NonIdealShockTube.py (Source)

# coding: utf-8
"""
Ignition delay time computations in a high-pressure reflected shock tube
reactor, comparing ideal gas and Redlich-Kwong real gas models.

In this example we illustrate how to setup and use a constant volume,
adiabatic reactor to simulate reflected shock tube experiments. This reactor
will then be used to compute the ignition delay of a gas at a specified
initial temperature and pressure. The example is written in a general way,
that is, no particular EoS is presumed and ideal and real gas EoS can be used
equally easily.

The reactor (system) is simply an 'insulated box,' and can technically be used
for any number of equations of state and constant-volume, adiabatic reactors.

Other than the typical Cantera dependencies, plotting functions require that
you have matplotlib (https://matplotlib.org/) installed.

Requires: cantera >= 2.5.0, matplotlib >= 2.0
Keywords: combustion, reactor network, non-ideal fluid, ignition delay, plotting
"""

# Dependencies: numpy, and matplotlib
import numpy as np
import matplotlib.pyplot as plt

import time

import cantera as ct
print('Running Cantera version: ' + ct.__version__)

# Define the ignition delay time (IDT). This function computes the ignition
# delay from the occurrence of the peak concentration for the specified
# species.
def ignitionDelay(states, species):
i_ign = states(species).Y.argmax()
return states.t[i_ign]

# Define the reactor temperature and pressure:
reactorTemperature = 1000  # Kelvin
reactorPressure = 40.0*101325.0  # Pascals

# Define the gas: In this example we will choose a stoichiometric mixture of
# n-dodecane and air as the gas. For a representative kinetic model, we use:
#
# H.Wang, Y.Ra,  M.Jia, R.Reitz, Development of a reduced n-dodecane-PAH
# mechanism. and its application for n-dodecane soot predictions., Fuel 136
# (2014) 25–36. doi:10.1016/j.fuel.2014.07.028

# R-K constants are calculated according to their critical temperature (Tc) and
# pressure (Pc):
#
#     a = 0.4275*(R^2)*(Tc^2.5)/(Pc)
#
# and
#
#     b = 0.08664*R*Tc/Pc
#
# where R is the gas constant.
#
# For stable species, the critical properties are readily available. For
# radicals and other short-lived intermediates, the Joback method is used to
# estimate critical properties. For details of the method, see: Joback and Reid,
# "Estimation of pure- component properties from group-contributions," Chem.
# Eng. Comm. 57 (1987) 233-243, doi: 10.1080/00986448708960487

# Real gas IDT calculation

# Load the real gas mechanism:
real_gas = ct.Solution('nDodecane_Reitz.yaml', 'nDodecane_RK')

# Set the state of the gas object:
real_gas.TP = reactorTemperature, reactorPressure

# Define the fuel, oxidizer and set the stoichiometry:
real_gas.set_equivalence_ratio(phi=1.0, fuel='c12h26',
oxidizer={'o2': 1.0, 'n2': 3.76})

# Create a reactor object and add it to a reactor network
# In this example, this will be the only reactor in the network
r = ct.Reactor(contents=real_gas)
reactorNetwork = ct.ReactorNet([r])
timeHistory_RG = ct.SolutionArray(real_gas, extra=['t'])
# Tic
t0 = time.time()

# This is a starting estimate. If you do not get an ignition within this time,
# increase it
estimatedIgnitionDelayTime = 0.005
t = 0

counter = 1
while t < estimatedIgnitionDelayTime:
t = reactorNetwork.step()
if counter % 20 == 0:
# We will save only every 20th value. Otherwise, this takes too long
# Note that the species concentrations are mass fractions
timeHistory_RG.append(r.thermo.state, t=t)
counter += 1

# We will use the 'oh' species to compute the ignition delay
tau_RG = ignitionDelay(timeHistory_RG, 'oh')

# Toc
t1 = time.time()
print("Computed Real Gas Ignition Delay: {:.3e} seconds. "
"Took {:3.2f}s to compute".format(tau_RG, t1-t0))

# Ideal gas IDT calculation
# Create the ideal gas object:
ideal_gas = ct.Solution('nDodecane_Reitz.yaml', 'nDodecane_IG')

# Set the state of the gas object:
ideal_gas.TP = reactorTemperature, reactorPressure

# Define the fuel, oxidizer and set the stoichiometry:
ideal_gas.set_equivalence_ratio(phi=1.0, fuel='c12h26',
oxidizer={'o2': 1.0, 'n2': 3.76})

r = ct.Reactor(contents=ideal_gas)
reactorNetwork = ct.ReactorNet([r])
timeHistory_IG = ct.SolutionArray(ideal_gas, extra=['t'])

# Tic
t0 = time.time()

t = 0

counter = 1
while t < estimatedIgnitionDelayTime:
t = reactorNetwork.step()
if counter % 20 == 0:
# We will save only every 20th value. Otherwise, this takes too long
# Note that the species concentrations are mass fractions
timeHistory_IG.append(r.thermo.state, t=t)
counter += 1

# We will use the 'oh' species to compute the ignition delay
tau_IG = ignitionDelay(timeHistory_IG, 'oh')

# Toc
t1 = time.time()

print("Computed Ideal Gas Ignition Delay: {:.3e} seconds. "
"Took {:3.2f}s to compute".format(tau_IG, t1-t0))
print('Ideal gas error: {:2.2f} %'.format(100*(tau_IG-tau_RG)/tau_RG))

# Plot the result

plt.rcParams['xtick.labelsize'] = 12
plt.rcParams['ytick.labelsize'] = 12
plt.rcParams['figure.autolayout'] = True
plt.rcParams['axes.labelsize'] = 14
plt.rcParams['font.family'] = 'serif'

# Figure illustrating the definition of ignition delay time (IDT).

plt.figure()
plt.plot(timeHistory_RG.t, timeHistory_RG('oh').Y, '-o', color='b', markersize=4)
plt.plot(timeHistory_IG.t, timeHistory_IG('oh').Y, '-o', color='r', markersize=4)
plt.xlabel('Time (s)')
plt.ylabel(r'OH mass fraction, $\mathdefault{Y_{OH}}$')

# Figure formatting:
plt.xlim([0, 0.00055])
ax = plt.gca()
ax.annotate("", xy=(tau_RG, 0.005), xytext=(0, 0.005),
arrowprops=dict(arrowstyle="<|-|>", color='k', linewidth=2.0),
fontsize=14)
plt.annotate('Ignition Delay Time (IDT)', xy=(0, 0), xytext=(0.00008, 0.00525),
fontsize=16)

plt.legend(['Real Gas', 'Ideal Gas'], frameon=False)

# If you want to save the plot, uncomment this line (and edit as you see fit):
# plt.savefig('IDT_nDodecane_1000K_40atm.pdf', dpi=350, format='pdf')

# Demonstration of NTC behavior
# Let us use the reactor model to demonstrate the impacts of non-ideal behavior on IDTs
# in the Negative Temperature Coefficient (NTC) region, where observed IDTs, counter
# to intuition, increase with increasing temperature.

# Make a list of all the temperatures at which we would like to run simulations:
T = np.array([1250, 1225, 1200, 1150, 1100, 1075, 1050, 1025, 1012.5, 1000, 987.5,
975, 962.5, 950, 937.5, 925, 912.5, 900, 875, 850, 825, 800])

# If we desire, we can define different IDT starting guesses for each temperature:
estimatedIgnitionDelayTimes = np.ones(len(T))
# But we won't, at least in this example :)
estimatedIgnitionDelayTimes[:] = 0.005

# Now, we simply run the code above for each temperature.
# Real Gas
ignitionDelays_RG = np.zeros(len(T))
for i, temperature in enumerate(T):
# Setup the gas and reactor
reactorTemperature = temperature
real_gas.TP = reactorTemperature, reactorPressure
real_gas.set_equivalence_ratio(phi=1.0, fuel='c12h26',
oxidizer={'o2': 1.0, 'n2': 3.76})
r = ct.Reactor(contents=real_gas)
reactorNetwork = ct.ReactorNet([r])

# create an array of solution states
timeHistory = ct.SolutionArray(real_gas, extra=['t'])

t0 = time.time()

t = 0
counter = 1
while t < estimatedIgnitionDelayTimes[i]:
t = reactorNetwork.step()
if counter % 20 == 0:
timeHistory.append(r.thermo.state, t=t)
counter += 1

tau = ignitionDelay(timeHistory, 'oh')
t1 = time.time()

print("Computed Real Gas Ignition Delay: {:.3e} seconds for T={}K. "
"Took {:3.2f}s to compute".format(tau, temperature, t1-t0))

ignitionDelays_RG[i] = tau

# Repeat for Ideal Gas
ignitionDelays_IG = np.zeros(len(T))
for i, temperature in enumerate(T):
# Setup the gas and reactor
reactorTemperature = temperature
ideal_gas.TP = reactorTemperature, reactorPressure
ideal_gas.set_equivalence_ratio(phi=1.0, fuel='c12h26',
oxidizer={'o2': 1.0, 'n2': 3.76})
r = ct.Reactor(contents=ideal_gas)
reactorNetwork = ct.ReactorNet([r])

# create an array of solution states
timeHistory = ct.SolutionArray(ideal_gas, extra=['t'])

t0 = time.time()

t = 0
counter = 1
while t < estimatedIgnitionDelayTimes[i]:
t = reactorNetwork.step()
if counter % 20 == 0:
timeHistory.append(r.thermo.state, t=t)
counter += 1

tau = ignitionDelay(timeHistory, 'oh')
t1 = time.time()

print("Computed Ideal Gas Ignition Delay: {:.3e} seconds for T={}K. "
"Took {:3.2f}s to compute".format(tau, temperature, t1-t0))

ignitionDelays_IG[i] = tau

# Figure: ignition delay (tau) vs. the inverse of temperature (1000/T).
fig = plt.figure()
ax.plot(1000/T, 1e6*ignitionDelays_RG, '-', linewidth=2.0, color='b')
ax.plot(1000/T, 1e6*ignitionDelays_IG, '-.', linewidth=2.0, color='r')
ax.set_ylabel(r'Ignition Delay ($\mathdefault{\mu s}$)', fontsize=14)
ax.set_xlabel(r'1000/T (K$^\mathdefault{-1}$)', fontsize=14)

ax.set_xlim([0.8, 1.2])

# Add a second axis on top to plot the temperature for better readability
ax2 = ax.twiny()
ticks = ax.get_xticks()
ax2.set_xticks(ticks)
ax2.set_xticklabels((1000/ticks).round(1))
ax2.set_xlim(ax.get_xlim())
ax2.set_xlabel('Temperature (K)', fontsize=14)

ax.legend(['Real Gas', 'Ideal Gas'], frameon=False, loc='upper left')

# If you want to save the plot, uncomment this line (and edit as you see fit):
# plt.savefig('NTC_nDodecane_40atm.pdf', dpi=350, format='pdf')

# Show the plots.
plt.show()