Equality dependent variables

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

For complex reactions more than one dependent variable is measured. The fitting procedure should take all the observed variables into account. When each of the variables has a normally distributed error, all data are equally precise, and there is no correlation between the variables measured, parameters can be estimated by minimizing the following function ... 

Introduce other relationships and balances such that the number of equations equals the number of dependent variables... 

Equation (3) in essence states that the rate of change of the concentration of A at time t is equal to that of B and that each of these changes at time t is proportional to the product of the concentrations of the reactants raised to the respective prowers. Note that CA[t) and CB( ) are time-dependent variables. As the reaction proceeds, both CA and CB( ) will decrease in magnitude. For simplicity, these concentrations can be denoted simply by Ca and CB, respectively. 

An important class of the constrained optimization problems is one in which the objective function, equality constraints and inequality constraints are all linear. A linear function is one in which the dependent variables appear only to the first power. For example, a linear function of two variables x and x2 would be of the general form ... 

In Section 1.5 we briefly discussed the relationships of equality and inequality constraints in the context of independent and dependent variables. Normally in design and control calculations, it is important to eliminate redundant information and equations before any calculations are performed. Modem multivariable optimization software, however, does not require that the user clearly identify independent, dependent, or superfluous variables, or active or redundant constraints. If the number of independent equations is larger than the number of decision variables, the software informs you that no solution exists because the problem is overspecified. Current codes have incorporated diagnostic tools that permit the user to include all possible variables and constraints in the original problem formulation so that you do not necessarily have to eliminate constraints and variables prior to using the software. Keep in mind, however, that the smaller the dimensionality of the problem introduced into the software, the less time it takes to solve the problem. 

Let the starting point be (1, 0), at which the objective value is 6.5 and the inequality is satisfied strictly, that is, its slack is positive (s = 1). At this point the bounds are also all satisfied, although y is at its lower bound. Because all of the constraints (except for bounds) are inactive at the starting point, there are no equalities that must be solved for values of dependent variables. Hence we proceed to minimize the objective subject only to the bounds on the nonbasic variables x and y. There are no basic variables. The reduced problem is simply the original problem ignoring the inequality constraint. In solving this reduced problem, we do keep track of the inequality. If it becomes active or violated, then the reduced problem changes. 

Because we now have reached an active constraint, use it to solve for one variable in terms of the other, as in the earlier equality constrained example. Let x be the basic, or dependent, variable, and y and s the nonbasic (independent) ones. Solving the constraint for x in terms of y and the slack s yields... 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

V. Introduce Other Relationships and Balances Such That the Number of Equations Equals the Number of Dependent Variables... 

We have considered in some detail in Section 4.2 the case where the random vector Y of n ancillary or dependent variables relates linearly to those of a vector X of n principal or independent variables (e.g., raw data) with covariance matrix L through the matrix equality... 

The relationship of the dependent variables for diffraction and absorption, I,/Io and n, is considered here. The transmitted intensity of X-ray photons is equal to their incident intensity minus the intensity of photoelectrons of wavelength X emitted ( scattered ) at all angles, z/,... 

The description of any operation or design problem in a multistage separation process requires assigning numerical values to, or setting, a certain number of independent variables. The number of variables to be set depends on the process, and is usually determined easily by the method the authors have called the description rule (HI). Alternatively the number of variables to be set may be determined by writing all of the independent equations which define the process, then counting the number of variables and the number of equations. In order to solve the equations, a sufficient number of independent variables must be set so that the number of dependent variables remaining equals the number of equations. 

Source-dispersion and receptor-oriented models have a common physical basis. Both assume that mass arriving at a receptor (sampling site) from source j was transported with conservation of mass by atmospheric dispersion of source emitted material. From the source-dispersion model point of view, the mass collected at the receptor from source j, Mj, Is the dependent variable which Is equal to the product of a dispersion factor, Dj (which depends on wind speed, wind direction, stability, etc.) and an emission rate factor, Ej, 1. e. , ... 

From the receptor model viewpoint, the total aerosol mass, M, collected on a filter at a receptor Is the dependent variable and equal to a linear sum of the mass contributed by p Individual sources,... 

Evans St Ablow (Ref 2) defined the steady-flow as "a flow in which all partial derivatives with.respect to time are equal to zero . The five equations listed in their, paper (p 131), together with. appropriate initial and boundary conditions, are sufficient to solve for the dependent variables q (material or particle velocity factor), P (pressure), p (density), e (specific internal energy) and s (specific entropy) in regions which.are free of discontinuities. When dissipative irreversible effects are present, appropriate additional terms are required in the equations... 

Recall that mole fractions are related to the mass fractions as X = YkW/Wk. When the dependent variables (i.e., Yk, have their correct values), the residual will equal zero. 

In this case the entropy and volume are dependent variables and are functions of the temperature, pressure, and mole numbers of the components. In addition, we have four derivatives that are all equal to the chemical potential, one from each of the four equations, so... 

Unconstrained Optimization Unconstrained optimization refers to the case where no inequality constraints are present and all equality constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objective function. Very few realistic problems in process optimization are unconstrained. However, the availability of efficient unconstrained optimization techniques is important because these techniques must be applied in real time, and iterative calculations may require excessive computer time. Two classes of unconstrained techniques are single-variable optimization and multivariable optimization. 

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. 

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