Satisfiability

For vapor-liquid equilibria, the equations of equilibrium which must be satisfied are of the form... 

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

King, 1971 Naphtali and Sandholm, 1971 Newman, 1963 and Tomich, 1970). Moreover the choice of appropriate computation procedures for distillation, absorption, and extraction is highly dependent on the system being separated, the conditions of separation, and the specifications to be satisfied (Friday and Smith, 1964 Seppala and Luus, 1972). The thermodynamic methods presented in Chapters 3, 4, and 5, particularly when combined to... 

The equilibrium ratios are not fixed in a separation calculation and, even for an isothermal system, they are functions of the phase compositions. Further, the enthalpy balance. Equation (7-3), must be simultaneously satisfied and, unless specified, the flash temperature simultaneously determined. 

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. 

To calculate the vapor load for a single column of a sequence, start by assuming a feed condition such that q can be fixed. Initially assume saturated liquid feed (i.e., q = 1). Equation (5.1) can be written for all NC components of the feed and solved for the necessary values of 0. There are (JVC - 1) real positive values of 0 which satisfy Eq. (5.1), and each lies between the a values of the... 

Analogous effects are caused by the inappropriate use of utilities. Utilities are appropriate if they are necessary to satisfy the enthalpy imbalance in that part of the process. Above the pinch in Fig. 6.7a, steam is needed to satisfy the enthalpy imbalance. Figure 6.86 illustrates what happens if inappropriate use of utilities is made and some cooling water is used to cool hot streams above the pinch, say, XP. To satisfy the enthalpy imbalance above the pinch, an import of (Q mjj,+XP) is needed from steam. Overall, (Qcmin+AP) of cooling water is used. ... 

An alternative inappropriate use of utilities involves heating of some of the cold streams below the pinch by steam. Below the pinch, cooling water is needed to satisfy the enthalpy imbalance. Figure... 

The shaded areas in Fig. 6.24, known as pockets, represent areas of additional process-to-process heat transfer. Remember that the profile of the grand composite curve represents residual heating and cooling demands after recovering heat within the shifted temperature intervals in the problem table algorithm. In these pockets in Fig. 6.24, a local surplus of heat in the process is used at temperature differences in excess of AT ,in to satisfy a local deficit. ... 

The process requires (Qup + Qlp) to satisfy its enthalpy imbalance above the pinch. If there were no losses from the boiler, then fuel W would be converted to shaftwork W at 100 percent efficiency. However, the boiler losses Qloss reduce this to below 100 percent conversion. In practice, in addition to the boiler losses, there also can be significant losses from the steam distribution system. Figure 6.336 shows how the grand composite curve can be used to size steam turbine cycles. ... 

Xp is chosen to satisfy the minimum allowable Ft (e.g., for Ft > 0.75, Xp = 0.9 is used). Once the real (noninteger) number of shells is calculated from Eq. (7.14), this is rounded up to the next largest number to obtain the number of shells. 

By comparison, Fig. 13.36 shows an exothermic reactor integrated below the pinch. Although heat is being recovered, it is being recovered into part of the process which is a heat source. The hot utility requirement cannot be reduced because the process above the pinch needs at least Q//m-,n to satisfy its enthalpy imbalance. 

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...

The tick-off heuristic. Once the matches around the pinch have been chosen to satisfy the criteria for minimum energy, the design should be continued in such a manner as to keep capital costs to a minimum. One important criterion in the capital cost is the number of units (there are others, of course, which shall be addressed later). Keeping the number of units to a minimum can be achieved using the tick-off heuristic. To tick off a stream, individual units are made as... 

Turning now to the cold-end design, Fig. 16.6a shows the pinch design with the streams ticked off. If there are any cold streams below the pinch for which the duties eu e not satisfied by the pinch matches, additional process-to-process heat recovery must be used, since hot utility must not be used. Figure 16.66 shows an additional match to satisfy the residual heating of the cold streams below the pinch. Again, the duty on the unit is maximized. Finally, below the pinch the residual cooling duty on the hot streams must be satisfied. Since there are no cold streams left below the pinch, cold utility must be used (Fig. 16.6c). 

Before any matches are placed, the target indicates that the number of units needed is equal to the number of streams (including utility streams) minus one. The tick-off heuristic satisfied the heat duty on one stream every time one of the units was used. The stream that has been ticked off is no longer part of the remaining design problem. The tick-off heuristic ensures that having placed a unit (and used up one of our available units), a stream is removed from the problem. Thus Eq. (7.2) is satisfied if eveiy match satisfies the heat duty on a stream or a utility. 

Temperature feasibility requires constraints on the CP values to be satisfied for matches between streams at the pinch. 

The pinch design method developed earlier followed several rules and guidelines to allow design for minimum utility (or maximum energy recovery) in the minimum number of units. Occasionally, it appears not to be possible to create the appropriate matches because one or other of the design criteria cannot be satisfied. 

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). 

Clearly, in designs different from those in Figs. 16.13 and 16.14 when streams are split to satisfy the CP inequality, this might create a problem with the number of streams at the pinch such that Eqs. (16.3) and (16.4) are no longer satisfied. This would then require further stream splits to satisfy the stream number criterion. Figure 16.15 presents algorithms for the overall approach. ... 

The cold-utility target for the problem shown in Fig. 16.22 is 4 MW. If the design is started at the pinch with stream 3, then stream 3 must be split to satisfy the CP inequality (Fig. 16.22a). Matching one of the branches against stream 1 and ticking off stream 1 results in a duty of 8 MW. 

If the number of hot streams at the pinch above the pinch is greater than the number of cold streams, the cold streams must be split to satisfy the constraint. If the number of cold streams at... 

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).

The value of Pi 2 required in each 1-2 shell to satisfy a chosen value of Xp is defined by... 

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