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McCabe-Thiele Method of Determination of Number of Theoretical Plates

McCabe-Thiele Method of Determination of Number of Theoretical Plates. McCabe-Thiele Method of Calculation for Number of Theoretical Stages. Introduction and assumptions.

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McCabe-Thiele Method of Determination of Number of Theoretical Plates

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  1. McCabe-Thiele Method of Determination of Number of Theoretical Plates

  2. McCabe-Thiele Method of Calculation for Number of Theoretical Stages Introduction and assumptions • A mathematical – graphical method for determining the number of theoretical trays or stages needed for a given separation of a binary mixture of A and B has been developed by McCabe and Thiele. • The method uses material balances around certain parts of the tower, which give operating lines and the xy equilibrium curve for the system. • Main assumption • Equimolar overflow through the tower between the feed inlet and the top tray and the feed inlet and bottom tray. • Liquid and vapor streams enter a tray, are equilibrated, and leave. 2

  3. A total material balance: A component A balance: Where Vn+1 is mol/h of vapor from tray n+1 Ln is mol/h liquid from tray n yn+1 is mole fraction of A in Vn+1 and so on. 3

  4. Equation for enriching section A total material balance: (1) A component A balance: (2) Where F is the entering feed (mol/h) D is the distillate (mol/h) W is the bottoms (mol/h) 4

  5. (3) Material balance over dashed-line section: (4) A balance on component A: 5

  6. Solving for yn+1, the enriching-section operating line is (5) Since and equation becomes (6) where = reflux ratio = constant. The eqn. (1) is a straight line on a plot of vapor composition versus liquid composition. 6

  7. The slope is or . It intersects the y=x line (45º diagonal line) at . The intercept of the operating line at x = 0 is . The theoretical stages are determined by starting at xD and stepping off the first plate to x1. Then y2 is the composition of the vapor passing the liquid x1. In a similar manner, the other theoretical trays are stepped off down the tower in the enriching section to the feed tray. 7

  8. Equation for stripping section Material balance over dashed-line section: (7) A component A balance: (8) 8

  9. Solving for ym+1, the enriching-section operating line is (9) Again, since equimolal flow flow is assumed, = constant and = constant, eqn. (2) is a straight line when plotted as y versus x, with a slope of . It intersects the y = x line at x = xw. The intercept at x = 0 is . 9

  10. The theoretical stages for the stripping section are determined by starting at xW, going up to yW, and then across to the operating line, etc. 10

  11. Effect of feed conditions The condition of feed stream is represented by the quantity q, which is the mole fraction of liquid in feed. (10) (11) The enriching and striping operating-line equations on an xy diagram can be derived as follows: (12) (13) Where the y and x values are the point of intersection of the two operating lines. Subtracting eqn.(3) from eqn.(4), (14) 11

  12. Effect of feed conditions Substituting eqn.(2), (10), and (11) into eqn.(14) and rearranging, (15) Cold-liquid feed Superheated vapor where CpL, CpV = specific heats of liquid and vapor, respectively TF = temperature of feed Tb, Td = bubble point and dew point of feed respectively λ = heat of vaporization 12

  13. Location of the feed tray in a tower and number of trays. From eqn.(15), the q-line equation and is the locus of the intersection of the two operating lines. Setting y = x in eqn(15), the intersection of the q-line equation with the 45º line is y=x=xF, where xF is the overall composition of the feed. In given below the figure, the q line is plotted for various feed conditions. The slope of the q line is q/(q-1). q = 0 (saturated vapor) q = 1 (saturated liquid) q > 1 (subcooled liquid) q < 0 (superheated vapor) 0 < q < 1 (mix of liquid and vapor) 13

  14. 1st point 2nd point 3rd point Number of stages and trays n = 7 =number of tray + reboiler Number of tray = 6

  15. Using Operating Lines and the Feed Line in McCabe-Thiele Design Slope = R/(R+1) Slope = q/(1-q) Slope = L/ V

  16. Exemple:A continuous fractioning column is to be designed to separate 30,000 kg/h of a mixture of 40 percent benzene and 60 percent toluene into an overhead product containing 97 percent benzene and a bottom product containing 98 percent toluene. These percentages are by weight. A reflux ratio of 3.5 mol to 1 mol of product is to be used. The molal latent heats of benzene and toluene are 7,360 and 7,960 cal/g mol, respectively. Benzene and toluene from a nearly ideal system with a relative volatility of about 2.5. The feed has a boiling point of 95ºC at a pressure of 1 atm. a) Calculate the moles of overhead product and bottom product per hour. b) Determine the number of ideal plates and the position of the feed plate (i) if the feed is liquid and at its boiling point; (ii) if the feed is liquid and at 20ºC (specific heat 0.44 cal/g.ºC); (iii) if the feed is a mixture of two-thirds vapor and one-third liquid.

  17. Solution (a) The average molecular weight of the feed is The average of heat vaporization is The feed rate F is 30,000/85.8 = 350 kg mol/h. By an overall benzene balance, using Eq. below

  18. Solution (b) (i), We determine the number of ideal plates and position of the feed plate. 1) Plot the equilibrium diagram, erect verticals at xD, xF, and xB. 2) Draw the feed line. Here q=1, and the feed line is vertical. 3) Plot the operating lines. The intercept of the rectifying lie on the y axis is, xD/(R+1) = 0.974/(3.5+1) = 0.216 (eqn (6)). From the intersection of the rectifying operating line and the feed line, the stripping line is drawn. 4) Draw the rectangular steps between the two operating lines and the equilibrium curve. The stripping line is at the seventh step. By counting steps it is found that, besides the reboiler, 11 ideal plates are needed and feed should be introduced on the seventh plate from the top.

  19. Solution (b) (ii), The latent heat of vaporization of the feed λ is 7,696/85.8 = 98.7 cal/g. The slope of the feed line is -1.37/(1-1.37) = 3.70. When steps are drawn for this case, as shown in Fig. below, it is found that a reboiler and 10 ideal plates are needed and that the feed should be introduced on the sixth plate.

  20. Solution (b) (iii), From the definition of q it follows that for this case q = 1/3 and the slope of the feed line is -0.5. The solution is shown in Fig. below. It calls for a reboiler and 12 plates, with the feed entering on the seventh plate.

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