Less than

The accuracy of our calculations is strongly dependent on the accuracy of the experimental data used to obtain the necessary parameters. While we cannot make any general quantitative statement about the accuracy of our calculations for multicomponent vapor-liquid equilibria, our experience leads us to believe that the calculated results for ternary or quarternary mixtures have an accuracy only slightly less than that of the binary data upon which the calculations are based. For multicomponent liquid-liquid equilibria, the accuracy of prediction is dependent not only upon the accuracy of the binary data, but also on the method used to obtain binary parameters. While there are always exceptions, in typical cases the technique used for binary-data reduction is of some, but not major, importance for vapor-liquid equilibria. However, for liquid-liquid equilibria, the method of data reduction plays a crucial role, as discussed in Chapters 4 and 6. 

If the vapor mixture contains only ideal gases, the integrals in Equations (3) and (6) are zero, z is unity for all compositions, and ()i equals 1 for each component i. At low pressures, typically less than 1 bar, it is frequently a good assumption to set ( ) = 1, but even at moderately low pressures, say in the vicinity of 1 to 10 bars, (f) is often significantly different from unity, especially if i is a polar component. 

At moderate densities. Equation (3-lOb) provides a very good approximation. This approximation should be used only for densities less than (about) one half the critical density. As a rough rule, the virial equation truncated after the second term is valid for the present range... 

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be... 

Convergence of the iteration requires the norm of the objective vector 1g to be less than the convergence criterion, e. The initial estimates used, if not provided externally, are, in addition to Equation (7-28)... 

The criterion used for "too near the plait point" is that ratio of K s for the two "solvent" components is less than seven with the feed composition in the two-phase region. 

ERF error flag, integer variable normally zero ERF= 1 indicates parameters are not available for one or more binary pairs in the mixture ERF = 2 indicates no solution was obtained ERF = 3 or 4 indicates the specified flash temperature is less than the bubble-point temperature or greater than the dew-point temperature respectively ERF = 5 indicates bad input arguments. 

PARCH reads update information from formatted cards, or other input file containing card images. Input having existing component indices replace old data with those indices. Otherwise the new component data is added to the library, with the restriction that the total number of components is always less than or equal to 100. 

If a reaction is reversible, there is a maximum conversion that can be achieved, the equilibrium conversion, which is less than 1.0. Fixing the mole ratio of reactants, temperature, and pressure fixes the equilibrium conversion. ... 

Figure 3.1a shows a flash drum used to separate by gravity a vapor-liquid mixture. The velocity of the vapor through the flash drum must be less than the settling velocity of the liquid drops. Figure 3.11) shows a simple gravity settler for removing a... 

On the other hand, if Ki is very small (Douglas has suggested a value less than 0.1) in Eq. (4.9), then... 

This means that all of component i entering with the feed Fz leaves with the liquid phase as Lxi. Thus, if a component is required to leave in the liquid phase, its K value should be small (typically less than 0.1). 

Ideally, the K value for the light key component in the phase separation should be greater than 10, and at the same time, the K value for the heavy key should be less than 0.1. Having such circumstances leads to a good separation in a single stage. However, use of phase separators might still be effective in the flowsheet if the K values for the key components are not so extreme. Under such circumstances a more crude separation must be accepted. 

This is an exothermic, reversible, homogeneous reaction taking place in a single liquid phase. The liquid butadiene feed contains 0.5 percent normal butane as an impurity. The sulfur dioxide is essentially pure. The mole ratio of sulfur dioxide to butadiene must be kept above 1 to prevent unwanted polymerization reactions. A value of 1.2 is assumed. The temperature in the process must be kept above 65°C to prevent crystallization of the butadiene sulfone but below lOO C to prevent its decomposition. The product must contain less than 0.5 wt% butadiene and less thM 0.3 wt% sulfur dioxide. 

If the feed is dominated by the middle product (typically more than 50 percent) and the lightest product is present in small quantities (typically less than 5 percent), then the arrangement shown in Fig. 5.106 can be an attractive option. This time the light product must find its way up the column past the sidestream. Again, unless the light product is a small flow and the middle product a high flow, a... 

Equation (7.2) put in words states that the minimum number of units required is one less than the number of streams (including utility streams). 

By constrast, Fig. 7.46 shows a diflFerent arrangement. Hot stream A with a low coefficient is matched with cold stream D, which also has a low coefficient but uses temperature diflferences greater than vertical separation. Hot stream B is matched with cold stream C, both with high heat transfer coefficients but with temperature differences less than vertical. This arrangement requires 1250 m of area overall, less than the vertical arrangement. 

Providing film coefficients vary by less than one order of magnitude, then Eq. (7.6) has been found to predict network area to within 10 percent of the actual minimum. ... 

The area target is usually slightly less than the area observed in design. 

These small positive and negative errors partially cancel each other. The result is that capital cost targets predicted by the methods described in this chapter are usually within 5 percent of the final design, providing heat transfer coefficients vary by less than one order of magnitude. If heat transfer coefficients vary by more than one order of magnitude, then a more sophisticated approach can sometimes be justified. ... 

It was also noted in Sec. 4.4 that the hatch nature of a process can lead to less than full utilization of the equipment. Let us consider how utilization of equipment can be improved. 

Distillation capital costs. The classic optimization in distillation is to tradeoff capital cost of the column against energy cost for the distillation, as shown in Fig. 3.7. This wpuld be carried out with distillation columns operating on utilities and not integrated with the rest of the process. Typically, the optimal ratio of actual to minimum reflux ratio lies in the range 1.05 to 1.1. Practical considerations often prevent a ratio of less than 1.1 being used, as discussed in Chap. 3. 

It is not only the stream number that creates the need to split streams at the pinch. Sometimes the CP inequality criteria [Eqs. (16.1) and (16.2)] CEmnot be met at the pinch without a stream split. Consider the above-pinch part of a problem in Fig. 16.13a. The number of hot streams is less than the number of cold, and hence Eq. (16.3) is satisfied. However, the CP inequality also must be satisfied, i.e., Eq. (16.1). Neither of the two cold streams has a large enough CP. The hot stream can be made smaller by splitting it into two parallel branches (Fig. 16.136). 

The problem with this approach is that if the steam generated in the boilers is at a very high pressure and/or the ratio of power to fuel costs is high, then the value of low-pressure steam can be extremely low or even negative. This is not sensible and discourages efficient use of low-pressure steam, since it leads to low-pressure steam with a value considerably less than its fuel value. 

Franck-Condon principle According to this principle the time required for an electronic transition in a molecule is very much less than the period of vibration of the constituent nuclei of the molecule. Consequently, it may be assumed that during the electronic transition the nuclei do not change their positions or momenta. This principle is of great importance in discussing the energy changes and spectra of molecules. 

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