Cross-Flow Model

The permeate product of the two-stage cross-flow model is obtained by combining the permeate streams from both stages. The cross-flow residue is the same as stage 2 residue  

If the film resistances are neglected on both sides of the membrane, the fluid composition on each side may be assumed constant across a plane perpendicular to the membrane. On the residue side the composition changes continuously in the direction of flow (parallel to the membrane) as in plug flow. On the permeate side the composition also changes continuously in the direction parallel to the membrane due to the changing composition on the residue side. 

Molar flow rates f are the variable rates inside the separator, with subscripts P andf designating, respectively, the permeate side and residue side of the membrane. A material balance around the differential volume is written for component i. The component differential permeate rate equals the differential change in the component residue rate  

Mole fractions y represent the variable compositions inside the separator. The above equation is combined with Equation 18.30 to eliminate dfp, and then rearranged  

The above differential equations along with the summation Equation 18.25 may be solved numerically. 

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Residence time distribution measurements, together with a theoretical model, provide a method to calculate the rate of mass transfer between the liquid flowing through the column, the dynamic holdup, and the stagnant pockets of liquid in between the particles. We have chosen the cross flow model (10). It has to be noted that the model starts from the assumption that the flow pattern has a steady-state character, which is in conflict with reality. Nevertheless, average values of the number of mass transfer units can be calculated as well as the part of the liquid being in the stagnant situation. 

Three-parameter PDE model (Van Swaaij et aL106) This model is largely used to correlate the RTD curves from a trickle-bed reactor. The model is based on the same concept as the crossflow or modified mixing-cell model, except that axial dispersion in the mobile phase is also considered. The model, therefore, contains three arbitrary parameters, two of which are the same as those used in the cross-flow model and the third one is the axial dispersion coefficient (or the Peclet number in dimensionless form) in the mobile phase (see Fig. 3-11). 

The solution of the nonlinear optimization problem (PIO) gives us a lower bound on the objective function for the flowsheet. However, the cross-flow model may not be sufficient for the network, and we need to check for reactor extensions that improve our objective function beyond those available from the cross-flow reactor. We have already considered nonisothermal systems in the previous section. However, for simultaneous reactor energy synthesis, the dimensionality of the problem increases with each iteration of the algorithm in Fig. 8 because the heat effects in the reactor affect the heat integration of the process streams. Here, we check for CSTR extensions from the convex hull of the cross-flow reactor model, in much the same spirit as the illustration in Fig. 5, except that all the flowsheet constraints are included in each iteration. A CSTR extension to the convex hull of the cross-flow reactor constitutes the addition of the following terms to (PIO) in order to maximize (2) instead of [Pg.279]

The feed is sent to the residue side of the first stage with a membrane area of 1.5 X 102 cm . The value of 0 is determined to satisfy the summation equations, and the product rates and compositions are calculated. The residue from the first stage is sent to the residue side of the second stage which also has a membrane area of 1.5 x I02 cm . The residue from the second stage is the residue product of the cross-flow model, and the combined stream of the permeates from the two stages constitutes the permeate product of the model. The results are tabulated as follows ... 

The procedures described above can be applied to multicomponent systems and are not limited to binary systems. Although the multistage approach is only an approximation to the rigorous cross-flow model, it is a useful tool for preliminary studies. It could show the enrichment and depletion trends of the components in the mixture and their dependence on the number of stages. 

The countercurrent and cocurrent flow models may also be approximated by a series of perfect mixing blocks as described for the cross-flow model. The counter-current flow model would require an additional iterative loop to converge the recycle created by the counterflowing permeate stream. 

FIGURE 20-6-5 Schematic diagram used for cross flow model development. 

Equations 12.S.b-l, 2 reduce to a large variety of special cases. As written, they are the standard equations for interfacial mass transfer used in Chapters 6 and 14 (also see Pavlica and Olson [60]. They are also the basis of the cross-flow models for fluidized beds (see Chapter 13) or other multiphase reactors, and have been used for heat transfer studies. 

Daud, W.R.W., A cross-flow model for continuous plug flow fluidized-bed cross-flow dryers. Drying Technol, 25, 1229-1235, 2007. 

In the simulation of the HMD process, the gas-permeation membrane used is assumed to be an asymmetric hollow-fiber membrane. For this type of membrane, gas permeation does not depend on the flow pattern on the permeate side as the porous supporting layer prevents mixing of the permeate fluxes (Pan, 1986). A schematic of the flow pattern in an asymmetric hollow-fiber membrane is shown in Figure 10.2. Hence, a simple cross-flow model is sufficient to describe the membrane behavior. 

Here, F, Zf and h are, respectively, the molar flow rate, mole fraction of component of i and total enthalpy, all in cell k their subscripts, ret and perm, refer to retentate and permeate streams. Equations (10.4) and (10.5) are mass balances and mass-transfer equations for each of the components present in the membrane feed. The cross-flow model [Equations (10.3)-(10.7)] was implemented in ACM v8.4 and validated against the experimental data in Pan (1986) and the predicted values of Davis (2002). The Joule-Thompson effect was validated by simulating adiabatic throttling of permeate gas through a valve in Aspen Hysys. Both these validations are described in detail in Appendix lOA. 

Dynamic analysis of a trickle bed reactor is carried out with a soluble tracer. The impulse response of the tracer is given at the inlet of the column to the gas phase and the tracer concentration distributions are obtained at the effluent both from the gas phase and the liquid phase simultaneously. The overall rate process consists the rates of mass transfer between the phases, the rate of diffusion through the catalyst pores and the rate of adsorption on the solid surface. The theoretical expressions of the zero reduced and first absolute moments which are obtained for plug flow model are compared with the expressions obtained for two different liquid phase hydrodynamic models such as cross flow model and axially dispersed plug flow model. The effect of liquid phase hydrodynamic model parameters on the estimation of intraparticle and interphase transport rates by moment analysis technique are discussed. 

Dynamic analysis of TBR by sitimules response technique has been succesfully applied to determine the extent of liquid axial mixing. There are number of learning and predictive models proposed in literature 2. Among them the ones having less number of parameters such as cross-flow model and axially dispersed plug flow ADPF model are the most adequate ones. A more realistic model profound for a TBR can be the one which includes the simultaneous effect of interphase and intraparticle transport rates, and the adequate hydrodynamic model, to minimize the relative importance of liquid mixing on these rates. 

The aim of this research is to show the influence of cross-flow model and ADPF model on the estimation of mass transfer coefficient, liquid hold up and adsorption factor by dynamic analysis of a TBR. 

Solutions of and Cq in Laplace domain are different for Case I and Case II. However the limits of C and Cq as s- O and the limits of dCj /ds and dCQ/ds as s- 0 are same as plug flow model. Therefore consideration of cross-flow model to represent the mixing in the liquid phase does not bring any improvement on which are derived for plug flow model. 

The moment analysis has shown that the cross flow model parameters and k do not affect and yj However the axially... 

A detailed flow diagram for the cross-flow model derived by Weller and Steiner (W3, W4) is shown in Fig. 13.6-1. In this case the longitudinal velocity of the high-pressure or reject stream is large enough so that this gas stream is in plug flow and flows parallel to the membrane. On the low-pressure side the permeate stream is almost pulled into vacuum, so that the flow is essentially perpendicular to the membrane. 

In the design for the complete-mixing model there are seven variables and two of the most common cases were discussed in Section 13.4B. Similarly, for the cross-flow model these same common cases occur. 

The cross-flow model for reverse osmosis is similar to that for gas separation by membranes discussed in Section 13.6. Because of the small solute concentration, the permeate side acts as if completely mixed. Hence, even if the module is designed for countercurrent or cocurrent flow, the cross-flow model is valid. This is discussed in detail elsewhere (HI). 

Design Using Cross-Flow Model for Membrane. Use the same conditions for the... 

The cross flow model may also be approximated by a series of stages each of which is represented by a perfect mixing model as depicted in Figure 18.5. The accuracy of the model increases with the number of stages used. By a total material balance on each stage. 

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