Fenske equation for minimum equilibrium stages

• For a specified separation between two key componentsof a multi component mixture, an exact expression is easilydeveloped for the required minimum no of equilibrium stageswhich corresponds to total reflux.•This condition can be achieved in practice by charging thecolumn with feedstock and operating it with no further input of feed and no withdrawal of distillate or bottoms as illustrated in figure .•For S.S. operation within the column, heat input to the reboiler and heat output from the condenser are made equal (assumingno heat losses).•Then, by material balance, vapor and liquid streams passing between any pair of stages have equal flow rates and compositions,

•However, molar vapour & liquid flow rates will change from stage to stage unless the assumption of constant molar overflow is valid.•Derivation of an exact equation for the minimun nos of equilibrium stages involves only the definition of the K value & the mole fraction equality between stages.For component i at stage 1 in above figure.

-----1-----1

----2

------3

-----4

Combining 3 & 4, we have,

-----5

Equation 5 is readily extended in this fashion to give;

Combining 6 & 7 we find that

-------6

-------7

or-----9

Where,the relative volatility between component i& j

Eq.9 relates the relative enrichments of any 2 components i & j

over a cascades of N theoretical stages to the relative volatilities

between the two components.

Limitation to eq.9 is that the conditions of each stage must be

known to compute the set of relative volatilities. Therefore

equaiton 9 is rarely used.

However if the relative volatilities is assumed constant, eq 9

simplifies to

----10

-----11

Eq. 11 is extremely useful. It is referred to as Fenske equation

When i= the light key (LK) & j= the heavy component (HK), the

minimum no of equilibrium stages is influenced by the non key

components only by their effect (if any) on the value of the

relative volatility between the key components.

A more convenient form of equation 11 is obtained by replacing the product of mole fraction ratios by the equivalent product of mole distribution ratios in term of component distillate & bottom flow rates d & b, respectively, and by replacing the relative volatility by a geometric mean of the top stage and bottom stages values. Thus,

Where the mean relative volatility is approximated by,

------------------(12)

------------------(13)

Fenske’s equation is reliable except α varies appreciably over the column and/or when the mixture forms non ideal liquid solutions.

Distribution of non key components at total reflux

The fenske equation is not restricted to the two components.

Once Nmin is known eq. 12 can be used to calculate molar flow

rates of d & b for all non key components. These values provide a

first approximation to the actual product distribution when more

than the minimum number of stages are employed.

Let i = a non key component and j = the heavy key or reference

components denoted by r.

Then eq 12 becomes,

------------------------(14)

-----15

-----16

Equations 15 & 16 gives the distribution of non key components at total reflux.

In 14 gives