Countercurrent and Cocurrent Flow Models

In these models parallel flow to the membrane is assumed on both sides of it. If the film resistances are neglected the composition is considered constant on each side of the membrane across a plane peipendicular to it, and the streams on both sides are in plug flow. 

FIGURE 18.6 (a) Countercurrent flow and (b) cocurrent flow membrane separators. 

A material balance on component i around the differential volume on the residue 

The flow through the membrane is determined by the flux equation  

The above differential and algebraic equations may be solved numerically to calculate the separation in these flow pattern models. 

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The models are represented in Figure 18.6. The following derivations apply to both the countercurrent and cocurrent flow models. A total material balance around the differential volume surrounding membrane area dA on each side of it gives... 

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. 

The equations derived in the previous section represent penneation rates and flux, and Examples 18.1 and 18.2 are apphcations to a speciflc model. The model assumes the fluid on each side of the membrane to have a constant composition parallel to the membrane. The compositions normal to the membrane are also assumed constant, with the possible exception of composition gradients in the films adjacent to the membrane. The bulk phase compositions on both sides of the membrane had to be given since no material balances were considered. The flow pattern implied in this model is that of perfect mixing (if the film resistances next to the membrane are neglected). Other flow patterns include cross flow, countercurrent flow, and cocurrent flow. 

Countercurrent and Cocurrent Plug Flows. The model equations for these flow patterns cannot be solved analytically. Oishi and coworkers first derived the general model eqnstions for a binary-component system with porous media.19 Walawenderand Stem,16 Blaisdell and Kammermeyer,1 and Pan and Habgood17 later reported solutions for similar membrane separators. The cocurrent-counteicurrent combiner inu flow pattern also lies been studied by Pen ned Habgood.17... 

UDah A, Wang W, Li J Evaluation of drag models for cocurrent and countercurrent gas-soUd flows, Chem Eng Sci 92 89—104, 2013a. 

Theoretical profiles calculated by the cell model described previously are also shown in Fig. 4 through 6. The solid lines are obtained with a plug flow model, cind the dotted lines are obtained with a cell model. The simulated profiles fit the average values of the temperature and conversions calculated for each monolith and chaimber. A seven x seven grid used to simulate the two dimensional profiles in the cocurrent cind countercurrent runs produced the best agreement between experiment and theory. 

A schematic diagram of a 1-2 S T exchanger is shown in Figure 15.12. It can be seen that the basic flow patterns of the shell and tube fluids are not countercurrent. Indeed, the flow patterns of the two fluids are very complicated. However, for 1-2 S T exchangers a reasonable assunption is that the shell-side fluid flows cocurrently with the tube-side fluid in one direction and counter-currently in the other direction. Using this idealized model of the fluid flow, an analytical expression may be conputed for the F factor... 

Fixed-bed reactors are used for testing commercial catalysts of larger particle sizes and to collect data for scale-up (validation of mathematical models, studying the influence of transport processes on overall reactor performance, etc.). Catalyst particles with a size ranging from 1 to 10 mm are tested using reactors of 20 to 100 mm ID. The reactor diameter can be decreased if the catalyst is diluted by fine inert particles the ratio of the reactor diameter to the size of catalyst particles then can be decreased to 3 1 (instead of the 10 to 20 recommended for fixed-bed catalytic reactors). This leads to a lower consumption of reactants. Very important for proper operation of fixed-bed reactors, both in cocurrent and countercurrent mode, is a uniform distribution of both phases over the entire cross-section of the reactor. If this is not the case, reactor performance will be significantly falsified by flow maldistribution. 

If flow is cocurrent the lower sign is used if countercurrent the upper sign is used. Since the mass flowrate of the cooling fluid is based upon the cross-sectional area of the reactor tube the ratio G Ip Gq SpC(= H is a measure of the capacities of the two streams to exchange heat. In terms of the limitations imposed by the onedimensional model, the system is fully described by equations 3.9S and 3.96 together with the mass balance equation ... 

Cova (3 ) measured the solid concentration profiles of a Raney nickel catalyst with an average diameter of 15.7 ym in a h.6 cm id reactor, using water and acetone as the liquids. He developed a sedimentation diffusion model, assuming solid and liquid dispersion coefficients were equal, and slurry settling velocities were independent of solid concentration. The model was then applied to data for Raney nickel in 6.35 and kk.J cm id bubble columns, in both cocurrent and countercurrent flow. 

Govindarao10 also postulated generalized nonisothermal (constant reactor wall temperature) models for batch as well as cocurrent- and countercurrent-flow three-phase gas-liquid-solid systems carrying out a first-order reaction. 

Same as above, with the exclusion of the surface flow contribution, owing to the high operating temperatures (about 500°C). An analytically solved model is also proposed under the hypothesis of very fast kinetics, which shrink the reaction zone to a surface. Cocurrent and countercurrent operation are considered... 

Incremental models track the local conditions of the gas and particles through the dryer, mainly in one dimension. They are especially suit le for cocurrent and countercurrent dryers, e.g., flash (pneumatic conveying) and rotary dryers. The air conditions are usually treated as uniform across the cross-section and dependent only on axial position. This method can also be used to determine local conditions (e.g., temperature) where a simpler model has been used to find the overall drying rate. A two- or three-dimensional grid can also be used, e.g., modeling vertical and horizontal variations in a band dryer or plug-flow fluidized bed. 

Cascading Rotary Dryers In design mode, the required gas flow rate can be obtained from a heat and mass balance. Bed cross-sectional area is found from the scoping design calculation (a typical gas velocity is 3 m/s for cocurrent and 2 m/s for countercurrent units). Length is normally between 5 and 10 times drum diameter (an L/D value of 8 can be used for initial estimation) or can be calculated by using an incremental model (see worked example). 

Figure 13.20 shows how a cocurrent absorber may be analyzed. Note the equilibrium limitation, which limits the degree of recovery of solute. Basic models for estimating hydraulics and mass transfer are based on countercurrent flow, which is normally more efficient, and must be adjusted if the gas flow direction is reversed. 

Stelmaszek and coworkers have developed models for liquid membranes. Stelmaszek (100) modeled ELM droplets. Gladek et al. (101) modeled ELMs in cocurrent and countercurrent flow processes. Gladek et al. ( ) used an advancing front approach to model ELM extraction. 

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