallel to the axes of the fuel rods, suitable spacing devices are used, which also provide rigidity to the fuel element. Moreover, this flow arrangement is also found significant in the fluidized bed drying of fibrous materials, filtration of paper and pulp suspensions and biological systems, etc. (Duli et al., 1995; Martin et al., 1998; Vijaysri et al., 1999; Mandhani et al., 2002; Mangadoddy et al., 2004). The above flow geometries are oriented in different geometrical arrangements in the process equipment such as heat exchangers, evaporators, boilers and condensers, etc. Typically, tube banks consist of square, triangular, rectangular and hexagonal array of cylinders. Amongst all these arrangements, the square array of cylinders is most popular and widely used because of its simple geometrical orientations. Because the physical geometry of interest, the expected patterns of the flow through an array of cylinders have a periodically repeating nature. The present work is devoted to the periodic flow of fluids across a square array of circular cylinders in cross-flow. Understanding of the flow and thermal features from such a system is always challenging, because the boundary layers are periodic and continuous which cause a great resistance to the flow of fluids. The kind of flow whether longitudinal or transverse, laminar or turbulent and the porosity of the cylinders (cylinder arrays are treated as porous media), are the other factors which further make the problem more stimulating to explore and investigate. For instance, in the case of transverse flow, the buoyancy acts perpendicular to flow direction, which enhances the rate of heat transfer. Few such examples are the air flowing horizontally over a heated pipe, steam leaving a boiler passes through a pipe where a fan blowing over it, etc. Other examples include the applications in solar thermal extraction system as well as some parts of electronic equipment cooling, etc. (Gowda et al., 1998; Soares et al., 2009; Daniel and Dhiman, 2013). Furthermore, the non-Newtonian fluid (shear-thinning and shear-thickening) nature and the buoyancy induced flow are the additional factors which make the problems more intricate and ambiguous due to the direct impact of these parameters on the flow and thermal fields, drag iv force and Nusselt number, etc. In fact, a thorough and in-depth understanding of the detailed kinematics of such a flow is germane to the development of a reliable methodology for the process equipment design. In other words, the flow and heat transfer characteristics across a periodic array of cylinders are one of the starving areas to be studied in fluid mechanics. Owing to such an excellent industrial relevance, still very little is known about the flow and heat transfer characteristics across a periodic array of cylinders even with simple Newtonian kind of fluids (Koch and Ladd, 1997; Martin et al., 1998; Spelt et al., 2005b). Notwithstanding, in the literature, two distinct approaches/schemes are available to investigate such type of flow and thermal problems. In the first scheme, the field equations are solved for the different geometrical arrangements of a periodic array of cylinders/spheres with known geometrical configuration, whereas, in the second scheme, the assemblage of cylinders/spheres is modeled by using the approximate cell models where the modeling depends upon particle-particle interactions. For instance, a frictional pressure gradient over an assemblages of long cylinders was estimated by employing a most common velocity and stress-variational principle (Slattery, 1972). Further, free surface cell model (Happel, 1964) was used to determine the interferences amongst the cylinders. The drag on cylinders was described to gradually reduce under the analogous value of Newtonian fluids with an increasing shear-thinning behavior. Subsequently, the model equations were solved by using free surface cell models, zero vorticity cell model, etc. (Vijaysri et al., 1999; Shibu et al. 2001; Mandhani et al., 2002; Mangadoddy et al., 2004, etc.) In summary, the critical review of the available literature on the periodic flow across an array of cylinders for Newtonian and non-Newtonian fluids suggests that insights for global engineering parameters such as the drag coefficients, pressure loss, permeability, Nusselt numbers, etc. using various cell models are available, but they are limited to lemianr creeping flow and low porosity of the cylinders. In fact, there is no literature which could reveal the flow and thermal features by using the direct periodic array of cylinders. Additionally, in case of mixed convection flow problems, negligible literature/studies are available to such flow geometries. These gaps in the literature motivated us to investigate the problems of forced and mixed convection across an array of cylinders to explore the features of Newtonian and non-Newtonian power-law fluids flow through such an industrially important flow geometry. Therefore, with the aim of fulfilling the gap in the literature, this thesis is concerned with the numerical investigation of steady forced/mixed convection flow and heat transfer characteristics of Newtonian and non-Newtonian power-law fluids across an array of circular cylinders in a square arrangement. v Objectives of the thesis The objectives of this thesis are to supplement the available knowledge through a CFD investigation for the wide ranges of pertinent dimensionless parameters such as fluid volume fraction, power-law index, Reynolds, Prandtl and Richardson numbers to explore the momentum and heat transfer characteristics of Newtonian and non-Newtonian fluids across a periodic array of circular cylinders in square configuration. Particularly, this dissertation has focused on an investigation of the five problems as mentioned in the following Table. Table: Problems studied in this dissertation S. No. Problems studied across a periodic array of cylinders in a square configuration Ranges of physical parameters 1. Forced convection flow and heat transfer characteristics of Newtonian fluids 0.70 ≤ f  ≤ 0.99 0.70 ≤ Pr ≤ 100 1 ≤ Re ≤ 40 2. Forced convection flow characteristics of non- Newtonian power-law fluids 0.70 ≤ f  ≤ 0.99 1 ≤ Re ≤ 40 1 ≤ Pr ≤ 100 0.4 ≤ n ≤ 1.8 3. Forced convection heat transfer characteristics of non-Newtonian power-law fluids 0.70 ≤ f  ≤ 0.99 1 ≤ Re ≤ 40 1 ≤ Pr ≤ 100 0.4 ≤ n ≤ 1.8 4. Mixed convection flow and heat transfer characteristics of Newtonian fluids 0.70 ≤ f  ≤ 0.99 0.70 ≤ Pr ≤50 1 ≤ Re ≤ 40 0 ≤ Ri ≤ 2 5. Mixed convection flow and heat transfer characteristics of power-law fluids 0.70 ≤ f  ≤ 0.99 1 ≤ Pr ≤ 50 1 ≤ Re ≤ 40 0 ≤ Ri ≤ 2 0.4 ≤ n ≤ 1.8 ( f  : fluid volume fraction, Pr: Prandtl number, Re: Reynolds number, Ri: Richardson number and n: power-law index) The solution of aforementioned problems has been obtained by solving the modified form of Navier-Stokes equations with appropriate boundary conditions for the various cases and conditions with and without the use of Boussinesq approximations by using commercial CFD solver ANSYS Fluent (2009). An unstructured non-uniform grid consisting of triangular cells was generated by using a commercial grid tool GAMBIT. A finer mesh was generated near the cylinder surfaces to better resolve the sharper gradients. The 2-D, laminar and segregated solver vi was utilized to simulate the incompressible flow on the collocated grid arrangement. A second order upwind scheme was used to discretize the convective terms appearing in both flow and thermal equations. A SIMPLE scheme was used to handle the pressure and velocity coupling. The double precision solver was used to improve the accuracy of solutions and the converged results were obtained when each of the continuity, momentum residuals reached in the order of 10-10 and energy residuals in the order of 10-14. In view of all these facts, the above five problems are discussed herein for a brief overview of this dissertation: 1. Forced convection flow and heat transfer characteristics of Newtonian fluids across periodic array of circular cylinders The forced convection flow and heat transfer characteristics of Newtonian fluids have been studied for the ranges of parameters as mentioned in Table. Particularly, the influences of flow governing parameters (Re, Pr and f ) on the local and global characteristics have been revealed. The numerical results suggest that for a given value of Reynolds number, the total drag coefficients decrease drastically as the fluid volume fraction increases. The importance of inertia diminishes with an increasing compactness of cylinders (i.e., the decreasing f  ) because of the largest contribution of viscous dissipation in this limit in small gaps between the cylinders. Therefore, the friction drag is dominating over pressure drag. These features reveal that there is the strong dependence of drag coefficient on both fluid volume fraction as well as Reynolds number. In the case of heat transfer, for the fixed values of Reynolds and Prandtl numbers, the average Nusselt number increases with decreasing fluid volume fraction. This increase in average Nusselt number is greatly influenced by the interactions between the cylinders. Attempts are also made to interpret the average Nusselt number values in terms of the Colburn heat transfer factor ( Hj ) for their easy use in process engineering and design calculations. Further, the numerical results of individual and total drag coefficients and the average Nusselt number have been used to develop the simple correlations as a function of pertinent dimensionless variables (i.e., Re, Pr and f  ) and results have been compared with the available literature which displayed an excellent agreement. 2. Forced convection momentum transfer characteristics of power-law fluids across periodic array of circular cylinders Here, the flow features of power-law fluids have been displayed for the ranges of parameters as listed in Table. The results are presented in terms of streamlines, pressure coefficient and individual and total drag coefficients for the governing parameters. The dependence of individual vii and total drag coefficients on the power-law index, fluid volume fraction and Reynolds number shows the non-monotonous behavior. For shear-thinning fluids (n < 1), the pressure drag coefficient dominates over friction drag coefficient; whereas, an opposite behavior was seen for the shear-thickening fluids (n > 1) except at Re = 40. Further, both of the individual and total drag coefficients were seen to increase and decrease with the increase in the power-law index over the range of fluid volume fraction (0.70 ≤ f  ≤ 0.90) and (0.92 ≤ f  ≤ 0.99), respectively. Strong interactions between the periodic cylinders were observed at the lower values of fluid volume fractions which diminish with the increasing value of fluid volume fractions. The results were further used to develop an empirical correlation for the pressure, friction and total drag coefficients to give an additional physical insight of this study. The results have been compared with the available literature which shows an excellent agreement. 3. Forced convection heat transfer characteristics of power-law fluids across periodic array of circular cylinders In this part, the results have been discussed in terms of local and global characteristics of heat transfer such as isotherm patterns, local and average Nusselt numbers for the broad range of governing parameters (Table). The numerical results revealed the heat transfer enhancement of approximately 97% in the shear-thinning fluids among the lowest and the highest fluid volume fractions for the highest value of Pr and the lowest value of Re and n. Under the identical conditions, the enhancement was about 83% in the shear-thickening region. For the ranges examined herein, different levels of improvement in the average Nusselt number were noticed because of the shear-thinning and shear-thickening natures. The results were further used to develop an empirical correlation for the average Nusselt number and the Colburn heat transfer factor to give additional physical insight. Additionally, the present results have been compared with the available literature which displayed a good agreement. 4. Aiding buoyancy mixed convection characteristics of Newtonian fluids across periodic array of circular cylinders Mixed convection features of Newtonian fluids have been studied under the aiding buoyancy conditions for the ranges of parameters stated in Table. The numerical results were observed to be the strongly dependent on the governing parameters ( f  , Re, Pr and Ri). The drag coefficients were observed to be diminished with an upturn in Reynolds number and fluid volume fractions, whereas an opposite behavior was noticed with the rise in Prandtl number and buoyancy parameter (Ri). Further, the local and average Nusselt numbers were improved with increased viii value of Prandtl and Reynolds numbers and surprisingly with the fluid volume fractions ( f ) also, as opposed to decrease in forced convection case. Additionally, aiding buoyancy enhances both flow as well as heat transfer features. The strong influence of f on results was observed relative to other parameters (Re, Pr and Ri). Moreover, at the higher values of f , Re and Pr, a transient behavior was also noticed. Statistical correlations were developed for the total drag coefficient and average Nusselt number to gain the further physical insights. Finally, the results have been compared with the scant available literature which displayed an excellent agreement. 5. Aiding buoyancy mixed convection characteristics of power-law fluids across periodic array of circular cylinders The mixed convection flow and heat transfer characteristics to non-Newtonian fluids across a periodic array of circular cylinders have been investigated numerically for a wide range of governing parameters (n, f  , Re, Pr and Ri) under the aiding buoyancy conditions (Table). The influences of these parameters on the streamlines, pressure coefficient, isotherm patterns and individual and total drag coefficients, local and average Nusselt numbers were explored and presented. The local flow phenomenon (streamlines, pressure coefficient and isotherm patterns) describes that the dense arrays offer higher resistances to flow of fluids and hence sparse array is required to minimize the flow resistances. Further, the drag coefficients decrease gradually with increasing value of Re for all the values of fluid volume fractions and power-law index. However, an increase in drag coefficients was observed with increasing value of Prandtl number in the mixed convection (Ri > 0) case as opposed to forced convection case (Ri = 0). The isotherm patterns reveal that the increasing inertial effects enhance the rate of heat transfer due to the dense clustering of the isotherms near the cylinder surfaces. The average Nusselt number was observed to be increased with the increasing values of f  , Re, Pr and Ri and the decreasing value of n (increasing shear-thinning behavior). A transient behavior was observed for both of the drag coefficients as well as average Nusselt number at higher fluid volume fractions and Reynolds numbers. Additionally, aiding buoyancy enhances both of the flow and thermal parameters in the vicinity of periodic cylinders. Moreover, statistical correlations for the drag coefficients and average Nusselt numbers are developed for gaining the more physical insight of the results. In summary, the detailed insights of the forced and mixed (aiding buoyancy) convection flow and heat transfer characteristics have been gained and presented for both Newtonian and non-Newtonian power-law fluids for the wide ranges of flow governing parameters across a periodic array of circular cylinders in a square geometrical configuration.