Equilibrium Stage Processes - Distillation.

Introduction

This course is, in general, concerned with the processes effecting the separation in the outline of a chemical process shown below.

Fig 1: Typical Chemical Process

One example of a separation process, liquid-liquid extraction, may be used to seperate propionic acid from a mixture with kerosine as follows:

Fig 2: Batch liquid-liquid extraction.

For continuous operation,

Fig 3: Continuous liquid-liquid extraction.

The 'Ideal Equilibrium Stage', also known as a 'Theoretical Stage', 'Theoretical Plate' or 'Ideal Stage', is one which has the exit phases/streams in thermodynamic equilibrium, each phase/stream being removed from the stage without entraining any of the other phase / stream.

Binary Distillation

Distillation is a process involving an equilibrium between two phases - liquid and vapour. For a pure compound, in particular a pure ionic compound, a sharp boiling point usually exists. For a mixture, however, a phase equilibrium exists over a range of temperature, as shown below.

Isobaric temp-composition diagram
Fig 4: Isobaric (Constant Pressure) Temperature Composition Diagram.

The above diagram applies to the system:

Fig 5: Liquid and vapour space in equilibrium.

The first bubble of vapour formed when a liquid (subcooled) is heated is formed at the bubble curve.

The first drop of liquid formed when a vapour (superheated) is cooled is formed at the dew curve.

x = mole fraction m.v.c. in liquid
y = mole fraction m.v.c. in vapour

The temperature - composition diagram is used for isobaric conditions. For isothermal (constant temperature) conditions a pressure - composition diagram is used.

Isothermal Pressure-Composition Diagram
Fig 6: Isothermal (constant temperature) Pressure-Composition Diagram.

X-Y Diagrams for Constant Pressure

If the isobaric temperature-composition diagram is drawn or plotted, the x and y data for different temperatures may be utilised to form the x-y diagram (i.e. the plot of y against x). The x-y diagram will correspond to the pressure for which the temperature-composition diagram was plotted.

Fig 7: X-Y Diagram for Constant Pressure.

X-Y Diagrams for Constant Temperature

Fig 8: X-Y Diagram for Constant Temperature.

P_mvc = partial pressure of more volatile component
P_lvc = partial pressure of less volatile component

Azeotropes

Type A

(e.g. Acetone - CS2, Chloroform - methanol)

Fig 9
Fig 9:

Fig 10
Fig 10:

Fig 11
Fig 11:

Type B

(e.g. Acetone-Chloroform)

Fig 12
Fig 12:

Fig 13
Fig 13:

Fig 14
Fig 14:

Fig 15
Fig 15:

Fig 16
Fig 16:

Flash Distillation

Flash distillation is a process typically used to effect seperation of crude oil. The process involves heating a feed stream and then allowing it to expand into a vessel maintained at low pressure. Partial vaporisation then occurs, and a phase equilibrium is (ideally) reached.

Fig 17: Flash Distillation.

A material balance gives:

F = L + V

...1

An m.v.c. balance gives:

FZ_F = Lx_e + Vy_e

...2

Now from 2,

Vy_e = FZ_F - Lx_e

y_e = (FZ_F / V) - (Lx_e/V)

y = c + mx

Putting x_e = Z_F gives

y_e = Z_F[ (F / V) - (L/V) ]

but from 1, (F / V) = (L / V) + 1, so (F / V) - (L / V) = 1

thus y_e = Z_F when x_e = Z_F

(this is not surprising considering the fact that with a feed of composition Z_F and a liquid bottom product of composition Z_F the top product cannot be of any composition other than Z_F).

For the constant pressure at which the flash vessel is operated, a y against x diagram may be plotted and the material balance line (which is straight) is seen to pass through Z_F, Z_F and x_e, y_e (corresponding to the temperature T_e).

Fig 18:

Simple Differential Distillation

Fig 19: Differential Distillation

The above diagrams represent classical simple laboratory distillation, attributed to Rayleigh, 1903. Heat is applied to vapourise some of the solution. The vapour is condensed and found to contain a high m.v.c. composition (depending on amount vapourised).

Now in a small time increment dt, vapour of m.v.c. composition y is given off. The amount of vapour given off is V kmol. Assuming x and y are equilibrium values throughout the process, In time increment dt,

dV = -dS

m.v.c. balance

ydV = -d(Sx) = -Sdx - xdS

-ydS = -Sdx - xdS

xdS - ydS = -Sdx

(x - y) dS = -Sdx

Rayleigh Equation
Fig 20:

The last equation above is known as Rayleigh’s Equation, where

S₁ = total kmol solution to start with
S₂ = total kmol solution left in bottoms
x₁ = starting m.v.c. composition in liquid
x₂ = finishing m.v.c. composition in liquid

Rayleighs
Fig 21:

Continuous Fractionation

The system typically adopted for continuous fractionation is shown below.

Fig 22: Continuous Fractionation.

where,

F = Feed flow rate (kmols/hr)

x_f = m.v.c. composition of feed (mole fraction or mol percentage)
V = Vapour flow rate (kmols/hr)
L = Reflux flow rate (kmols/hr)
D = Top Product flow rate (kmols/hr)
x_D = m.v.c. composition of top vapour stream, top product, and reflux (mole fraction or mol percentage)
V" = Reboiler exit stream flow rate (kmols/hr)
W = Bottom-product flow rate (kmols/hr)
x_W = m.v.c. composition of bottom product and feed to reboiler (mole fraction or mol percentage)

Comparison of Continuous Fractionation with Flash and Rayleigh Distillation

Flash Distillation

Fig 23: Flash Distillation.

Rayleigh Distillation (Simple Differential Distillation)

Fig 24: Rayleigh Distillation.

A single stage of the continuous fractionation column is now considered for comparison.

Continuous Simple Distillation

Fig 25: Continuous Distillation.

Multiple Continuous Simple Distillation

Fig 26: Multiple Continuous Distillation.

Consider a fractionating column of N plates, where the condenser and reboiler are counted as 'plates'. A typical 'n^th' plate has the streams shown below associated with it:

Column
Fig 27: Column.

Fig 28: Temperature - Composition Diagram for nth Plate.

M.v.c balance for N^th plate

Fig 29: mvc balance.

V_N-1Y_N-1 + L_N+1X_N+1 = V_NY_N + L_NX_N

But X_N+1 = X_D

V_N-1Y_N-1 + L_N+1X_D = V_NY_N + L_NX_N

M.v.c. balance for plates n to N (where n,N in rectifying section)

Fig 30: M.v.c. balance for plates n to N.

(The balance is for the solid red line area)

V_n-1Y_n-1 + L_N+1X_N+1 = V_NY_N + L_nX_n

But X_N+1 = X_D

V_n-1Y_n-1 + L_N+1X_D = V_NY_N + L_nX_n

M.v.c. balance as above incorporating condenser.

(balance as above + dotted red line area)

V_n-1Y_n-1 = L_nX_n + DX_D

Conditions for Constant Molal Overflow

Heat losses negligable (achieved more easily in industrial columns)
Negligable heat of mixing
Equal or close heats of vaporisation

In general, values of V and L very from stage to stage, and an enthalpy balance over each stage is required to calculate L,V.

With constant molal overflow assumption,
L_n-1 = L_n = L_n+1 = ... etc.
V_n-1 = V_n = V_n+1 = ... etc.

M.v.c. balance for plate n to condenser continued.

V_n-1Y_n-1 = L_nX_n + DX_D
Assuming constant molal overflow,

VY_n-1 = LX_n + DX_D

Note: V=Vapour from top of column
L = reflux
D = top-product

Dividing through by V gives

Y_n-1 = (L/V)X_n + (D/V)X_D
Y   =  m   X  +  c

This material balance equation is called the Upper Operating Line. Note that (L/V) and (D/V) are constants. This linear relationship links the compositions of passing streams between stages.

The Lewis-Sorel Method

This uses the above equilibrium relationship and the operating line equation alternately to step up or down the column.

e.g. at the top of the column:

Y_N = X_D = (known)

Equilibrium X_N

Operating Line Y_N-1 = (L/V)X_N + (D/V)X_D

Equilibrium X_N-1

Operating Line Y_N-2 = (L/V)X_N-1 + (D/V)X_D

etc.

McCabe-Thiele

Recognised the fact that the operating line is straight simple graphical construction.

On an x-y diagram, the operating line is a straight line of gradient (L/V) and passes through X_D, X_D

Fig 31:

Fig 32:

Reflux Ratio, R = L / D

V = L + D

L / V = R / (R + 1)

D / V = 1 - (L / V)

= (R + 1 - R) / (R + 1)

1 / (R + 1)

Fig 33:

Fig 34:

Fig 35:

Lower Operating Line

L', V' may be different from L, V

m.v.c. balance

L' x_m = V' Y_m-1 + W X_w

Y_m-1 = (L' / V') X_m - (W/V') X_w

This material balance is a straight line passing through the point (X_w,X_w)

The intersection of the upper and lower operating lines is determined by the feed.

Importance of the feed

The feed should be introduced where the appropriate stream in the column has the same composition as the feed.

The thermodynamic state of the feed determines the relationships between L' and L and V' and V

Different States of the Feed

Saturated Liquid, i.e. at bubble temperature
Saturated Vapour, i.e. at dew temperature
Two-Phase Feed

Summary of McCabe Thiele Construction

Fig 39:

Plot equilibrium line
Draw 45^o line
Locate Distillate (X_D,X_D)
Draw Upper Operating Line (gradient R / (R + 1)) between (X_D,X_D) and (D,X_D/(R + 1) )
Locate bottom product (X_W,X_W)
Locate (X_F,X_F)
Draw "q-line" with gradient q / (q - 1)
Draw lower operating line (from q-line / upper operating line intersection to (X_W,X_W) )