Energy Computations for Saline Water Conversion by Idealized

Energy Computations for Saline Water Conversion by Idealized Freezing Processes Downloaded by UNIV OF NEW SOUTH WALES on September 5, 2015 | http://pubs.acs.org Publication Date: January 1, 1960 | doi: 10.1021/ba-1960-0027.ch008

HENRY M. CURRAN St. Edward's University, Austin 4, Tex.

Theoretical energy equations are developed for saline water conversion by means of idealized freezing processes utilizing reversible refrigeration machines. These equations may be used to compute the minimum energy required for saline water conversion by a freezing process at specified values of extraction ratio, ambient saline water temperature, and heat exchange temperature differentials. Minimum energy values so computed provide criteria for feasibility analyses and for comparison of actual freezing processes.

Among

t h e m o r e i m p o r t a n t e c o n o m i c f a c t o r s i n v o l v e d i n large-scale c o n v e r s i o n o f saline w a t e r are t h e a m o u n t a n d cost o f t h e e n e r g y necessary f o r o p e r a t i o n o f the c o n v e r s i o n processes. T h e p u r p o s e o f t h i s p a p e r i s t o e s t a b l i s h t h e t h e o r e t i c a l e n e r g y r e q u i r e d b y c e r t a i n i d e a l i z e d c o n v e r s i o n processes u t i l i z i n g f r e e z i n g . T h e theoretical amount of energy required f o r the reversible p a r t i t i o n of a m u l t i c o m p o n e n t s y s t e m h a v i n g a p a r t i c u l a r c o m p o n e n t r a t i o i s e q u a l t o t h e c h a n g e i n free e n e r g y c o r r e s p o n d i n g t o t h i s p a r t i t i o n . I n t h e case o f saline w a t e r c o n v e r s i o n , t h e t h e o r e t i c a l a m o u n t o f e n e r g y r e q u i r e d i s e q u a l t o t h e change i n free e n e r g y c o r r e s p o n d i n g to the reversible p a r t i t i o n of saline water i n t o potable w a t e r a n d concentrated b r i n e . T h i s t h e o r e t i c a l e n e r g y is a f u n c t i o n of t h e e x t r a c t i o n r a t i o — i . e . , t h e p o t a b l e w a t e r o u t p u t d i v i d e d b y t h e p o t a b l e w a t e r f r a c t i o n o f t h e saline w a t e r i n p u t — a n d a p p r o a c h e s i t s a b s o l u t e m i n i m u m v a l u e as t h e e x t r a c t i o n r a t i o a p p r o a c h e s zero. B e c a u s e a n y a c t u a l c o n v e r s i o n process m u s t n e c e s s a r i l y b e t h e r m o d y n a m i c a l l y i r r e v e r s i b l e , t h e a c t u a l e n e r g y r e q u i r e m e n t f o r a specified e x t r a c t i o n r a t i o m u s t a l w a y s exceed t h e t h e o r e t i c a l e n e r g y r e q u i r e m e n t f o r a r e v e r s i b l e process a t t h e same e x t r a c t i o n r a t i o . I n t h i s p a p e r i d e a l i z e d c o n v e r s i o n processes u t i l i z i n g r e v e r s i b l e r e f r i g e r a t i o n m a chines a r e a n a l y z e d w i t h respect t o e n e r g y r e q u i r e m e n t s . F i r s t , e n e r g y e q u a t i o n s a r e e s t a b l i s h e d f o r a c o m p l e t e l y r e v e r s i b l e process. S e v e r a l i r r e v e r s i b l e m o d i f i c a t i o n s o f t h i s process are t h e n a n a l y z e d .

The Freezing of Saline Water T h e use o f f r e e z i n g f o r s a l i n e w a t e r c o n v e r s i o n i s b a s e d o n t h e p h y s i c a l p r i n c i p l e t h a t ice c r y s t a l s o b t a i n e d b y f r e e z i n g a n a q u e o u s s a l t s o l u t i o n are p u r e w a t e r i n t h e s o l i d 56

In SALINE WATER CONVERSION; Advances in Chemistry; American Chemical Society: Washington, DC, 1960.

57

CURRAN—COMPUTATIONS FOR IDEALIZED FREEZING

phase. B e c a u s e of the c o m m e n s u r a t e densities of t h e i c e c r y s t a l s a n d the r e s i d u a l b r i n e , there is n o a u t o m a t i c separation of the pure water phase f r o m the b r i n e corresponding t o t h e s e p a r a t i o n o f v a p o r i n d i s t i l l a t i o n processes. T h u s saline w a t e r c o n v e r s i o n b y f r e e z i n g i n v o l v e s t h e t w o b a s i c o p e r a t i o n s o f f o r m a t i o n o f ice c r y s t a l s a n d s e p a r a t i o n o l these c r y s t a l s f r o m t h e r e s i d u a l b r i n e . I n g e n e r a l , these a r e d i s t i n c t o p e r a t i o n s , b u t i t i s p h y s i c a l l y possible t o effect t h e m s i m u l t a n e o u s l y . P e r f e c t s e p a r a t i o n of i c e a n d b r i n e i s a s s u m e d i n e a c h o f t h e processes discussed i n t h i s p a p e r .

Downloaded by UNIV OF NEW SOUTH WALES on September 5, 2015 | http://pubs.acs.org Publication Date: January 1, 1960 | doi: 10.1021/ba-1960-0027.ch008

D e p e n d i n g o n t h e f r e e z i n g process, t h e f r e e z i n g of saline w a t e r p r o d u c e s one o r t h e o t h e r of t w o basic ice f o r m a t i o n s : 1. I c e - b r i n e s l u s h , e s s e n t i a l l y a s l u r r y o f ice c r y s t a l s i n b r i n e , t h e o v e r - a l l s a l i n i t y r a n g i n g f r o m t h a t o f t h e feed w a t e r d o w n t o a b o u t one h a l f o f t h i s v a l u e . 2. I c e - b r i n e solids c o n s i s t i n g of c o n n e c t e d c r y s t a l l a t t i c e s c o n t a i n i n g b r i n e i n i n t e r c r y s t a l l i n e spaces, t h e o v e r - a l l s a l i n i t y r a n g i n g f r o m t h a t of t h e feed w a t e r d o w n t o p o table values. T w o basic freezing methods m a y be distinguished : 1. I n d i r e c t f r e e z i n g i n w h i c h t h e h e a t o f c r y s t a l l i z a t i o n i s r e m o v e d f r o m t h e saline solutions t h r o u g h a solid barrier. 2. D i r e c t f r e e z i n g i n w h i c h t h e h e a t o f c r y s t a l l i z a t i o n i s r e m o v e d b y p a r t i a l e v a p o r a t i o n of t h e s o l v e n t , o r b y c o n t a c t i n g t h e s o l u t i o n w i t h a n i m m i s c i b l e r e f r i g e r a n t . D e p r e s s i o n of F r e e z i n g T e m p e r a t u r e . O n e of t h e c o l l i g a t i v e p r o p e r t i e s of s o l u t i o n s o f n o n v o l a t i l e solutes i s t h a t t h e f r e e z i n g t e m p e r a t u r e i s l o w e r t h a n t h a t of t h e p u r e s o l v e n t . T h e d e p r e s s i o n of t h e f r e e z i n g t e m p e r a t u r e i s a p p r o x i m a t e l y p r o p o r t i o n a l t o the mass ratio of solute t o s o l v e n t — t h a t is, AT =

-KZ

(1)

i n w h i c h Κ i s a c o n s t a n t o f p r o p o r t i o n a l i t y a n d Ζ i s t h e m a s s r a t i o o f solute t o s o l v e n t . F o r sea w a t e r , w h i c h i s t h e p r i n c i p a l saline w a t e r b e i n g c o n s i d e r e d f o r c o n v e r s i o n , t h e d e p r e s s i o n o f t h e f r e e z i n g t e m p e r a t u r e i s closely a p p r o x i m a t e d b y t a k i n g Κ = 52.41 f o r t h e K e l v i n scale (1 ) o r Κ = 94.34 f o r t h e R a n k i n e scale. T a k i n g s as t h e m a s s f r a c t i o n of salt i n t h e saline w a t e r , w e m a y w r i t e

w h e r e Τ i s t h e e q u i l i b r i u m t e m p e r a t u r e f o r i c e i n c o n t a c t w i t h saline w a t e r h a v i n g a s a l t m a s s f r a c t i o n s, a n d T i s t h e f r e e z i n g t e m p e r a t u r e o f p u r e w a t e r . Ice F r a c t i o n . T h e m a s s f r a c t i o n , / , of i c e p r o d u c e d b y c o o l i n g a u n i t m a s s of saline w a t e r h a v i n g a n i n i t i a l s a l t m a s s f r a c t i o n s f r o m i t s f r e e z i n g t e m p e r a t u r e , T t o a l o w e r t e m p e r a t u r e , T, i s 0

i

s

i}

(4)

w h e r e s is the solute mass f r a c t i o n i n t h e r e s i d u a l s o l u t i o n a t t e m p e r a t u r e Τ. B y c o m b i n a t i o n of E q u a t i o n s 3 a n d 4, / i s o b t a i n e d as a f u n c t i o n of Τ : (5) E x t r a c t i o n R a t i o . I n t h e i d e a l i z e d processes discussed t h e s a l i n e w a t e r i s a s s u m e d c o o l e d t o a final t e m p e r a t u r e , T , a n d t h e r e s u l t i n g i c e i s a s s u m e d t o b e p e r f e c t l y s e p a r a t e d f r o m t h e r e s i d u a l b r i n e . W h e n m e l t e d , t h i s ice b e c o m e s t h e f r e s h w a t e r p r o d u c t . f


58

ADVANCES IN CHEMISTRY SERIES

T h u s t h e e x t r a c t i o n r a t i o , r, defined as t h e m a s s r a t i o o f f r e s h w a t e r p r o d u c t t o t h e w a t e r c o n t e n t of t h e saline w a t e r i n p u t , i s d i r e c t l y p r o p o r t i o n a l t o t h e final ice f r a c t i o n ,

1 -

si

T 0

(6)

T

f

Reversible Refrigeration Machines


A l l of t h e idealized conversion reversible refrigeration machines.

processes discussed

are assumed

to incorporate

q+w _T -T q

c

Figure I. Type I reversible refrigeration machine T y p e I . F i g u r e 1 i l l u s t r a t e s r e v e r s i b l e r e f r i g e r a t i o n m a c h i n e s of T y p e I . T h i s m a c h i n e receives a n a m o u n t o f h e a t q r e v e r s i b l y a t t e m p e r a t u r e T a n d discharges i t r e v e r s i b l y a t a h i g h e r t e m p e r a t u r e , T . B y t h e first p r i n c i p l e o f t h e r m o d y n a m i c s t h e necessary e n e r g y i n p u t , w, m u s t also b e d i s c h a r g e d r e v e r s i b l y a t T . T h e coefficient o f p e r f o r m a n c e of t h i s m a c h i n e i s t h e same as t h a t of a C a r n o t r e f r i g e r a t i o n m a c h i n e . T h u s we have c

a

a

cp.

T T - T c

W

a

(7) c

w h i c h yields w — q

Τg - T T

c

(8)

c

q+w

For uniform discharge of ^ over Tjjto Τ , α

Figure 2.

Type II reversible refrigeration machine


59


T y p e I I . F i g u r e 2 i l l u s t r a t e s r e v e r s i b l e r e f r i g e r a t i o n m a c h i n e s of T y p e I I . T h i s m a c h i n e receives a n a m o u n t o f h e a t q r e v e r s i b l y a t t e m p e r a t u r e T a n d discharges i t r e v e r s i b l y o v e r a h i g h e r t e m p e r a t u r e r a n g e f r o m T t o T . T a k i n g Τ as a n y t e m p e r ature between T and T a n d assuming a T y p e I machine operating between T a n d Τ w h i c h receives a d i f f e r e n t i a l a m o u n t o f h e a t dq a t T , w e h a v e t h e d i f f e r e n t i a l e q u a t i o n , c

b

b

a

a

c

c

dw = dq

Τ -

T

c

(9)

Tc

M a k i n g the f u r t h e r a s s u m p t i o n t h a t the discharge of q over the range T t o T is e x p r e s s i b l e as a f u n c t i o n q(T), t h e e n e r g y i n p u t is b

a

Τ - T dT T

(10)


c

c

I f q is a l i n e a r f u n c t i o n o f Τ—that — t h e energy is _

9 T . - n J

is, q is discharged u n i f o r m l y over the range T t o T b

a

Γ*a Τ - T dT T c

T

(ID

c

b

E q u a t i o n I I differs f r o m E q u a t i o n 8 f o r a T y p e I m a c h i n e o n l y i n h a v i n g t h e a v e r age o f T a n d T i n t h e p l a c e o f T . T y p e I I I . F i g u r e 3 i l l u s t r a t e s r e v e r s i b l e r e f r i g e r a t i o n m a c h i n e s of T y p e I I I . T h i s a

b

a

q'(T)

For uniform reception of q over

Figure 3.

toT t c

Type III reversible refrigeration machine

m a c h i n e receives a n a m o u n t o f h e a t q r e v e r s i b l y o v e r a t e m p e r a t u r e r a n g e f r o m T t o T a n d discharges i t r e v e r s i b l y a t t e m p e r a t u r e T . T a k i n g Τ as a t e m p e r a t u r e i n t h e r a n g e T t o T , a n d a s s u m i n g a T y p e I m a c h i n e o p e r a t i n g b e t w e e n Τ a n d T w h i c h receives a d i f f e r e n t i a l a m o u n t o f h e a t dq a t T, t h e d i f f e r e n t i a l e q u a t i o n f o r e n e r g y i n p u t i s d

c

a

d

c

a

T - Τ

dw =

(12)

a

A s s u m i n g t h a t t h e r e c e p t i o n o f η o v e r t h e range Τ t o T i s expressible as a f u n c t i o n q(T), t h e e n e r g y i n p u t i s d

c

(13)

%) Τ d I f g i s a l i n e a r f u n c t i o n o f Τ—that — t h e energy required is

=

τ^τγ,

J*j/

'^Ύ

1

is, q is received u n i f o r m l y over the range T t o T d

d

T

-

T^TTt

[

T

°

l

n

S

-

(

T

c

-

T d )

]


c

(14)



For uniform reception α d i s c h a r g e of q ,

Figure 4. Type IV reversible refrigeration machine

w,+w

2

Te-Tc «ι

-

Ά

Capacity =

w.-

Figure 5. Combination of two Type I re versible refrigeration machines in series to utilize a limited isothermal reservoir In SALINE WATER CONVERSION; Advances in Chemistry; American Chemical Society: Washington, DC, 1960.

61


T y p e I V . F i g u r e 4 i l l u s t r a t e s r e v e r s i b l e r e f r i g e r a t i o n m a c h i n e s of T y p e I V . T h i s t y p e o f m a c h i n e receives a n a m o u n t o f h e a t q r e v e r s i b l y o v e r a t e m p e r a t u r e range T t o T a n d d i s c h a r g e s i t r e v e r s i b l y o v e r a h i g h e r r a n g e , T t o T . T h i s may be c o n s i d e r e d as a c o m b i n a t i o n o f a T y p e I I m a c h i n e w i t h a T y p e I I I m a c h i n e . Limiting the present d i s c u s s i o n t o t h e s p e c i a l case o f u n i f o r m d i s c h a r g e o f o v e r t h e r a n g e T t o T , i t m a y also b e c o n s i d e r e d as a T y p e I I I machine w i t h d i s c h a r g e at /^{T + T ) i n place o f T . T h e e n e r g y i s t h e n g i v e n b y t h e c o r r e s p o n d i n g m o d i f i c a t i o n o f E q u a t i o n 13 as d

c

b

a

b

x

a

a

b

a

^

'

m

«

^

h

i

f

as)

f

F o r t h e case o f q r e c e i v e d u n i f o r m l y o v e r t h e range T t o T , m o d i f i c a t i o n of E q u a t i o n 14


d

c

gives =

w

T

c

Î

T

[

l/2(Ta

d

+

T

b

) l

n

Y " d

{

T

c

"

]

Td)

(

1

6

)

C o m b i n a t i o n s of Reversible R e f r i g e r a t i o n M a c h i n e s . R e v e r s i b l e r e f r i g e r a t i o n m a c h i n e s m a y also b e u s e d i n c o m b i n a t i o n i n o r d e r t o effect t h e u t i l i z a t i o n o f h e a t r e s e r v o i r s o f l i m i t e d c a p a c i t y . F i g u r e 5 i l l u s t r a t e s t w o m a c h i n e s o f T y p e I i n series t o u t i l i z e a n i s o t h e r m a l r e s e r v o i r a t T having a c a p a c i t y j u s t e q u a l t o q,. T h e first machine receives q a t T a n d d i s c h a r g e s q at T , t h e e n e r g y i n p u t b e i n g w . B e c a u s e the heat r e j e c t i o n c o r r e s p o n d i n g t o w c a n n o t be a c c o m p l i s h e d a t T , a s e c o n d m a c h i n e receives w a t T a n d discharges i t a t T . F r o m E q u a t i o n 8 w e h a v e e

c

e

x

Y

Y

e

e

a

W l

ΎΓ

=

5

Έ±

«

W

l

(

1

7

)

and W 2

(18)

E q u a t i o n 18 also a p p l i e s w h e n t h e first m a c h i n e i s o f T y p e I I I , w t h e n b e i n g o b t a i n e d f r o m E q u a t i o n 13 or 14. [ A n o t h e r w a y of o b t a i n i n g t h e s a m e r e s u l t w o u l d b e b y t h e use o f T y p e I m a c h i n e s i n p a r a l l e l , o n e r e c e i v i n g q(T /T ) a t T a n d discharging at T , t h e other receiving (1 - T /T ) a t T a n d d i s c h a r g i n g a t T .] l

c

c

e

e

c

c

e

a

First Idealized Process T h e first i d e a l i z e d process, i l l u s t r a t e d i n F i g u r e 6, d e v i a t e s f r o m c o m p l e t e r e v e r s i b i l i t y o n l y i n m i n o r s i m p l i f y i n g a s s u m p t i o n s . S a l i n e w a t e r enters t h e process a t t e m p e r a t u r e T . ( T h i s i s a s s u m e d t o b e t h e l o w e s t t e m p e r a t u r e t o w h i c h t h e saline w a t e r c a n be c o o l e d b y h e a t exchange w i t h t h e coldest m e d i u m i n t h e e n v i r o n m e n t . T h e c o n d i t i o n T > T i s also assumed.) T h e i n c o m i n g saline water is t h e n cooled b y perfect c o u n t e r c u r r e n t h e a t exchange w i t h t h e f r e s h w a t e r p r o d u c t a n d t h e w a s t e b r i n e d o w n t o t h e m e l t i n g t e m p e r a t u r e o f ice, T , t h e m e a n specific h e a t - t e m p e r a t u r e f u n c t i o n o f t h e effluent s t r e a m s b e i n g a s s u m e d e q u a l t o t h e specific h e a t - t e m p e r a t u r e f u n c t i o n o f t h e i n c o m i n g saline water. ( T h i s a s s u m p t i o n introduces a n error of negligible magnitude i n t h e e n e r g y e q u a t i o n s t o follow.) F u r t h e r c o o l i n g t o t h e i n i t i a l f r e e z i n g t e m p e r a t u r e , T i s a c c o m p l i s h e d b y p e r f e c t c o u n t e r c u r r e n t h e a t exchange w i t h i c e a n d w a s t e b r i n e , supplemented b y a refrigeration machine, R , this a u x i l i a r y refrigeration being necessary because t h e specific h e a t o f ice i s o n l y a b o u t one h a l f t h a t o f w a t e r . u

u

0

0

iy

z

B e l o w t e m p e r a t u r e T changes o f e n t h a l p y m a y b e c o n v e n i e n t l y a s s u m e d t o b e c o m p o s e d o f t w o c o m p o n e n t s , o n e c o r r e s p o n d i n g t o t h e sensible c o o l i n g o f t h e s a l i n e water, a n d t h e other c o r r e s p o n d i n g to the removal o f l a t e n t h e a t o f c r y s t a l l i z a t i o n t o f o r m ice c r y s t a l s . T h u s , the sensible c o o l i n g b e l o w T i s a c c o m p l i s h e d b y f u r t h e r p e r fect c o u n t e r c u r r e n t h e a t exchange s u p p l e m e n t e d b y r e f r i g e r a t i o n m a c h i n e R , a n d h e a t of c r y s t a l l i z a t i o n i s r e c e i v e d b y a n o t h e r r e f r i g e r a t i o n m a c h i n e , R . x

t

s

{



62 Brine

Saline Water

Fresh Water

y-

I


Relative Mass: ~ - -1

Figure 6.

Schematic diagram of first idealized process

A s c r y s t a l s o f ice are f o r m e d , t h e y are p e r f e c t l y s e p a r a t e d f r o m t h e r e s i d u a l b r i n e a n d passed t o a m e l t i n g t a n k , b e i n g r a i s e d t o t h e m e l t i n g t e m p e r a t u r e , T , e n r o u t e b y t h e p r e v i o u s l y m e n t i o n e d c o u n t e r c u r r e n t h e a t e x c h a n g e w i t h t h e i n c o m i n g saline w a t e r . F r e e z i n g a n d r e m o v a l of ice are continued t o a final temperature T , a t w h i c h p o i n t t h e b r i n e a t t a i n s i t s h i g h e s t c o n c e n t r a t i o n a n d i s d i s p o s e d o f as w a s t e , a f t e r b e i n g raised back t o temperature T b y absorbing heat f r o m the i n c o m i n g saline water. R e f r i g e r a t i o n m a c h i n e iuj t r a n s f e r s t h e h e a t o f c r y s t a l l i z a t i o n f r o m t h e f r e e z i n g t e m p e r a t u r e r a n g e t o T , w h e r e i t i s a b s o r b e d b y t h e i c e a c t i n g as a n i s o t h e r m a l h e a t sink. T h e water obtained b y melting t h e ice is then raised t o temperature T b y a b s o r b i n g h e a t f r o m t h e i n c o m i n g saline w a t e r b e f o r e b e i n g d i s c h a r g e d as t h e f r e s h water product. F o r steady-state operation, the capacity of the ice i n t h e m e l t i n g unit t o absorb heat i s o t h e r m a l l y i s j u s t e q u a l t o t h e h e a t o f c r y s t a l l i z a t i o n r e m o v e d i n f r e e z i n g t h e ice. T h u s a n o t h e r r e f r i g e r a t i o n m a c h i n e , R , i s e m p l o y e d t o t r a n s f e r t h e a d d i t i o n a l heat r e j e c t e d b y R t o t h e saline w a t e r m a s s , w h i c h , i n t h i s i d e a l i z a t i o n , m a y b e c o n s i d e r e d as h a v i n g i n f i n i t e h e a t - a b s o r b i n g c a p a c i t y a t i t s a m b i e n t t e m p e r a t u r e , T . R e f r i g e r a t i o n m a c h i n e R i s o f T y p e I I I , r e c e i v i n g heat o f c r y s t a l l i z a t i o n o v e r t h e r a n g e Τ τ t o T a n d d i s c h a r g i n g i t i s o t h e r m a l l y a t T . I n o r d e r t o u t i l i z e E q u a t i o n 13 for the energy i n p u t , the rate of heat reception w i t h respect t o temperature m u s t be known. F o r a u n i t m a s s o f ice f o r m e d o v e r t h e t e m p e r a t u r e r a n g e f r o m T t o T t h e a m o u n t of h e a t o f c r y s t a l l i z a t i o n r e c e i v e d b y R b e t w e e n some i n t e r m e d i a t e t e m p e r a t u r e Τ a n d 0

f

u

0

u

2

1

u

x

t

0

f

i

Y

TVs


63


if i n w h i c h L i s t h e h e a t of c r y s t a l l i z a t i o n p e r u n i t m a s s of ice. S i n c e / i s a f u n c t i o n of T, w i t h a n e g a t i v e first d e r i v a t i v e , w i t h i n t h e r a n g e T t o T t h e d i f f e r e n t i a l e q u a t i o n f o r the r a t e o f r e c e p t i o n o f h e a t o f c r y s t a l l i z a t i o n i s c

f

dL = -

i

(20)

^T(T)dT if


E q u a t i o n 13 m a y n o w b e m o d i f i e d t o g i v e f? Ι ' JTf

Wi = -

I XT) ^

™

7

(21)

dT

T

Wi b e i n g t h e e n e r g y i n p u t t o R p e r u n i t m a s s of i c e f o r m e d o v e r t h e r a n g e Tt t o T . E q u a t i o n 5 gives I(T). S u b s t i t u t i o n of t h e d e r i v a t i v e i n t o E q u a t i o n 21 y i e l d s {

x

_

L

K

si

C

C

% TT i i

(T

dT -

T)

0

(22) β

L Sj

K

Ti(T -

c

T)

0

I T.

f

Tf(T

f

-

0

Ti)

A t t h e condition of t h e extraction ratio a p p r o a c h i n g zero, corresponding t o t h e p r o d u c t i o n of a u n i t mass of fresh water f r o m a n infinite mass of saline w a t e r , t h e v a l u e o f I also a p p r o a c h e s z e r o , so t h a t E q u a t i o n s 21 a n d 2 2 m a y n o t b e u s e d . I n t h i s case, h o w e v e r , R b e c o m e s a T y p e I m a c h i n e a n d t h e e n e r g y i n p u t i s g i v e n b y m o d i f i c a t i o n o f E q u a t i o n 8, f

x

Wt -

L

c

T

o

~ \r^0,

(23)

If-^0

T

R e f r i g e r a t i o n m a c h i n e R i s o f T y p e I , so t h a t t h e e n e r g y i n p u t i s 2

Wi - W,

T

(24)

~ °

u

T

A s p r e v i o u s l y n o t e d , r e f r i g e r a t i o n m a c h i n e R is n e e d e d t o a u g m e n t t h e sensible c o o l i n g o b t a i n e d b y h e a t e x c h a n g e . T a k i n g as a b a s i s t h e p r o d u c t i o n o f a u n i t m a s s o f ice, t h e f r a c t i o n o f a m a s s u n i t of s a l i n e w a t e r w h i c h c a n n o t b e c o o l e d b y c o u n t e r c u r r e n t h e a t exchange i s (1 — Cj/C ) i n t h e r a n g e T t o T a n d (1 — 0 /0 ) ( 1 — / / / / ) i n t h e r a n g e T t o T C a n d C b e i n g t h e specific h e a t s o f i c e a n d s a l i n e w a t e r , r e s p e c t i v e l y . T h e s e specific heats m a y b e a s s u m e d c o n s t a n t w i t h n e g l i g i b l e e r r o r . T h e f a c t o r (1 — / / / / ) a c c o u n t s f o r t h e decrease i n t h e b r i n e m a s s as i c e i s f o r m e d . s

9

f

if

§

i

0

§

8

s

T h e d i f f e r e n t i a l e q u a t i o n f o r sensible h e a t r e c e i v e d b y R i s 3

dHs = (1 = (Cs -

C,-/C.)(l -

/ / / / ) C dT s

C,-)(l - I/If) dT, I = 0 when Τ > Ti

(25)

a n d t h e e n e r g y i n p u t is

(C, - C,)

τ , 1η ψ -

(Γ. -

Τ,) -

ι r:

Γ^ί τ \_ ΚΤ) ^

— τ Ϊ Γ ^

d

T

I

(2 ) β

T h e i n t e g r a l o n t h e r i g h t of E q u a t i o n 2 6 m a y b e e v a l u a t e d b y s u b s t i t u t i n g t h e f u n c t i o n o f E q u a t i o n 5 f o r I(T). Thus,


64


- (Τ, -

(1 - «,) [ ΐ ' . g

Κ „ [ £ t o j i g ' l g

Γ,)] -

-

In

(27)

Substitution of this i n E q u a t i o n 26 gives W


3

- (C. -

6',) £ r . l n ^

-

T/) -

(Γ. -

i |(1 - « , ) ( ϊ ' « In J - ' -

(Γ, -

Γ,)) -

T h e total energy required for the operation of the first idealized process is the sum W = TTi + W + Wz

(29)

2

In this idealization the formation and removal of ice are assumed to be simultane ous, so that the mass flow rate decreases throughout the transition from the initial freezing temperature, T to the final temperature, T . T h e above computation of the oretical energy is not dependent upon removal of the ice in this manner, however. Ice formed at temperatures above T could, for example, remain i n contact with the brine and be cooled to T before separation, the cooling being accomplished b y perfect countercurrent heat exchange with previously formed ice. Because this process is essentially reversible, the equations above provide a means of computing the minimum theoretical energy as a function of extraction ratio for saline water conversion. iy

f

f

f

Second Idealized Process T h e second idealized process, illustrated i n Figure 7, differs from the preceding process only i n the freezing arrangement. I n this case the incoming saline water is cooled to temperature T as before ; then it is mixed with a large volume of brine at the exit concentration and the corresponding freezing temperature, Tr. Refrigeration machine R which is of T y p e I, receives heat of crystallization at Tt so as to form ice at a rate corresponding to the mass flow rate of the incoming saline water and the ex traction ratio. A s R and R are T y p e I machines i n series, the theoretical energy inputs per unit mass of ice formed are t

h

x

2

Wi - L

°T

T

c

f

T

(30)

/

and W

= Wi

2

T u

~

(31)

T o

o

Addition of these two equations yields the theoretical energy input for freez ing part of the process, Wi + W = L 2

c

T u ( T

f T o

f

(32)

T f )

Because the ice is formed isothermally in this process, the mass requiring auxiliary refrigeration over the range T to T is (1 — Cj/C ) per unit mass of ice formed, and the differential of sensible heat to be received b y R is f

0

8

s

dH

z

-

Refrigeration machine #

(1 3

Cj/OC,

dT = ( C . -

Ci) dT

(33)

is of T y p e III with uniform reception of sensible heat


65


Brine

Saline Water I

Relative Mass: τ - - I


Fresh Water I

4w,+w 2

W

9

R

W

2

2

W" 3

w, LJw, h-w,

V Tj-

Brine at T

f

Ice Schematic diagram of second idealized process

Figure 7. over the range Tt to Τ

so modification of Equation 14 gives the theoretical energy

ω

input to R as s

= ( f t - ft) [ r „ l n

ψ

-

{T 0

T )j

(34)

f

T h e total theoretical energy input for the second idealized process, per unit mass of fresh water product, is the sum of Equations 32 and 34: T (T T Tf U

W = L

e

0

T) f

+

( f t - ft) [ t

u

In ^

-

(T. -

Tf)]

(35)

0

T h e theoretical energy at all extraction ratios, except r = 0, is greater for this process than for the first idealized process. T h i s is due to the irreversibility introduced into the second idealized process b y the mixing of saline water with brine at the exit concentration, and the consequent necessity of separating fresh water, in the form of ice, at the exit concentration instead of throughout the range from the inlet concen tration to the exit concentration. M o r e precisely, this irreversible internal mixing is accompanied b y an increase in entropy, requiring a proportionately larger amount of energy input to effect the separation, in accordance with the second principle of thermo dynamics.

Third Idealized Process T h e third idealized process, illustrated in Figure 8, is a modification of the first to include irreversible heat transfer across finite temperature differences, and discharge of



waste heat to a finite stream of saline cooling water. ferences are assumed: t, t, t, t, t, t x e

m w v

Vf

T h e following temperature dif

for countercurrent heat exchange for reception of heat of crystallization for discharge of heat of fusion to melting ice for discharge of waste heat to saline cooling water for temperature rise of saline cooling water between temperatures of input and effluent streams

T h e general equation for theoretical energy input to R per unit mass of ice, ob tained b y modification of Equation 21 to account for the increased difference between reception and discharge temperatures is lf

Wi -

-

j-

f

f

jTf

c

I

T

i

i m

( « + *»> y

(Τ -

I) T d

Τ - te


(36)

67


B y substitution of Γ(Τ) obtained from Equation 5 into Equation 36, the theo retical energy input to R is t

.

_ ΚL

C

if

jT

(T -

Ti

eSi

T) + t + t

a

m

KjjçSi Γ < Γ

Κ LcSj -


0

T

a

i

t + u

ι

τ

If(T

,

e

{Το-ΤηΤ-Q

f

\(

t + t\ m

tc) LV

Ί ,

m

e

T.-

(Tj - t )(T -

Tf)

(Tf -

Ti)

c

t)

m

c

0

tc)(T 0

T

(t + t )(Tj m

e

(To -

Ti)(T

0

Tf) 1

-

Tf)]

When t = t = 0, Equation 37 reduces to Equation 22. Refrigeration machine R is of T y p e II with uniform discharge over the range (T + t ) to (T + t + t ), and operating i n series with R Modification of E q u a tions 11 and 18 yields c

m

2

u

w

u

w

y

v

W = Wi [ 2

~ *]

Vo^tm*"

Γμ

(

3

8

)

I n this process two refrigeration machines are employed for sensible cooling. Machine R provides the supplementary cooling of the incoming saline water neces sitated by the difference between the specific heats of ice and saline water, and machine # provides the supplementary cooling necessitated b y the temperature differential, t . R receives heat over the range T to T from incoming saline water i n the range (T + t ) to (T + t ), and discharges heat uniformly over the same range as R . F o r each unit mass of ice produced, the mass fraction of incoming saline water between (Ti + t ) and (T + t ) which cannot be cooled b y heat exchange with the unit mass of ice between T and T is (1 — Cj/C ). Below T the mass fraction of ice is a function of temperature, [1 — I(T)/I ] so that the mass fraction of saline water below ( T + t ) which cannot be cooled b y heat exchange with ice is [1 — (Cj/C ) (1 — I(T)/I )]. If the condition (T + t ) < T exists, the mass of saline water to be cooled b y R d e creases from Ti to (T + t ) and the expression for the mass fraction requiring cooling becomes [(1 - C / C , ) (1 - I(T)/I ) - I(T + t )/I ). T h i s latter expression m a y be used as the general expression throughout the operative range of R if the following restrictions are observed: I(T) = 0 when Τ > T and I(T + t ) = 0 when (T + t ) > T^ When t = 0, the expression reduces to that used i n Equation 25. T h e differential equation for sensible heat received b y R is 3

x

4

3

f

f

x

0

0

x

x

2

0

x

0

4

8

f

%

y

f

x

f

3

{

f

x

{

8

x

f

x

f

3f

{

x

x

x

3

-

[ f t - ft + jj I(T) -

I(T + t )]dT

(39)

x

in which I(T) = 0 when Τ > T I(T + t ) = 0 when (T + t ) > T . Refrigeration machine R is of T y p e I V , so substitution into Equation 15 gives the theoretical energy as i}

x

t

m

3

IT,

= (ft -ft)

J

CTo φ ^

_

τ

dT

+f.g j

CTi

τ — τ j(T) 5 y (7 jj

u dr -

Γ^* — tx J

t

T„ — Τ I(T + t )

dT

x

(40)

in which T = T + t + / t , and the last integral exists only when ( T — t ) > T . When ^ = t = 0, Equation 40 reduces to Equation 26. Substitution of the function of Equation 5 for I(T) into Equation 40 gives the value of W as n

u

w

x

2

v

{

w

3


x

f

68


=

W

z

(C, -

C,)

[r. In |?

-

(Γ. -

2V)]

g [d - «) (r. m J; - (τ, -

g [d

- « ) ( Γ . in ^

-

(

Γ

-

Τ / . Refrigeration machine # receives sensible heat over the range (T — t ) to T from the total mass of saline water i n the range T to (T + t ), and discharges heat uniformly over the range (T 4- t ) to ( T 4- t + 2 ί ) into the effluent steams of brine and fresh water over the range (T — t ) to (T + 1 — Κ Τ 4- £ ) Per unit mass of ice formed, the mass of brine to be cooled b y R is ^ — Λ

f


4

f

u

v

x

w

u

x

f

x

f

x

υ

u

é

the variable being (Τ 4- ί ) because the range of integration is (T — t ) to T . If ( T + y > T the mass to be cooled between Γ« and (J , 4- t,) is 1 / / ; . T h e differen tial equation for reception of sensible heat b y i £ is thus f

Λ

x

f

1

if

r

4

dH

A

=

~

1

/

(

Γ

Γ +

(42)

C dT

t x )

8

if

with 1 = 0 when (T + £ ) > T . T h i s is also a machine of T y p e I V , so the theoretical energy is 7

W

C.

A

I

f

p

Γ

L J T

f

(

- t

{

f

r, +

VA)

- r

d

T

_

Γ* τ

+

+

JTf-t,

T

s

J

T

(43) in which T * is the lesser of T and (Τ* - ί , ) . W h e n t = 0, I F = 0, b y virtue of a reduction of the range of integration to 0. Since t is not arbitrary, but dependent on a simultaneous solution of energy and heat balance equations, it is omitted from E q u a tion 43 for convenience. Since t is relatively small, this introduces only a negligible error i n W . 7

f

x

4

v

v

4

Substitution of the function of E q u a t i o n 5 into E q u a t i o n 43 gives W

A

( r

= ψ [ ( Γ . + / t ) In ψ^ΓΤ 1

. ^ ,+ T

t x )

)

2

x

χ

+ K s i

-

t x

(*-±^ι»

T o

(1 -

·,) ( ( T

+ V A ) In fy^rj,

t t

,f, *

~

-m ^ï-V)]

r

r

( 4 4 )

in which T * is the lesser of T and ( T — ί ) . T h e effluent exit temperature differential, t , may be computed b y means of a heat balance. T h e heat capacity of the effluent streams between (T — t ) and (T + T ), per unit mass of product water, is equal to the heat discharged b y # . T h u s f

4

Λ

v

u

x

u

v

4

«· + t ) x

[ft

Q-

- l) + l] -

(45)

W< + H

Energy Computations for Saline Water Conversion by Idealized

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