Inlet Stagnation Conditions

The term /R Tot is proportional to the speed of sound at the inlet stagnation conditions, and is a velocity term. Hence, bearing in mind the definition of the Reynolds number ... 

Figure 8-2. NsDs diagram for a turbine stage. Efficiency is on a total-to-total basis that is, it is related to inlet and exit stagnation conditions. Diagram values are suitable for machine Reynolds number Re > 10 . (Balje, O.E., A Study of Reynolds Number Effects in Turbomachinery, Journal of Engineering for Power, ASME Trans., Vol. 86, Series A, p. 227.)...

Here, the subscript 0 indicates the stagnation conditions (M 0) corresponding to a specific flow state. In practice, these are the conditions of the gaseous products of chemical decomposition or combustion that exist upstream of the nozzle inlet and, therefore, may be regarded as known states for the purposes of the nozzle modeling. 

For a compressible flow, this is the thermodynamic state that would exist if a flow were brought to rest isenfropi-cally. In practice, this would correspond to the thermodynamic state of a very low-speed flow entering the nozzle inlet from an upstream combustion chamber or a pressurized reservoir. For this reason, stagnation conditions are also sometimes referred to as chamber or reservoir conditions. 

The gas flow (nitrogen N2) through conventional straight bore holes or apertures during expansion of gas from an atomizer nozzle is that of under-expanded jets if the atomization pressure is above the critical value 0.189 MPa. In this case the gas exits the nozzle at a Mach number Ma = UJc = 1. The state of gas at the nozzle exit, that is at the Inlet in Figs. 18.15 and 18.16, can be determined by the stagnation conditions in the atomizer as follows ... 

Figure 26.4 Surface temperature and surface fuel mole fraction (a), and NO (6) as functions of inlet composition, along the adiabatic curve, for the stagnation reactor (solid curves) and the PSR (dashed curves). The fuel-lean and fuel-rich regions are indicated. The conditions are pressure of 1 atm, inlet temperature of 25 °C, a strain rate of 1000 s (stagnation reactor), and a residence time of 1 ms (PSR)...

Figure 26.5 Surface mole fractions of fuel and NO as functions of stagnation surface (solid curves) and PSR temper-atirre (dashed curves), for 28% inlet H2 in air (a) and 12% inlet H2 in air (b). The maximum temperature indicates adiabatic operation. The conditions are the same as in Fig. 26.4...

Figure 26.6 Wall conductive flux and surface fuel mole fraction (a) and NOa, near the surface (6) vs. the inverse of the strain rate for a stagnation reactor with the surface at temperatures of 500 K (dashed curves) and 1000 K (solid curves). The conditions of pressure and inlet temperature are the same as in Fig. 26.4...

Conditions favoring efficient separation of flocculent metal precipitates in a basin include a low surface overflow rate, adequate depth, and inlet and outlet designs for a uniform velocity field with minimal short circuiting or stagnation. Solids separation is likely to limit the overall efficiency of metals removal in treatment, so pilot studies include batch and continuous flow settling column studies.18... 

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. 

This difference formula propagates axial-velocity u information from the lower boundary (e.g., stagnation surface) up toward the inlet-manifold boundary. At the stagnation surface, a boundary value of the axial velocity is known, u = 0. A dilemma occurs at the upper boundary, however. At the upper boundary, Eq. 6.106 can be evaluated to determine a value for the inlet velocity u j. However, in the finite-gap problem, the inlet velocity is specified as a boundary condition. In general, the velocity evaluated from the discrete continuity equation is not equal to the known boundary condition, which is a temporary contradiction. The dilemma must be resolved through the eigenvalue, which is coupled to the continuity equation through the V velocity and the radial-momentum equation. 

The eigenvalue equation (Eq. 6.108) and the continuity equation (Eq. 6.105) are both first-order equations, which together demand two boundary conditions. The two required boundary conditions are the axial velocities at the boundaries, the inlet and the stagnation surface. This specification requires that the residual equations at the upper boundary do not have a direct correspondence with the dependent variables. At the stagnation surface ( / = 1), the residual equations are stated as... 

In these equations, as in the finite-gap stagnation flow, the Reynolds number is based on the gap distance and the inlet velocity, Rcy = p aUL/n, the Prandtl number is evaluated at the inlet conditions, and the nondimensional eigenvalue is given as Ar = ArL2 / pmU2. 

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. 

The pressure-gradient term Ar requires either a further boundary condition or a determination of the domain size. In the semi-infinite cases, Ar is a constant that is specified in terms of the outer potential-flow characteristics. However, the extent of the domain end must be determined in such a way that the viscous boundary layer is entirely contained within the domain. In the finite-gap cases, the inlet velocity n(zend) is specified at a specified inlet position. Since the continuity equation is first order, another degree of freedom must be introduced to accommodate the two boundary conditions on u, namely u — 0 at the stagnation surface and u specified at the inlet manifold. The value of the constant Ar is taken as a variable (an eigenvalue) that must be determined in such a way that the two velocity boundary conditions for u are satisfied. 

The equations for the opposed-flow situation are exactly those of the finite-gap stagnation flow. The boundary conditions are altered to represent the inlet velocities at both boundaries. For example,... 

As an illustration, consider the stagnation flow over a catalytic surface during an ignition event. The inlet flow is steady, but the surface temperature increases as power through the platinum-foil surface increases. At a certain temperature the catalytic ignition occurs very rapidly. The flow configuration and conditions, which are taken from Deutschmann [101], are u n — 8 cm/s, Tln = 300 K, with an inlet mixture of 3% CH4, 3% O2, and 94% N2. The inlet-to-surface separation is L = 5 cm, and the surface Pt sites are initially covered... 

Hence the ratio of the throat pressure to the stagnation inlet pressure at critical conditions is ... 

The inlet pipe has a diameter of 8 inches, and an area Ao = 50.27 in, which is between 33 and 66 times larger than the combined throat areas of the nozzle groups. On using equation (A6.8) and the inequality of (A6.7), it is found that the upper limiting value of the inlet velocity, cq, must very low for each group of nozzles less than 540/33= 16.3 m/s. It follows from equation (14.40), repeated below that the local inlet temperature must be almost identical with the stagnation temperature under all conditions ... 

Figure 16.13 Schlieren images of supersonic flow over various aspect-ratio cavities under off-design inlet conditions. Inlet flow stagnation pressure is 35 psi (a) and 120 psi (6).

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). 

Table 2.2 Stagnation disc temperature and inlet conditions...

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