Excess Enthalpies of Some Binary Steam Mixtures - American

22 Excess Enthalpies of Some Binary Steam Mixtures C. J. W O R M A L D and C . N . C O L L I N G

Downloaded by HARVARD UNIV on May 28, 2014 | http://pubs.acs.org Publication Date: October 29, 1980 | doi: 10.1021/bk-1980-0133.ch022

School of Chemistry, The University, Bristol BS8 1TS, U K

Despite the importance of mixtures containing steam as a component there is a shortage of thermodynamic data for such systems. At low densities the solubility of water in compressed gases has been used (1,2) to obtain cross term second virial coefficients Β . At high densities the phase boundaries of several water + hydrocarbon systems have been determined (3,4). Data which would be of greatest value, pVT measurements, do not exist. Adsorption on the walls of a pVT apparatus causes such large errors that it has been a difficult task to determine the equation of state of pure steam, particularly at low densities. Flow calorimetric measurements, which are free from adsorption errors, offer an alternative route to thermodynamic information. Flow calorimetric measurements of the isothermal enthalpy -pressure coefficient φ (5) extrapolated to zero pressure yield the quantity φ = Β - TdB/dT where Β is the second virial coefficient. From values of φ it is possible to obtain values of Β without recourse to pVT measurements. 12

p

o

o

As w i t h pure steam the p r o p e r t i e s o f b i n a r y steam mixtures can be obtained from f l o w c a l o r i m e t r i c measurements of the enthalpy of the mixture. With steam + n-alkane b i n a r i e s , f o r which the e n t h a l p i e s of both components are known, i t i s more s e n s i b l e to measure the excess enthalpy d i r e c t l y r a t h e r than measure the l a r g e t o t a l enthalpy of the mixture t o determine a^ small excess q u a n t i t y . E x t r a p o l a t i o n of the excess enthalpy Hp at pressure ρ t o zero pressure y i e l d s = χ ^ Χ £ ρ ( 2 φ ^ ~ Φ\\ ~ $22^ and from t h i s q u a n t i t y B ^ f o r a steam + n-alkane be obtained.

i n t e r a c t i o n can

0-8412-0569-8/80/47-133-435$05.00/0 © 1980 American Chemical Society

In Thermodynamics of Aqueous Systems with Industrial Applications; Newman, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

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O u t l i n e of the Flow C a l o r i m e t r i c Apparatus A mixing c a l o r i m e t e r s u i t a b l e f o r measurements at low d e n s i t i e s i s d e s c r i b e d i n the l i t e r a t u r e ( 6 ) , and t h i s c a l o r i m e t e r has been used to make measurements on steam mixtures a t temperatures around 373 K. Although h i g h pressure c a l o r i m e t e r s have been d e s c r i b e d (]_,&>9) °f these designs i s s u i t a b l e f o r work a t h i g h temperature. D e t a i l s of the c a l o r i m e t e r used i n t h i s work w i l l be p u b l i s h e d s h o r t l y (10). The f l o w system i s shown i n o u t l i n e i n f i g u r e 1. The system i s p r e s s u r i z e d w i t h n i t r o g e n which enters at 1. Pumps 2 and 3 supply l i q u i d components to f l a s h b o i l e r s 4 where the l i q u i d s are v a p o r i z e d . The vapours mix i n c a l o r i m e t e r 5 which i s contained i n a p r e s s u r i z e d v e s s e l immersed i n a f l u i d i z e d bed thermostat. The f a l l i n temperature produced on mixing i s sensed by four platinum r e s i s t a n c e thermometers and a heater i n the centre o f the c a l o r i m e t e r i s used to o b t a i n i s o t h e r m a l c o n d i t i o n s . The mixture i s next passed i n t o a t o t a l enthalpy b o i l - o f f c a l o r i m e t e r 6 where the enthalpy change of the f l u i d v a p o r i z e s some n-pentane. The n-pentane vapour i s condensed at 7 and the r a t e of b o i l - o f f i s measured u s i n g c a l i b r a t e d bulbs 8. The l i q u i d condensate i s c o l l e c t e d i n v e s s e l 9. The apparatus has been used t o make measurements a t pressures up t o 16 MPa and a t temperatures up t o 698 K. With any flow mixing c a l o r i m e t e r i t i s important to t e s t f o r the presence of heat l e a k s . T h i s can be done by measuring the enthalpy of mixing at constant composition, temperature and pressure over a wide range of f l o w r a t e . Tests on our c a l o r i m e t e r were done on steam + n i t r o g e n a t χ = 0.5. The r e s u l t s of measurements a t 698 Κ and 12.3 MPa are shown i n f i g u r e 2, and demonstrate that even under these extreme c o n d i t i o n s heat leaks are n e g l i g i b l e . The r e s u l t s a l s o show the good r e p r o d u c i b i l i t y (one percent) of which the apparatus i s capable. Measurements have so f a r been made on mixtures of steam + hydrogen, n i t r o g e n , argon, methane, c a r b o n - d i o x i d e , n-hexane, η-heptane, benzene and cyclohexane. The measurements cover the range 373 t o 698 Κ at pressures from 0.1 MPa t o s a t u r a t i o n or 12.5 MPa. The o n l y e x c e p t i o n t o t h i s i s steam + carbon d i o x i d e for which the measurements extend up to 5.5 MPa. The accuracy of the measurements i s around ±2 percent.


n

o

n

e

R e s u l t s f o r the M i x t u r e Steam + n-Heptane Some r e s u l t s f o r the mixture steam + η-heptane a t χ = 0.5 are shown i n f i g u r e 3. The r e s u l t s f o r steam + n-hexane, + c y c l o hexane, and + benzene are s i m i l a r . The measurements a t 548 and 598 Κ are above the c r i t i c a l temperature of n-heptane (540 K) and below that of steam (647 K ) . The measurements a t 648 and 698 Κ are above the c r i t i c a l temperature o f both components. A l l the r e s u l t s which are below the c r i t i c a l temperature of one of the components show a maximum and terminate at the s a t u r a t i o n pressure



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Binary Steam Mixtures

IIOO Λ 900 ε ^ χ

700YIOO ( F /mol s"' Figure 2.

)-'

I50

200

Test of the flow mixing calorimeter on steam + nitrogen at χ = (measurements were made at 698 Κ and 12.3 MPa)


0.5


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8 h

Figure 3.

n>

Enthalpy of mixing H for steam + η-heptane at\ = 0.5


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of the s u b c r i t i c a l component. A n a l y s i s of the r e s u l t s at low pressures can be done using the v i r i a l equation of s t a t e . F o l l o w i n g Lambert (11) we separate the second v i r i a l c o e f f i c i e n t Β i n t o a p h y s i c a l B° and a chemical c o n t r i b u t i o n Β = B° - RTK

(1)

where Κ i s the e q u i l i b r i u m constant f o r dimer formation. The excess enthalpy H of a steam (1) and n-heptane (2) mixture can be w r i t t e n (12) E

H

E

= p[ct> - x ^ ° + ΚΔΗ) 2


-χ φ ]

m

-(p /RT)[B^

2

2

- ( B ° - RTK)^° + ΚΔΗ)

m

X l

-

^Φ] 2

2

where

Φ =

(Φ° + ΚΔΗ) + 2 χ Φ

πι

B

m

=

Χ

?

2

Χ]

( Β

R

° "

T

K

)+

2 χ

Χ

Β

1 2 12

+

Χ

12

+χ Φ 2

2

Β

( 2 )

2 2

and ΔΗ i s the enthalpy of dimer formation. B° can be taken as the second v i r i a l c o e f f i c i e n t of methyl f l u o r i d e which has the same d i p o l e moment as steam (1.85D). B j 2 and φ|2 can be c a l c u l a t e d from B° and B 2 u s i n g a Kihara-Stockmayer p o t e n t i a l . With 1

1

ΔΗ = -16.426 k J mol" and Κ = 0.385 MPa" (at 298 K) equation (2) f i t s H f o r steam + n-alkane mixtures at low d e n s i t i e s to w i t h i n experimental e r r o r . F i g u r e 4 shows the f i t to the r e s u l t s at standard atmospheric pressure. To f i t the r e s u l t s at higher pressures r e q u i r e s f u r t h e r v i r i a l c o e f f i c i e n t s , and the method runs i n t o d i f f i c u l t i e s . E

A n a l y s i s of the R e s u l t s at High Pressures The enthalpy of mixing H Η

m

=

*

*

m

i s given by the equation

*

- XjHj - x H 2

, ^ ν

(3)

2

where^Hj and H are the r e s i d u a l e n t h a l p i e s of components 1 and 2 and H i s the r e s i d u a l enthalpy of the mixture. For f l u i d s which are s l i g h t l y p o l a r we might expect the Peng-Robinson (P-R) equation of s t a t e (13) to give a reasonable estimate of H . While we would not expect i t to work w e l l f o r mixtures c o n t a i n i n g steam i t i s i n s t r u c t i v e to see what i t g i v e s . The r e s i d u a l enthalpy Η i s given by 2

M

m

H*( ,T) - H Z V-b

RT-

V

V(V + b) + b(V - b)

2

a (

l K) ln

V + (l-2^)b

+

5

V + (l+2 )b (4)



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T / K

i:

Excess enthalpy H of steam + η-heptane at χ = 0.5 and standard atmospheric pressure (( ) calculated using Equation 2)


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To c a l c u l a t e H we use the mixing r u l e s M

a», = Σ Σ x.x.a.. M . . ι j IJ 1 J a.. = k..(a.a.)^ ij i JιJ

; '

b = Σ x.b. M . 1 1 w

1

;

κ = Σ χ.κ. Μ ιι

(5)

£


P u t t i n g k^j = 1, and using c r i t i c a l i t y c o n d i t i o n s t o c a l c u l a t e a and b f o r steam + η-heptane, the e n t h a l p i e s of mixing are found to be only about h a l f as b i g as the experimental r e s u l t s and show only the r i g h t q u a l i t a t i v e behaviour. I t i s i n t e r e s t i n g to see i f a value of k ^ which b r i n g s the c a l c u l a t e d e n t h a l p i e s i n t o agree ment w i t h experiment can be found. The best k ^ turns out to be -0.3, and curves c a l c u l a t e d u s i n g t h i s value are shown i n f i g u r e 5. To o b t a i n a f i t to the r e s u l t s a t 548 Κ r e q u i r e s k.. = -0.5, and ij to f i t the r e s u l t s a t 698 Κ r e q u i r e s k.. = 0.0. Now the P-R iJ equation i s not a good f i t to the r e s i d u a l e n t h a l p i e s of e i t h e r steam or η-heptane. A t temperatures up to 623 Κ values of the enthalpy of η-heptane are a v a i l a b l e . I f P-R parameters are chosen to f i t the r e s i d u a l e n t h a l p i e s of steam and η-heptane, i t i s found that the best value of k ^ i s -0.2. A temperature dependent value of k ^ j i s s t i l l r e q u i r e d , although the change of k^. w i t h temperature i s l e s s than was r e q u i r e d when pure component e n t h a l p i e s were obtained from c r i t i c a l i t y c o n d i t i o n s . The l a r g e negative values of k ^ c l e a r l y i n d i c a t e that a d i f f e r e n t approach to the c a l c u l a t i o n of mixture p r o p e r t i e s i s needed. The Separated A s s o c i a t e d F l u i d I n t e r a c t i o n Model f o r P o l a r + Nonpolar Mixtures Woolley (14) has developed the equation of s t a t e f o r an a s s o c i a t e d f l u i d i n terms of the formation of dimer, t r i m e r , tetramer, e t c . c h a r a c t e r i s e d by e q u i l i b r i u m constants K^, K^, K^, etc. The molecules i n the model have no s i z e , and are simply p o i n t s between which i n t e r a c t i o n s occur. Lambert (11) developed a s i m i l a r approach f o r low d e n s i t y gases, and regarded the observed second v i r i a l c o e f f i c i e n t as the sum of a " p h y s i c a l term" which he obtained from the B e r t h e l o t equation of s t a t e and which i m p l i e s that the molecules have f i n i t e s i z e and a t t r a c t i v e f o r c e s , and a "chemical" term c h a r a c t e r i s e d by a d i m e r i s a t i o n constant K^. The i n c l u s i o n of the p h y s i c a l term ensures that Β can be p o s i t i v e at high temperatures without having t o change s i g n . A disadvantage of the Woolley treatment i s that extension t o mixtures of a s s o c i a t e d f l u i d s i s i m p o s s i b l e , as cross term v i r i a l c o e f f i c i e n t s cannot be c a l c u l a t e d . The i n c l u s i o n of a " p h y s i c a l


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Figure 5. Comparison of the Peng-Robinson equation using kjj = — 0.3 with the results for steam + τι-heptane (( ) calculated using Equations 3 and 4)



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term" i n the v i r i a l s overcomes t h i s problem. With steam i n mind V u k a l o v i t c h (15) developed a model s i m i l a r to that o f Woolley, i n which he regarded each c l u s t e r as obeying the van der Waals equation of s t a t e . As there i s no easy way of a s s i g n i n g van der Waals parameters a and b to the c l u s t e r s , the V u k a l o v i t c h equations are almost unusable. The model presented here develops these ideas and introduces f e a t u r e s which make the c a l c u l a t i o n of mixture p r o p e r t i e s simple. For a p o l a r f l u i d w i t h approximately c e n t r a l d i s p e r s i o n f o r c e s together w i t h a strong angle dependent e l e c t r o s t a t i c f o r c e we may separate the i n t e r m o l e c u l a r p o t e n t i a l i n t o two p a r t s so that the v i r i a l c o e f f i c i e n t s , B, C, D, e t c . of the f l u i d can be w r i t t e n as the sum of two terms. The f i r s t terms B°, C°, D°, e t c , a r i s e from d i s p e r s i o n f o r c e s and may i n c l u d e a c o n t r i b u t i o n a r i s i n g from the permanent d i p o l e of the molecule. The second terms c o n t a i n e q u i l i b r i u m constants K^, K^, K^, e t c . which describe the formation of dimer, t r i m e r , e t c . by hydrogen bonding. The f i r s t v i r i a l c o e f f i c i e n t s f o r the p o l a r f l u i d can be w r i t t e n

three

Β = B° - K (RT) 2

2

C = C° - ( 2 K - 4 K ) ( R T )

2

3

D = D° - ( 3 K - 18K K 4

2

3

+ 20K )(RT)

3

(6)

2

We can i n p r i n c i p l e c a l c u l a t e B°, C°, D°, e t c . from a s u i t a b l e p a i r p o t e n t i a l , and so o b t a i n K , and K^. F o r gas mixtures a t 2

low d e n s i t i e s where Β and C terms are s u f f i c i e n t t h i s approach can be used s u c c e s s f u l l y . However, the c a l c u l a t i o n of D° and higher c o e f f i c i e n t s becomes p r o h i b i t i v e l y d i f f i c u l t , and f o r mixtures the c a l c u l a t i o n of cross c o e f f i c i e n t s i s an a d d i t i o n a l problem. An a l t e r n a t i v e approach i s p o s s i b l e . J u s t a t the c o e f f i c i e n t s B, C, D, e t c . d e f i n e the thermodynamic p r o p e r t i e s of the r e a l f l u i d so c o e f f i c i e n t s B°, C°, D°, e t c . d e f i n e thermodynamic p r o p e r t i e s f o r a h y p o t h e t i c a l f l u i d which we w i l l c a l l the primary f l u i d . The primary f l u i d can be regarded as having the p r o p e r t i e s which the r e a l f l u i d might have i n the absence of a s s o c i a t i o n . I t i s assumed that when secondary i n t e r a c t i o n s such as hydrogen bonding are imposed on the primary f l u i d the r e a l f l u i d w i l l be simulated. This assumption i s an acceptable approximation a t low d e n s i t i e s , but i s u n l i k e l y t o hold a t high d e n s i t i e s where the a d d i t i o n of hydrogen bonds may produce new s t r u c t u r a l f e a t u r e s . At moderate d e n s i t i e s we can make the not unreasonable approximation that any property which i s a f u n c t i o n f of the v i r i a l c o e f f i c i e n t s can be separated i n t o two c o n t r i b u t i o n s f(B,

C, D ...) = f(B°, C°, D° ...) + f ( K , K , K

real fluid

2

primary f l u i d

3

4

...)

(7)

secondary e q u i l i b r i a



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Our present concern i s t o f i n d a model f o r steam which can be used to c a l c u l a t e mixture p r o p e r t i e s . An estimate of B°, the second v i r i a l c o e f f i c i e n t which steam might have i n the absence of a s s o c i a t i o n , could be obtained by s e l e c t i n g a molecule which has about the same d i s p e r s i o n f o r c e as steam, such as argon, t a k i n g the Lennard-Jones 12-6 parameters which f i t Β f o r argon together w i t h the d i p o l e moment of steam 1.85 D, and u s i n g the Stockmayer p o t e n t i a l to c a l c u l a t e values of B°. I f we do t h i s i t i s found that the c a l c u l a t e d values of B° l i e c l o s e t o the second v i r i a l c o e f f i c i e n t of methyl f l u o r i d e which a l s o has a d i p o l e moment of 1.85 D, and which would a l s o be a reasonable model f o r unassociated steam. The f o l l o w i n g four steps are now taken. 1. The primary f l u i d i s replaced by the r e a l f l u i d methyl f l u o r i d e . T h i s removes the need t o evaluate f(B°,C°,D°,etc.), as the thermodynamic f u n c t i o n X r e q u i r e d i s simply that f o r methyl f l u o r i d e . 2. The c o n t r i b u t i o n of the secondary e q u i l i b r i a ί^,Κ^,Κ^ ...) i s obtained by s u b t r a c t i n g the thermodynamic property of the primary f l u i d from that of the r e a l f l u i d . X ( a s s o c i a t i o n ) = X ( r e a l f l u i d ) - X(primary f l u i d ) 3.

4.

(8)

An equation of s t a t e which w i l l represent the thermodynamic property X 2 of the non p o l a r component of the mixture i s chosen. T h i s same equation of s t a t e i s used t o f i t the property of the primary f l u i d , component 1. Using combining r u l e s appropriate t o the equation of s t a t e the thermodynamic property X of the non p o l a r component + primary f l u i d mixture i s c a l c u l a t e d . The change i n property Χ on forming the mixture X i s given by an equation s i m i l a r to 3 M

m

X

m

=

- XjX(primary f l u i d ) - X j X ( a s s o c i a t i o n ) - x^X . 2

(9)

Equation 9 was used to c a l c u l a t e f o r steam + η-heptane as f o l l o w s . The Peng-Robinson equation w i t h parameters obtained from c r i t i c a l i t y c o n d i t i o n s was used t o c a l c u l a t e the r e s i d u a l enthalpy Hj of methyl f l u o r i d e . Peng-Robinson parameters f o r η-heptane were obtained by f i t t i n g to the r e s i d u a l enthalpy of the f l u i d at temperatures below the c r i t i c a l , and by u s i n g c r i t i c a l i t y c o n d i t i o n s a t higher temperatures. The mixing r u l e s given i n equation 4 w i t h k.. = 1 were used t o c a l c u l a t e H^. As hydrogen bonding occurs only f o r H^O + ^ 0 i n t e r a c t i o n s and not f o r H 0 + η-heptane i n t e r a c t i o n s ?

H^.

At the temperatures

H (ass) makes no c o n t r i b u t i o n to

at which mixing experiments had been done,

the r e s i d u a l enthalpy of steam c a l c u l a t e d from steam t a b l e s was f i t t e d t o polynomial equations i n powers of the pressure, and Η ( a s s o c i a t i o n ) was obtained by s u b t r a c t i n g H j .



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Figure 6. Comparison of the Separated Associated Fluid Interaction Model using no adjustable parameters with the results for steam + n-heptane (( ) calculated from the model)


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m

H f o r steam + η-heptane c a l c u l a t e d by the above method i s shown by the dashed l i n e s i n f i g u r e 6. C o n s i d e r i n g the s i m p l i c i t y of the model and the f a c t that no a d j u s t a b l e parameters have been used, agreement w i t h experiment i s remarkable. For mixtures of steam + n-hexane, benzene and cyclohexane agreement w i t h experiment i s much the same. At low d e n s i t i e s the model reproduces the curvature of the l i n e s through the r e s u l t s b e t t e r than the v i r i a l equation of s t a t e . The method f a i l s t o f u l l y reproduce the down ward t u r n o f the experimental curves a t pressures near s a t u r a t i o n , but does m a r g i n a l l y b e t t e r i n t h i s r e g i o n than the P-R equation w i t h k ^ j = -0.3. At s u p e r c r i t i c a l temperatures the model seems t o work w e l l . F o r steam + methane, + n i t r o g e n , + argon, the model g i v e s values of H which are too l a r g e , and does no b e t t e r than the v i r i a l equation of s t a t e . The choice o f methyl f l u o r i d e as the primary f l u i d f o r steam i s e v i d e n t l y reasonable but i s not n e c e s s a r i l y the best. There i s no reason why T , P and ω f o r the primary f l u i d should not be t r e a t e d as a d j u s t a b l e parameters so that a h y p o t h e t i c a l primary f l u i d which g i v e s best agreement w i t h r e s u l t s on a l l the above steam + hydrocarbon mixtures can be defined. There are c l e a r l y many ways i n which the model can be m o d i f i e d . As i t i s formulated above, the model can be used without f u r t h e r m o d i f i c a t i o n f o r the e s t i m a t i o n of the thermodynamic p r o p e r t i e s o f mixtures of steam w i t h hydrocarbons i n the C^ t o Cg


m

c

c

range. Below the c r i t i c a l temperature of steam the model can be used a t pressures up t o 10 MPa w i t h reasonable confidence. At temperatures above the c r i t i c a l p o i n t of steam the model can be used a t higher p r e s s u r e s . Acknowledgment We are g r a t e f u l t o the B r i t i s h Gas C o r p o r a t i o n f o r f i n a n c i a l support which made t h i s work p o s s i b l e . Abstract

Flow calorimetric measurements of the excess enthalpy of a steam + n-heptane mixture over the temperature range 373 to 698 Κ and at pressures up to 12.3 MPa are reported. The low pressure measurements are analysed in terms of the virial equation of state using an association model. An extension of this approach, the Separated Associated Fluid Interaction Model, fits the measure ments at high pressures reasonably well. Literature Cited

1. Rigby, M; Prausnitz, J.M. J.Phys.Chem. (1968), 72., 330. 2. Coan, C.R.; King, A.D. J.Amer.Chem. Soc. (1971), 93, 1857.


22. WORMALD AND COLLING

3. 4. 5. 6. 7. 8. 9.


10. 11. 12. 13. 14. 15.


447

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RECEIVED

January 31, 1980.


Excess Enthalpies of Some Binary Steam Mixtures - American

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