Minimum number

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

To understand the minimum number of matches or units in a heat exchanger network, some basic results oigraph theory can be used. A graph is any collection of points in which some pairs of points are... 

In general, the final network design should be achieved in the minimum number of units to keep down the capital cost (although this is not the only consideration to keep down the capital cost). To minimize the number of imits in Eq. (7.1), L should be zero and C should be a maximum. Assuming L to be zero in the final design is a reasonable assumption. However, what should be assumed about C Consider the network in Fig. 7.16, which has two components. For there to be two components, the heat duties for streams A and B must exactly balance the duties for streams E and F. Also, the heat duties for streams C and D must exactly balance the duties for streams G and H. Such balemces are likely to be unusual and not easy to predict. The safest assumption for C thus appears to be that there will be one component only, i.e., C = 1. This leads to an important special case when the network has a single component and is loop-free. In this case, ... 

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). 

This is a useful result, since if the network is assumed to be loop-free and has a single component, the minimum number of units can be predicted simply by knowing the number of streams. If the problem does not have a pinch, then Eq. (7.2) predicts the minimum number of units. If the problem has a pinch, then Eq. (7.2) is applied on each side of the pinch separately ... 

Before suggesting an approach for predicting the minimum number of shells for an entire network, a more convenient method for determining the number of shells in a single unit must first be found. Adopting the design criterion given by Eq. (7.13) as the basis, then any need for trial and error can be eliminated, since an explicit... 

In practice, the integer number of shells is evaluated from Eq. (7.18) for each side of the pinch. This maintains consistency between achieving maximum energy recovery and the corresponding minimum number of units target Nu- ixs- In summary, the number of shells target can be calculated from the basic stream data and an assumed value of Xp (or equivalently,... 

Ahmad, S., and Smith, R., Targets and Design for Minimum Number of Shells in Heat Exchanger Networks, IChemE, ChERD, 67 481, 1989. 

Designing for a minimum number of shutdowns and product changeover s. 

Note one further point fi om Fig. 16.7. The number of units is 7 in total (including the heater and cooler). Referring back to Example 7.1, the target for the minimum number of units was calculated to be 7. It therefore appears that there was something in our procedure... 

It is in fact the tick-off heuristic that steered the design toward the minimum number of units. The target for the minimum number of units was given by Eq. (7.2) ... 

Example 16.1 The process stream data for a heat recovery network problem are given in Table 16.1. A problem table analysis on these data reveals that the minimum hot utility requirement for the process is 15 MW and the minimum cold utility requirement is 26 MW for a minimum allowable temperature diflFerence of 20°C. The analysis also reveals that the pinch is located at a temperature of 120°C for hot streams and 100°C for cold streams. Design a heat exchanger network for maximum energy recovery in the minimum number of units. 

The completed design is shown in Fig. 16.8c. The minimum number of units for this problem is given by... 

The design in Fig. 16.8 is seen to achieve the minimum number of units target. 

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. 

Wood, R. M., Wilcox, R. J., Euid Grossmann, I. E., A Note on the Minimum Number of Units for Heat Exchanger Network Synthesis, Chem. Eng. Commun., 39 371, 1985. 

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).

As each match is successively placed in the interval, the minimum number of matches can be achieved because there is one fewer stream to match and the CP ratio of the remaining streams (i.e., ratio of 2 CPh and 2 CPc of the remaining streams) in the interval still satifies the CP ratio between the composite curves. 

Expression for the Minimum Number of 1-2 Shell-and-Tube Heat Exchangers for a... 

Once specifications for the input stream and end product are known theprocess engineer must determine the minimum number of steps required to achieve the transformation. 

The results showed that some yards used ultrasonics to examine thousands of checkpoints for quality control purposes whilst others were content to examine the almost the minimum number that would be accepted. It is cleeu from the results that the minimum extent of NDE recommended by Classification Societies falls some way short of that applied for quality control purposes by some shipbuilders. 

To define the thennodynamic state of a system one must specify fhe values of a minimum number of variables, enough to reproduce the system with all its macroscopic properties. If special forces (surface effecls, external fields—electric, magnetic, gravitational, etc) are absent, or if the bulk properties are insensitive to these forces, e.g. the weak terrestrial magnetic field, it ordinarily suffices—for a one-component system—to specify fliree variables, e.g. fhe femperature T, the pressure p and the number of moles n, or an equivalent set. For example, if the volume of a surface layer is negligible in comparison with the total volume, surface effects usually contribute negligibly to bulk thennodynamic properties. 

C2.5.3.6 MINIMUM NUMBER OF RESIDUES FOR OBTAINING FOLDABLE PROTEIN STRUCTURES... 

Find the minimum number of rings to describe a ring system. 

The minimum number of cycles is given by the nullity or Frerejacque number ( ) according to Eq. (5). It is the difference between the number of nodes a = atoms) and the number of edges h = bonds). The value of 1 stands for the number of compounds considered (here, one compound). This minimum number corresponds to the munber of chords. These are defined as nodes that turn a cyclic graph or structure into an acyclic one. 

As mentioned above, HMO theory is not used much any more except to illustrate the principles involved in MO theory. However, a variation of HMO theory, extended Huckel theory (EHT), was introduced by Roald Hof nann in 1963 [10]. EHT is a one-electron theory just Hke HMO theory. It is, however, three-dimensional. The AOs used now correspond to a minimal basis set (the minimum number of AOs necessary to accommodate the electrons of the neutral atom and retain spherical symmetry) for the valence shell of the element. This means, for instance, for carbon a 2s-, and three 2p-orbitals (2p, 2p, 2p ). Because EHT deals with three-dimensional structures, we need better approximations for the Huckel matrix than... 

In order to consider the 3D structure but make the chirality code independent of a specific conformer, r- is taken as the sum of the bond lengths between atoms i and j on the path with a minimum number of bond counts. 

Each combination of four atoms (A, B. C. and D) is characterized by two parameters, e and e.. As for the CICC, is a parameter that depends on atomic properties and on distances, and is calculated by Eq. (27), with r, again being the sum of bond lengths between atoms on the path with the minimum number of bond counts. However c is now a geometric parameter (dependent on the conformation)... 

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