Yield and Selectivity: PFR vs CSTR

Multiple Reactions

• Yield is calculated by dividing the moles of desired product formed by the moles of reactant consumed: $Y_{desired} = \frac{moles desired product formed}{moles of reactant consumed}$
• Selectivity is determined by dividing the moles of desired product formed by the moles of undesired product formed: $S_{desired} = \frac{moles desired product formed}{moles of undesired product formed}$
• The goal is to maximize both yield and selectivity by manipulating reactor conditions to favor the production of desired products.
• Reactor conditions that can be manipulated include:
  • Reactor type (batch, CSTR, PFR)
  • Temperature
  • Concentration

PFR vs. CSTR

• For the reactions $A \rightarrow D$ (desired, with rate $r_D = k_1 C_A^{\alpha_1}$) and $A \rightarrow U$ (undesired, with rate $r_U = k_2 C_A^{\alpha_2}$), selectivity, $S$, is given by $\frac{r_D}{r_U} = \frac{k_1}{k_2} C_A^{\alpha_1 - \alpha_2}$
• If $\alpha_1 > \alpha_2$, a PFR should be used to keep $C_A$ high.
• If $\alpha_1 < \alpha_2$, a CSTR should be used to keep $C_A$ low.
• If $\alpha_1 = \alpha_2$, the reactor type is irrelevant because $S$ is independent of $C_A$.
• When $\alpha_1 > \alpha_2$ and $k_1 > k_2$, operate at the highest possible $C_A$.

Temperature

• Temperature affects reaction rates, with $S = \frac{r_D}{r_U} = \frac{k_1}{k_2} = \frac{A_1 e^{-E_1/RT}}{A_2 e^{-E_2/RT}} = \frac{A_1}{A_2} e^{(E_2 - E_1)/RT}$
• High temperatures should be used if $E_1 > E_2$.
• Low temperatures should be used if $E_2 > E_1$.

Optimum Temperature Profile

• For exothermic reactions in an adiabatic reactor, the optimum temperature profile usually decreases along the reactor length.

Non-Ideal Reactors: Residence Time Distribution (RTD)

• RTD is useful for:
  • Troubleshooting (diagnosing "dead volume")
  • Scale-up
  • Reactor Modeling

RTD Properties

• The exit age distribution function is given by: $E(t) = \frac{C(t)}{\int_0^\infty C(t) dt}$.
• The integral of $E(t)$ over all time equals 1: $\int_0^\infty E(t) dt = 1$
• The cumulative distribution function is: $F(t) = \int_0^t E(t') dt'$.
• $F(t)$ represents the fraction of effluent that has spent time less than $t$ in the reactor.

Mean Residence Time

• The mean residence time is: $\overline{t} = \int_0^\infty t E(t) dt$
• The variance is: $\sigma^2 = \int_0^\infty (t - \overline{t})^2 E(t) dt$

Models for Non-Ideal Reactors

• Models for non-ideal reactors include:
  • Series of CSTRs
  • PFR with Dispersion
  • PFR with Recycle

Series of CSTRs

• The total space time is: $\tau = \frac{V}{v_0}$
• The space time for each reactor in series is: $\tau_i = \frac{\tau}{n}$
• The exit age distribution for n tanks in series is: $E(t) = \frac{t^{n-1}}{(n-1)! (\tau/n)^n} e^{-t/(\tau/n)}$
• As n increases, the series of CSTRs approaches a PFR.

Tanks-in-Series

• The normalized variance is: $\sigma_t^2 = \frac{\sigma^2}{\overline{t}^2} = \frac{1}{N}$
• $N$ represents the number of tanks

Dispersion Model

• The dispersion model is represented by the equation: $-\frac{d}{dz} (D_a \frac{dC}{dz}) + u \frac{dC}{dz} + r_A = 0$
  • $D_a$ is the dispersion coefficient
  • $u$ is the velocity
• Open-Open Boundary Conditions:
  • $C|{z=0^-} = C|{z=0^+}$
  • $-D_a \frac{dC}{dz}|{z=0^-} + uC|{z=0^-} = uC_0$
  • $\frac{dC}{dz}|_{z=L} = 0$
• Closed-Closed Boundary Conditions:
  • $-D_a \frac{dC}{dz}|_{z=0} = 0$
  • $-D_a \frac{dC}{dz}|_{z=L} = 0$

Dispersion Number

• The Peclet number is: $Pe = \frac{uL}{D_a}$
• $Pe \rightarrow 0$ indicates a well-mixed reactor.
• $Pe \rightarrow \infty$ indicates plug flow.

Recycle Reactor

• The recycle ratio, $R$, is the ratio of the volume of fluid recycled to the volume leaving the system.

Heterogeneous Catalysis

• Steps in heterogeneous catalysis:
  1. Mass Transfer (Bulk $\rightarrow$ Surface)
  2. Adsorption
  3. Surface Reaction
  4. Desorption
  5. Mass Transfer (Surface $\rightarrow$ Bulk)

Adsorption Isotherms

• The fraction of surface covered is: $\theta = \frac{V}{V_m}$
  • $V$ is the volume adsorbed
  • $V_m$ is the volume adsorbed in a monolayer

Langmuir Isotherm

• For the reaction $A(g) + S \rightleftharpoons A \cdot S$, the rates of adsorption and desorption are:
  • $r_{ads} = k_a P_A (1 - \theta_A)$
  • $r_{des} = k_d \theta_A$
• At equilibrium, $r_{ads} = r_{des}$, so $k_a P_A (1 - \theta_A) = k_d \theta_A$
• The fraction of surface covered by A is: $\theta_A = \frac{K_A P_A}{1 + K_A P_A}$, where $K_A = \frac{k_a}{k_d}$

Competitive Langmuir Isotherm

• For the reactions $A(g) + S \rightleftharpoons A \cdot S$ and $B(g) + S \rightleftharpoons B \cdot S$:
  • $\theta_A = \frac{K_A P_A}{1 + K_A P_A + K_B P_B}$
  • $\theta_B = \frac{K_B P_B}{1 + K_A P_A + K_B P_B}$
  • $1 = \theta_V + \theta_A + \theta_B$, where $\theta_V$ is the fraction of vacant sites.

Steps for Determining Rate Law

• The steps for determining a rate law:
  1. Adsorption
  2. Surface Reaction
  3. Desorption
• One of these steps is usually rate limiting.
• Derive the rate law in terms of $\theta_i$ and eliminate $\theta_i$ using adsorption isotherms.