Three types of binary equilibrium curves are shown in Fig. 13-27. The y-x diagram is almost always plotted for the component that is the more volatile (denoted by the subscript 1) in the region where distillation is to take place. Curve A shows the most usual case, in which component 1 remains more volatile over the entire composition range. Curve B is typical of many systems (ethanol-water, for example) in which the component that is more volatile at low values of x1 becomes less volatile than the other component at high values of x1. The vapor and liquid compositions are identical for the homogeneous azeotrope where curve B crosses the 45° diagonal. A heterogeneous azeotrope is formed with two liquid phases by curve C.
An azeotrope limits the separation that can be obtained between components by simple distillation. For the system described by curve B, the maximum overhead-product concentration that could be obtained from a feed with xi = 0.25 is the azeotropic composition. Similarly, a feed with x1 = 0.9 could produce a bottom-product composition no lower than the azeotrope.
The phase rule permits only two variables to be specified arbitrarily in a binary two-phase system at equilibrium. Consequently, the curves in Fig. 13-27 can be plotted at either constant temperature or constant pressure but not both. The latter is more common, and data in Table 13-1 are for that case. The y-x diagram can be plotted in either mole, weight, or volume fractions. The units used later for the phase flow rates must, of course, agree with those used for the equilibrium data. Mole fractions, which are almost always used, are applied here.
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, a = K1/K2 = y1x2/x1y2
which, since x2 = 1 - x1 and y2 = 1 - y1, can be rewritten as
for use in calculating points for the equilibrium curve.
Operating Lines The McCabe-Thiele method is based upon representation of the material-balance equations as operating lines on the y-x diagram. The lines are made straight (and the need for the energy balance obviated) by the assumption of constant molar overflow. The liquid-phase flow rate is assumed to be constant from tray to tray in each section of the column between addition (feed) and withdrawal (product) points. If the liquid rate is constant, the vapor rate must also be constant.
The constant-molar-overflow assumption represents several prior assumptions. The most important one is equal molar heats of vaporization for the two components. The other assumptions are adiabatic operation (no heat leaks) and no heat of mixing or sensible heat effects. These assumptions are most closely approximated for close-boiling isomers. The result of these assumptions on the calculation method can be illustrated with Fig. 13-28, which shows two materialbalance envelopes cutting through the top section (above the top feed stream or sidestream) of the column. If Ln +1 is assumed to be identical to Ln _ 1 in rate, then Vn = Vn _ 2 and the component material balance for both envelopes 1 and 2 can be represented by
where y and x have a stage subscript n or n + 1, but L and V need be identified only with the section of the column to which they apply. Equation (13-21) has the analytical form of a straight line where L/V is the slope and DxD /V is the y intercept at x1 = 0.
The effect of a sidestream withdrawal point is illustrated by Fig. 13-29. The material-balance equation for the column section below the sidestream is
where the primes designate the L and V below the sidestream. Since the sidestream must be a saturated phase, V = V if a liquid side stream is withdrawn and L = L' if it is a vapor.
If the sidestream in Fig. 13-29 had been a feed, the balance for the section below the feed would be
Similar equations can be written for the bottom section of the column. For the envelope shown in Fig. 13-30, y„ = (L"/V") xm+1 - (BXb/V)
where the subscript m is used to identify the stage number in the bottom section.
Equations such as (13-21) through (13-24) when plotted on the y-x diagram furnish a set of operating lines. A point on an operating line represents two passing streams, and the operating line itself is the locus of all possible pairs of passing streams within the column section to which the line applies.
An operating line can be located on the y-x diagram if (1) two points on the line are known or (2) one point and the slope are known. The known points on an operating line are usually its intersection with the y-x diagonal and/or its intersection with another operating line.
The slope L/V of the operating line is termed the internal-reflux ratio. This ratio in the operating-line equation for the top section of the column [see Eq. (13-21)] is related to the external-reflux ratio R = Ln+1/D by
V = Vn = (1 + R)D = 1 + R when the reflux stream LN +1 is a saturated liquid.
Thermal Condition of the Feed The slope of the operating line changes whenever a feed stream or a sidestream is passed. To calculate this change, it is convenient to introduce a quantity q which is defined by the following equations for a feed stream F:
The primes denote the streams below the stage to which the feed is introduced. The q is a measure of the thermal condition of the feed and represents the moles of saturated liquid formed in the feed stage
per mole of feed. It takes on the following values for various possible feed thermal conditions.
Subcooled-liquid feed: q > 1 Saturated-liquid feed: q = 1 Partially flashed feed: 1 > q > 0 Saturated-vapor feed: q = 0 Superheated-vapor feed: q < 0 The q value for a particular feed can be estimated from q =
energy to convert 1 mol of feed to saturated vapor molar heat of vaporization Equations analogous to (13-26) and (13-27) can be written for a sidestream, but the q will be either 1 or 0 depending upon whether the sidestream is taken from the liquid or the vapor stream.
The q can be used to derive the "q-line equation" for a feed stream or a sidestream. The q line is the locus of all points of intersection of the two operating lines, which meet at the feed-stream or sidestream stage. This intersection must occur along that section of the q line between the equilibrium curve and the y = x diagonal. At the point of intersection, the same y, x point must satisfy both the operating-line equation above the feed-stream (or sidestream) stage and the one below the feed-stream (or sidestream) stage. Subtracting one equation from the other gives for a feed stage
(V - V)y = (L - L')x + Fxf which when combined with Eqs. (13-26) and (13-27) gives the q-line equation q y = 1- x -
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