Equation of Motion for the Gas

The flow pattern of the gas can be determined by solving the Navier-Stokes equations if there is no influence from the particles. If we assume a Newtonian viscosity in Eq. (2.A.2), it becomes  

Using a characteristic velocity Vch, along with D and p as scaling parameters, we get the following dimensionless parameters  

Introducing these in Eq. (8.A.1), we get, after some work (Bird et ah, 2002) 

The dimensionless velocity and pressure of the gas are thus determined by Re and Fr. As mentioned in the main text, gravity only affects a flow pattern if the system contains free surfaces or stratification layers. Thus, Reynolds mun-ber similarity with geometric similarity is enough to ensure dynamic similarity for a gas cyclone with low solids loading. 

This is the formal requirement for dynamic similarity, and is consistent with the results of the classical dimensional analysis in the main text. As we mentioned there, experience teaches us that over a wide range of operating conditions Reynolds number similarity is not all that critical for Stokes number similarity between cyclones, and this indicates that, in this range, it is not all that critical for dynamic similarity. 

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In a) we list the variables influencing the cyclone performance, and arrange them in dimensionless groups. In b) we arrive at the groups by making the equations of motion for the gas and the particles dimensionless. Both lines of enquiry are enlightening in their own way, so we shall follow both, the latter in Appendix 8.A. 

This is as far as classical dimensional analysis can take us. However, in Appendix 8. A we obtain more physical insight by inspecting the equations of motion for the gas and the particles. One important result of this is that the density ratio in (8.1.5) need not appear separately, as the effect of the particle density is accounted for in Stk. This fact allows us to simplify (8.1.5) even further so that it becomes ... 

Conservation of momentum may be used to simplify the equation of motion for the Brownian gas for long time behavior t /I At this regime, the Brownian gas will reach an internal equilibrium with the heat bath. From Eq. (7.188) and the mean velocity in Eq. (7.190), the equation of motion for the mean velocity becomes... 

Differential Equations for Fluidized Bed Gasifier Model. In a hydrodynamical sense, the processes in fluidized bed gasifiers involve the interaction of a system of particles with flowing gas. The motion of these particles and gas is, at least in principle, completely described by the Navier-Stokes equations for the gas and by the Newtonian equations of motion for the particles. Solution of these equations together with... 

Various mathematical models have been put forth to describe the rate of bubble growth and the threshold pressure for rectified diffusion.f ° The most widely used model quantifies the extent of rectified diffusion (i.e., the convection effect and bubble wall motion) by separately solving the equation of motion, the equation of state for the gas, and the diffusion equation. To further simplify the derivation, Crum and others made two assumptions 1) the amplitude of the pressure oscillation is small, i.e., the solution is restricted to small sinusoidal oscillations, and 2) the gas in the bubble remains isothermal throughout the oscillations.Given these assumptions, the wall motion of a bubble in an ultrasonic field with an angular frequency of co = 2nf can be described by the Rayleigh-Plesset equation ... 

It is explained in chap 2 that the Boltzmann equation is an equation of motion for the one-particle distribution function and is appropriate to a rare gas [86, 50]. In this particular case appropriate expressions for the collision... 

Derivation of a modified equation of motion for the particle that accounts approximately for the non-Stokesian motion of the particles is based on the general expression for the drag on a fixed spherical particle in a gas of uniform velocity, U (Brun et al., 1955) ... 

Answer by Author The procedure used to compute sonic velocities under these conditions is essentially that employed by Heinrich P] and modified by Holzman [ ] to treat the liquid density as a variable. The basic laws of continuity, energy, and momentum, and the ideal-gas equations of state are applied to a mixture in such a way as to derive an equation of motion for the mixture in terms of the properties of the gaseous and liquid constituents. The resulting equation is greatly simplified by the following assumptions ... 

Iordansky, S.V. On equation of motion for the liquids containing gas bubbles. PMTF(1960),3. 

It is very satisfactory from a macroscopic point of view that the Boltzmann equation, through the H-theorem, predicts the approach to equilibrium of an initial nonequilibrium state of the gas. However, one can raise serious objections to the //-theorem, and to the Boltzmann equation, from a microscopic point of view. The fundamental difficulty is that the Boltzmann equation is inconsistent with the laws of mechanics. The laws of mechanics require that any equation of motion describing the gas be invariant under time reversal if the particles make specular collisions with the walls. Otherwise any dynamical processes that do not involve collisions with the walls must be time reversal invariant. That is, the form of the equations of motion must be invariant if v-> —V and t —t. It is clear from an inspection of the Boltzmann equation for points far from the walls. 

In many industrial applications such as bubble columns and stirred tank reactors, it is of interest to know the local concentration of bubbles. In general, this is an extremely difficult problem because the bubbles modify the flow and one must compute shape and velocity of the bubbles simultaneously with the motion of the liquid phase. However, for dilute flows, one may be able to obtain some progress with the so-called one-way coupling approximation. In this approach, one ignores the effect of the bubbles on the motion of the liquid. This is reasonable provided that the gas volume fraction is very small and if the bubbles are smaller than the energy-containing eddies so that the turbulence created by the bubbles is unimportant. One then integrates an approximate equation of motion for the bubble. 

In this model the potential interaetion between the molecule and the surface atoms Pgg is a function of the co-ordinates (x and z) of the gas moleeule, the primary zone atom y and the equilibrium values for the atoms in the seeondary zone obeying the classical equation of motion for the molecule ... 

The solution of the dynamical problem for the gas and surface atoms requires in principle solution of the quantum mechanical equations of motion for the system. Since this problem has been solved only for 3-4 atomic systems we need to incorporate some approximations. One obvious suggestion is to treat the dynamics of the heavy solid atoms by classical rather than quantum dynamical equations. As far as the lattice is concerned we may furthermore take advantage of the periodicity of the atom positions. At the surface this periodicity is, however, broken in one direction and special techniques for handling this situation are needed. Lattice dynamics deals with the solution of the equations of motion for the atoms in the crystal. As a simple example we consider first a one-dimensional crystal of atoms with identical masses. If we include only the nearest neighbor interaction, the hamiltonian is given by ... 

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

Example 2.10. Probably the best contemporary example of a variable-mass system would be the equations of motion for a space rocket whose mass decreases as fuel is consumed. However, to stick with chemical engineering systems, let us consider the problem sketched in Fig. 2.8. Petroleum pipelines are sometimes used for transferring several products from one location to another on a batch basis, i.e., one product at a time. To reduce product contamination at the end of a batch transfer, a leather ball or pig that just fits the pipe is inserted in one end of the hne. Inert gas is introduced behind the pig to push it through the hne, thus purging the hne of whatever hquid is in it. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

In the dynamical approach, one attempts to solve directly the quantum-mechanical or classical equations of motion for a system, Such a direct approach is practicable, for example, for treating the binary collisions between molecules in a gas, by either classical or quantum-mechanical methods.3 However, in a dense system such as a liquid, only the classical equations are tractable,4 even with high-speed computers,... 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

The trajectory of a particle moving in a gas can be estimated by integrating the equation of motion for a particle over a time period given by increments of the ratio of the radial distance traveled divided by the particle velocity, that is, r/q. Interpreting the equation of motion, of course, requires knowledge of the flow field of the suspending gas one can assume that the particle velocity equals the fluid velocity at some distance r far from the collecting body. 

If the linear gas velocity approaches the speed of sound, the simple mathematical model used in equation (4) breaks down. The acceleration term must be taken into account, and the steady-state equation of motion for a straight pipeline with constant diameter may be written (8) ... 

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