F6. Distillation Formulas

Vapor Liquid Equilibrium: Vapor liquid equilibrium ratio, K, is defined by

Where γ_i is the activity coefficient of the ith component in the liquid phase, f^o_i is fugacity of pure liquid i at system temperature T and pressure P, φ_i is fugacity coefficient of the ith species in the vapor phase. Fugacity takes the place of vapor pressure when the vapor fails to show ideal gas behavior, either because of high pressure or as a result of vapor phase association. The vapor phase fugacity coefficient can be neglected when the system pressure is low (e.g. < 100 psi) and the system temperature is not below a reduced temperature of 0.8. The pure-liquid fugacity is essentially equal to the vapor pressure at system temperature up to reduced temperature of 0.7.

Fugacity of pure liquid:

For components whose critical temperature is greater than the system temperature

Where f^o is pure liquid fugacity, υ is fugacity coefficient for pure vapor at the system temperature T, P^o is vapor pressure at that temperature, V is liquid molar volume, P is system pressure, and T is temperature. The exponential term is known as the "Poynting Correction". It is greater than unity if system pressure is greater than the vapor pressure. When the system temperature is above the critical temperature, generalized fugacity coefficient graphs can be used. The Henry's constant is γ⁴f^o where γ⁴ is activity coefficient at infinite dilution.

K-value for Ideal Liquid Phase, Non-Ideal Vapor Phase:

When the liquid phase is ideal,

K-value for Ideal Vapor Phase, Non-Ideal Liquid Phase: When vapor phase is ideal,

Where P_i is the vapor pressure of ith component.

Dew Point:

The dew point of a system at pressure P whose vapor composition is given by mole fraction y_i is that temperature at which there is onset of condensation. Mathematically, it is then temperature at which

And for a binary mixture

The procedure for determining the dew point is as follows: (1) Guess a temperature, (2) Calculate the K_i, and (3) Check if the dew point equation is satisfied. (4) If not, repeat the procedure with a different guess.

Bubble Point: The bubble point of a system at pressure P and whose liquid composition is given by mole fraction x_i is that temperature at which there is onset of vaporization. Mathematically it is the temperature such that

For a binary mixture, this equation can be rearranged to yield

Relative volatility: It is a ratio of volatility of component A to volatility of component B.

Binary Phase Diagram:

Vapor liquid equilibrium behavior of binary systems can be represented by a temperature composition diagram at a given constant pressure. The bubble- and dew point- curves converge at the two ends, which represent the saturation points of the two components. Equilibrium composition of any mixture in two-phase region can be found by drawing a horizontal line. Certain non-ideal systems deviate so much from these as to form maxima or minima at an intermediate composition rather than at on end or the other of the diagram. Such a composition is called an "azeotropic composition". Phase equilibrium is important in design of distillation columns. Such design is commonly based on use of a xy diagram, a plot of equilibrium vapor composition y versus liquid composition x for a given binary system at a given pressure.

Flash Vaporization: If a liquid of known composition is vaporized in one stage, then the composition of the liquid and vapor after separation can be found as

A component balance gives

Which can be rearranged to give

This line passes through point 1(x_F,x_F) and has a slope of -W/D. Intersection of this line with the equilibrium curve yields point 2(x,y) that gives composition of vapor y, and a liquid of a composition x leaving the separator.

Knowing the equilibrium composition, we can find the temperature in the separator. Both liquid and vapor are at same temperature. We can either use bubble- or dew-point equation to find the temperature.

Batch Distillation:

If a solution is distilled in a batch operation, then the amount of residue left after its concentration is reduced from x_F to x_W can be found from the following equation.

If the equilibrium relationship is given by constant relative volatility (α), then the above equation can be integrated and the result is given as

Where F and W are moles of feed and residue and x_F and x_W are mole fractions of more-volatile component in the feed and residue, respectively. This equation is non-linear in x_W. This equation can be rearranged into a simpler form and solved by using Newton's method.

Its derivative is given as

Where c₁ and c₂ are given as

Once x_W is known, x_D can be found as

Continuous Distillation: In a typical distillation column, feed enters the distillation tower at the boiling point. The feed gets in contact with the liquid reflux traveling down the rectification section. The vapors leave the column and are condensed in a total condenser. Liquid is withdrawn and a part of it is used as reflux. The residue from the bottom of the column is withdrawn and is reboiled to strip off the more-volatile product. The section above the feed plate is called rectification section where more-volatile component is fractionated. The section below the feed plate is called stripping section where less-volatile component is fractionated. Heating steam is provided to strip more-volatile component.

The McCabe-Thiele method is used to calculate the number of stages. Typically, the composition of the distillate and residue are available. If the amount and composition of streams are given in mass basis, then we can obtain the corresponding values in molar basis as follows:

And the mole fraction of feed is given as

Where X_F is mass fraction of component A (more-volatile) in the feed, F' is mass flow rate of feed, x_F is mole fraction of A in the feed, and F is molar flow rate of feed. Molecular weight of the feed can be computed as follows:

Quality of the Feed: The thermal condition of the feed plays an important role in the feed location and the number of plates required to achieve a particular level of separation. For a given feed composition, find the bubble point and dew point.

Top Operating Line: The top operating line or rectification line is a material balance for a section above the feed plate. Total material balance is given as

Whereas individual (more-volatile component) balance is as follows:

Now these equations can be combined to give the top section operating line.

This line passes through (x_D, x_D) and has a slope of x_D/(R+1).

Feed quality line:

The mass balance at the feed plate is represented by feed quality (q) line.

This line passes through (x_F, x_F) and has a slope of q/(q-1). It is more convenient to draw q-line by joining (x_F, x_F) and (x_F/q, 0). There may be five different conditions of the feed.

(1) If feed is cold, q > 1, q-line is tilted to the right,

(2) If feed is boiling, q = 1, q-line is vertical,

(3) If feed is partially vapor, 1>q>0, q-line is tilted to the left,

(4) If feed is saturated vapor, q = 0, q-line is horizontal,

(5) If feed is superheated vapor, q<0, q-line is directing south-west.

Intersection of q-line and top-operating line defines the rectification section below the rectification section is stripping section.

Intersection of q-line with the rectification-line:

The intersection of these two lines can occur at any of the following points:

(1) At diagonal line: This is a situation when the tower is operating at total reflux. This gives us a way to determine the minimum number of stages that are required when no product is withdrawn.

(2) Above diagonal line and below equilibrium curve: This corresponds to normal operation of a distillation column.

(3) On the equilibrium curve: This is a situation when the tower is operating at minimum reflux. An infinite number of stages are required to achieve the desired separation.

(4) Above the equilibrium curve: This is a situation when an infinite number of stages will be insufficient to achieve the desired separation.

Minimum Reflux:

One of the important information about a column could be the value of minimum reflux. Knowing its value we can get an idea of the operating reflux and an estimate of the number of stages. The easiest method is to draw the equilibrium curve and locate the intersection of the q-line and the equilibrium curve. Draw a line originating from (x_D, x_D) and passing through this intersection point. Extend this line to y-axis. The steps involved in finding the minimum reflux ratio are

(1) Find the slope of the q-line = q/(q - 1).

(2) Draw point (1) having coordinates (x_F/q, 0).

(3) Draw point (2) having coordinates (x_F, x_F).

(4) Join points (1) and (2) and extend the line (1-2) to intersect the equilibrium curve and obtain point (3).

(5) Draw point (4) having coordinates (x_D, x_D).

(6) Now join point (4) with (3) and extend the line (4-3) to intersect at y-axis at point (5). Point (5) has the coordinates (0, x_D/(R_m+1)).

(7) Calculate R_m given by