Algebraic relationships

This distinction between Kd and D is important. The partition coefficient is an equilibrium constant and has a fixed value for the solute s partitioning between the two phases. The value of the distribution ratio, however, changes with solution conditions if the relative amounts of forms A and B change. If we know the equilibrium reactions taking place within each phase and between the phases, we can derive an algebraic relationship between Kd and D. 

An equation of state is an algebraic relationship that relates the intensive... 

If both the model and commercial bed are in the region where the respective Reynolds numbers based on particle diameter and gas density are very low, then a single algebraic relationship can be developed. In that region... 

It is easy to write down an algebraic relationship that describes the simpler forms of rheological behaviour. For example... 

The internal structure of an observer is based on the model of the considered system. Of course, the model can be extremely simple or reduced to a simple algebraic relationship binding available measurements. However, when the model is of the dynamical type, the value of a variable is no longer influenced uniquely by the inputs at the considered moment but also by the former values of the inputs as well as by other system variables. These phenomena are then described by differential equations. Since these models carry information on the interactions between the inputs and the state variables, they are used to estimate unmeasured variables from the readily available measurements. 

Intrinsic viscosity is related to the relative viscosity via a logarithmic function and to the specific viscosity by a simple algebraic relationship. Both of these functions can be plotted on the same graph, and when the data are extrapolated to zero concentration they both should predict the same intrinsic viscosity. The specific viscosity function has a positive slope and the relative viscosity function has a negative slope, as shown in Fig. 3.7. The molecular weight of the polymer can be determined from the intrinsic viscosity, the intercept of either function, using the Mark-Houwink-Sakurada equation. 

Figure 1.2 Genera) system theory view of the algebraic relationship y = x + 2. 

Just as there are many different types of systems, there are many different types of transforms. In the algebraic system pictured in Figure 1.2, the system transform is the algebraic relationship y = a + 2. In the wine-making system shown in Figure 1.3, the transform is the microbial colony that converts raw materials into a Hnished wine. Transforms in the chemical process industry are usually sets of chemical reactions that transform raw materials into finished products. 

Plot the algebraic relationship y = lOx-p. How can the response y be controlled to have the value 9 Plot the algebraic relationship x = lOy-y. Can the response y be controlled to have the value 9 ... 

The problem of calculating the relationship between wall shear stress and 8 V/D from shear-stress-shear-rate data depends on the assumption of an algebraic relationship between the shear stress and shear rate unless graphical procedures are employed. For the very important case of fluids which follow the power-law relationship between shear stress and shear rate, Eq. (3), it has been shown (Mil) that... 

The spectral profile is given by a simple algebraic relationship involving... 

Figure 3.2 illustrates the relatively complex nature of the compressibility factor s dependence on temperature and pressure. It is evident that there can be very substantial departures from ideal-gas behavior. Whenever possible, it is useful to represent the equation of state as an algebraic relationship of pressure, temperature, and volume (density). Certainly, when applied in computational modeling, the benefits of a compact equation-of-state representation are evident. There are many ways that are used to accomplish this objective [332], most of which are beyond our scope here. 

The advantage of defining a transfer function in terms of Laplace transforms of input and output is that the differential equations developed to describe the unsteady-state behaviour of the system are reduced to simple algebraic relationships (e.g. cf. equations 7.17 and 7.19). Such relationships are much easier to deal with, and normal algebraic laws can be used to relate the various transfer functions of each component in the control loop (see Section 7.9). Furthermore, the output (or response) of the system to a variety of inputs may be obtained without classical integration. 

From a quantitative viewpoint, it is evident that the relative intensities of the resonances from carbons associated with branches and end groups can be compared to the intensity for the major methylene resonance, "n", at 30.00 ppm to determine branch concentrations and number average molecular weight or carbon number. The following definitions are useful in formulating the appropriate algebraic relationships ... 

In this way, the diffusion/reaction equations are reduced to trial and error algebraic relationships which are solved at each integration step. The progress of conversion can therefore be predicted for a particular semi-batch experiment, and also the interfacial conditions of A,B and T are known along with the associated influence of the film/bulk reaction upon the overall stirred cell reactor behaviour. It is important to formulate the diffusion reaction equations incorporating depletion of B in the film, because although the reaction is close to pseudo first order initially, as B is consumed as conversion proceeds, consumption of B in the film becomes significant. 

Rate equations such as Eq. 6 are converted to algebraic relationships to give the AN of a controlling component by substituting differences, A, for differentials, d. For example, in the case of Cu-AsS- reacting according to Eq. 3,... 

Jacket-Side Coefficient Here the calculations are shown for a jacket equipped with agitation nozzles that greatly increase the jacket fluid velocity. The jacket swirl velocity v, is calculated (iteratively) from the nonlinear algebraic relationship [17]... 

In essence, the application of any one of the three principal tools allows one or several rate equations to be replaced by algebraic equations which, in turn, can be used to eliminate concentrations of intermediates from the set. In the best of all worlds, mathematics can be reduced to a single rate equation and simple algebraic relationships between the concentrations of the reactants and products. More often, several simultaneous rate equations remain, but the reduced set is nevertheless much easier to handle for network elucidation and more convenient for modeling. 

This example has shown how the procedures developed in earlier chapters can be used effectively for modeling. The reaction system has seventeen participants olefin, paraffin, aldehyde, alcohol, H2, CO, HCo(CO)3Ph, HCo(CO)2Ph, and nine intermediates. "Brute force" modeling would require one rate equation for each, four of which could be replaced by stoichiometric constraints (in addition to the constraints 11.2 to 11.4, the brute-force model can use that of conservation of cobalt). Such a model would have 22 rate coefficients (arrowheads in network 11.1, not counting those to and from co-reactants and co-products), whose values and activation energies would have to be determined. This has been reduced to two rate equations and nine simple algebraic relationships (stoichiometric constraints, yield ration equations, and equations for the A coefficients) with eight coefficients. Most impressive here is the reduction from thirteen to two rate equations because these may be differential equations. 

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 ... 

Start by using these correlations to develop algebraic relationships, for geometrically similar systems, between the impeller power per unit volume (P/V) required as a function of the vessel volume for various process results. The culmination of that analysis is presented as Figure 10.49, which presents (P/V)2/ (P/ F) versusVi for the various process results. 

The process of matrix inversion is analogous to obtaining the reciprocal of a number fl. The matrix relationship that corresponds to the algebraic relationship a X (1/a) = 1 is... 

The equations in Table 12-8 are insufficient on their own. Some algebraic relationships are needed to formulate a complete problem, as illustrated in Example 16. Equations for the mass- and heat-transfer coefficients are also needed for the boundary conditions presented in Table 12-8. These require the physical properties of the air, the object geometry, and Reynolds number. Example 16 shows the solution for a problem using numerical modeling. This example shows some of the important qualitative characteristics of drying. 

A useful algebraic relationship is coned r coned Tail X Cj ]. 

In this section, we describe two types of volumetric calculations. The first involves computing the molarity of solutions that have been standardized against either a primary-standard or another standard solution. The second involves calculating the amount of analyte in a sample from titration data. Both types are based on three algebraic relationships. Two of these are Equations 13-1 and 13-3, both of which are based on millimoles and milliliters. The third relationship is the stoichiometric ratio of the number of millimoles of the analyte to the number of millimoles of titrant. 

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