rigourous methods

7/29/2019 Rigourous Methods

1/12

Rigorous methods for multicomponent

distillation process

General representation of an equilibrium stage

as a thermodynamic system:

Stage j

F

j

F

j

F

jjij TPhzF ,,,, , jQ

11

11,1

,

,,,

jj

L

jjij

TP

hxL

11

11,1

,

,,,

jj

V

jjij

TP

hyV

jj

L

jjij

TP

hxL

,

,,, ,

jj

V

jjij

TP

hyV

,

,,, ,

jU

jW


2/12



1. M: Material balance equations:

0)()( ,,,1,11,1 jijjjijjjijjijjijij yWVxULzFyVxLM

- There should be C equations for each stage

2. E: Equilibrium relations:

0,,

jiijjiij

xKyE

- There should be C equations for each stage


3/12



1. S: Summation equations:

011

,

C

i

jiij ySy

- There should be 2 equations for each stage

2. H: Enthalpy Hbalance:

- There should be 1 equation for each stage

011

,

C

i

jiij xSx

0)()(1111 j

V

jjj

L

jjj

F

jj

V

jj

L

jjj QhWVhULhFhVhLH

MESH equations


4/12



Degrees of freedom analysis:

Total number of equations: N (2C+3)

Unknowns:jijiji yxz ,,, ,, jjjjj

F

j

F

jjjj QWUTPTPFLV ,,,,,,,,,

3C 10

N(3C+10)+1

N itself


5/12



Thermodynamics relations:

jijj

V

j

jijj

L

j

jijijjij

yPTfh

xPTfh

yxPTfK

,

,

,,

,,

,,

,,,

(Once we know temperature, pressure

and composition of a stream we can

calculate thermodynamic properties of the

stream using P-v-T equation of states and

activity coefficients models)


6/12



Design specifications:

jjjj

F

j

F

jjji QWUPTPFzN ,,,,,,,,

C

N(2C+3)Remaining unknowns:


7/12



General solution strategy:

The idea is to rewrite the first 3 of MESH equations so to eliminate

y and L:

jjijjijjij DxCxBxA ,,1,

where

NjzFD

NjKVC

NjKWVUVUWFVB

NjVUWFVA

jijj

jijj

ijjjjmmm

j

m

jj

j

m

mmmjj

1

11

1])()([

2)(

,

1,1

1

1

1

1

1

1

Note that these equations now

contain only V and x as unknowns


8/12




In matrix form:

NNi

i

i

i

NN D

D

D

D

x

x

x

x

BA

CBA

CBA

BA

3

2

1

,

3,

2,

1,

333

222

11

0000...0

0

0

0...000

0...00

0...00

Given V and Tvalues for each stage we can solve

this matrix for each component to obtain

mole fraction of component i in liquid phase

at each stagej (xi,j)

C

i ji

jicorrected

ji

jijj

x

xx

xTV

1 ,

,

,

,),(


9/12




In a similar way we can compose another set of equations using the enthalpy

balance:

jjjjj VV 1where

L

j

V

jj

V

j

L

jj

j

L

j

V

jj

F

j

L

jj

L

j

L

j

j

mmmmj

hh

hh

QhhWhhFhhVUWF

1

1

1

1

11 )()()()(


10/12




In a similar way we can compose another set of equations using the enthalpy

balance:

NNNN

V

V

V

V

V

4

3

222

5

4

3

11

44

23

2

0000.. .0

0

0

0...0000

0...000

0.. .000 Given T, (xi,j), (yi,j)values for each stage we can solve

this matrix for each stage to calculate

vapour flow rate for the stage Vj

jjijij VyxT ),,( ,,


11/12



Complete design strategy:

11, ,,,,,,,,,, LVQWUPTPFzN jjjjF

j

F

jjjiSpecify

1. Guess ),( jj TV

C

i

ji

jicorrected

ji

jijj

x

xx

xTV

1

,

,

,

,),(2. Solvematrix 3. Compute Tj from

the bubble point

calculation.

Calculate yij

jjijij VyxT ),,( ,,4. Solve thesecond matrix

5. New Lj6. Check convergence

If no -> repeat the cycle


12/12

Stage efficiency analysis

Step 1: Thermodynamics data and methods to predict equilibrium

phase compositions

Step 2: Design of equilibrium stage separation

Step 3: Develop an actual design by applying the stage efficiency analysis

to equilibrium stage design

rigourous methods

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