Nonideal Flow and RTD in Chemical Reactors

• Lecture notes cover nonideal flow in chemical reactors.
• Real reactors often deviate from ideal conditions of perfect mixing or plug flow.
• Residence Time Distribution (RTD) describes flow patterns inside a reactor.

Residence Time Distribution (RTD)

• A tracer is injected to measure its concentration in the effluent stream.
• Types of RTD input include pulse, step, and arbitrary.

RTD in Ideal Reactors

• Formulas for RTD are examined in Continuos Stirred Tank Reactors (CSTRs) and Plug Flow Reactors (PFRs).

CSTR

• The equation is: $V \frac{dC(t)}{dt} = vC_0(t) - vC(t)$.
• For pulse input, $C_0(t) = C_0 \delta (t)$.
• Solving with initial condition $C(0^-) = 0$ and resulting gives $C(0^+) = C_0 v/V$.
• The concentration is $C(t) = C(0^+) e^{-t/\tau} = C_0 \frac{v}{V} e^{-t/\tau}$.
• $E(t) = \frac{1}{\tau} e^{-t/\tau}$ and $\int_0^\infty E(t) dt =1$

PFR

• The RTD is described by $E(t) = \delta (t - \tau)$.
• $\int_0^\infty E(t) dt = 1$

Characteristics of the RTD

Measurement of the RTD

• Pulse and Step Inputs are examined.

Pulse Input

• $E(t) = \frac{C(t)}{\int_0^\infty C(t) dt}$.
• The injected tracer mass calculation is $M = \int_0^\infty v C(t) dt$.
• $E(t) = \frac{v C(t)}{M}$

Step Input

• $F(t) = \int_0^t E(t) dt$.
• $F(t) = \frac{C(t)}{C_0}$

Relationship between $E(t)$ and $F(t)$

• $E(t) = \frac{dF(t)}{dt}$.

Properties of the RTD

• These notes cover normalization, mean residence time, variance, and dimensionless RTD.

Normalization

• $\int_0^\infty E(t) dt = 1$ and $\int_0^\infty F(t) dt = \infty$.

Mean Residence Time

• $t_m = \int_0^\infty t E(t) dt$ and $t_m = \frac{\sum t_i C_i \Delta t_i}{\sum C_i \Delta t_i}$.

Variance

• $\sigma^2 = \int_0^\infty (t - t_m)^2 E(t) dt$ and $\sigma^2 = \int_0^\infty t^2 E(t) dt - t_m^2$.

Dimensionless RTD

• $\Theta = \frac{t}{t_m}$, $E_\theta (\Theta) = t_m E(t)$, and $\int_0^\infty E_\theta (\Theta) d\Theta = 1$.
• $\sigma_\theta ^2 = \frac{\sigma^2}{t_m ^2}$.

RTD and Chemical Reactions

• RTD reveals molecular time spent in a reactor but not mixing details.

Models for Nonideal Reactors

• Three models are examined for predictable behavior.
• Those models include segregation, tanks-in-series, and dispersion.