Density transport equation

Consider a thin layer solid bowl centrifuge as shown in Figure 4.20. In this device, particles are flung to the wall of the vessel by centrifugal force while liquor either remains stationary in batch operation or overflows a weir in continuous operation. Separation of solid from liquid will be a function of several quantities including particle and fluid densities, particle size, flowrate of slurry, and machine size and design (speed, diameter, separation distance, etc.). A relationship between them can be derived using the transport equations that were derived in Chapter 3, as follows. 

The limiting current density in equation 20.77 has been derived on the assumption that transport is solely by diffusion, but if migration also occurs then for a cathodic process... 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

Via Eq. (136) the kinematic condition Eq. (131) is fulfilled automatically. Furthermore, a conservative discretization of the transport equation such as achieved with the FVM method guarantees local mass conservation for the two phases separately. With a description based on the volume fraction fimction, the two fluids can be regarded as a single fluid with spatially varying density and viscosity, according to... 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

In fact, since the unsteady-state transport equation for forced convection is linear, it is possible in principle to derive solutions for time-dependent boundary conditions, starting from the available step response solutions, by applying the superposition (Duhamel) theorem. If the applied current density varies with time as i(t), then the local surface concentration at any time c0(x, t) is given by... 

Their G//OST -codc essentially consisted of a mass balance for the gas, a transport equation for the bubble number density nb, and a force balance for a single bubble, respectively, which run as... 

Note that although the density is constant, we have included it in the transport equations to be consistent with the formulation used in commercial CFD codes. 

For variable-density flows, the transport equation for the density-weighted PDF is used as the starting point. The resulting PDF codes use the particle mass as an intrinsic random variable. The particle density and specific volume can be computed based on the particle properties. 

Note that the RANS formulation used in (B.44) and (B.45) can easily be extended to the LES, as outlined in Section 5.10. Moreover, by following the same steps as outlined above, DQMOM can be used with the joint velocity, composition PDF transport equation. Finally, the reader can observe that the same methodology is applicable to more general distribution functions than probability density functions. Indeed, DQMOM can be applied to general population balance equations such as those used to describe multi-phase flows. 

Janicka, J., W. Kolbe, and W. Kollmann (1979). Closure of the transport equation for the probability density function of turbulent scalar fields. Journal of Non-Equilibrium Thermodynamics 4, 47-66. 

Fig. 13 Experimental (symbols) and theoretical (lines) data for the current-density as a function of applied voltage for a polymer film of a derivative of PPV under the condition of space-charge-limited current flow. Full curves are the solution of a transport equation that includes DOS filling (see text), dashed lines show the prediction of Child s law for space-charge-limited current flow assuming a constant charge carrier mobility. From [96] with permission. Copyright (2005) by the American Institute of Physics...

The alternative is the use of a descriptive mathematical model without any relation with the solution of the transport equation. On the analog of the characterization of statistical probability density functions a peak shape f(t) can be characterized by moments, defined by ... 

To find a principle which so narrows down the statistics (e.g., to a subclass of probability densities S ) that 9 relations in addition to the 5 transport equations will hold between our macroscopic quantities, so that the initial value problem for (p, ua, E), with the basic boundary conditions, will become determinate. In actuality, this will amount to finding a principle from which the various phenomenological relations of two paragraphs back are (approximate) deductions. 

BOLTZMANN TRANSPORT EQUATION. The fundamental equa tion describing the conservation of particles which are diffusing in a scattering, absorbing, and multiplying medium. It states that the lime rate of change of particle density is equal to the rate of production, minus the rate of leakage and the rate of absorption, in the form of a partial differential equation sucli as... 

Using IT lb/h, for fluid flow allows usage of this same equation for gas or liquid the density term p also applies to either. The velocity variable and density transport property factor must stay reasonably near a constant value for the Darcy equation to be valid. For a liquid fluid, p and v remain nearly constant. Thus, most any pipeline length may be calculated using a single application of Eq. (6.6). 

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