Local sensitivity coefficient

Let us consider the solution of the induced kinetic differential equation at a fixed time point as the function of the rate constants and let us consider the Taylor series of the solution with respect to the parameters around the nominal value of the parameters. The coefficients of this Taylor series are the local sensitivity coefficients. In general only the first order coefficients are considered. 

Here the partial derivative dYildxj is called the first-order local sensitivity coeffi-dent, the second-order partial derivative d YJdx dxj is called the second-order local sensitivity coefficient, etc. Commonly only the first-order linear sensitivity coefficients dYJdxj are calculated and interpreted, although we will see in Sect. 5.6.5 that this may cause problems in some cases. The local sensitivity... 

If the local sensitivity coefficients are calculated using Eq. (5.6) (this is called the direct method), then the error of calculation can be estimated and limited. The... 

Here are the local sensitivity coefficients (Turanyi 1990). This approach has been used, for example, for the uncertainty analysis of the RADM2 tropospheric chemical mechanism (Gao et al. 1995). 

Atherton et al. (1975) provided an early example of the application of local uncertainty analysis in the chemical engineering literature. They calculated the variance of the output of dynamic models from the variance of the parameters cP ixj) and the local sensitivity coefficients dYJdxf. 

Fig. 5.23 Local sensitivity coefficients of the laminar flame velocity of a stoichiometric methane-air flame. Grey stripes refer to the local sensitivity coefficients at the nominal parameter set During the Monte Carlo analysis, the local sensitivity coefficients were calculated for each parameter set, which allowed the calculation of the standard deviation of the sensitivity coefficients small bars interconnected with a horizontal line) and the attainable minimum and maximum sensitivity coefficients at any parameter set within the uncertainty limits of parameters (outer larger bars). Adapted with permission from Z or et al. (2005b). Copyright (2005) American Chemical Society...

The summation refers to all the Nr reaction steps. For the calculation of lifetime not only the chemical lifetime t,- is taken into account but also the residence time in a reactor and the species rate of diffusion. The half-normalised local sensitivity coefficient 3T,/3 In A shows the effect of perturbing the A-factor of reaction step I on concentration T and Vy is the corresponding stoichiometric coefficient. The index (LOI)jy estimates the error of the calculadcHi of the concentration of species j due to the application of the QSSA on species i. 

Here T,(x) is the stored exact value, Ax is the deviation of the queried point from the stored point, and d YJdxj and d YJdx dxj are the first-order and second-order local sensitivity coefficients, respectively. There are several efficient numerical methods for the calculation of the first-order local sensitivity coefficients (see Sect. 5.2). The second-order local sensitivity coefficients can be calculated from the first-order coefficients using a finite-difference approximation. The Taylor series approximations have the general disadvantage that the accuracy significantly decreases further from the central point. 

MCA distinguishes between local and global (systemic) properties of a reaction network. Local properties are characterized by sensitivity coefficients, denoted as elasticities, of a reaction rate v,(S,p) toward a perturbation in substrate concentrations (e-elasticities) or kinetic parameters ( -elasticities). The elasticities measure the local response of a reaction in isolation and are defined as the partial derivatives at a reference state S°... 

The sensitivity analysis of a system of chemical reactions consist of the problem of determining the effect of uncertainties in parameters and initial conditions on the solution of a set of ordinary differential equations [22, 23], Sensitivity analysis procedures may be classified as deterministic or stochastic in nature. The interpretation of system sensitivities in terms of first-order elementary sensitivity coefficients is called a local sensitivity analysis and typifies the deterministic approach to sensitivity analysis. Here, the first-order elementary sensitivity coefficient is defined as the gradient... 

For a local sensitivity analysis, Eq. (2.69) may be differentiated with respect to the parameters a to yield a set of linear coupled equations in terms of the elementary sensitivity coefficients, cXM /daj. 

According to (5.44) the semi-logarithmic sensitivity coefficients show the local change in the solutions when the given parameter is perturbed by unity on the logarithmic scale and are invariant under the scaling of the parameters. 

To illustrate the method, the water molecule is taken as a case study. Given the geometry (Rqh = 0.9700 A and Rhh = 1.5288 A) and the hardness parameters (r ), it is possible to construct the hardness matrix as is shown in Table 3. After diagonalization the eigenvalues h and eigenvectors U e are obtained the softness matrix is obtained by inverting q. The softness kernels can also be calculated by using Eq. (22) and local softnesses are obtained via Eq. (7), (23) or (27). Table 3 also summarizes some results for the other sensitivity coefficients. 

Clearly the charges and local softnesses contain different information since they sometimes change in the opposite direction. Even without proton jump, the sensitivity coefficients of the pyridine molecule are drastically affected. In this model, the role of the active surface is explicitly taken into account and can help us to better understand its influence on the activation of substrates. 

In the general case, we have to investigate how a small change of rate coefficient kj changes the production rate dyjdt of product T,. This effect appears in the local rate sensitivity coefficient d(AcJAt)ldkj (see Sect. 5.2). If this coefficient is much higher for reaction j than for the other reaction steps, then reaction j is the ratedetermining step of the production of species i (Turanyi 1990). 

If the sensitivity functions are locally similar at time t, then the ratios of any two sensitivity coefficients are independent of the selection of the modified parameter. Let us select another variable and substitute Eq. (8.2) that defines the local similarity into Eq. (8.20) ... 

Equation (8.24) shows that the ratio of two sensitivity coefficients at any time t > fi is independent of the selection of the model result T, and time. Therefore, the corresponding sensitivity functions are globally similar. The meaning of Eqs. (8.16) to (8.24) can be summarised as follows. If the sensitivity differential equations are pseudo-homogeneous in the time interval (tj, 12) and the sensitivity coefficients are locally similar at time ti, then the sensitivity functions are globally similar in the time interval (ti, 12). The ratio fi/cm is independent of the selection of model ouqjut 7, and therefore Eq. (8.21) implies the presence of global and local similarity at the same time. 

Zsely et al. (2003) performed numerical experiments to investigate this consequence of global similarity. Initially the concentration profiles were calculated for simulations of the adiabatic explosion of a stoichiometric hydrogen-air mixture using a nominal parameter set based on the values recommended by Baulch et al. (2005). Local sensitivity analysis was then used to select those parameters with the largest influence oti the simulated species concentrations based on a study of A-factors for the reaction rate coefficients. Five reactions were selected as dominating the influence on the calculated concentrations. At the next stage, the... 

---