Minimum
Number of Stages, Total Reflux:
It is very useful to get an idea of minimum number of stages required to
achieve a desired separation. This can
be achieved by using the diagonal line as the operating line. The top- and bottom-operating lines merge
when the column is operating under total reflux conditions. No product is withdrawn.
We
can draw the number of steps as follows:
C
Draw a vertical line from point (x_{W}, x_{W})
till it touches the equilibrium curve.
C
Draw a horizontal line till it touches the diagonal line.
C
Repeat the steps (1) and (2) till we reach point (x_{D},
x_{D}).
C
The number of triangles represents the number of stages.
We
may not get a whole number.
If
relative volatility is fairly constant, then minimum number of stages can be
calculated as
_{} |
Where
the subscript m, W, and D stand for minimum, waste, and
distillate, respectively. And α_{WD}
is average relative volatility between the residue- and
distillate-sections. The number of
stages computed in this way includes 1/2 stage for condenser and 1/2 reboiler.
The
minimum number of stages above the feed point can also be calculated by
replacing x_{W} with x_{F} in the above equation
as follows:
_{} |
Where
the subscript m, F, and D stand for minimum, feed, and
distillate, respectively. And α_{FD}
is average relative volatility between the feed- and distillate-sections.
A
ratio of N_{mWD}/N_{mFD} gives a measure of the
percentage of rectification section above the feed location.
Theoretical
Number of Stages:
If
the reflux ratio at which the distillation column is to operate is known, then
we can determine the number of stages required to achieve the desired
separation using the following steps.
C
Draw the equilibrium curve from the data.
C
Determine the value of the intercept, I = x_{D}/(R+1).
C
Locate point (1) having coordinates (0, I).
C
Locate point (2) having coordinates (x_{D}, x_{D}).
C
Draw the top-operating line (TOL) by connecting points (1)
and (2).
C
Locate point (3) having coordinates (x_{F}, x_{F}).
C
Locate point (4) having coordinates (x_{F}/q, 0).
C
Draw q-line by connecting points (4) and (3).
C
Extend the q-line 4-3 to intersect the rectification
line 1-2 at point (5).
C
Locate point (6) having coordinates (x_{W}, x_{W}).
C
Draw bottom-operating line (BOL) by connecting points (6)
and (5).
Now
we can draw the number of steps as follows:
C
Draw a vertical line from point 6 till it touches the
equilibrium curve.
C
Draw a horizontal line till it touches the BOL.
C
Repeat the steps (1) and (2) till we cross point (5).
C
Draw a vertical line from the point obtained from previous
step till touches the equilibrium line.
C
Draw a horizontal line till it touches the TOL.
C
Repeat the steps (4) and (5) till we cross point (2).
C
The number of triangles represents the number of stages.
C
Do not round off the number of triangles.
We
can avoid drawing the number of steps, and find the number of stages directly
by using a graphical presentation of Molokanov equation.
_{} |
Z
is plotted against ψ, where
_{} |
R
is
the reflux ratio and subscript m represents the minimum.
Example
6.10: Calculate the number of theoretical stages
when feed to a column contains a binary mixture with a composition of 36 %
more-volatile component on a molar basis.
Distillate withdrawn is 91.4 % rich on a molar basis. The residue is 0.565 % impure. The column is operated at a reflux ratio of
1.5 times the minimum reflux ratio. Use
the following information:
Feed quality, q
= 1.04
Relative
volatility, α = 4.12
Minimum reflux
ratio, R_{m} = 0.606
Minimum Number
of Stages, N_{m} = 5.33
Solution: R = 1.5 ´ 0.606 = 0.909.
Using
the technique of Molokanov, we can find ψ equal to 0.159.
This
gives a value of Ζ to be 0.497.
Finally,
the number of stages, N, are found to be 11.6.
Actual
Number of Stages: If the efficiency
of the stages is given, then the actual number of stages can be found by
dividing the total number of stages by the efficiency of each stage. However, the condenser and reboiler are not
equilibrium stages. We must reduce the
number of stages by one and then correct for the efficiency of the stages.
Liquid
and Vapor Quantities Inside the Tower: The liquid flow rates in the
rectification- and stripping- section are given by the following relationships.
_{} |
Where
subscript n and m represent rectification- and stripping- section
respectively. Vapor flow rates in the
rectification- and stripping-sections are given by the following equations.
_{} |
Condenser
Load: If the condenser utilized in a distillation
column to condense the vapors is a total condenser, then the composition of the
condensate, x_{D}, is the same as the composition of vapors, y_{D},
entering the condenser. Heat removed in
the condenser can be calculated by subtracting the enthalpy of the condensate
from the enthalpy of the vapors. And
the enthalpy of the condensate is comprised of enthalpy of the distillate and
reflux. Enthalpy of the vapors can be
calculated by adding sensible- and latent- heats. Sensible heat of the vapors, H_{GL}, is given by
the following equation:
_{} |
Where
T_{d} is the dew point and T_{o} is the reference
temperature. The latent heat of
vaporization of the vapors, H_{Gλ}, is given by the
following equation
_{} |
The
enthalpy of the condensate, H_{DL}, can be found by knowing the
bubble point of mixture having composition of the distillate and is given by
the following equation.
_{} |
The
condenser load, Q_{c}, can be found to be
_{} |