Studies on flow through and around a porous permeable sphere: II. Heat Transfer

Studies on flow through and around a porous permeable sphere: II. Heat Transfer A. K. Jain and S. Basu 1 Department of Chemical Engineering Indian Institute of Technology Delhi New Delhi 110016, India ABSTRACT Heat transfer from a porous-permeable sphere due to flowing of fluid through and around it is studied for a wide range of Reynolds number (0.02 to 2000) and Pradtl number (0.7 to 7) using standard CFD software. CFD simulated results for impermeable sphere predicts the Whitaker correlation quite well. The Simulation is then extended for heat transfer from an isothermal porous-permeable sphere. The results are presented in terms of four dimensionless parameters - particle Reynolds number (Re) based on the free stream velocity and diameter of the porous sphere, permeability ratio, Pradtl number (Pr) and Nusselt number (Nu). The results show that the heat transfer rate from porous permeable sphere increases with the increase in permeability. At low Re, the Nu for a permeable sphere is higher than that for solid sphere when the Pr is low, whereas, for high Pr, permeability has only a weak effect on the Nu. At high Re, the Nu for permeable sphere is much higher than that for solid sphere irrespective Pr values. The correlation obtained from the CFD simulation data for heat transfer from porous permeable sphere is useful in predicting Nu for porous permeable sphere for a wide range of Re from 0.02 to 2000 and Pr from 0.7 to 7 at different permeability ratios. Keywords: Porous permeable sphere, Heat transfer, Permeability Ratio INTRODUCTION Heat transfer from an object due to flowing of surrounding fluid is a topic of industrial importance. The object concerned may be relatively simple, such as a cylinder or sphere, or it may be more complex, such as a tube bundle made up of a set of cylindrical tubes with a stream of gas or liquid flowing between them. In the present investigation, spherical aggregates forming porous permeable sphere is considered because of 1 Corresponding author; e-mail: sbasu@chemical.iitd.ac.in; Fax +011 91 26581120 1

simplicity in analyzing symmetric flow condition, availability of data and for the purpose of comparing results for solid sphere. The results may be applicable to many processes that involve spray of bubbles or droplets (Bird et al., 2002). Also, it may be useful in understanding the geothermal energy transportation under varied geological structures (Vasudeviah and Balamurugan, 1998). Acrivos and Taylor (1962) studied the problem of forced convection from an isothermal sphere for small and large Peclet number (Pe). Their analysis is valid when Reynolds number (Re) and Pe ( = Re. Pr) is less than one, with no restriction on the Prandtl number (Pr). Different empirical heat transfer correlations were given by different investigators for single impermeable sphere along with experimental data e.g., Ranz and Marshall (1952), Whitaker (1972), Achenbach (1978) and Romkes et al. (2003). The different empirical correlations are given below. Ranz and Marshall (1952) 1 2 1 3 Nu = 2.0 + 0.66 Re Pr for 3.5 < Re < 7.6 x 10 4 (1) Whitaker (1972) 1 2 2 3 2.0+ 0.4Re + 0.06Re Nu = Pr (2) for 3.5 < Re < 7.6 x 10 4 and 0.7 < Pr < 380 Achenbach (1978) 1 6 Nu = 2.0 + + 3*10 4 Re 1. for 10 2 < Re < 2 x 10 5 (3) 4 2 3 It is well known fact that the Nusselt number (Nu) is equal to 2 for a sphere immersed in an infinite medium as the steady state conduction solution prevails (Whitaker, 1972). Johnson and Smet (1984) examined the heat transfer from a permeable sphere in uniform flow and at low Re. They considered the case when conduction is dominant heat transfer mechanism in the exterior fluid, i.e. small Pe, and the case when convection was dominant, i.e. large Pe. They found from their study that at small Peclet numbers the heat transfer rate from a permeable sphere differ at leading-order from that found for an impermeable sphere, whereas, for large Peclet numbers permeability has only a weak or 2

second-order effect on the heat transfer rate. But their theoretical work was only limited to Stokes flow. They did not study the system for high Reynolds number. In the present work heat transfer study on flow through and around a porous permeable sphere for a wide range of Re (0.02 to 2000) and Pr (0.7 to 7) is studied using standard CFD software. The main objective is to study the effect of permeability and to examine how the heat transfer process is influenced by fluid flow not only around but also through the porous permeable sphere. A generalized correlation is suggested based on all simulation results obtained for a wide range of Reynolds number, Prandtl number and permeability ratio. COMPUTATIONAL FLUID DYNAMICS SIMULATION Porous permeable sphere is assumed to consist of small spherical particles (grains) with some specific void fraction. Mass and energy conservation expressions and the Navier- Stokes equations are used for the outer region of porous permeable sphere. Whereas, in the inner region, mass and energy conservation and Darcy s law of Brinkman s extension are applied in order to solve the temperature, pressure and velocity fields. Thus, the flow regions through and around a porous permeable sphere are divided into two parts, namely, internal flow and external flow. The flow regions and the coordinate systems for flow through and around a porous permeable sphere are shown in figure 1 of Jain et al. (submitted). External flow: Equation of continuity, Reynolds-average Navier-Stokes (RANS) given in Jain et al. (submitted) were used along with the energy equation. The energy equation is given below. Equation of energy t ( E) + v ( ρe + P) ( ) = λeff T Σ H jj j + ( τ eff v) + Sh ρ (1) j Where E is total energy, ρ is the density, P is the pressure, T is the temperature, H is the enthalpy, τ eff is the effective shear stress, J j is the diffusion flux of species j and λ eff is the 3

effective thermal conductivity. λ eff = λ + C p µ t / Pr t, where Pr t is the turbulent Prandtl number, C p is the specific heat of fluid and λ is the thermal conductivity of the flowing fluid. The first three terms on the right-hand side of eq. (9), represent energy transfer due to conduction, species diffusion, and viscous dissipation, respectively. S h includes the heat of chemical reaction, and any other volumetric heat sources have been defined. In the present study S h is zero. Internal flow: (i) Momentum equations for porous media: Porous media are modeled by the addition of a momentum source term to the Reynoldsaverage Navier-Stokes equation. The source term is composed of two parts, a viscous loss term and an inertial loss term. For simple homogeneous porous media, the source term, the Kozeny equation for relating relating permeability and porosity, the inertial resistance factor, Darcy s equation and intertial loss term given in Jain et al. (submitted) were used for the modeling of internal flow region. It should be noted that isothermal condition is assumed inside of the porous permeable sphere. Meshing of the system and setting of tolerance limit was done following same procedure as described in Jain et al (submitted). Laminar model was used for low Reynolds number and k-ε model is used at high Reynolds number. Inside the porous permeable sphere laminar model is used. Calculation of heat transfer coefficient (h) and Nusselt number (Nu): The temperature of the surface of the particle, constitutes the porous permeable sphere, was set to a surface temperature, T s, whereas the fluid at the boundary of the domain (inlet condition) was set to a lower bulk temperature, T. As a result of the temperature difference between the particle surface and the fluid, heat is transferred from the particle to the fluid. The particle-to-fluid heat transfer rate, q, for flow around sphere of diameter d, and heat transfer coefficient, h, is expressed as, q = h π d 2 (T s -T ) and Nusselt number is expressed as, Nu = h d / λ. 4

RESULT AND DISCUSSION Solid sphere: At first, simulations are carried out for solid sphere for different Reynolds number ranging from 0.02 to 2000 and different Prandtl number varying from 0.7 to 7. The simulation results are plotted in the form of Whitaker correlation (Whitaker, 1972) and compared with the experimental data. In figure 1, the symbols show experimental data whereas dashed line indicates the Whitaker correlation (eq. 1). It is well known fact that the correlations available in the literature (Holman, 1986) slightly under predict the experimental value for Nu at low Re. It is seen in figure 1 that the experimental data are well predicted by the CFD simulations, as shown by solid line at low Re. The simulated Nu values for high Reynolds number are also in good agreement with the experimental data available in literature. Porous permeable sphere: In figure 2, Nu, is plotted against Re Pr 2/3 at different permeability ratio, K/d 2 for flowing of air through and around the porous permeable sphere. The solid line indicates the Nu for impermeable sphere whereas different dashed lines indicate Nu for permeable sphere of different K/d 2 values. Nu for solid sphere approaches to the asymptotic value of 2 with the decrease in Re. Nu increases with the increase in Re for all K/d 2 values. The increase in heat transfer coefficient with the increase in Re is due to the increase in velocity field inside the porous permeable sphere, which in turn increased the heat transfer rate. The increase in Nu with K/d 2 values for porous permeable sphere is significant for high Reynolds number. The heat transfer coefficient increases with the increase in K/d 2 value because of the lower resistance to flow offered by highly permeable sphere. At low Reynolds number, value of Nu for permeable sphere approaches to that for solid sphere. Present simulation result on Nu for porous permeable sphere is verified with the analytical solution given by Johnson and Smet (1984) at low Re (Table 1). They concluded that the Nu for porous permeable sphere is higher than that for solid sphere at low Pe ( = Re. Pr) of 0.015. On the other hand, Nu has weak dependence on permeability at a slightly higher Pe (say, 0.14) and hence heat transfer rate from permeable and solid 5

sphere is almost the same. It is seen in table 1 that the Nu for porous permeable sphere differs by 15 % from that for impermeable sphere at low Re and Pr. However, Nu for porous permeable sphere differs only by 0.09 % from that for impermeable sphere at low Re and high Pr. Further, at high Re, permeability has strong effect on Nu irrespective of the Prandtl number. Thus, Nu for porous permeable sphere is higher than that for solid sphere at high Reynolds number. The Nu increases with the increase in Pr for both permeable and impermeable sphere at high Re, whereas no significant increase in Nu is with the increase in Pr at low Re. Based on simulated results a correlation for heat transfer from porous permeable sphere is generated using non-linear regression analyses. The regression coefficient of 0.91 was obtained. The correlation for heat transfer from porous permeable sphere is given by, 1 13.1 2 2 3 ( ) ( 2 ) 1 2 2 = 3.7 Re Pr 1 + K Nu (15) d valid for, 0.02 < Re < 2000, 0.7 < Pr < 7 and 10-5 < K/d 2 < 10-2 When predicting the Whitaker correlation (for solid sphere) using above equation (eq. 15), an average error of 15 % was obtained at low Re with zero permeability value (K/d 2 = 0). This correlation can be extended for mass transfer from porous permeable sphere since the phenomena of mass transfer is analogous to heat transfer. In eq. (15), Nu and Pr have to be replaced by Sherwood number and Schimdt number. This would work fine for no change in size of porous permeable sphere due to mass transfer. CONCLUSION Heat transfer study on flow through and around a porous permeable sphere is investigated with the help of standard CFD software for wide ranges of Reynolds number and Prandtl number. Experimental data and Whitaker correlation (1972) for heat transfer from solid sphere are excellently predicted by present CFD simulation. The CFD simulation then extended for heat transfer from isothermal porous permeable sphere by considering mass, momentum and energy equations outside of the porous permeable sphere and Darcy s law with Brinkman s extension inside the porous permeable sphere. Present simulated results show that the heat transfer rate increases with the increase in permeability. At low 6

Re, the Nu for a permeable sphere is higher than that for solid sphere when the Pr is low, whereas, for high Pr, permeability has only a weak effect on the Nu. At high Re, the Nu for permeable sphere is much higher than that for solid sphere irrespective Pr values. The correlation obtained from the CFD simulation data for heat transfer from porous permeable sphere is useful in predicting Nu for porous permeable sphere for a wide range of Re from 0.02 to 2000 and Pr from 0.7 to 7 at different permeability ratios. 7

Notation C p specific heat of fluid, J kg -1 K -1 d E diameter of sphere, m total energy, N m h heat transfer coefficient, Wm -2 K -1 H enthalpy, N m J j diffusion flux of species j, mole m -2 s -1 K permeability, m 2 K/d 2 Nu permeability ratio, dimensionless Nusselt number (= h D / λ), dimensionless P pressure, N/m 2 Pe Pr Pr t q Re Peclet number (= Re. Pr), dimensionless Prandtl number (= µ C p / λ), dimensionless turbulent Prandtl number, dimensionless heat transfer rate, Watt Reynolds number (= d v ρ / µ), dimensionless S h heat source term in eq. 1 t T T s T v i v j time, s temperature, K sphere temperature, K bulk temperature, K velocity component in x i direction velocity component in x j direction 8

v l v velocity component in x l direction approaching velocity, m/s Greek letters λ λ eff thermal conductivity, W m -1 K effective thermal conductivity, W m -1 K µ viscosity of fluid, Pa s µ t turbulent viscosity of fluid, Pa s ρ density of fluid, kg/m 3 τ eff Effective shear stress, N m -2 9

References Achenbach, E., in: Proceeding of the 6 th international heat transfer conference on heat transfer from sphere up to Re = 6 x 10 6, Vol. 5, Hemisphere 1978 Washington DC. Acrivos, A., and T. D. Taylor, Heat and mass transfer from single sphere in stokes flow, J. Phys. Fluids, (1962) 5(4), 387-394. Bird R. B., Stewart E. W., Lightfoot N. E., Transport Phenomena, John Wiley and Sons 2002, India. FLUENT User s Guide, Fluent Inc., 1998. Hinze, J. O., Turbulence, McGraw-Hill, 1975 New york. Holman, J. P., Heat transfer, McGraw-Hill, 1986 New York. Jain, A. K., C. Sirker, and S. Basu, Studies on flow through and around a porous permeable sphere: I. Hydrodynamics (Submitted) Johnson, R. E., and R. P. Smet, On the heat transfer from a permeable sphere in stokes flow, Chem. Eng. Sci., (1984) 22 (7) 947-958. McCabe L. W., and C. J. Smith, Unit operation of chemical engineering, 3 rd edition, 10

McGraw-Hill Kogakusha, Ltd, 1980, Japan. Ranz, W. E., and W. R. Marshall, Evaporation from drops, Chem. Eng. Prog., (1952) 48, 141-146. Romkes, S. J. P., F. M. Dautzenberg, and C. M. van den Bleek, and H. P. A. Calis, CFD modeling and experimental validation of particle-to-fluid mass and heat transfer in a packed bed at very low channel to particle diameter ratio, Chem. Eng. J., (2003) 96, 3-13. Vasudeviah, M., and K. Balamurugan, Heat transfer from a porous sphere in a slow viscous flow, Int. J. Non-linear Mechanics, (1998) 33(1), 111-124. Whitaker S., Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed beds and tube bundles, AIChE J. (1972) 18 (2), 361-370. 11

Figure captions Fig. 1. Comparison of simulated results with Whitaker (1972) correlation and experimental results for heat transfer from impermeable sphere Fig. 2. Forced-convection heat transfer from a porous permeable sphere Table caption Table 1: Nusselt number values for flow through and around sphere at different Reynolds number 12

100 10 Whitaker correlation (Nu-2)/Pr 0.4 =0.4Re 0.5 +0.06Re 2/3 Experimental Present simulation (Nu-2)/Pr 0.4 1 0.1 0.01 0.01 0.1 1 10 100 1000 10000 Re Fig. 1. Comparison of simulated results with Whitaker (1972) correlation and experimental results for heat transfer from a impermeale sphere 13

1000 100 k/d 2 = 1e-2 k/d 2 = 1e-3 k/d 2 = 1e-4 k/d 2 = 1.82e-5 solid sphere Nu 10 1 0.001 0.01 0.1 1 10 100 1000 10000 Re Pr 2/3 Fig. 2. Forced-convection heat transfer from a porous permeable sphere 14

Table 1: Nusselt number values for flow through and around sphere at different Reynolds number Peclet Number (Pe=Re. Pr) Reynolds number (Re) Prandtl number (Pr) Nu for permeable sphere (K/d 2 = 1e-2) Nu for impermeable sphere (K/d 2 = 0) Variation between Nu for permeable and impermeable sphere 0.0149 0.02 0.7442 2.5371 2.2062 14.99 % 0.1398 0.02 6.9909 2.2428 2.2408 0.09 % 1488 2000 0.7442 265.7559 26.4973 902.95 % 13980 2000 6.9909 2410.7440 69.6281 3361.32 % 15