Rate laws CSTRs

To determine the form of the rate law, values of (-rA) as a function of cA may be obtained from a series of such experiments operated at various conditions. For a given reactor (V) operated at a given % conditions are changed by varying either cAo or q. For a rate law given by (—rA) = kAcA, the parameter-estimation procedure is die same as that in the differential method for a BR in the use of equation 3.4-2 (linearized form of the rate law) to determine kA and n. The use of a CSTR generates point ( -rA) data directly without the need to differentiate cA data (unlike the differential method with aBR). 

The experimental investigation of the form of the rate law, including determination of the rate constants kf and kr, can be done using various types of reactors and methods, as discussed in Chapters 3 and 4 for a simple system. Use of a batch reactor is illustrated here and in Example 5-4, and use of a CSTR in problem 5-2. 

Suppose the reaction in Example 5-4 was studied in a CSTR operated at steady-state, and the results given below were obtained. Calculate the values of kf and kr, and hence write the rate law. Assume T to be the same, constant density, and no D in the feed. 

These rate laws are coupled through the concentrations. When combined with the material-balance equations in the context of a particular reactor, they lead to uncoupled equations for calculating the product distribution. For a constant-density system in a CSTR operated at steady-state, they lead to algebraic equations, and in a BR or a PFR at steady-state, to simultaneous nonlinear ordinary differential equations. We demonstrate here the results for the CSTR case. 

An unusual feature of a CSTR is the possibility of multiple stationary states for a reaction with certain nonlinear kinetics (rate law) in operation at a specified T, or for an exothermic reaction which produces a difference in temperature between the inlet and outlet of the reactor, including adiabatic operation. We treat these in turn in the next two sections. 

We can, however, consider the stability of each of the three operating points in Example 14-7 with respect to the inevitable small random fluctuations in operating conditions, including cA, in steady-state operation. Before doing this, we note some features of the rate law as revealed in Figure 14.4. There is a maximum value of (- rA) at cA = 1.166 mol m-3. For cA < 1.166, the rate law represents normal kinetics ( rA) increases as cA increases for cA > 1.166, we have abnormal kinetics (—rA) decreases as cA increases. We also note that (-rA) in equation (C), the rate law, represents the (positive) rate of disappearance of A by reaction within the CSTR, and that (—rA) in equation (D), the material balance, represents the (positive) net rate of appearance of A by flow into and out of the reactor. As noted above, in steady-state operation, these two rates balance. 

As for a single-stage CSTR, the volume of each stage is obtained from the material balance around that stage, together with the rate law for the system. Thus, for the ith stage, the material-balance equation (14.3-5) becomes... 

The liquid-phase reaction A - B + C takes place in a single-stage CSTR The rate law for... 

Consider the following gas-phase reaction that occurs at 350 K and a constant pressure of 200 kPa (Lynch, 1986) A- B + C, for which the rate law is (-/a) = 0.253cA/(l + 0.429cA)2, where (- rA) has units of mol m-3 s-1 cA has units of mol m-3. Pure A is fed to a reactor at a rate of 8 mol s-1. The desired fractional conversion, fA, is 0.99. A recycle PFR is proposed for the reaction. When the recycle ratio, R, is zero, the recycle reactor is equivalent to a PFR. As R approaches infinity, the system is equivalent to a CSTR. However, it is generally stated that the recycle reactor behavior is close to that of a CSTR once R reaches approximately 10 to 20. Furthermore, it is often stated that, for an equivalent fractional conversion, the volume... 

The CRE approach for modeling chemical reactors is based on mole and energy balances, chemical rate laws, and idealized flow models.2 The latter are usually constructed (Wen and Fan 1975) using some combination of plug-flow reactors (PFRs) and continuous-stirred-tank reactors (CSTRs). (We review both types of reactors below.) The CRE approach thus avoids solving a detailed flow model based on the momentum balance equation. However, this simplification comes at the cost of introducing unknown model parameters to describe the flow rates between various sub-regions inside the reactor. The choice of a particular model is far from unique,3 but can result in very different predictions for product yields with complex chemistry. 

CSTR. The cell growth rate law for this system is... 

To obtain a plot of heat generated, G(T), as a function of temperature, we must solve for X as a function of T using the CSTR mole balance, the rate law, and stoichiometry. For example, for a first-order liquid-phase reaction, the CSTR mole balance becomes... 

This rate law can now be used in the design of an industrial CSTR. 

Liquid Phase. For liquid-phase reactions in which there is no volume change, concentration is the preferred variable. The mole balances are shown in Table 4-5 in terms of concentration for the four reactor types we have been discussing. We see from Table 4-5 that we have only to specify the parameter values for the system (CAo,Uo,etc.) and for the rate law (i.e., ifcyv. .3) to solve the coupled ordiaaiy differential equations for either PFR, PBR, or batch reactors or to solve the coupled algebraic equations for a CSTR. 

In a similar fashion one can solve the combined CSTR mole balances and rate laws, that is,... 

A Second-Order Reaction in a CSTR. For a second-order liquid-phase reaction being carried out in a CSTR, the combination of the rate law and the design equation yields... 

The preceeding four sections conclude our discussion of elementary heterogeneous catalysis mechanisms and design equations. In the following section we shall work through an example problem using experimental data to (1) deduce a rate law, (2) determine a mechanism consistent with experimental data, (3) evaluate the rate law parameters, and (4) design a CSTR and packed-bed reactor. 

Mechanism Evaluate Rate law parameters Design PBR CSTR... 

For a packed-bed reactor, the approach is quite similar to that described for a CSTR. For a first-order reaction, the combined mole balance and rate law is... 

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