Mathematical modeling of flow processes

Mathematical models of flow processes are non-linear, coupled partial differential equations. Analytical solutions are possible only for some simple cases. For most flow processes which are of interest to a reactor engineer, the governing equations need to be solved numerically. A brief overview of basic steps involved in the numerical solution of model equations is given in Section 1.2. In this chapter, details of the numerical solution of model equations are discussed. 

I have made an attempt to provide sufficient information to understand and to define the specific role of computational flow modeling in reactor engineering applications. Discussions on the main features of reactor engineering, computational flow modeling and their interrelationship will help to select appropriate models, and to apply these computational models to link reactor hardware to reactor performance. Mathematical modeling of flow processes (including turbulent flows, multiphase flows and reactive flows) and corresponding numerical methods to solve these model... 

Once the flowsheet structure has been defined, a simulation of the process can be carried out. A simulation is a mathematical model of the process which attempts to predict how the process would behave if it was constructed (see Fig. 1.1b). Having created a model of the process, we assume the flow rates, compositions, temperatures, and pressures of the feeds. The simulation model then predicts the flow rates, compositions, temperatures, and pressures of the products. It also allows the individual items of equipment in the process to be sized and predicts how much raw material is being used, how much energy is being consumed, etc. The performance of the design can then be evaluated. 

Based upon the work of Evans et al. (1983), Thorpe (1987) describes a semi-commercial scale continuous-flow fluidized bed disinfestor capable of handling up to 150 th of wheat. He gives a detailed description of the thermodynamic performance of the plant together with a mathematical model of the process which is validated by experimental results obtained from the plant. The plant consists of a single fluidized... 

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. 

We denote by 07 = Hi/HijS the dimensionless variables corresponding to the energy flow rates Hiy i = 1,..., N (the subscript s denotes steady-state values). Appending a generic representation of the overall and component material-balance equations, with xtfc IRm being the material-balance variables, the overall mathematical model of the process in Figure 6.1 becomes... 

Descriptive model and its division into parts. The first steps in the model construction are related to Fig. 3.7. The pump PA assures simultaneously the suspension transport and the necessary transmembrane pressure. The excessive accumulation of the solid in the retentate is controlled by its permanent removal as a concentrated suspension from the reservoir RZ. The clear liquid (permeate) flow rate and the solid concentration in the exit suspension are permanently measured and these values are transferred to the control and command computer CE. The instantaneous values of the operation pressure and input rate of fresh suspension are established by the computer (this works with software based on the mathematical model of the process) and corrected with the command execution system CSE. 

Once the descriptive model has been realized, we need to make the mathematical model of the process, which can be used to identify the mean pore radius of the membrane pores and the associated tortuosity. Before starting with the establishment of the model, we consider that the elementary processes allowdng the gas flow through the membrane are a combination of Knudsen diffusion with convective flow. If we only take into account the linear part of the curve of the pressure increase with time then we can write ... 

Subsequent studies (6) produced a mathematical model of the process which defined the importance of tube diameter on pressure drop. The design flow in a channel was found to be proportional to the 2.0 to 2.5 power of the tube diameter. Thus, a prototype utilizing 1 1/4-inch diameter tubes was constructed and operated stably at 85% juice yields with much lower pressure drops (250-350 psi). This same study (6) also addressed optimization of viscosity reduction of the puree and membrane flux utilizing commercial liquefaction enzymes. Viscosity reduction was readily obtained with even small amounts of liquefaction enzymes, and further increases in enzyme concentration did not appreciably affect viscosity reduction. However, steady state flux was proportional to the level of enzyme used up to 0.044%. Membrane flux correlated very well, as expected (3), with reduction of total pectin. It was evident that enzyme pretreatment should be further developed with the goal of enhancing flux rather than reducing viscosity, especially since increased tube diameter could be used to overcome pressure drops imparted by the viscosity of the retentate at high juice recoveries. 

The mathematical models of such processes in SOFC as charge, heat and mass transfer, diffusion, viscous flow of gases in channels were elaborated. The models gave necessary information for optimizing the SOFC design. 

Analysis of conditions of creation of the most important for production quasi-plug-flow mode and mathematical modelling of mixing process of two reacting liquids in the framework of K-e turbulence model have shown [31], that quasi-plug-flow mode formation in turbulent flows under fast chemical reactions is reached at the expense of production of initial components mixture and/or... 

In paper [10] the first mathematical model of the process was developed. It was based on single-phase flow model coupled with population dynamics equation. The bacterial population was considered in the average and various forms of its existence were reflected in nonlinear kinetics of population growth. 

Much more general findings and recommendations can be made using mathematical models of hydrodynamical processes in the separator. Creating a mafliematical model of the motion of a particle of dust in the swirling flow will evaluate the impact of various factors on the collection efficiency of dust in the separators, as well as to create a methodology to assess the effectiveness of the dust eollector. 

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