Model, mathematical state variables

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

Parameter estimation problems result when we attempt to match a model of known form to experimental data by an optimal determination of unknown model parameters. The exact nature of the parameter estimation problem will depend on the mathematical model. An important distinction has to be made at this point. A model will contain both state variables (concentrations, temperatures, pressures, etc.) and parameters (rate constants, dispersion coefficients, activation energies, etc.). 

State estimators are basically just mathematical models of the system that are solved on-line. These models usually assume linear DDEs, but nonlinear equations can be incorporated. The actual measured inputs to the process (manipulated variables) are fed into the model equations, and the model equations are integrated. Then the available measured output variables are compared with the predictions of the model. The differences between the actual measured output variables and the predictions of the model for these same variables are used to change the model estimates through some sort of feedback. As these differences between the predicted and measured variables are driven to zero, the model predictions of all the state variables are changed. 

State variables appear very naturally in the differential equations describing chemical engineering systems because our mathematical models are based on a number of first-order differential equations component balances, energy equations, etc. If there are N such equations, they can be linearized (if necessary) and written in matrix form... 

The simplest and often most suitable modeling tool is the one-box model. One-box models describe the system as a single spatially homogeneous entity. Homogeneous means that no further spatial variation is considered. However, one-box models can have one or several state variables, for instance, the mean concentration of one or several compounds i which are influenced both by external forces (or inputs) and by internal processes (removal or transformation). A particular example, the model of the well-mixed reactor with one state variable, has been discussed in Section 12.4 (see Fig. 12.7). The mathematical solution of the model has been given for the special case that the model equation is linear (Box 12.1). It will be the starting point for our discussion on box models. 

Writing mass-, heat-, energy-, and/or momentum-balance equations to obtain the model equations that relate the system input and output to the state variables and the physico-chemical parameters. These mathematical equations describe the state variables with respect to time and/or space. 

In the next sections, the multiscale modeling methods are presented from the different disciplines perspectives. Clearly one could argue that overlaps occur, but the idea here is to present the multiscale methods from the paradigm from which they started. For example, the solid mechanics internal state variable theory includes mathematics, materials science, and numerical methods. However, it clearly started from a solid mechanics perspective and the starting point for mathematics, materials science, and numerical methods has led to other different multiscale methods. 

The proposed mathematical modeling to fit the experimental data is based on Bree et al. (1988), and is represented by a set of differential Equations A.l to A.9, consisting of seven state variables (Table A2) and 20 fitted parameters (Table A3). 

In the steady state simulation and design, the state variables are the flowrates and the pressure drops or terminal pressures for each branch of the net. Each of the branches between two nodepoints are described by a mathematical model of the hydrodynamics relating the pressure drop to the fluid flow between the nodes. The material balance sums up the flow into and out of a node no is... 

Biomass concentration is of paramount importance to scientists as well as engineers. It is a simple measure of the available quantity of a biocatalyst and is definitely an important key variable because it determines - simplifying - the rates of growth and/or product formation. Almost all mathematical models used to describe growth or product formation contain biomass as a most important state variable. Many control strategies involve the objective of maximizing biomass concentration it remains to be decided whether this is always wise. 

A flow model may be considered to be a mathematical equation that can describe rheological data, such as shear rate versus shear stress, in a basic shear diagram, and that provides a convenient and concise manner of describing the data. Occasionally, such as for the viscosity versus temperature data during starch gelatinization, more than one equation may be necessary to describe the rheological data. In addition to mathematical convenience, it is important to quantify how magnitudes of model parameters are affected by state variables, such as temperature, and the effect of structure/composition (e.g., concentration of solids) of foods and establish widely applicable relationships that may be called functional models. 

In this section, classical state-space models are discussed first. They provide a versatile modeling framework that can be linear or nonlinear, continuous- or discrete-time, to describe a wide variety of processes. State variables can be defined based on physical variables, mathematical solution convenience or ordered importance of describing the process. Subspace models are discussed in the second part of this section. They order state variables according to the magnitude of their contributions in explaining the variation in data. State-space models also provide the structure for... 

Choose scales, state variables, processes, and parameters. We produce a written formulation of the model. Producing and updating this formulation is essential for the entire modeling process, including final publication or delivery to clients (Grimm et al. 2006). With mathematical models, we use equations to formulate the model with simulation models, we use a mixture of verbal descriptions, pseudocode, model rules, and the equations that are implemented in the computer programs. 

Determination of the quantitative laws governing the rates of the processes in terms of the state variables. These quantitative laws can be obtained from the literature and/or throu an experimental research program coupled with a mathematical modelling program. 

The steady states which are unstable using the static analysis discussed above are always unstable. However, steady states that are stable from a static point of view may prove to be unstable when the full dynamic analysis is performed. That is to say simply that branch 2 in Figure 4.8 is always unstable, while branches 1,3 in Figure 4.8 and branch 4 in Figure 4.8 can be stable or unstable depending upon the dynamic stability analysis of the system. As mentioned earlier, the analysis for the CSTR presented here is mathematically equivalent to that of a catalyst pellet using lumped parameter models or a distributed parameter model made discrete by a technique such as the orthogonal collocation technique. However, in the latter case, the system dimensionality will increase considerably, with n dimensions for each state variable, where n is the number of internal collocation points. 

Mathematically, the dynamic behaviour of the system is represented by ordinary differential equations in the state variables which are pellet temperature and concentration of reactants (and products in certain cases), with time as the independent variable. This is in contrast to the more realistic distributed models which account for intraparticle temperature and concentration gradients. 

Once programmed, the dynamic simulation will be used to understand the various processes going on inside a complex plant and to make usable predictions of the behaviour that will result from any changes or disturbances that may occur on the real plant, represented on the simulation by forcing functions or alterations to the chosen starting conditions. A basic first step is to characterize the condition of the plant at any given instant in time, and it is the state vector that, taken in conjunction with its associated mathematical model, allows us to do this. The state vector is an ordered collection of all the state variables. For a typical chemical plant, the state vector will consist of a number of temperatures, pressures, levels and valve positions, and the total number of state variables will be the dimension or order of the plant. For those... 

In view of the fundamental importance of the state vector to the way in which we look at the plant, it might be supposed that only one set of state variables could emerge from a valid mathematical description of the plant, and that the composition of the state vector would have to be unique. In fact, this is not so. It will normally be possible to choose several different ways of describing a process plant, and each description will lead to a different set of variables making up the state vector, and a different associated mathematical model. 

We changed from one set of state variables to another in Section 2.4 and, although the meanings and values of the state variables changed, the number of state variables remained the same. Intuitively, this is not surprising, since we had introduced no new physical phenomena into our modelling, and the two descriptions of the plant were based on different manipulations of the same descriptive equations. The fact that different mathematical descriptions based on the same set of modelled phenomena give rise to the same number of state variables leads us to look on the dimension of our model as a measure of its complexity. 

The state equations with the associated state variables constitute the mathematical model of a process, which yields the dynamic or static behavior of the process. The application of the conservation principle... 

The thermodynamic equilibrium between the vapor and liquid phases [imposes certain restrictions on the state variables of the system, and I must be included in the mathematical model of the flash drum if it is to be consistent and correct. These equilibrium relationships, as known from chemical thermodynamics, are ... 

Equation (5.1) is the mathematical model of the stirred tank heater with T the state variable, while T, and Ta are the input variables. Let us see how we can develop the corresponding input-output model. 

Furthermore, assuming that from all possible disturbances only the feed compositions cA, and cA2 are expected to change significantly whereas the feed flow rates F, and F2 and feed temperatures Tt and T2 are expected to remain almost the same, we can omit from the mathematical model the total mass and energy balances and from the set of state variables volume V and temperature T2. Thus the simplified model is given only by the balance on component A [eq. (4.13a)]. 

The state equations (4.4a) and (4.5b), with the state variables, the inputs, and the parameters, constitute the mathematical model of the stirred tank heater. We need only solve them in order to find the tank s dynamic or steady-state behavior. 

The mathematical models we learned to develop in Chapter 4 using state variables are not of the direct input-output type. Nevertheless, they constitute the basis for the development of an input-output model. This is particularly easy and straightforward when the state variables coincide completely with the output variables of a process. In such a case we can integrate the state model to produce the input-output model of the process. 

The external disturbances which are expected to appear and affect the operation of a process will influence the mathematical model that we need to develop. Furthermore, disturbances with a very small impact on the operation of the process can be neglected, whereas disturbances with significant impact on the process must be included in the model. This will determine the complexity of the model needed that is, what balances and what state variables should be included in the model. 

Let us return to the stirred tank heater (Example 4.4). If the feed flow rate (disturbance) is not expected to vary significantly, the volume of the liquid in the tank will remain almost constant. In this case dV/dt = A dh/dt = 0 and we can neglect the total mass balance and the associated state variable h. The mathematical model of interest for control purposes is given by the total energy balance alone [eq. (4.5b)], with temperature the only state variable. 

II.2 Develop the mathematical model for the system shown in Figure PII.2. What are the state variables for this system and what type of balance equations have you used All the flow rates are volumetric, and the cross-sectional areas of the three tanks are A i, A2, and A 3 (ft2), respectively. The flow rate F5 is constant and does not depend on hi, while all other effluent flow rates are proportional to the corresponding hydrostatic liquid pressures that cause the flow. 

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