Figure 6.13:

Eigenvalues of Jacobian matrix vs. reactor temperature.

Code for Figure 6.13

Text of the GNU GPL.

main.py

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# Converted from st_st_eig.m - Steady state with eigenvalues (single DeltaH=-3e5)
import numpy as np
from scipy.optimize import fsolve
from misc import save_ascii

p = dict(
    k_m      = 0.001,
    T_m      = 298.,
    E        = 8000.,
    c_Af     = 2.,
    C_p      = 4.,
    rho      = 1000.,
    T_f      = 298.,
    T_a      = 298.,
    U        = 0.,
    DeltaH_R = -3e5,
)
p['C_ps'] = p['rho'] * p['C_p']

def st_st_cA(x, c_A, p):
    theta = x[0]; T = x[1]
    k     = p['k_m'] * np.exp(-p['E'] * (1./T - 1./p['T_m']))
    return [p['c_Af'] - (1 + k*theta)*c_A,
            p['U']*theta*(p['T_a']-T) + p['C_ps']*(p['T_f']-T) - k*theta*c_A*p['DeltaH_R']]

x0     = [1., p['T_f']]
nc_As  = 250
c_Avect = np.linspace(0.999*p['c_Af'], 0.003*p['c_Af'], nc_As)
tmp_table = []

for c_A in c_Avect:
    X, info_d, ier, msg = fsolve(lambda x: st_st_cA(x, c_A, p), x0, full_output=True)
    theta = X[0]; T = X[1]
    conv  = (p['c_Af'] - c_A) / p['c_Af']
    k     = p['k_m'] * np.exp(-p['E'] * (1./T - 1./p['T_m']))
    Jac   = np.array([[-1./theta - k,  -k*c_A*p['E']/(T*T)],
                      [-k*p['DeltaH_R']/p['C_ps'],
                       -p['U']/p['C_ps'] - 1./theta - k*c_A*p['DeltaH_R']/p['C_ps']*p['E']/(T*T)]])
    lam   = np.linalg.eigvals(Jac)
    a = lam[0].real; b = lam[0].imag
    c = lam[1].real; d = lam[1].imag
    if a >= c:
        lamrow = [a, b, c, d]
    else:
        lamrow = [c, d, a, b]
    tmp_table.append([theta, T, conv] + lamrow + [float(ier)])
    x0 = X

table = np.array(tmp_table)
save_ascii('st_st_eig.dat', table)