Mathematical models, flow, emulsion

A comprehensive review of the important factors that affect the flow of emulsions in porous media is presented with particular emphasis on petroleum emulsions. The nature, characteristics, and properties of porous media are discussed. Darcy s law for the flow of a single fluid through a homogeneous porous medium is introduced and then extended for multiphase flow. The concepts of relative permeability and wettability and their influence on fluid flow are discussed. The flow of oil-in-water (OfW) and water-in-oil (W/O) emulsions in porous media and the mechanisms involved are presented. The effects of emulsion characteristics, porous medium characteristics, and the flow velocity are examined. Finally, the mathematical models of emulsion flow in porous media are also reviewed. 

When using stable, dilute Newtonian emulsions through porous media, the flowing permeability, fcf, must be used in Darcy s law to describe its behavior instead of the initial or conventional permeability. When plugging due to the flow of Newtonian macroemulsions occurs, only the permeability of the porous medium should be adjusted. Emulsion rheology with respect to Newtonian and non-Newtonian behavior will be reviewed under the section Mathematical Models of Emulsion Flow in Porous Media . 

Mathematical Models of Emulsion Flow in Porous Media... 

The analytical predictor, as well as the other dead-time compensation techniques, requires a mathematical model of the process for implementation. The block diagram of the analytical predictor control strategy, applied to the problem of conversion control in an emulsion polymerization, is illustrated in Figure 2(a). In this application, the current measured values of monomer conversion and initiator feed rate are input into the mathematical model which then calculates the value of conversion T units of time in the future assuming no changes in initiator flow or reactor conditions occur during this time. 

A recent paper by Kiparissides, et al. (8) details a mathematical model for the continuous polymerization of vinyl acetate in a single CSTR. Operating conditions were shown to exist in which either steady-state operation or sustained conversion oscillations would occur for vinyl acetate. Experimental results for both cases were successfully simulated by their model. In addition, regulatory conversion control policies were considered in which both initiator feed rate and emulsifier feed rate were used as manipulated variables (Kiparissides (9)). The problem of conversion control in the operating region in which sustained conversion oscillations occur is one of significant commercial importance. Most commonly, however, a uniform concentration of emulsifier is required in the emulsion recipe and, hence, emulsifier flow rate cannot be used as a manipulated variable. 

The flow of emulsions in porous media is very complex, and to model it mathematically has been a challenge. It requires an understanding of the emulsion formation, its behavior, and its rheology inside the reservoir. Factors that affect the flow of emulsions through porous media were discussed earlier in this chapter, and the available mathematical models will be reviewed here. 

In a simulation study, Leffew and Deshpande [62] have evaluated the use of a dead-time compensation algorithm in the control of a train of CSTRs for flie emulsion polymerization of vinyl acetate. In this study, monomer conv ion was controlled by manipulating the initiator flow rate. Experiments indicate that there is a period of no response (dead-time) between the time of increase in the flow of initiator and the response of monomer conversion. Dead-time compensation attempts to correct for this dead-time by using a mathematical model of the polymerization system. Reported results indicate that if the reactor is operated at low surfoctant concentration (where oscillations are observed), the control algorithm is incapable of controlling monomer conversion by the manipulation of either initiator flow rate or reactor temperature. The inability of the controller to eliminate oscillations is most probably due to the choice of manipulated variable (initiator flow rate) rather than to the perfotmance of the control algorithm (deadtime conq)ensation). 

The counter-current backmixing model of Fryer and Potter has been modified by assuming mixed flow in the emulsion phase. The terminal conversions obtained with the present model are compared with those of the original model and found to agree well except at very low values of bubble diameter. The assumption of complete mixing in the emulsion phase converts the original two-point boundary value problem into a simpler initial value problem, thereby considerably reducing the mathematical complexity. 

The commercially used emulsion polymerization reactors (stirred-tank and continuous-loop) are designed to achieve perfect mixing. As will be discussed in Section 6.4.5, perfect mixing is not always achieved. Nevertheless, this flow model allows a good prediction of the emulsion polymerization reactor performance with a moderate mathematical effort, and it will be used here. Macroscopic balances (i.e., considering the reactor as a whole) are used. For the sake of generality, inlet and outlet streams are included in the balances. Both terms should be removed for batch operation, the outlet term should be eliminated in semibatch and both maintained in continuous processes. 

The two-phase theory of fluidization was first proposed by Toomey and Johnstone (1952). The model assumed that the aggregative fluidization eonsists of two phases, i.e., the particulate (or emulsion) phase and the bubble phase. The flow rate through the emulsion phase is equal to the flow rate for minimum fluidization, and the voidage is essentially constant at Sn,f. Any flow in excess of that required for minimum fluidization appears as bubbles in the separate bubble phase. Mathematically, the two-phase theory can be expressed as... 

The Ferry model with a flow index of n = 1 does not provide a suitable mathematical description of the experimental data. The region of adequate description for the Haven model is presented in Figxire 7. It should be noted that as the content of dispersed phase in the emulsion increases, the region of adequate description for the emulsion viscosity expands. 

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