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

2. 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,

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

4. 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),

5. 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,

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

7. x xD

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

9. 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

10. 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)

11. 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

12. 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),

13. Component material balance of the feed:
(15)

14. 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 = –

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

16. xB xF xD

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

18. MINIMUM REFLUX xB xF xD

19. MINIMUM REFLUX xB xF xD

20. TOTAL REFLUX xB xF xD

21. 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:

22. N = 11 y x

23. (b)
Intercept =
Rmin = 1,18

24. (c)
N = 8

25. SIDE PRODUCT

26. 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,

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

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

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

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

32. Vm Lm S

33. 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,

34. MATERIAL BALANCE IN STRIPPING SECTION
F, xF
B, xB

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

36. F

37. xB xD

38. MULTIPLE-FEED

39. Rectifying section Lm Vm Middle section Stripping section D, xD L0
F1, xF1
Middle section
F2, xF2
Stripping section
B, xB

40. MATERIAL BALANCE IN RECTIFYING SECTION
Assuming constant molar overflow, then for the rectifying section the operating line is given by:
D, xD L0 Lm m
Vm+1
F1, xF1
(16)

41. MATERIAL BALANCE IN MIDDLE SECTION
D, xD
F2, xF2 L0
F1, xF1
Rectifying section
Middle section

42. Overall:
(25)
Component:
(26)
(27)
For constant molal overflow:
(29)
Equation (29) represents a line of slope

43. Material balance around feed plate F1 :Vm Lm
(30) F1
(31)

44. MATERIAL BALANCE IN STRIPPING SECTION
F2, xF2
B, xB

45. Overall:
(32)
Component:
(33)
(34)
For constant molal overflow:
(35)
Equation (24) represents a line of slope , which passes through the point (xB, xB)

46. Material balance around feed plate F2 :
(36) F2
(37)

47. EXAMPLE 3 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:

48. EXAMPLE 3
A feed mixture composed of 42 mole % heptane, 58 mole % ethyl benzene is to be fractionated at 760 mm Hg to produce distillate containing 97 mole % heptane and a residue containing 99 mole % ethyl benzene. Using (L/D) = 2.5, determine the number of equilibrium stages needed for a saturated liquid feed and bubble-point reflux.

49. EXAMPLE 3
Benzene (1) is to be separated from toluene (2) in a distillation column, Three streams are available:
No, of stream 1 2 3
Lb-mol/h
200
500
300
Benzene mole fraction
0,7
0,5
0,2
Fraction vaporized
0,25
0,0 q
0,75
1,0
A high-purity product containing 98-mol% benzene is required at the rate of 200 lb-mol/h, Toluene is to be recovered at 95 mol % purity, A total overhead condenser is to be used, nser yang digunakan adalah kondenser total, The column is designed to operate at R = 1,25 Rmin, How many stages is required, and where should be the feed stream?

50. T (K) x1 y1
385
0,000
380
0,100
0,210
375
0,230
0,410
370
0,365
0,575
365
0,525
0,725
360
0,700
0,850
355
0,910
0,965
353
1,000

51. SOLUTION
Slope of feed lines:
Feed 1:
Feed 2:
Feed 3: