Differential mass balance models

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... 

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. 

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... 

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

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 pattern of flow through a packed adsorbent bed can generally be described by the axial dispersed plug flow model. To predict the dynamic response of the column therefore requires the simultaneous solution, subject to the appropriate initial and boundary conditions, of the differential mass balance equations for an element of the column,... 

Implementation of a more sophisticated description of the emission into this simple model and taking air exchange of the room with adjacent spaces and other sinks into account lead to a more realistic description of the modelled situation. Such a model requires solving a differential mass balance equation ... 

In 1987 Valdes104 developed a model for composite deposition at a RDE taking into account the various ways in which a particle is transported to the cathode surface. As starting point an equation of continuity for the particle number concentration, C p, based on a differential mass balance was chosen, that is ... 

These basic rate models were Incorporated Into a differential mass balance In a tubular, plug-flow reaction. This gives a set of coupled, non-llnear differential equations which, when Integrated, will provide a simulation model. This model corresponds to the Integral reactor data provided by experimentation. A material balance Is written for each of the four components In our system ... 

It is essential that while setting the conditions for the differential mass-balance equation we did not define the function of the excess adsorption isotherm. We can now use the expression (2-46) for measurement of the model independent excess adsorption values. It is convenient to use it for the study of the adsorption behavior of binary eluents [22]. 

The simplest model describing this mode of drug delivery applies to the low volumetric flow range for small molecules — for example/ cisplatin delivered at 0.9 pL/hr (23). The model is a differential mass balance for a typical shell volume surrounding the cannula tip. Deriving it in the same fashion as in Equation 9.1/ except taking the spherical geometry of the distribution into account/ it is... 

At its core, the distributed model for high-flow microinfusion is once again a differential mass balance for the drug solute in the infusate. However, because the pumps used in this method generate relatively high fluid velocities, transport of molecules... 

The distributed model is completed by forming a differential mass balance for the drug solute in a manner completely analogous to that shown previously in deriving Equation 9.14, except for the inclusion of an additional term describing convective flow ... 

The model includes six differential mass-balance equations... 

Clearly, at one extreme—when q(a) is zero throughout the reactor and we have a general j a)—we have the equations for a segregated-flow model. On the other extreme—when (a) is a Dirac delta exactly at one point and we have a general nonzero q oi)—this model reduces to the Zwietering (1959) model of maximum mixedness. Also, we define Q(a) as the flow of molecules at point a. Based on this nomenclature, a differential mass balance on an element Aa leads to... 

Peak profiles can be calculated with a proper column model, the differential mass balance equation of the compound(s), the adsorption isotherm, the mass transfer kinetics of the compound(s) and the boundary and initial conditions [13], When a suitable column model has been chosen, the proper parameters (isotherm and mass transfer parameters and experimental conditions) are entered into the calculations. The results from these calculations can have great predictive value [13, 114], The most important of the column models are the ideal model , the equilibrium-dispersive (ED) model , the... 

All mathematical models of chromatography consist of a differential mass balance equation for each component involved and the equation expresses mass conservation in the process [13, 109], In the ED model the mass balance equation for a single component is expressed as follows ... 

A common modeling approach for chromatographic batch reactors is the equilibrium-transport dispersive model (Chapter 6.2.4.1). Therefore, only the equations for this approach are discussed here. The differential mass balance (Fig. 8.6) takes into account axial dispersion as well as mass transfer between fluid and both solid phases. 

The ideal TMBR model is based on a differential mass balance of each section (Fig. 8.7). Only convection and homogeneous or quasi-heterogeneous reaction is taken into account. Axial dispersion and mass transfer resistances are neglected. 

The transport approach has been used very early, and most extensively, to calculate the chromatographic response to a given input function (injection condition). This approach is based on the use of an equation of motion. In this method, we search for the mathematical solution of the set of partial differential equations describing the chromatographic process, or rather the differential mass balance of the solute in a slice of column and its kinetics of mass transfer in the column. Various mathematical models have been developed to describe the chromatographic process. The most important of these models are the equilibrium-dispersive (ED) model, the lumped kinetic model, and the general rate model (GRM) of chromatography. We discuss these three models successively. 

In the ideal model, we assume that the column efficiency is infinite, hence the rate of the mass transfer kinetics is infinite and the axial dispersion coefficient in the mass balance equation (Eq. 2.2) is zero. The differential mass balances for the two components are written ... 

There are N columns in the SMB unit (here, we will simplify the presentation by considering the simple case when N = 4). These columns are numbered 1, - rj, - , N. We consider for these coliunns the ideal model (see Chapter 7). However, we need to accoxmt for the movement of the solid phase, so the differential mass balance in column j, for component i is written ... 

The basis of the flow models are ordinary differential mass balances for each component on either side of the membrane. The mechanism for permeation is substituted in the mass balances. When reaction occurs the kinetic expression is also added to the balances. The chemical reaction is assumed to take place in... 

In order to develop a continuous separation process, Kataoka et al. [54] simulated permeation of metal ion in continuous countercurrent column. They developed the material balance equation considering back mixing only in the continuous phase and steady-state diffusion in the dispersed emulsion drops which is similar to the Hquid extraction situation. Bart et al. [55] also modeled the extraction of copper in a continuous countercurrent column. They considered only the continuous phase back mixing in the model and assumed that the reaction between copper ions and carrier is slow, so that the differential mass balance equation for external phase in their model is... 

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