ODEs and DAEs

We saw in Section 4.5 that it was convenient to specify reactor models for nonconstant -density cases by using DAEs instead of ODEs. Compare, for example, the complexity of the ODE and DAE models appearing in Table 4.2. Some care should be exercised, however, to avoid creating a DAE that is difficult to solve. The DAEs in Table 4.2 are of the following form in which the time derivatives of only the x variables appear explicitly in the model. 

Many methods for linear system analysis are based on explicit linear ODEs. Constrained linear systems have therefore to be reduced first to an explicit linear ODE. In Sec. 1.4 such a reduction was obtained for tree structured systems by formulating the system in relative coordinates. For general systems this reduction has to be performed numerically. The reduction to this so-called state space form will be the topic of the first part of this chapter. Then, the exact solution of linear ODEs and DAEs is discussed. 

The solution of these equations can be obtained explicitly by modal transformation, i.e. computing eigenvalues and eigenvectors. Prom the computation of the solution the main differences between explicit ODEs and DAEs are figured out. 

The main differences between explicit ODEs and DAEs can be seen directly from the above computations ... 

For these time periods, the ODEs and active algebraic constraints influence the state and control variables. For these active sets, we therefore need to be able to analyze and implicitly solve the DAE system. To represent the control profiles at the same level of approximation as for the state profiles, approximation and stability properties for DAE (rather than ODE) solvers must be considered. Moreover, the variational conditions for problem (16), with different active constraint sets over time, lead to a multizone set of DAE systems. Consequently, the analogous Kuhn-Tucker conditions from (27) must have stability and approximation properties capable of handling ail of these DAE systems. 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

In this approach, the ODE or DAE process models are discretised into a set of algebraic equations (AEs) using collocation or other suitable methods and are solved simultaneously with the optimisation problem. Application of the collocation techniques to ODEs or DAEs results in a large system of algebraic equations which appear as constraints in the optimisation problem. This approach results in a large sparse optimisation problem. 

When the ODE or DAE process models and the control policy are both discretized, as described in section 5.8, the gradients that are required for the NLP problems can be obtained directly from the resulting set of algebraic equations. 

The above set of odes is now solved, choosing some algorithm. Nothing has been specified about the homogeneous chemical reaction function F(C), but it will add terms to the matrix W when specified. After the time derivative is discretised in some way, the equation can be rearranged into the same form as described in Chap. 8 and solved using the same methods or, as mentioned above, solved using a professional ode or DAE solver. 

Limiting the discussion to the use of first-principle models, the behavior of the desired equipment is represented by either (i) a set of algebraic equations or (ii) a system of ordinary differential equations (ODEs), or DAEs, or (iii) partial differential equation (PDE) and partial differential-algebraic equation (PDAE), including... 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

A chemical process plant consists of many unit operations connected by process streams. Each process unit may be modelled by a set of equations (ODEs, PDEs, DAEs, algebraic equations), which include material, energy and momentum balances, phase and chemical equilibrium relations, rate equations and physical property correlations. These equations relate the outlet stream variables to the inlet stream variables for a given set of equipment parameters. At present, there are three approaches of flowsheet calculations the sequential modular, the equation oriented approach and the simultaneous modular strategy. 

Simultaneous solution of the so-called differential-algebraic equation (DAE) set requires coupling of the ODE and algebraic equation solvers, the latter which are not discussed here, but can be found in detail elsewhere [ 1 ]. Description of a DAE set and its solution in the context of a one-dimensional (ID) heterogeneous packed-bed reactor model for autothermal conversion of methane to hydrogen is available in the literature [7]. It is also worth noting that packages such as DASSL and DAEPACK are also available for the solution of coupled DAE sets. 

The dynamic model consists of a set of 33 DAEs (14 of them are ODEs) and 44 variables. The value of three flowrates qr2, grj and qp) are fixed at their steady-state values corresponding to a certain nominal operational conditions. Therefore, this leaves 8 design variables for the integrated design problem, namely the volume of the aeration tanks (v and v ), the areas of the settlers (adj and ad2), the aeration factors (fki and fk2), the gain and the integral time of a PI controller. 

There are algebraic equations describing the constraints, so that the system (1.3.11) is a system of differential-algebraic equations (DAE). These systems are principally different from ODEs and require special numerical methods for their solution. 

We will start by discussing the smooth ODE case. DAEs and discontinuous system dynamics will be considered in Secs. 7.3.1, 7.3.2, respectively. 

Once the PDE has been semi-discretized (i.e., discretize the spatial derivatives but not the timelike derivatives) to form a system of ODEs, the ODEs can be solved by high-level software packages. In the standard form there are many such packages available, with relatively fewer for DAEs (see Section 15.3.3). In the method of lines, the spatial differencing must be done by the user, while time discretization and error control is handled by the ODE software. Overall, the effort to develop a new simulation is reduced, since a good deal of existing high-level software can be used. 

Problems like plug flow can be posed as standard-form ODEs, but it is much more convenient to pose them as DAEs. Other situations, such as boundary-layer flow (Chapters 7 and 17) are difficult to pose as standard-form ODEs, but a DAE formulation works well. The Dassl family of software [46] is designed for solving DAEs and is used extensively in the Chemkin software. 

Taken together, the system of equations represents a set of stiff ordinary differential equations, which can be solved numerically. Because more than one dependent-variable derivative can appear in a single equation (e.g., the momentum equation has velocity and pressure derivatives), it is usually more convenient to use differential-algebraic equation (DAE) software (e.g., Dassl) for the solution rather than standard-form ODE software. 

Remark 2.1. For a standard singularly perturbed model, the DAE system (2.10) has an index v=l, i.e., the variables x2 can be solved for directly from the algebraic equations (2.9) and the reduced-order (equivalent ODE) representation (2.13) is obtained directly. For systems that are in the nonstandard singularly perturbed form, the DAE system (2.10) obtained in the limit as —> 0 has an index v > 1 and an equivalent ODE representation for the slow dynamics is not always readily available. 

If the matrix Lbf(x) is invertible (which is typically true, as will be shown in the following examples), the index of the DAE system (4.27) is two (i.e., a solution for z is obtained directly from Equation (4.29)), and in this case the dimension of the underlying ODE system describing the slow dynamics is 1. 

Once the large internal flow rates have been set via appropriate control laws, the index of the DAE system (7.21) is well defined, and a state-space realization (ODE representation) of the slow subsystem can be derived. This representation of the slow dynamics of the column can be used for the derivation of a model-based nonlinear controller to govern the input-output behavior of the column, namely to address the control of the product purity and of the overall material balance. To this end, the small distillate and bottoms flow rates as well as the setpoints of the level controllers are available as manipulated inputs. 

The common underlying principle in the approaches for characterizing the solvability of a DAE system is to obtain, either explicitly, or implicitly, a local representation of an equivalent ODE system, for which available results on existence and uniqueness of solutions are applicable. The derivation of the underlying ODE system involves the repeated differentiation of the algebraic constraints of the DAE, and it is this differentiation process that leads to the concept of a DAE index that is widely used in the literature. For the semi-explicit DAE systems (A. 10) that are of interest to us here, the index has the following definition. 

The index yj provides a measure of the singularity of the algebraic equations and the resulting differences from ODE systems. More specifically consider the DAE system of Equation (A.10) in the case in which the matrix L(x)... 

Unlike continuous distillation, batch distillation is inherently an unsteady state process. Dynamics in continuous distillation are usually in the form of relatively small upsets from steady state operation, whereas in batch distillation individual species can completely disappear from the column, first from the reboiler (in the case of CBD columns) and then from the entire column. Therefore the model describing a batch column is always dynamic in nature and results in a system of Ordinary Differential Equations (ODEs) or a coupled system of Differential and Algebraic Equations (DAEs) (model types III, IV and V). 

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