Equation variables

The differential equations are often highly non-linear and the equation variables are often highly interrelated. In the above formulation, yj represents any one of the dependent system variables and, fi is the general function relationship, relating the derivative, dyi/dt, with the other related dependent variables. Tbe system independent variable, t, will usually correspond to time, but may also represent distance, for example, in the simulation of steady-state models of tubular and column devices. 

The phase space of eqn. (73) is the space of vectors c. Its points are specified by the coordinates cx,. . . , cn. The set of phase space points is the set of all possible states of the system. Phase space can be not only the whole vector space but also a certain part. Thus in chemical kinetic equations, variables are either concentrations or quantities of substances in the system. Their values cannot be negative. It is therefore natural to restrict ourselves to the set of those c all the components of which are not negative, i.e. Ci > 0. In what follows we shall refer to these d values as non-negative. Hence positive are those c values all the components of which are positive, i.e. Cj > 0. 

In most simulations, it will be advantageous to transform the given equation variables into dimensionless ones. This is done by expressing them each as a... 

FIG. 7-1 Constants of the power law and Arrhenius equations by linearization a) integrated equation, h) integrated first order, (c) differential equation, d) half-time method, e) Arrhenius equation, (/) variable activation energy, and (g) change of mechanism with temperature (T in K). 

The third level of complexity in airshed modeling involves the solution of the partial differential equations of conservation of mass. While the computational requirements for this class of models are much greater than for the box model or the plume and puff models, this approach permits the inclusion of chemical reactions, time-varying meteorological conditions, and complex source emissions patterns. However, since this model consists only of the conservation equations, variables associated with the momentum and energy equations—e.g., wind fields and the vertical temperature structure—must be treated as inputs to the model. The solution of this class of models will be examined here. 

Both (/i,/pD) and (dpupf p.) are dimensionless groups kg is a coefficient that expresses the rate at which a particular species transfers from the gas to the solid particles and the coefficients 2.00 and 0.600 are dimensionless constants obtained by fitting experimental data covering a wide range of values of the equation variables. 

Solution The energy balance relates the temperature to conversion, providing a method of eliminating T from the rate equation. Variables other than x and T are not involved in the energy balance because the operation is adiabatic. In Eq. (3-18) the second term is zero, so that the volume of the reactor does not enter into the problem. Thus... 

The extraction model is shown in Figure 11.3. The equation variables Lq and are the solvent and extract component flows, and are the feed and raffinate component flows, and N is the number of equilibrium stages. Flow rates are on a mass or mole basis. The extraction factors E are defined as... 

The time evolution of the concentrations of the substrate ATP (a), of intracellular (j8) and extracellular (y) cAMP, and of the different complexes formed by adenylate cyclase and by the cAMP receptor is then governed by a system of nine differential equations, as in the slightly different model studied in chapter 5. When a quasi-steady-state hypothesis is adopted for the enzyme-substrate complexes formed by adenylate cyclase in its free (C) and activated (E) states, the dynamics is described by the system of seven differential equations (6.2). In these equations, variables and parameters are defined as in eqns (5.6) (see table 5.3), but for dimensional reasons, /3 and y represent the concentrations of intracellular and extracellular cAMP divided by moreover, c - ( Cr/A d) and - (1 + a) (Martiel Goldbeter, 1984 Goldbeter, Decroly Martiel, 1984). 

Equation Variable Case 1 Case 2 Equation Variable Case 1 Case 2... 

A series of papers concerning the use of immobilized enzymes in industrial reactors has been published.The operational effectiveness factors of immobilized enzyme systems have been described.Analytical expressions have been developed that allow the generation of effectiveness graphs for immobilized whole-cell hollow-fibre reactors. A theoretical method of determining the kinetic constants of immobilized enzymes in continuous stirred tank and plug-flow reactors by transformation of rate-equation variables has been presented. 

To give a more general solution to the expressions, it is common to cast the equation variables in a dimensionless form. For example, the concentration, c, is cast dimensionless using... 

Empirical parameter in the Frumkin adsorption equation Variables defined in Eqs. (145-147)... 

It will, in most simulations, be advantageous to transform the given equation variables into dimensionless ones. This is done by expressing them each as a multiple of a chosen reference value, so that they no longer have dimensions. The time variable t, for example, becomes the dimensionless T via the relation... 

In this equation variables r and t have been suppressed and integration is over the first Brillouin zone. Evaluation of transition probabilities P(k, k ) generally forms the most tedious part of any transport calculation, though if the relatively simple deformation potential aproximation can be applied its calculation (as we have seen previously) is not so difficult. In thermal equilibrium the collisions alone do not change the total density of representative points in the phase space and equation (9.29) vanishes. Thus we arrive at the principle of detailed balance ... 

Process modeling is both an art and a science. Creativity is required to make simplifying assumptions that result in an appropriate model. The model should incorporate all of the important dynamic behavior while being no more complex than is necessary. Thus, less important phenomena are omitted in order to keep the number of model equations, variables, and parameters at reasonable levels. The failure to choose an appropriate set of simplifying assumptions invariably leads to either (1) rigorous but excessively complicated models or (2) overly simplistic models. Both extremes should be avoided. Fortunately, modeling is also a science, and predictions of process behavior from alternative models can be compared, both qualitatively and quantitatively. This chapter provides an introduction to the subject of theoretical dynamic models and shows how they can be developed from first principles such as conservation laws. Additional information is available in the books by Bequette (1998), Aris (1999), and Cameron and Hangos (2001). 

Enter the following equations, variables, initial parameters, and constraints into the appropriate windows. 

Part One of the book introduces the mathematical equations step by step with increasing complexity. The six chapters in Part One follow a natural logic flow. It is useful to point out that Section 6.6 provides an overview on the conditions in which the different models can be used. The mathematical models involve a large number of equations, variables and parameters. Some variable or parameter can take a subtly different meaning in a different context. A complete set of eqnations with the definitions of all the variables and parameters are always summarised after some new issnes or factors are introduced in each chapter. Consequently, readers do not have to go back... 

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