Increments, compositing

If the dependence on temperature as well as on composition is known for a solution, enthalpies and entropies of adsorption may be calculated from the appropriate thermodynamic relationships [82]. Neam and Spaull [147] have, for example, calculated the enthalpies of surface adsorption for a series of straight-chain alcohols. They find an increment in enthalpy of about 1.96 kJ/mol per CH2 group. 

Equation (A2.1.23) can be mtegrated by the following trick One keeps T, p, and all the chemical potentials p. constant and increases the number of moles n. of each species by an amount n. d where d is the same fractional increment for each. Obviously one is increasing the size of the system by a factor (1 + dQ, increasing all the extensive properties U, S, V, nl) by this factor and leaving the relative compositions (as measured by the mole fractions) and all other intensive properties unchanged. Therefore, d.S =. S d, V=V d, dn. = n. d, etc, and... 

The initial sample is called the primary, or gross sample and may be a single increment drawn from the target population, or a composite of several increments. In many cases the gross sample cannot be analyzed without further treatment. Processing the gross sample may be used to reduce the sample s particle size, to transfer the sample into a more readily analyzable form, or to improve its homogeneity. 

Samples from municipal wastewater treatment plants and samples of industrial discharges often are collected as 24-h composites. Samples are obtained using an automatic sampler that periodically removes individual grab samples. The volume of each sample increment and the frequency of sampling may be constant or may vary in response to changes in flow rate. 

Plot a family of curves, each of different n, with composition as the y axis and O2 absorbed as the x axis. Evaluate by Eq. (5.30) for n = 1, 2, 3, and 4 and 0.1 < p < 0.9 in increments of 0.1. Plot these results on y axis) on a separate graph drawn to the same scale as the experimental results. Compare your calculated curves with the experimental curves with respect to each of the following points (1) coordinates used, (2) general shape of curves, and (3) labeling of curves. 

The foam stabiUty of hand-dishwashing compositions can also be measured more directly and more quantitatively using mechanical means to whip up a foam and adding increments of food soil to a predeterrnined no-foam end point (112). 

Composite Samples Obtained by Multiple Sample Extractions Material flow streams are sampled in practice by combining extractions taken at successive time intei vals into a composite sample. Multiple increment collection to obtain representative composite sampfes for specified bulk-material flows is performed according to a... 

The calculations are made as follows. The exchanger is divided into small increments to allow numerical integrations. A tube wall temperature is first calculated and then QAV. The gas temperature and composition from an increment can then be calculated. If the gas composition is above saturation for the temperature, any excess condensation can occur as a fog. This allows the degree of fogging tendency to be quantified. Whenever possible, experimental data should be used to determine the ratio of heat transfer to m.ass transfer coefficients. This can be done with a simple wet and dry bulb temperature measurement using the components involved. 

When all of the ISS spectra are plotted in a three-dimensional manner, such as the z- plot shown in Figure 3, the changes in surface composition with depth are much more obvious. In this figure, each spectrum represents the composition at a different cross section of the total depth sputtered, hence the spectra are plotted at different depths. Note that the spectra are not recorded at identical incremental depths. 

Assume xi values of bottoms compositions of light key for approximate equal increments from final bottoms to initial feed charge. Calculate L/V values corresponding to the assmned xi values by inserting the various xi values in the Fenske equation for minimum reflux ratio of l-(d). The xi values replace the x b of this relation as the various assumptions are calculated. The actual (L/D) are calculated as in l-(d) keeping the minimmn number of trays constant. Complete the table values. 

The existence of an azeotropic composition has some practical significance. By conducting a polymerization with the monomer feed ratio equal to the azeotropic composition, a high conversion batch copolymer can be prepared that has no compositional heterogeneity caused by drift in copolymer composition with conversion. Thus, the complex incremental addition protocols that arc otherwise required to achieve this end, are unnecessary. Composition equations and conditions for azeotropic compositions in ternary and quaternary eopolymerizations have also been defined.211,21... 

Flynn and Dickens [142] have translated the relaxation methods of fluid kinetics into terms applicable to solid phase thermogravimetry. The rate-determining variables such as temperature, pressure, gas flow rate, gas composition, radiant energy, electrical and magnetic fields are incremented in discrete steps or oscillated between extreme values and the effect on reaction rate determined. 

One reactant is charged to the reactor in small increments to control the composition distribution of the product. Vinyl copolymerizations discussed in Chapter 13 are typical examples. Incremental addition may also be used to control the reaction exotherm. 

The solution of Equations (5.23) or (5.24) is more straightforward when temperature and the component concentrations can be used directly as the dependent variables rather than enthalpy and the component fluxes. In any case, however, the initial values, Ti , Pi , Ui , bj ,... must be known at z = 0. Reaction rates and physical properties can then be calculated at = 0 so that the right-hand side of Equations (5.23) or (5.24) can be evaluated. This gives AT, and thus T z + Az), directly in the case of Equation (5.24) and imphcitly via the enthalpy in the case of Equation (5.23). The component equations are evaluated similarly to give a(z + Az), b(z + Az),... either directly or via the concentration fluxes as described in Section 3.1. The pressure equation is evaluated to give P(z + Az). The various auxiliary equations are used as necessary to determine quantities such as u and Ac at the new axial location. Thus, T,a,b,. .. and other necessary variables are determined at the next axial position along the tubular reactor. The axial position variable z can then be incremented and the entire procedure repeated to give temperatures and compositions at yet the next point. Thus, we march down the tube. 

The compositions of the monomer feed and of the polymer formed may be expressed as mole fractions instead of the mole ratios used above. To this end we let Fi represent the fraction of monomer Mi in the increment of copolymer formed at a given stage in the polymerization. Then... 

The composition of the increment of polymer formed at a monomer composition specified by /i(= 1 —/2) is readily calculated from Eq. (8) if the monomer reactivity ratios ri and V2 are known. Again it is apparent that the mole fraction Fi in general will not equal /i hence both /i and Fi will change as the polymerization progresses. The polymer obtained over a finite range of conversion will consist of the summation of increments of polymer differing progressively in their mole fractions F. ... 

Curves calculated from Eq. (8) for various values of the parameters ri and are shown in Figs. 24 and 25. The ordinate (Ei) represents the composition of the increment of copolymer formed from the monomer mixture having the composition (/i) given along the abscissa axis. Fig. 24 treats the case in which the two radicals display the same preference for one of the monomers over the other. That is... 

Fig. 24.—Incremental polymer composition (mole fraction Fi) plotted against the monomer composition (mole fraction/i) for ideal copolymerizations (ri — X/r F). Values of r are indicated.

Fig. 25.—Incremental polymer compositions Fi as functions of the monomer composition fi for the values of the reactivity ratios indicated ri/r ). The broken straight line represents Fi—fi (i.e., ri=r2 = l).

The successive fractions may be obtained by lowering the temperature in suitable increments rather than by varying the solvent composition isothermally. If a poor solvent is found from which the polymer precipitates in a convenient temperature range, it may be used alone without addition of another component. 

Mole fractions of monomer units 1 and 2 in the increment of polymer formed from monomer mixture having the composition/i (Chap. V). 

Multiple developments can be classifled into nniform mnltiple developments (UMD, also called nnidimensional multiple developments) for methods in which the migration distance and mobile phase composition are constant and incremental multiple development (IMD) with mostly ascending migration distances. However,... 

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