Stagnation flow derivation

Deriving the axisymmetric stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.60, and 3.60), but considering flow only in the z-r plane. In general, there may be a circumferential velocity component ui, but there cannot be variations of any variable in the circumferential direction 0. The derivation depends on two principal conjectures. First, the velocity field is presumed to be described in terms of a streamfunction that has the separable form... 

The equations for stagnation flow are quite general. However, it is instructive initially to restrict attention to incompressible, isothermal flows. Here the general features can be demonstrated and a concrete connection can be made to historical literature. With these further restrictions on the equations derived in the previous section, the equations governing the fluid flow become... 

As with semi-infinite stagnation flow, there are no natural physically observable length and velocity scales that form the basis for nondimensionalization. Rather, mathematically derived length and velocity scales lead to a nondimensional system of equations that are parameter free. These scales are... 

Derive the nondimensional thermal-energy equation for an axisymmetric, semi-infinite stagnation flow of a constant-property incompressible fluid. 

Derive the relationship between the vorticity-transport equation (Eq. 6.47) and the radial-momentum equation for semi-infinite stagnation flow. In other words, carefully fill in all the missing pieces in the derivation leading to Eq. 6.55. In the course of this exercise, review how to apply integration by parts. 

Following the general approach discussed in Section 6.2 for axisymmetric flows, derive the general equations for planar stagnation flow. The planar equations are summarized, but not derived, in Section 6.9. Discuss the differences and similarities between the two stagnation flows. 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

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. 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

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.

A brief explanation of differential-algebraic equations (DAE) facilitates a further mathematical discussion of the stagnation-flow equations. In general, DAEs are stated as a vector residual equation, where w is the dependent-variable vector and the prime denotes a time derivative. For the discussion here, it is convenient to consider a restricted class of DAEs called semi-explicit nonlinear DAEs, which are represented as... 

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. 

In retrospect, the effect of the change of variables has been to deform the velocity field from axisymmetric stagnation flow over a sphere to linear shear flow along a flat plate. The main advantage of the new coordinates is that the coefficients of the derivatives in (32) are independent of X, and consequently Duhamel s theorem can be applied. Thus, the following procedure can be used (1) solve equation (32) subject to a uniform surface concentration (2) extend this solution to one valid for an arbitrary, nonuniform surface concentration by applying Duhamel s theorem (3) select the surface concentration which satisfies (33a). 

For the heated vertical plate and horizontal cylinder, the flow results from natural convection. The stagnation configuration is a forced flow. In each case the flow is of the boimdai7 Kiyer type. Simple analytical solutions can be obtained when the thickness of the du.st-free space is much smaller than that of the boundary layer. In this case the gas velocity distribution can be approximated by the first term in an expansion in the distance norroal to the surface. Expressions for the thickness of the dust-free space for a heated vertical surface and a plane stagnation flow are derived below. 

The study of reaction kinetics in flow reactors to derive microkinetic expressions also rehes on an adequate description of the flow field and well-defined inlet and boundary conditions. The stagnation flow on a catalytic plate represents such a simple flow system, in which the catalytic surface is zero dimensional and the species and temperature profiles of the estabhshed boundary layer depend only on the distance from the catalytic plate. This configuration consequently allows the application of simple measurement and modehng approaches (Sidwell et al., 2002 Wamatz et al., 1994a). SFRs are also of significant technical importance because they have extensively been used for CVD to produce homogeneous deposits. In this deposition technique, the disk is often additionally forced to spin to achieve a thick and uniform deposition across the substrate (Houtman et al., 1986a Oh et al., 1991). 

The theoretical results that have been discussed here allow k to be of order unity, a condition that sometimes has been referred to as one of strong strain, although the term moderate strain seems better, with strong strain reserved for large values of k. Small values of k are identified as conditions of weak strain under these conditions the reaction sheet remains far to the reactant side of the stagnation point, and by integrating across the convective-diffusive zone, a formulation in terms of the location of the reaction sheet can be derived [102], like that discussed at the end of Section 9.5.1. By combining the approximations of weak strain and weak curvature, convenient approaches to analyses of wrinkled flames in turbulent flows can be obtained [38]. 

Compressible ID stagnation-point flow analysis forms the basis of the equation system presented below. It was found that the prediction of the effect of internal mass transfer limitations in the catalytic washcoat of the SFR configuration is crucial to derive microkinetic data from SFR experiments (Karadeniz, 2014 Karadeniz et al., 2013) our model is extended to include the diffusion limitations due to a porous layer. It should be noted that the CFiEMKIN code has no abihty to account for internal mass transport in the catalytic coating. 

In this derivation we have ignored the difference between the static and the stagnation temperatures. This is important when the gas velocity is high. However, in the fuel cell itself the kinetic energy of the gas is not important. Even in diesel engines, whose flow rates after the compressor are very unsteady and often rapid, the equations given here are usually used. 

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