Homogeneous mass balance

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

A homogeneous flow basis must be used when thermodynamic equilibrium is assumed. For furtl er simplification it is assumed there will be no reaction occurring in the pipeline. The vapor and liquid contents of the reactor are assumed to be a homogeneous mass as they enter the vent line. The model assumes adiabatic conditions in the vent line and maintains constant stagnation enthalpy for the energy balance. 

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... 

Lumping compartments 1, 2, and 3 into a single homogeneous tissue compartment implies the blood flow-limited model. The tissue mass balance equation for a noneliminating organ is... 

Equations 12.7.28 and 12.7.29 provide a two-dimensional pseudo homogeneous model of a fixed bed reactor. The one-dimensional model is obtained by omitting the radial dispersion terms in the mass balance equation and replacing the radial heat transfer term by one that accounts for thermal losses through the tube wall. Thus the material balance becomes... 

Table 18.2 lists 30 of the molecules used in this study that are known to be substrates for active transport or active efflux. The mechanistic ACAT model was modified to accommodate saturable uptake and saturable efflux using standard Michaelis-Menten equations. It was assumed that enzymes responsible for active uptake of drug molecules from the lumen and active efflux from the enterocytes to the lumen were homogeneously dispersed within each luminal compartment and each corresponding enterocyte compartment, respectively. Equation (5) is the overall mass balance for drug in the enterocyte compartment lining the intestinal wall. 

Here the pseudo-homogeneous rate r is related to the surface reaction rate r" through the area of active catalyst per unit volume of reactor. Assuming further a plug-flow regime, the integration of the mass balance equation for this simple rate expression gives an expression for CO conversion ... 

The simplest approach considers a perfectly mixed bioturbated layer of thickness L and homogeneous concentration C. If v is the sedimentation rate, the mass balance condition for element i reads... 

The present study shows that the asymptotic salt rejection, r, is determined by the top skin layer of a membrane. This is a result of the steady-state mass balance and the boundary conditions. Although there are no experimental data to support this, it has been shown theoretically that the asymptotic salt rejection is identical to the reflection coefficient for the homogeneous membrane, r = 0. 

In catalytic reactors we assume that there is no reaction in the fluid phase, and all reaction occurs on the surface of the catalyst. The surface reaction rate has the units of moles per unit area of catalyst per unit time, which we will call r". We need a homogeneous rate r to insert in the mass balances, and we can write this as... 

Thus we see that environmental modeling involves solving transient mass-balance equations with appropriate flow patterns and kinetics to predict the concentrations of various species versus time for specific emission patterns. The reaction chemistry and flow patterns of these systems are sufficiently complex that we must use approximate methods and use several models to try to bound the possible range of observed responses. For example, the chemical reactions consist of many homogeneous and catalytic reactions, photoassisted reactions, and adsorption and desorption on surfaces of hquids and sohds. Is global warming real [Minnesotans hope so.] How much of smog and ozone depletion are manmade [There is considerable debate on this issue.]... 

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. 

In the above derivation of the mass balance equation, we assume that the column is radially homogeneous, the compressibility of the mobile phase is negligible, the axial dispersion coefficient is constant, and the temperature is unchanged. Furthermore, no diffusion in the stationary phase is assumed. 

The composition profile is approximated by a step profile, with a uniform composition xf in the surface layer (0bulk phase x, at z>L. It is assumed that the total amount of liquid can be divided into two parts with the first constituting the homogeneous bulk phase (mole numbers in it n° = til -I- 2) and the remainder standing under the influence of the forces emanating from the solid surface causing adsorption (mole numbers, referred to unit mass of adsorbent, = n, -i- 2 the superscript a referring to adsorption) [17]. Simple mass balance considerations lead to the following expressions [12] ... 

Let us assume that turbulence in the tank keeps the suspended particle concentration homogeneous, but that at the bottom of the tank the particles can sink through some screen below which no water currents exist (Fig. 23.2 b). In the absence of any external particle fluxes or in situ production/removal of particles, the mass balance equation for suspended particle mass is given by equating the rate of change of particle mass in the water volume V with time with the rate of loss due to settling ... 

The isotope dilution results in Table II are on fuel source samples obtained from NBS which were considered homogeneous. The results in Table III are from the sampling points indicated in Figure 4. These summarized results are mostly by the SSMS general scan technique which has an estimated accuracy of better than 50%. The isotope dilution measurements are limited by the emulsion detector to 3-5%. The results are in grams of metal flow per minute. The mass balance for the various elements was computed by the following equations ... 

For a homogeneous chemical reaction of a chemical substance A (without mass transfer limitation and without a segregation effect) carried out in a CSTR the mass balance results in do = a + rkda when n is the chemical reaction order. (In this case the chemical reaction order is the same as the over-all conversion order.) It now follows that... 

Derive the steady-state mass balance equation for an isothermal CSTR in which a consecutive homogeneous reaction... 

To illustrate we consider a homogeneous tubular reactor. The simplest model is given by plug flow and the design equations are obtained from the mass-balance equations by taking the mass balance over an element of length Al. This is expressed in the formula... 

In the case of a distributed system we need to establish the mass-balance for a small element AV and then take liniAV o to arrive at a DE, as detailed for distributed homogeneous systems earlier in Chapter 4 and depicted in Figure 6.11. 

In this section we develop the heat-balance design equations for heterogeneous systems. Based on the previous sections it is clear how to use the heat-balance and heat-balance design equations that were developed earlier for homogeneous systems, as well as the principles that were used to develop the mass-balance and mass-balance design equations for heterogeneous systems for our purpose. We will start with lumped systems. 

The mass balance equations of the traditional multicomponent rate-based model (see, e.g., Refs. 57 and 58) are written separately for each phase. In order to give a common description to all three considered RSPs (where it is possible, of course) we will use the notion of two contacting fluid phases. The first one is always the liquid phase, whereas the second fluid phase represents the gas phase for RA, the vapor phase for RD and the liquid phase for RE. Considering homogeneous chemical reactions taking place in the fluid phases, the steady-state balance equations should include the reaction source terms ... 

The mass balance resulting is very similar to the one discussed above, but a term incorporating the reaction is added, yielding a non-homogeneous PDE ... 

These approaches place particular emphasis on the spatial aspects of the drug absorption from the gastrointestinal tract. The small intestine is assumed to be a cylindrical tube with fixed dimensions where the drug solution or suspension follows a homogeneous flow. Mass balance relationships under steady-state as-... 

For the production of chemicals, the rate of the reaction is a key parameter for the productivity defined in Equation (5) as the number of molecules produced per time. In homogeneous systems, the reaction rate depends on temperature, pressure, and composition [1]. In the case of solarthermal cycles, a metal oxide is used for the C02-splitting reaction rendering the reaction medium a heterogeneous two-phase system consisting of a solid (metal, metal oxide) and a fluid (CO2, CO, or carrier gas with O2). Therefore, the reaction kinetics becomes much more complex. Whereas microscopic kinetics only deals with time-dependent progress of the reaction, macroscopic kinetics additionally takes the heat- and mass-transport phenomena in heterogeneous systems into account. The transfer of species from one phase to the other must be considered in the overall mass balance [1]. The reaction of a gas with a porous solid consists of seven steps ... 

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