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Revise flame speed sensitivity example based on Jupyter version
Co-authored-by: Santosh Shanbhogue <santosh.shanbhogue@gmail.com>
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Ray Speth
Ingmar Schoegl
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doc/sphinx/_static/images/samples/flame-speed.svg
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doc/sphinx/_static/images/samples/flame-speed.svg
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"""
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r"""
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Laminar flame speed sensitivity analysis
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========================================
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Sensitivity analysis for a freely-propagating, premixed methane-air
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flame. Computes the sensitivity of the laminar flame speed with respect
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to each reaction rate constant.
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In this example we simulate a freely-propagating, adiabatic, premixed methane-air flame,
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calculate its laminar burning velocity and perform a sensitivity analysis of its
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kinetics with respect to each reaction rate constant.
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Requires: cantera >= 2.5.0
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Requires: cantera >= 3.0.0, pandas
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.. tags:: Python, combustion, 1D flow, flame speed, premixed flame,
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sensitivity analysis, plotting
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The figure below illustrates the setup, in a flame-fixed coordinate system. The
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reactants enter with density :math:`\rho_u`, temperature :math:`T_u` and speed
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:math:`S_u`. The products exit the flame at speed :math:`S_b`, density :math:`\rho_b`
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and temperature :math:`T_b`.
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.. image:: /_static/images/samples/flame-speed.svg
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:width: 50%
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:alt: Freely Propagating Flame
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:align: center
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.. tags:: Python, combustion, 1D flow, flame speed, premixed flame, sensitivity analysis
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"""
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import cantera as ct
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import numpy as np
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import pandas as pd
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import matplotlib.pyplot as plt
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plt.rcParams['figure.constrained_layout.use'] = True
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# %%
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# Simulation parameters
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p = ct.one_atm # pressure [Pa]
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Tin = 300.0 # unburned gas temperature [K]
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reactants = 'CH4:0.45, O2:1.0, N2:3.76'
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# ---------------------
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width = 0.03 # m
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# Define the gas mixture and kinetics used to compute mixture properties
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# In this case, we are using the GRI 3.0 model with methane as the fuel
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mech_file = "gri30.yaml"
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fuel_comp = {"CH4": 1.0}
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# Solution object used to compute mixture properties
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gas = ct.Solution('gri30.yaml')
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gas.TPX = Tin, p, reactants
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# Inlet temperature in kelvin and inlet pressure in pascals
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# In this case we are setting the inlet T and P to room temperature conditions
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To = 300
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Po = ct.one_atm
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# Flame object
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f = ct.FreeFlame(gas, width=width)
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f.set_refine_criteria(ratio=3, slope=0.07, curve=0.14)
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# Domain width in metres
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width = 0.014
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f.solve(loglevel=1, auto=True)
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print(f"\nmixture-averaged flamespeed = {f.velocity[0]:7f} m/s\n")
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# %%
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# Simulation setup
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# ----------------
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# Create the object representing the gas and set its state to match the inlet conditions
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gas = ct.Solution(mech_file)
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gas.TP = To, Po
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gas.set_equivalence_ratio(1.0, fuel_comp, {"O2": 1.0, "N2": 3.76})
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flame = ct.FreeFlame(gas, width=width)
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flame.set_refine_criteria(ratio=3, slope=0.07, curve=0.14)
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# %%
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# Solve the flame
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# ---------------
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flame.solve(loglevel=1, auto=True)
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# %%
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print(f"\nmixture-averaged flame speed = {flame.velocity[0]:7f} m/s\n")
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# %%
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# Plot temperature and major species profiles
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# -------------------------------------------
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#
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# Check and see if all has gone well.
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# Extract the spatial profiles as a SolutionArray to simplify plotting specific species
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profile = flame.to_array()
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fig, ax = plt.subplots()
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ax.plot(profile.grid * 100, profile.T, ".-")
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_ = ax.set(xlabel="Distance [cm]", ylabel="Temperature [K]")
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# %%
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fig, ax = plt.subplots()
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ax.plot(profile.grid * 100, profile("CH4").X, "--", label="CH$_4$")
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ax.plot(profile.grid * 100, profile("O2").X, label="O$_2$")
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ax.plot(profile.grid * 100, profile("CO2").X, "--", label="CO$_2$")
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ax.plot(profile.grid * 100, profile("H2O").X, label="H$_2$O")
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ax.legend(loc="best")
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plt.xlabel("Distance (cm)")
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_ = ax.set(xlabel="Distance [cm]", ylabel="Mole fraction [-]")
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# %%
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# Sensitivity Analysis
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# --------------------
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#
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# See which reactions affect the flame speed the most. The sensitivities can be
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# efficiently computed using the adjoint method. In the general case, we consider a
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# system :math:`f(x, p) = 0` where :math:`x` is the system's state vector and :math:`p`
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# is a vector of parameters. To compute the sensitivities of a scalar function
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# :math:`g(x, p)` to the parameters, we solve the system:
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#
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# .. math::
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# \left(\frac{\partial f}{\partial x}\right)^T \lambda =
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# \left(\frac{\partial g}{\partial x}\right)^T
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#
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# for the Lagrange multiplier vector :math:`\lambda`. The sensitivities are then
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# computed as
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#
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# .. math::
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# \left.\frac{dg}{dp}\right|_{f=0} =
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# \frac{\partial g}{\partial p} - \lambda^T \frac{\partial f}{\partial p}
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#
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# In the case of flame speed sensitivity to the reaction rate coefficients,
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# :math:`g = S_u` and :math:`\partial g/\partial p = 0`. Since
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# :math:`\partial f/\partial x` is already computed as part of the normal solution
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# process, computing the sensitivities for :math:`N` reactions only requires :math:`N`
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# additional evaluations of the residual function to obtain
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# :math:`\partial f/\partial p`, plus the solution of the linear system for
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# :math:`\lambda`.
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#
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# Calculation of flame speed sensitivities with respect to rate coefficients is
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# implemented by the method `FreeFlame.get_flame_speed_reaction_sensitivities`. For
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# other sensitivities, the method `Sim1D.solve_adjoint` can be used.
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# Create a DataFrame to store sensitivity-analysis data
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sens = pd.DataFrame(index=gas.reaction_equations(), columns=["sensitivity"])
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# Use the adjoint method to calculate sensitivities
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sens = f.get_flame_speed_reaction_sensitivities()
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sens.sensitivity = flame.get_flame_speed_reaction_sensitivities()
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print()
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print('Rxn # k/S*dS/dk Reaction Equation')
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print('----- ---------- ----------------------------------')
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for m in range(gas.n_reactions):
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print(f"{m: 5d} {sens[m]: 10.3e} {gas.reaction(m).equation}")
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# Show the first 10 sensitivities
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sens.head(10)
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# %%
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# Sort the sensitivities in order of descending magnitude
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sens = sens.iloc[(-sens['sensitivity'].abs()).argsort()]
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fig, ax = plt.subplots()
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# Reaction mechanisms can contains thousands of elementary steps. Limit the plot
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# to the top 15
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sens.head(15).plot.barh(ax=ax, legend=None)
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ax.invert_yaxis() # put the largest sensitivity on top
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ax.set_title("Sensitivities for GRI 3.0")
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ax.set_xlabel(r"Sensitivity: $\frac{\partial\:\ln S_u}{\partial\:\ln k}$")
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ax.grid(axis='x')
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#sphinx_gallery_thumbnail_number = -1
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