Nonisothermal reactions steady-state

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. 

For isothermal systems this equation, together with an appropriate expression for rv, is sufficient to predict the concentration profiles through the reactor. For nonisothermal systems, this equation is coupled to an energy balance equation (e.g., the steady-state form of equation 12.7.16) by the dependence of the reaction rate on temperature. 

X(T) still has the same qualitative shape, and if the reaction is reversible and exothermic, X(T) decreases at sufficiently high temperature as the equihhrium X decreases. We can generalize therefore to say that multiple steady states may exist in ary exothermic reaction in a nonisothermal CSTR. 

In this and the previous chapters we considered the effects of nonisothermal operation on reactor behavior. The effects of nonisothermal operation can be dramatic, especially for exothermic reactions, often leading to reactor volumes many times smaller than if isothermal and often leading to the possibility of multiple steady states. Further, in nonisothermal operation, the CSTR can require a smaller volume for a given conversion than a PFTR. In this section we summarize some of these characteristics and modes of operation. For endothermic reactions, nonisothermal operation cools the reactor, and this reduces the rate, so that these reactors are inherently stable. The modes of operation can be classified as follows ... 

Write the steady-state mass and heat balance equations for this system, assuming constant physical properties and constant heat of reaction. (Note Concentrate your modeling effort on the adiabatic nonisothermal reactor, and for the rest of the units, carry through a simple mass and heat balance in order to define the feed conditions for the reactor.)... 

A theory has been developed which translates observed coke-conversion selectivity, or dynamic activity, from widely-used MAT or fixed fluidized bed laboratory catalyst characterization tests to steady state risers. The analysis accounts for nonsteady state reactor operation and poor gas-phase hydrodynamics typical of small fluid bed reactors as well as the nonisothermal nature of the MAT test. Variations in catalyst type (e.g. REY versus USY) are accounted for by postulating different coke deactivation rates, activation energies and heats of reaction. For accurate translation, these parameters must be determined from independent experiments. 

An effective approach to help address the issues involved in controlling catalyst performance is to formulate and analyze reaction schemes that describe the essential chemistry taking place on the catalyst surface. This approach has been used successfully in catalysis research for many years. We suggest that this approach will see increased use in catalysis research. Specifically, continuing improvements in computer capabilities allow rapid analysis of complex reaction schemes for all common reactor configurations (e.g., reactors operating at steady state as well as under transient reaction conditions and nonisothermal reactors). Moreover, recent advances in quantum... 

Assuming a steady state, for first-order reaction-diffusion system A -> B under nonisothermal catalyst pellet conditions, the mass and energy balances are... 

Example 2—Unstable CSTR with bounded output. Consider the reaction R P occurring in a nonisothermal jacket-cooled continuous stirred tank reactor (CSTR) with three steady states. A, B, C, corresponding to the intersection points of the two lines shown in Fig. 3 (Stephanopoulos,... 

Multiple steady states as discussed in the previous subsection are related to the nonisothermicity of the CSTR. However, even in the isothermal case, a CSTR is known to be able to exhibit multiple steady states, periodic orbits, and chaotic behavior for sufficiently complex reaction network structures (see, e.g.. Gray and Scott, 1990). When the number of reactions is very large, the problem becomes a formidable one. In a series of papers (Feinberg 1987, 1988, and the literature quoted therein), Feinberg and his coworkers have developed a procedure for CSTRs that can be applied to systems with arbitrarily large numbers of reactants and reactions. The procedure is based on the deficiency concept discussed in Appendix C. 

Example 10.2. Nonisothermal Reaction in a Catalyst Pellet - Multiple Steady States... 

Falling Off the Steady State 623 Nonisothermal Multiple Reactions 625 Unsteady Operation of Plug-Flow Reactors... 

Mass transfer in combination with even quite "normal" reaction kinetics can produce a wealth of phenomena including multiple steady states, instabilities, and oscillations. An example is the behavior of nonisothermal catalyst particles outlined in Section 9.5.2. Such phenomena are covered in detail in standard texts on reaction engineering, to which the reader is referred. The examination in this section will remain restricted to effects produced by vagaries of multistep or multiple simultaneous reactions. 

Example 14. /. Multiple steady states and hysteresis in a nonisothermal continuous stirred-tank reactor (CSTR) [1,2]. In a CSTR, the curve for the temperature dependence of heat loss to the cooling coil is linear (loss proportional to temperature difference) while that for heat generation by the reaction is S-shaped (Arrhenius ex-... 

Instability typically arises from the interaction of two phenomena with different dependences on a reaction parameter In a nonisothermal reaction, the dependence on temperature is exponential for heat generation by the reaction, but linear for heat loss to the cooling coil or environment in a reaction with chain branching, the dependence on radical population is exponential for acceleration by branching, but quadratic for chain termination. A reaction is unstable if acceleration outruns retardation. This can cause an explosion or, in a CSTR, lead to multiple steady states. Feinberg s network theory can help to assess whether an isothermal reaction admits multiple steady states in a CSTR. 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation 5.27. The same basic procedure can be used for Equation 5.24. The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from //out after each time step. The examples that follow assume constant physical properties and use Equation 5.27. Their purpose is to explore nonisothermal reaction phenomena rather than to present detailed design calculations. 

Nonisothermal stirred tanks are governed by an enthalpy balance that contains the heat of reaction as a significant term. If the heat of reaction is unimportant so that a desired Tout can be imposed on the system regardless of the extent of reaction, then the reactor dynamics can be analyzed by the methods of the previous section. This section focuses on situations where Equation 14.3 must be considered as part of the design. Even for these situations, it is usually possible to control a steady-state CSTR at a desired temperature. If temperature control can be achieved rapidly, then isothermal design techniques again become applicable. Rapid means on a time scale that is fast compared to reaction times and composition changes. 

For the single-reaction, nonisothermal problem, we solved the so-called Weisz-Hicks problem, and determined the temperature and concentration profiles within the pellet. We showed the effectiveness factor can be greater than unity for this case. Multiple steady-state solutions also are possible for this problem, but for realistic values of the... 

Multiple steady-state behavior is a classic chemical engineering phenomenon in the analysis of nonisothermal continuous-stirred tank reactors. Inlet temperatures and flow rates of the reactive and cooling fluids represent key design parameters that determine the number of operating points allowed when coupled heat and mass transfer are addressed, and the chemical reaction is exothermic. One steady-state operating point is most common in CSTRs, and two steady states occur most infrequently. Three stationary states are also possible, and their analysis is most interesting because two of them are stable whereas the other operating point is unstable. 

Figure 5-1 Numerical and graphical example for a nonisothermal CSTR with exothermic chemical reaction, illustrating the phenomenon of three steady-state op ting points as dictated by three intersections of the rates of thermal energy gen tion and removal vs. temperature curves.

Figure 5-2 Effect of the inlet cooling fluid tonperature on the rate of thermal enragy removal and the number of allowed steady-state operating points for a nonisothermal CSTR with exothermic chemical reaction. All of the other parameters are the same as those in Figure 5-1.

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