Primary Variable Specifications

The separation may be defined by specifying C of the primary variables (Xj. X T,., Yc V)- The g functions representing these constraints would take the form 

Note that specifying only X, i = 1,..C or only F, z = 1,..C does not constitute a complete dehnition of the separation because the mole fractions are interrelated by their summation equations. Providing all the JQs or all the F,s does not result in C independent specihcations. tn order to uniquely dehne the separation, the C specihed variables could be a mix of X,s, F,s, and y. 

If for a given component z both and F are specihed, then xp is determined by Equation 4.1. Consequently, xp may not be given as an additional independent specihcation. Also, there could be only one component for which both X, and F, are specihed. Otherwise, the column would be over specihed since each pair of X, and F determines xp. Because of these considerations, the allowable combinations of primary variable specihcations are 

Another restriction applies to situations where mole fractions of a given component are specihed in both the overhead and bottoms. If both Xj and F are specihed, then one of the following inequalities should hold  

That is, if a component gets concentrated in the overhead, it must get diluted in 

---

The C specifications required to define the separation could include a combination of g functions and primary variable specifications. It is important that the specifications be independent and feasible. Specifying both D, and B /or instance, is redundant since they are not independent they are related by material balance,... 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

Cellular activities such as those of enzymes, DNA, RNA and other components are the primary variables which determine the performance of microbial or cellular cultures. The development of specific analytical tools for measurement of these activities in vivo is therefore of essential importance in order to achieve direct analytical access to these primary variables. The focus needs to be the minimization of relevant disturbances of cultures by measurements, i. e. rapid, non-invasive concepts should be promoted in bioprocess engineering science [110,402]. What we can measure routinely today are the operating and secondary variables such as the concentrations of metabolites which fully depend on primary and operating variables. 

In order to solve the equations, C specifications, or constraints, are required, making the total number of equations equal to the number of dependent variables. The specifications define the component separation taking place in the column. In general, these specifications may be written as a function of the primary variables in the form... 

The g functions (Equation 4.3) that define the column performance may be general functions of the primary variables designed to meet special separation specifications, or they may directly specify the values of the primary variables. Combinations of different types of specifications are possible. 

Although column separation constraints usually consist of primary and derived variable specifications as discussed in Sections 4.2.1 and 4.2.2, in general, specifications could be any function of these variables. One could define the g functions of Equation 4.3 as, for instance, sums, differences, or ratios of component rates, recoveries, and so on. The only restrictions on the specification functions, at least from the mathematical standpoint, are that they be independent and feasible. 

In Subsections 5.1, 5.2, a form of Noether s Theorem has been applied in order to derive the associated weak statements of the conserved currents. This implementation led to Equations 14, 16, which correspond to the conservation of energy and momentum, respectively. These equations express in a clear manner the participation of each primary variable in the statements of conserved currents, a task that proves to be not trivial. To be more specific, in the case of linear and angular momentum-conservation statement 16, only the weak velocities and not, as someone may expect, the momentum type variables enter. Moreover, in the case of energy conservation, it is shown in (Eq. 14) that the weak velocities and not the strong time derivatives of displacement determine the kinetic energy. 

For the spatial discretization, the Finite Element Method is employed. To be more specific, the same spatial interpolation basis has been selected for the whole set of primary variables as well as for their variations, albeit this is not required. In case of linearly elastic constitutive equations, and since variations of the primary variables are independent, the following discrete system of equations is obtained ... 

The hydrolysis of alkylaluminum compounds is discussed in considerable detail by A. R. Barron [28], who also provides some alternative routes to synthesize alkylalumoxane compounds. Reaction temperature (ambient to low temperature) and Al/water ratio are the primary variables to control the specific composition of the alkylalumoxane mixture obtained. Note that Barron strongly recommends, as a safety precaution, that any alkylalumoxane synthesis NOT be carried out at an elevated temperature. 

These are seven equations with seven unknowns Xi, X2, X3, Y, Y2, F3, and y/. The last specification was converted to a function of the primary variables using Equation 4.4. The first two specifications and the summation equation define the bottoms composition ... 

This is the energy source for hydraulic systems. It converts electrical energy into dynamic, hydraulic pressure. In almost all cases, hydraulic systems utilize positive displacement pumps as their primary power source. These are broken down into two primary sub-classifications constant-volume or variable-volume. In the former, the pumps are designed to deliver a fixed output (i.e. both volume and pressure) of hydraulic fluid. In the later, the pump delivers only the volume or pressure required for specific functions of the system or its components. 

The ET cover cannot be tested at every landfill site so it is necessary to extrapolate the results from sites of known performance to specific landfill sites. The factors that affect the hydrologic design of ET covers encompass several scientific disciplines and there are numerous interactions between factors. As a consequence, a comprehensive computer model is needed to evaluate the ET cover for a site.48 The model should effectively incorporate soil, plant, and climate variables, and include their interactions and the resultant effect on hydrology and water balance. An important function of the model is to simulate the variability of performance in response to climate variability and to evaluate cover response to extreme events. Because the expected life of the cover is decades, possibly centuries, the model should be capable of estimating long-term performance. In addition to a complete water balance, the model should be capable of estimating long-term plant biomass production, need for fertilizer, wind and water erosion, and possible loss of primary plant nutrients from the ecosystem. 

---