Differential Equation Models

Let us first concentrate on dynamic systems described by a set of ordinary differential equations (ODEs). In certain occasions the governing ordinary differential equations can be solved analytically and as far as parameter estimation is concerned, the problem is described by a set of algebraic equations. If however, the ODEs cannot be solved analytically, the mathematical model is more complex. In general, the model equations can be written in the form 

C is the mxn observation matrix which indicates the state variables (or linear combinations of state variables) that are measured experimentally. 

The state variables are the minimal set of dependent variables that are needed in order to describe fully the state of the system. The output vector represents normally a subset of the state variables or combinations of them that are measured. For example, if we consider the dynamics of a distillation column, in order to describe the condition of the column at any point in time we need to know the prevailing temperature and concentrations at each tray (the state variables). On the other hand, typically very few variables are measured, e.g., the concentration at the top and bottom of the column, the temperature in a few trays and in some occasions the concentrations at a particular tray where a side stream is taken. In other words, for this case the observation matrix C will have zeros everywhere except in very few locations where there will be 1 s indicating which state variables are being measured. 

In the distillation column example, the manipulated variables correspond to all the process parameters that affect its dynamic behavior and they are normally set by the operator, for example, reflux ratio, column pressure, feed rate, etc. These variables could be constant or time varying. In both cases however, it is assumed that their values are known precisely. 

As another example let us consider a complex batch reactor. We may have to consider the concentration of many intermediates and final products in order to describe the system over time. However, it is quite plausible that only very few species are measured. In several cases, the measurements could even be pools of several species present in the reactor. For example, this would reflect an output variable which is the summation of two state variables, i.e., yi=xi+x2 

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There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

Nonlinear versus Linear Models If F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives only appear to the first power. If the rate of reac tion were second order, then the resiilting dynamic mass balance woiild be ... 

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

The primary classification that is employed throughout this book is algebraic versus differential equation models. Namely, the mathematical model is comprised of a set of algebraic equations or by a set of ordinary (ODE) or partial differential equations (PDE). The majority of mathematical models for physical or engineered systems can be classified in one of these two categories. 

Hence, the differential equation model takes the form,... 

If we wish to avoid the additional objective function evaluation at p=qa/2, we can use the extra information that is available at p=0. This approach is preferable for differential equation models where evaluation of the objective function requires the integration of the state equations. It is presented later in Section 8.7 where we discuss the implementation of Gauss-Newton method for ODE models. 

We strongly suggest the use of the reduced sensitivity whenever we are dealing with differential equation models. Even if the system of differential equations is non-stiff at the optimum (when k=k ), when the parameters are far from their optimal values, the equations may become stiff temporarily for a few iterations of the Gauss-Newton method. Furthermore, since this transformation also results in better conditioning of the normal equations, we propose its use at all times. This transformation has been implemented in the program for ODE systems provided with this book. 

Our experience with algebraic and differential equation models has shown that this is indeed the easiest and most effective approach to use. It has been implemented in the computer programs provided with this book. 

This approach is useful when dealing with relatively simple partial differential equation models. Seinfeld and Lapidus (1974) have provided a couple of numerical examples for the estimation of a single parameter by the steepest descent algorithm for systems described by one or two simultaneous PDEs with simple boundary conditions. 

ORDINARY DIFFERENTIAL EQUATION MODELS 17.3.1 Contact Inhibition in Microcarrier Cultures of MRC-5 Cells... 

Another important quantity is the steady state gain.3 With reference to a general differential equation model (2-28) and its Laplace transform in (2-29), the steady state gain is defined as the... 

This section is a review of the properties of a first order differential equation model. Our Chapter 2 examples of mixed vessels, stined-tank heater, and homework problems of isothermal stirred-tank chemical reactors all fall into this category. Furthermore, the differential equation may represent either a process or a control system. What we cover here applies to any problem or situation as long as it can be described by a linear first order differential equation. 

Just as you are feeling comfortable with transfer functions, we now switch gears totally. Nevertheless, we are still working with linearized differential equation models in this chapter. Whether we have a high order differential equation or multiple equations, we can always rearrange them into a set of first order differential equations. Bold statements indeed We will see that when we go over the examples. 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

For optimization problems that are derived from (ordinary or partial) differential equation models, a number of advanced optimization strategies can be applied. Most of these problems are posed as NLPs, although recent work has also extended these models to MINLPs and global optimization formulations. For the optimization of profiles in time and space, indirect methods can be applied based on the optimality conditions of the infinite-dimensional problem using, for instance, the calculus of variations. However, these methods become difficult to apply if inequality constraints and discrete decisions are part of the optimization problem. Instead, current methods are based on NLP and MINLP formulations and can be divided into two classes ... 

Instead of the partial differential equation model presented above, the model is developed here in dynamic difference equation form, which is suitable for solution by dynamic simulation packages, such as MADONNA. Analogous to the previous development for tubular reactors and extraction columns, the development of the dynamic dispersion model starts by considering an element of tube... 

Villadsen J, Michelsen ML (1978) Solution of differential equation models by polynomial approximation. Prentice-Hall, Englewood Cliffs, N.J. 

Cuthrell, J. E., and Biegler, L. T., Simultaneous solution and optimization of process flowsheets with differential equation models, Chem. Eng. Res. Des. 64, 341 (1986). 

Villadsen, J., and Michelsen, M. L., Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall Inc., New York, 1978. 

A comparison of the benefits and drawbacks of common numerical solution techniques for complex, nonlinear partial differential equation models is given in Table II. Note that it is common and in some cases necessary to use a combination of the techniques in the different dimensions of the model. 

Valstar, J. M., Van Den Berg, P. J., and Oyserman, J., Chem. Eng. Sci. 30,723-728 (1975). Vatcha, S. R., Analysis and Design of Methanation Processes in the Production of Substitute Natural Gas from Coal, Ph.D. thesis, California Institute of Technology, Pasadena (1976). Villadsen, J. V., and Michelsen, M. L., Solution of Differential Equation Models by Polynomial Approximation." Prentice-Hall, New York, 1978. 

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