Stagnation flow similarity

Not all reactor designs and operating conditions lead to the stagnation-flow similarity regimes illustrated in Fig. 6.1. Buoyancy-induced flow, owing to large temperature gra-... 

The discussion of stagnation flow usually considers flow that impinges on a solid surface. In genera], however, the surface itself is not needed for the stagnation-flow similarity to be valid. The opposed-flow situation illustrated in Fig. 6.19 is one in which the viscous boundary layer is in the interior of the domain, bounded by regions of inviscid flow on the top and the bottom. 

Flat flames can be made to impinge onto surfaces. Such strained flames can be used for a variety of purposes. On the one hand, these flames can be used in the laboratory to study the effects of strain on flame structure, and thus improve understanding of the fluid-mechanical effects encountered in turbulent flows. It may also be interesting to discover how a cool surface (e.g., an engine or furnace wall) affects flame structure. Even though the stagnation-flow situation is two-dimensional in the sense that there are two velocity components, the problem can be reduced to a one-dimensional model by similarity, as addressed in the book. 

Stagnation flows can be viewed either as a similarity reduction of the flow equations in a boundary-layer region or as an exact reduction of the Navier-Stokes equations under certain simplifying assumptions. Depending on the circumstances of a particular problem of interest, one or the other view may be more natural. In either case, the same governing equations emerge, with the differences being in boundary conditions. The alternatives are explored in later sections, where particular problems and boundary conditions are discussed. 

A principal assumption for similarity is that there exists a viscous boundary layer in which the temperature and species composition depend on only one independent variable. The velocity distribution, however, may be two- or even three-dimensional, although in a very special way that requires some scaled velocities to have only one-dimensional content. The fact that there is only one independent variable implies an infinite domain in directions orthogonal to the remaining independent variable. Of course, no real problems have infinite extent. Therefore to be of practical value, it is important that there be real situations for which the assumptions are sufficiently valid. Essentially the assumptions are valid in situations where the viscous boundary-layer thickness is small relative to the lateral extent of the problem. There will always be regions where edge effects interrupt the similarity. The following section provides some physical evidence that supports the notion that there are situations in which the stagnation-flow assumptions are valid. 

The similarity behavior for stagnation flows requires that V = v/r be a function of z alone. Usually the most practical condition for inlet radial velocity is that v = 0. However, a radial velocity that varies linearly with r is also an acceptable boundary condition as far as the similarity is concerned. An inlet boundary that specifies V equals a constant is mathematically acceptable, although manufacturing a manifold to deliver such a flow might be difficult. The axial velocity at the inlet must be independent of r under any circumstances for similarity to hold. 

A great many stagnation-flow solutions have been shown in the previous sections. The solution algorithms are similar to those discussed in Chapter 16. However, there are some differences and some complications that arise to deal with the special needs of stagnation flows. One issue has to do with computing the pressure-gradient eigenvalue in finite-gap problems. Another has to do with velocity reversal in opposed flows (Section 6.10). 

Fig. 6.20 Experimental particle paths in an opposed stagnation flow. A mixture of 25% methane and 75% nitrogen issues upward from the bottom porous-plate manifold and a mixture of 50% oxygen and 50% nitrogen issues downward from the top porous-plate manifold. The inlet velocity of both streams is 5.4 cm/s. Both streams are seeded with small titania particles that are illuminated to visualize the flow patterns. The upper panel shows cold nonreacting flow that is, the flame is not burning. In the lower panel, a nonpremixed flame is established between the two streams. Thermal phoresis forces the particles away from the flame zone. The fact that the flame region is flat (i.e., independent of radius) illustrates the similarity of the flow. Photographs courtesy of Prof. Tadao Takeno, Meijo University, Nagoya, Japan, and Prof. Makihito Nishioka, Tsukuba University, Tsukuba, Japan.

As with the axisymmetric stagnation-flow case, deriving the tubular stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.59, and 3.60). The approach depends on essentially the same assumptions as the axial stagnation flows described earlier, albeit with the similarity requiring no variation in the axial coordinate. The velocity field is presumed to be described in terms of a stream function that has the form... 

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. 

Note that the radial-flux term now involves a factor V = v/r, which in the similarity formulation of the stagnation-flow problem is a function only of z. Thus, per unit-area, this mass balance is independent of r, so long as the similarity assumptions apply. 

Because of the similarity behavior of the stagnation flow, the UI definition does not depend on the reactor size (radius). That is, it is valid per unit deposition surface area. Note also that the definition does depend on the inlet-to-surface dimension L, which can be an important design variable to improve the reactor efficiency. 

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. 

While our primary interest in this text is internal flow, there are certain similarities with the classic aerodynamics-motivated external flows. Broadly speaking, the stagnation flows discussed in Chapter 6 are classified as boundary layers where the outer flow that establishes the stagnation flow has a principal flow direction that is normal to the solid surface. Outside the boundary layer, there is typically an outer region in which viscous effects are negligible. Even in confined flows (e.g., a stagnation-flow chemical-vapor-deposition reactor), it is the existence of an inviscid outer region that is responsible for some of the relatively simple correlations of diffusive behavior in the boundary layer, like heat and mass transfer to the deposition surface. 

From (a) and (b), the stagnation pressure and temperature can thus be calculated at exit from the cooled row they can then be used to study the flow through the next (rotor) row. From there on a similar procedure may be followed (for a rotating row the relative (7 o)r, i and (po)k replace the absolute stagnation properties). In this way, the work output from the complete cooled turbine can be obtained for use within the cycle calculation, given the cooling quantities ip. 

Interestingly, the shape of the wake is similar to that developed behind a hypersonic blunt body where the flow converges to form a narrow recompression neck region several body diameters downstream of the rear stagnation point due to strong lateral pressure gradients. The liquid material, that is continuously stripped off from the droplet surface, is accelerated almost instantaneously to the particle velocity behind the wave front and follows the streamline pattern of the wake, suggesting that the droplet is reduced to a fine micromist. 

The entrance and exit regions in the spiral dam were also modified to eliminate the stagnant sections of the channel. The modification Is shown In Fig. 11.40. This modification allowed a relatively small amount of resin to flow Into the smaller channel at the entry such that stagnation of the resin cannot occur. A similar modification was made at the exit to allow a small amount of resin to flow out of the smaller channel into the main flow channel. To eliminate the unmelted particles or the particles that appeared to be more viscous because they were at a lower temperature, the clearance to the spiral dam was decreased from 0.76 to 0.25 mm. Since the meter channel depth was unchanged, the specific rotational flow rate for the modified screw was unchanged at 0.94 kg/(h-rpm). 

Similarity Relations for One-Dimensional, Constant-Area Channel Flow with Chemical Reactions. Similarity relations between stagnation temperature and mass fractions obtain during flow in a channel of constant cross section, provided a binary mixture approximation is used for the diffusion coefficient, the Lewis number is set equal to unity, the Prandtl number is set equal to 3/4, and a constant value is employed for the species and average isobaric specific heats. [The assumption that the species (cPii) and average (cp = 2YiCp,i) isobaric specific heats are... 

Fig. 6.13 Comparison of streamlines from rotating-disk solutions at two rotation rates. Both cases are for air flow at atmospheric pressure and T = 300 K. The induced inlet velocity is greater for the higher rotation rate. In both cases the streamlines axe separated by 27tA4< = 1.0 x 10-6 kg/s. The solutions are illustrated for a 2 cm interval above the stagnation plane and a 3 cm radius rotation plane. The similarity solutions themselves apply for the semi-infinite half plane above the surface.

The disk rotation is specified by a boundary condition for W at z = 0. In principal, a nonzero circumferential velocity could also be specified at the inlet. Physically, however, inlet swirl can lead to difficulties. When the flow swirls and the stagnation surface is stationary, a tomadolike circumstance is created. Fluid tends to be drawn radially inward near the stationary surface, which has deleterious consequences that are similar to starved flow. 

To illustrate the behavior of a stagnation flame impinging into a wall, consider the following example based on an atmospheric-pressure, stoichiometric, premixed, methane-air flame [271]. Geometrically the situation is similar to that shown in Fig. 17.1. The manifold-to-surface separation distance is one centimeter, the inlet mixture is at 300 K, and the surface temperature is maintained at Ts = 800 K. Figure 17.4 shows the flow field and flame structure for two inlet velocities. The flow is from right to left, with the inlet manifold on the right-hand side and the surface on the left. 

Vertical CVD Reactors. Models of vertical reactors fall into two broad groups. In the first group, the flow field is assumed to be described by the one-dimensional similarity solution to one of the classical axisymmetric flows rotating-disk flow, impinging-jet flow, or stagnation point flow (222). A detailed chemical mechanism is included in the model. In the second category, the finite dimension of the susceptor and the presence of the reactor walls are included in a detailed treatment of axisymmetric flow phenomena, including inertia- and buoyancy-driven recirculations, whereas the chemical mechanism is simplified to a few surface and gas-phase reactions. 

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