Quasi-stationary flow

The terminus quasi-stationary flow means that the flow of particles Jj changing their aggregation number from j to j-1 does not depend on j in the minimum region (Jj = J = -Jm at 1 < j < s ). Then in analogy to Eq. (5.176) we get for the region of the minimum... 

When we consider again the time interval essentially exceeding ti, the condition of the quasi-stationary flow of monomers in the minimum region of the size distribution of aggregates is justified and the relations (5.216) to (5.220) hold. Then, substituting expressions (5.217) for Sc, 8Cp, and (5.218) for 8cj into equation (5.225), summing up from j = s to j = jmax and using relations (5.221), (5.222) we get... 

Io(x) and l2(x) are modified Bessel functions of zero and second order, Bb and pe are the bulk elasticity modulus and the density of phase B, / is the capillary length (/ ac). Substituting Eqs. (12) and (14) into (8) one can obtain the general expressions for the bubble (drop) volume variation 6V ,(ico), the pressure variations inside the bubble (drop) 5P, (ici)), and inside the cell 6P (ico) at the given bP, and 6Vp . The effect of both the flow mobility and the compressibility of the medium in the capillary on the oscillating bubble or drop measurements is analysed in details in [26]. Here only the simpler case of quasi-stationary flow of incompressible media in the capillary is described. 

With account for Eqs. (10), (11) and (20) one obtains the equations for the meniscus volume and pressure variation in the cell which are valid for the case of quasi-stationary flow of an incompressible medium in the capillary... 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

The auxiliary conditions are electroneutrality, no electric current flow and a quasi-stationary state. 

It is reasonable to assume that in the vast majority of cases encountered in reactive processing Re < < 1 and (H/L)Re < < 1. Thus, we can consider the flow to be quasi-stationary and that temperature changes occur quickly after alterations in the temperature and degree of conversion distributions.202 Now we can rewrite the system of balance equations in the following dimensionless form ... 

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

Note that equation (5.1) follows immediately from Fick s laws on the assumption of a quasi-stationary distribution of the concentration of components within the diffusion boundary layer. Indeed, if in this layer cAl t 0, then the second Fick law yields cA x const. It means that the distribution of the concentration of component A within this layer is close to linear (Fig. 5.1). Anywhere outside of this layer, the concentration of A is assumed to be the same and equal to an instantaneous value, c. This implies sufficiently intensive agitation of the liquid. In such a case, the flow of A atoms across the diffusion boundary layer under the condition of constancy of the surface area of the dissolving solid is... 

Some typical disturbance patterns and control difficulties are summarized here. A detailed discussion is made in (1). Hydraulic disturbances are significant in amplitude. Diurnal variations as well as shock loads from rain storms or melting snow may cause major upsets. Significant disturbances also appear from internal sources like primary pumps, back-washing of deep bed filters or return sludge flow rate changes. The amplitudes are such, that quasi-stationary of linear control methods are seldom adequate. 

Secondly, it is worth noting the simplification of enthalpy balances which occurs using the quasi-stationary state approximation. The writing of the energy balance causes a reaction enthalpy flow ti to appear, where... 

To illustrate the concepts of determining, non-determining and negligible processes, the mechanism of the pyrolysis of neopentane will be discussed briefly here. Neopentane pyrolysis has been chosen because it has been studied by various techniques batch reactor [105— 108], continuous flow stirred tank reactor [74, 109], tubular reactor [110], very low pressure pyrolysis [111], wall-less reactor [112, 113], non-quasi-stationary state pyrolysis [114, 115], single pulse shock tube [93, 116] amongst others, and over a large range of temperature, from... 

The measurements on the research facility were carried out at stationary or quasi stationary conditions. The measurements of air flows, gas temperatures, gas composition, and heat output were analysed continously and monitored online. The gas composition was analyzed in the flue gas after the boiler exit with industrial gas analyzers. For the analysis of the hot gas in the reduction zone a suction pyrometer combined with a probe for detection was used. With this probe also short fluctuations could be monitored with extremely short delay. Besides, a hot gas sampling line with different analyzers for measuring the gas in the reburn zone was installed. Table 2 gives on overview over the gas analysis equipment. 

Reactions also occur with the carbon parts of the furnace and the graphite boats, but these are of minor importance. Excessive carbon losses occur in flowing hydrogen if the carburization is carried out in open graphite boats. These losses can be prevented by using graphite covers, thus forming a quasi-stationary atmosphere above the W + C powder mixture. 

The data on initiated styrene polymerization were used in calculations. The reaction system was regarded as quasi-stationary from a hydrodynamic point of view, meaning that the flow velocity profiles instantaneously adjust themselves to variations of viscosity through temperature and conversion. 

In our experiments, the compression plasma flows were obtained in a quasi-stationary plasma accelerator known as the magnetoplasma compressor (MPC)... 

Once again, other dimensionless numbers are used to represent the shifting boundary conditions. The periodic behavior of SMB processes is induced by the stepwise switching of the inlet and outlet ports. To describe this operation as a quasi-stationary process comparable to the TMB process the following flow rates are introduced ... 

The action of compression plasma flows (CPF), generated by quasi-stationary plasma accelerators, upon solid surfaces leads to a substantial modification of surface properties of exposed materials [1-3]. It was found that exposure of silicon crystals to CPF causes formation of sub-micron bulk periodic structures on its surface. These structures are of great interest for development of nanoelectronic devices. 

Compression plasma flow was obtained using a quasi-stationary plasma accelerator of the magneto-plasma compressor of compact geometry whose energy storage system consisted of a capacitor bank of 1200 pF at an initial... 

The first parameter, the Strouhal s number, is a measure of non-stationarity of the process. At St 1 or t L/U the first term in the left part of (5.107)-(5.109) can be neglected. At that, the flow can he considered as stationary. In this connection it is necessary to notice, that there are problems in which the time-dependence can exist at the boundary, for example in problems with time-varying interface (drops or bubbles deformation, surface waves in liquid etc.). In this case the bulk flow can be assumed stationary everywhere except for a thin layer near a surface. This kind of a problem is called sometimes the quasi-stationary problem. 

Assume that in the first approximation, the flow is quasi-stationary with respect to the mobile system of coordinates. Actually, it corresponds to the asymptotic solution at t R /D. Another assumption is that the longitudinal component of concentration gradient is constant, that is, 8C/dx = const. Then, in this approximation, concentration depends only on r and is described by the equation... 

If drop sizes are small enough, and the difference in densities of the internal and external liquid is sufficiently small, the flow of the liquid can be considered as slow, and the motion of drops - as inertialess. This is the case when we separate emulsions of the w/o (water-oil) type. Then equations (11.32) are reduced to the equations of inertialess motion of drops in the quasi-stationary approximation... 

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