Mass conservation equation solid phase

Note 5.4 (On the permeability andflow in a porous medium). The seepage equation can be obtained by substituting Darcy s law into the mass conservation equations of fluid and solid phases, as described above. The effects of the micro-structure and microscale material property are put into the hydraulic conductivity k, which is fundamentally specified through experiments. It is not possible to specify the true velocity field by this theory, whereas by applying a homogenization technique, we can determine the velocity field that will be affected by the microscale characteristics. In Chap. 8 we will outline the homogenization theory, which is applied to the problem of water flow in a porous medium, where the microscale flow field is specified. 

Mass conservation equation for the dense-phase solid ... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Compartmental soil modeling is a new concept and can apply to both modules. For the solute fate module, for example, it consists of the application of the law of pollutant mass conservation to a representative user specified soil element. The mass conservation principle is applied over a specific time step, either to the entire soil matrix or to the subelements of the matrix such as the soil-solids, the soil-moisture and the soil-air. These phases can be assumed in equilibrium at all times thus once the concentration in one phase is known, the concentration in the other phases can be calculated. Single or multiple soil compartments can be considered whereas phases and subcompartments can be interrelated (Figure 2) with transport, transformation and interactive equations. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

In equation 13, C1 and Cs are the total concentrations in the liquid and solid phases, respectively. This statement of the problem assumes that the convective flux due to the moving boundary (growing surface) is small, the diffusion coefficients are mutual and independent of concentration, the area of the substrate is equal to the area of the solution, the liquid density is constant, and no transport occurs in the solid phase. Further, the conservation equations are uncoupled from the equations for the conservation of energy and momentum. Mass flows resulting from other forces (e.g., thermal diffusion and Marangoni or slider-motion-induced convective flow) are neglected. 

Agar gel is a hygroscopic medium for w < 60% [7], In this range, the equation (10) allows the determination of de from the chemical potential of water given by the desorption isotherm. Assuming that the two-phase structure of the medium is conserved, the coefficient (3 can be expressed by (3 = 1/3 (a I w), where a is the ratio between the specific mass of the water and the specific mass of the solid. This expression is confirmed by experimental tests (fig. 4). The coefficient Kw (fig. 5) was measured for a plate of gel placed in a PEG solution [6]. Dw and De (fig. 6) are deduced from ... 

Heterogeneous Models. The two-phase character of a packed-bed is preserved in a heterogeneous model. Thus mass and energy conservation equations are written separately for the fluid and solid phases. These equations are linked together by mass and heat transport between the phases. 

The coordinate system is selected so that the solid-gas interface, where conditions are identified by the subscript i, is maintained at x — 0. By mass conservation, m is independent of x everywhere, but it varies with t in this coordinate system. Boundary conditions for equation (56) are T = Tq, the initial temperature of the propellant, at x = — oo and T = 7 - at x = 0. The interface Arrhenius law, given by equation (7-6) but also interpretable in terms of a distributed solid-phase reaction in a thin zone (as indicated at the end of Section 7.4), is written here in the form... 

In the general case where the active material is dispersed through the pellet and the catalyst is porous, internal diffusion of the species within the pores of the pellet must be included. In fact, for many cases diffusion through catalyst pores represents the main resistance to mass transfer. Therefore, the concentration and temperature profiles inside the catalyst particles are usually not flat and the reaction rates in the solid phase are not constant. As there is a continuous variation in concentration and temperature inside the pellet, differential conservation equations are required to describe the concentration and temperature profiles. These profiles are used with intrinsic rate equations to integrate through the pellet and to obtain the overall rate of reaction for the pellet. The differential equations for the catalyst pellet are two point boundary value differential equations and besides the intrinsic kinetics they require the effective diffusivity and thermal conductivity of the porous pellet. 

Transport Equations. In a system with fluid-filled porosity f/, A p reactive solid phases, and aqueous-phase species consisting of N, basis species (the smallest set of species needed to define the chemical system in the aqueous phase) and Nc complexes, the conservation equation for the mass of a component i is ... 

It is the first one that will be emphasized, and can be broken into conservation of mass and energy, which are coupled with Einstein s mass-energy equivalence (E=mc ). As such, the accumulation terms of the conservation of mass are not affected. Also, we could neglect forced convection effects in the system. The resulting mass diffusion equation would be similar to that in Eq. (1.5.2), except that a so-called elastic strain energy could be added to the potential function to take into account crystal lattice differences between solid phases (De Fontaine, 1967). 

Following equations are the basis of the model the conservation of mass for each component, the continuity equation, the conservation of energy for the gaseous phase as well as for the solid phase and - concerning the two-dimensional case - one equation of conservation of momentum for each of the two velocity components. These differential equations are solved numerically by a commercial CFD code using the SIMPLER algorythm. The source terms in the differential equations are due to heat transfer between the two phases, heat loss to the environment and - most important - due to the occuring chemical reactions. The calculation of these source terms is one of the main features of REBOS. 

Thus, an analysis method with a much better estimation accuracy is necessary. For a more detailed analysis, a method using the diffusion equation or the geochemical mass transfer analysis must be used. The biggest difference in these two methods is the chemical reaction model. The method is frequently used for predicting neutralization or salt attack of concrete structures in the civil engineering fields. Regarding mass transfer, the law of conservation of mass relating to the solid-phase element concentration Cp and... 

In the continuum (Euler-Euler)-type formulation, the gas, liquid, and solid phases are assumed to be continuum and the volume-averaged mass and momentum equations (see Table 6.10) are solved for each phase separately to predict the pressure, phase holdup, and phase velocity distributions. As a result of time and volume averaging, additional terms appear in the momentum conservation equations. These additional terms need closure models and such unclosed terms are highlighted in Table 6.10. 

We here introduce a process of diffusion and reaction in the solid phase similar to that in the fluid phase. By drawing an analogy to the analysis of the fluid phase, we have the following equation of mass conservation for the ath component ... 

Using an equation of equilibrium or motion, which determines the deformation of the solid skeleton, and (5.58), a system of differential equations for specifying the mean velocity v (i.e., the conventional consolidation problem) is achieved. Note that in (5.58)

total head excluding the velocity potential), k is the hydraulic conductivity tensor, p is the pore pressure of the fluid, g is the gravity constant, and is the datum potential. Thus by starting with the mass conservation laws for both fluid and solid phases, we can simultaneously obtain the diffusion equation and the seepage equation which includes a term that accounts for the volumetric deformation of the porous skeleton. 

Here, and x f refer to the mass fractions of species w in the solids of the dense and the dilute phases, and x h is the mass fiaction of species w in the phase where T, comes from. And the source term ofthe speciesj or w caused by the heterogeneous reaction in the dilute and dense phase in the above species conservation equations are Rjc, Rj(, Rji -Rwo - wf. and R. ... 

In contrast to the DPM, the TFM does not longer distinguish single particles. Instead, a continuum description for the soUds phase is employed, resulting in a second set of conservation equations for mass and momentum (similar to the gas-phase Navier Stokes equations of the DPM model, but then for the solids phase) ... 

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