Divide

There is no sharp dividing line between "slightly" supercritical and "highly" supercritical. Experience has shown, however, that for most practical purposes the dividing line is in the region T/T, 1.8. Using this criterion as a guide, it... 

The critical temperature of methane is 191°K. At 25°C, therefore, the reduced temperature is 1.56. If the dividing line is taken at T/T = 1.8, methane should be considered condensable at temperatures below (about) 70°C and noncondensable at higher temperatures. However, in process design calculations, it is often inconvenient to switch from one method of normalization to the other. In this monograph, since we consider only equilibria at low or moderate pressures in the region 200-600°K, we elect to consider methane as a noncondensable component. 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

Many different manipulations of these equations have been used to obtain solutions. As discussed by King (1971), many of the older approaches work in terms of V/L, which has the disadvantage of being unbounded and which, in the classical implementation, leads to poorly convergent iterative calculations. A preferable arrangement of this equation system for solution is based on the ratio V/F, which must lie between 0 and 1. If we substitute in Equation (7-1) for L from Equation (7-2) and for y from Equation (7-4), and then divide by F, we obtain... 

This value determines the amount the step-size is reduced to satisfy the criteria of a SSQ which decreases from one iteration to the next. The amount of the decrease is equal to the previous value of the step-limiting parameter divided by RP. 

ENERGY parameter DIVIDED BY BOLTXMAN CONSTANT. CONTROL PARAMETER NORMALLY ZERO WHICH IS SET EQUAL TO 1 WHEN ORGANIC ACIDS ARE PRESENT (ANY ETA( IJ).GE.4.S ). 

Flotation. Flotation is a gravity separation process which exploits differences in the surface properties of particles. Gas bubbles are generated in a liquid and become attached to solid particles or immiscible liquid droplets, causing the particles or droplets to rise to the surface. This is used to separate mixtures of solid-solid particles and liquid-liquid mixtures of finely divided immiscible droplets. It is an important technique in mineral processing, where it is used to separate different types of ore. 

This technique is useful not only when the mixture is impossible to separate by conventional distillation because of an azeotrope but also when the mixture is difficult to separate because of a particularly low relative volatility. Such distillation operations in which an extraneous mass-separating agent is used can be divided into two broad classes. 

With this rule in mind, divide the process at the pinch as shown in... 

Not all problems have a pinch to divide the process into two parts. Consider the composite curves in Fig. 6.10a. At this setting, both steam and cooling water are required. As the composite curves are moved closer together, both the steam and cooling water requirements decrease until the setting shown in Fig. 6.106 results. At this setting, the composite curves are in alignment at the hot end,... 

Example 6.2 A process is to be divided into two operationally independent areas of integrity, area A and area B. The stream data for the two areas are given in Table 6.6." Calculate the penalty in utility consumption to maintain the two areas of integrity for = 20°C. 

Solution Figure 7.2 shows the stream grid with the pinch in place dividing the process into two parts. Above the pinch there are five streams, including the steam. Below the pinch there are four streams, including the cooling water. Applying Eq. (7.3),... 

Figure 7.3 To determine the network area, the balanced composite curves are divided into enthalpy intervals.

Solution First, we must construct the balanced composite curves using the complete set of data from Table 7.1. Figure 7.5 shows the balanced composite curves. Note that the steam has been incorporated within the construction of the hot composite curve to maintain the monotonic nature of composite curves. The same is true of the cooling water in the cold composite curve. Figure 7.5 also shows the curves divided into enthalpy intervals where there is either a... 

A simple algorithm can be developed (see App. E) to target the minimum total number of shells (as a real, i.e., noninteger number) for a stream set based on the temperature distribution of the composite curves. The algorithm starts by dividing the composite... 

Total heat transfer area is assumed to be divided equally between exchangers. This tends to overestimate the capital cost. 

Distillation. There is a large inventory of boiling liquid, sometimes under pressure, in a distillation column, both in the base and held up in the column. If a sequence of columns is involved, then, as discussed in Chap. 5, the sequence can be chosen to minimize the inventory of hazardous material. If all materials are equally hazardous, then choosing the sequence that tends to minimize the flow rate of nonkey components also will tend to minimize the inventory. Use of the dividing-wall column shown in Fig. 5.17c will reduce considerably the inventory relative to two simple columns. Dividing-wall columns are inherently safer than conventional arrangements because they lower not only the inventory but also the number of items of equipment and hence lower the potential for leaks. 

Use the divided wall column shown in Fig. 5.17c to reduce the inventory relative to two simple columns, and reduce the number of items of equipment and hence lower the potential for leaks. 

Ultrafiltration. Ultrafiltration was described under pretreatment methods. It is used to remove finely divided suspended solids, and when used as a tertiary treatment, it can remove virtually all the BOD remaining after secondary treatment. 

Figure 13.3 shows a process represented simply as a heat sink and heat source divided hy the pinch. Figure 13.3a shows the process with an exothermic reactor integrated above the pinch. The minimum hot utility can be reduced by the heat released by reaction, Qreact-... 

Figure 13.7 The problem can be divided into two parts, one associated with the reactor and the other with the rest of the process (AT i = 10°C), and then superimposed.

Fig. 14.1a. The background process (which does not include the reboiler and condenser) is represented simply as a heat sink and heat source divided by the pinch. Heat Qreb is taken into the reboiler above pinch temperature and rejected from the condenser at a lower temperature, which is in this case below pinch temperature. Because the process sink above the pinch requires at least Q min to satisfy its...

If complex distillation columns are being considered, then these also can bring about significant reductions in capital cost. The dividing-wall column shown in Fig. 5.17 not only requires typically 20 to 30 percent less energy than a conventional arrangement but also can be typically 30 percent lower in capital cost than a conventional two-column arrangement. ... 

Divide the problem at the pinch into separate problems. 

Figure 16.10 shows another threshold problem that requires only hot utility. This problem is different in characteristic from the one in Fig. 16.9. Now the minimum temperature difference is in the middle of the problem, causing a pseudopinch. The best strategy to deal with this type of threshold problem is to treat it as a pinched problem. For the problem in Fig. 16.10, the problem is divided into two parts at the pseudopinch, and the pinch design method is followed. The only complication in applying the pinch design method for such problems is that one-half of the problem (the cold end in Fig. 16.10) will not feature the flexibility offered by matching against utility.

Figure 16.26 shows a pair of composite curves divided into enthalpy intervals with a possible superstructure shown for one of the intervals. The structure is created by splitting each hot stream... 

Figure B.l shows a pair of composite curves divided into vertical enthalpy intervals. Also shown in Fig. B.l is a heat exchanger network for one of the enthalpy intervals which will satisfy all the heating and cooling requirements. The network shown in Fig. B.l for the enthalpy interval is in grid diagram form. The network arrangement in Fig. B.l has been placed such that each match experiences the ATlm of the interval. The network also uses the minimum number of matches (S - 1). Such a network can be developed for any interval, providing each match within the interval (1) satisfies completely the enthalpy change of a strearh in the interval and (2) achieves the same ratio of CP values as exists between the composite curves (by stream splitting if necessary).

One particularly important property of the relationships for multipass exchangers is illustrated by the two streams shown in Fig. E.l. The problem overall is predicted to require 3.889 shells (4 shells in practice). If the problem is divided arbitrarily into two parts S and T as shown in Fig. El, then part S requires 2.899 and Part T requires 0.990, giving a total of precisely 3.889. It does not matter how many vertical sections the problem is divided into or how big the sections are, the same identical result is obtained, provided fractional (noninteger) numbers of shells are used. When the problem is divided into four arbitrary parts A, B, C, and D (Fig. E.l), adding up the individual shell requirements gives precisely 3.889 again. 

To establish the shells target, the composite curves are first divided into vertical enthalpy intervals as done for the area target algorithm. It was shown in App. B that it is always possible to design a network for an enthalpy interval with (5, -1) matches, with each match having the same temperature profile as the enthalpy interval. 

Figure E.l. If the real (noninteger) number of shells is calculated, the heat exchange profiles Ccm be divided in any way and the sum is always the same, (From Ahmad, Linnhoff, and Smith, Trans. ASME, J. Heat Transfer, 110 304, 1988 reproduced by permission of the American Society of Mechanical Engineers.)...

The large number of matches assumed in Eq. (E.2) is not a complication in establishing the target. This is so because the additive property shows that the total fractional number of shells is independent of how many vertical sections are used to divide a given heat exchange profile. 

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