Model equations, CSTR

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples, THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG and STABIL. 

VR/n is the volume of the CSTR used in the model Equations of this form were treated in Section 8.3.2.2. For a constant density system,... 

The CSTR model, on the other hand, is based on a stirred vessel with continuous inflow and outflow (see Fig. 1.2). The principal assumption made when deriving the model is that the vessel is stirred vigorously enough to eliminate all concentration gradients inside the reactor (i.e., the assumption of well stirred). The outlet concentrations will then be identical to the reactor concentrations, and a simple mole balance yields the CSTR model equation ... 

Eq.(l) and (9) can be utilized for modeling a CSTR considering the material and energy balances as well as the expression for the rate flow of heat removed, Q. This heat rate is obtained from the overall heat transfer coefficient U and the transmission area A by the equation Q = UA(T — Tj) [1], [9], [13], [14], [18], [22],... 

Derive the model equation for this reaction taking place in a liquid phase CSTR. Put the model equation into dimensionless form. Assume that the feed concentrations of B and C are zero. If = 0.6, = 1.2, 71 = 18, and 72 = 27 and the ratio... 

In this section we replace the CSTR by a plug-flow reactor and consider the conventional control structure. Section 4.5 presents the model equations. The energy balance equations can be discarded when the heat of reaction is negligible or when a control loop keeps constant reactor temperature manipulating, for example, the coolant flow rate. The model of the reactor/separation/recycle system can be solved analytically to obtain (the reader is encouraged to prove this) ... 

Similarly to the CSTR, case Eq. (4.14), X = 0 is a trivial solution satisfying the model equations irrespective of the Da value. However, it is unfeasible, corresponding to infinite flow rates. 

Equation may be used to model a CSTR by modifying the vector c to allow for the removal of latex particles in the effluent stream while incorporating the particle creation terms of the type discussed above. The form of the term describing latex particle removal is given simply for all i by... 

The model for the CSTR (see Example 4.10) is given by eqs. (4.8a), (4.9a), and (4.10b). These constitute a set of nonlinear equations for which there is no analytic solution available. Therefore, in order to study the dynamic behavior of the CSTR, we must solve the modeling equations numerically using a computer. 

We have no way of knowing how long it will take a given reaction or set of reactions to achieve a steady state in the CSTR before we either do an experiment or solve the time-dependent model equations. If we choose to do experiments as a means to assessing this, then we need to be prepared to do many of them. But if we already know the kinetics, then we do the analysis and the math instead. If we do it correctly, then it is fast and it provides us with insights that complement the experiments and in many cases provides interpretations that a purely experimental approach cannot yield. Therefore, in this problem we will consider just such a case with a more complex set of kinetics. 

The CSTR, operated at steady state, can be a useful tool for kinetic studies. Conditions in a batch or semi-batch reactor often change rapidly with time and slopes of experimental data must be used in model equations. Rqiid phenomena, such as particle nucleation, are difiicult to study. Also particle size distributions can be narrow and therefore batch experiments may not be sensitive enough to choose between alternate particle growth models. 

I.3 Conversion to an LP Problem Once the connectivity model equations have been established, the system of linear equations may be transformed into an LP problem. This is achieved by introducing non-negativity constraints on all flow rates and reactor volumes for each CSTR. An objective function must be supplied in the linear program. Two vectors, aj and a2, are introduced to act as objective functions for the LP problem. The following LP may then be formulated ... 

There are three basic homogeneous reactor models (DCSTR, CSTR, and CPFR, Fig. 3.30) that can be considered ideal cases for calculating conversion. The equations for balancing all reactor models derive from the conservation of mass equation, Equ. 2.3. The equation for a balancing all types of ideal continuous stirred tank reactors (idCSTR) (c = Cr) can uniformly be based on a consideration of the following (see Fig. 3.36) ... 

Equation 3.329 is the axial dispersion model equation and the Peclet number Pe is the model parameter. Pe = for an ideal PFR and Pe = 0 for an ideal CSTR. Pe is a finife value greater than 0 for any non-ideal PFR wifh axial mixing. 

It can be seen that the plot of E(9) versus 9 shifts away from the ideal CSTR plot and moves towards the ideal PFR plot as Pe increases. Given a plot of E(9) versus 9 obtained from the tracer experiment, the value of the parameter Pe is estimated as the value for which the experimental plot fits well with the theoretical plot of E(9) versus 9 shown in Figure 3.62. But one cannot derive a theoretical expression for E(9) as it is not possible to obtain an analytical solution to the model Equation 3.329 with Danckwarts boundary conditions (3.331) and (3.334). Flowever, an explicit equation relating the variance and mean 9 of the RTD to the Peclet number Pe has been derived using the method of moments without actually solving the model equation. This equation... 

A typical example of dynamic optimization in ch ical engineering is the change between steady states in a continuous-stured tank reactor (CSTR) in which the irreversible reaction A B takes place ([21,22]) (Figure 14.4). The reaction is first order and exothermic and follows Arrhenius rate law. The reactor is equipped with a cooling jacket with refrigerant fluid at constant temperature T . To develop model equations, we formulate mass and energy balances. 

Models of BCR can be developed on the basis of various view points. The mathematical structure of the model equations is mainly determined by the residence time distribution of the phases, the reaction kinetics, the number of reactive species involved in the process, and the absorption-reaction regime (slow or fast reaction in comparison to mass transfer rate). One can anticipate that the gas phase as well as the liquid phase can be either completely backmixed (CSTR), partially mixed, as described by the axial dispersion model (ADM), or unmixed (PFR). Thus, it is possible to construct a model matrix as shown in Fig. 3. This matrix refers only to the gaseous key reactant (A) which is subjected to interphase mass transfer and undergoes chemical reaction in the liquid phase. The mass balances of the gaseous reactant A are the starting point of the model development. By solving the mass balances for A alone, it is often possible to calculate conversions and space-time-yields of the other reactive species which are only present in the liquid phase. Heat effects can be estimated, as well. It is, however, assumed that the temperature is constant throughout the reactor volume. Hence, isothermal models can be applied. 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

Multistage CSTR This model has a particular importance because its RTD curve is beU-shaped hke those of many experimental RTDs of packed beds and some empty tubes. The RTD is found by induction by solving the equations of one stage, two stages, and so on, with the result. 

The value of n is the only parameter in this equation. Several procedures can be used to find its value when the RTD is known experiment or calculation from the variance, as in /i = 1/C (t ) = 1/ t C t), or from a suitable loglog plot or the peak of the curve as explained for the CSTR battery model. The Peclet number for dispersion is also related to n, and may be obtainable from correlations of operating variables. 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

Analytical solutions also are possible when T is constant and m = 0, V2, or 2. More complex chemical rate equations will require numerical solutions. Such rate equations are apphed to the sizing of plug flow, CSTR, and dispersion reactor models by Ramachandran and Chaud-hari (Three-Pha.se Chemical Reactors, Gordon and Breach, 1983). 

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