Local equilibrium isothermal flows

Example 5.1 Isothermal Compressible Flow For local equilibrium, isothermal flows in the absence of external forces, the conservation of mass and momentum are, respectively. 

The equations for a local equilibrium cell model of pressure swing adsorption processes with linear isotherms have been derived. These equations may be used to describe any PSA cycle composed of pressurization and blowdown steps and steps with flow at constant pressure. The use of the equations was illustrated by obtaining solutions for a single-column recovery process and a two-column recovery and purification process. The single-column process was superior in enrichment and recovery of the light component at large product cuts. The two-column process was superior at small cuts ... 

Furthermore, for non-isothermal situations we need to be able to calculate the thermodynamics of fluid dynamics. However, thermodynamics deals with relatively permanent states, called equilibrium states, within uniform fields of matter [7] [145] [42] [54]. Any changes are assumed to be extremely slow. On the other hand, the fluid motions of interest in fluid mechanics are not necessary slow. Nevertheless, it has been assumed that the classical thermodynamics can be directly applied to any flow system provided that an instantaneous local thermodynamic state is considered and that the rates of change are not too large [168]. A more common statement is that the thermodynamics require that the fluids are close to local equilibrium, but may not be in global equilibrium. However, all systems are supposed to be relaxing towards a state of global thermodynamic equilibrium. 

D13. We wish to use the local equilibrium model to estimate reasonable flow rates for the separation of dextran and fructose using an SMB. The isotherms are linear and both q and c are in g L. The linear equilibrium constants are dextran, 0.23 and fructose, 0.69. The interparticle void fraction = 0.4 and the intraparticle void fraction = 0.0. The columns are 40.0 cm in diameter. We want a feed flow rate of 1.0 L/min. The feed has 50.0 of each component. The desorbent is water and the adsorbent is silica gel. The columns are each 60.0 cm long. The limped parameter mass transfer coefficients using fluid concentration differences as the driving force are 2.84 l/min for both dextran and fructose. Operation is isothermal. Use multiplier values (see notation in Figure 18-14i of M] = 0.97, M2 = 0.99, M3 = 1.01, and M4 = 1.03. Determine the flow rates of desorbent, dextran product, fructose product, and recycle rate and find the ratio D/F. 

We estimated the N2 desorption characteristics from these zeolites under two idealized but common concepts of operation of PSA processes. They are (a) isothermal evacuation of an adsorbent column which is initially equilibrated with a binary gas mixture of N2 (yi=0,79) and O2 (y2=0.21), and (b) isothermal and isobaric desorption of pure N2 from an adsorbent column by flowing a stream of pure O2 (called purging) through the column. The adsorbers are initially at a pressure of one atmosphere and at a temperature of 30 C in both cases. Analytical model solutions are available for the above described desorption processes when they are carried out under local equilibrium conditions and when the adsorbates follow Langmuir isotherms [1,6]. 

Let us describe the mathematical model of a three-phase non-isothermal compressible flow in porous media taking into account capillary effects. It is assumed that the movement of phases obeys the generalized Darcy s law. We assume that the phases are in the local thermal equilibrium, so that in any elementary volume the fluids saturating the porous medium and the rock have the same temperature. Furthermore, oil is assumed to be homogeneous non-evaporable fluid and oil reservoir consists of one type of rock. In this case, three-phase non-isothermal flow in a bounded domain 2 c M (d = 1, 2, 3) taking into account capillary forces and the phase transitions between the phases of water and heat transfer is described by the following system of equations ... 

Adapted to trickle-bed geometry assuming Newtonian unidirectional ID (streamwise) and isothermal (local thermal equilibrium) immiscible (no mass transfer exchange between bulk phases) gas-liquid flows in a nondeformable and homogeneous... 

Some of these studies focused on the analysis of equilibrium-limited reactions, namely those in which the conditions of the respective conversion could be enhanced relatively to the value obtained in a conventional reactor, the so-called thermodynamic equilibrium conversion.i i The developed models considered generic equilibrium-limited reactions carried on in membrane reactors with perfectly mixed or plug-flow pattems. In all these studies, the main assumptions considered consisted in isothermal and steady-state operation, Fickian transport across a non-porous membrane with a homogeneously distributed nanosized catalyst with constant diffusion coefficients, Henry s law for describing the equilibrium condition at the interfaces membrane/gas, and equality of local concentrations at the interface polymer phase/catalyst surface. 

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