Quasi-steady model

The pressure-time variation in the driver section can be approximated by a quasi-steady model consisting of a single lumped-mass system with choked outflow at the minimum throat area when the ratio of the driver volume to the throat area is equal to 20 ft3/ft2 (6.1 m3/m2) or more. For smaller values of the ratio, the representation by a quasi-steady model becomes less accurate, and the numerical wave analysis approach is more suitable. 

Dispersed phase resistances are increased when surface contaminants reduce interfacial mobility. Huang and Kintner (H9) used Savic s stagnant-cap theory in a semiempirical model for this resistance. A simpler quasi-steady model is proposed here, analogous to that for continuous phase resistance. The Sherwood... 

The preceding expressions can easily be extended to non-Newtonian fluids by using, for the region near the wall, the quasi-steady model [67], For a power law fluid... 

In case (a), turbulence can be described as in a pipe, with the difference that stirring ensures the uniformity of temperature in the core of the fluid vessel. Near the wall, the process is described by the quasi-steady model. This leads to... 

The use of the compressibility term can be described as follows. The greater the stiffness a system model has, the more quickly the flow reacts to a change in pressure, and vice versa. For instance, if all fluids in the system are incompressible, and quasi-steady assumptions are used, then a step change to a valve should result in an instantaneous equilibrium of flows and pressures throughout the entire system. This makes for a stiff numerical solution, and is thus computationally intense. This pressure-flow solution technique allows for some compressibility to relax the problem. The equilibrium time of a quasi-steady model can be modified by changing this parameter, for instance this term could be set such that equilibrium occurs after 2 to 3 seconds for the entire model. However, quantitative results less than this timescale would then potentially not be captured accurately. As a final note, this technique can also incorporate flow elements that use the momentum equation (non-quasi-steady), but its strength is more suited by quasi-steady flow assumptions. 

The analysis of the conditions within a gas channel can also be assumed to be onedimensional given that the changes in properties in the direction transverse to the streamwise direction are relatively small in comparison to the changes in the stream-wise direction. In this section, we examine the transport in a fixed cross-sectional area gas channel. The principle conserved quantities needed in fuel cell performance modeling are energy and mass. A dynamic equation for the conservation of momentum is not often of interest given the relatively low pressure drops seen in fuel cell operation, and the relatively slow fluid dynamics employed. Hence, momentum, if of interest, is normally given by a quasi-steady model,... 

Fig. 25 In-phase component of pressure-coupled response function for propellant-N (50 atm). Shaded region represents T-bumer measure-ments Horton and Price [35] NWC [36]. Model calculations are for quasi-steady model using parameters of Table 2. Condensed-phase reaction zone characteristic frequency is estimated to be // = 10,000 Hz quasi-steady assumption should be valid up to at least 3000 Hz.

STABILITY ANALYSIS WITH QUASI-STEADY MODELLING OF SHEAR STRESSES, 327... 

Inspection of the K-H stability condition indicates that the structure of Equation 18 is invariant with the specific modelling of the wall and interfacial shear stresses and evolves essentially from the continuity equations and the left hand side of the momentum equations. On the other hand. Equation 19 for is directly related to the quasi-steady models adopted for the various shear stresses terms (the rhs of the two-fluid momentum Equation 8). In this sense, the form of 18 is general and is affected by the specific modelling of shear stresses only indirectly through the value. Thus, given different correlations for the shear stresses, the general form of 19 provides the corresponding values for... 

The general structure of stability Equation 18 remains unchanged when different quasi-steady models are applied for the various shear stresses terms. Moreover, even when the viscous effects are completely ignored, resorting to an inviscid K-H stability type of analysis, the structure of the resulting stability condition. Equation 18, is still maintained while Equation 19 for attains different expression. For instance, the long wave K-H stability analysis on two inviscid layers (rectangular channel) yields ... 

Figure 2. Typical neutral stability boundary obtained with quasi-steady modelling of the interfacial shear.

The stability conditions in Equations 18-19 correspond to a quasi-steady modelling of the various shear stresses hence, the effect of axial convection of the wave-... 

Temporal stability analysis is carried out on the linearized set of Equations 1, 2 and 7, 8 with Equation 23 replacing the quasi-steady model for x.. The resulting dispersion equation obtained for the complex wave celerity, C = oVk, as a function of the real wave number, k, is of identical form to Equation 16, except that d, includes now an additional term, which evolves from the dynamic component of the interfacial shear ... 

It is to be noted that the criterion for ill-posedness is affected only by the terms which are proportional to the gradients of h, u, u (derivatives with respect to time and space) and, therefore, apparently unaffected by the quasi-steady modelling of the shear stresses. However, the test for well-posedness is carried out on a stratified wavy configuration, which is represented by the averaged values of H, U, (obtained from the solution of AF = 0, Equation 11). Obviously, their values depend on the models used for the wall and interfacial shear stresses. In particular, the modelling of x. deserves a special attention since in the wavy regime the augmentation of the interfacial friction factor, due to the interfacial waviness is to be considered. 

Figures 10-13 represent some typical comparisons between the analytical boundaries and experiments. The figures include both the zero neutral stability line (ZNS) obtained with quasi-steady modelling of the interfacial shear stress, = 0, and the corresponding modified ZNS line obtained with as evolved from Equation 27. Along the ZNS, ZNS lines, -> 0, and, therefore, the destabilizing inertia terms...

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