CSTRs, unsteady

The ODEs governing the unsteady CSTR are obtained by adding accumulation terms to Equations (4.1). The simulation holds the volume constant, and... 

Equations (14.1)-(14.3) are a set of simultaneous ODEs that govern the performance of an unsteady CSTR. The minimum set is just Equation (14.2), which governs the reaction of a single component with time-varying inlet concentration. The maximum set has separate ODEs for each of the variables... 

Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

Hold volume constant and set 5 = dout in Equation 1.6 to obtain the ODEs governing an unsteady CSTR ... 

Ideal CSTR Witt a step input of magnitude Cy the unsteady material balance is... 

Such a differential equation together with a rate equation for the main reactant constitutes a pair that must be solved simultaneously. Take the example of a CSTR for which the unsteady material balance is... 

The inlet monomer concentration was varied sinusoidally to determine the effect of these changes on Dp, the time-averaged polydispersity, when compared with the steady-state case. For the unsteady state CSTR, the pseudo steady-state assumption for active centres was used to simplify computations. In both of the mechanisms considered, D increases with respect to the steady-state value (for constant conversion and number average chain length y ) as the frequency of the oscillation in the monomer feed concentration is decreased. The maximum deviation in D thus occurs as lo 0. However, it was predicted that the value of D could only be increased by 10-325S with respect to the steady state depending on reaction mechanism and the amplitude of the oscillating feed. Laurence and Vasudevan (12) considered a reaction with combination termination and no chain transfer. 

A well-mixed stirred tank (which we will continue to call a CSTR despite possibly discontinuous flow) has p = Pout- The unsteady-state balance for total mass is obtained just by including the accumulation term ... 

As in the case of a batch reactor for commercial operation, a CSTR is normally used for a liquid-phase reaction. In the laboratory, it may also be used for a gas-phase reaction for experimental measurements, particularly for a solid-catalyzed reaction, as in Figure 1.2. The operation is normally one of steady-state, except for startup, shutdown, and operational disturbances or upsets, in which cases unsteady-state operation has to be taken into account. 

For unsteady-state operation of a CSTR, the full form of the material balance, equation 14.3-2 or its equivalent, must be used. 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

Similarly the unsteady material balance of a CSTR has an accumulation term added to it, for example,... 

Conditions that may give rise to unsteadiness are changes in feed rate, composition or temperature. In problem P4.09.34, sinusoidal variations of these three properties are forced on the CSTR. The resulting outlet conditions... 

An irreversible first order reaction reaches steady state in a three stage CSTR. Then the second stage is bypassed. Find the unsteady performance of the third stage. 

Automatic controllers can produce small oscillations of the controlled variable. The effect of sinsoidal variations in concentration, temperature or feed rate on the effluent concentration of a second order reaction in a CSTR will be examined. The unsteady material balance is... 

Tracer input to a CSTR is sinusoidal with equation C/Cf - l+sin((Jt). Transient and steady output concentrations are to be found. The unsteady material balance is... 

Batch [well-mixed (CSTR), closed system] Ordinary differential equations (unsteady state) Algebraic equation (steady state)... 

Remark 1. Eq.(9) can be used on a CSTR in unsteady state operation or in a semibatch reactor when there is no phase change. 

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