The Model Equations

The mass balance on the inactive bubble phase for the monomer is given by 

With the dimensionless parameters X b = CMb/Co, X o = Gmo/Gq, andXi equation (7.99) takes on the dimensionless form 

Dense emulsion-phase monomer mass-balances Here the mass-balance equation is 

By substituting (7.102) into equation (7.101) and using the known previous hydrodynamic relations, equation (7.101) becomes 

Put into dimensionless form, this equation becomes dX i 

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The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

As described in Chapter 3, Section 5.1 the application of the VOF scheme in an Eulerian framework depends on the solution of the continuity equation for the free boundary (Equation (3.69)) with the model equations. The developed algorithm for the solution of the described model equations and updating of the free surface boundaries is as follows ... 

Here /, r and, v are unequal integers in the set 1, 2, 3. As already mentioned, in the thin-layer approach the fluid is assumed to be non-elastic and hence the stress tensor here is given in ternis of the rate of deforaiation tensor as r(p) = riD(ij), where, in the present analysis, viscosity p is defined using the power law equation. The model equations are non-dimensionalized using... 

After substitution of the leading terms of the expanded variables into the model equations and equating coefficients of equal powers of e from their sides, they are divided by common factors to obtain the following set ... 

Step 2 General structure of stiffness matrices derived for the model equations of Stokes flow in (x, 3O and (r, z) formulations (see Chapter 4) are compared. 

Here [ ] is the jump of a function across the crack faces and v is the normal to the surface describing the shape of the crack. Thus, we have to find a solution to the model equations of a thermoelastic plate in a domain with nonsmooth boundary and boundary conditions of the inequality type. 

Solution of the model equations shows that, for a linear isothermal system and a pulse injection, the height equivalent to a theoretical plate (HETP) is given by... 

Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate rate equations. The equations are used to andve at an expression that relates measurable parameters of the reactions to constants and to concentration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a... 

Using the developed eomputer program PROGl, the model equation 1/Y = A + B 1/X represents the Lineweaver-Burk plot represented by Equation 11-17 as ... 

Rewrite the model equations for the time-dependent case to handle this complication and solveusingthesameparametervaluesasbefore. 

The model equations are determined by writing the balance equations based on the conservation of mass and energy. Tlie balance equations have the following basic form ... 

Wynn (1992, 1993) have presented sliort-eut studies of solids proeess flowslieet-ing. Custom-written software to solve the model equations has been used by a number of authors. It has also been proposed that symbolie manipulation paekages, like Mathematiea, ean be used to solve the population balanee equations as a stand-alone modelling environment or as a set of modules to be linked to a generie proeess simulation paekage sueh as SPEEDUP (Hounslow, 1989 Sheikh and Jones, 1996). 

The calculated values y of the dependent variable are then found, for jc, corresponding to the experimental observations, from the model equation (2-71). The quantity ct, the variance of the observations y is calculated with Eq. (2-90), where the denominator is the degrees of freedom of a system with n observations and four parameters. 

Introducing the dimensionless variables x = and 6 = t/t, where t is the switch time interval, and is the length of one SMB column, the model equations become ... 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Important features of the modelling work are the means of integration of the model equations and the method of regenerating the dynamic polymer distribution from its moments. The framework provided by this approach makes it possible to produce models with few assumptions about the model behaviour. 

For large values of z a fully developed case is reached in which the velocities are only functions of r and 0. In the fully developed case the weight fraction polymer increases linearly in z with the same slope for all r and 0. An implicit finite difference scheme was used to solve the model equations, and for the fully developed case the finite difference method was combined with a continuation method in order to efficiently obtain solutions as a function of the parameters (see Reference 14). It was determined that except for very large Grashof... 

The Emulsion Polymerization Model (EPM) described in this paper will be presented without a detailed discussion of the model equations due to space limitations. The complete set of equations has been presented in a formal publication (Richards, J. R. et al. J. AppI. Poly. Sci . in press). Model results will then be compared to experimental data for styrene and styrene-methyl methacrylate (MMA) copolymers published by various workers. 

In the model equations, A represents the cross sectional area of reactor, a is the mole fraction of combustor fuel gas, C is the molar concentration of component gas, Cp the heat capacity of insulation and F is the molar flow rate of feed. The AH denotes the heat of reaction, L is the reactor length, P is the reactor pressure, R is the gas constant, T represents the temperature of gas, U is the overall heat transfer coefficient, v represents velocity of gas, W is the reactor width, and z denotes the reactor distance from the inlet. The Greek letters, e is the void fraction of catalyst bed, p the molar density of gas, and rj is the stoichiometric coefficient of reaction. The subscript, c, cat, r, b and a represent the combustor, catalyst, reformer, the insulation, and ambient, respectively. The obtained PDE model is solved using Finite Difference Method (FDM). 

QUANTITATIVE-PROPAGATECq, e, F) comment propagate the quantitative values of the variables through the model equations ... 

Mixer-2-input-flow-i = positive i = HCN, H2SO4, HCHO Propagating these values through the modeling equations, we can complete the definition of the goal state (see Fig. 14b). Comparing initial and... 

If, for example, the route from parent carcinogen to the actual biologically-active metabolite is considered as a multi-step pathway, the terms that appear in the model equation can be thought of as representing the rate-determining steps. 

The model equations for each component of this system are =-K Ca dCB... 

INITIAL specifies the start of the INITIAL region specify the initial concentrations DYNAMIC specifies the start of the DYNAMIC region represent the model equations is a check on the total mass balance... 

When substituted into the model equation, the result is... 

The parameter term (k x) which is called Damkohler Number I, is dimensionless and is now the single governing parameter in the model. This results in a model simplification because originally the three parameters, x, k and Cao. all appeared in the model equation. 

A second-order reaction takes place in a two-phase batch system. Reactant A is supplied by gas-liquid transfer and reactant B supplied by liquid feed. The model equations are... 

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