Moment-transport equation disperse phase

Many disperse-phase systems involve collisions between particles, and the archetypical example is hard-sphere collisions. Thus, Chapter 6 is devoted to the topic of hard-sphere collision models in the context of QBMM. In particular, because the moment-transport equations for a GBPE with hard-sphere collisions contain a source term for the rate of change of the NDF during a collision, it is necessary to derive analytical expressions for these source terms (Fox Vedula, 2010). In Chapter 6, the exact source terms are derived... 

These cases would suggest that the best averaged velocity to use depends on which internal coordinates are held constant. However, because conservation of momentum is of fundamental importance, it is always best to use the mass-average velocity in Eq. (4.85). For this reason, Eq. (4.85) should always be included in the set of moment-transport equations used to model the disperse phase. The disperse-phase momentum source terms appearing on the right-hand side of Eq. (4.85) are defined as follows. The first term. 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

As mentioned above, macroscale models are written in terms of transport equations for the lower-order moments of the NDF. The different types of moments will be discussed in Chapters 2 and 4. However, the lower-order moments that usually appear in macroscale models for monodisperse particles are the disperse-phase volume fraction, the disperse-phase mean velocity, and the disperse-phase granular temperature. When the particles are polydisperse, a description of the PSD requires (at a minimum) the mean and standard deviation of the particle size, or in other words the first three moments of the PSD. However, a more complete description of the PSD will require a larger set of particle-size moments. 

The transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields... 

The major goal of The direct quadrature method of moments (DQMOM) was to derive transport equations for the weights w and abscissas that can be solved directly and which yield the same moments nk without resorting to the ill-conditioned PD algorithm. Another novel concept imposed is that each phase can be characterized by a weight w and a property vector )i, thus the DQMOM can be employed solving the multi-fluid model describing multi-phase systems. Moreover, since each phase has its own momentum balance in the multi-fluid model, the nodes of the DQMOM quadrature approximation are convected with their own velocities. The DQMOM was proposed by Marchisio and Fox [143] and Fan et al. [53] in order to handle poly-dispersed multi-variate systems. 

The moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

The dimensionless mean retention time, Hi/to is independent of the carrier gas velocity and is only a function of the thermodynamic properties of the polymer-solute system. The dimensionless variance, i2 /tc2. is a function of the thermodynamic and transport properties of the system. The first term of Equation 30 represents the contribution of the slow stationary phase diffusion to peak dispersion. The second term represents the contribution of axial molecular diffusion in the gas phase. At high carrier gas velocities, the dimensionless second moment is a linear function of velocity with the slope inversely proportional to the diffusion coefficient. 

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