Total hold

Figure 9-42. Total hold-up for ceramic tower packings air-water. Used by permission of Shulman, H. L, Ullrich, C. F., and Weils, H. ACh.E. Jour., V. 1, No. 2 (1955) p. 247 all rights reserved.

Total hold-up, ht, of water is represented for Raschig rings and Berl saddles [66]. 

Figures 9-42 A, B, C, D, E, F present the graphical interpretation of the total hold-up equation. These are more...

Ref. 64 and 66, all rights reserved. Table 9-35 Total Hold-up Constants ... 

These data are t aluable for determining the total weight of liquid held in the packing, and also the void fraction, in an operating column, e is the void fraction of the dry packing minus the total hold-up, hj. 

Total hold-up, ft3 liquid/ft packing volume Enthalpy of air at any temperature higher than inlet, Btu/lb dry air note hj = exit air Enthalpy of inlet air to tower, equivalent to enthalpy of saturated air at wet bulb temperature, Btu/lb dry air from Moist Air Tables, ASHVTE Guide... 

Void fraction of packing under operating conditions = void fraction of dry packing minus the total hold-up (not the free volume of dry-packing)... 

Fig. 123 shows the typical construction of a mixer-settler extractor. The main parameters usually required for the design of an extractor are maximum output, total holding capacity, organic reagent capacity, mixer capacity, phase contact time, settler surface area and specific settler output. 

Finding the optimal design according to the above definition amounts to finding the minimum of the total holding time, which is a function of all dr 3 s. Mathematically formulated, this involves finding the intermediate dr j-values subject to the following equation ... 

Table 11.2 Dimensionless total holding times for optimal and equal-sized mixed reactors, for a =0.01 and two values of k.

Table 11.2 gives the total holding times for two values of K, both for a series of CSTRs with minimal total volume and for a series of equal-sized mixed reactors. Total holding times for equal-sized mixed reactors have been calculated using a zero finding routine. The last value in Table 11.2 is the dimensionless holding time for a PFR reactor with Michaelis-Menten kinetics, calculated by means of the following equation ... 

An important observation from Table 11.2 is the considerable difference going from one to two or more CSTRs. For the conditions studied, there is only a minor difference (less than 10%) between the total holding time for optimal and equal-sized mixed reactors. Even in extreme cases, i.e., for very low values of k and this difference remains relatively small (37% for n=3, k=10 and df=10 ). 

Furthermore, it can be shown that, in the limiting cases of first-order kinetics [Equation (11.35) also holds for this case] and zero-order kinetics, the equal and optimal sizes are exactly the same. As shown, the optimal holding times can be calculated very simply by means of Equation (11.40) and the sum of these can thus be used as a good approximation for the total holding time of equal-sized CSTRs. This makes Equation (11.31) an even more valuable tool for design equations. The restrictions are imposed by the assumption that the biocatalytic activity is constant in the reactors. Especially in the case of soluble enzymes, for which ordinary Michaelis-Menten kinetics in particular apply, special measures have to be taken. Continuous supply of relatively stable enzyme to the first tank in the series is a possibility, though in general expensive. A more attractive alternative is the application of a series of membrane reactors. 

In the absence of liquid flow the gas hold-up is 0.58. This value is the porosity of the packing determined with a 10% error by a volumetric method. For constant gas velocity at 0.037 m/s measurement of the total hold-up in the gas phase have shown that the pressure had no effect on this hydrodynamic parameter in the range 0.1 to 1.3 MPa. This result can be explained by the fact that the hold-up only depends on the residence time, which is constant for a given gas velocity in the pressure range studied, as shown in table 1. 

Suppose that the length L of the tubular reactor is equivalent to R small stirred tanks for the total holding times to be the same, L = Rdv. If the exit concentrations are to be the same, then, by (4) and (8),... 

The objective function to be considered is of one of the simpler types, either stoichiometric or material. In the first case it is required to maximize the final extent of reaction, Ci in the second case this will also be required if the objective function is monotonically increasing with Cl. Here the interesting problem is to find the optimal policy subject to some restriction on the total holding time, say... 

Let us consider first the choice of temperature and holding time such that a given conversion, Ci, is achieved from a given feed state, Cr+, in the least total holding time. This total holding time will be... 

As before, let Jr cr+i) denote the minimum total holding time... 

The difference between a disjoint policy and one that is not is clearly seen by comparing Figs. 7.4 and 7.10. With a disjoint policy a given feed condition requires the same inlet temperature whatever the total holding time of the process. Variation of holding time merely takes one farther along the profile. With a policy which is not disjoint, however, the same feed condition will demand different... 

After all, we have formulated mathematically the problem as a mixed-integer programming problem stated below. The objective function is composed of the total transportation cost between every facility, the total production cost at plant, and the total operational cost at each facility, the total holding cost at DC over the planning horizon, and the total fixed-charge for the open DCs. 

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