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### Chemical Reaction Engineering #### Fall 2003 **Prof. K. Dane Wittrup** **Lecture 39: Review III: Review of Multiple Reactions, Non-Ideal Reactors, and Heterogeneous Catalysis** ##### Multiple Reactions * Yield: $Y_{desired} = \frac{moles desired product formed}{moles of reactant consumed}$ * Selectivity: $S_{desired} = \frac{moles desired product formed}{moles of undesired product formed}$ Maximize yield and selectivity. Manipulate reactor conditions to favor desired products. * Reactor Type (batch, CSTR, PFR) * Temperature * Concentration ##### PFR vs. CSTR Consider: $A \rightarrow D$ (desired) $r_D = k_1 C_A^{\alpha_1}$ $A \rightarrow U$ (undesired) $r_U = k_2 C_A^{\alpha_2}$ Maximize $S = \frac{r_D}{r_U} = \frac{k_1}{k_2} C_A^{\alpha_1 - \alpha_2}$ * $\alpha_1 > \alpha_2$: Keep $C_A$ high $\rightarrow$ PFR * $\alpha_1 < \alpha_2$: Keep $C_A$ low $\rightarrow$ CSTR * $\alpha_1 = \alpha_2$: $S$ is independent of $C_A \rightarrow$ reactor type doesn't matter * If $\alpha_1 > \alpha_2$ and $k_1 > k_2$, operate at highest possible $C_A$ ##### Temperature $S = \frac{r_D}{r_U} = \frac{k_1}{k_2} = \frac{A_1 e^{-E_1/RT}}{A_2 e^{-E_2/RT}} = \frac{A_1}{A_2} e^{(E_2 - E_1)/RT}$ If $E_1 > E_2$, run at high $T$ If $E_2 > E_1$, run at low $T$ ##### Optimum Temperature Profile Adiabatic Reactor: For an exothermic reaction, the optimum temperature profile usually decreases along the length of the reactor. ##### Non-Ideal Reactors: Residence Time Distribution (RTD) **Uses:** * Troubleshooting (i.e. diagnosing "dead volume") * Scale-up * Reactor Modeling **C(t) Measurement:** ##### RTD Properties $E(t) = \frac{C(t)}{\int_0^\infty C(t) dt}$ $\int_0^\infty E(t) dt = 1$ $F(t) = \int_0^t E(t') dt'$ $F(t)$ = fraction of effluent that has spent time less than $t$ in the reactor. ##### Mean Residence Time $\overline{t} = \int_0^\infty t E(t) dt$ $\sigma^2 = \int_0^\infty (t - \overline{t})^2 E(t) dt$ $\sigma^2$ is the variance. ##### Models for Non-Ideal Reactors * Series of CSTRs * PFR with Dispersion * PFR with Recycle ##### Series of CSTRs $\tau = \frac{V}{v_0}$ (total) $\tau_i = \frac{\tau}{n}$ $E(t) = \frac{t^{n-1}}{(n-1)! (\tau/n)^n} e^{-t/(\tau/n)}$ As **n** increases, approaches a PFR. ##### Tanks-in-Series $\sigma_t^2 = \frac{\sigma^2}{\overline{t}^2} = \frac{1}{N}$ $N$ = number of tanks ##### Dispersion Model $-\frac{d}{dz} (D_a \frac{dC}{dz}) + u \frac{dC}{dz} + r_A = 0$ $D_a$ = dispersion coefficient $u$ = velocity **Boundary Conditions:** Open-Open: $C|_{z=0^-} = C|_{z=0^+}$ $-D_a \frac{dC}{dz}|_{z=0^-} + uC|_{z=0^-} = uC_0$ $\frac{dC}{dz}|_{z=L} = 0$ Closed-Closed: $-D_a \frac{dC}{dz}|_{z=0} = 0$ $-D_a \frac{dC}{dz}|_{z=L} = 0$ ##### Dispersion Number $Pe = \frac{uL}{D_a}$ $Pe \rightarrow 0$: Well-mixed $Pe \rightarrow \infty$: Plug flow ##### Recycle Reactor $R = \frac{volume \; of \; fluid \; recycled}{volume \; leaving \; the \; system}$ ### Heterogeneous Catalysis 1. Mass Transfer (Bulk $\rightarrow$ Surface) 2. Adsorption 3. Surface Reaction 4. Desorption 5. Mass Transfer (Surface $\rightarrow$ Bulk) ##### Adsorption Isotherms $\theta = \frac{V}{V_m}$ $\theta$ = fraction of surface covered $V$ = volume adsorbed $V_m$ = volume adsorbed in a monolayer ##### Langmuir Isotherm $A(g) + S \rightleftharpoons A \cdot S$ $r_{ads} = k_a P_A (1 - \theta_A)$ $r_{des} = k_d \theta_A$ At equilibrium, $r_{ads} = r_{des}$ $k_a P_A (1 - \theta_A) = k_d \theta_A$ $\theta_A = \frac{K_A P_A}{1 + K_A P_A}$ where $K_A = \frac{k_a}{k_d}$ ##### Competitive Langmuir Isotherm $A(g) + S \rightleftharpoons A \cdot S$ $B(g) + S \rightleftharpoons B \cdot S$ $\theta_A = \frac{K_A P_A}{1 + K_A P_A + K_B P_B}$ $\theta_B = \frac{K_B P_B}{1 + K_A P_A + K_B P_B}$ $1 = \theta_V + \theta_A + \theta_B$ where $\theta_V$ is the fraction of vacant sites. ##### Steps for Determining Rate Law 1. Adsorption 2. Surface Reaction 3. Desorption (one step is usually rate limiting) Derive rate law in terms of $\theta_i$ then eliminate $\theta_i$ using adsorption isotherms.

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