2.18 Perform a degrees of freedom analysis for the model in Eqs. 2-64 through 2-68. Identify parameters, output variables, and inputs (manipulated and disturbance variables). tions in the inlet Mixture of A and B mics of the pro- omatic control is 9, CA, T V, p, T 2.4 Dyna reaction. In other words, the heat of mixing is neg- ligible compared to the heat of reaction. 8. Shaft work and heat losses to the ambient can be neglected. decrease in lelllllld displayed in centration as The following form of the CSTR energy balance is convenient for analysis and can be derived from Eqs. 2-62 and 2-63 and Assumptions 1-8 (Fogler, 2006; Russell and Denn, 1972), TR) have wide- embody many STR models tend types of continu- and packed-bed odel provides a ing principles for Table 2.3 Nc Cooling medium at temperature T. Parameter VpcdT = wC(T; - T) + (-AHR)Vkc, Figure 2.6 A nonisothermal continuous stirred-tank reactor. dt + UA(T. – T) (2-68) For these assumptions, the unsteady-state mass balance { for the CSTR is where AHR is the heat of reaction per mole of A that is reacted. In summary, the dynamic model of the CSTR consists of Eqs. 2-62 to 2-64, 2-66, 2-67, and 2-68. This model is nonlinear as a result of the many product terms and Tne exponential temperature dependence of k in Eq. 2-63. Consequently, it must be solved by numerical integra- tion techniques (Fogler, 2006). The CSTR model will become considerably more complex if -AHR versible chemical ets to form species B. We assume that d(pV) (2-64) = pq, - pg dt Because V and p are constant, Eq. 2-64 reduces to 450 respect to compo- (2-65) q = 9i Thus, even though the inlet and outlet flow rates may change due to upstream or downstream conditions, Eq. 2-65 must be satisfied at all times. In Fig. 2.6, both flow rates are denoted by the symbol q. For the stated assumptions, the unsteady-state com- ponent balances for species A (in molar concentration units) is (2-62) e 400- 1. More complicated rate expressions are considered. For example, a mass action kinetics model for a second-order, irreversible reaction, 2A B, is given by per unit volume, units of reciprocal tion of species A. onstant is typically ature given by the 1 = kzc 350 ---. (2-69) 2. Additional species or chemical reactions are involved. If the reaction mechanism involved pro- duction of an intermediate species, 2A B → B, then unsteady-state component balances for both A and B* would be necessary (to calculate c, and CR), or balances for both A and B could be written (to calculate c, and cB). Information concerning the reaction mechanisms would also be required. y dea = q(CAi - CA) – VkcA (2-66) 300 dt (2-63) This balance is a special case of the general component balance in Eq. 2-7. Next, balance for the CSTR. But first we make five additional Figure 2.7 Rea. changes in cool and from 300 to E is the activation The expressions in theoretical consid- and E are usually consider an unsteady-state energy we Reactions involving multiple species are described by high-order, highly coupled, nonlinear reaction models, because several component balances must be written. assumptions: 1.0 4. The thermal capacitances of the coolant and the cooling coil wall are negligible compared to the thermal capacitance of the liquid in the tank. 5. All of the coolant is at a uniform temperature, T- (That is, the increase in coolant temperature as the coolant passes through the coil is neglected.) 0.9 data. Thus, these obe semi-empirical in Section 2.2. CSTR is shown in 0.8 EXAMPLE 2.5 0.7 To illustrate how the CSTR can exhibit nonlinear dynamic behavior, we simulate the effect of a step change in the coolant temperature T, in positive and negative directions. Table 2.3 shows the parameters and nominal operating condition for the CSTR based on Eqs. 2-66 and 2-68 for the exothermic, irreversible first-order reaction A two state variables of the ODES are the concentration of A (c.) and the reactor temperature T. The manipulated input variable is the jacket water temperature, T Two cases are simulated, one based on increased cool- ing by changing T, from 300 to 290 K and one reducing the cooling rate by increasing T, from 300 to 305 K. These model equations are solved in MATLAB with a numerical integrator (ode15s) over a 10-min horizon. The 0.6 of pure component cooling coil is used t the desired oper- cat that is released 0.5 0.4 6. The rate of heat transfer from the reactor contents to the coolant is given by 0.3 0.2- 0.1 B. The nitial CSTR model (2-67) Q = UA(T- T) mptions: where U is the overall heat transfer coefficient and A is the heat transfer area. Both of these model parameters are assumed to be constant. 7. The enthalpy change associated with the mixing of the feed and the liquid in the tank is negligible com- pared with the enthalpy change for the chemical 0. Figure 2.8 React. changes in coolin and product streams are denoted by p. ctor is kept constant Reactant A concentration (mol/L) Reactor temperature (K):