Algebraic equations continuation

Nonlinear equations, discrete algebraic equations, continuation and stability analysis, parameter analysis, and parameter estimation... 

The resulting finite difference equations constitute a set of nonho-mogeneous linear algebraic equations. Because there are three dependent variables, the number of equations in the set is three times the number of material points. Obviously, if a large number of points is required to accurately represent the continuous elastic body, a computer is essential. 

We continue by substituting the eigenvalues, in turn, into the algebraic equations (3-110). Because these equations are not independent, it is not possible to solve uniquely for the individual 7a, 7z values only ratios can be obtained, as follows ... 

Just as a known root of an algebraic equation can be divided out, and the equation reduced to one of lower order, so a known root and the vector belonging to it can be used to reduce the matrix to one of lower order whose roots are the yet unknown roots. In principle this can be continued until the matrix reduces to a scalar, which is the last remaining root. The process is known as deflation. Quite generally, in fact, let P be a matrix of, say, p linearly independent columns such that each column of AP is a linear combination of columns of P itself. In particular, this will be true if the columns of P are characteristic vectors. Then... 

A material balance analysis taking into account inputs and outputs by flow and reaction, and accumulation, as appropriate. This results in a proper number of continuity equations expressing, fa- example, molar flow rates of species in terms of process parameters (volumetric flow rate, rate constants, volume, initial concentrations, etc.). These are differential equations or algebraic equations. 

When the rate data come from a CSTR, the analysis was indicated in Section 3.3.4. Other data of (Ca, t) can be fitted by an algebraic equation and differentiated to obtain the derivative. Then the analysis is continued by Equation 3.10. 

For continuous processes, checking the steady-state results is very useful. Algebraic equations for this can be added to the program, such that both sides became equal at steady state. For batch systems, all the initial mass must equal all the final mass, not always in mols but in kg. Expressed in mols the stoichiometry must be satisfied. 

A sound decomposition strategy should be applicable to any type of mathematical model of a physical process. Therefore, the set of system equations might include linear or nonlinear equations algebraic, differential, difference, or integral equations continuous or discrete variables with the following restrictions ... 

Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. The principal difficulties are two (1) solution of simultaneous nonlinear algebraic equations (usually done by some iterative method), and (2) numerical integration of ordinary differential equations (using discrete finite-difference equations to approximate continuous differential equations). 

The digital simulation of a distillation column is fairly straightforward. The main complication is the large number of ODEs and algebraic equations that must be solved. We will illustrate the procedure first with the simplified binary distillation column for which we developed the equations in Chap. 3 (Sec. 3.11). Equimolal overflow, constant relative volatility, and theoretical plates have been assumed. There are two ODEs per tray (a total continuity equation and a light component continuity equation) and two algebraic equations per tray (a vapor-liquid phase equilibrium relationship and a liquid-hydraulic relationship). 

In this chapter we consider the performance of isothermal batch and continuous reactors with multiple reactions. Recall that for a single reaction the single differential equation describing the mass balance for batch or PETR was always separable and the algebraic equation for the CSTR was a simple polynomial. In contrast to single-reaction systems, the mathematics of solving for performance rapidly becomes so complex that analytical solutions are not possible. We will first consider simple multiple-reaction systems where analytical solutions are possible. Then we will discuss more complex systems where we can only obtain numerical solutions. 

Many reviews and several books [61,62] have appeared on the theoretical and experimental aspects of the continuous, stirred tank reactor - the so-called chemostat. Properties of the chemostat are not discussed here. The concentrations of the reagents and products can not be calculated by the algebraic equations obtained for steady-state conditions, when ji = D (the left-hand sides of Eqs. 27-29 are equal to zero), because of the double-substrate-limitation model (Eq. 26) used. These values were obtained from the time course of the concentrations obtained by simulation of the fermentation. It was assumed that the dispersed organic phase remains in the reactor and the dispersed phase holdup does not change during the process. The inlet liquid phase does not contain either organic phase or biomass. 

It should be noted that the importance of the continuity equation is in evaluating actual velocities within the reactor bed as influenced by the mole, temperature, and pressure changes. Because of the use of mass velocities (pgug), the importance of the actual velocities is really restricted to cases where pressure relationships such as the Blake-Kozeny equation or velocity effects on heat transfer parameters are considered. As will be shown later, very little increased computational effort is introduced by retaining the continuity equation, since it is solved as a set of algebraic equations. 

This same analysis is then performed for the energy balance for the gas, the energy balance for the thermal well, the two mass balances, and the continuity equation. The final coupled system of algebraic and differential equations consists of 5N ordinary differential equations and N + 6(NE + 1) algebraic equations, where... 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

The algebraic equations for the end-point gas temperatures are then substituted into the linearized continuity equations, which are then solved for the velocities. The linearized reaction rate expressions and the linearized expressions for the velocities and for the concentrations and temperatures at the axial boundary points are substituted into the ordinary differential equations. 

Understanding the order of the hydrodynamics equations, continuity and momentum, can be somewhat confusing and possibly not the same from problem to problem. The continuity and momentum equations must be viewed as a closely coupled system. Again, it is clear that the momentum equations are second order in velocity and first order in pressure. The continuity equation is first order in density. However, an equation of state requires that density be a function of pressure, and vice versa. Density and pressure must be dependent on each other through an algebraic equation. Therefore a substitution could be done to eliminate either pressure or density. As a result the coupled system is third order, which can present some practical issues for boundary-condition assignment. The first-order behavior must carry information from some portions of the boundary into the domain, but it does not communicate information back. Therefore, over some portions of a problem... 

At this point, we can proceed with applying the boundary conditions. For the case of a smooth surface with a continuous normal vector, the values of the normal heat flux will be continuous and matrix G will be calculated in the same way that matrix H was found, which will reduce the system to N equations with 2N unknowns. This requires N boundary conditions. As with the constant element, the columns will be exchanged, forming a linear system of algebraic equations. For the case of surfaces with a discontinuous normal vector, we will have the following situations ... 

This section discusses treatment of experimental data, especially for conditions where state variables change over time. These are the most difficult data to treat and correspond to cultures from batch, fed-batch, or any continuous transient phase. In continuous steady state, the state variables and rates values do not alter with time, and the rate calculation results from the algebraic equation solution. 

Equations (3) and (4) represent the component mass balances for continuous systems (packed columns). For discrete systems (tray columns), the differential terms transform to finite differences, and the balances are reduced to 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). 

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. 

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