Mass balances differential

V is the stoichiometric coefficient of species r in the reaction, a differential mass balance on this substance gives... 

Differential and Integral Balances. Two types of material balances, differential and integral, are applied in analyzing chemical processes. The differential mass balance is valid at any instant in time, with each term representing a rate (i.e., mass per unit time). A general differential material balance may be written on any material involved in any transient process, including semibatch and unsteady-state continuous flow processes ... 

In the present study, the UASB reactor was modeled in terms of the dispersed plug flow and the Monod type of rate equations to constmct the differential mass balance equations fcs- the anaerobic biodegradation of single and multiple substrates components of the volatile fetty acids. 

The system of differential mass balance equations (2)- (5) should be solved, provided that the "in" time functions were known, with the initial conditions ... 

Under normal circumstances, the use of a characteristic velocity equation of the type shown above can cause difficulties in computation, owing to the existence of an implicit algebraic loop, which must be solved, at every integration step length. In this the appropriate value of L or G satisfying the value of h generated in the differential mass balance equation, must be found as shown in the information flow diagram of Fig. 3.54. 

Figure 2A illustrates a noneliminating tissue compartment, i, divided into three anatomically relevant subcompartments, each homogeneous with respect to drug concentration. The corresponding differential mass balance equations are... 

The three-compartment tissue model is ordinarily simplified by lumping all three subcompartments, lumping subcompartments 1 and 2, or lumping subcompartments 2 and 3. These simplifications result in the blood flow-limited (i.e., lumping all three subcompartments) and the membrane-limited (i.e., lumping any two subcompartments) tissue models. Differential mass balance equations for a noneliminating membrane-limited compartment are... 

We assume all reactions to be first order and irreversible within the range of the experimental conditions. The governing differential mass balance equations and their solutions have been reported [9J. The values of the constants through at 450°C are shown in Table I. A comparison of the experimental data with the theoretical predictions is shown in Figures 2 through 4 the above assumption of a first order reaction appears reasonable. 

Stirred tank reactor (ST/ ). The differential mass balance referred to the azo-dye converted by bacteria (assuming unstructured model for the biophase, i.e., that it is characterized only by cell mass or concentration X) yields... 

In the case of a chromatographic column packed with porous adsorbent particles showing a unimodal pore size distribution, the differential mass balance of the solute in a porous adsorbent particle can be described by... 

The transport-dispersive model consists of one differential mass balance equation for each component, i, in the mobile phase ... 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

Although convection, axial diffusion, and radial diffusion actually occur simultaneously, a multistep procedure was adopted in the finite-difference calculation. For each 5-cm increment in tidal volume and for each time increment At, the differential mass-balance equations were solved for convection, axial difihision, and radial diffusion in that order. This method may slightly underestimate the dosage for weakly soluble gases, because the concentration gradient in the airway may be decreased. 

We must develop a differential mass balance of composition versus position and then solve the resulting differential equation for Ca(z) and Ca(L) (Figure 3-2). We consider a tube of length L with position z going fiom 0 to L. The molar flow rate of species j is Fjo at the inlet (z = 0), Fj (z) at position z, and Fj (L) at the exit L. 

The distribution of the solute between the mobile and the stationary phases is continuous. A differential equation that describes the travel of a zone along the column is composed. Then the band profile is calculated by the integration of the differential mass balance equation under proper initial and boundary conditions. Throughout this chapter, we assume that both the chemistry and the packing density of the stationary phase are radially homogeneous. Thus, the mobile and stationary phase concentrations as well as the flow velocities are radially uniform, and a one-dimensional mass balance equation can be considered. 

As illustrated in Figure 10.1, an elementary slice of the column is regarded, and a mass balance is determined for that slice. The differential mass balance equation states that the rate of accumulation in a slice of Az length is the sum of the accumulation by convection and the accumulation by diffusion. 

Several mass balance equations are written for the kinetics of each step as the analyte is passing through the porous stationary phase. For the bulk mobile phase in the interstitial volume, the following differential mass balance equation is written... 

In order to estimate the real effect of a dispersed organic phase on the productivity of fermentation, the oxygen absorption rate has to be defined. Knowing its mathematical expression, the rate equation can then be incorporated into the differential mass balance equations of fermentation and, after solving these equations, the concentration of components in a fermentation... 

The set of differential mass balance equations for the heterogeneous part of the... 

During fermentation, the enhanced absorption rate of oxygen increases the bulk concentration and, as a consequence, the production rate of cells can be increased as well. To predict this effect, the enhanced transfer rate has to be incorporated into the differential mass balance equations of fermentation processes studied. If you know the mathematical expression of the biochemical reactions and their dependence on oxygen concentration as well as the enhanced absorption rates due to the dispersed organic phase,you can calculate the fermentation exactly after solving the equation system obtained. 

In a review paper, LeVan [7] studied constant pattern models for gas adsorption. The model is based on the differential mass balance for the solute ... 

The differential mass-balance equations for methane and carbon dioxide are... 

The conversion and yield of carbon dioxide and methane are defined in the opposite sense of the yield and conversion used in steam reformers in equations (7.125) and (7.126) and the same differential mass balance equations (7.127) and (7.128) are used. The mole fractions of all components are computed from similar relations as given before. 

Solvent transport in organic polymer matrices is usually depicted as a two-step mechanism. The first step is the dissolution of the solvent in the superficial polymer layer. This process, which can be considered almost instantaneous in the case of water, creates a concentration gradient. The second step is the diffusion of the solvent in the direction of the concentration gradient. This process may be described by a differential mass balance (often called Fick s second law), which, in the unidimensional case, may be written as... 

Equations (A 10), which are generally valid for both liquid and second fluid phases, represent nothing but differential mass balances for the film region, with the... 

The change in the solid width is obtained by a differential mass balance as follows... 

The differential mass-balance equation for state-state conditions, for cylindrical coordinate and for the mth sublayer is... 

The Equation of Continuity by Differential Mass Balance Derive the equation of continuity in cylindrical coordinates by making a mass balance over the differential volume Ar(rA6)Az. 

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