Quantification of Auto Ignition Time and Ignition Delay for Methane Combustion Under Varying Initial Conditions of Pressure and Temperature

INTRODUCTION

• Auto-ignition occurs when a mixture of gases or vapors ignites spontaneously in absence of any external ignition source such as a flame or a spark.
• The temperature at which auto-ignition occurs is called the auto-ignition temperature.
• It is the lowest temperature in a system where the rate of heat evolved from the gases or vapors increases beyond the rate of heat loss to the surroundings, resulting in the ignition.
• The auto-ignition temperature is not an intrinsic property of the gases or vapors and is affected by the pressure, vessel shape and volume, surface activity, contaminants, flow rate, reaction rate, gravity, reactant concentration, etc.

BACKGROUND OF CANTERA REACTOR NETWORK

• A Cantera reactor network represents the simplest form of a chemically reacting system.
• It corresponds to an extensive thermodynamic control volume, in which all state variables are homogeneously distributed.
• Transient state changes due to chemical reactions are possible, however, thermodynamic (and not chemical) equilibrium is assumed to be present throughout the reactor at all instants of time.
• The ReactorNet() object is used to model the reactor class.
• The advance(time) function is used to advance the reactor to the specified time.
• This thermodynamic state of the reactor can then be extracted at each time-step to study the transient behavior of the reactor.
• Even for a single reaction, a reactor network must be created in Cantera to visualize the transient behavior.
• For this problem, we are defining a reactor network for an ideal gas reactor whose state variables are:

1. m [Kg]: Reactor’s Content Mass
2. V [m³]: Volume of the Reactor
3. T [K]: Temperature to describe the energy of the system
4. Y_k: Mass Fraction of Each Species

The following conservation laws are applicable for the reactor:

1. Conservation of Mass
2. Conservation of Species
3. Conservation of Energy

CORROSPONDING GOVERNING EQUATIONS FOR THE REACTOR AS DERRIVED FROM THE CONSERVATION LAWS

1. `(dm)/dt=sum_(i n)dotm_(i n) - sum_(out)dotm_(out) + dotm_(wall)`

2. `(d(mY_k))/dt = sum_(i n)dotm_(i n)Y_(k, i n ) - sum_(out)dotm_(out)Y_k + dotm_(k, g en)`

3. `(dU)/dt = u(dm)/dt + mC_v(dT)/dt + msum_(k)u_(k)(dY_k)/dt`

STEPS TO BE FOLLOWED:

1. Define the initial temperature and pressure condition and the mass fractions of each species (CH₄, O₂ and N₂).
2. Define the solution time (10s for this case).
3. Initialize the Solution object using the initial conditions.
4. Define the type of reactor (Ideal gas reactor for this case).
5. Create the reactor network from the single reactor.
6. Initialize the thermodynamic state array of the reactor for output.
7. Advance the reactor network through the desired time and adding the thermodynamic state at each time step to the output.
8. Post-process the output data.

The auto-ignition temperature for this case is defined as:

`Time_(ignition de lay) = Time_(i nitial) + 400 K`

The time at which this temperature is reached is called the ignition delay.

Part I

The auto-ignition of methane is studied under different initial conditions:

• A constant temperature of 1250 K and varying pressure from 1 to 5 atm.
• A constant pressure of 5 atm and varying temperature from 950 to 1450 K.

Part II

The rate of change of mole fractions of and is studied for cases:

• Temperature of 1000 K and Pressure of 5 bar.
• Temperature of 500 K and Pressure of 5 bar.

A function is created to output the state of the reactor network for the specified initial conditions and time. This function is called through the main code which makes it easier to implement it for all the different cases.

PYTHON CODE

\"\"\"
Program to study the auto-ignition of methane under various conditions.
Program by: Shouvik Bandopadhyay
\"\"\"

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

# Defining the function to output the thermodynamic states at each time-step
def auto_ignition(T, P, species_dict, t):
	gas = ct.Solution(\'gri30.xml\')
	gas.TPX = T, P, species_dict
	r = ct.IdealGasReactor(gas)
	sim = ct.ReactorNet([r])
	states = ct.SolutionArray(gas, extra=[\'time_in_ms\'])
	for tp in t:
		sim.advance(tp)
		states.append(r.thermo.state, time_in_ms = tp*1e3)
	return states

# Defining the various initial conditions
T = np.linspace(950,1450,11)
P = 101325*np.linspace(1,5,9)
species_dict = {\'CH4\':0.095, \'O2\':0.19, \'N2\':0.715}
t = np.linspace(0,10,10001)

# Initializing the output parameters
T_ai_T = [[0 for time in t] for temp in T]
T_ai_P = [[0 for time in t] for pres in P]
T_mole = [1000, 500]
T_comp = [[0 for time in t] for i in range(0,2)]
time_ai_T = []
time_ai_P = []
mole_frac1 = [[0 for time in t] for i in range(0,3)]
mole_frac2 = [[0 for time in t] for i in range(0,3)]

# Auto-ignition for constant temperature (1250 K)
for i in range(0,len(P)):
	states = auto_ignition(T[6], P[i], species_dict, t)
	T_ai_P[i] = states.T
	for j in range(0,len(states.T)):
		if states.T[j] >= states.T[0] + 400:
			time_ai_P.append(states.time_in_ms[j])
			break

# Auto-ignition for constant pressure (5 atm)
for i in range(0,len(T)):
	states = auto_ignition(T[i], P[-1], species_dict, t)
	T_ai_T[i] = states.T
	for j in range(0,len(states.T)):
		if states.T[j] >= states.T[0] + 400:
			time_ai_T.append(states.time_in_ms[j])
			break

# Mole Fractions for temperature of 1000 K
states = auto_ignition(T_mole[0], 5*100000, species_dict, t)
mole_frac1[0] = states.X[:, states.species_index(\'H2O\')]
mole_frac1[1] = states.X[:, states.species_index(\'O2\')]
mole_frac1[2] = states.X[:, states.species_index(\'OH\')]
T_comp[0] = states.T

# Mole Fractions for temperature of 500 K
states = auto_ignition(T_mole[1], 5*100000, species_dict, t)
mole_frac2[0] = states.X[:, states.species_index(\'H2O\')]
mole_frac2[1] = states.X[:, states.species_index(\'O2\')]
mole_frac2[2] = states.X[:, states.species_index(\'OH\')]
T_comp[1] = states.T

# Plotting all the results
fig1, ax1 = plt.subplots()
for i in range(0,len(P)):
	plt.plot(t, T_ai_P[i], linewidth=2, label=str(P[i]/101325)+\' atm\')
plt.xlabel(\'Time [sec]\', fontsize=14)
plt.ylabel(\'Temperature [K]\', fontsize=14)
plt.title(\'Reaction Temperature with varying Pressure\', fontsize=14, fontweight=\'bold\')
plt.grid(\'both\', linestyle=\'--\', linewidth=0.8)
plt.legend(fontsize=12)

fig2, ax2 = plt.subplots()
plt.plot(P/101325, time_ai_P, \'o-\', linewidth=2, markersize=10)
plt.xlabel(\'Pressure [atm]\', fontsize=14)
plt.ylabel(\'Ignition Delay [ms]\', fontsize=14)
plt.title(\'Ignition Delay with varying Pressure\', fontsize=14, fontweight=\'bold\')
plt.grid(\'both\', linestyle=\'--\', linewidth=0.8)

fig3, ax3 = plt.subplots()
for i in range(0,len(T)):
	plt.plot(t, T_ai_T[i], linewidth=2, label=str(int(T[i]))+\' K\')
plt.xlabel(\'Time [sec]\', fontsize=14)
plt.ylabel(\'Initial Temperature [K]\', fontsize=14)
plt.title(\'Reaction Temperature with varying Initial Temperature\', fontsize=14, fontweight=\'bold\')
plt.grid(\'both\', linestyle=\'--\', linewidth=0.8)
plt.legend(fontsize=12)

fig4, ax4 = plt.subplots()
plt.plot(T, time_ai_T, \'o-\', linewidth=2, markersize=10)
plt.xlabel(\'Initial Temperature [K]\', fontsize=14)
plt.ylabel(\'Ignition Delay [ms]\', fontsize=14)
plt.title(\'Ignition Delay with varying Initial Temperature\', fontsize=14, fontweight=\'bold\')
plt.grid(\'both\', linestyle=\'--\', linewidth=0.8)

fig5, ax5 = plt.subplots(3)
ax5[0].plot(t, mole_frac1[0], linewidth=2)
plt.suptitle(\'Mole Fractions for Reaction at 1000 K and 5 bar\', fontsize=14, fontweight=\'bold\')
ax5[0].grid(\'both\', linestyle=\'--\', linewidth=0.8)
ax5[0].set_xticklabels([])
ax5[1].plot(t, mole_frac1[1], linewidth=2)
ax5[1].grid(\'both\', linestyle=\'--\', linewidth=0.8)
ax5[1].set_xticklabels([])
ax5[2].plot(t, mole_frac1[2], linewidth=2)
ax5[0].set_ylabel(\'$X_{H_2O}$\', fontsize=14)
ax5[1].set_ylabel(\'$X_{O_2}$\', fontsize=14)
ax5[2].set_ylabel(\'$X_{OH}$\', fontsize=14)
ax5[2].set_xlabel(\'Time [sec]\', fontsize=14)
ax5[2].grid(\'both\', linestyle=\'--\', linewidth=0.8)

fig6, ax6 = plt.subplots(3)
plt.suptitle(\'Mole Fractions for Reaction at 500 K and 5 bar\', fontsize=14, fontweight=\'bold\')
ax6[0].plot(t, mole_frac2[0], linewidth=2)
ax6[0].grid(\'both\', linestyle=\'--\', linewidth=0.8)
ax6[0].set_xticklabels([])
ax6[1].plot(t, mole_frac2[1], linewidth=2)
ax6[1].grid(\'both\', linestyle=\'--\', linewidth=0.8)
ax6[1].set_xticklabels([])
ax6[2].plot(t, mole_frac2[2], linewidth=2)
ax6[0].set_ylabel(\'$X_{H_2O}$\', fontsize=14)
ax6[1].set_ylabel(\'$X_{O_2}$\', fontsize=14)
ax6[2].set_ylabel(\'$X_{OH}$\', fontsize=14)
ax6[2].set_xlabel(\'Time [sec]\', fontsize=14)
ax6[2].grid(\'both\', linestyle=\'--\', linewidth=0.8)

fig7, ax7 = plt.subplots()
for i in range(0,2):
	plt.plot(t, T_comp[i], linewidth=2, label=str(T_mole[i])+\' K\')
plt.xlabel(\'Time [sec]\', fontsize=14)
plt.ylabel(\'Temperature [K]\', fontsize=14)
plt.title(\'Reaction Temperature with varying Initial Temperature\', fontsize=14, fontweight=\'bold\')
plt.grid(\'both\', linestyle=\'--\', linewidth=0.8)
plt.legend(fontsize=12)

plt.show()

RESULTS FOR PART I

PART I CONCLUSIONS

• The ignition delay asymptitotically reduces to 0 ms as initial temperature or pressure increases.
• As observed from the above plots, the final temperature is also dependent on the changes made in the initial conditions. 
• Rate of change of ignition delay is higher with varying temperatures when compared to varying pressures.

RESULTS FOR PART II

PART II CONCLUSIONS

FOR 1000 K

Auto ignition occurs with some amount of ignition delay which increases the mole fraction of the products namely wayer and hydroxyl and at the same time decreases the mole fraction of reactant namely oxygen.

Spike for the change in mole fraction is consistent for all 3 species which indicates a spontaneous ignition of methane.

FOR 500 K

Mole fraction of the products is nearly 0 compared to the 1000 K case while the reactant oxygen\'s mole fraction remains the same as 0.19 implying that no reaction has taken place.

500 K is too low temperature to ignite methane without an external spark or flame as demonstrated in the temperature VS time curve above.