Stirred-tank reactor linear equations

If the points lie close to a straight line, this is taken as confirmation that a second-order equation satisfactorily describes the kinetics, and the value of the rate constant k2 is found by fitting the best straight line to the points by linear regression. Experiments using tubular and continuous stirred-tank reactors to determine kinetic constants are discussed in the sections describing these reactors (Sections 1.7.4 and 1.8.S). 

Many chemical and biological processes are multistage. Multistage processes include absorption towers, distillation columns, and batteries of continuous stirred tank reactors (CSTRs). These processes may be either cocurrent or countercurrent. The steady state of a multistage process is usually described by a set of linear equations that can be treated via matrices. On the other hand, the unsteady-state dynamic behavior of a multistage process is usually described by a set of ordinary differential equations that gives rise to a matrix differential equation. 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

In Example 6.4 we developed the linearized model of a continuous stirred tank reactor in terms of deviation variables, given by eqs. (6.36) and (6.37). After rearranging the terms in these equations, we take... 

The reactor is modelled as a continuous stirred tank reactor (CSTR) (cf. [22] and Sect. 3.2.2). The equations on the basis of the reaction Eq. (4.9) are given below values and explanations of the quantities used are found in Table 4.4. The system of non-linear equations of first order is solved using the Rimge-Kutta method. 

Solves a set of simultaneous linear algebraic equations that represent the material balances for a set of continuous stirred tank reactors using the Jacobi Iterative method (Jacobi.m). 

This section is a review of the properties of a first order differential equation model. Our Chapter 2 examples of mixed vessels, stined-tank heater, and homework problems of isothermal stirred-tank chemical reactors all fall into this category. Furthermore, the differential equation may represent either a process or a control system. What we cover here applies to any problem or situation as long as it can be described by a linear first order differential equation. 

The equations for the two-phase fully mixed system are thus reduced to the equations for a single stirred tank by the physically motivated notion of only using the available fraction of the feed. This has been made possible by the uniformity of the dense phase and the linearity of the transfer process in the bubble. This allows us to see how the rather implausible assumption that the bubble phase is really well mixed can be made more realistic. Let us go to the other extreme, and suppose that the bubbles ascend with uniform velocity U. The surface area per unit length of reactor is SIH, where 5 is, as before, the total interphase area and H the height of the bed. If h is the transfer coefficient and z the height of a given point, a balance over the interval (z, z + dz) gives the equation for the concentration in the bubble phase b(z)... 

The mathematical treatment of FMC data can be accomplished by standard procedures via the solution of mass balance equations, on condition that the data were converted to reaction rate data with Eq. (21). As mentioned above, this requires the determination of the transformation parameter a. Two approaches based on calibration were developed and tested. In the first approach, thermometric signals are combined with the absolute activity of IMB, which had been determined by a separate measurement using an independent analytical technique. Figure 5 shows a calibration for the cephalosporin C transformation catalyzed by D-amino acid oxidase. The activity of the IMB was determined by the reaction rate measurement in a stirred-tank batch reactor. The reaction rate was determined as the initial rate of consumption of cephalosporin C monitored by HPLC analysis. The thermometric response was measured for each IMB packed in the FMC column, and plotted against the corresponding reaction rate. From the calibration results shown in Fig. 5 it can be concluded, independently of the type of immobilized biocatalyst, that the data fall to the same line and that there is a linear correlation between the heat response and the activity of the catalyst packed in the column. The transformation parameter a was determined from... 

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