Residence Time Distribution (RTD) for CSTR and PFR - Python Code Explanation

 

Explanation of the Code: Residence Time Distribution (RTD) for CSTR and PFR

This Python script models the Residence Time Distribution (RTD) for two types of chemical reactors: Continuous Stirred Tank Reactor (CSTR) and Plug Flow Reactor (PFR). It visualizes the response of these reactors to a pulse tracer input, helping to understand their behavior over time.

---

Key Concepts and Formulas

1. Residence Time Distribution (RTD):
   • RTD, $E(t)$, describes how long fluid elements spend in a reactor.
   • The mean residence time, $\tau$, is the average time the material stays in the reactor.

---

RTD for CSTR

Formula:

$$E(t) = \frac{1}{\tau} e^{-\frac{t}{\tau}}$$

Explanation:

• The RTD of a CSTR follows an exponential decay.
• It assumes perfect mixing, meaning all fluid elements have an equal chance of leaving the reactor at any moment.
• $E(t)$ represents the probability density function of residence times.

Implementation in Code:

def rtd_cstr(t, tau):
    return (1 / tau) * np.exp(-t / tau)

---

RTD for PFR

Formula:

$$E(t) = \delta(t - \tau)$$

Where $\delta(t - \tau)$ is the Dirac delta function.

Explanation:

• For a PFR, all fluid elements spend exactly the same time, $\tau$, in the reactor.
• The Dirac delta function represents this instantaneous residence time.
• In numerical simulations, the delta function is approximated as a sharp peak.

Implementation in Code:

def rtd_pfr(t, tau):
    E = np.zeros_like(t)
    E[np.argmin(np.abs(t - tau))] = 1  # Approximation of delta function
    return E

---

Pulse Tracer Input

Concept:

• A pulse tracer is a sharp input signal at $t = 0$ used to study RTD.
• In the code, the pulse tracer is modeled as an array with a value of 1 at $t = 0$ and 0 elsewhere.

Implementation in Code:

def pulse_tracer(t):
    pulse = np.zeros_like(t)
    pulse[0] = 1  # Sharp input at t=0
    return pulse

---

Visualization

The script uses Matplotlib to create plots comparing the RTD and pulse tracer input for both CSTR and PFR.

Key Elements:

1. Subplots:

   • Two side-by-side plots for CSTR and PFR.
   • Pulse tracer input and RTD responses are plotted on the same axes.

2. Styling:

   • Different colors and line styles for clarity.
   • Labels, titles, and legends for better interpretation.

3. Time Range:

   • The time array spans from $0$ to $3\tau$, capturing significant parts of the RTD curve.

---

Code Walkthrough

1. Import Libraries:

   • math, numpy, and matplotlib for mathematical operations and plotting.

2. Define Parameters:

   • Mean residence time, $\tau = 1$.
   • Time array, t, created using np.linspace.

3. Define Functions:

   • rtd_cstr for CSTR RTD.
   • rtd_pfr for PFR RTD.
   • pulse_tracer for the pulse input.

4. Plotting:

   • Two subplots for CSTR and PFR.
   • RTD curves and pulse input are overlaid for comparison.

5. Display Results:

   • The plots are displayed using plt.show().

---

Output

1. CSTR Plot:

   • Exponential decay RTD.
   • Smooth curve indicating a range of residence times due to perfect mixing.

2. PFR Plot:

• A sharp peak at $t = \tau$.
• Indicates all fluid elements exit the reactor simultaneously.

---

Significance of RTD Analysis

1. Reactor Design:

   • RTD helps in understanding flow behavior, mixing, and reaction performance.

2. Comparison:

   • CSTR: Suitable for reactions needing long residence times due to mixing.
   • PFR: Ideal for plug flow with uniform reaction time.

3. Applications:

   • Chemical reaction engineering.
   • Troubleshooting flow maldistribution or bypassing in reactors.

By analyzing RTD, engineers can optimize reactor performance and ensure better process control.