Mathematical modelling of the

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. 

Lakestani, F., Validation of mathematical models of the ultrasonic inspection of steel components, PISC III report 6, IRC Inst. Adv. Mater., Petten, 1992. 

The solution adopted by us is the use of computer simulations of mathematical models of the process and the mock-up situations. Eventually, simulation techniques will become so accurate, that the mock-up step can be discarded. For the time being it is reasonable to use such models to generate corrections for smaller differences between mock-up and process. 

Mathematical Model of the Nucleic Acids Conformational Transitions with Hysteresis over Hydration-Dehydration Cycle... 

An R-matrix has a series of interesting matheinatical properties that directly reflect chemical laws. Thus, the sum of all the entries in an R-matrix must be zero, as no electrons can be generated or annihilated in a chemical reaction. Furthermore, the sum of the entries in each row or column of an R-matrix must also he zero as long as there is not a change in formal charges on the corresponding atom. An elaborate mathematical model of the constitutional aspects of organic chemistry has been built on the basis of BE- and R-matriccs [17. ... 

Another approach to optimizing a method is to develop a mathematical model of the response surface. Such models can be theoretical, in that they are derived from a known chemical and... 

The production of acetic acid from butane is a complex process. Nonetheless, sufficient information on product sequences and rates has been obtained to permit development of a mathematical model of the system. The relationships of the intermediates throw significant light on LPO mechanisms in general (22). Surprisingly, ca 25% of the carbon in the consumed butane is converted to ethanol in the first reaction step. Most of the ethanol is consumed by subsequent reaction. 

A mathematical model of the operating characteristics of a modem HLW storage tank has been developed (60). This model correlates experimental data for the rate of radiolytic destmction of nitric acid, the rate of hydrogen generation owing to radiolysis of water, and cooling coil heat transfer. These are all functions of nitric acid concentration and air-lift circulator operation. 

Some of the inherent advantages of the feedback control strategy are as follows regardless of the source or nature of the disturbance, the manipulated variable(s) adjusts to correct for the deviation from the setpoint when the deviation is detected the proper values of the manipulated variables are continually sought to balance the system by a trial-and-error approach no mathematical model of the process is required and the most often used feedback control algorithm (some form of proportional—integral—derivative control) is both robust and versatile. 

Considerable work has been done on mathematic models of the extmsion process, with particular emphasis on screw design. Good results are claimed for extmsion of styrene-based resins using these mathematical methods (229,232). With the advent of low cost computers, closed-loop control of... 

The relationship between output variables, called the response, and the input variables is called the response function and is associated with a response surface. When the precise mathematical model of the response surface is not known, it is still possible to use sequential procedures to optimize the system. One of the most popular algorithms for this purpose is the simplex method and its many variations (63,64). 

Distillation Columns. Distillation is by far the most common separation technique in the chemical process industries. Tray and packed columns are employed as strippers, absorbers, and their combinations in a wide range of diverse appHcations. Although the components to be separated and distillation equipment may be different, the mathematical model of the material and energy balances and of the vapor—Hquid equiUbria are similar and equally appHcable to all distillation operations. Computation of multicomponent systems are extremely complex. Computers, right from their eadiest avadabihties, have been used for making plate-to-plate calculations. 

Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a Rill plant simulation can be made, various alternatives can be put through the computer. Such an operation is called jlowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-HiU, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. 

Once fundamental data have been obtained, the goal is to develop a mathematical model of the process and to utilize it to explore such possibilities as produc t selectivity, start-up and shut-down behavior, vessel configuration, temperature, pressure, and conversion profiles, and so on. 

One mathematical model of the oxidation of nickel spheres was confirmed when it took into account the decrease in the reaction surface as the reaction proceeded. 

Measurement Selection The identification of which measurements to make is an often overlooked aspect of plant-performance analysis. The end use of the data interpretation must be understood (i.e., the purpose for which the data, the parameters, or the resultant model will be used). For example, building a mathematical model of the process to explore other regions of operation is an end use. Another is to use the data to troubleshoot an operating problem. The level of data accuracy, the amount of data, and the sophistication of the interpretation depends upon the accuracy with which the result of the analysis needs to oe known. Daily measurements to a great extent and special plant measurements to a lesser extent are rarelv planned with the end use in mind. The result is typically too little data of too low accuracy or an inordinate amount with the resultant misuse in resources. 

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

MATHEMATICAL MODELING OF THE PROCESS TAKEN PLACE IN THE SOLID SUPPORT - SOLUTION , TYPE INDICATOR PIPES... 

For field-oriented controls, a mathematical model of the machine is developed in terms of rotating field to represent its operating parameters such as /V 4, 7, and 0 and all parameters that can inlluence the performance of the machine. The actual operating quantities arc then computed in terms of rotating field and corrected to the required level through open- or closed-loop control schemes to achieve very precise speed control. To make the model similar to that lor a d.c. machine, equation (6.2) is further resolved into two components, one direct axis and the other quadrature axis, as di.sciis.sed later. Now it is possible to monitor and vary these components individually, as with a d.c. machine. With this phasor control we can now achieve a high dynamic performance and accuracy of speed control in an a.c. machine, similar to a separately excited d.c. machine. A d.c. machine provides extremely accurate speed control due to the independent controls of its field and armature currents. 

Mathematical modelling of the machine is a complex subject and is not discussed here. For this, research and development works carried out by engineers and the textbooks available on the subject may be consulted. A few references are provided in the Further reading at the end of this chapter. In the above analysis we have considered the rotor flux as the reference frame. In fact any of the following may be fixed as the reference frame and accordingly the motor s mathematical model can be developed ... 

Air pollution meteorology came of age and, by 1980, mathematical models of the pollution of the atmosphere were being energetically developed. A start had been made in elucidating the photochemistry of air pollution. Air quality monitoring systems became operational throughout the world. A wide variety of measuring instruments became available. 

If the dynamic behaviour of a physical system can be represented by an equation, or a set of equations, this is referred to as the mathematical model of the system. Such models can be constructed from knowledge of the physical characteristics of the system, i.e. mass for a mechanical system or resistance for an electrical system. Alternatively, a mathematical model may be determined by experimentation, by measuring how the system output responds to known inputs. 

In a continuous reaction process, the true residence time of the reaction partners in the reactor plays a major role. It is governed by the residence time distribution characteristic of the reactor, which gives information on backmixing (macromixing) of the throughput. The principal objectives of studies into the macrokinetics of a process are to estimate the coefficients of a mathematical model of the process and to validate the model for adequacy. For this purpose, a pilot plant should provide the following ... 

Develop a mathematical model of the system, which is tested and fitted to the system with the aid of a computer. 

In general, the optimum conditions cannot be precisely attained in real reactors. Therefore, the selection of the reactor type is made to approximate the optimum conditions as closely as possible. For this purpose, mathematical models of the process in several different types of reactors are derived. The optimum condition for selected parameters (e.g., temperature profile) is then compared with those obtained from the mathematical expressions for different reactors. Consequently, selection is based on the reactor type that most closely approaches the optimum. 

The natural frequency, co associated with the mode shape that exhibits a large displacement of the pump is compared with the fundamental frequency, of the wall. If co is much less than ru, then the dynamic interaction between the wall and the loop may be neglected, but the kinematic constraint on the pump imposed by the lateral bracing is retained. If nearly equals nr , the wall and steam supply systems are dynamically coupled. In which case it may be sufficient to model the wall as a one-mass system such that the fundamental frequency, Wo is retained. The mathematical model of the piping systems should be capable of revealing the response to the anticipated ground motion (dominantly translational). The mathematics necessary to analyze the damped spring mass. system become quite formidable, and the reader is referred to Berkowitz (1969),... 

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. 

Traditional control systems are in general based on mathematical models that describe the control system using one or more differential equations that define the system response to its inputs. In many cases, the mathematical model of the control process may not exist or may be too expensive in terms of computer processing power and memory. In these cases a system based on empirical rules may be more effective. In many cases, fuzzy control can be used to improve existing controller systems by adding an extra layer of intelligence to the current control method. 

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

Many elements of a mathematical model of the catalytic converter are available in the classical chemical reactor engineering literature. There are also many novel features in the automotive catalytic converter that need further analysis or even new formulations the transient analysis of catalytic beds, the shallow pellet bed, the monolith and the stacked and rolled screens, the negative order kinetics of CO oxidation over platinum,... 

A comprehensive mathematical model of the pellet bed was developed in the IIEC program, and described by Wei 127) and by Kuo ei al. 21, H0). This model seeks to replace the catalytic bed by a series of cells with uniform temperatures and concentrations. The heat balance of the solid in cell i is given by... 

Gas-liquid-particle operations are of a comparatively complicated physical nature Three phases are present, the flow patterns are extremely complex, and the number of elementary process steps may be quite large. Exact mathematical models of the fluid flow and the mass and heat transport in these operations probably cannot be developed at the present time. Descriptions of these systems will be based upon simplified concepts. 

Although a dynamic mathematical model of the polymerization system has been developed (17) it is not capable of providing the necessary operating policies for the reactor in order to preselect the time-averaged MWD in the product. Hence the flow policies for the reagents were selected empirically and for experimental convenience. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

It has been possible to obtain a good measure of agreement between the experimental results, and those predicted by even a simple mathematical model of the system, assuming ideal stirred tank behaviour. One typical result is presented here. 

Mathematical models of the reaction system have been developed, enabling prediction of the molecular weight... 

In order to understand the effect of each process variable, a fundamental understanding of the heat transfer and polymer curing kinetics is needed. A systematic experimental approach to optimize the process would be expensive and time consuming. This motivated the authors to use a mathematical model of the filament winding process to optimize processing conditions. 

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