Differential and Algebraic Equation Systems

Manenti, F., Dones, I., Buzzi-Ferraris, G., and Preisig, H.A. (2009) Efficient numerical solver for partially structured differential and algebraic equation systems. Ind. Eng. Chem. Res., 48,9979 9984. 

A 5-point finite difference scheme along with method of lines was used to transform the partial differential Equations 4-6 into a system of first-order differential and algebraic equations. The final form of the governing equations is given below with the terms defined in the notation section. 

Those requiring solution of a system of differential and algebraic equations... 

When solving systems of differential and algebraic equations, you must list the differential equations first. If a variable is first referred to in an algebraic equation,... 

Tjoa and Biegler (1991) used this formulation within a simultaneous strategy for data reconciliation and gross error detection on nonlinear systems. Albuquerque and Biegler (1996) used the same approach within the context of solving an error-in-all-variable-parameter estimation problem constrained by differential and algebraic equations. 

To rationalize his observations Teorell suggested the following model in terms of ordinary differential and algebraic equations for the average dynamic characteristics of the system concerned. 

A system of differential and algebraic equations (DAE system) is obtained from the model balances. The developed set of equations consists of the ordinary differential equations of first order and of partial differential equations. An analytical solution of the coupled equations is not possible. Therefore, a numeric procedure is used. 

Most of the research on the analysis and numerical simulation of nonlinear DAEs has focused on systems in the fully implicit form of Equation (A.9). However, the generality of the form of the system in Equation (A.9) does not allow the development of explicit controller synthesis results. Also, the majority of chemical process applications (see examples throughout this book), as well as other engineering applications, are modeled by DAEs in a semi-explicit form, such that there is a distinct separation of the differential and algebraic equations ... 

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). 

Morison, K.R., Optimal Control of Processes Described by Systems of Differential and Algebraic Equations. PhD. Thesis, (Imperial College, University of London, 1984). 

Egly et al. (1979), Cuille and Reklaitis (1986), Mujtaba (1989), Reuter et al. (1989), Albet et al. (1991), Basualdo and Ruiz (1995) and Wajge and Reklaitis (1999) considered the development of mathematical models to simulate BREAD processes. In most cases, the model was posed as a system of Differential and Algebraic Equations (DAEs) and a stiff solution method was employed for integration. 

Tran and Mujtaba (1997), Mujtaba et al. (1997) and Mujtaba (1999) have used an extension of the Type IV- CMH model described in Chapter 4 and in Mujtaba and Macchietto (1998) in which few extra equations related to the solvent feed plate are added. The model accounts for detailed mass and energy balances with rigorous thermophysical properties calculations and results to a system of Differential and Algebraic Equations (DAEs). For the solution of the optimisation problem the method outlined in Chapter 5 is used which uses CVP techniques. Mujtaba (1999) used both reflux ratio and solvent feed rate (in semi-continuous feeding mode) as the optimisation variables. Piecewise constant values of these variables over the time intervals concerned are assumed. Both the values of these variables and the interval switching times (including the final time) are optimised in all the SDO problems mentioned in the previous section. 

The state trajectory u t) is computed by the implicit integrator DDASSL (Petzold 1982 Brenan, Campbell, and Petzold 1989). updated here to handle the initial condition of Eq. (B.1-2). The DDASSL integrator is especially designed to handle stiff, coupled systems of ordinary differential and algebraic equations. It employs a variable-order, variable-step predictor-corrector approach initiated by Gear (1971). The derivative vector applicable at t +i. is approximated in the corrector stage by a... 

The application of the QSSA to the above scheme leads to a simple set of differential and algebraic equations describing the system and to an algebraic relationship between the QSSA and the non-QSSA species. As above we choose O and OH as the QSSA species (HO2 is no longer in the scheme it is considered a stable product) and set d[0]/dr = d[OH]/dr = 0. This results in the following algebraic equations ... 

The last important evolution of PrODHyS is the integration of a dynamic hybrid simulation kernel (Ferret et al., 2004 Olivier et al., 2006, 2007). Indeed, the nature of the studied phenomena involves a rigorous description of the continuous and discrete dynamic. The use of Differential and Algebraic Equations (DAE) systems seems obvious for the description of continuous aspects. Moreover the high sequential aspect of the considered systems justifies the use of Petri nets model. This is why the Object Differential Petri Nets (ODPN) formalism is used to describe the simulation model associated with each component. It combines in the same structure a set of DAE systems and high level Petri nets (defining the legal sequences of commutation between states) and has the ability to detect state and time events. More details about the formalism ODPN can be found in previous papers (Ferret et al., 2004). 

Some simple reaction kinetics are amenable to analytical solutions and graphical linearized analysis to calculate the kinetic parameters from rate data. More complex systems require numerical solution of nonlinear systems of differential and algebraic equations coupled with nonlinear parameter estimation or regression methods. 

Performing Simulations As illustrated earlier, the mathematical representation of a PBPK model generally comprises a system of coupled ordinary differential and algebraic equations. These equations are normally amenable to solution via numerical integration methods on the computer. 

Mathematically, all models form a system of (partial) differential and algebraic equations. For the solution of those systems initial and boundary conditions for the chromatographic column are necessary. The initial conditions for the concentration and the loading specify their values at time t = 0. Generally, zero values are assumed ... 

It is also possible to have differential-algebraic equations. There will be both differential and algebraic equations, with all the variables possibly occurring in all the equations. This essentially means that the variables determined from the algebraic equations are acting infinitely fast this is merely a system of stiff equations taken to an extreme. Special methods exist for these problems, too. Because some of the variables are supposed to change immediately, they cannot be included in the error tests used to select the time step. 

Thus, the reaction model is represented by a system of differential and algebraic equations. Although they may be linear, in the general case they are nonlinear. The parameters of the model are of two different types constant and variable. A constant parameter may be either known or unknown, and may appear as a rate or equilibrium constant or within a rate expression. A variable parameter is the concentration of a reactant. A variable parameter is known if sufficient data are given to describe its behavior over the period of time of the reaction. 

The model is formulated in terms of mixed differential and algebraic equations. It is suitable for liquid-solid and gas-liquid-solid reactors. The main assumptions are that the liquid volume and solid mass inside the reactor are constant and that the effects of mass and heat transfer can be neglected. It was generated from balances on both the liquid and solid systems. 

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