Coefficient fractional

Although normally the coefficients in a balanced chemical equation are the smallest possible whole numbers, a chemical equation can be multiplied through by a factor and still be a valid equation. At times it is convenient to use fractional coefficients for example, we could write... 

Later, we show that it can be convenient to use fractional coefficients such as. When this is the case, the... 

Uj i i = the fraction of the total flow of component k entering unit i that leaves in the outlet stream connected to the unit j the split-fraction coefficient , gi 0 lc = any fresh feed of component k into unit i flow from outside the system (from unit 0). 

The flow of any component from unit i to unit j will equal the flow into unit i multiplied by the split-fraction coefficient. 

The value of the split-fraction coefficient will depend on the nature of the unit and the inlet stream composition. 

For practical processes most of the split-fraction coefficients are zero and the matrix is... 

In general, the equations will be non-linear, as the split-fractions coefficients (a s) will be functions of the inlet flows, as well as the unit function. However, many of the coefficients will be fixed by the process constraints, and the remainder can usually be taken as independent of the inlet flows (A s) as a first approximation. 

The fresh feeds will be known from the process specification so if the split-fraction coefficients can be estimated, the equations can be solved to determine the flows of each component to each unit. Where the split-fractions are strongly dependent on the inlet flows, the values can be adjusted and the calculation repeated until a satisfactory convergence between the estimated values and those required by the calculated inlet flows is reached. 

The procedure for setting up the equations and assigning suitable values to the split-fraction coefficients is best illustrated by considering a short problem the manufacture of acetone from isopropyl alcohol. 

Figure 4.12 is redrawn in Figure 4.13, showing the fresh feeds, split-fraction coefficients and component flows. Note that the fresh feed g2ok represents the acetone and hydrogen generated in the reactor. There are 5 units so there will be 5 simultaneous equations. The equations can be written out in matrix form (Figure 4.14) by inspection of Figure 4.13. The fresh feed vector contains three terms.

The values of the split-fraction coefficients will depend on the function of the processing unit and the constraints on the stream flow-rates and compositions. Listed below are suggested first trial values, and the basis for selecting the particular value for each component. 

Substitution of the values of the split-fraction coefficients for the other components will give the sets of equations for the component flows to each unit. The values of the split-fraction coefficients and fresh feeds are summarised in Table 4.2. 

Table 4.3 shows the feed of each component and the total flow to each unit. The composition of any other stream of interest can be calculated from these values and the split-fraction coefficients. The compositions and flows should be checked for compliance with the process constraints, the split-fraction values adjusted, and the calculation repeated, as necessary, until a satisfactory fit is obtained. Some of the constraints to check in this example are discussed below. 

This should approximate to the azeotropic composition (9 per cent alcohol, 91 per cent water). The flow of any component in this stream is given by multiplying the feed to the column (X5k) by the split-fraction coefficient for the recycle stream (a 15/.). The calculated flows for each component are shown in Table 4.4. 

These compositions should be checked against the vapour-liquid equilibrium data for acetone-water and the values of the split-fraction coefficients adjusted, as necessary. 

Guide rules for estimating split-fraction coefficients... 

The split-fraction coefficients can be estimated by considering the function of the process unit, and by making use of any constraints on the stream flows and compositions that arise from considerations of product quality, safety, phase equilibria, other thermodynamic relationships and general process and mechanical design considerations. The procedure is similar to the techniques used for the manual calculation of material balances discussed in Section 4.3. 

Suggested techniques for use in estimating the split-fraction coefficients for some of the more common unit operations are given below. 

For a unit that simply combines several inlet streams into one outlet stream, the split-fraction coefficients for each component will be equal to 1. = 1. 

If the unit simply divides the inlet stream into two or more outlet streams, each with the same composition as the inlet stream, then the split-fraction coefficient for each component will have the same value as the fractional division of the total stream. A purge stream is an example of this simple division of a process stream into two streams the main stream and the purge. For example, for a purge rate of 10 per cent the split-fraction coefficients for the purge stream would be 0.1. 

A distillation column divides the feed stream components between the top and bottom streams, and any side streams. The product compositions are often known they may be specified, or fixed by process constraints, such as product specifications, effluent limits or an azeotropic composition. For a particular stream, 5 , the split-fraction coefficient is given by ... 

The split-fraction coefficients are not very dependent on the feed composition, providing the reflux flow-rate is adjusted so that the ratio of reflux to feed flow is held constant Vela (1961), Hachmuth (1952). 

Then the split-fraction coefficients can be calculated from a material balance. 

Ctjik Split-fraction coefficient fraction of component k flowing from unit i to unit j —... 

True wt. fraction of carbamazepine Experimentally determined wt. fraction of carbamazepine, mean SD Percent of true wt. fraction Coefficient of variation, %... 

Use smallest whole number coefficients. However, it may be useful to temporarily use a fractional coefficient, then for the last step, multiply all the terms by a factor to change the fractions to whole numbers. 

When plotted in a In (Al203/Mg0) vs ln(FeO/MgO) (Figure 1.12), the clinopyroxene data define a unique trend which is reasonably linear, thereby supporting the hypothesis of fractional melting made by Johnson et al. (1990). In addition, the linear array supports a homogeneous source and rather constant 0MgoFeO = 0-3 and >MgoAl2° fractionation coefficients. 

Figure 9.16 Kinetic fractionation during crystal growth. Steady-state distribution of melt concentrations in the vicinity of a solid growing at the rate v for trace elements with different solid-liquid fractionation coefficients [equation (9.6.5), Tiller et al. (1953)]. The stippled area indicates the steady-state chemical boundary-layer with thickness <5 = <5>/v.

Let us consider the influence of a solid-liquid interface advancing at a constant velocity on the solid-liquid fractionation of an element i. In the case of unidirectional solidification, it is convenient to consider that liquid crosses the immobile interface with an absolute constant velocity v, while a solid-liquid fractionation coefficient K is applied to the fractionation of element i. Let us assume that the interface is at x=0, the medium being solid for x<0. Liquid fills the half-space 0[Pg.442]

Indirect evidence of isotopic fractionation among different complexes was obtained by Marechal et al. (1999) and Marechal and Albarede (2002) who observed different elution rates of Cu and Cu on anion-exchange columns (Fig. 11). These experiments were confirmed by Zhu et al. (2002) and Rouxel (2002) with similar results on fractionation coefficients. Figure 11 shows that, in HCl medium, the heavier isotope 65 is less well retained on the column than the lighter isotope 63. Marechal and Albarede (2002) used an error function approximation to the elution curve to derive the ratio of fractionation coefficients for the 63 and 65 isotopes between the resin and the eluent. From the relationship between the elution volume (position... 

The rate of sedimentation is defined by the sedimentation constant 5, which is directly proportional to the polymer mass m, solution density p, and specific volume of the polymer V, and inversely proportional to the square of the angular velocity of rotation o), the distance from the center of rotation to the point of observation in the cell r, and the fractional coefficient /, which is inversely related to the diffusion coefficient D extrapolated to infinite dilution. These relationships are shown in the following equations in which (1 — Vp) is called the buoyancy factor since it determines the direction of macromolecular transport in the cell. 

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