The steady-state numerical method described in Steady-state Solver for Homogeneous Systems and Steady-state 1-D Solution Methods facilitates sensitivity analysis by having already computed the solution Jacobians. Once the Jacobian has been computed for the purposes of solving the nonlinear equations, the sensitivity coefficients are easily calculated. These techniques have been developed over many years and reported in the chemical engineering literature. [129], [130], [131]

To specify the sensitivity coefficients for steady-state computations, we begin by rewriting Equation 16–8 , which represents the vector of governing equations, as

(17–1)

Here, we have introduced the idea that the equation may depend not only on the solution variables but also on a set of model parameters, . The residual vector depends, both explicitly and implicitly, on the solution vector . For reaction-rate sensitivity, we consider the ’s to represent the pre-exponential "A-factors" in the Arrhenius reaction-rate expressions. For heat-of-formation sensitivity, represents the vector of heats of formation for all the species in the system (available for Premixed Laminar Flame, Flame-speed Calculator, Opposed-flow Flames, Stagnation-flow and Rotating-disk CVD reactor models). By differentiating Equation 17–1 with respect to an we obtain a matrix equation for the sensitivity coefficients:

(17–2)

The matrix is the Jacobian of the original system and is the matrix of partial derivatives of with respect to the parameters. The sensitivity coefficients are defined as . It is helpful to think of the matrix column by column, with each column indicating the dependence of the residual vector on each parameter. There are as many columns as there are parameters, that is, the number of gas-phase and surface reactions. The sensitivity coefficient matrix contains quantitative information on how each reaction-rate coefficient affects the temperature(s) and species fractions. The sensitivity coefficient matrix has a structure similar to that of the matrix. That is, each column contains the dependence of the solution vector on a particular chemical reaction.

The Jacobian and its LU factorization are already available from the solution of the original system by Newton’s method. Parameter derivatives are computed in a manner similar to the computation of the Jacobian. We can therefore readily solve the linear system of equations represented by Equation 17–2 for each column of the sensitivity coefficient matrix corresponding to the sensitivities of the solution vector to each of the reaction rates in the Gas-phase Kinetics and Surface Kinetics mechanisms. Twopnt employs the LINPACK [132] software to perform these computations. We factor the Jacobian only once, and compute each column of the sensitivity coefficient matrix by back substitution, such that the calculation is relatively inexpensive computationally.