Stagnation flow INDEX

It is probably clear that any number of performance indexes can be written by comparing the various mass fluxes. The important point is that for the stagnation-flow geometries, all the mass fluxes can be written per unit surface area. Thus the indexes, which are ratios of fluxes, are independent of reactor size, so long as the reactor preserves the desirable similarity behavior. It is also important to note that these effectiveness indexes can be derived from the one-dimensional similarity simulations that consider the detailed chemical reaction behavior. 

Fig. 17.14 Finite-volume, staggered-grid, spatial-difference stencil for the transient compressible stagnation-flow equations. Grid points, which are at control-volume centers, are used to represent all dependent variables except axial velocity, which is represented at the control-volume faces. The grid indexes are shown on the left and the face indexes on the right. The right-facing protuberance on the stencils indicates where the time derivative is evaluated. For the pressure-eigenvalue equation there is no time derivative.

With the brief discussion of index, it is now possible to identify and compare some aspects of the high-index behavior of the constant-pressure and the compressible stagnation-flow equations. To understand the structure of the DAE system, it is first necessary to identify all variables that are not time differentiated (i.e., the x vector). In the constant-pressure formulation, neither the axial velocity u nor the pressure curvature A has time derivatives. By introducing the axial momentum equation, the compressible formulation introduces du/dt. To be of value in reducing the index, however, the momentum equation must be coupled to the other equations. The coupling is accomplished through pressure, which is included as a dependent variable. The variable A is not time differentiated in either formulation. 

It may be shown that the DAE system corresponding to the discrete form of the compressible stagnation-flow equations is of the so-called Hessenberg-index-two structure [46], which is represented by Eq. 17.29. The constraints g do not depend on x, and the matrix... 

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