Distillation Calculations with Nonideal Mixtures

8 Distillation Calculations with Nonideal Mixtures a

Downloaded by MICHIGAN STATE UNIV on September 27, 2013 | http://pubs.acs.org Publication Date: August 1, 1974 | doi: 10.1021/ba-1972-0115.ch008

JOSEPH A. BRUNO, JOHN L . YANOSIK, and JOHN W. TIERNEY Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pa. 15213

A new iteration sequence is presented for solving distillation problems with composition dependent equilibrium and enthalpy relations. For a steady state, non-reacting system with m components and n stages, the procedure requires simultaneous iteration of n(m + 1) variables. These are the vapor flow rates, the stage temperatures, and all but one of the liquid compositions. The basic equations for an equilibrium stage system are presented using matrix notation. The calculation sequence is outlined, and a correction algorithm based on Newton's method is derived. This requires the calculation of the Jacobian matrix of partial derivatives, and an analytical method for obtaining these derivatives by vector differentiation of the system equations is presented. This method is much simpler than those used previously. Modification of the iteration process to hold selected vapor flows constant is described, and a method of obtaining starting values for the first iteration is presented. Results obtained from solution of a sample extractive distillation problem are presented. Quadratic convergence is obtained near the solution, indicating that the equations derived for the Jacobian matrix are correct.

^ p h e m a t h e m a t i c a l m o d e l for a steady state e q u i l i b r i u m stage s e p a r a t i o n process consists of a l a r g e set o f s i m u l t a n e o u s n o n l i n e a r e q u a t i o n s w h i c h m u s t b e s o l v e d to d e t e r m i n e the phase flow rates, the stage t e m peratures, a n d t h e phase c o m p o s i t i o n . A m a t r i x n o t a t i o n w a s p r e v i o u s l y p r e s e n t e d (1, 2) w h i c h p e r m i t s w r i t i n g t h e e q u a t i o n s i n a concise f o r m "Present address: A M O C O Production Co., Tulsa, Okla.

122

In Extractive and Azeotropic Distillation; Tassios, D.; Advances in Chemistry; American Chemical Society: Washington, DC, 1974.

8.

BRUNO,

YANOsiK,

A N D

Nonideal

TiERNEY

a n d p r o v i d e s for a n y interstage

flow

123

Mixtures

pattern.

It w a s also s h o w n that

these equations c a n b e a n a l y t i c a l l y differentiated t o o b t a i n iterative algo r i t h m s w i t h q u a d r a t i c convergence rates near the s o l u t i o n . F o r t h e case w h e r e the e q u i l i b r i u m ratios are functions o n l y of stage

temperatures

a n d not of c o m p o s i t i o n s , it w a s s h o w n t h a t q u a d r a t i c convergence

can

be o b t a i n e d b y s i m u l t a n e o u s i t e r a t i o n of o n l y the stage temperatures a n d the v a p o r

flow

rates.

A l l other variables can be obtained b y

solving

n o t h i n g m o r e c o m p l i c a t e d t h a n sets of s i m u l t a n e o u s l i n e a r equations.


O n c e a set of temperatures a n d v a p o r flows are a s s u m e d , t h e equations b e c o m e l i n e a r i n the r e m a i n i n g variables.

T h u s , for a system w i t h η

stages a n d m c o m p o n e n t s , i t is o n l y necessary to iterate o n 2 n v a r i a b l e s rather t h a n the c o m p l e t e set o f n ( 2 m + 3 ) u n k n o w n s . If the e q u i l i b r i u m ratios are functions of phase c o m p o s i t i o n s as occurs i n l i q u i d e x t r a c t i o n or extractive d i s t i l l a t i o n , it is necessary to i n c l u d e m o r e v a r i a b l e s i n the iterative process.

It w a s later s h o w n ( 3 )

t h a t for

l i q u i d e x t r a c t i o n p r o b l e m s w i t h k n o w n stage temperatures, t h e m i n i m u m n u m b e r of i t e r a t i o n variables for q u a d r a t i c convergence is ran, t h e η v a p o r flow rates, a n d n(m v a r i a b l e s is η(2m

— 1) of the phase compositions. T h e t o t a l n u m b e r of + 2 ) because t h e temperatures are k n o w n . T h e i t e r a

t i o n sequence is c o m p l e t e l y different for this case as c o m p a r e d w i t h the p r e v i o u s case w i t h c o m p o s i t i o n i n d e p e n d e n t e q u i l i b r i u m ratios. D e v e l o p e d here is a c o r r e c t i o n process w i t h q u a d r a t i c

convergence

near t h e s o l u t i o n for p r o b l e m s i n w h i c h the e q u i l i b r i u m ratios are c o m p o s i t i o n d e p e n d e n t a n d i n w h i c h the stage temperatures are n o t fixed b u t must be determined.

It is necessary to i n t r o d u c e t h e energy

equations to g i v e the a d d i t i o n a l equations n e e d e d .

balance

T h e derivation fol

l o w s t h e g e n e r a l lines o f that for the l i q u i d e x t r a c t i o n p r o b l e m , b u t t h e extension is not t r i v i a l . T h e m e t h o d r e q u i r e s the s i m u l t a n e o u s c o r r e c t i o n of n(m

-f- 1) variables. W e w i l l also present a s i m p l i f i e d m e t h o d of a n a

l y t i c a l l y differentiating the m a t r i x equations w h i c h g r e a t l y reduces

the

w o r k necessary to d e r i v e a c o r r e c t i o n a l g o r i t h m . I n related convergence

investigations

Roche

(4)

for c o m p o s i t i o n d e p e n d e n t

constant temperature.

has d e m o n s t r a t e d

quadratic

l i q u i d extraction problems

H e essentially iterates o n a l l η(2m

at

+ 2) variables.

N e l s o n ( 5 ) has u s e d the m a t r i x n o t a t i o n of References I a n d 2 t o i n v e s t i gate systems w h e r e c h e m i c a l r e a c t i o n occurs i n the stages. Equations for an Equilibrium The

Separation System

equations d e s c r i b i n g a steady-state

e q u i l i b r i u m stage system

w i t h o u t c h e m i c a l r e a c t i o n are s u m m a r i z e d here. has b e e n p r e v i o u s l y d i s c u s s e d (1,2,3),

T h e matrix notation

a n d a l l s y m b o l s are defined b e l o w .


124

E X T R A C T I V E AND

AZEOTROPIC DISTILLATION

T h e o v e r a l l m a t e r i a l b a l a n c e e q u a t i o n gives a n i m p o r t a n t r e l a t i o n b e t w e e n t h e l i q u i d a n d v a p o r flow rate vectors, V a n d L . BL

+ AV

+ Σ (F*) 3

= Ο

(1)

T h e m a t r i c e s A a n d Β a r e fixed b y t h e interstage flow p a t t e r n a n d together w i t h t h e f e e d m a t r i x F are a s s u m e d to b e g i v e n i n t h e p r o b l e m statement. Downloaded by MICHIGAN STATE UNIV on September 27, 2013 | http://pubs.acs.org Publication Date: August 1, 1974 | doi: 10.1021/ba-1972-0115.ch008

T h e fact that t h e l i q u i d c o m p o s i t i o n s m u s t s u m to a constant (1.0 i f m o l e fractions are u s e d ) i n e a c h stage is g i v e n b y : Σ (ZO = U

(2)

A s i m i l a r e q u a t i o n c a n b e w r i t t e n f o r t h e v a p o r phase, b u t i t is n o t i n d e pendent a n d can be derived b y combining Equations ( 1 ) , ( 2 ) , a n d (3). A m a t e r i a l b a l a n c e c a n b e w r i t t e n f o r e a c h c o m p o n e n t i n e a c h stage, Β L X> + A V F> +

= 0; 1 < j < m

(3)

X a n d Y are vectors c o n t a i n i n g l i q u i d a n d v a p o r c o m p o s i t i o n s f o r c o m j

j

p o n e n t /'. T h e matrices L a n d V are d i a g o n a l a n d h a v e t h e same elements o n t h e d i a g o n a l as t h e vectors L a n d V. A n energy b a l a n c e a r o u n d e a c h stage gives BLH

+ AY

G +

Q +

Q

f

=

(4)

0

H a n d G are vectors c o n t a i n i n g t h e specific enthalpies of l i q u i d a n d v a p o r phases i n e a c h stage.

Q is t h e vector o f stage heat duties, a n d Q is t h e

f e e d e n t h a l p y vector.

Q a n d Q are a s s u m e d to b e g i v e n i n t h e p r o b l e m

f

f

statement. T h e e q u i l i b r i u m r e l a t i o n b e t w e e n l i q u i d a n d v a p o r c o m p o s i t i o n s is given by Λ> Υ

3

-

Γ ' K>" X> = Ο; 1 < j < m ?

(5)

Λ/ a n d Ρ a r e t h e a c t i v i t y coefficient m a t r i c e s for v a p o r a n d l i q u i d phases a n d are d i a g o n a l . F o r i d e a l solutions, t h e y b e c o m e t h e i d e n t i t y m a t r i x . K

;

is t h e f u g a c i t y ratio m a t r i x a n d is also d i a g o n a l . T h e enthalpies a n d e q u i l i b r i u m d a t a are p h y s i c a l properties o f t h e

m i x t u r e s b e i n g separated a n d are a s s u m e d to b e k n o w n e x p l i c i t f u n c t i o n s of c o m p o s i t i o n s a n d temperatures as f o l l o w s :


8.

B R U N O ,

YANOSIK,

A N D

Nonideal

T I E R N E Y

H{X\

G

G(Y ,

Y,

. . Y,

T)

(7)

Γ

Γ(Χ , X,

. . X,

T)

(8)

Λ

Λ ( F , 7», . . F » ,

T)

(9)

3

2

3

. . X",

T)

H

2

X\

125

Mixtures

m

m

2

(6)

K(T)

K

(10)

E a c h of the a b o v e functions is a s s u m e d to b e c o n t i n u o u s a n d to possess


first derivatives w i t h respect to c o m p o s i t i o n a n d t e m p e r a t u r e .

Also, the

c o m p o s i t i o n of c o m p o n e n t 1 is n o t to b e c o n s i d e r e d as a n i n d e p e n d e n t v a r i a b l e i n the d e v e l o p m e n t omitting X The

1

and Y

Calculation

1

w h i c h f o l l o w s , a n d this is e m p h a s i z e d

by

as arguments i n E q u a t i o n s 6 - 9 .

Sequence

E q u a t i o n s 1-10 constitute a set of ( 5 m + 5 ) m a t r i x equations w h i c h are to b e s o l v e d s i m u l t a n e o u s l y to d e t e r m i n e 2 m c o m p o s i t i o n

vectors,

t w o flow vectors, a t e m p e r a t u r e vector, t w o e n t h a l p y vectors, 2 m a c t i v i t y coefficient vectors, a n d m f u g a c i t y r a t i o vectors.

T h e solution must be

iterative because the equations are n o n l i n e a r . H o w e v e r , there are v a r i o u s m e t h o d s of o r g a n i z i n g the c a l c u l a t i o n s , a n d one of o u r p r i m a r y objects here is to d e v e l o p a n efficient c a l c u l a t i o n order. A t one extreme i t w o u l d b e possible to c o n s i d e r a l l 5 m +

5 vectors as i t e r a t i o n v a r i a b l e s a n d thus

to h a v e a process w i t h ( 5 m + 5 ) η v a r i a b l e s of i t e r a t i o n . It seems o b v i o u s that a m o r e efficient m e t h o d is to r e d u c e the size of t h e iterative process b y e l i m i n a t i n g as m a n y of t h e v a r i a b l e s as possible.

I n p r a c t i c e , it is

u n d e s i r a b l e a c t u a l l y to e l i m i n a t e t h e v a r i a b l e s ; i n s t e a d the v a r i a b l e s w i l l b e d i v i d e d i n t o t w o groups.

T h e first g r o u p is the v a r i a b l e s w h i c h w i l l

b e i t e r a t e d a n d w i l l b e c a l l e d t h e i t e r a t e d or i n d e p e n d e n t v a r i a b l e s ; t h e second group w i l l be called dependent

variables.

One equation must

b e u s e d t o define e a c h d e p e n d e n t v a r i a b l e , a n d these equations w i l l b e c a l l e d the d e f i n i n g equations. T h e r e m a i n i n g equations, e q u a l i n n u m b e r to t h e i n d e p e n d e n t variables, w i l l b e c a l l e d the error equations.

The

i t e r a t i o n sequence w i l l t h e n b e to assume first a set of values for the i n d e p e n d e n t variables a n d t h e n to solve t h e d e f i n i n g equations for the de p e n d e n t variables.

T h e error equations are t h e n u s e d to correct

the

i n d e p e n d e n t v a r i a b l e s , a n d the process is repeated. T h e n u m b e r of i n d e p e n d e n t variables s h o u l d g e n e r a l l y b e as s m a l l as possible. A l s o , the d e f i n i n g equations s h o u l d p e r m i t s i m p l e c a l c u l a t i o n of the d e p e n d e n t variables, a n d t h e c o r r e c t i o n process itself s h o u l d b e s i m p l e a n d efficient.

H e r e the set of i n d e p e n d e n t v a r i a b l e s has b e e n

c h o s e n to b e as s m a l l as possible subject t o t h e l i m i t a t i o n t h a t the d e f i n i n g equations

r e m a i n l i n e a r i n the

dependent

variables.

The

correction


126

EXTRACTIVE

AND AZEOTROPIC

DISTILLATION

a l g o r i t h m is also l i n e a r a n d is t h e m u l t i v a r i a b l e f o r m o f t h e N e w t o n Raphson method. T h e set o f i n d e p e n d e n t v a r i a b l e s w h i c h satisfies these r e q u i r e m e n t s is the v a p o r flow v e c t o r V , the l i q u i d p h a s e c o m p o s i t i o n s , X t o X , a n d 2

m

the t e m p e r a t u r e v e c t o r T . T h e c a l c u l a t i o n sequence is, a. V a l u e s are a s s u m e d f o r ( V , X , X , . . X , T ) . 2

3

W

b. T h e l i q u i d flow vector L is c a l c u l a t e d u s i n g E q u a t i o n 1. c. E q u a t i o n 2 is u s e d to c a l c u l a t e X . 1

d . E q u a t i o n 3 is u s e d m times t o c a l c u l a t e Y ^ ,.. Y . e. E q u a t i o n s 6 - 1 0 a r e u s e d t o c a l c u l a t e t h e d e p e n d e n t v a r i a b l e s H, G, Λ, Γ, a n d K . A l l v a r i a b l e s h a v e n o w b e e n e v a l u a t e d f o r the c u r r e n t set o f i n d e p e n d e n t v a r i a b l e s . f. C u r r e n t values f o r a l l v a r i a b l e s are s u b s t i t u t e d i n t o the error e q u a tions, w h i c h a r e E q u a t i o n s 4 a n d 5. I f t h e equations a r e satisfied, t h e i t e r a t i v e process is t e r m i n a t e d . O t h e r w i s e , t h e i n d e p e n d e n t v a r i a b l e s m u s t b e c o r r e c t e d , a n d the c a l c u l a t i o n r e p e a t e d f r o m Step b .


1

The Jacobian

Correction

2

m

Matrix

T h e o n l y u n d e f i n e d step i n t h e c a l c u l a t i o n sequence p r o p o s e d a b o v e is t h e c o r r e c t i o n o f the i n d e p e n d e n t v a r i a b l e s i n step f. A l i n e a r correc t i o n process is g i v e n b y ( J ) v { ( C )

v

+

1

-

(C)v}

=

-

(11)

(D)v

w h e r e C is t h e d i r e c t s u m o f t h e i n d e p e n d e n t v a r i a b l e s i n t h e o r d e r ( V , X , X , . . X , T ) , a n d D is the d i r e c t s u m o f the error vectors, defined as 2

3

m

E

k

= AY k

k

-

Γ*Κ*Χ*; 1 < k < m

= BL H + AVG T h e most r a p i d convergence

+ Q+ Q

f

(12) (13)

o f E q u a t i o n 11 t o t h e s o l u t i o n is o b

t a i n e d w h e n J is t h e J a c o b i a n m a t r i x , defined as t h e m a t r i x i n w h i c h e a c h element is t h e p a r t i a l d e r i v a t i v e o f o n e of t h e errors w i t h respect to o n e o f t h e i t e r a t i o n v a r i a b l e s w i t h a l l other i t e r a t i o n v a r i a b l e s h e l d constant. T h u s , a n estimate o f the effect o f e a c h c h a n g e i n a n i t e r a t i o n v a r i a b l e o n e a c h o f t h e errors i s i n c l u d e d i n t h e c o r r e c t i o n . T h e d i s a d v a n t a g e i n u s i n g t h e J a c o b i a n m a t r i x is t h e large n u m b e r o f d e r i v a t i v e s n e e d e d , (mn + n ) , a n d a s i m p l e means o f o b t a i n i n g these d e r i v a t i v e s is 2

needed.

W e h a v e f o u n d that v e c t o r differentiation o f t h e m a t r i x e q u a

tions g i v e n a b o v e does g r e a t l y s i m p l i f y t h e d e r i v a t i o n of t h e e q u a t i o n s for c a l c u l a t i o n o f t h e J a c o b i a n . B y vector differentiation w e m e a n t h e o p e r a t i o n o f differentiating one v e c t o r w i t h respect to a s e c o n d vector.

T h e result is a m a t r i x i n

w h i c h e a c h c o l u m n is t h e d e r i v a t i v e o f t h e first v e c t o r w i t h respect t o


8.

B R U N O ,

YANOSiK,

Nonideal

A N D T B E R N E Y

127

Mixtures

one o f t h e elements o f t h e s e c o n d vector. F o r e x a m p l e ( dL/dV ) is a n η b y η m a t r i x i n w h i c h t h e j t h c o l u m n is (dL/dVj),

a n d t h e ij element is

(dk/dv,). J c a n n o w b e defined b y v e c t o r derivatives. p a r t i t i o n e d into (m + l )

2

F i r s t , h o w e v e r , i t is

s u b m a t r i c e s , e a c h o f size η b y n . S u b s c r i p t s

are u s e d t o designate t h e submatrices.

Thus, from the definition of the


Jacobian matrix, w e obtain

w ' ' * J* =

{ ~ ; 2

f^;k

1

(

4

)

(15)

< k 1

Distillation Calculations with Nonideal Mixtures

Recommend Documents