Solving the Dynamic Model Equations

The differential and algebraic equations that make up the dynamic distillation model are solved by numerical methods. If the differential equations cannot be solved analytically, they are first approximated in a number of ways as described below. For convenience, the left hand sides of Equations 13.65,13.66 and 13.68 are abbreviated and represented simply as functions of the model variables Vj and Vj-. 

In order to demonstrate the methods, a simple model is used, consisting of a tank 

At time f = 0, the holdup Af = 100 kmol. The feed flow rate is constant, F = 10 kmol/min. The exit stream flow rate, L, is proportional to the holdup, L = cM, where c = 0.2 min. It follows that at t = 0, L° = 0.2 x 100 = 20 kmol/min. Applying the equivalent of Equation 13.66 to this model, the material balance is expressed as 

This model being simple enough, its differential equation can be solved analytically  

Thus M, and L = cM, are determined at any time t. The following are the results at time increments of 1 min, over a span of 5 minutes  

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Program THERM solves the dynamic model equations. The initial values of concentration and temperature in the reactor can be changed after each run using the ISIM interactive commands. The plot statement causes a composite phase-plane graph of concentration versus temperature to be drawn. Note that for comparison both programs should be used with the same parameter values. 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

The computational strategy for solving the dynamic model is shown in Figure 5.27. It involves three basic sections that solve for the equilibrium tray temperatures, the simultaneous solutions of the algebraic equations giving the vapor and liquid flow rates, and the solution of the diflferential equations giving the liquid compositions. 

To study the response of the kiln to transient conditions and to different control schemes, the set of partial differential equations (39), (40), (44), and (48) were solved using a hybrid analog-digital computer, the EAI Hydac 2000. A description of the computer and of the methods used in the solution are given in the paper by Weekman et al. (1967). The kiln conditions used in the simulation to be discussed are given in Table V. The dynamic model was first used to study the effect of fast coke on kiln stability. 

As written, this set of equations provide the dynamic model for the Petl3mk column. Since a design for the column must first be obtained, the same set of equations but written for steady state conditions provides the basis for such a design problem. In any case, the model shows a coupled structure because of the recycle streams between the two columns such that the full set of equations must be solved simultaneously. 

In this section we deal with estimating the parameters p in the dynamical model of the form (5.37). As we noticed, methods of Chapter 3 directly apply to this problem only if the solution of the differential equation is available in analytical form. Otherwise one can follow the same algorithms, but solving differential equations numerically whenever the computed responses are needed. The partial derivations required by the Gauss - Newton type algorithms can be obtained by solving the sensitivity equations. While this indirect method is... 

In contrast to the sequential solution method, the simultaneous strategy solves the dynamic process model and the optimization problem at one step. This avoids solving the model equations at each iteration in the optimization algorithm as in the sequential approach. In this approach, the dynamic process model constraints in the optimal control problem are transformed to a set of algebraic equations which is treated as equality constraints in NLP problem [20], To apply the simultaneous strategy, both state and control variable profiles are discretized by approximating functions and treated as the decision variables in optimization algorithms. 

Correlations by Computation of Molecular Dynamics. The power of modem computing systems has made it possible to solve the dynamical equations of motion of a model system of several hundred molecules, with fairly realistic interaction potentials, and hence by direct calculation obtain correlation functions for linear velocity, angular velodty, dipole orientation, etc. Rahman s classic paper on the motion of 864 atoms of model argon has stimulated a great amount of further work, of which we cite particularly that of Beme and Harpon nitrogen and carbon monoxide, and that of Rahman himself and Stillinger on water. ... 

Recently, computational fluid dynamics (CFD) models have been developed to guide the development of new BO designs [62-67]. Baker et al. developed a two-dimensional finite-difference model to solve the Navier-Stokes equation and to predict... 

The prediction horizon is discretized in cycles, where a cycle is a switching time tshift multiplied by the total number of columns. Equation 9.1 constitutes a dynamic optimization problem with the transient behavior of the process as a constraint f describes the continuous dynamics of the columns based on the general rate model (GRM) as well as the discrete switching from period to period. To solve the PDE models of columns, a Galerkin method on finite elements is used for the liquid... 

The thermal effects of solar and atmospheric radiation have been studied using mathematical models, generally by solving the radiative transfer equations which we have just described (e.g., Ramanathan, 1976). The most elaborate models, particularly those with fine spectral resolution, require rather large computing resources, and cannot be coupled with chemical or dynamical models using present computers unless they are greatly simplified. 

To first gain insights into the properties of this system, a simulation study is performed. Therefore, all parameters are set to 1, and an artificial Epo-receptor time course is chosen. The dynamical model is solved numerically and the observation equations are evaluated. The resulting time courses for the phosphorylated STAT-5 in the cytoplasm yi and the total amount of STAT-5 in the cytoplasm y2 are displayed in Fig. 17.1-2. 

In more quantitative applications, one must also solve the -dynamics for the classical part of the model. Using the GED ansatz, an eigenvalue equation emerges after first performing the integration over the electronic configuration q-space ... 

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