Blowers-Masel reaction rates

A simple example to demonstrate the difference between Blowers-Masel reaction and elementary reaction.

The first two reactions have the same reaction equations with Arrhenius and Blowers-Masel rate parameters, respectively. The Blowers-Masel parameters are the same as the Arrhenius parameters with an additional value, bond energy.

First we show that the forward rate constants of the first 2 different reactions are different because of the different rate expression, then we print the forward rate constants for reaction 2 and reaction 3 to show that even two reactions that have the same Blowers-Masel parameters can have different forward rate constants.

The first plot generated shows how the rate constant changes with respect to temperature for elementary and Blower-Masel reactions. The second plot shows the activation energy change of a Blowers-Masel reaction with respect to the delta enthalpy of the reaction.

Requires: cantera >= 2.6.0, matplotlib >= 2.0

Tags: Python kinetics sensitivity analysis

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The first and second reactions have same reaction equation, but they have different reaction types, so the forward rate constant of the first reaction is 5195.447 kmol/(m^3.s), the forward rate constant of the second reaction is 946.051 kmol/(m^3.s).
The rate parameters of second and the third reactions are same, but the forward rate constant of second reaction is 946.051 kmol/(m^3.s), the forward rate constant of the third reaction is 2533.800 kmol/(m^3.s).

import cantera as ct
import numpy as np
import matplotlib.pyplot as plt

#Create an elementary reaction O+H2<=>H+OH
r1 = ct.Reaction({"O":1, "H2":1}, {"H":1, "OH":1},
                 ct.ArrheniusRate(3.87e1, 2.7, 6260*1000*4.184))

#Create a Blowers-Masel reaction O+H2<=>H+OH
r2 = ct.Reaction({"O":1, "H2":1}, {"H":1, "OH":1},
                 ct.BlowersMaselRate(3.87e1, 2.7, 6260*1000*4.184, 1e9))

#Create a Blowers-Masel reaction with same parameters with r2
#reaction equation is H+CH4<=>CH3+H2
r3 = ct.Reaction({"H":1, "CH4":1}, {"CH3":1, "H2":1},
                 ct.BlowersMaselRate(3.87e1, 2.7, 6260*1000*4.184, 1e9))

gas = ct.Solution(thermo='ideal-gas', kinetics='gas',
                  species=ct.Solution('gri30.yaml').species(), reactions=[r1, r2, r3])

gas.TP = 300, ct.one_atm

r1_rc = gas.forward_rate_constants[0]
r2_rc = gas.forward_rate_constants[1]
r3_rc = gas.forward_rate_constants[2]

print("The first and second reactions have same reaction equation,"
      " but they have different reaction types, so the forward rate"
      " constant of the first reaction is {0:.3f} kmol/(m^3.s),"
      " the forward rate constant of the second reaction is {1:.3f} kmol/(m^3.s).".format(r1_rc, r2_rc))

print("The rate parameters of second and the third reactions are same,"
      " but the forward rate constant of second reaction is {0:.3f} kmol/(m^3.s),"
      " the forward rate constant of the third reaction is"
      " {1:.3f} kmol/(m^3.s).".format(r2_rc, r3_rc))

# Comparing the reaction forward rate constant change of
# Blowers-Masel reaction and elementary reaction with
# respect to the temperature.
r1_kf = []
r2_kf = []
T_range = np.arange(300, 3500, 100)
for temp in T_range:
    gas.TP = temp, ct.one_atm
    r1_kf.append(gas.forward_rate_constants[0])
    r2_kf.append(gas.forward_rate_constants[1])
plt.plot(T_range, r1_kf, label='Reaction 1 (Elementary)')
plt.plot(T_range, r2_kf, label='Reaction 2 (Blowers-Masel)')
plt.xlabel("Temperature(K)")
plt.ylabel(r"Forward Rate Constant (kmol/(m$^3\cdot$ s))")
plt.title("Comparison of $k_f$ vs. Temperature For Reaction 1 and 2",y=1.1)
plt.legend()
plt.savefig("kf_to_T.png")

# This is the function to change the enthalpy of a species
# so that the enthalpy change of reactions involving this species can be changed
def change_species_enthalpy(gas, species_name, dH):
    """
    Find the species by name and change it's enthalpy by dH (in J/kmol)
    """
    index = gas.species_index(species_name)

    species = gas.species(index)
    # dx is in fact (delta H / R). Note that R in cantera is 8314.462 J/kmol/K
    dx = dH / ct.gas_constant
    perturbed_coeffs = species.thermo.coeffs.copy()
    perturbed_coeffs[6] += dx
    perturbed_coeffs[13] += dx

    species.thermo = ct.NasaPoly2(species.thermo.min_temp, species.thermo.max_temp,
                                  species.thermo.reference_pressure, perturbed_coeffs)

    gas.modify_species(index, species)
    return gas.delta_enthalpy[1]

# Plot the activation energy change of reaction 2 with respect to the
# enthalpy change
E0 = gas.reaction(1).rate.activation_energy
upper_limit_enthalpy = 5 * E0
lower_limit_enthalpy = -5 * E0

Ea_list = []
deltaH_list = np.linspace(lower_limit_enthalpy, upper_limit_enthalpy, 100)
for deltaH in deltaH_list:
    delta_enthalpy = change_species_enthalpy(gas, "H", deltaH - gas.delta_enthalpy[1])
    gas.reaction(1).rate.delta_enthalpy = delta_enthalpy
    Ea_list.append(gas.reaction(1).rate.activation_energy)

plt.figure()
plt.plot(deltaH_list, Ea_list)
plt.xlabel("Enthalpy Change (J/kmol)")
plt.ylabel("Activation Energy Change (J/kmol)")
plt.title(r"$E_a$ vs. $\Delta H$ For O+H2<=>H+OH", y=1.1)
plt.savefig("Ea_to_H.png")
plt.show()