Transformation of the Model Equations

A suitable transformation of the model equations can simplify the structure of the model considerably and thus, initial guess generation becomes a trivial task. The most interesting case which is also often encountered in engineering applications, is that of transformably linear models. These are nonlinear models that reduce to simple linear models after a suitable transformation is performed. These models have been extensively used in engineering particularly before the wide availability of computers so that the parameters could easily be obtained with linear least squares estimation. Even today such models are also used to reveal characteristics of the behavior of the model in a graphical form. 

If we have a transformably linear system, our chances are outstanding to get quickly very good initial guesses for the parameters. Let us consider a few typical examples. 

The integrated form of substrate utilization in an enzyme catalyzed batch bioreactor is given by the implicit equation 

Initial estimates for the parameters can be readily obtained using linear least squares estimation with the transformed model. 

The famous Michaelis-Menten kinetics expression shown below 

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The output equation remains the same as in (E4-40). Laplace transform of the model equations and rearrangement lead us to... 

There are still other causes of nonlinearities than (apparent or real) higher-order transformation kinetics. In Section 12.3 we discussed catalyzed reactions, especially the enzyme kinetics of the Michaelis-Menten type (see Box 12.2). We may also be interested in the modeling of chemicals which are produced by a nonlinear autocatalytic reaction, that is, by a production rate function, p(Q, which depends on the product concentration, C,. Such a production rate can be combined with an elimination rate function, r(C,), which may be linear or nonlinear and include different processes such as flushing and chemical transformations. Then the model equation has the general form ... 

To check the validity of their model, Brunauer, Emmett and Teller proposed the transformation of the original equation (1) into the linear form ... 

Our equation for the orbital x been obtained by a transformation of the HF equation and so only has the range of validity of the single-determinant model of electronic structure. Thus far, we have written the equation as ... 

Mechanical forces, stresses, strains, and velocities play a critical role in many important aspects of cell physiology, such as cell adhesion, motility, and signal transduction. The modeling of cell mechanics is a challenging task because of the interconnection of mechanical, electrical, and biochemical processes involvement of different structural cellular components and multiple timescales. It can involve nonlinear mechanics and thermodynamics, and because of its complexity, it is most hkely that it will require the use of computational techniques. Typical steps in the development of a cell modeling include constitutive relations describing the state or evolution of the cell or its components, mathematical solution or transformation of the corresponding equations and boundary conditions, and computational implementation of the model. 

Another kind of approach has been initiated in the recent investigations by Toth and Hars (1986a). They studied linear transforms of the Lorenz equation (1968) and of the Rossler model (1976) in order to obtain kinetic models. The failure of their efforts underline the importance of negative cross-effects. Based upon these results the following conjecture can be formulated if a nonkinetic polynomial differential equation shows chaotic behaviour then it cannot be transformed into a kinetic one. 

The solutions given in Table 8.1 were all obtained by Laplace transformation. To obtain the solution of the model equations in the Laplace domain is straightforward but inversion of the transform to obtain an analytic expression for the breakthrough curve or pulse response is difficult. Simple analytic expressions for the moments of the pulse response may, however, be derived rather easily directly from the solution in Laplace form by the application of van der Laan s theorem... 

The key principle for obtaining the inverse model from the direct one is to successively differentiate the outputs with respect to time until the inputs appear in the expression of the output derivatives. Then, from this transformation of the model, the aim is to express the inputs in terms of the outputs by inverting these equations if possible. The condition for the existence of this inversion will also be discussed in the following sections. 

Numerically, the solution of the model equations (PDEs subject to initial and boundary conditions) corresponds to an integration with respect to the space and time coordinates. In general, this is an approximation to the mathematical model s exact solution. In simple cases, often restricted models, analytical solutions given by some, even complex, mathematical function are available. Additional work, e.g., Laplace transformation of the original mathematical model, may be required. Generating an analytical solution is commonly not termed simulation ( modelling... without... simulation [15]). If such solutions are not practical, several techniques are applied, among these ... 

Taking the Laplace transforms of the above equation and rearrangement gives the required transfer-function model ... 

Here c is the local tracer concentration in environmental i,Ui,and are the velocity vector and diffusivity in environment i, and t is the residence time in environment i. Proper boundary conditions require no accumulation of tracer at the boundaries and continuity of tracer flux. Closedness of the system on the boundaries with the inlet and exit environment is required also, i.e. the net input and output of tracer occurs by flow only. To obtain directly from the model the joint p.d.f. a normalized unit impulse input is required. First of all it is readily apparent that if we take the Laplace transform of the above equation we get ... 

Pseudopotentials have been derived from different points of view —transformation of the Schrodinger equation, transformation of KKR theorythe fitting of free-ion term values, etc. The last of these possibilities, which introduces a judicious amount of empiricism, in that it sidesteps various difficulties concerning exchange and correlation on the core, is called the model potential method. Whichever method is used, the results are much the same if the pseudopotential is chosen to be as weak as possible within the chosen framework. A typical pseudopotential is shown in Figure 10. Note the remarkable consistency of the theory and experiment (see also Stafleu and de Vroomen 3- "). 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

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. 

On the debit side, the linearization method is quite sensitive to the form of the network element model. Jeppson and Tavallaee (J2) reported that convergence rate was slow when the usual pump and reservoir models were incorporated, but they obtained significant improvements after the models had been suitably transformed. Although the number of iterations required is small using formulations A and B, the dimension of the matrix equation is substantial. Hence, it becomes essential to use sparse computation techniques if the method is to retain its competitive edge in larger problems. 

There appears to be a more adequate approach when a local polarization characteristic is obtained as a result of analysis of the processes in the elementary cell and the local section of the electrode. This characteristic depends on the state transformation of the solid reagents and the concentrations of the electrolyte components. It further may be introduced into the equations describing the macrokinetic processes in an electrode, and may be used to model the behaviour of the system as a whole. 

Equation 23.4-6 is one form of the performance equation for the bubbling-bed reactor model. It can be transformed to determine the amount of solid (e.g., catalyst) holdup to achieve a specified /A or cA ... 

The hydraulic performance of sewer pipes can be described at different levels. In the case of nonstationary, nonuniform flow, the Saint Venant Equations should be applied. However, under dry-weather conditions, the Manning Equation is an adequate description of the wastewater flow in a gravity sewer pipe when considering the prediction of wastewater quality changes under transport. There are no grounds for using advanced hydraulic models because of the uncertainties in the prediction of the microbial transformations of the wastewater. 

The mathematics underlying transformation of the data from different experiments can be applied to simple models. In the case of the relationship between G (a>) and G(t) it is straightforward. To give an example, consider a Maxwell model. It has an exponentially decaying modulus with time. We have indicated that the relationship between the complex modulus and the relaxation function is given by Equation (4.117). So if we substitute the relaxation function into this expression we get... 

If kinetic data are to be used, it is necessary to transform the variables to conform with those of the partial equilibrium model. The units used in the model equations for and nj are moles formed/kg of solution. Thus the mass of solution in the reacting system from which the kinetic data comes must be known. Frequently, one will know the volume and have to approximate the density. A relation between and t is also needed. For this, the mass of solid originally present must be known. The amount of solid reacing, -ANg, for a time interval At can be obtained from rate curves or calculated from an integrated rate equation. The fraction of the original mass reacting in the time interval gives an approximate value of 5, e.g.,... 

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