Optimization of Pressure-Swing Distillation by Evolutionary Techniques: Separation of Ethanol-Water and Acetonitrile-Water Mixtures

Adjay Sagar S; Imran Rahman

doi:10.1515/cppm-2017-0007

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Optimization of Pressure-Swing Distillation by Evolutionary Techniques: Separation of Ethanol-Water and Acetonitrile-Water Mixtures

Adjay Sagar S and Imran Rahman

Published/Copyright: October 13, 2017

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From the journal Chemical Product and Process Modeling Volume 13 Issue 2

Abstract

Complete separation can be achieved in selective homogeneous azeotropic mixtures by exploiting the pressure sensitive nature of the system. In the present work the optimal number of trays, feed location and reflux ratio for sequential column systems encountered in continuous pressure swing distillation (PSD) have been determined by use of two evolutionary techniques. Two industrially relevant systems: ethanol-water and acetonitrile-water have been considered. The Napthali-Sandholm model is solved to obtain the concentration and temperature profiles. The objective is to minimize the total cost using Genetic Algorithm (GA) and Differential Evolution (DE) for the two azeotropic systems. The techniques offer attractive features like applicability to discontinuous and non-differentiable search spaces.

Keywords: Differential Evolution; Genetic Algorithm; Napthali-Sandholm method; Pressure Swing Distillation

Appendix

A.1 Costing and design of the Column

TotalCost=(EnergyCost,C1)+(Depreciation+Interest+Maintenance)×(FixedCost,C2)

The energy costs were calculated according to:

C1=Qr×Cs×ND×24λsteam

Qr Is the reboiler duty, λsteam is the latent heat of evaporation of steam, Cs refers to the steam cost, ND denotes the number of yearly working days.

Cpack=Ac×N×HETP×Cpack0

N is the number of stages, Ac Represents the column area, for area calculations it is necessary to know the velocity.

We choose the velocity to be 80% of the flooding velocity.

uf=K1ρl−ρvρv

FLV=LwVwρvρl

LW=Liquid mass flow rate, kg/s

VW=Vapour mass flow rate, kg/s

uf=Flooding vapour velocity, m/s

K1 Can be found from K1 v/s FLV diagram in Coulson Richardson Chemical engineering, Vol. 6, Page 585 [24].

Ccol=π×1.4×dc×Nst×HETP×Ws×ρs×Csteel

dc, Ws ,ρs and Csteel. Refer to the column diameter, column thickness, density and cost of steel respectively.

Cvacuumpump=1.5Lakhs

A heuristic was taken to consider the fixed cost of the pump, energy calculations were done considering the power necessity of the pump to be 8 units of electricity and at a nominal cost of Rs. 6/unit of Electricity.

Assuming depreciation, interest, and maintenance costs of 18%, 15% and 2%, respectively.

CT=C1+0.35×C2

A.2 Napthali-Sandholm Method

The Napthali-Sandholm method is one of the most robust techniques and is superior in convergence compared to the successive substitution methods but constraints have to put on the iterated variables for smooth convergence. The simple distillation flow diagram is shown in Figure 14.

Figure 14:

Schematic of simple distillation column.

Material Balance Equations:

F(1,j)+L2×x(2,j)+V0×x(1,j)−L1×x(1,j)−V1×y(1,j)=m(1,j) ∀j=1,2....c

F(i,j)+Li+1×x(i+1,j)+Vi−1×y(i−1,j)−Li×x(i,j)−Vi×y(i,j)=m(i,j)

∀j=1,2....c∀i=1,2....n−1

F(n,j)+Ln+1×y(n,j)+Vn−1×x(n−1,j)−Ln×x(n,j)−Vn×y(n,j)=m(n,j)

∀j=1,2....c

Summation Equations:

∑j=1c(K(i,j)−1)×x(i,j)=0(Rachford−RiceEquation)

y(i,j)=K(i,j)×x(i,j)∀i=1,2...nj=1,2...c

K values are predicted using the UNIQUAC model.

The Jacobian of the Method forms a Block-Tridiagonal Matrix of the form:

J=B1C1000A2B2C2000A3⋅⋯000⋅⋯Cn−1000An−1Bn

Where Ai=∂M(i)∂X(i−1), X(i−1)=[x(i,1),x(i−1,2)⋯x(i−1,c),T(i−1)]

M(i)=[m(i,1),m(i−1,2)⋯m(i−1,c),m(i−1,c+1)]

WhereAi=∂m(i,1)∂x(i−1,1)∂m(i,1)∂x(i−1,2)⋯∂m(i,1)∂T(i−1))∂m(i,2)∂x(i−1,1)⋅⋯∂m(i,2)∂T(i−1))⋮⋮⋮⋮ ∂m(i,c)∂x(i−1,1)⋅⋯∂m(i,c)∂T(i−1))

Bi=∂M(i)∂X(i)Ci=∂M(i)∂X(i+1)

JΔX=−m

ΔX=Xnew−Xold

If, the vapour flow rates are assumed to be the constant, the energy equations need not be solved for, The MES or the n×(c+1) form of equations is used for simulations. This system of Non-Linear equations is solved using Thomas Algorithm for Block Tridiagonal matrices. In, general thomas algorithm for tridiagonal matrices has a time complexity O(n) compared to Gauss Elimination with a time complexity of O(n3). Iterations are carried out until tolerance≤10−3.

This iterative scheme is sensitive to the initial Guesses and fails to converge if the initial values are too far from the feasible solution. All simulations have been carried out on MATLAB 2015b. The method discussed before is applicable to simple columns. If the columns are interlinked, then the system no longer remains Tridiagonal [15].

A.3 Summary of fitness functions

Table 7:

Summary of fitness functions tested.

S.No	P	G	w1	w2	w3	w4	r1	r2	Fitness function	Cost	Distillate	Bottom	Converged fitness value
1	15	50	1	10−6	10	7.5	w1∗(1−w4∗(1+ecost∗w2)−1)	w3∗\|dist(1)−0.99\|+w4∗\|bot(2)−0.99\|	r1+r2	3.68∗107	0.897	0.813	1.65
2	30	60	10	10−8	10	-	w1∗(1−(1+ecost∗w2)−1)	w3∗(1+\|dist(1)−0.99\|+\|bot(2)−0.99\|)−1	r1+r2	3.73∗107	0.804	0.731	12.87
3	30	60	10	-	7.5	-	cost	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r1*r2	4.81∗107	0.986	0.827	6.04
4	30	60	-	-	-	-	cost	\|dist(1)−0.99\|∗\|bot(2)−0.99\|	r1*r2	7.80∗107	0.989	0.866	0
5	50	15	10	10−8	7.5	-	ecost∗w2	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r1*r2	8.10∗107	0.989	0.866	0
6	60	30	10	10−8	7.5	-	ecost∗w2	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r1*r2	8.19∗107	0.989	0.866	0
7	60	30	10	10−8	7.5	-	ecost∗w2	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r1+10*r2	3.72∗107	0.987	0.731	1.68
8	30	60	10	10−8	7.5	-	ecost∗w2	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r110r2	7.84∗107	0.989	0.867	0
9	15	30	10	10−8	7.5	-	ecost∗w2	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r110r2	7.93∗107	0.989	0.867	0
10	30	45	10	10−8	7.5	-	ecost∗w2	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r110r2	7.41∗107	0.989	0.865	0
11	45	60	10	10−8	7.5	-	ecost∗w2	w1∗\|dist(1)−0.99\|+w3∗\|bot(2)−0.99\|	r110r2	7.52∗107	0.989	0.865	0

P = Population Size, G = Generation Size

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Received: 2017-3-15

Revised: 2017-7-25

Accepted: 2017-7-25

Published Online: 2017-10-13

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Articles in the same Issue

https://doi.org/10.1515/cppm-2017-0007

Keywords for this article

Differential Evolution; Genetic Algorithm; Napthali-Sandholm method; Pressure Swing Distillation