Feeding mathematical 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. 

Non-ideal reactors are described by RTD functions between these two extremes and can be approximated by a network of ideal plug flow and continuously stirred reactors. In order to determine the RTD of a non-ideal reactor experimentally, a tracer is introduced into the feed stream. The tracer signal at the output then gives information about the RTD of the reactor. It is thus possible to develop a mathematical model of the system that gives information about flow patterns and mixing. 

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit functionOptimisation and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of a numerical optimisation algorithms. The first method is easily applied with ISIM, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method however becomes extremely cumbersome. 

There are two objectives of setting up a kinetic mathematical model for chiral products. The first is the elucidation of the reaction mechanism with identification of the rate-controlhng step. The second is to derive a mathematical expression for the selectivity in terms of the ratio of the major product to the minor product. Then, based upon this expression, the reaction conditions such as pressure or feed ratio are changed to increase the selectivity. However, when the enantiomeric purity is over 99%, the selectivity is extremely high hence, the reaction mechanism for the major manifold can be neglected to simplify the establishment of the kinetic model. 

In some cases a variable can be independent and in others the same variable can be dependent, but the usual independent variables are pressure, temperature, and flow rate or concentration of a feed. We cannot provide examples for all of these criteria, but have selected a few to show how they mesh with the optimization methods described in earlier chapters and mathematical models listed in Table 14.1. 

Develop a mathematical model for the three-column train of distillation columns sketched below. The feed to the first column is 400 kg mol/h and contains four components (1, 2, 3, and 4), each at 25 mol %. Most of the lightest component is removed in the distillate of the first column, most of the next lightest in the second column distillate and the final column separates the final two heavy components. Assume constant relative volatilities throughout the system ai, CI2, and a3. The condensers are total condensers and the reboilers are partial. Trays, column bases, and reflux drums are perfectly mixed. Distillate flow rates are set by reflux drum... 

Assuming each reaction is linearily dependent on the concentrations of each reacr-tant, derive a dynamic mathematical model of the system. There are two feed streams, one pure benzene and one concentrated nitric acid (98 wt %). Assume constant densities and complete miscibility. 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

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. 

Those based on strictly empirical descriptions Mathematical models based on physical and cnemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component... 

The control of a fed-batch alcoholic fermentation process can be obtained by controlling the substrate concentration in the medium by manipulation of the feed flow. The fermentation process presents complicated kinetic mechanisms. In addition, there is the absence of accurate and reliable mathematical models as well as the difficulty of obtaining direct measurements of the process variables owing to a lack of appropriate on-line analyzers and sensors. Control systems are formed by a set of instruments and control mechanisms connected through electrical signals in the... 

A simple way to circumvent the problem and improve the control is to use predictive control, often called feed forward. The idea is to use a control signal, CCff, which in some way reflects the expected time-course of CC that is needed to keep CO close to CL In technical systems this control is frequently called model predictive control (MPC [12]), because CCff is typically derived from a mathematical model. In biological systems, the feed forward is mostly delivered by nerve signals, more rarely by hormones. 

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