CHAPTER FIVE PARTICLE-TO-FLUID MASS & HEAT TRANSFER Convective transport at low & high pressure. Free convection effects in Supercritical Fluids « Não há normas. Todos os homens são excepção a uma regra que não existe » Fernando Pessoa (1888 – 1935) Image: Henri Cartier-Bresson, « Praying » Chapter Five One of the most important parameters needed in the design of packed bed systems is the particle-to-fluid transport coefficients. In the past six decades, a substantial amount of work has been devoted to the study of these parameters. Particle-to-fluid mass transfer studies were first carried out by Gamson et al. (1943), and Hurt (1943). They obtained mass transfer coefficients from measurements of the rates of evaporation of water from wet porous particles. Hurt (1943) also reported mass transfer coefficients derived from the measurement of rates of naphthalene sublimation. Since their pioneering work, a large number of experimental studies have been carried out on mass transfer coefficients in packed bed systems. Theoretical work has also been in progress. Pfeffer (1964) and Pfeffer and Happel (1964) applied a free surface cell model to the creeping flow region. LeClair and Hamielec (1968, 1970) proposed a zero vorticity cell model, and El-Kaissy and Homsy (1973) applied the free surface cell model, zero vorticity cell model and distorted cell model to a multiparticle system at low Reynolds number. Nishimura and Ishii (1980) also applied the free surface cell model to the study of mass transfer at high Reynolds numbers. These models which are based on different assumptions, generally give different and inconsistent values of particle-to-fluid mass transfer coefficients. Therefore, theoretical prediction of mass transfer coefficients is far from satisfactory. For the design and analysis of packed bed catalytic reactors, it is necessary to know the temperatures of the fluid and the catalyst particles in which the chemical reactions are taking place. In general, fluid temperature is measured with little difficulty, but the measurement of the solid surface temperature is not easy. This is particularly true of packed bed reactors. The particle temperature or temperature drop at the particle surface then has to be estimated in terms of the heat transfer coefficient between the particle and the fluid. Because of the importance of the particle-to-fluid heat transfer coefficient in packed bed reactors, a considerable effort has been made to evaluate this parameter. Experimental determinations of heat transfer coefficients for a wide variety of systems have been made using various experimental techniques, under either steady-state or unsteady-state conditions. An extensive review on experimental/theoretical works on particle-to-fluid mass and heat transfer can be found in Wakao and Kaguei (1982). Although there is broad information available of published experimental data for particle-to- fluid forced convection heat and mass transfer at low pressure (see e.g. Wakao and Kaguei, 1982), scarce data are available for high pressure situations or supercritical packed bed reactors. Many authors have studied catalytic reactions using a supercritical fluid as a solvent (Poliakoff et al., 1996; Ramirez et al., 2004), but main interest has been centered in kinetics of chemical reactions under supercritical conditions. Other authors have studied mass transfer in packed beds under supercritical conditions applied to supercritical extraction (Debenedetti and Reid, 1986; Stüber et al., 1996). The recent study and development of new compact packed bed supercritical nuclear reactors opens a branch for the study of heat transfer phenomena under high pressure conditions (see e.g. Oka et al., 1994). In the following sections, analytical solutions of the steady-state temperature and composition profiles in a packed bed will be shown. In addition, a CFD simulation strategy for the estimation of forced convection mass and heat transfer coefficients at low pressure is discussed. Estimated transport parameters are compared to broadly accepted correlations. Mixed convection mass and heat transfer at high pressure is modeled and analyzed, and numerical results obtained are compared to previously published experimental data (Stüber et al., 1996), and a correlation for predicting heat transfer parameters in a supercritical packed bed reactor is presented (Guardo et al., 2006). The obtained numerical results validates the idea that the modified correlation presented by Guardo et al. (2006) can be used to describe heat transfer - 93 - Chapter Five phenomena in a supercritical packed bed reactor under mixed convection regime at high pressures. 5.1. Geometry, mesh design and CFD modeling To properly design a mesh capable of capturing the transport mechanisms present in the study in detail, a dimensionless analysis of Navier-Stokes equations under simulation conditions was developed. The dimensionless equations corresponding to continuity, mass, momentum, and energy balances are detailed in Chapter two. The orders of magnitude of the dimensionless groups were estimated by taking physical-chemical property values for air from experimental data and empirical correlations available in the literature (Reid et al., 1987; Yaws, 1999; Poling et al., 2000). Reynolds number was calculated using particle diameter as characteristic length. Reynolds analogy was used to estimate values of Prt from Ret (White, 1991). For first analysis purposes, Prt was assumed as a constant value within the bed, justified in the fact that majority of experimental results shows a range of variations between 0.75 < Prt < 2 for air and water (Kays, 1994). Boundary (operating) conditions for each analyzed situation are shown in Table 5.1. Details on the dimensionless numbers used can be found in Appendix C. Results of the order of magnitude of the dimensionless groups are shown in Table 5.2. Boundary condition Low pressure High pressure Mass transfer simulations Circulating fluid CO2 3 C7H8 concentration at inlet, mol/m 0 C7H8 concentration at particle 5.95 120 – 190 surface (equilibria), mol/m3 Pressure, Pa 101325 9 – 9.2 x 106 Mass flow at the inlet, kg/m2· s - 0.015 – 0.100 Velocity at the inlet, m/s 7.5 x 10-4 – 7.5 x 10-1 - Heat transfer simulations Circulating fluid Air CO2 Temperature at the inlet, K 298 330 Temperature at the particle surface, K 423 340 Pressure, Pa 101325 1 x 107 Mass flow at the inlet, kg/m2· s - 0.013 – 0.132 Velocity at the inlet, m/s 3 x 10-4 – 7.5 x 10-1 - Table 5.1. Boundary (operating) conditions for analyzed cases In the case of particle-to-fluid transport at low pressure, dimensionless analysis allows identifying the problem as forced convection in laminar or turbulent flow. For the momentum balance it becomes clear that viscous forces decrease their contribution as Re increases. Inertial gravity forces increase their contribution as Re decreases, and pressure drop together with turbulent forces become the most important terms in the momentum balance at high Re. In energy and species balances, the convective and the diffusive term become important for the balances. For both balances, steady state analysis can be used. - 94 - Chapter Five When analyzing a mass/heat transfer case at high pressure, the problem is identified as mixed (free + forced) convection in laminar or transitional flow. Body forces and pressure drop become the most important terms in the momentum balance. Turbulence forces contribution to all balance equations is negligible. In order to couple species/energy and momentum balance, unsteady state analysis is required. Low Pressure High Pressure Re 100 101 102 103 100 101 Sr 102 101 100 10-1 105 104 Ma 10-6 10-5 10-4 10-3 10-6 10-5 Eu 103 - 104 101 - 103 10-1 - 101 10-1 106 104 Fr 10-4 10-4 - 10-2 10-2 - 100 100 10-10 10-8 Pr 10-1 10-1 10-1 10-1 100 100 Sc 100 100 100 100 100 100 Ec 10-10 10-10 - 10-8 10-8 - 10-6 10-6 10-16 10-14 ReT 101 100 100 100 101 100 PrT 10-1 10-1 10-1 10-1 10-1 10-1 ScT 103 102 101 100 103 102 Table 5.2. Dimensionless groups’ magnitude orders for analyzed cases In order to define the computational domain, a 44 spheres stacking with a sphere-to-tube diameters ratio of 3.923 was chosen for the geometrical model. Modeled geometry was constructed following the bottom-up technique (generating surfaces and volumes from nodes and edges) in order to control mesh size around critical points (i.e. particle-to-particle and particle-to-wall contact points). In this study, to include real contact points, the spheres were modeled overlapping by 0.5 % of their diameters with the adjacent surfaces in the geometric model. For further details on the geometrical model please refer to Guardo et al., (2004) and Chapter Four of this thesis. 5.2. Model setup and analysis Navier-Stokes equations together with species/energy balance were solved using commercially available finite volume code software Fluent® 5.x/6.x. For the low pressure models the fluid was taken to be incompressible, Newtonian, and in a laminar or turbulent flow regime. CO2, air and toluene at standard conditions were chosen as the simulation fluids. Incompressible ideal gas law for density, power law for viscosity and appropriated mixing rules were applied to the model for making these variables temperature and composition dependent. For the high pressure simulations, the fluid was taken to be Newtonian, in laminar flow regime and with variable density. CO2 and toluene were chosen in this case as the simulation fluids, and its - 95 - Chapter Five properties at high pressure were incorporated to the solver code through User Defined Functions (UDF) and User Defined Equations (UDE). Peng-Robinson equation of state was used to calculate fluid’s density and heat capacity, Lucas method was used to calculate the viscosity, and thermal excess method was used to calculate the thermal conductivity (Reid et al., 1987).