NUMBER OF EQUILIBRIUM STAGES IN BINARY DISTILLATION GRAPHICAL METHOD McCABE-THIELE METHOD

This method is based on the Lewis modification of the Sorel method. It assumes equimolal overflow in the rectifying section, in the stripping section, and equimolal latent heats. L0 is a saturated liquid Column pressure and reflux ratio are fixed.

Lm L0 D xD Vm+1 qD m B qB p Overall mass balance: F = D + B F

ENVELOPE A (1) Vm+1 = Lm + D Vm+1 ym+1 = Lm xm + D xD (2) (3) F Lm Vm+1 qD m (1) Vm+1 = Lm + D Vm+1 ym+1 = Lm xm + D xD (2) (3) This is an equation of a straight line on a plot of vapor composition versus liquid composition, where (Lm/Vm+1) is the slope and (DxD/Vm+1) is the intercept which passes through the point (xD, xD) and (xm, ym+1).

Since all L values are equal and all V values are equal (due to constant molal overflow assumption: (4) Equation (4) is the operating line or material balance line for the rectifying section.

Since: Vm = Lm + D In term of R, equation (4) can be written as: (5)

x xD

ENVELOPE B (6) B qB p p+1 (7) (8)

Since all L values are equal and all V values are equal (due to constant molal overflow assumption: (9) Equation (9) is the operating line or material balance line for the stripping section. This is an equation of a straight line with slope and intercept passing through (xB, xB) and (xp, yp+1). This line can be drawn from point (xB, yB) to point or with slope

The problem is, how to calculate and ? and is calculated by material and enthalpy balance relationship around the feed plate. F Lm Vm (10) (11) (12) (13)

q is the number of moles of saturated liquid formed on the feed plate by the introduction of 1 mole of feed: q = 1 : saturated liquid feed, xF = xi q = 0 : saturated vapor feed, xF = yi q > 1 : cold liquid feed, xF < xi q < 1 : superheated vapor, xF > xi 0 < q < 1 : two-phase feed, xF  xi

Substituting eqs. (10) and (13) to eq. (9) yields: (14) This equation gives the slope of the operating line in the stripping section as There is an easier way to draw the operating line in the stripping section, i.e. by using the q-line, which started from point (xF, yF = xF).

Component material balance of the feed: (15)

Eq. (12) is the equation of the q line having a slope of q/(q – 1) and terminating at xF on the 45 line and at point (xi, yi). Saturated liquid feed : q = 1 : slope =  Saturated vapor feed : q = 0 : slope = 0 Cold liquid feed : q > 1 : slope = + Superheated vapor feed : q < 1 : slope = – Two-phase feed : 0 < q < 1 : slope = –

q = 1 q > 1 0 < q < 1 q = 0 xB xF xD

xB xF xD

x1, y1 x2, y2 x1, y2 x3, y3 x2, y3 x4, y4 x3, y4 xB xF xD

MINIMUM REFLUX xB xF xD

MINIMUM REFLUX xB xF xD

TOTAL REFLUX xB xF xD

EXAMPLE 2 SOLUTION Using the data of EXAMPLE 1, determine: The number of equilibrium stages needed for saturated-liquid feed and bubble-point reflux with R = 2.5 using McCabe-Thiele graphical method Rmin Minimum number of equilibrium stages at total reflux. SOLUTION (a) The slope of the operating line in the rectifying section:

N = 11 y x

(b) Intercept = Rmin = 1.18

(c) N = 8

SIDE PRODUCT

If a product of intermediate composition is required, a vapor or a liquid side stream can be withdrawn. This kind of column configuration is typical of the petrochemical plants, where the most common running unit operation is the fractional distillation. This consists in splitting a mixture of various components, the crude oil, into its components. Because of their different boiling temperatures, the components (or so- called fractions) of the crude oil are separated at different level (i.e. plate) of the column, where different boiling temperatures are present. The fractions are then withdrawn from the plate where they form, therefore the column presents numerous side streams.

Rectifying section Lm Vm Middle section Stripping section D, xD L0 S, xS Middle section F, xF Stripping section B, xB

MATERIAL BALANCE IN RECTIFYING SECTION D, xD L0 S, xS Vm+1 Lm m Assuming constant molar overflow, then for the rectifying section the operating line is given by: (16)

MATERIAL BALANCE IN MIDDLE SECTION D, xD F, xF L0 S, xS Rectifying section Middle section

Overall: (17) Component: (18) (19) For constant molal overflow: (20) Since the side stream is normally removed as a liquid:

Equation (20) represents a line of slope , which passes through the point which is the mean molar composition of the overhead product and side streams. Since xS < xD and , this additional operating line cuts the line y = x at a lower value than the operating line though it has a smaller slope.

MATERIAL BALANCE IN STRIPPING SECTION F, xF B, xB

Overall: (21) Component: (22) (23) For constant molal overflow: (24) Equation (24) represents a line of slope , which passes through the point (xB, xB)

xB xD