Note
Go to the end to download the full example code.
A pseudo stochastic scheme to simulate a stratified-charged compression-ignition engine#
For stratified-charge compression ignition (SCCI) engines, the multi-zone homogeneous charged compression ignition (HCCI) model can be applied to address the composition and temperature variations of the initial gas charge. The incylinder gas mixture at IVC is “grouped” to form several imaginary ‘fluid material zones’ based on the local gas temperature and composition. Note that the ‘fluid material zone’ does not have to form a single continuous volume of gas mass, rather it can be a collection of many small gas masses through out the cylinder.
The multi-zone HCCI engine model assumes that fluid flow would simply relocate, distort, and break down (into smaller volumes) a ‘fluid material zone’, and there is no mass or energy exchange between the zones. That is, the gas mass in a ‘fluid zone’ will remain in the same zone during the entirety of the simulation. The only interaction allowed between the ‘fluid zones’ is the pressure work that erases any pressure differences in the zones by changing the zonal volumes.
The pseudo-chastic SCCI (PSCCI) engine model described in this example assumes that the turbulence scalar mixing processes should act as the mechanism to facilitate the gas species and enthalpy exchange between different ‘fluid material zones’. That is, the zonal masses remain the same but the gas species are allowed to move across the zone boundaries by turbulence diffusion. The turbulence diffusion effect is imitated by the stochastic micro mixing process. The micro mixing process will be performed at preset ‘pause’ crank angles. The PSCCI engine model will stop at every ‘pause’ crank angle to perform the micro mixing process and restart the simulation till the last ‘pause’ crank angle (which should be the EVO) is reached. The micro mixing sub-model of the PSCCI engine has three model parameters: the micro mixing time step size (delta_time), the characteristic turbulence scalar mixing time scale (tau), and the mixing model parameter (Cmix). The PSCCI engine model is termed ‘pseudo-chastic’ instead of ‘stochastic’ because it performs the micro mixing events less frequently than what is typically required for a stochastic simulation to reduce the simulation time. By increasing the number of the ‘pause’ crank angles, the PSCCI engine model should behave more closely to a ‘true’ stochastic model.
This example is focused on describing the process of setting up the micro mixing model. You can refer to the multi-zone HCCI engine example for the steps to set up the initial zonal properties of the PSCCI model. You can also compare the results from the PSCCI engine model in this example to the results from the multi-zone HCCI engine model to ‘extract’ the impacts of the micro mixing process.
Import PyChemkin packages and start the logger#
import copy
from pathlib import Path
import ansys.chemkin.core as ck # Chemkin
from ansys.chemkin.core import Color
# Chemkin homogeneous charge compression ignition (HCCI) engine model (transient)
from ansys.chemkin.core.engines.HCCI import HCCIengine
from ansys.chemkin.core.logger import logger
from ansys.chemkin.core.microprocess import MicroMixing
from ansys.chemkin.core.utilities import random_pick_integers
import matplotlib.pyplot as plt # plotting
import numpy as np # number crunching
# check working directory
current_dir = str(Path.cwd())
logger.debug("working directory: " + current_dir)
# set verbose mode
ck.set_verbose(True)
# set interactive mode for plotting the results
# interactive = True: display plot
# interactive = False: save plot as a PNG file
global interactive
interactive = True
Create a chemistry set#
The mechanism to load is the GRI 3.0 mechanism for methane combustion.
This mechanism and its associated data files come with the standard Ansys Chemkin
installation in the /reaction/data directory.
# set mechanism directory (the default Chemkin mechanism data directory)
data_dir = Path(ck.ansys_dir) / "reaction" / "data"
mechanism_dir = data_dir
# create a chemistry set based on the GRI mechanism
MyGasMech = ck.Chemistry(label="GRI 3.0")
# set mechanism input files
# including the full file path is recommended
MyGasMech.chemfile = str(mechanism_dir / "grimech30_chem.inp")
MyGasMech.thermfile = str(mechanism_dir / "grimech30_thermo.dat")
MyGasMech.tranfile = str(mechanism_dir / "grimech30_transport.dat")
Preprocess the chemistry set#
# preprocess the mechanism files
iError = MyGasMech.preprocess()
Set up the fuel-air mixture#
You must set up the fuel-air mixture inside the engine cylinder
right after the intake valve is closed. Here the x_by_equivalence_ratio()
method is used. You create the fuelmixture and the air mixtures first.
You then define the complete combustion product species and provide the
additives composition if there is any. Finally, you set
the equivalenceratio value to create the fuel-air mixture. In this case,
the fuel mixture consists of methane, ethane, and propane as the simulated
natural gas. Because HCCI engines generally run on lean fuel-air mixtures,
the equivalence ratio is set to 0.8.
# create the fuel mixture
fuelmixture = ck.Mixture(MyGasMech)
# set fuel composition
fuelmixture.x = [("CH4", 0.9), ("C3H8", 0.05), ("C2H6", 0.05)]
# setting pressure and temperature is not required in this case
fuelmixture.pressure = 1.5 * ck.P_ATM
fuelmixture.temperature = 400.0
# create the oxidizer mixture: air
air = ck.Mixture(MyGasMech)
air.x = [("O2", 0.21), ("N2", 0.79)]
# setting pressure and temperature is not required in this case
air.pressure = 1.5 * ck.P_ATM
air.temperature = 400.0
# create the unburned fuel-air mixture
fresh = ck.Mixture(MyGasMech)
# products from the complete combustion of the fuel mixture and air
products = ["CO2", "H2O", "N2"]
# species mole fractions of added/inert mixture.
# can also create an additives mixture here
add_frac = np.zeros(MyGasMech.kk, dtype=np.double) # no additives: all zeros
# mean equivalence ratio
equiv = 0.8
ierror = fresh.x_by_equivalence_ratio(
MyGasMech, fuelmixture.x, air.x, add_frac, products, equivalenceratio=equiv
)
# check fuel-oxidizer mixture creation status
if ierror != 0:
print("Error: Failed to create the fuel-oxidizer mixture.")
exit()
# list the composition of the unburned fuel-air mixture
fresh.list_composition(mode="mole")
Specify pressure and temperature of the fuel-air mixture#
Since you are going to use the fresh fuel-air mixture to instantiate
the engine object later, setting the mixture pressure and temperature
is equivalent to setting the initial temperature and pressure of
the engine cylinder.
fresh.temperature = 447.0
fresh.pressure = 1.065 * ck.P_ATM
Add EGR to the fresh fuel-air mixture#
Many engines have the configuration for exhaust gas recirculation (EGR). Chemkin
engine models allow you to add the EGR mixture to the fresh fuel-air mixture entered
the cylinder. If the engine you are modeling has EGR, you should have the EGR ratio,
which is generally the volume ratio between the EGR mixture and the fresh fuel-air
ratio. However, because you know nothing about the composition of the exhaust gas,
you cannot simply combine these two mixtures. In this case, you use the
get_EGR_mole_fraction() method to estimate the major components of the exhaust
gas from the combustion of the fresh fuel-air mixture. The threshold=1.0e-8
parameter tells the method to ignore any species with a mole fraction below the
threshold value. Once you have the EGR mixture composition, use the
x_by_equivalence_ratio() method a second time to re-create the fresh
fuel-air mixture with the original fuelmixture and air mixtures along with
the EGR composition you just got as the “additives”.
egr_ratio = 0.3
# compute the EGR stream composition in mole fractions
add_frac = fresh.get_egr_mole_fraction(egr_ratio, threshold=1.0e-8)
# recreate the initial mixture with EGR
ierror = fresh.x_by_equivalence_ratio(
MyGasMech,
fuelmixture.x,
air.x,
add_frac,
products,
equivalenceratio=equiv,
threshold=1.0e-8,
)
# list the composition of the fuel+air+EGR mixture for verification
fresh.list_composition(mode="mole", bound=1.0e-8)
Set up the HCCI engine reactor#
Use the HCCIengine() method to create a multi-zone HCCI engine named
MyMZEngine and make the new fresh mixture as the initial incylinder
gas mixture at IVC. Set the nzones parameter to the number of zones in your
multi-zone HCCI engine model.
# create a five-zone HCCI engine
numbzones = 5
MyMZEngine = HCCIengine(reactor_condition=fresh, nzones=numbzones)
# show initial gas composition inside the reactor
MyMZEngine.list_composition(mode="mole", bound=1.0e-8)
Set basic engine parameters#
Set the required engine parameters as shown in the following code. These
engine parameters are used to describe the cylinder volume during the
simulation. The starting_ca argument should be the crank angle corresponding
to the cylinder IVC. The ending_ca is typically the EVC crank angle.
# cylinder bore diameter [cm]
MyMZEngine.bore = 12.065
# engine stroke [cm]
MyMZEngine.stroke = 14.005
# connecting rod length [cm]
MyMZEngine.connecting_rod_length = 26.0093
# compression ratio [-]
MyMZEngine.compression_ratio = 16.5
# engine speed [RPM]
MyMZEngine.rpm = 1000
# set other parameters
# simulation start CA [degree]
MyMZEngine.starting_ca = -142.0
# simulation end CA [degree]
MyMZEngine.ending_ca = 116.0
# list the engine parameters
MyMZEngine.list_engine_parameters()
print(f"engine displacement volume {MyMZEngine.get_displacement_volume()} [cm3]")
print(f"engine clearance volume {MyMZEngine.get_clearance_volume()} [cm3]")
print(f"number of zone(s) = {MyMZEngine.get_number_of_zones()}")
Set up the engine wall heat transfer model#
By default, the engine cylinder is adiabatic. You must set up a wall heat transfer model to include the heat loss effects in your engine simulation. Chemkin support three widely used engine wall heat transfer models. The models and their parameters follow.
dimensionless: [<a> <b> <c> <Twall>]dimensional: [<a> <b> <c> <Twall>]hohenburg: [<a> <b> <c> <d> <e> <Twall>]
There is also the in-cylinder gas velocity correlation (the Woschni correlation) that is associated with the engine wall heat transfer models. Here are the parameters of the Woschni correlation:
[<C11> <C12> <C2> <swirl ratio>]
You can also specify the surface areas of the piston head and the cylinder head for more precision heat transfer wall area. By default, both the piston head and the cylinder head surfaces are flat.
heattransferparameters = [0.035, 0.71, 0.0]
# set cylinder wall temperature [K]
t_wall = 400.0
MyMZEngine.set_wall_heat_transfer("dimensionless", heattransferparameters, t_wall)
# in-cylinder gas velocity correlation parameter (Woschni)
# [<C11> <C12> <C2> <swirl ratio>]
gv_parameters = [2.28, 0.308, 3.24, 0.0]
MyMZEngine.set_gas_velocity_correlation(gv_parameters)
# set piston head top surface area [cm2]
MyMZEngine.set_piston_head_area(area=124.75)
# set cylinder clearance surface area [cm2]
MyMZEngine.set_cylinder_head_area(area=123.5)
Set zonal properties#
By default, all zones in the multi-zone HCCI engine model have the same
properties. You can artificially stratify the temperature and/or the equivalence
ratio distribution in the cylinder at the IVC by utilizing the set_zonal
methods of the HCCI object.
# initial zonal temperatures [K]
z_temperature = [447.5, 447.5, 447, 447, 447]
MyMZEngine.set_zonal_temperature(zonetemp=z_temperature)
# initial zonal volume fractions
z_volumefrac = [0.3, 0.25, 0.2, 0.2, 0.05]
MyMZEngine.set_zonal_volume_fraction(zonevol=z_volumefrac)
# initial wall heat transfer area fractions
z_htarea = [0.0, 0.15, 0.2, 0.25, 0.4]
MyMZEngine.set_zonal_heat_transfer_area_fraction(zonearea=z_htarea)
# initial zonal equivalence ratios
z_phi = [equiv, equiv, equiv, equiv, equiv]
MyMZEngine.set_zonal_equivalence_ratio(zonephi=z_phi)
# zonal EGR ratios
z_egrr = [0.3, 0.3, 0.3, 0.35, 0.35]
MyMZEngine.set_zonal_egr_ratio(zoneegr=z_egrr)
# set fuel "molar" composition
MyMZEngine.define_fuel_composition([("CH4", 0.9), ("C3H8", 0.05), ("C2H6", 0.05)])
# set oxidizer "molar' composition
MyMZEngine.define_oxid_composition([("O2", 0.21), ("N2", 0.79)])
# set products
MyMZEngine.define_product_composition(["CO2", "H2O", "N2"])
# set EGR composition in mole fractions
z_add = [add_frac, add_frac, add_frac, add_frac, add_frac]
MyMZEngine.define_additive_fractions(addfrac=z_add)
Set output options#
You can turn on the adaptive solution saving to resolve the steep variations in
the solution profile. Here additional solution data points are saved for every
20 solver internal steps. You must include the set_ignition_delay() method for
the engine model to report the ignition delay crank angle after the simulation
is done. If method="T_inflection" is set, the reactor model treats the
inflection points in the predicted gas temperature profile as the indication of
an auto-ignition. You can choose a different auto-ignition definition.
Note
Type
ansys.chemkin.show_ignition_definitions()to get the list of all available ignition delay time definitions in Chemkin.The
set_ignition_delay()method is required for the engine model to report the ignition delay time for each zone as well as the cylinder averaged ignition delay time derived from the cylinder averaged temperature profile.By default, time/crank angle intervals for both print and save solution are 1/100 of the simulation duration, which in this case is \(dCA=(EVO-IVC)/100=2.58\). You can make the model report more frequently by using the
ca_step_for_saving_solution()or theca_step_for_printing_solution()method to set different interval values in crank angles.
# set the number of crank angles between saving solution
MyMZEngine.ca_step_for_saving_solution = 0.5
# set the number of crank angles between printing solution
MyMZEngine.ca_step_for_printing_solution = 10.0
# turn on adaptive solution saving
MyMZEngine.adaptive_solution_saving(mode=True, steps=20)
# specify the ignition definitions
MyMZEngine.set_ignition_delay(method="T_inflection")
Set solver controls#
You can overwrite the default solver controls by using solver-related methods, such as those for tolerances.
# set tolerances in tuple: (absolute tolerance, relative tolerance)
MyMZEngine.tolerances = (1.0e-12, 1.0e-10)
# get solver parameters
atol, rtol = MyMZEngine.tolerances
print(f"Default absolute tolerance = {atol}.")
print(f"Default relative tolerance = {rtol}.")
# turn on the force non-negative solutions option in the solver
MyMZEngine.force_nonnegative = True
# show solver and output options
# show the number of crank angles between printing solution
print(
f"Crank angles between solution printing: "
f"{MyMZEngine.ca_step_for_printing_solution}"
)
# show other transient solver setup
print(f"Forced non-negative solution values: {MyMZEngine.force_nonnegative}")
Display the added parameters (keywords)#
Use the showkeywordinputlines() method to verify that the preceding parameters
are correctly assigned to the engine model.
MyMZEngine.showkeywordinputlines()
Set up the ‘pseudo-cgastic’ micro mixing model#
The ‘pseudo-chastic’ micro mixing model option is turned on by setting the flag
do_micromixing to ‘True’. Then you should set the ‘pause’ points for performing
the ‘stochastic’ micro mixing process. This can be achieved by using either the
simulation intervals in crank angles or by specifying a series of ‘pause’ crank
angles. In this example, a series of ‘pause points’ are defined by the ca_stops
list. The second steps is to set the parameters of the micro mixing model. The micro
mixing model currently available in the PSCCI engine model is the
‘modified Curls model’ which requires four parameters. The micro mixing time step
size is the time during for the micro mixing process, delta_time. Typically,
this time step size should be kept comparable to the solver time step size.
Since Chemkin transient solver works differently from typical CFD solvers, the PSCCI
engine model allows delta_time to be set independently of the time step size
used by the solver. Subsequently the PSCCI engine model performs the micro mixing
process less frequently and hence introduces larger bias and errors. In this example,
delta_time is set to the crank angle interval between the ‘pause points’. The
second parameter is the number of statistical events/particles used in the
stochastic process numb_particles. The number of particles is related to the
‘statistical error’ of the stochastic process. Typically ‘statistical error’ is
inversely proportional to the square-root of the particle number. The third
parameter for the micro mixing model is the ‘scalar mixing time scale’
mixing_time_scale. This parameter controls the intensity (or the scope) of
the molecule-level mixing by flow turbulence, and its value should be set to the
‘turbulence scalar mixing time scale’ or the ‘characteristic turbulence time scale’
of the incylinder flow. The last parameter is the mixing model dependent parameters
c_mix. For the ‘modified Curls model’, the default c_mix value is 1.0.
This parameter controls the ‘completeness’ of the mixing of the particles.
The higher the c_mix value, the closer the particles resembling to each other
after mixing. When c_mix is large enough, the particles are well-mixed and
become identical after mixing.
When do_micromixing is set to ‘True’, the simulation will pause at each
‘pause point’. Once the simulation is stopped, the micro mixing process is initiated
by instantiating a micro mixing object MicroMixing. You will then specify the
number of particles to be used in the micro mixing stochastic process by the
set_numb_particles() method. You also have to extract the gas mixture properties
of each ‘fluid zone’ from the multi-zone solution at the ‘pause time’ by using
the get_last_zone_mixtures() method. You run the modified Curls micro mixing
model by using the modified_curls() method with the model parameters and the
zonal mixture objects obtained from the latest multi-zone simulation solution.
Afterwards, use the restart() method with the updated zonal mixture properties
from the micro mixing model to run the multi-zone simulation to the next
‘pause point’. Repeat these steps till the simulation reaches the designated
simulation end time (EVO).
Note
You can add more complexity to the model by including other processes during
the pause. For example, you can imitate the ever-changing heat transfer area
between the zones and the cylinder wall by randomly re-assigning the wall heat
transfer area fractions of the zones. See the random_wall_area sections below
for more details.
# set stopping/restarting CAs
# the last stop CA should be the actual simulation end time EVO
ca_stops = [-50.0, -20.0, -10.0, -5.0, 0.0, 5.0, 10.0, 20.0, 116.0]
# perform micro mixing process during restarts: set "do_micromixing = True"
do_micromixing = True
# do_micromixing = False
# total number of particles for the pseudo-chastic process
# the particles are shared among the zones in the HCCI engine
numb_particles = 400
# turbulence scalar mixing time scale [sec]
mixing_time_scale = 0.01
# mixing model parameter Cmix
mixing_model_parameter = 1.0
# randomly rearrange zonal wall heat transfer area: set "random_wall_area = True"
random_wall_area = False
if random_wall_area:
# create a list for random pick
area_list: list[int] = []
for i in range(numbzones):
area_list.append(i + 1)
# solution profile sets
crank_angle_data = []
pressure_data = []
temperature_data = []
volume_data = []
co_data = []
ch4_data = []
no_data = []
zone_temperature_data = []
zone_volume_data = []
zone_co_data = []
Run the simulation#
Use the run() method to start the multi-zone HCCI engine simulation.
runcount = 0
#
for end_ca in ca_stops:
# Run the simulation
if runcount == 0:
# simulation end CA [degree]
MyMZEngine.ending_ca = end_ca
runstatus = MyMZEngine.run()
# check run status
if runstatus != 0:
# Run failed.
print(Color.RED + ">>> Run failed. <<<", end=Color.END)
exit()
# Run succeeded.
print(Color.GREEN + ">>> Run completed. <<<", end=Color.END)
else:
if random_wall_area:
# re-assign the zonal wall heat transfer fractions
picked, unpicked = random_pick_integers(numbzones, area_list)
print(str(picked))
this_area = copy.deepcopy(z_htarea)
for i, a in enumerate(picked):
this_area[i] = z_htarea[a - 1]
print(str(this_area))
MyMZEngine.set_zonal_heat_transfer_area_fraction(zonearea=this_area)
if do_micromixing:
# perform random micro mixing between particles
# the mixing time step size is the time duration from the previous stop CA
delta_time = MyMZEngine.get_time(
MyMZEngine.ending_ca
) - MyMZEngine.get_time(MyMZEngine.starting_ca)
# create a mixing instance
zonemixing = MicroMixing()
# set the total number of particles
zonemixing.set_numb_particles(numb_particles)
# set the zone mixture
zonemixtures = MyMZEngine.get_last_zone_mixtures()
zonemixing.set_particle_mixtures(zonemixtures)
# use the modified Curl's mixing model
mixed_zones = zonemixing.modified_curls(
delta_time,
mixing_time_scale,
mixing_model_parameter,
)
# restart run(s)
# reset the simulation end CA [degree]
runstatus = MyMZEngine.restart(end_ca=end_ca, new_mixtures=mixed_zones)
else:
# restart run(s)
# reset the simulation end CA [degree]
runstatus = MyMZEngine.restart(end_ca=end_ca)
# check run status
if runstatus != 0:
# Run failed.
print(Color.RED + ">>> Restart Run failed. <<<", end=Color.END)
exit()
# Run succeeded.
print(Color.GREEN + ">>> Restart Run completed. <<<", end=Color.END)
####################################################
# Postprocess the solution profiles in selected zone
# ==================================================
# The solution of the multi-zone HCCI engine model contains the results of
# the individual zones plus the cylinder averaged results. This means that
# if there are n zones in the multi-zone engine model, there are (n+1) solution
# records: n zonal results and the cylinder averaged results.
#
# To process the result of the zone number :math:`j`\ , :math:`(1 \leq j \leq n)`\ ,
# set the parameter value of ``zoneID`` to :math:`j` when you call the engine
# postprocessor with the ``process_engine_solution()`` method. Otherwise, the
# cylinder averaged results are postprocessed by default, that is, when the ``zoneID``
# parameter is omitted.
#
# .. note ::
# Because The ``process_engine_solution()`` method can process only one set of
# results at a time (one zonal result or the cylinder averaged result), you must
# postprocess the zones one by one to obtain all solution data of the multi-zone
# simulation.
#
thiszone = 1
MyMZEngine.process_engine_solution(zone_id=thiszone)
plottitle = "Zone " + str(thiszone) + " Solution"
# get the number of solution time points
solutionpoints = MyMZEngine.getnumbersolutionpoints()
print(f"Number of solution points = {solutionpoints}.")
# get the time profile
timeprofile = MyMZEngine.get_solution_variable_profile("time")
# convert time to crank angle
ca_profile = np.zeros_like(timeprofile, dtype=np.double)
count = 0
for t in timeprofile:
ca_profile[count] = MyMZEngine.get_ca(timeprofile[count])
count += 1
# get the cylinder pressure profile
presprofile = MyMZEngine.get_solution_variable_profile("pressure")
presprofile *= 1.0e-6
# create arrays for cylinder-averaged mixture temperature
tempprofile = MyMZEngine.get_solution_variable_profile("temperature")
# get the zonal volume profile
volprofile = MyMZEngine.get_solution_variable_profile("volume")
# get the zonal CO mass fraction profile
co_profile = MyMZEngine.get_solution_variable_profile("CO")
# post-process cylinder-averged solution
# do NOT set the zoneID parameter
MyMZEngine.process_average_engine_solution()
# get the cylinder volume profile
cylinder_volprofile = MyMZEngine.get_solution_variable_profile("volume")
# create arrays for cylinder-averaged mixture temperature
cylinder_tempprofile = MyMZEngine.get_solution_variable_profile("temperature")
# get the cylinder-averaged CO mass fraction profile
cylinder_co_profile = MyMZEngine.get_solution_variable_profile("CO")
# get the cylinder-averaged CH4 mass fraction profile
cylinder_ch4_profile = MyMZEngine.get_solution_variable_profile("CH4")
# get the cylinder-averaged NO mass fraction profile
cylinder_no_profile = MyMZEngine.get_solution_variable_profile("NO")
# store solution profiles
crank_angle_data.append(copy.deepcopy(ca_profile))
pressure_data.append(copy.deepcopy(presprofile))
volume_data.append(copy.deepcopy(cylinder_volprofile))
temperature_data.append(copy.deepcopy(cylinder_tempprofile))
co_data.append(copy.deepcopy(cylinder_co_profile))
ch4_data.append(copy.deepcopy(cylinder_ch4_profile))
no_data.append(copy.deepcopy(cylinder_no_profile))
zone_volume_data.append(copy.deepcopy(volprofile))
zone_temperature_data.append(copy.deepcopy(tempprofile))
zone_co_data.append(copy.deepcopy(co_profile))
# increase the run count
runcount += 1
Plot the engine solution profiles#
Plot the zonal and the cylinder averaged profiles from the multi-zone HCCI engine simulation.
Note
You can get profiles of the thermodynamic and the transport properties
by applying Mixture utility methods to the solution mixtures.
plt.subplots(2, 2, sharex="col", figsize=(12, 6))
plt.suptitle(plottitle, fontsize=16)
for i in range(len(crank_angle_data)):
plt.subplot(221)
plt.plot(crank_angle_data[i], pressure_data[i], "r-")
plt.ylabel("Pressure [bar]")
plt.subplot(222)
plt.plot(crank_angle_data[i], zone_volume_data[i], "b-")
plt.plot(crank_angle_data[i], volume_data[i], "b--")
plt.ylabel("Volume [cm3]")
plt.legend(["Zone", "Cylinder"], loc="upper right")
plt.subplot(223)
plt.plot(crank_angle_data[i], zone_temperature_data[i], "g-")
plt.plot(crank_angle_data[i], temperature_data[i], "g--")
plt.xlabel("Crank Angle [degree]")
plt.ylabel("Temperature [K]")
plt.legend(["Zone", "Averaged"], loc="upper left")
plt.subplot(224)
plt.plot(crank_angle_data[i], zone_co_data[i], "m-")
plt.plot(crank_angle_data[i], co_data[i], "m--")
plt.xlabel("Crank Angle [degree]")
plt.ylabel("CO mass fraction [-]")
plt.legend(["Zone", "Averaged"], loc="upper left")
# clean up
ck.done()
# plot results
if interactive:
plt.show()
else:
plt.savefig("plot_pseudo_HCCI_engine.png", bbox_inches="tight")