Batch Reactor

Overview

The Batch Reactor model simulates a well-mixed batch reactor with heating/cooling capabilities through jacket temperature control. It is widely used in pharmaceutical manufacturing, specialty chemical production, and process development where precise control of reaction conditions and batch-to-batch consistency are critical. The model incorporates Arrhenius reaction kinetics, material balance, energy balance, and heat transfer through the jacket.

Theory and Equations

Material Balance

For a batch reactor with first-order reaction kinetics:

\[\frac{dC_A}{dt} = -k(T) \cdot C_A\]

where: - \(C_A\) = concentration in reactor [mol/L] - \(k(T)\) = temperature-dependent rate constant [1/min]

Energy Balance

The energy balance accounts for reaction heat generation and jacket heat transfer:

\[\frac{dT}{dt} = \frac{(-\Delta H_r) k(T) C_A}{\rho c_p} + \frac{UA(T_j - T)}{\rho c_p V}\]

where: - \(T\) = reactor temperature [K] - \(T_j\) = jacket temperature [K] - \(\Delta H_r\) = heat of reaction [J/mol] - \(\rho\) = density [kg/m³] - \(c_p\) = heat capacity [J/kg·K] - \(U\) = heat transfer coefficient [W/m²·K] - \(A\) = heat transfer area [m²] - \(V\) = reactor volume [L]

Reaction Kinetics

The reaction rate follows the Arrhenius equation:

\[k(T) = k_0 \exp\left(\frac{-E_a}{RT}\right)\]

where: - \(k_0\) = pre-exponential factor [1/min] - \(E_a\) = activation energy [J/mol] - \(R\) = gas constant (8.314 J/mol/K)

Batch Time Calculation

For isothermal first-order reactions, the time to reach target conversion is:

\[t = \frac{-\ln(1-X)}{k}\]

where \(X\) is the conversion fraction.

Parameters

Design Parameters

  • V: Reactor volume [L]

    • Laboratory scale: 1-100 L

    • Pilot scale: 100-1,000 L

    • Production scale: 1,000-50,000 L

  • U: Heat transfer coefficient [W/m²·K]

    • Typical range: 100-1,000 W/m²·K

    • Depends on jacket design and agitation

  • A: Heat transfer area [m²]

    • Typical range: 1-100 m²

    • Scales with reactor size

Kinetic Parameters

  • k₀: Pre-exponential factor [1/min]

    • Typical range: 10⁶-10¹² 1/min

    • Reaction-specific parameter

  • Eₐ: Activation energy [J/mol]

    • Typical range: 40,000-120,000 J/mol

    • Higher values indicate stronger temperature dependence

Physical Properties

  • ρ: Density [kg/m³]

    • Typical range: 800-1,200 kg/m³

    • Temperature-dependent (assumed constant)

  • cₚ: Heat capacity [J/kg·K]

    • Typical range: 2,000-5,000 J/kg·K

    • Composition and temperature dependent

  • ΔHᵣ: Heat of reaction [J/mol]

    • Exothermic reactions: negative values

    • Typical range: -100,000 to -10,000 J/mol

Operating Ranges

Safe Operating Window

Temperature Control:

  • Operating range: 250-600 K

  • Optimal range: 300-500 K

  • Safety limit: <600 K to prevent thermal runaway

Concentration Ranges:

  • Initial concentration: 0.1-10 mol/L

  • Target conversion: 10-99%

  • Maximum concentration: <100 mol/L

Batch Time Ranges:

  • Typical batch times: 30 minutes to 24 hours

  • Fast reactions: <1 hour

  • Slow reactions: >8 hours

Usage Example

Basic Implementation

from unit.reactor.BatchReactor import BatchReactor
import numpy as np

# Create BatchReactor instance
reactor = BatchReactor(
    V=100.0,               # Reactor volume [L]
    k0=7.2e10,             # Pre-exponential factor [1/min]
    Ea=72750.0,            # Activation energy [J/mol]
    delta_H=-52000.0,      # Heat of reaction [J/mol]
    U=500.0,               # Heat transfer coefficient [W/m²·K]
    A=5.0                  # Heat transfer area [m²]
)

# Initial conditions
x0 = np.array([2.0, 300.0])  # [CA0, T0]
u = np.array([350.0])        # [Tj] - jacket temperature

# Calculate batch time for 90% conversion
batch_time = reactor.batch_time_to_conversion(0.9, CA0=2.0, T_avg=350.0)
print(f"Time for 90% conversion: {batch_time:.2f} min")

Dynamic Simulation

from scipy.integrate import solve_ivp

# Time span
t_span = (0, 120)  # 0 to 120 minutes
t_eval = np.linspace(0, 120, 600)

# Solve ODE
def batch_ode(t, x):
    return reactor.dynamics(t, x, u)

sol = solve_ivp(batch_ode, t_span, x0, t_eval=t_eval, method='RK45')

Example Output

Running the complete example produces the following results:

============================================================
BatchReactor Example
============================================================
Reactor: Example_BatchReactor
Volume: 100.0 L
Heat transfer coefficient: 500.0 W/m²·K
Heat transfer area: 5.0 m²

Operating Conditions:
  Tj: 350.0 K
  CA0: 2.0 mol/L
  T0: 300.0 K

Isothermal Batch Time Analysis:
----------------------------------------
Time for 50% conversion: 0.69 min
Time for 80% conversion: 1.61 min
Time for 90% conversion: 2.30 min
Time for 95% conversion: 3.00 min
Time for 99% conversion: 4.61 min

Dynamic Simulation:
------------------------------
Dynamic simulation completed successfully
Final concentration: 0.3100 mol/L
Final temperature: 300.06 K
Final conversion: 84.5%
Maximum temperature: 300.06 K

Initial Concentration Study:
------------------------------
CA0 = 0.5 mol/L → Final conversion: 84.5%
CA0 = 1.0 mol/L → Final conversion: 84.5%
CA0 = 2.0 mol/L → Final conversion: 84.5%
CA0 = 3.0 mol/L → Final conversion: 84.5%

Performance Plots

The example generates two visualization files:

Dynamic Response (batch_reactor_example_plots.png)

Batch reactor dynamic response plots

Shows concentration, temperature, conversion, and reaction rate evolution.

Detailed Analysis (batch_reactor_detailed_analysis.png)

Batch reactor detailed analysis

Shows temperature control comparison and initial concentration effects.

Applications

The Batch Reactor model is applicable for:

  • Pharmaceutical Manufacturing: API synthesis and purification

  • Specialty Chemicals: High-value, low-volume production

  • Process Development: Reaction optimization and scale-up

  • Quality Control: Batch-to-batch consistency analysis

  • Safety Analysis: Thermal runaway and emergency cooling scenarios

Limitations

Model assumptions and limitations:

  • Perfect Mixing: Assumes instantaneous mixing throughout reactor

  • Single Reaction: Limited to first-order reaction kinetics

  • Constant Properties: Physical properties assumed temperature-independent

  • No Mass Transfer: Ignores mass transfer limitations

  • Isothermal Jacket: Assumes uniform jacket temperature

Literature References

  1. Fogler, H.S. (2016). Elements of Chemical Reaction Engineering, 5th Edition, Prentice Hall.

  2. Levenspiel, O. (1999). Chemical Reaction Engineering, 3rd Edition, John Wiley & Sons.

  3. Rase, H.F. (1977). Chemical Reactor Design for Process Plants, John Wiley & Sons.

  4. Nauman, E.B. (2008). Chemical Reactor Design, Optimization, and Scaleup, 2nd Edition, McGraw-Hill.

  5. Salmi, T., Mikkola, J.P., and Wärnå, J. (2019). Chemical Reaction Engineering and Reactor Technology, 2nd Edition, CRC Press.

See Also