Taylor Series Reaction Rates#

Taylor Series-like reaction rate constant equations are calculated as follows:

\[A e^{-\frac{E_a}{k_bT}} \left(\frac{T}{D}\right)^B (1.0 + E \cdot P) + \sum_{i=0}^{m} C_i T^i\]

where:

  • \(A\) is the pre-exponential factor (\((\mbox{mol}\,\mathrm{m}^{-3})^{-(n-1)}\,\mathrm{s}^{-1}\)),

  • \(n\) is the number of reactants,

  • \(E_a\) is the activation energy \((\mathrm{J})\),

  • \(k_b\) is the Boltzmann constant \((\mathrm{J}/\mathrm{K})\),

  • \(D\) \((\mathrm{K})\), \(B\) (unitless), and \(E\) (\(\mathrm{Pa}^{-1}\)) are reaction parameters,

  • \(T\) is the temperature \((\mathrm{K})\), and \(P\) is the pressure \((\mathrm{Pa})\).

  • \(C_i\) are the Taylor series coefficients, and \(m\) is the order of the Taylor series.

The first two terms are described in Finlayson-Pitts and Pitts (2000) [FPJ99]. The final term before the Taylor series is included to accommodate CMAQ EBI solver type 7 rate constants. The final term is the Taylor series expansion, which allows for more complex temperature dependencies beyond the Arrhenius form.

Input data for Taylor Series equations have the following format:

 type: TAYLOR_SERIES
 name: foo-taylor
 A: 123.45
 Ea: 123.45
 B: 1.3
 D: 300.0
 E: 6.0e-06
 gas phase: gas
 taylor coefficients:
   - 1.0
   - 2.0
   - 3.0
 reactants:
   - species name: foo
   - species name: bar
     coefficient: 2
   ...
 products:
   - species name: baz
   - species name: qux
     coefficient: 0.65
   ...

 {
     "type" : "TAYLOR_SERIES",
     "name" : "foo-taylor",
     "A" : 123.45,
     "Ea" : 123.45,
     "B"  : 1.3,
     "D"  : 300.0,
     "E"  : 0.6E-5,
     "gas phase" : "gas",
     "taylor coefficients" : [
         1.0,
         2.0,
         3.0
     ],
     "reactants" : [
       { "species name" : "spec1" },
       { "species name" : "spec2", "coefficient" : 2 },
       ...
     ],
     "products" : [
       { "species name" : "spec3" },
       { "species name" : "spec4", "coefficient" : 0.65 },
       ...
     ]
 }

The key-value pairs reactants and products are required. When a coefficient is not specified for a reactant or product, it is assumed to be 1.0.

Optionally, a parameter C may be included, and is taken to equal \(\frac{-E_a}{k_b}\). Note that either Ea or C may be included, but not both. When neither Ea nor C are included, they are assumed to be 0.0. When A is not included, it is assumed to be 1.0. When D is not included, it is assumed to be 300.0 K. When B is not included, it is assumed to be 0.0. When E is not included, it is assumed to be 0.0. When taylor coefficients are not included, the order of the Taylor series is assumed to be 0.

Rate constants are in units of \(\mathrm{(m^{3}\ mol^{-1})^{(n-1)}\ s^{-1}}\) where \(n\) is the total number of reactants.

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