Solid mass balance equation

Note that the scaling analysis hasn t involved the reaction timescale so far. This will be required to obtain a consistent balance at 0(1) from the steady state solid mass balance. Equation 3.32 can be written as... 

Mass balance equations can be developed to describe mass transfer in the stirred tank. The system is defined as the fluid within the tank, excluding any headspace above the fluid and the solid tank itself. The total mass balance on this sytem, given above in verbal form as Eq. (1), takes the following form when fluid density (p) is assumed to be constant ... 

Once the operating line is set, the equations that govern the thickener operation are determined from a solids mass balance as follows. At steady state (stable) operating conditions, the net solids flux is... 

The particle and bulk densities are commonly used in mass balance equations, since the mass and the external volume of the particles are involved. On the other hand, the hydraulic density should be preferably used in hydrodynamic calculations, because buoyancy forces are involved, and so the total mass of the particle should be taken into account, including the fluid in the open pores. It is obvious that the particle density is equal to the skeletal and hydrodynamic density in the case of nonporous particles. Moreover, in the case of a porous solid in a gas-solid system, the gas density is much lower than the particle density, and tlius... 

The gas energy and mass balance equations, unlike the corresponding solid balances, do not have a term for accumulation. This is because the high convective flow of gas through the channels of the monolith makes accumulation of heat or reactants in the gas phase negligible. In practice, the accumulation term in the solid mass balance could also be removed as, in general, it also tends to be small. However, it is included in our models as it enables the equations to be solved numerically more easily. 

Let us formulate the dynamic mass balance equation of the chemical in the SMSL. Fig.23.5 summarizes all processes. At this point we have to select the variable which shall characterize the SMSL. (Remember for the open water box we have chosen the total concentration Ctop.) Due to the large solid-to-water ratio of sediments rss, chemicals with moderate to large distribution coefficients (Kd > 0.1 m3kg ) are predominantly sorbed to the solid phase. Therefore, C offers itself as the natural choice for the second state variable. 

Wallis (1969) defined the dynamic wave in one-phase flow as being that which occurs whenever there is a net force on the flowing medium produced by a concentration gradient. For a two-phase flow, i.e., gas-solid flow, the flow medium refers to the gas phase and the concentration refers to the solids holdup. Thus, to analyze dynamic waves, one can examine the wave equation obtained from the perturbation of the momentum and mass balance equations for the gas and solid phases. The analyses given later for both 6.5.2.2 and 6.5.2.3 follow those of Rietema (1991). 

The mass balance equations for gas and solid phases yield... 

The mass balance equations for a gas-solid flow in a riser can be described as... 

Substituting this relationship into the solid mass balance, Eq. 2, one obtains a single equation for cA ... 

For a 10x10 array of unit cells, there is a total of 601 e-quations 200 mass balances (Equations 10 and 12), 300 energy balances (Equations 17, 21, 28), 100 electron balances (Equation 40) and one voltage equation (41). The corresponding 601 unknowns are 200 local fuel and oxidant conversions, 300 local fuel, oxidant and solid temperatures, 100 local current densities, and the dimensionless operating voltage. 

The equations for sizing rotary-drum filters are summarized in Table 6.18. Equation 6.18.1 is the liquid mass balance. In this procedure, y is a mass fraction. Because the cake is wet, the liquid entering the filter will be less then the liquid leaving. Equation 6.18.2 is the solids mass balance, assuming that all the solids in the slurry are removed. Solve Equation 6.18.2 for the cake formation rate, me. Then, solve Equation 6.18.1 for the filtrate volumetric flow rate, V2. Next, calculate the filtration area from Equation 6.18.5 and the dmm area from Equation 7.18.6. Finally, select a standard rotary filter from Table 6.20. The calculation procedure for sizing a rotary filter is outlined in Table 6.19. Example 6.5 illustrates the sizing procedure. 

The boundary conditions are zero stress at the gel surfaces that contact air (we neglect surface tension) and zero velocity at the surfaces that contact the flat solid surface. Thus all boundary conditions can be written so that the right sides are zero, and they rescale trivally. Likewise, the mass balance equation in Eq. (A3-23) rescales trivially. 

The distribution of species A and B in both the solid and solution phases can be determined by the above equations together with a mass-balance equation. When the calculation is conducted, the species can be treated as those in solution reactions with units of meq/L. If the ionic strength is not high, activity can be treated as concentration, which makes the calculation less complicated. 

The total mass balance equations, describing the relationship between the internal and external flow rates, are the same for both the TMBR and SMBR models (Eqs. 8.11, 8.13, 8.15 and 8.17). Component balances, however, have to take into account the solid streams. 

The most comprehensive study of the combined effects of axial dispersion and mass transfer resistance under constant pattern conditions has been done by Rhee and Amimdson [17,18] using the shock layer theory. These authors assumed a solid film linear driving force model (Eq. 14.3) and wrote the mass balance equation as... 

Although the solubilities of quartz, its polymorphs, and amorphous silica are fixed and pH-in-dependent in most natural waters, the dissociation of silicic acid at alkaline pH s leads to substantial increases in their solubilities above pH 9 to 10. The following calculations show how we can predict this effect. First, the solubility of any silica solid must equal the sum of the concentrations of all species of silica in solution at equilibrium. This summation is given by the mass-balance equation... 

Note that the same mass-balance equations apply whether the reaction in the liquid phase is homogeneous or catalyzed by solid particles as in a slurry reactor. The difference between catalyzed and noncatalytic systems is accounted for in the global rate. If the reactants are introduced only in the gas phase, a mass balance is needed only for that phase. This situation exists for some slurry reactors where the liquid phase is inert and its purpose is simply to suspend the.catalyst particles. 

The CONDOR code also takes possible condensates (both solid and liquid) into account. For example, Fe metal (or liquid Fe depending on the temperature) forms if the thermodynamic activity of Fe (aFe) is equal to or greater than unity. The code computes the temperature at which ape reaches unity (the Fe metal condensation temperature), resets ape to unity at all temperatures below this point, and adds a new term for the abundance of Fe metal to the mass balance equation. 

The same approach as for the fixed-bed reactor is employed here but by making allowance for the special features of the moving-bed reactor (Figure 11.42). The main difference is that the solid is also moving, and a mass balance equation for the solid phase is therefore needed—both for... 

The consideration of thermal effects and non-isothermal conditions is particularly important for reactions for which mass transport through the membrane is activated and, therefore, depends strongly on temperature. This is, typically, the case for dense membranes like, for example, solid oxide membranes, where the molecular transport is due to ionic diffusion. A theoretical study of the partial oxidation of CH4 to synthesis gas in a membrane reactor utilizing a dense solid oxide membrane has been reported by Tsai et al. [5.22, 5.36]. These authors considered the catalytic membrane to consist of three layers a macroporous support layer and a dense perovskite film (Lai.xSrxCoi.yFeyOs.s) permeable only to oxygen on the top of which a porous catalytic layer is placed. To model such a reactor Tsai et al. [5.22, 5.36] developed a two-dimensional model considering the appropriate mass balance equations for the three membrane layers and the two reactor compartments. For the tubeside and shellside the equations were similar to equations (5.1) and... 

When symmetric membranes are used or when enzymes are fed to the spongy part of asymmetric membranes, enzyme immobilization results in either a uniform fixation of enzymes throughout the membrane wall, or in the formation of a carrier-enzyme insoluble network in the sponge of the membrane. Mass transfer through this solid phase must therefore be taken into account. A theoretical model neglecting radial convective transport and the dense layer in asymmetric membranes is available in the literature.81 The reacting solution is still assumed to be fed to the core of the hollow fibers. Steady state, laminar flow, and isothermal conditions are assumed. Moreover, the enzymes are assumed to be uniformly distributed and the membrane wall curvature is neglected. Differential dimensionless mass balance equations can be written as follows ... 

Every time a new solid phase appears, its concentration must be added to the appropriate mass balance equations and a new equilibrium constant (for formation from its monatomic gaseous constituents) is included in the system of equations to be solved. 

Mass balance of solid Mass balance of water Mass balance of air Momentum balance for the medium Internal energy balance for the medium The resulting system of Partial Differential Equations is solved numerically. Finite element method is used for the spatial discretization while finite differences are used for the temporal discretization. The discretization in time is linear and the implicit scheme uses two intermediate points, t and t between the initial 1 and final t limes. Finally, since the problems are nonlinear, the Newton-Raphson method has been adopted following an iterative scheme. 

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