Nonisothermal conditions multiple steady states

An interesting feature of systems involving exothermic chemical reactions accompanied by external transport resistances is that of multiple steady states [9, 13, 19]. The simplest case exists 

The equations for Qr and Q define the temperature difference Ts- Tb) in terms of parameters including a, A, AH h, and Cas- The requirement that Qjj = Q at steady state leads to multiple steady-state solutions, and it is typical that the particular steady state which is physically realized during operation depends on the conditions under which the reaction is started. There are two possible situations to be considered, depending on whether external concentration gradients are (i) negligible or (ii) substantial. 

Negligible CAh Cas) This case is simpler, because the unknown surface concentration Cas is replaced by CAh, since Cas CAh fot slow surface reaction and fast mass transfer. The Qr equation is directly written in terms of CAh- 

Substantial (Cab Cas) In this case, the equality of rates of surface reaction and mass transfer given by Equation 2.40a and the equality of Qr and Q have to be solved simultaneously to determine the steady-state values of Cas and Ts- For the sake of simplicity, n = 1 is assumed for the reaction order  

An analytical solution for Qr=Q is not feasible however, when both Qr and Q are plotted versus the dimensionless temperature rise 0, the intersection point(s) of the two curves will determine the steady-state value of Ts for a given Cab and T.  

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This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Figure 11.9.a-l shows the relation between the effectiveness factors rj and tjo and the modulus [104, 108]. This relation can only be obtained by numerical integration of the system, Eqs. 11.9.a-l to 11.9.a-8, except for the cases already mentioned. With isothermal situations ri tends to a limit of 1 as 0 increases, with nonisothermal situations, however, r/ or tic, may exceed 1. Curve 1 corresponds to the t] concept, curves 2, 3, and 4 to r/o. The dotted part of curve 4 corresponds to a region of conditions within which multiple steady states inside the catalyst are... 

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