Equilibrium curve

If we apply the simplifying assumption that negligible evaporation occurs in the system, then L2 — L, and our general balance becomes 

Note that Ntu is determined from process conditions imposed on the operation. It is not based on the performance of the tower itself. The water temperature unfortunately is not a simple function of Hvap0I, hence Equation 5.33 must be solved either numerically or via graphic solution. 

The area of the plot bound between the equilibrium curve (saturation curve) and the operating line represents the potential that drives the total 

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Two other types of equilibrium curves are occasionally encountered with the system of two components forming a continuous series of solid solutions. These are shown in Figs. 1,16, 3 and 1,16, 4. In the former the freezing or melting curve passes through a minimum (examples p-chloroiodobenzene, m.p. 57° - p-dichlorobenzene, m.p. 53° naphtha-... 

When the equilibrium curve is not straight, there is no strictly logical basis for the use of an overall transfer coefficient, since the value of m will be a function of position in the apparatus, as can be seen from Fig. 5-27. In such cases the rate of transfer must be calculated by solving for the interfacial compositions as described above. 

These equations are strictly valid only when m, the slope of the equilibrium curve, is constant, as noted previously. 

For the liquid-phase mass-transfer coefficient /cl, the effects of total system pressure can be ignored for all practical purposes. Thus, when using Kq and /cl for the design of gas absorbers or strippers, the primary pressure effects to consider will be those which affect the equilibrium curves and the values of m. If the pressure changes affect the hydrodynamics, then Icq, and a can all change significantly. 

E] Use s BoUes Fair (Ref. 75) data base to determine new effective area to use with Onda et al. (Ref. 126) correlation. Same definitions as 5-28-D. P = total pressure, atm Mq = gas, molecular weight m = local slope of equilibrium curve Lf /Gf = slope operating line Z = height of packing in feet. 

It is sometimes permissible to assume constant relative volatility in order to approximate the equilibrium curve quickly. Then by applying Eq. (13-2) to components 1 and 2,... 

FIG. 13-34 Illustration of how equdihrium stages can he located on the x-q diagram through the alternating use of the equilibrium curve and the operating line. 

FIG. 13-101 Determination of minimum reflux for equilibrium curve with inflection. 

Importance of Design Diagrams One of the first things a designer should tiy to do is lay out a carefully constructed equilibrium curve, y° = Fix), on an xy diagram, as shown in Fig. 14-2. A horizontal line corresponding to the inlet-gas composition yi is then the locus of feasible outlet-liquor compositions, and a vertical hue corresponding to the inlet-solvent-liquor composition X9 is the locus of feasible out-... 

Once the design recovery for an absorber has been established, the operating curve can be constructed by first locating the point X9, y2 ou the diagram. The intersection of the horizontal hue corresponding to the inlet gas composition yi with the equilibrium curve y° = F x)... 

For stripping one begins by using the design specification to locate the point Xi, y. Then the iutersecfiou of the vertical huex = X9 with the equilibrium curve y° = F x) defines the theoretical minimum gas-to-liquid ratio. The actual value of is chosen to be about 20 to 50... 

Figure 14-4 shows the relationship between the operating curve and the equilibrium curve = F(x,) for a typical example involving solvent recovery, where yj and Xj are the interfacial compositions (assumed to be in equilibrium). Once y is known as a function of x along the operating curve, y can be found at corresponding points on the equilibrium curve by... 

Algebraic Method for Dilute Gases By assuming that the operating and equilibrium curves are straight hues and that heat effects are negligible. Senders and Brown [Ind. Eng. Chem., 24, 519 (1932)] developed the following equation ... 

Stripping Equations When the hquid feed is dilute and the operating and equilibrium curves are straight lines, the stripping equations analogous to Eqs. (14-31) and (14-33) are... 

Classical Adiabatic Design Method The classical adiabatic method assumes that the heat of solution serves only to heat up the liquid stream and that there is no vaporization of solvent. This assumption makes it feasible to relate increases in the hquid-phase temperature to the solute concentration x by a simple eutnalpy balance. The equihbrium curve can then be adjusted to account For the corresponding temperature rise on an xy diagram. The adjusted equilibrium curve will become more concave upward as the concentration increases, tending to decrease the driving forces near the bottom of the tower, as illustrated in Fig. 14-8 in Example 6. 

For dilute systems in countercurrent absorption towers in which the equilibrium curve is a straight line (i.e., yj = mXi) the differential relation of Eq. (14-60) is formulated as... 

Aside from the lack of an explicitly defined liquid-phase-resistance term, the limitations on the use of Eq. (14-66) are related to the fact that its derivation implicitly assumes that the system is dilute iysM = 1) and that the operating and equilibrium curves are straight lines over the range of tower operation. Also, Eq. (14-66) is strictly vahd only for the temperature and pressure at which the original test was run even though the total pressure pf appears in the denominator. 

F Weight (or mass flow rate) m cD Slope of equilibrium curve. (kmol/uA)/ [(lb-mol)/ft ]/... 

The end points of the operating line on an XY plot (Fig. 15-13) are X., Y, andXy, Y., and the number of theoretical stages can be stepped off graphically. The equilibrium curve is taken from the Hand type of correlation shown earlier (Fig. 15-9). When the equilibrium line is straight, its intercept is zero, and the operating line is straight, the number of theoretical stages can be calculated with one of the Kremser equations [Eqs. (l5-14) and (15-15)]. When the intercept of the eqnihbrinm line is not zero, the value of YJK, should be used... 

The local equilibrium curve is in approximate agreement with the numerically calculated profiles except at very low concentrations when the isotherm becomes linear and near the peak apex. This occurs because band-spreading, in this case, is dominated by adsorption equilibrium, even if the number of transfer units is not very high. A similar treatment based on local eqnihbrinm for a two-component mixture is given by Golshau-Shirazi and Gniochou [J. Phys. Chem., 93, 4143(1989)]. 

Detect pinched regions where the operating lines approach the equilibrium curve. 

Identify mislocated feed points where the feed, for binary mixtures, is not where the q-line intersects the equilibrium curve. This is not necessarily the case for multicomponent feed, however. 

Identify excessive reflux and reboil when there is too wide a gap between both operating lines and the equilibrium curve. 

Identify cases where feed or intermediate heat exchangers would be helpful, when only one of the operating lines is too far from the equilibrium curve. A large gap for the bottom section could indicate potential for feed preheat or an interreboiler. A large gap for the top section could portend precooler or intercondenser need. 

For liquid/liquid extraction, data on mass transfer rate of the system at typical operating conditions are required. Also required are an applicable liquid/liquid equilibrium curve and data on chemical reactions occurring after mass transfer in the mixer. 

An important system in distillation is an azeotropic mixture. An azeotrope is a liquid mixture which when vaporized, produces the same composition as the liquid. The VLE plots illustrated in Figure 11 show two different azeotropic systems one with a minimum boiling point and one with a maximum boiling point. In both plots, the equilibrium curves cross the diagonal lines. 

These are azeotropic points where the azeotropes occur. In other words, azeotropic systems give rise to VLE plots where the equilibrium curves crosses the diagonals. Both plots are however, obtained from homogenous azeotropic systems. An azeotrope that contains one liquid phase in contact with vapor is called a homogenous azeotrope. A homogenous azeotrope carmot be separated by conventional distillation. However, vacuum distillation may be used as the lower pressures can shift the azeotropic point. Alternatively, an additional substance may added to shift the azeotropic point to a more favorable position. When this additional component appears in appreciable amounts at the top of the column, the operation is referred to as an azeotropic distillation. When the additional component appears mostly at the bottom of the column, the operation is called extractive distillation. 

The design of a distillation column is based on information derived from the VLE diagram describing the mixtures to be separated. The vapor-liquid equilibrium characteristics are indicated by the characteristic shapes of the equilibrium curves. This is what determines the number of stages, and hence the number of trays needed for a separation. Although column designs are often proprietary, the classical method of McCabe-Thiele for binary columns is instructive on the principles of design. 

Lj and are the pure liquid and inert gas loading rates, respectively, in units of Ib-moles/hr-ft. The second expression is the operating line on an equilibrium diagram. In all scrubbing application, where the transfer of solute is from the gas to the liquid, the operating line will lie above the equilibrium curve. When the mass transfer is from the liquid to the gas phase, the operating line will lie below the equilibrium curve. The latter case is known as stripping . 

Figure 43. Equilibrium curve on solvent free basis.

In the extreme case where both solvents B and S are completely insoluble, then there are only four variables, namely, T, P, and and the more familiar equilibrium curve illustrated in Figure 44 can be constructed. 

The procedure to calculate the points for an equilibrium curve is to start with x = 0, and calculate the value of y for a given value of a. A second point is calculated by incrementing the value of x and repeating the calculation for y. This calculation is repeated until the value of x = 1. 

Note that this equation holds for any component in a multi-component mixture. The integral on the right-hand side can only be evaluated if the vapor mole fraction y is known as a function of the mole fraction Xr in the still. Assuming phase equilibrium between liquid and vapor in the still, the vapor mole fraction y x ) is defined by the equilibrium curve. Agitation of the liquid in tire still and low boilup rates tend to improve the validity of this assumption. 

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