Page 139 - Distillation theory

P1: JPJ/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
0521820928c05 CB644-Petlyuk-v1 June 11, 2004 20:15

5.2 Calculation of Distillation at Minimum Reflux for Ideal Mixtures 113

of constant concentration to the output of top product, taking into consideration
the conditions of phase equilibrium between the incoming vapor flow and the
outgoing liquid flow in this zone:
V r y i = α i d i /(α i − L/VK n )

Designating ϕ = L/VK n and summing up by all the components, we obtain one
of the main equations of Underwood:

min
α i d i /(α i − ϕ) = V r (5.1)
i
The analogous equation for the bottom section is as follows:

min
α i b i /(α i − ψ) =−V (5.2)
s
i
The main achievement of Underwood consists in the proof of equality of pa-
rameters φ and ψ in Eqs. (5.1) and (5.2) in the mode of minimum reflux. Sum up
these equations is following main equation:

α i f i /(α i − θ) = (1 − q)F (5.3)
i
where q is a portion of liquid in the feed and θ is a common parameter (root) of
Eqs. (5.1) and (5.2).
For the set composition of the top product and for the set reflux number, the
number of roots of Eq. (5.1) equals that of the components in top product (k):

0 <ϕ 1 <α k <ϕ 2 <α k−1 < ··· <α 2 <ϕ k <α 1
Similarly, the number of roots of Eq. (5.2) equals that of components in the
bottom product (m):

α m <ψ 1 <α m−1 < ··· <α 2 <ψ m−1 <α 1 <ψ m
Correspondingly, the number of roots of Eq. (5.3) is less by one than that of
the components, present in the top and in the bottom products (i.e., the number
of distributed components).
The Underwood equation system determines separation product compositions
and internal liquid and vapor flows in the sections for the set values of two param-
eters, characterizing the separation process. The reflux number R and withdrawal
of one of the products D/F or recoveries of some two components into the top
product ξ i = d i /f i and ξ j = d j /f j , etc., can be chosen as such two parameters. For
example, at direct split of three-component ideal mixture 1(2) : (1)2,3 (here the
top product contains component 1 and small admixture of component 2 and the
bottom product contain components 2,3 and small admixture of component 1),
Eq. (5.3) has only one common for both section root α 2 <θ <α 1 .If ξ 1 and ξ 2
are set, then d 1 and d 2 can be defined and V min can be obtained from Eq. (5.1).
r
The rest of internal flows in the column section can be defined with the help of
the material balance equations.