Degree of Rate Control
The degree of rate control (DRC), introduced by Campbell, quantifies how much each elementary step (or each intermediate, in its thermodynamic form) controls a net reaction rate. It is defined as the derivative of the log of the rate with respect to the log of a step’s forward rate constant, with that step’s equilibrium constant and all other rate constants held fixed, and it generalizes the idea of a single rate-determining step to complex networks, including transient and selectivity variants.
Equilibrium-preserving sensitivity analysis for NH3 decomposition on Ru (2004)
To identify the rate determining step in NH3 decomposition on Ru near 50% conversion, Mhadeshwar and co-workers perturbed the forward and reverse pre-exponential factors of each reversible step together, so that equilibrium constants and partial equilibria were unchanged, and reported normalized sensitivity coefficients. Without N-N interactions N2 desorption dominated; with them NH2* decomposition (NH2* + * to NH* + H*) controlled the rate because it is far from partial equilibrium and NH2* coverage is tiny (1e-5 to 1e-8) (Mhadeshwar et al. 2004). Summary: mhadeshwar2004.
Degree of rate control by automatic differentiation (2022)
Yang, Achar and Kitchin computed steady-state and transient DRCs with forward-mode automatic differentiation through stiff ODE solvers (Julia ForwardDiff with DifferentialEquations) instead of finite differences (Yang et al. 2022). AD reproduced the analytical transient DRCs of a hypothetical two-step reaction and the published water-gas shift result (OH* dissociation 0.88, CO2* formation 0.12). For a 17-step propylene partial oxidation mechanism, finite differences with perturbations of 1e-4, 1e-11 and 1e-14 deviated from AD by up to 7%, about 1% and up to 845% respectively, showing the truncation and rounding error trade-off that AD avoids, and AD was about ten times faster (0.1 versus 1.0 s). Combining AD with DELSA gave distributions of step importance over an uncertain parameter space, identifying the same key steps (3, 14 and 17) as the DRC but revealing a bimodal importance for step 14. Summary: yang2022b.