Depletion isothermal

Floatabilities of francolite and dolomite using oleate as collector are shown in Figs. 4.28 and 4.29 as a function of oleate concentration and pH. It can be seen that the floatabilities of dolomite and francolite are quite similar with a sudden increase at an oleate concentration of 1.6 X 10 mol/1. Selective flotation of francolite from dolomite might be expected only above pH 9 or below pH 5. However, flotation of a binary mixture of dolomite and francolite (50 50) with 1.7 x 10 mol/1 oleate at pH 5.2 and 9.0 yielded poor separation. The loss of selectivity during flotation may be attributed to the interactions among the dissolved mineral species from minerals and oleate as well as the formation of oleate precipitates (Amankonah and Somasundaran, 1985). Depletion isotherms for labeled oleate on francolite and dolomite are shown in Fig. 4.30. Both francolite and dolomite show a two-region (II and III) linear isotherms with a break at a residual concentration of... 

Fig. 4.30. Depletion isotherms of 14C labeled oleic acid on francolite and dolomite.

In the production of hydrocarbon reservoirs, the process of isothermal depletion is normally assumed, that is reducing the pressure of the system while maintaining a constant temperature. Hence, a more realistic movement on the pressure-temperature plot is from point A to A . 

Starting at condition A with the ethane in the liquid phase, and assuming isothermal depletion, then as the pressure is reduced so the specific volume increases as the molecules move further apart. The relationship between pressure and volume is governed by the compressibility of the liquid ethane. 

The four vertical lines on the diagram show the isothermal depletion loci for the main types of hydrocarbon gas (incorporating dry gas and wet gas), gas condensate, volatile oil and black oil. The starting point, or initial conditions of temperature and pressure, relative to the two-phase envelope are different for each fluid type. 

Pressure depletion in the reservoir can normally be assumed to be isothermal, such that the isothermal compressibility is defined as the fractional change in volume per unit change in pressure, or... 

Figure 6 shows the temperature proflle that should be used with the initiator monomer system described in the caption to reduce the monomer concentration from 0.47 mol/L to 0.047 mol/L. The optimal nonisothermal policy consists of decreasing temperature from a temperature above the optimal isothermal temperature to one below it. The rate of polymerization could be increased, as expected, by an initially higher temperature, but the temperature must be decreased to avoid depletion of initiator and depolymerization. However, the amount of time saved by this policy does not seem to be significant in comparison to the isothermal policy for this case. 

A closer look at the nonisothermal and isothermal policy results reveals some additional interesting features with regard to optimization. As mentioned earlier, isothermal policies were determined by two factors. One was the M, value and the other was the dead end polymerization caused by depletion of initiator. It was also observed that the minimum time from a nonisothermal policy was considerably less than the minimum time due to the isothermal policy whenever H>, was the controlling factor in the isothermal policy when the isothermal policy was controlled by initiator depletion, a nonisothermal policy did not show significant improvement in minimum time relative to the isothermal one. 

In this paper we formulated and solved the time optimal problem for a batch reactor in its final stage for isothermal and nonisothermal policies. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and optimum time was studied. It was shown that the optimum isothermal policy was influenced by two factors the equilibrium monomer concentration, and the dead end polymerization caused by the depletion of the initiator. When values determine optimum temperature, a faster initiator or higher initiator concentration should be used to reduce reaction time. 

A full development of the rate law for the bimolecular reaction of MDI to yield carbodiimide and CO indicates that the reaction should truly be 2nd-order in MDI. This would be observed experimentally under conditions in which MDI is at limiting concentrations. This is not the case for these experimements MDI is present in considerable excess (usually 5.5-6 g of MDI (4.7-5.1 ml) are used in an 8.8 ml vessel). So at least at the early stages of reaction, the carbon dioxide evolution would be expected to display pseudo-zero order kinetics. As the amount of MDI is depleted, then 2nd-order kinetics should be observed. In fact, the asymptotic portion of the 225 C Isotherm can be fitted to a 2nd-order rate law. This kinetic analysis is consistent with a more detailed mechanism for the decomposition, in which 2 molecules of MDI form a cyclic intermediate through a thermally allowed [2+2] cycloaddition, which is formed at steady state concentrations and may then decompose to carbodiimide and carbon dioxide. Isocyanates and other related compounds have been reported to participate in [2 + 2] and [4 + 2] cycloaddition reactions (8.91. 

Adsorption is determined by the depletion method using a Dohrmann DC 80 carbon analyzer. The mineral is contacted with the polymer solution and agitated with a mechanical tumbler for 24 hours, a time which has been verified to be sufficient for adsorption to be complete (9). A more detailed description of experimental procedures is given elsewhere (10). All the data reported in this study are taken in the plateau region of the adsorption isotherm. 

Fig. 6. Plateau-values, I"P1 /mg m 2, of adsorption isotherms of lysozyme (LSZ), ribonuclease (RNase), a -lactalbumin (aLA), calcium-depleted (X -lactalbumin (aLA(-Ca )) and bovine serum albumin (BSA) on hydrophobic polystyrene (PS) and hydrophilic hematite (a — Fe203) and silica (Si02) surfaces. An indication of the charge density of the surface is given by the zeta-potential, C, and of the proteins by + and signs. Ionic strength 0.05 M T = 25°C. (Derived from Currie et al. 2003).

FIGURE 2.7. Depletion of Inhibitor Stability DSC Curve (A) and Isothermal Curves (B) for an Inhibited Material. 

DSC can be used effectively in the isothermal mode as well. In this case, the container with the sample is inserted into the DSC preheated to the desired test temperature. This type of experiment should be performed to examine systems for induction periods that occur with autocatalytic reactions and with inhibitor depletion reactions. (Reactions with induction periods can give misleading results in the DSC operated with increasing temperature scans.) Autocatalytic reactions are those whose rates are proportional to the concentration of one or more of the reaction products. Some hydroperoxides and peroxy esters exhibit autocatalytic decomposition. Inhibitor depletion can be a serious problem with certain vinyl monomers, such as styrene and acrylic acid, that can initiate polymerization at ambient temperatures and then selfheat into runaways. Isothermal DSC tests can be used to determine a time to runaway that is related to the inhibitor concentration. 

This simplification is not possible for some CVD systems in which large density changes are associated with the deposition process. The growth of CdHgTe is a typical example that shows how the depletion of Hg next to the substrate creates an unstable density gradient that drives recirculations (205), as discussed earlier and illustrated in Figure 14. LPCVD processes use little or no diluent and often involve several species, and multicomponent diffusion may be an important factor (21). Fortunately, these reactors are isothermal, and the relative insensitivity of reactor performance to details of the fluid flow greatly simplifies the analysis. 

Encapsulation of the gas decreases the pressure to the three-phase (Lw-H-V) condition. The system pressure may be controlled by an external reservoir for addition or withdrawal of gas, aqueous liquid, or some other fluid such as mercury. After hydrate formation, the pressure is reduced gradually, the equilibrium pressure is observed by the visual observation of hydrate crystal disappearance. Upon isothermal dissociation, the pressure will remain constant for a simple hydrate former until the hydrate phase is depleted. 

Adsorption isotherms were measured by the solution depletion technique. Enzyme solutions were equilibrated with powdered quartz for varying lengths of time. The solid was removed by centrifugation and the supernatants were analyzed. Total protein content of the solutions, before and after equilibration with the quartz, was measured via absorbance at 280 nm and the difference calculated as adsorbed protein. All adsorption isotherms were measured at 25.0°C in Tris buffer (0.1M Tris[hydroxymethyl]aminomethane and 10 mM CaCl2 adjusted to pH 8.6 with HCl). 

After fuel cell isothermal chemical oxidation (Figures 5.3 and 5.4), the depleted fuel, steam and CO2 exhaust is partially recirculated to... 

Heated fuel and water are reformed and isothermally oxidised at the MCFC anode. The unused fuel and depleted air from the anode are burnt with added air in a catalytic oxidiser, the output of which heats the cathode and supplies it with oxygen. The cathode exhaust heats the incoming fuel and water in a heat exchanger. The latter exhausts to desired users, for example steam generation or thermal process. 

Again, the degree of deactivation can be expected to depend on the temperature and conversion level, as expressed by Equations 4 and 5. Indeed, coke profiles over isothermal laboratory reactors (72) show such differences, primarily due to a reduction in hydrogen partial pressure. Metals deposition over residue catalysts beds show a decrease with conversion simply because of depletion of the reactant 2,13,14). 

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