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We will consider Mc-Cabe Thiele method for developing relationship between number of trays, liquid/vapor rratio and product compositions

• Enriching Section
• Stripping Section
• Feed Plate
• Total Reflux
• Minimum Reflux Ratio

Enriching Section

The condensor removes all the latent heat from the overhead vapor but does not cool the resulting liquid further. The reflux and distillate product are therfore liquids at the bubble point and y1 = xd = x_o . Also liquid, L mol/h falling from each tray and vapor V mol/h rising from each tray are constant

The trays are theoretical, so that composition y_n of vapor from the n^th tray is in equilibrium with the liquid of composition x_n leaving the same tray.

• A total Material balance gives
  V = L + D
• Material balance for more volatile component A gives:
  Vy_n+1 = Lx_n + Dx_d ......(for all plates from 1 to f-1)
• As we know that :
  L/D = R
• From above two equations we get:
  Y_n+1 = R/(R+1)x_n + Dx_d......Equation of I^st Operating Line for Enriching section

Stripping Section

Trays are again theoretical stages

• A total Material balance geves:
  L' = V' + B
• A material balance on A gives:
  V'y_n + Bx_b = L'x_m-1
• From above two equations we get :
  y_m = L'/(L'-B)x_m-1 - B/(L'-B)x _b ....II^nd Operating line for Stripping Section

Feed Plate

(q = Fraction of Feed which is liquid)
We will consider balance over Feed Plate

• Balance on Liquid part of Feed gives :
  L' - L = qF
• Balance on vapor part of Feed :
  V - V' = (1 - q)F

Now we will look how value of q varies with different types of feed

• q = 1 ....... Feed is saturated with Liquid
• q = 0 ....... Saturated Vapor
• q > 1 ....... For subcooled Liquid
  

  q = 1 + C_pL(T_BP - T_F)/ dH_v
  

• q < 1 ....... for SuperHeated Vapor

  q = 0 - C_pV(T_F - T_DP) /dH_v

Locus of point of intersection of two operating lines is called Q-Line

• y = q/(q-1)x - x_F/(q-1)

Total Reflux

As reflux ratio R = L/D is increased the ratio L/V increases until ultimately when R = infinity, L/V becomes 1 and operating lines of both sections of column coincide with the diagonal y=x

This state will give us minimum number of stages . It can be achieved practically by returning all overhead product back to the column as reflux and reboiling all residue product, whereupon forward flow of feed must be returned to zero.

Minimum Reflux Ratio

Minimum Reflux ratio R_m is the maximum ratio which will recquire an infinite number of trays for the separation desired and it corresponds to minimum reboiler heat and condensor cooling capacity for separation. When R=0 no separation is possible. As we increase R point of intersection of two operating lines come close to equilibrium curve. Number of stages will begin to apear finite when point of intersection lies on equilibrium curve.

In actual distillation processes we add actual reflux 1.5 or 2 times the minimum reflux ratio.