Columns material balance relations

The oscillating jet method is not suitable for the study of liquid-air interfaces whose ages are in the range of tenths of a second, and an alternative method is based on the dependence of the shape of a falling column of liquid on its surface tension. Since the hydrostatic head, and hence the linear velocity, increases with h, the distance away from the nozzle, the cross-sectional area of the column must correspondingly decrease as a material balance requirement. The effect of surface tension is to oppose this shrinkage in cross section. The method is discussed in Refs. 110 and 111. A related method makes use of a falling sheet of liquid [112]. 

A further (independent) material balance around the entire column enables a relation to be established between, for example, pKout and cA our... 

Figure El2.2a shows the boundary conditions X0 and Yx. Given values for m, Nox, and the length of the column, a solution for Y0 in terms of vx and vY can be obtained Xx is related to Y0 and F via a material balance Xx = 1 - (Yq/F). Hartland and Meck-lenburgh (1975) list the solutions for the plug flow model (and also the axial dispersion model) for a linear equilibrium relationship, in terms of F ...

The performance of a given column or the equipment requirements for a given separation are established by solution of certain mathematical relations. These relations comprise, at every tray, heat and material balances, vapor-liquid equilibrium relations, and mol fraction constraints. In a later section, these equations will be stated in detail. For now, it can be said that for a separation of C components in a column of n trays, there still remain a number, C + 6, of variables besides those involved in the dted equations. These must be fixed in order to define the separation problem completely. Several different combinations of these C + 6 variables may be feasible, but the ones commonly fixed in column operation are the following ... 

Mass-Transfer Units The mass-transfer unit concept follows directly from mass-transfer coefficients. The choice of one or the other as a basis for analyzing a given application often is one of preference. Colburn [Ind. Eng. Chem., 33(4), pp. 450-467 (1941)] provides an early review of the relationship between the height of a transfer unit and volumetric mass-transfer coefficients (k a). From a differential material balance and application of the flux equations, the required contacting height of an extraction column is related to the height of a transfer unit and the number of transfer units... 

To accomplish this analysis, an overall material balance is written for the condenser as V = L + D, which relates the vapor (V) leaving the top stage, the liquid reflux returning (L) to the column from the condenser (reflux), and the distillate (D) collected. A material balance for component A is written as... 

The separation of a multi-component mixture into products with different compositions in a multistage process is governed by phase equilibrium relations and energy and material balances. It is not uncommon in simulation studies to require certain column product rates, compositions, or component recoveries to satisfy given specifications with no concern for conditions within the column. Such would be the case when downstream processing of the products is of primary interest. In these instances, one would be concerned only with overall component balances around the column but not necessarily with heat balances or equilibrium relations. Separation would thus be arbitrarily defined, and the problem would be to calculate product rates and compositions. The actual performance of the separation process is analyzed independently in all the following chapters. 

The equations describing total column operation include vapor-liquid equilibrium relations. Equation 5.12 component balances in the rectifying and stripping sections, Equations 5.13 and 5.14 feed stage component balance. Equation 5.15 feed stage energy balance and overall material balance, expressed as Equations 5.16 and 5.17 and overall column component balance. Equation 5.18 ... 

Section 3.3.3). The product rates and compositions in such a column are determined by the energy and material balances and the vapor-liquid equilibrium relations. Under these circumstances the operator has no control over the column performance. 

These relations apply throughout the column, that is, for j between 1 and A-1. A material balance on component i around trays 1 through j takes the form... 

In general, the performance of a packed column can be represented by an equilibrium curve, y =jYX, and an operating line. The latter relates crossing vapor and liquid compositions at any point in the column, and is based on material balance for the transferred component. For constant vapor and liquid flows, L and V, if at some reference point the concentrations of that component in the liquid and vapor are X-j-and Yf, and at another point in the column they are X and T, then by material balance, L X - Xj-) = KTj. - T), or, Y = Tj- -H (L/y)(Xj- - X). 

If the column is to be operated isothermally, then the typical problem is to use the energy balance (12.4.23), together with material balances and phase-equilibrium relations, to compute Q and the composition x< for the liquid leaving the column. If the column is to be operated adiabatically, then the typical problem is to determine both the temperature T and the composition x< at the liquid outlet. We illustrate both problems using absorption of ammonia from air into water the following problem was originally analyzed by Sherwood and Pigford [13]. 

In order to relate compositions of liquid and vapour streams on successive plates it is necessary to carry out material balances around the top and bottom sections of the column. Consider a balance around the top of the column down to plane n (above the feed plate) as shown in Figure 7.35. 

For the countercurrent contact with multiple stages in Fig. 10.3-2, the material-balance or operating-line equation (10.3-13) was derived which relates the concentrations of the vapor and liquid streams passing each other in each stage. In a distillation column the stages (referred to as sieve plates or trays) in a distillation tower are arranged vertically, as shown schematically in Fig. 11.4-1. 

Chihara et al. (1978) showed the moment solutions for this case. The basic equations for a particle of monodisperse pore structure are used for material balances in the column and a macroparticle (Eqs. (6-1) to (6-3)), but the following relations are introduced instead of Eq. (6-4) to take into account the diffusion in microparticles of the radius, a. 

Because the ratio of any feed or product rate to internal liquid or vapor flow is infinitesimal at total reflux. Equation 12.7, derived for two-product columns, applies to multiproduct columns as well. This conclusion may be ascertained by carrying out component material balances over different column sections, similar to the material balances represented by Equations 12.5 and 12.6. The equilibrium relation. Equation 12.2, holds regardless of the number of products. The derivations that lead to Equation 12.9 can therefore be generalized to multiproduct columns at total reflux, and that equation may be rewritten for component i in any column section as follows ... 

In addition to the basic continuous column model assumptions of equilibrium stages and adiabatic operation, dynamics related assumptions are made for the batch model. Distefano (1968) assumed constant volnme of liquid holdup, negligible vapor holdup, and negligible fluid dynamic lag. Although different solntion strategies may be employed, the fundamental model equations are the same. Condenser total material balance ... 

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