Approximations in practice

This chapter treats the effects of temperature on the three types of ideal reactors batch, piston flow, and continuous-flow stirred tank. Three major questions in reactor design are addressed. What is the optimal temperature for a reaction How can this temperature be achieved or at least approximated in practice How can results from the laboratory or pilot plant be scaled up ... 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

The optimal distribution of silver catalyst in a-Al203 pellets is investigated experimentally for the ethylene epoxidation reaction network, using a novel single-pellet reactor. Previous theoretical work suggests that a Dirac-delta type distribution of the catalyst is optimal. This distribution is approximated in practice by a step-distribution of narrow width. The effect of the location and width of the active layer on the conversion of ethylene and the selectivity to ethylene oxide, for various ethylene feed concentrations and reaction temperatures, is discussed. The results clearly demonstrate that for optimum selectivity, the silver catalyst should be placed in a thin layer at the external surface of the pellet. 

Equations (3.1) and (3.2) are, however, only approximations. In practice the bond valence falls to zero at a finite distance somewhere between 300 and 400 pm depending on the bond type, and at short distances the curve becomes very steep as can be seen for H-O bonds in Fig. 7.1. As a result, eqns (3.1) and (3.2) both overestimate the valences of very weak bonds and underestimate the valences of very strong bonds. 

In this chapter, we first present (Section 10b) some qualitative aspects of MPE processes based on their visualization via the configuration coordinate model. We next discuss the problem more quantitatively (Section 10c). It is also to be noted that although the configuration coordinate model (i.e., visualization in terms of a single effective frequency) follows from the quantitative treatment only in certain approximations, in practice most of the literature does use such approximations in one form or another. Finally (Section lOd), we review the literature. 

The application of general thermodynamic theory can be considered, first, in linear approximation. In practice, it is sufficient for the most part of applications. We can use our usual notations for symmetric and antisymmetric tensors of the velocity gradients... 

A liquid-liquid system can be created by coating a particulate matter with a thin layer of a liquid phase, similar to the way packed columns are used in GLC. To maintain such an LLC column, the stationary phase should be insoluble in the mobile phase, just as GLC phases need to be involatile at the temperature of operation. Unfortunately, insolubility is an absolute demand that can at best be approximated in practice. The solubility of the stationary phase in the mobile phase becomes even more critical once some flexibility is desired with regard to the choice of the mobile phase. For example, mixtures of several pure solvents are usually required in order to adapt the eluotropic strength (polarity) of the mobile phase such that the capacity factors fall in the optimum range. 

Theoretically, there is no limit on the number n of points used in the approximations. In practice, however, a limit is set by roundoff in the computations, making an increase in n useless, and the factorials in the matrix H will increase to impractical levels. In any case, there seems little point in n values greater than about 8, although for the usual 32-bit computers in use today, up to 12-point formulas can be accommodated. 

The relationship between the Thiele-modulus and effectiveness factor represented here for a first order reaction can also be applied to other reaction orders for approximations in practical use. This makes the model simpler and easier to use and develop quantitatively. 

Finally we draw attention to the fact that the single configurational coordinate diagram is only an approximation. In practice there is more than one vibrational mode involved and the system is not harmonic. Therefore the value of S is not so easy to determine as suggested above. However, for a general understanding the simple model is extremely useful, as we will see below. 

The solutions in Eqs. 4-42 and 4—44 correspond to the case when the temperature of the exposed surface of the medium is suddenly raised (or lowered) to at f f= 0 and is maintained at that value at all times. The specified surface temperatiite case is closely approximated in practice when condensation or boiling takes place on the surface. Using a similar approach or the l.aplace transfont/technique, analytical solutions can be obtained for other boundary condition s on the surface, with the following results. 

The idealizations stated above are closely approximated in practice, and they greatly simplify the analysis of a heat exchanger with little sacrifice from accuracy. Therefore, they are commonly used. Under these assumptions, the first law of iheniwdyitamics requires that the rate of heat transfer from the hot fluid be equal to the rate of heat transfer to the cold one. That is,... 

Most chemical engineers relate the term incompressible flow to incompressible fluid systems. For non-reactive ideal liquid mixtures operated at nearly constant temperatures, the incompressible flow limit is obviously a reasonable approximation in practice. 

Theoretically, one pound of pure methanol requires 26.7 cu. ft. of dry air measured at 32° F. and 760 mm. of mercury pressure for oxidation to formaldehyde. This proportion can be attained only approximately in practice by bubbling air through methanol since even if the liquid methanol is maintained at the temperature to give the correct vapor pressure, complete equilibrium betweeu air and methanol is not easily obtained without the use of long gas travel. It has been found, however, that considerable variation of this theoretical ratio does not seriously alter the conversion. 

As we stated above, SAPT is formulated in a top down manner. Eq. (6) then forms the top going down to workable equations, one is forced to introduce a multitude of approximations. In practice, i is restricted to the values 1 and 2 interactions of first and second order in Different truncation levels for j + k are applied, depending on the importance of the term (and the degree of complexity of the formula). Working out the equations to the level of one- and two-electron integrals is a far from trivial job. This has been done in a long series of papers that use techniques from coupled cluster theory and many-body PT see Refs. [147,148] for references to this work and a concise summary of the formulas resulting from it. 

Summation over the single-particle states is restricted to positive energy contributions, which has become known as the no virtual pair approximation. In practice, the interaction vu may be the complete energy-dependent, covariant interaction, vf2, the Coulomb interaction, gi2, or the transverse-gauge Breit operator, 312 -I- 612-... 

For scientific theories, being exact in principle seems to be a nice euphemism for being approximate in practice. Density-functional theory suffers from the same fate, and any DFT calculation can only be as reliable as the incorporated parametrization scheme for exchange and correlation. Indeed, the search for reliable exchange-correlation functionals is the greatest challenge to DFT. 

This process is extremely demanding as far as information processing at the related sites and information transfer between distant sites are concerned. Although the available processing power and transmission speed increase exponentially with time, the environment around us is so complex that full copying remains impossible. Thus, virtual environments are only approximated in practice. 

The general rate model was used as a basis for the development of a computer program for the simulation of chromatographic processes by Gu [53]. The solution of the partial differential equations in the nonlinear range of the adsorption isotherms can be obtained by application of numerical methods. One drawback for the modeling of real chromatographic separations with this model is the multitude of physical parameters, which cannot be determined experimentally and have to be estimated by approximations. In practice, these parameters are often only inaccurately fittable, so that a reasonable calculation is impossible. This model is rather applicable for theoretical studies [54]. 

Brandt s calculations are only approximate in practice, the observed charges deviate from the theoretically possible values. For conductive particles, the charge acquired in the field of a corona discharge is 10-20% below the theoretical value. For nonconductive particles, the reverse is observed [130]. 

As we wish to avoid calculating the second derivatives with respect to the free energy due to the complexity and computational cost we can approximate the two second derivative matrices by the static-only components. Because one matrix is multiplied by the inverse of the other there will be a significant cancellation of errors and this turns out to be a good approximation in practice. 

If the underflow line of N versus y is straight and horizontal, the amount of liquid associated with the solid in the slurry is constant for all concentrations. This would mean that the underflow liquid rate is constant throughout the various stages as well as the overflow stream. This is a special case which is sometimes approximated in practice. 

We discuss now the static GW approximation that is the most often used approximation in practice in the BSE approach. 

This equation has a simple solution when the electrical contribution is small com pared to the thermal one. Ozie f/ < kT)—the Debye-Hiickel approximation In practice, when water is used as a liquid medium, at room temperature, this is valid for potentials below- 25 mV. This approximation allows to obtain the simple relationship between the electric potential and position (SI) (Fig. 10b) ... 

These general equations are not limited by any multipolar approximation. In practical implementations, as mentioned above, the reaction field response function can be calculated from atomic or group dipolar polarizabilities, while the solvent charge density operator can be approximated by atomic multipoles. 

Notice the important clause in the question if the energy of the reservoir is used at maximum efficiency. This requires a reversible separation process such as has been discussed above. How could such a process be approximated in practice ... 

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