Perfect mixing, model for

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...

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. 

The classical CRE model for a perfectly macromixed reactor is the continuous stirred tank reactor (CSTR). Thus, to fix our ideas, let us consider a stirred tank with two inlet streams and one outlet stream. The CFD model for this system would compute the flow field inside of the stirred tank given the inlet flow velocities and concentrations, the geometry of the reactor (including baffles and impellers), and the angular velocity of the stirrer. For liquid-phase flow with uniform density, the CFD model for the flow field can be developed independently from the mixing model. For simplicity, we will consider this case. Nevertheless, the SGS models are easily extendable to flows with variable density. 

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]. 

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 ... 

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 ... 

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. 

We shall examine some simple models of population growth to see what consequences imperfect mixing might have. Consider first a set of species with populations P, P2, - - competing under perfectly mixed conditions for an unlimited food supply. Assume that each species grows at a rate g,(p,), which depends upon its population. It has a death rate growth rate g, might be also be proportional to Pi, so that... 

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

In this chapter the most important operation modes of reactors are considered. Models are developed by combining simple reaction kinetics for single-phase reactions with mass balances for five ideal model reactors the ideal batch reactor the semi-batch reactor the plug flow reactor the perfectly mixed continuous reactor and the cascade of perfectly mixed reactors. For isothermal conditions, conversions can be calculated on the basis of chemical kinetics only. 

Sastri et al. (1983) modeled a three-phase noncatalytic but reactive system to produce industrial concentrations of zinc hydrosulfite (ZnS204) in an SBR. Three different approaches were proposed plug-flow, axial diffusion, and perfect mixing mathematical models. The authors compared the numerical solutions for the three models and noticed that the experimental data are well predicted by the axially dispersed plug-flow (diffusion) model, moderately predicted with the plug-flow model, and poorly predicted with the perfect mixing model. 

In the second model (Fig. 2.16) the continuous well-stirred model, feed and product takeoff are continuous, and the reactor contents are assumed to he perfectly mixed. This leads to uniform composition and temperature throughout. Because of the perfect mixing, a fluid element can leave at the instant it enters the reactor or stay for an extended period. The residence time of individual fluid elements in the reactor varies. 

The name continuous flow-stirred tank reactor is nicely descriptive of a type of reactor that frequently for both production and fundamental kinetic studies. Unfortunately, this name, abbreviated as CSTR, misses the essence of the idealization completely. The ideality arises from the assumption in the analysis that the reactor is perfectly mixed, and that it is homogeneous. A better name for this model might be continuous perfectly mixed reactor (CPMR). 

Most room models contain only one zone air node, thus assuming perfect mixing of the zone air and a homogenous temperature distribution in the space. Spatial temperature variations, such as vertical temperature gradients, are not considered. For specific applications such as displacement ventilation or atria, models with several zone air nodes in the vertical direction have been developed. ... 

As outlined earlier, in multizone models, perfect mixing is assumed in the individual zone. The spatial distribution of velocities, contaminant concentrations, and air temperatures in a zone can be determined only by using CFD. On the other hand, wind effects are easily accounted for in multizone models, and unsteady-state simulation is normally performed. On the combined use of the two methods, see Schaelin et al.--... 

For the models described, the usual assumption for air nodes in regard to the room air distribution is still valid. This means that each air node represents a volume of perfectly mixed air. Thus, the same limitations as for thermal and airflow models apply Local air temperatures and air velocities as well as local contaminant concentrations can he neither considered nor determined. This also means that thermal comfort evaluations in terms of draft risk cannot be performed. 

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

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