Simultaneous Equations Models

Since nothing is excluded from either equation and there are no other restrictions, neither equation passes the order condition for identification. 

All reduced form parameters are estimable directly by using least squares, so the reduced form is identified in all cases. Now, 71 = n /n2. On is the residual variance in the euqation (yi - yiy2) = Sj, so G must be estimable (identified) if 71 is. Now, with a bit of manipulation, we find that con - ron = -On/A. Therefore, with On and Yi known (identified), the only remaining unknown is y2. which is therefore identified. With 71 and y2 in hand, P may be deduced from n2. With y2 and P in hand, ct22 is the residual variance in the equation (y2 - Px -YiVi) 2, which is dfrectly estimable, therefore, identified. 

Verify the rank and order conditions for identification of the second and thud behavioral equation in Klein s Model I. [Hint See Example 15.6.] 

Following the method in Example 15.6, for identification of the investment equation, we require that 

Identification requires that the rank of each matrix be M-l =3. The second is obviously not identified. In (1), none of the three columns can be written as a linear combination of the other two, so it has rank 3. (Although the second and last columns have nonzero elements in the same positions, for the matrix to have short rank, we would require that the third column be a multiple of the second, since the first cannot appear in the linear combination which is to replicate the second column.) By the same logic, (3) and (4) are identified. 

Obtain the reduced form for the model in Exercise 1 under each of the assumptions made in parts (a) and (bl), (b6), and (b9). 

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Vol. 312 J. Krishnakumar, Estimation of Simultaneous Equation Models with Error Coniponents Structure. X, 357 pages. 1988. 

Traditionally, least-squares methods have been used to refine protein crystal structures. In this method, a set of simultaneous equations is set up whose solutions correspond to a minimum of the R factor with respect to each of the atomic coordinates. Least-squares refinement requires an N x N matrix to be inverted, where N is the number of parameters. It is usually necessary to examine an evolving model visually every few cycles of the refinement to check that the structure looks reasonable. During visual examination it may be necessary to alter a model to give a better fit to the electron density and prevent the refinement falling into an incorrect local minimum. X-ray refinement is time consuming, requires substantial human involvement and is a skill which usually takes several years to acquire. 

In what immediately follows, we will obtain eigenvalues i and 2 for //v / = Ei ) from the simultaneous equation set (6-38). Each eigenvalue gives a n-election energy for the model we used to generate the secular equation set. In the next chapter, we shall apply an additional equation of constr aint on the minimization parameters ai, 2 so as to obtain their unique solution set. 

Lagrangian trajectory models can be viewed as foUowing a column of air as it is advected in the air basin at the local wind velocity. Simultaneously, the model describes the vertical diffusion of poUutants, deposition, and emissions into the air parcel as shown in Eigure 4. The underlying equation being solved is a simplification of equation 5 ... 

We turn now to the numerical solution of Equations (9.1) and (9.3). The solutions are necessarily simultaneous. Equation (9.1) is not needed for an isothermal reactor since, with a flat velocity profile and in the absence of a temperature profile, radial gradients in concentration do not arise and the model is equivalent to piston flow. Unmixed feed streams are an exception to this statement. By writing versions of Equation (9.1) for each component, we can model reactors with unmixed feed provided radial symmetry is preserved. Problem 9.1 describes a situation where this is possible. 

This approach is useful when dealing with relatively simple partial differential equation models. Seinfeld and Lapidus (1974) have provided a couple of numerical examples for the estimation of a single parameter by the steepest descent algorithm for systems described by one or two simultaneous PDEs with simple boundary conditions. 

Equations (22), and (25)-(29) constitute a set of simultaneous equations from which the IAS model calculation can be made. 

There is a problem with this approach, however - a problem with the residuals. The residuals are neither parameters of the model nor parameters associated with the uncertainty. They are quantities related to a parameter that expresses the variance of the residuals, The problem, then, is that the simultaneous equations approach in Equation 5.25 would attempt to uniquely calculate three items (P, r, and r,2) using only two experiments, clearly an impossible task. What is needed is an additional constraint to reduce the number of items that need to be estimated. A unique solution will then exist. 

Cuthrell, J. E., and Biegler, L. T., Simultaneous solution and optimization of process flowsheets with differential equation models, Chem. Eng. Res. Des. 64, 341 (1986). 

The emphasis in the previous sections has been on the accuracy with which the Gibbs energy, particularly the entropy component above T , can be calculated. However, as the number of components, C, and the number of atoms in the chosen cluster, n, increases, the number of simultaneous equations that have to be solved is of the order of C". This number is not materially reduced by redefining the equations in terms of multi-site correlation functions (Kikuchi and Sato 1974). The position may be eased as extra computing power becomes available, but a choice will inevitably have to be made between supporting a more complex model or extending a simpler model to a greater number of components. 

These models describe the development of surface charge and potential together with ion adsorption in a quantitative manner. They have in common a set of simultaneous equations that can be solved by numerical methods using the appropriate... 

At some point in most processes, a detailed model of performance is needed to evaluate the effects of changing feedstocks, added capacity needs, changing costs of materials and operations, etc. For this, we need to solve the complete equations with detailed chemistry and reactor flow patterns. This is a problem of solving the R simultaneous equations for S chemical species, as we have discussed. However, the real process is seldom isothermal, and the flow pattern involves partial mixing. Therefore, in formulating a complete simulation, we need to add many additional complexities to the ideas developed thus far. We will consider each of these complexities in successive chapters temperature variations in Chapters 5 and 6, catalytic processes in Chapter 7, and nonideal flow patterns in Chapter 8. In Chapter 8 we will return to the issue of detailed modeling of chemical reactors, which include all these effects. 

This set of linear simultaneous equations is a mathematical model of the blending operation. Given the volumes of the components, vh and the... 

We first examine the final state of the reaction, i.e. the chemical equilibrium composition. This is not of great relevance to oscillatory behaviour but is an important first check that the model is chemically reasonable . Equilibrium arises when all three rates of change become zero simultaneously. Equations (2.1)—(2.3) have a unique point satisfying this condition, as required chemically, given by... 

Equations (5.68-5.72) and (5.61) form a set of simultaneous equations for the unknown temperatures, Tc, Ta, T°ut, Tf, 7y ut, and the heat distribution factor a for one cell of the stack. Writing similar equations for all cells in the stack will result in a larger system of simultaneous equations. The equations for neighboring cells are coupled through heat conduction terms. Cell power, voltage, heat generation factor, utilizations of hydrogen and methane and inlet temperatures and concentrations of fuel and air for each cell are the input parameters for the model. 

Physical Models versus Empirical Models In developing a dynamic process model, there are two distinct approaches that can be taken. The first involves models based on first principles, called physical or first principles models, and the second involves empirical models. The conservation laws of mass, energy, and momentum form the basis for developing physical models. The resulting models typically involve sets of differential and algebraic equations that must be solved simultaneously. Empirical models, by contrast, involve postulating the form of a dynamic model, usually as a transfer function, which is discussed below. This transfer function contains a number of parameters that need to be estimated from data. For the development of both physical and empirical models, the most expensive step normally involves verification of their accuracy in predicting plant behavior. 

An alternative to the sequential modular approach is to solve the equations modeling all of the units in a process flowsheet simultaneously this is known as the equation-based approach. Advantages to the sequential modular approach include (1) specialized numerical techniques tailored to each unit operation can be used, and (2) the numerical failure of one unit operation may still yield usable flowsheet information. Advantages to the equation-based... 

In the computer model the simultaneous equations (5), (6) and (8) are combined into a single third-order equation for Ce, which is solved iterately for each integration step. For relevant values of the various parameters, the equation has a single positive solution. 

In the modeling of solid fuel conversion reactors differential equations arise for the description of particle temperatures and gas-solid reactions among others. These equations are coupled and they must be solved simultaneously. Because of the usually wide range of particle sizes the time constants for thermal transients of solids differ considerably. This causes stiffness in the differential equation model. Depending on the type of the gas-solid reactions stiffness may also be introduced by the variation of reaction rates with individual reaction type and with temperature. 

It is impossible or impractical to model processes of this type with simultaneous equations. 

We let a beam of atomic hydrogen and a beam of methyl radicals interact with the film surface simultaneously. This can be considered the simplest of all multi-species experiments first, H and CH3 are the simplest radicalic hydrocarbon species. Second, by restricting ourselves to radicals, the interaction of the beams with the film is purely chemical and expected to be limited to the very surface. Third, the effect of each species separately is already known the previous section described the temperature dependent interaction of CH3 radicals with the a-C H surface. The interaction of atomic hydrogen with carbonaceous materials has been studied extensively in the past by various groups [53,56,57]. A rate equation model describing chemical erosion by atomic hydrogen is well-established [53,58]. 

The number m of observations exceeds (usually by an order of magnitude or more) the number n of adjustable parameters. Thus the mathematical problem is overdetermined. (For a situation in which m= n, the corresponding set of simultaneous equations could in principle be solved exactly for the parameters aj. But this exact fit to the data would provide no test at all of the validity of the model.) In least squares as properly applied, the number of observations is made large compared to the number of parameters in order (1) to sample adequately a domain of respectable size for testing the validity of the model, (2) to increase the accuracy and precision of the parameter determinations, and (3) to obtain statistical information as to the quality of the parameter determination and the applicability of the model. 

Rate equations rK for the end members of simple networks can be given either in explicit form or as an algorithm. Such equations are useful for mathematical modeling in that they are much simpler to handle than the sets of simultaneous rate equations for all participants in a reaction. This is because the concentrations of all intermediates have been eliminated and the number of simultaneous equations is much smaller. However, even for networks of only moderate complexity, the concentration dependence of the rates is rather involved. The principal application... 

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