# Sets

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i [c.25]

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. [c.67]

The method described here is based on the high degree of correlation of model parameters, in this case, UNIQUAC parameters. Thus, although a certain set of binary parameters may be best for VLE data, we are able to find other sets of binary parameters for the miscible binaries which significantly improve ternary LLE prediction while only slightly decreasing accuracy of representation of the binary VLE. Fitting ternary LLE data only, may yield unrealistic parameters that predict grossly erroneous results when used in regions not identical to those employed in data reduction. By contrast, fitting ternary LLE data simultaneously with binary VLE data, effectively provides constraints on the binary parameters, preventing them from attaining arbitrary values of little physical significance. Determination of a single set of parameters which can adequately represent both VLE and LLE is particularly important in three-phase distillation. [c.69]

Using the ternary tie-line data and the binary VLE data for the miscible binary pairs, the optimum binary parameters are obtained for each ternary of the type 1-2-i for i = 3. .. m. This results in multiple sets of the parameters for the 1-2 binary, since this binary occurs in each of the ternaries containing two liquid phases. To determine a single set of parameters to represent the 1-2 binary system, the values obtained from initial data reduction of each of the ternary systems are plotted with their approximate confidence ellipses. We choose a single optimum set from the intersection of the confidence ellipses. Finally, with the parameters for the 1-2 binary set at their optimum value, the parameters are adjusted for the remaining miscible binary in each ternary, i.e. the parameters for the 2-i binary system in each ternary of the type 1-2-i for i = 3. .. m. This adjustment is made, again, using the ternary tie-line data and binary VLE data. [c.74]

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). [c.108]

The correlations were generated by first choosing from the literature the best sets of vapor-pressure data for each fluid. [c.138]

A. The first four data cards contain control parameters which are read only once for a series of binary VLE data sets. [c.220]

Multiple sets of binary VLE data may be correlated by continuing with another set of cards starting at part B. The last set of cards must be followed with a blank card to end the program. [c.227]

If selectivity increases as conversion increases, the initial setting for reactor conversion should he on the order of 95 percent, and that for reversible reactions should be on the order of 95 percent of the equilibrium conversion. If selectivity decreases with increasing conversion, then it is much more difficult to give guidance. An initial setting of 50 percent for the conversion for irreversible reactions or 50 percent of the equilibrium conversion for reversible reactions is as reasonable as can be guessed at this stage. However, these are only initial guesses and will almost certainly be changed later. [c.27]

Again, it is difficult to select the initial setting of the reactor conversion with systems of reactions in series. A conversion of 50 percent for irreversible reactions or 50 percent of the equilibrium conversion for reversible reactions is as reasonable as can be guessed at this stage. [c.27]

The next requirement is to achieve the initial setting for the [c.27]

Single reactions. For single reactions, a good initial setting is 95 percent conversion for irreversible reactions and 95 percent of the equilibrium conversion for reversible reactions. Figure 2.9 summarizes the influence of feed mole ratio, inert concentration, temperature, and pressure on equilibrium conversion. [c.63]

Multiple reactions. For multiple reactions in which the byproduct is formed in parallel, the selectivity may increase or decrease as conversion increases. If the byproduct reaction is a higher order than the primary reaction, selectivity increases for increasing reactor conversion. In this case, the same initial setting as single reactions should be used. If the byproduct reaction of the parallel system is a [c.63]

It should be emphasized that these recommendations for the initial settings of the reactor conversion will almost certainly change at a later stage, since reactor conversion is an extremely important optimization variable. When dealing with multiple reactions, selectivity is maximized for the chosen conversion. Thus a reactor type, temperature, pressure, and catalyst are chosen to this end. Figure 2.10 summarizes the basic decisions which must be made to maximize selectivity. [c.64]

Partially vaporized feed reverses these effects. For a given separation, the feed conditions can be optimized. No attempt should be made to do this at this stage in the design, since heat integration is likely to change the optimal setting later in the design. It is usually adequate to set the feed to saturated liquid conditions. This tends to equalize the vapor rate below and above the feed. [c.78]

Reactor conversion. In Chap. 2 an initial choice was made of reactor type, operating conditions, and conversion. Only in extreme cases would the reactor be operated close to complete conversion. The initial setting for the conversion varies according to whether there are single reactions or multiple reactions producing byproducts and whether reactions are reversible. [c.95]

Given the possibilities for changing the sequence of simple columns with the introduction of prefractionators, side-strippers, side-rectifiers, and fully thermally coupled arrangements, it is apparent that the problem is extremely complex with many structural alternatives. The problem can be tackled using the approach based on optimization of a reducible structure. As discussed in Chap. 1, this approach starts by setting up a grand flowsheet in which all candidates for an optimal solution are embedded. A reducible structure for a four-component mixture is shown in Fig. 5.19a. This structure is then subjected to optimization, and during the optimization, some features of the design are discarded, as shown in Fig. 5.19b. Methods also have been developed which consider sequencing, thermal coupling, and heat integration. [c.155]

Figure 6.6 The correct setting for is fixed by econo- |

As discussed earlier, the correct setting for the composite curves is determined by an economic tradeoff between energy and capital corresponding to an economic minimum temperature difference AT in between the curves. Accepting for the moment that the correct economic ATn,i is known, this fixes the relative position of the composite curves and hence the energy target. The AT in for the composite curves and its location have important implications for design if the energy target is to be achieved in the design of a heat exchanger network. The A7 nin is normally observed at only one point between the hot and the cold composite curves, called the heat recovery pinch. The pinch point has a special significance. [c.166]

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, [c.169]

Figure 6.12 The optimal setting of the capital/energy tradeoff for threshold problems. |

The total annualized cost at the optimal setting of the capital/energy tradeoff is [c.236]

BYPRODUCT is series in nature, which leads to the selectivity becoming very poor at high conversions. We recall that in Chap. 2 the initial setting for reactor conversion was to be 0.5 for such reaction systems. Figure 8.4 shows why a high setting for reactor conversion would be inappropriate. The byproduct formation cost forces the optimum to lower values of conversion. Again, if the primary reaction had been reversible, then a similar picture would have been obtained. However, instead of being limited by a reactor conversion of 1, the curves would have been limited by the equilibrium conversion. [c.245]

Now there are two global variables in the optimization. These are reactor conversion (as before) but now also the concentration of IMPURITY in the recycle. For each setting of the IMPURITY concentration in the recycle, a set of tradeoffs can be produced analogous to those shown in Figs. 8.3 and 8.4. [c.246]

Figure 8.6 shows the component costs combined to give a total cost which varies with both reactor conversion and recycle inert concentration. Each setting of the recycle inert concentration shows a cost profile with an optimal reactor conversion. As the recycle inert concentration is increased, the total cost initially decreases but then [c.247]

In carrying out these optimizations, targets should be used for the energy and capital cost of the heat exchanger network. This is the only practical way to carry out these optimizations, since changes in reactor conversion and recycle inert concentration change the material and energy balance of the process, which, in turn, change the heat recovery problem. Each change in the material and energy balance in principle calls for a different heat exchanger network design. Furnishing a new heat exchanger network design for each setting of reactor conversion and recycle inert concentration is just not practical. On the other hand, targets for energy and capital cost of the heat exchanger network are, by comparison, easily generated. [c.252]

One further important point needs to be made regarding stream splitting. In Fig. 16.13 the hot stream is split into two branches with CP values of 3 and 2 to satisfy the CP inequality criteria. However, a different split could have been chosen. For example, the split could have been into branch CP values of 4 and 1, or 2.5 and 2.5, or 2 and 3 (or any setting between 4 and 1 and 2 and 3). Each of these also would have satisfied the CP inequalities. Thus there is a degree of freedom in the design to choose the branch flow rates. By fixing the heat duties on the two units in Fig. 6.136 and changing the branch flow rates, the temperature differences across each unit are changed. The best choice can only be made by sizing and costing the various [c.378]

Figure 16.246 shows the network with another loop marked. Figure 16.246 shows the effect of shifting heat duty V 2U ound the loop. Again, the heat balance is maintained, but the temperatures as well as the duties around the loop change. As before, the value of V can be optimized by costing the network at different settings of V. If V is optimized to 7.0 MW (the original duty on unit A), then the duty on unit A becomes zero, and this unit is removed from the design. [c.392]

Figure 16.25a shows the network with a utility path highlighted. Heat duty can be shifted along utility paths in a similar way to that for loops. Figure 16.25a shows the effect of shifting heat duty W along the path. This time the heat balance changes because the loads imported from the hot utility and exported to the cold utility both change by W. The supply and target temperatures are maintained. If W is optimized to 7.0 MW, this will result in unit A being removed from the design. Different values of W can be taken and the network sized and costed at each value to find the optimal setting for W. Figure 16.256 through d shows other utility paths which can be exploited for optimization. [c.392]

In carrying out these optimizations, targets should be used for the energy and capital cost of the heat exchanger network. This is the only practical way to carry out these optimizations, since changes in reactor conversion and recycle inert concentration change the material and energy balance of the process, which in turn, changes the heat recovery problem. Each change in the material and energy balance, in principle, calls for a different heat exchanger network design. Furnishing a new heat exchanger network design for each setting of reactor conversion and recycle inert concentration is just not practical. On the other hand, targets for energy and capital cost of the heat exchanger network are by comparison easily generated. [c.402]

The preceding definitions of economic potential and total annual cost can be simplified if it is accepted that they will be used to compare the relative merits of difierent structural options in the flowsheet and difierent settings of the operating parameters. Thus items which will be common to the options being compared can be neglected. [c.407]

This additive property takes on fundamental practical significance when the problem of setting a target for the number of shells in a network from the composite curves is considered. [c.437]

When a condensable solute is present, the activity coefficient of a solvent is given by Equation (15) provided that all composition variables (x, 9, and ) are taicen on an (all) solute-free basis. Composition variables 9 and 4 are automatically on a solute-free basis by setting q = q = r = 0 for every solute. [c.57]

Example 7.5 For the process in Fig. 6.2, determine the value of and the total cost of the heat exchsinger network at the optimal setting of the [c.233]

The change in heat duties around the loop maintains the network heat balance and the supply Euid target temperatures of the streams. However, the temperatures around the loop change, and hence the temperature differences of the exchangers in the loop change in addition to their duties. The magnitude of U could be changed to different values and the network sized and costed at each value to find the optimal setting for U. If the optimal setting for U turns out to be 6.5 MW (the original duty on unit E), then the duty on unit E becomes zero, and this unit should be removed from the design. [c.392]

Targets also can be set for total heat exchange area, number of units, and number of shells for 1-2 shell-and-tube heat exchangers. These can be combined to establish a targej for capital costs, taking into account mixed materials of construction, pressure rating, and equipment type. Furthermore, the targets for energy and capital cost can be optimized to produce an optimal setting for the capital/energy tradeoff" before any network design is carried out. [c.401]

See pages that mention the term

**Sets**:

**[c.56] [c.69] [c.75] [c.83] [c.217] [c.224] [c.226] [c.26] [c.170] [c.192] [c.240] [c.390] [c.409]**

Modelling molecular structures (2000) -- [ c.175 , c.201 ]