The Dynamic Model

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

After simple manipulations the unsteady-state heat-balance equations for the reactor can be written as follows. 

The unsteady thermal behavior of the dense phase is described by the nonlinear ordinary differential equation 

The assumption of pseudosteady state in the bubble phase and the general assumptions allow us to describe the bubble-phase temperature profile using the algebraic equation (7.35). 

Simple manipulations similar to those used for the reactor give the following heat-balance equations for the regenerator. 

---

Where Ui denotes input number i and there is an implied summation over all the inputs in the expression above A, Bj, C, D, and F are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic models trom u to y, as well as of the noise model from e to y). Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system close to the input). Likewise Fj(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise N(t). 

The above FF controller can be implemented using analog elements or more commonly by a digital computer. Figure 8-33 compares typical responses for PID FB control, steady-state FF control (.s = 0), dynamic FF control, and combined FF/FB control. In practice, the engineer can tune K, and Tl in the field to improve the performance oTthe FF controller. The feedforward controller can also be simplified to provide steady-state feedforward control. This is done by setting. s = 0 in Gj. s). This might be appropriate if there is uncertainty in the dynamic models for Gl and Gp. 

Note that the characteristic equation wiU be unchanged for the FF + FB system, hence system stability wiU be unaffected by the presence of the FF controller. In general, the tuning of the FB controller can be less conservative than wr the case of FB alone, since smaller excursions from the set point will residt. This in turn woidd make the dynamic model Gp(.s) more accurate. 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

In a short time period, the dynamic model shown in Equation (3.13.1.1) at quasi-steady-state condition, OTR to microbial cells would be equal to oxygen molar flow transfer to the liquid phase.4... 

Solution The dynamic model governing the flow of an inert tracer through an unsteady PFR is Equation (14.13) with =0 ... 

The numerical experiment started at a steady-state value of 200 C for both temperature nodes with an output of 16.89% for both heaters output number 1 was then stepped to 19.00%. If both outputs had been stepped to 19%, then both nodes would have gone to 220 C. The temperature of node 5 does not go as high, and the temperature of node 55 goes too high. In the reduced order model, the time constant x represents the effect of radial heat conduction, while the time constant X2 represents the effect of axial heat conduction. SimuSolv estimates these two parameters of the dynamic model as ... 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

In general, the form of the solution to the dynamic model equations will be in the form... 

The principle of the perfectly-mixed stirred tank has been discussed previously in Sec. 1.2.2, and this provides essential building block for modelling applications. In this section, the concept is applied to tank type reactor systems and stagewise mass transfer applications, such that the resulting model equations often appear in the form of linked sets of first-order difference differential equations. Solution by digital simulation works well for small problems, in which the number of equations are relatively small and where the problem is not compounded by stiffness or by the need for iterative procedures. For these reasons, the dynamic modelling of the continuous distillation columns in this section is intended only as a demonstration of method, rather than as a realistic attempt at solution. For the solution of complex distillation problems, the reader is referred to commercial dynamic simulation packages. 

The component mass balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

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 dynamic model involves a component mass balance, an energy balance, the kinetics and the Arrhenius relationship. Hence... 

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. 

Revise the program to include nth-order reaction. Evaluate the conversion for first-order reaction from the dynamic model. Compare this with the conversion calculated from the E curve. Show by simulation that the two are equal only if the reaction is first order. 

The separate phase balances form the dynamic model. 

The design procedures depend heavily on the dynamic model of the process to be controlled. In more advanced model-based control systems, the action taken by the controller actually depends on the model. Under circumstances where we do not have a precise model, we perform our analysis with approximate models. This is the basis of a field called "system identification and parameter estimation." Physical insight that we may acquire in the act of model building is invaluable in problem solving. 

For real physical processes, the orders of polynomials are such that n > m. A simple explanation is to look at a so-called lead-lag element when n = m and y(L + y = x(L + x. The LHS, which is the dynamic model, must have enough complexity to reflect the change of the forcing on the RHS. Thus if the forcing includes a rate of change, the model must have the same capability too. 

The dynamical model described in Figure 9 indicates that the trajectories may recross the central barrier several times if the Cintra R Cintra p transition is faster... 

The dynamic model and track existence pdfs are updated. If the target does not exist it produces no measurement if it does and is detected the expected measurement pdf,dynamical model and track existence pdfs are using the LMIPDA-IMM filter. 

As in the previous experiments, at each epoch we would like to select a waveform (or really the error covariance matrix associated with a measurement using this waveform) so that the measurement will minimize the uncertainty of the dynamic model of the target. We study two possible measures entropy of the a posteriori pdf of the models and mutual information between the dynamic model pdf and measurement history. Both of these involve making modifications to the LMIPDA-IMM approach that are described in [5]. Since we want to minimize the entropy before taking the measurement, we need to consider the expected value of the cost. To do this we replace the measurement z in the IMM equations by its expected value. In the case of the second measure, for a model we have... 

The initial structure of the program is then followed by statements reflecting the dynamic model equations. These are also provided with comment lines with surrounding braces to distinguish them from the executable program lines. Note that the kinetic rate equations are expressed separately apart from the balance equations, to provide additional simplicity and additional flexibility. The kinetic rates are now additional variables in the simulation and the rates can... 

We now proceed to demonstrate the application of the NDDR technique using a simulated CSTR with a first-order, exothermic reaction. The example was taken from Liebman et al. (1992). The dynamic model is given by... 

Based on the experimental data and some speculations on detailed elementary steps taking place over the catalyst, one can propose the dynamic model. The model discriminates between adsorption of carbon monoxide on catalyst inert sites as well as on oxidized and reduced catalyst active sites. Apart from that, the diffusion of the subsurface species in the catalyst and the reoxidation of reduced catalyst sites by subsurface lattice oxygen species is considered in the model. The model allows us to calculate activation energies of all elementary steps considered, as well as the bulk... 

In Section 3.1., we shall show that the dynamic model leads to an unambiguous determination of the type of nonbonded interactions involved while the static model may lead to erroneous predictions as a result of an ambiguous definition of the nature of a nonbonded interaction. The superiority of the dynamic model is due to the fact that nonbonded interactions affect bonded interactions and, thus, the change in an overall overlap population rather than the change of a specific overlap population between nonbonded atoms or groups is the most appropriate index of a nonbonded interaction. Accordingly, we shall employ the dynamic model in all subsequent discussions of molecular structure, unless otherwise stated. 

---