Non-Ideal Shock Tube Example¶

Ignition delay time computations in a high-pressure reflected shock tube reactor¶

In this example we will 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 any temperature and pressure. The example very explicitly follows the form set in batch_reactor_ignition_delay_NTC.pynb, which does very similar calculations, but with an IdealGasReactor. All credit is due to the developer of that example. This example generalizes that work to use a Reactor with no pre-assumed EoS. One can also run ideal gas phases through this simulation, simply by specifying a cti file with that thermodynamic EoS.

Other than the typical Cantera dependencies, plotting functions require that you have matplotlib installed, and data storing and analysis requires pandas. See https://matplotlib.org/ and http://pandas.pydata.org/index.html, respectively, for additional info.

The example here demonstrates the calculations carried out by G. Kogekar, et al., "Impact of non-ideal behavior on ignition delay and chemical kinetics in high-pressure shock tube reactors," Combust. Flame., 2017.

The reflected shock tube reactor is modeled as a constant-volume, adiabatic reactor. The heat transfer and the work rates are therefore both zero. With no mass inlets or exits, the 1st law energy balance reduces to:

\begin{equation*} \frac{dU}{dt} = \dot{Q} - \dot{W} = 0. \end{equation*}

Because of the constant-mass and constant-volume assumptions, the density is also therefore constant:

\begin{equation*} \frac{d\rho}{dt} = 0. \end{equation*}

Along with the evolving gas composition, then, the thermodynamic state of the gas is defined by the initial total internal energy $U = mu = m\sum_k\left(Y_ku_k\right)$, where $u_k$ and $Y_k$ are the specific internal energy (kJ/kg) and mass fraction of species $k$, respectively.

The species mass fractions evolve according to the nety chemical production rates due to homogeneous gas-phase reactions:

\begin{equation*} \frac{dY_k}{dt} = \frac{W_k}{\rho}\dot{\omega}_k, \end{equation*}

where $W_k$ is the molecular weight of species $k$ $\left({\rm kg}\,{\rm kmol}^{-3}\right)$, $\rho$ is the (constant) gas-phase density $\left({\rm kg}\,{\rm m^{-3}}\right)$, and $\dot{\omega}_k$ is the net production rate of species $k$ $\left({\rm kmol}\,{\rm m^{-3}}\,{\rm s^{-1}}\right)$.

In [1]:
from __future__ import division
from __future__ import print_function

import pandas as pd
import numpy as np

import time

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

Runnning Cantera version: 2.5.0a2

In [2]:
%matplotlib notebook
import matplotlib.pyplot as plt

plt.rcParams['axes.labelsize'] = 16
plt.rcParams['xtick.labelsize'] = 12
plt.rcParams['ytick.labelsize'] = 12
plt.rcParams['figure.autolayout'] = True


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 that developed by Wang, Ra, Jia, and Reitz (https://www.erc.wisc.edu/chem_mech/nC12-PAH_mech.zip) by [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].

To fun a different model or use a different EoS, simply replace this cti file with a different mechanism file.

In [3]:
gas = ct.Solution('data/WangMechanismRK.cti')


Define reactor conditions : temperature, pressure, fuel, stoichiometry¶

In [4]:
# Define the reactor temperature and pressure:
reactorTemperature = 1000 #Kelvin
reactorPressure = 40.0*101325.0 #Pascals

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

# Define the fuel, oxidizer and set the stoichiometry:
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=gas)
reactorNetwork = ct.ReactorNet([r])

# Now compile a list of all variables for which we will store data
stateVariableNames = [r.component_name(item) for item in range(r.n_vars)]

# Use the above list to create a DataFrame
timeHistory = pd.DataFrame(columns=stateVariableNames)


Define useful functions¶

In [5]:
def ignitionDelay(df, species):
"""
This function computes the ignition delay from the occurence of the
peak in species' concentration.
"""
return df[species].idxmax()

In [6]:
# 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.loc[t] = reactorNetwork.get_state()
counter+=1

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

# Toc
t1 = time.time()

print('Computed Ignition Delay: {:.3e} seconds. Took {:3.2f}s to compute'.format(tau, t1-t0))

# If you want to save all the data - molefractions, temperature, pressure, etc
# uncomment the next line
# timeHistory.to_csv("time_history.csv")

Computed Ignition Delay: 4.093e-04 seconds. Took 3.98s to compute


Plot the result¶

Figure illustrating the definition of ignition delay¶

In [7]:
plt.figure()
plt.plot(timeHistory.index, timeHistory['oh'],'-o',color='b',markersize=4)
plt.xlabel('Time (s)',fontname='Times New Roman')
plt.ylabel('$\mathdefault{OH\, mass\, fraction,}\, Y_{OH}}$',fontname='Times New Roman')

# Figure formatting:
plt.xlim([0,0.00075])
ax = plt.gca()
font = plt.matplotlib.font_manager.FontProperties(family='Times New Roman',size=14)
ax.annotate("",xy=(tau,0.005), xytext=(0,0.005),arrowprops=dict(arrowstyle="<|-|>",color='r',linewidth=2.0),fontsize=14,)
plt.annotate('Ignition Delay Time (IDT)', xy=(0,0), xytext=(0.00004, 0.00525), family='Times New Roman',fontsize=16);

for tick in ax.xaxis.get_major_ticks():
tick.label1.set_fontsize(12)
tick.label1.set_fontname('Times New Roman')
for tick in ax.yaxis.get_major_ticks():
tick.label1.set_fontsize(12)
tick.label1.set_fontname('Times New Roman')


Illustration : NTC behavior¶

In the paper by Kogekar, et al., the reactor model is used to demonstrate the impacts of non-ideal behavior on IDTs in the Negative Temperature Coefficient region, where observed IDTs, counter to intuition, increase with increasing temperature.

Define the temperatures for which we will run the simulations¶

In [8]:
# Make a list of all the temperatures we would like to run simulations at
T = [1800, 1600, 1400, 1200, 1100, 1075, 1050, 1025, 1000, 975, 950, 925, 900, 850, 825, 800,
750, 700]

estimatedIgnitionDelayTimes = np.ones(len(T))

# Set the initial guesses to a common value.  We could probably speed up simulations
# by tuning this guess, but as seen in the figure above, the 'extra' time after igntion
# does not add many data points or simulation steps.  The time savings would be small.
estimatedIgnitionDelayTimes[:] = 0.005

# Now create a dataFrame out of these
ignitionDelays = pd.DataFrame(data={'T':T})
ignitionDelays['ignDelay'] = np.nan


Run the code above for each temperature, and save the IDT for each.

In [9]:
for i, temperature in enumerate(T):
# Set up the gas and reactor
reactorTemperature = temperature
reactorPressure = 40.0*101325.0
gas.TP = reactorTemperature, reactorPressure
gas.set_equivalence_ratio(phi=1.0, fuel='c12h26', oxidizer={'o2':1.0, 'n2':3.76})
r = ct.Reactor(contents=gas)
reactorNetwork = ct.ReactorNet([r])

# Create an empty data frame
timeHistory = pd.DataFrame(columns=timeHistory.columns)

t0 = time.time()

t = 0
counter = 0
while t < estimatedIgnitionDelayTimes[i]:
t = reactorNetwork.step()
if not counter % 20:
timeHistory.loc[t] = r.get_state()
counter += 1

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

print('Computed Ignition Delay: {:.3e} seconds for T={}K. Took {:3.2f}s to compute'.format(tau, temperature, t1-t0))

ignitionDelays.at[i, 'ignDelay'] = tau

Computed Ignition Delay: 6.357e-07 seconds for T=1800K. Took 2.36s to compute
Computed Ignition Delay: 1.585e-06 seconds for T=1600K. Took 2.67s to compute
Computed Ignition Delay: 5.785e-06 seconds for T=1400K. Took 2.71s to compute
Computed Ignition Delay: 3.911e-05 seconds for T=1200K. Took 3.00s to compute
Computed Ignition Delay: 1.326e-04 seconds for T=1100K. Took 3.14s to compute
Computed Ignition Delay: 1.839e-04 seconds for T=1075K. Took 3.32s to compute
Computed Ignition Delay: 2.533e-04 seconds for T=1050K. Took 3.45s to compute
Computed Ignition Delay: 3.365e-04 seconds for T=1025K. Took 3.97s to compute
Computed Ignition Delay: 4.093e-04 seconds for T=1000K. Took 4.06s to compute
Computed Ignition Delay: 4.289e-04 seconds for T=975K. Took 4.01s to compute
Computed Ignition Delay: 3.911e-04 seconds for T=950K. Took 3.94s to compute
Computed Ignition Delay: 3.407e-04 seconds for T=925K. Took 4.10s to compute
Computed Ignition Delay: 3.145e-04 seconds for T=900K. Took 3.74s to compute
Computed Ignition Delay: 3.233e-04 seconds for T=850K. Took 4.12s to compute
Computed Ignition Delay: 3.439e-04 seconds for T=825K. Took 4.26s to compute
Computed Ignition Delay: 3.852e-04 seconds for T=800K. Took 4.51s to compute
Computed Ignition Delay: 6.824e-04 seconds for T=750K. Took 4.37s to compute
Computed Ignition Delay: 2.056e-03 seconds for T=700K. Took 4.67s to compute


Figure: ignition delay ($\tau$) vs. the inverse of temperature ($\frac{1000}{T}$).¶

In [10]:
fig = plt.figure()
ax.semilogy(1000/ignitionDelays['T'], ignitionDelays['ignDelay'],'o-',color='b')
ax.set_ylabel('Ignition Delay (s)',fontname='Times New Roman',fontsize=16)
ax.set_xlabel(r'$\mathdefault{1000/T\, (K^{-1})}$', fontsize=16,fontname='Times New Roman')

# 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)',fontname='Times New Roman',fontsize=16);

for tick in ax.xaxis.get_major_ticks():
tick.label1.set_fontsize(12)
tick.label1.set_fontname('Times New Roman')
for tick in ax.yaxis.get_major_ticks():
tick.label1.set_fontsize(12)
tick.label1.set_fontname('Times New Roman')
for tick in ax2.xaxis.get_major_ticks():
tick.label1.set_fontsize(12)
tick.label1.set_fontname('Times New Roman')