SciPy Central

# This code is an extension of http://scpyce.org/12/

import numpy as np
from scipy import integrate
from matplotlib.pylab import *

def tank(t, y):
    """
    Dynamic balance for a CSTR

    C_A = y[0] = the concentration of A in the tank, [mol/L]
    T   = y[1] = the tank temperature, [K]

    Returns dy/dt = [F/V*(C_{A,in} - C_A) - k*C_A^2       ]
                    [F/V*(T_in - T) - k*C_A^2*HR/(rho*Cp) ]
    """
    F = 20.1     # L/min
    CA_in = 2.5  # mol/L
    V = 100.0    # L
    k0 = 0.15    # L/(mol.min)
    Ea = 5000    # J/mol
    R = 8.314    # J/(mol.K)
    Hr = -590    # J/mol
    T_in = 288   # K
    rho = 1.050  # kg/L

    # Assign some variables for convenience of notation
    CA = y[0]
    T  = y[1]

    # Algebraic equations
    k = k0 * np.exp(-Ea/(R*T))  # L/(mol.min)
    Cp = 4.184 - 0.002*(T-273)  # J/(kg.K)

    # Output from ODE function must be a COLUMN vector, with n rows
    n = len(y)      # 2: implies we have two ODEs
    dydt = np.zeros((n,1))
    dydt[0] = F/V*(CA_in - CA) - k*CA**2
    dydt[1] = F/V*(T_in - T) - (Hr*k*CA**2)/(rho*Cp)
    return dydt

# The "driver" that will integrate the ODE(s):
# -----------

# Start by specifying the integrator:
# use ``vode`` with "backward differentiation formula"
r = integrate.ode(tank).set_integrator('vode', method='bdf')

# Set the time range
t_start = 0.0
t_final = 45.0
delta_t = 0.1
# Number of time steps: 1 extra for initial condition
num_steps = np.floor((t_final - t_start)/delta_t) + 1

# Set initial condition(s): for integrating variable and time!
CA_t_zero = 0.5
T_t_zero = 295.0
r.set_initial_value([CA_t_zero, T_t_zero], t_start)

# Additional Python step: create vectors to store trajectories
t = np.zeros((num_steps, 1))
CA = np.zeros((num_steps, 1))
temp = np.zeros((num_steps, 1))
t[0] = t_start
CA[0] = CA_t_zero
temp[0] = T_t_zero

# Integrate the ODE(s) across each delta_t timestep
k = 1
while r.successful() and k < num_steps:
    r.integrate(r.t + delta_t)

    # Store the results to plot later
    t[k] = r.t
    CA[k] = r.y[0]
    temp[k] = r.y[1]
    k += 1

# All done!  Plot the trajectories in two separate plots:
fig = figure()
ax1 = subplot(211)
ax1.plot(t, CA)
ax1.set_xlim(t_start, t_final)
ax1.set_xlabel('Time [minutes]')
ax1.set_ylabel('Concentration [mol/L]')
ax1.grid('on')

ax2 = plt.subplot(212)
ax2.plot(t, temp, 'r')
ax2.set_xlim(t_start, t_final)
ax2.set_xlabel('Time [minutes]')
ax2.set_ylabel('Temperaturere [K]')
ax2.grid('on')

fig.savefig('coupled-ode-plot.png')