Added input file documentation for Chebyshev and P-log rate expressions
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Ray Speth
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2 changed files with 115 additions and 4 deletions
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@ -53,7 +53,7 @@ Arrhenius function
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k_f(T) = A T^n \exp(-E/\hat{R}T)
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whish is defined with an :class:`Arrhenius` entry::
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which is defined with an :class:`Arrhenius` entry::
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rate_coeff = Arrhenius(A=1.0e13, n=0, E=(7.3, 'kcal/mol'))
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rate_coeff = Arrhenius(1.0e13, 0, (7.3, 'kcal/mol'))
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@ -93,7 +93,7 @@ Options
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-------
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Certain conditions are normally flagged as errors by Cantera. In some cases,
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theey may not be errors, and the options field can be used to specify how they
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they may not be errors, and the options field can be used to specify how they
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should be handled.
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``skip``
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@ -267,7 +267,7 @@ al. [#Gilbert1983]_:
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N = 0.75 - 1.27\; \log_{10} F_{cent}
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The :class:`Troe` directive requires specifying the first three parameters
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:math:`(A, T_3, T_1)`. The fourth paramteter, :math:`T_2`, is optional, defaulting to 0.0.
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:math:`(A, T_3, T_1)`. The fourth parameter, :math:`T_2`, is optional, defaulting to 0.0.
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The SRI Falloff Function
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------------------------
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@ -289,6 +289,88 @@ the "SRI" falloff function. It is implemented by the :class:`SRI` directive.
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function, which has a C++ implementation, but doesn't appear to be implemented
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in the CTI or CTML parsers.
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Pressure-Dependent Arrhenius Rate Expressions (P-Log)
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=====================================================
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The :class:`pdep_arrhenius` class represents pressure-dependent reaction rates
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by logarithmically interpolating between Arrhenius rate expressions at various
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pressures. Given two rate expressions at two specific pressures:
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.. math::
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P_1: k_1(T) = A_1 T^{b_1} e^{E_1 / RT}
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P_2: k_2(T) = A_2 T^{b_2} e^{E_2 / RT}
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The rate at an intermediate pressure :math:`P_1 < P < P_2` is computed as
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.. math::
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\log k(T,P) = \log k_1(T) + \bigl(\log k_2(T) - \log k_1(T)\bigr)
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\frac{\log P - \log P_1}{\log P_2 - \log P_1}
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Multiple rate expressions may be given at the same pressure, in which case the
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rate used in the interpolation formula is the sum of all the rates given at that
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pressure. For pressures outside the given range, the rate expression at the nearest
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pressure is used.
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An example of a reaction specified in this format::
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pdep_arrhenius('R1 + R2 <=> P1 + P2',
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[(0.001315789, 'atm'), 2.440000e+10, 1.04, 3980.0],
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[(0.039473684, 'atm'), 3.890000e+10, 0.989, 4114.0],
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[(1.0, 'atm'), 3.460000e+12, 0.442, 5463.0],
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[(10.0, 'atm'), 1.720000e+14, -0.01, 7134.0],
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[(100.0, 'atm'), -7.410000e+30, -5.54, 12108.0],
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[(100.0, 'atm'), 1.900000e+15, -0.29, 8306.0])
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The first argument is the reaction equation. Each subsequent argument is a
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sequence of four elements specifying a pressure and the Arrhenius parameters at
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that pressure.
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Chebyshev Reaction Rate Expressions
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===================================
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Class :class:`chebyshev` represents a phenomenological rate coefficient
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:math:`k(T,P)` in terms of a bivariate Chebyshev polynomial. The rate constant
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can be written as:
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.. math:: \log k(T,P) = \sum_{t=1}^{N_T} \sum_{p=1}^{N_P} \alpha_{tp}
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\phi_t(\tilde{T}) \phi_p(\tilde{P})
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where :math:`\alpha_{tp}` are the constants defining the rate, :math:`\phi_n(x)`
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is the Chebyshev polynomial of the first kind of degree :math:`n` evaluated at
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:math:`x`, and
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.. math::
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\tilde{T} \equiv \frac{2T^{-1} - T_\mathrm{min}^{-1} - T_\mathrm{max}^{-1}}
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{T_\mathrm{max}^{-1} - T_\mathrm{min}^{-1}}
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\tilde{P} \equiv \frac{2 \log P - \log P_\mathrm{min} - \log P_\mathrm{max}}
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{\log P_\mathrm{max} - \log P_\mathrm{min}}
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are reduced temperature and reduced pressures which map the ranges
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:math:`(T_\mathrm{min}, T_\mathrm{max})` and :math:`(P_\mathrm{min},
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P_\mathrm{max})` to :math:`(-1, 1)`.
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A Chebyshev rate expression is specified in terms of the coefficient matrix
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:math:`\alpha` and the temperature and pressure ranges. An example of a
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Chebyshev rate expression where :math:`N_T = 6` and :math:`N_P = 4` is::
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chebyshev_reaction('R1 + R2 <=> P1 + P2',
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Tmin=290.0, Tmax=3000.0,
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Pmin=(0.001, 'atm'), Pmax=(100.0, 'atm'),
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coeffs=[[-1.44280e+01, 2.59970e-01, -2.24320e-02, -2.78700e-03],
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[ 2.20630e+01, 4.88090e-01, -3.96430e-02, -5.48110e-03],
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[-2.32940e-01, 4.01900e-01, -2.60730e-02, -5.04860e-03],
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[-2.93660e-01, 2.85680e-01, -9.33730e-03, -4.01020e-03],
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[-2.26210e-01, 1.69190e-01, 4.85810e-03, -2.38030e-03],
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[-1.43220e-01, 7.71110e-02, 1.27080e-02, -6.41540e-04]])
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Note that the Chebyshev polynomials are not defined outside the interval
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:math:`(-1,1)`, and therefore extrapolation of rates outside the range of
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temperatures and pressure for which they are defined is strongly discouraged.
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.. rubric:: References
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@ -307,5 +389,5 @@ the "SRI" falloff function. It is implemented by the :class:`SRI` directive.
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*Combustion and Flame*, 75:25, 1989.
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.. [#Kee1989] R. J. Kee, F. M. Rupley, and J. A. Miller. Chemkin-II: A Fortran
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chemical kinetics package for the analysis of gasphase chemical
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chemical kinetics package for the analysis of gas-phase chemical
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kinetics. Technical Report SAND89-8009, Sandia National Laboratories, 1989.
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@ -1309,6 +1309,16 @@ class falloff_reaction(reaction):
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class pdep_arrhenius(reaction):
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"""
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Pressure-dependent rate calculated by interpolating between Arrhenius
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expressions at different pressures.
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:param equation:
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A string specifying the chemical equation.
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:param args:
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Each additiona argument is a sequence of four elements specifying the
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pressure and the Arrhenius parameters at that pressure.
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"""
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def __init__(self, equation='', *args, **kwargs):
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self.pressures = []
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self.arrhenius = []
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@ -1328,6 +1338,25 @@ class pdep_arrhenius(reaction):
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class chebyshev_reaction(reaction):
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"""
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Pressure-dependent rate calculated in terms of a bivariate Chebyshev
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polynomial.
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:param equation:
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A string specifying the chemical equation.
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:param Tmin:
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The minimum temperature at which the rate expression is defined
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:param Tmax:
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the maximum temperature at which the rate expression is defined
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:param Pmin:
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The minimum pressure at which the rate expression is defined
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:param Pmax:
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The maximum pressure at which the rate expression is defined
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:param coeffs:
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A 2D array of the coefficients defining the rate expression. For a
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polynomial with M points in temperature and N points in pressure, this
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should be a list of M lists each with N elements.
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"""
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def __init__(self, equation='', Tmin=300.0, Tmax=2500.0,
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Pmin=(0.001, 'atm'), Pmax=(100.0, 'atm'),
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coeffs=[[]], **kwargs):
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