The method used for transient sensitivity analysis (for 0-D Homogeneous and Plug-flow models) also takes advantage of the fact that the sensitivity equations are described in terms of the Jacobian of the model problem. For backwards-differencing methods, such as that used in DASPK, [123], [124] the Jacobian is required for solution of the model problem, so it is available for the sensitivity computation. In Reaction Design's modified version of DASPK, the sensitivity equations described below are solved simultaneously with the dependent variables of the solution itself.

The system of ordinary differential equations that describe the physical problem are of the general form

(17–3)

where, in our case, is the vector of temperature(s), mass fractions, surface site fractions, and bulk activities, as given for example in Equation 16–3 of Modified Damped Newton’s Method for 0-D Reactors . Here, the parameter vector is defined as in Equation 17–1 .

The first-order sensitivity coefficient matrix is defined as

(17–4)

where the indices and refer to the dependent variables and reactions, respectively. Differentiating Equation 17–4 with respect to the parameters yields

(17–5)

Note that this equation for the sensitivity coefficients is linear, even though the model problem may be nonlinear. Of course, when coupled with a nonlinear model problem, the whole system is still nonlinear. Nevertheless, when solved via the same backwards-differentiation formula method as the model problem, the sensitivity solution is efficient because of the linearity. The Newton iteration for the corrector step converges in one iteration.

The Jacobian matrix that appears in Equation 17–5 is exactly the one that is required by the backwards-differentiation formula method in solving the original model problem, so it is readily available for the sensitivity computation. Each column corresponds to the sensitivities with respect to one of the reaction pre-exponential constants. The solution proceeds column by column. Note that the Jacobian matrix is the same for each column of the . However, since the matrix describes the explicit dependence on each of the reaction parameters , each of its columns must be formed prior to solving for a column of .

In addition to determining time-integrated sensitivity of solution variables to reaction rate coefficients, derived sensitivity of growth or etch rates to the reaction parameters can also be calculated. Growth-rate sensitivity analysis for the transient calculations uses the same local-sensitivity calculations described for the steady-state calculations, as described in Sensitivity Analysis for Steady-state Solutions . In this case, the sensitivities are not integrated over time but are instead based only on local conditions at each time-step.