Discrete phases balance equations

Sha et al [104, 105] developed a similar multi-fluid model for simulating gas-liquid bubbly flows in CFX4.4. However, slightly different distributions of the velocity fields and the particle size classes were allowed in these two codes. To guarantee the conservation of mass the population balance solution method presented by Hagesaether et al [29, 30] was adopted. For the same bubble size distribution and feed rate at the inlet, the simulations were run as two, three, six, and eleven phase flow. The number of the discrete population balance equations was ten for all the simulations. 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

The CNMMR model with laminar flow liquid stream in the annular region consists of three ordinary differential equations for the gas in the tube core and two partial differential equations for the liquid in the annular region. These equations are coupled through the diffusion-reaction equations inside the membrane and boundary conditions. The model can be solved by first discretizing the liquid-phase mass balance equations in the radial direction by the orthogonal collocation technique. The resulting equations are then solved by a semi-implicit integration procedure [Harold etal., 1989]. 

A simplified homogeneous dispersed-phase mixing model was proposed by Curl (C16). Uniform drops are assumed, coalescence occurs at random and redispersion occurs immediately to yield equal-size drops of the same concentration, and the dispersion is assumed to be homogeneous. Irreversible reaction of general order s was assumed to occur in the drops. The population balance equations of total number over species concentration in the drop were derived for the discrete and continuous cases for a continuous-fiow well-mixed vessel. The population balance equation could be obtained from Eq. (102) by taking the internal coordinate to be drop concentration and writing the population balance equation in terms of number to yield... 

In an early attempt to calculate the phase fractions in an approximate implicit volume fraction-velocity-pressure correction procedure, Spalding [176, 177, 178, 180] calculated the phase fractions from the respective phase continuity equations. However, experience did show that it was difficult to conserve mass simultaneously for both phases when the algorithm mentioned above was used. For this reason, Spalding [179] suggested that the volume fraction of the dispersed phase may rather be calculated from a discrete equation that is derived from a combination of the two continuity equations. An alternative form of the latter volume fraction equation, particularly designed for fluids with large density differences, was later proposed by Carver [26]. In this method the continuity equations for each phase were normalized by a reference mass density to balance the weight of the error for each phase. 

Solving discrete phase particle track under particle effect balance equation of lagrangian coordinates. The form of particle effect balance equation under cartesian coordinate system (x direction) is (Morsi, S.A. Alexander, A.J. 1972) ... 

Unlike the aforementioned models, Fyhr and Rasmuson [41,42] and Cartaxo and Rocha [43] used an Eulerian-Lagrangian approach, in which the gas phase is assumed as the continuous phase and the solids particles are occupying discrete points in the computational domain. As a consequence, mass, momentum, and energy balance equations were solved for each particle within the computational domain. 

To solve the model A, the solid phase mass balance equation was discretized using second order central finite differences. For both models, the differential-algebraic equations were solved using a Backward-Difference Formula (BDF algorithm). 

The balance equations. The discrete model is based on assumptions Al to AlO of section 2.2. Let k be the index of the mixing-cell. The mass balance in the bulk gas and solid phases are respectively... 

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. 

For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more refined model for the bubble-bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of fluidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in large-scale fluidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors. 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

Model. A difference equation for the material balance was obtained from a discrete reactor model which was devised by dividing the annulus into a two dimensional array of cells, each taken to be a well stirred batch reactor. The model supposes that axial motion of the mobile phase and bed rotation occur by instantaneous discontinuous jumps, between cells. Reaction occurs only on the solid surface, and for the reaction type A B + C used in this work, -dn /dt = K n - n n. Linear isotherms, n = BiC, were used, and while dispersion was not explicitly included, it could be simulated by adjusting the number of cells. The balance is given by Eq. 2, where subscript n is the cell index in the axial direction, and subscript m is the index in the circumferential direction. 

Equations (10-98) through (10 100) constitute 7+1 governing equations for 7+1 variables Xj (/= ,...-/) andp/. They can be solved numerically, for example, by a discretization technique where a set of coupled differential equations is replaced by a set of NxM finite difference equations on a grid consisting of M mesh points. The necessary boundary conditions can be established by requiring the reaction equilibrium (i.e.. Equation (10 99)) and the sum of the mole fractions equal to one (i.e.. Equation (10 100) at the membrane interface and equality of the pressure at the membrane interface and the pressure in the adjacent gas phase. Additional boundary conditions can be obtained from mass balances coupling the molar fluxes from the gas phase to the membrane interface with those at the interface. Details can be found elsewhere [Sloot et al.. 1990]. 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

Table 2 shows the mass balance of the main components of the process in the fluid and solid phase. The whole set of equations is solved coupling the Orthogonal Collocation Method to discretize the radial ordinates with the Method of Lines to integrated the system of equations by a stiff integrator. This work is only a theoretical study, as a predictive tool, and all presented data were simulated. 

The mass balance differential equations for the gas phase and the surface species were solved simultaneously as a function of time and discrete position, using a variable step integration algorithm (Gear) [8]. 

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