Initiator mass balance equation

The SimuSolv program (Program B) which was written to simulate the reaction finishing process with extra initiator addition is similar to Program A and uses the monomer and initiators mass balance equations with optimized values of the kinetic parameters. The semibatch step had been experimentally optimized for obtaining... 

Besides equilibrium constant equations, two other types of equations are used in the systematic approach to solving equilibrium problems. The first of these is a mass balance equation, which is simply a statement of the conservation of matter. In a solution of a monoprotic weak acid, for example, the combined concentrations of the conjugate weak acid, HA, and the conjugate weak base, A , must equal the weak acid s initial concentration, Cha- ... 

The mechanism involved the overall conversion of [5] to [P], The reverse reaction is insignificant because only the initial velocity in one of the forward direction is concerned. The mass balance equation expressing the distribution of the total enzyme is ... 

This mass balance equation shows that material that is initially at radial position rin will move to radial position r for some downstream location, >0. A worked example of radial velocities and curved streamlines is given in Chapter 13, Example 13.10. 

The experiments were carried out with two initiators. According to published data (B.), at the base temperature, Tb, the fast initiator, II, has a half-life of 3.5 minutes, and the slow initiator, 12, has a half-life of 95 minutes. A minor modification of the monomer mass balance (Equation 7) is required for the case of two initiators. 

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

This section describes the continuous flux melting model used in Bourdon et al. (2003) and has many similarities with the model of Thomas et al. (2002). A significant difference is that the model described here keeps track of the composition of the slab as it dehydrates. This model is based on mass balance equations for both the mantle wedge and the slab. We assume secular equilibrium in the U-series decay chain initially ... 

Now we want to determine the time to reach the steady state. To answer this question, we return to the mass balance equation, Eq. (116), subject to the following initial and boundary conditions ... 

Table 7.9 Initial and Boundary Conditions Corresponding to Mass Balance Equations Given in Table 7.8...

Note The parameters are obtained in the temperature range 125 to 250°C. The mass balance equations are given in Table 7.8, while the corresponding initial and boundary conditions are given in Table 7.9. 

Various munerical techniques are used to indirectly obtain solutions to large systems of equations with too many imknowns to solve explicitly. One approach is to solve the equations iteratively. This is done by first assuming that all of the anions are unbound and, hence, their free ion concentrations are equal to their total (stoichiometric) concentrations. By substituting these assumed anion concentrations into the cation mass balance equations, an initial estimate is obtained for the free cation concentrations. These cation concentrations are substituted into the anion mass balance equations to obtain a first estimate of the free anion concentrations. These free anion concentrations are then used to recompute the free cation concentrations. The recalculations are continued imtil the resulting free ion concentrations exhibit little change with further iterations. The computer programs used to perform speciation calculations perform these iterations in a matter of seconds. 

Another key point of differentiation is the fact that nearly all PSA separations are bulk separations and any investigator interested in a high fidelity description of the problem of adsorption must solve a mass balance equation such as Eq. (9.9), the bulk separation equation, together with the uptake rate model and a set of thermal balance equations of similar form. In addition to the more complicated pde and its attendant boundary and initial conditions the investigator must also solve some approximate form of a momentum balance on the fluid flow as a whole. 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

As a simple example, consider the concentration versus time when a pure solvent initially in a tank (Cai = 0) is replaced by a solute at concentration Cao such us replacing pure water in a tank by a brine solution. Since there is no reaction, the mass-balance equation is... 

These come from simple application of the mass-balance equation dCj jdx = Yli which you should verify. Differential equations always need initial or boundary conditions, and for the batch reactor these are the initial concentrations of A, B, and C. For this system, the feed may be expected to be pure A, so Ca = Cao and Cbo Cco 0-... 

These are two simultaneous differential equations with two initial conditions for a single reaction. For R simultaneous reactions we have to solve R + 1 simultaneous differential equations with R + 1 initial equations because there are R independent mass-balance equations and one temperature equatiorr... 

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 this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

In accordance with the usual process conditions, the initial temperature of the reactive mixture To and the upper cap temperature Tw are constant during filling, and the temperature of the insert Ti equals the ambient temperature (20°C). The model takes into account that during filling the temperature of the insert increases due to heat transfer from the reactive mix. It is assumed that the thermal properties and density of both the reactive mass and the insert are constant. It is reasonable to neglect molecular diffusion, because the coefficient of diffusion is very small 264 therefore, the diffusion term is negligible in comparison with the other terms in the mass balance equation. 

The theory and verification of the mixing-cell mass balance equation has been reported previously (2). For a cell with initial concentration of tracer, C, flushed with tracer-free... 

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,... 

Since a composition is initially assumed, the mass balance equations may be written as ... 

As a strong acid mixes with a strong base (i.e., titration), the pH of the solution changes with increasing amounts of a strong acid added to an initial volume of a strong base. The units for the mass balance equation should be moles, not moles/liter (i.e., concentration unit). Suppose a volume Vb of NaOH with concentration Cb is titrated with a volume Va of HC1 with concentration Ca. Then,... 

ATP]0 denote the initial concentration of bound hydrogen ions and ATP, respectively. The third term on the right-hand side of the mass balance equation computes the number of times the reference reaction has turned over, generating hydrogen ions. Given a set of initial concentrations, Equation (2.20) can be numerically solved for pH at any ATP concentration. 

The preceding results may be converted to the basis of the new product flow rate. Mass-balance equations are linear, so that if the product flow rate is doubled, for example, all flow rates in the process are doubled. In this example then, the new flow rates are obtained by multiplying the previously calculated values by the ratio of the new-to-previous product-stream flow rates (880/99.99) = 8.801. This ratio can be calculated in the scratch pad (say location eh), the new CO feed rate, 880.1, entered into a 3, and the iterative calculations repeated until convergence is attained. The iteration could be avoided, however, if each mass balance had been entered as shown above, but multiplied by a coefficient e3 (the scratch-pad location). Initially, e3 would have been set equal to 1 and the calculations would have proceeded just as shown above. The user then enters 88Q/c8 for location e3, repeats the calculation, and obtains the final flow rates without iteration. 

When only the feed side and permeate side mass balance equations are considered under the isothermal condition, the resulting equations arc a set of first-order ordinary differential equations. Furthermore, a co-current purge stream renders the set of equations an initial value problem and well established procedures such as the... 

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... 

As for all partial differential equations, it is also necessary to complement the mass balance equation with initial and boundary conditions, as explained in detail by Guiochon et al. [13]. The initial condition describes the state of the column when the experiment begins, i.e., at t = 0. In this case, the initial condition corresponds to a column empty of sample and containing only mobile and stationary phases in equilibrium ... 

The mathematical treatment of FMC data can be accomplished by standard procedures via the solution of mass balance equations, on condition that the data were converted to reaction rate data with Eq. (21). As mentioned above, this requires the determination of the transformation parameter a. Two approaches based on calibration were developed and tested. In the first approach, thermometric signals are combined with the absolute activity of IMB, which had been determined by a separate measurement using an independent analytical technique. Figure 5 shows a calibration for the cephalosporin C transformation catalyzed by D-amino acid oxidase. The activity of the IMB was determined by the reaction rate measurement in a stirred-tank batch reactor. The reaction rate was determined as the initial rate of consumption of cephalosporin C monitored by HPLC analysis. The thermometric response was measured for each IMB packed in the FMC column, and plotted against the corresponding reaction rate. From the calibration results shown in Fig. 5 it can be concluded, independently of the type of immobilized biocatalyst, that the data fall to the same line and that there is a linear correlation between the heat response and the activity of the catalyst packed in the column. The transformation parameter a was determined from... 

The band profiles will be obtained as the solution of the relevant system of mass balance equations (Eq. 2.2), completed with a relationship between each stationary phase concentration, Csj, and the mobile phase concentrations, C (Eq. 2.4 or 2.5), and with the proper set of initial (Eq. 2.6) and boundary conditions (Eqs. 2.8 to 2.15). There is an equation of each type for each component of the feed and of the mobile phase, except for the weak solvent in the mobile phase. However, the mass balance equations of the additives or strong solvents, whose retention factors... 

For mass balance reasons, Cbfl = Cafi. The problem is to find a relationship between the isotherm parameters and the parameters and of the f-th harmonic. In the calculations made to relate the equilibrium isotherm and the response of the system (Eq. 3.102), the equilibrium-dispersive model is used (Chapter 2, Section 2.2.2) and the mass balance equation is integrated with the Danck-werts boundary conditions (Chapter 2, Section 2.1.4.3) and with the initial conditions C = Ca,o, q = qiCa,o). 

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