Stiffness

The concept of stiffness is described for a system of linear equations.

Let l_i be the eigenvalues of the matrix A. The solution to this set of equations can be written in the following form.

where the l_i are the eigenvalues of the matrix A_ij, defined by the determinant:

.

This equation gives the eigenvalues

^.

Thus a set of equations has a set of eigenvalues. The stability conditions must be met for every eigenvalue [Finlayson, 1980]. The smallest eigenvalue is the one that determines how long one integrates to steady state, since that eigenvalue controls the slowest component. The largest eigenvalue is the one that restricts the time step the most, since it is the time step times the eigenvalue that must be below a critical value. Thus if there is a wide variation in the eigenvalues, it is necessary to integrate to a long time but with a small time step, thus leading to a slow, cumbersome calculation. This effect is usually measured in terms of the stiffness ration, defined as [Lambert, 1973]:

SR = 20 is not stiff, SR= 10³ is stiff and SR= 10⁶ is very stiff.

If the problem is nonlinear then the solution is expanded about the current state.

The question of stiffness then depends on the solution at the current time. Consequently, for nonlinear problems the problem can be stiff during one time period and not stiff during another time period. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stability and accuracy of the numerical scheme and the step-size used.

Problems are stiff when the time constants for different phenomena have very different magnitudes. Consider flow through a packed bed reactor.

The time constants for different phenomena are: (1) time for flow through the device

where Q is the volumetric flow rate, A is the cross sectional area, L is the length of the packed bed, and f is the void fraction;

(2) time for reaction

where k is a rate constant (time^-1);

(3) time for diffusion inside the catalyst

where e is the porosity of the catalyst, R is the catalyst radius, and D_e is the effective diffusion coefficient inside the catalyst;

(4) time for heat transfer is

where r_s is the catalyst density, C_s is the catalyst heat capacity per unit mass, k_e is the effective thermal conductivity of the catalyst, and a is the thermal diffusivity. For example, in the model of a catalytic converter for an automobile [Ferguson and Finlayson, 1974] the time constants for internal diffusion was 0.3 seconds, for internal heat transfer was 21 seconds and for flow through the device was 0.003 seconds. The flow through the device is so fast that it might as well be instantaneous. The stiffness is approximately 7,000. Implicit methods must be used to integrate the equations. Alternatively a quasi-state model can be developed [Ramirez, 1989].

Take Home Message: A problem is stiff if the eigenvalues of the linearized equation are of different orders of magnitude. Alternatively, if some phenomena being modeled happen very fast, and other phenomena happen very slowly, then the problem is stiff.