Stokes number small

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

Note that we have used the fluid velocity U to describe convection of particles, which is valid for small Stokes number. In most practical applications, / is a highly nonlinear function of c. Thus, in a turbulent flow the average nucleation rate will depend strongly on the local micromixing conditions. In contrast, the growth rate G is often weakly nonlinear and therefore less influenced by turbulent mixing. 

By analyzing the motion of a small panicle in the region near the stagnation point, it can be shown that for an inviscid How, theory predicts that impaction does not occur until a critical Stokes number is reached. For an inviscid flow, the first term in an expansion of the velocity along the streamline in the plane of symmetry which leads to the stagnation point is... 

Note that, if the Stokes number is very small due to the particle diameter being sub-micron, velocity fluctuations due to Brownian motion will be important. 

While this expression does not contain terms for momenrnm transfer between phases (and, hence, would seem to be much simpler), it is of little practical value because the spatial flux term on the left-hand side cannot be written in terms of U x. Only in the limiting case where Umx is very close to Uf (i.e. very small Stokes numbers) would it be possible to model accurately a disperse multiphase flow using a closed form of Eq. (4.106). For finite Stokes numbers, it is best to solve the separate momentum equations for the disperse and fluid phases. 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

If the NDF is a function of the particle velocity then the solution of the GPBE provides the modeler with the essential information for calculating the real-space advection term. This approach is used whenever the particle Stokes number is not small, and will result in the development of a particle-velocity distribution. More details on this topic can be found in Chapter 8. An alternative approach consists of integrating the NDF with respect to the particle velocity. Let us consider, for example, a generic NDF n(t,x, p, p), which is a function of the time t, space x, particle velocity Vp, and internal coordinates p. By integrating out the particle velocity the following NDF is obtained ... 

When the particle Stokes number is not small, the truncated expansion for the particle velocity mean particle velocity must be calculated from the disperse-phase momentum equation described in Section 4.3.7. Let us for the time being consider a very dilute population of identical particles. The mean velocity of these particles can be found by solving Eq. (4.91). For small particle... 

In summary, the Eulerian two-fluid model is represented by Eqs. (5.112) and (5.113) in addition to a constitutive model for the fluid stress tensor Tf. As already mentioned, Eq. (5.112) was derived under the assumption that the particle-velocity distribution is very narrow (i.e. small particle Stokes number), and the particles must have the same internal coordinates. If these simplifications do not hold, for example under dense conditions when particle-particle collisions become important, then particle-velocity fluctuations must be taken into account, as discussed at the end of Chapter 4. 

The pseudo-homogeneous or dusty-gas model very small particle Stokes number and limited polydispersity (momentum-balance equation only for the continuous phase if the system is dilute or for the mixture of continuous and disperse phases if the system is dense). 

The Eulerian two-fluid model with particle-phase velocity that is based on the mean particle size small particle Stokes number and limited polydispersity (both in dilute and in dense systems). 

In this section, we consider the advection and diffusion of a univariafe NDF n(t, x, ), where R+ is a passive scalar (i.e. the velocity u t, x) does not depend on ). For example, could denote the particle mass for fine particles in a liquid with very small Stokes number. The diffusion coefficient, F(4, is assumed to be a function of For example, using the Stokes-Einstein formula for diffusivity reported in Eq. (5.116) on page 187, F( ) = Fq/, where is the particle mass. The PBE for this case reads... 

In particular, the left-hand side can be interpreted as the particle velocity conditioned on the particle size, and then the right-hand side is a local (in real space) second-order approximation of the conditional velocity. However, in order to accurately reproduce the moment fluxes, the approximation in Eq. (8.107) must be close to the true velocity. Or, to put it a different way, the conditional variance of the particle velocity must be small so that the conditional velocity distribution function is tightly centered at u(t,x,f). This will be true, for example, when the particle Stokes number is below the critical value at which PTC begin to occur (i.e. particles with very small inertia). 

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