Infinite plate number

Fig. 2.3.2-9 Balance lines and staircase construction, a) Minimum reflux ratio, infinite plate number, b) Finite reflux ratio, resp. finite plate number, c) Total reflux, minimum plate number.

G The minimum amount of vapor required For separation in a column with infinite plate number (mol)... 

The minimum amount of absorption liquid (amount of solvent that is needed to just achieve the separation at infinite plate number) from the intersection of the balance line with the point (2ij), Ye) (Figure 2.3.3-3a). 

The minimum amount of solvent is the quantity that just fulfills the given task (enrichment from Rq to Rsoii) at infinite plate number (Figure 2.3.4-6). Infinite theoretical plate number occurs when the pole P coincides with the innermost intersection of a tie line. This pole is determined by extending several tie lines. The straight line Po fmin intersects the binodal line at Lemin- The two straight lines Po-fo and Pj-Lj min then intersect at the point Mmin- The minumum feed ratio results from the lever rule as Po/Io = (Po-Mmin)/(Mmin-fo)-... 

Equation 10.115 has a considerable fundamental and practical importance. It combines parameters of fimdamentally different origins, the plate number at infinite dilution, N, which characterizes the intensity of axial dispersion taking place in the column and two parameters of thermod5mamic origin, the retention factor at infinite dilution, ICg, related to the initial slope of the isotherm, and the loading factor, proportional to the sample size and related to the saturation capacity of the isotherm. Accordingly, Eq. 10.115 indicates the extent to which the self-sharpening effect on the band profile due to the nonlinear thermodynamics is balanced by the dispersive effect of axial and eddy diffusion and of the mass transfer resistances. 

Internal one-dimensional transient conduction within infinite plates, infinite circular cylinders, and spheres is the subject of this section. The dimensionless temperature < ) = 0/0/ is a function of three dimensionless parameters (1) dimensionless position C, = xlZF, (2) dimensionless time Fo = otr/i 2, and (3) the Biot number Bi = hiElk, which depends on the convective boundary condition. The characteristic length IF, is the half-thickness L of the plate and the radius a of the cylinder or the sphere. The thermophysical properties k, a, the thermal conductivity and the thermal diffusivity, are constant. 

The final value, due to Lienhard and Dhir, is probably closest to experimental data for flat plates. For a small plate, the number of jets may not be representative of those for an infinite plate, and this effect can lead to either higher or lower critical heat fluxes for small plates, depending on the relationship between X and the size of the plate (Lienhard and Dhir [161]). 

Solutions to Pick s equations for a variety of boundary conditions and grain geometries, i.e. infinite plate, cylinder, sphere, etc., have been derived by numerous workers (e g. lost 1960, Crank 1975). Many of these solutions demonstrate how the degree of equilibration, F (see e g. Eqn. 97), is related to the diffusion coefficient, as well as grain size, grain geometry and solution to solid volume ratio. Diffusion rates are sensitive to a number of factors which can be broadly divided into (a) environmental and... 

Of course, the above presentation of arithmetic methods is not exhaustive. Fohl [157] published numerical methods for ideal mixtures and batch as well as continuous operation at infinite and finite reflux ratios which make possible a rapid and relatively simple determination of the plate number. The contributions of Stage and Juilfs [71] should also be mentioned in which further accurate and approximate methods are summarized. The same applies to the book of Rose et al. [153]. Zuiderweg [158] reports a procedure which considers the operating hold-up (see chap. 4.10.5) and the magnitude of the transition fraction in batch distillation. 

The actual amount of vapor required at individual points in the column can easily be calculated with balance equations by assuming an infinitely high number of plates [Kaibel 1989a]. For a two-component mixture the minimum vapor quyntity G at any position in a distillation column with a contration x of the light boiler in the liquid is given by Equation (2.3.2-38), where the quantities are in moles ... 

This equation has been integrated for a number of geometries [2]. For the case of two parallel infinite plates of the same material at a distance H, the attraction energy per unit area is,... 

For B/h = o(E/X = oo [a 00), the surface temperature Ts equals the temperature of the surrounding Tju, at all times after the plate is brought into contact with the surrounding fluid. With the respective values of Cl and C2 for an infinite Bi number, Eq. (3.2.59) leads to ... 

Looking back to Packie s work, it is known that for a given separation requirement expressed as (5-95) Gap, a given material balance expressed in terms of ASTM 50 volume percent temperature difference and a given number of trays in the separation section, there is a vdue, called F, which is the product of the number of actual trays and the volumetric reflux ratio in the section. Thus, a minimum allowable reflux falling from draw trays can be calculated. This is not minimum reflux in the sense that infinite plates are required for the separation. It is minimum allowable operating reflux for the specified number of trays and the required separation. 

An examination of Fig. 22-11 also shows the relation between absorption factor and the percentage of component that is absorbed. Consider the line for five theoretical plates. This line approaches the line for infinite plates as the absorption factor is decreased, until at 50 per cent absorption, or ( n+i — y-d/iVn+i — l/o), the value of A is nearly 0.5 (about 0.505). The number of plates that are used also affect the location of the point at which A is substantially equal to the percentage absorbed. Likewise the number of plates is related to the absorption factor that must be used to absorb substantially all of a component. 

The required number of actual plates, A/p, is larger than the number of theoretical plates, because it would take an infinite contacting time at each stage to estabhsh equihbrium. The ratio is called the overall column efficiency. This parameter is difficult to predict from theoretical... 

The more volatile (i.e., less soluble) components will be only partially absorbed even though the effluent liquid becomes completely saturated with respecd to these lighter substances. When a condition of saturation exists, the value of will remain finite even for an infinite number of plates or transfer units. This can be seen in Fig. 14-9, in which the asymptotes become vertical for values of greater... 

The conditions of total liquid reflux in a column also represent the minimum number of plates required for a given separation. Under such conditions the column has zero production of product, and infinite heat requirements, and Lj/Vs = 1.0 as shown in Figure 8-15. This is the limiting condition for the number of trays and is a convenient measure of the complexity or difficulty of separation. 

The minimum number of plates [129], for infinite time for separation ... 

In other words, both summation procedures are valid. They describe different physical situations. In the c-summation the size of the dipolar array is much smaller than the size of the capacitor, while in the p-approach both are equal. One can imagine an infinite number of different summation orders corresponding to different relations between A and Aj, leading to results between the p- and c-limits. To solve the problem unambiguously, one must consider a finite dipolar lattice between finite plates, which requires accounting for edge effects and makes the whole problem much more complicated. 

Under conditions of minimum reflux, a column has to have an infinite number of plates, or alternatively the composition on plate n is equal to that on plate n + 1. Dividing equation 11.51 by equation 11.52 and using the relations x(n+i)A = xnA and X( +i)b = xnB, then ... 

It is important to note that, if L m/mG m is less than 1, then a very large number of plates are required to achieve a high recovery, and even an infinite number will not give complete recovery. L m/mG m is the ratio of the slope of the operating line Lm/Gm to the slope of the equilibrium curve m, so that if L m/G m < m, or L m/mG m < 1, then the operating line will never cut the equilibrium curve and the gas leaving the top of the column will not therefore reach equilibrium with the entering liquid. 

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