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 (\((\mbox{molecule}\,\mathrm{cm}^{-3})^{-(n-1)}\,\mathrm{s}^{-1}\)),

  • \(n\) 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 has the following format:

 type: TAYLOR_SERIES
 A: 123.45
 Ea: 123.45
 B: 1.3
 D: 300.0
 E: 6.0e-06
 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
   ...

 {
     "type" : "TAYLOR_SERIES",
     "A" : 123.45,
     "Ea" : 123.45,
     "B"  : 1.3,
     "D"  : 300.0,
     "E"  : 0.6E-5,
     "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. Reactants without a qty value are assumed to appear once in the reaction equation. Products without a specified yield are assumed to have a yield of 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.

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