Perfect Mixing Model

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. 

Fp enters the separator on the residue side of the membrane. It separates into a 

By overall material balance on the separator at steady state, 

The flow rate of component i in the permeate equals the product of the flux of that 

The partial pressures at the film interface are replaced by the bulk partial pressures because the film resistances are neglected. Also, subscripts 1 and 2 are replaced by R and P to indicate the residue side and the permeate side. The partial pressures are now replaced by the total pressure and mole fractions of the products  

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The liquid and solid phases are often supposed to be well-mixed, and a perfect mixing model can be applied to model the species-concentration profile. However, many expressions for the description of axial dispersion in the liquid phase in a BSCR are given in the literature [37,38]. 

Figure 1. Schematic presentation of the two reactor models a, perfectly mixed model, and b, imperfectly mixed model.

For the computation of E(t) and T(p), in the case of a perfect mixing model, we use the representation and notation given in Fig. 3.24. Including the mass balance of the species in the signal, we derive the following differential equation ... 

Figure 10.3 Schematic diagram of a single cell or perfect mixing model for a packed-bed membrane shcll-and-tube reactor...

Fig. 9.11. A perfect mixing model for gas separation, x and y are mole fractions, Q is the molctr flux...

A membrane separator using 1 mil thickness low-density polyethylene membrane is to be designed for concentrating hydrogen in a hydrogen-methane-carbon monoxide gas mixture. The separator performance may be approximated by a perfect mixing model. The feed flow rate is 1.0 x 10" cnf (STP)/s, and its composition and component permeabilities in polyethylene membrane are given below ... 

The driving force in gas permeation may be expressed in terms of the difference between a component partial pressure on the residue side and the permeate side of the membrane. The feed is introduced to the separator at a high pressure, while the permeate side is controlled at a low pressure. Examples 18.3 and 18.4 use the perfect mixing model for the performance evaluation and the design of two gas permeation processes. 

It is required to design a reverse osmosis unit to process 2500 mVh of seawater at 25°C containing 3.5 wt% dissolved salts, and produce purified water with 0.05 wt% dissolved salts. The pressure will be maintained at 135 atm on the residue side and 3.5 atm on the permeate side, and the temperature on both sides at 25°C. The dissolved salts may be assumed to be NaCl. With the proposed membrane, the salt permeance is 8.0 x 10 m/h and the water permeance is 0.085 kg/rn-.h.atrn. The density of the feed seawater is 1020 kg/m ( of the permeate, 997.5 kg/nv and of the residue (with an estimated salt content of 5 wt%), 1035 kg/rnc Assuming a perfect mixing model and neglecting the mass transfer resistances, determine the required membrane area and calculate the product flow rates and compositions. 

The pressure is 3500 kPa on the residue side and 140 kPa on the permeate side. The composition on the permeate side is specified at 95 mol%. Assuming a perfect-mixing model, calculate the required membrane area. Use a fraction of feed permeated, 0 = 0.15. Calculate the product rates and compositions, and check if 0 needs to be modified. 

When dispersion is complete and uniform, the contents of the vessel are perfectly mixed with respect to both phases. In that case, the concentration of the solute in each of the two phases in the vessel is uniform and equal to the concentrations in the two-phase emulsion leaving the mixing tank. This is called the ideal CFSTR (continuous-flow-stirred-tank-reactor) model, sometimes called the perfectly mixed model. Next we develop an equation to estimate the Murphree-stage efficiency for liquid-liquid extraction in a perfectly mixed vessel. 

The perfect mixing model frequently provides a poor fit to the distribution of petroleum properties in vertically stacked reservoirs and fails to account for the compositional grading within the individual reservoirs of the stack. Conversely, the second end-member model often fits the observed data surprisingly well, considering that it implies no mixing at all. This supports the contention that the petroleum in many, if not all, fields is poorly mixed. We show two examples here. 

The PFR and CSTR models encompass the extremes of the residence-time distributions shown in Figure 4.3 however the batch reactor and the laminar-flow reactor, both of which we have already mentioned in this chapter, are also types exhibiting a well-defined mixing behavior. The batch reactor is straightforward, since it is simply represented by the perfect mixing model with no flow into or out of the system, and has been treated extensively in Chapter 1. 

Modeling of mass transfer has been carried out in the field of chemical engineering and environmental engineering (e.g., Takamatsu et al. 1977). Models commonly used are (1) batch model, (2) perfectly mixing model (Fig. 3.7), (3) piston flow model (Fig. 3.8), and (4) tank model (multi-step model). 

Concentration in aqueous solution is homogeneous in perfectly mixing model. However, in fact, concentration varies in a system. According to piston flow model the variation in concentration is expressed as... 

The change in concentration with time for perfectly mixing model is represented by... 

Fig. 6.16 Variation in concentration of lake water due to dilution (perfectly mixing model) (Morita Y in Hanya T (ed) 1979)...

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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