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A method to establish equations of state exactly representing all saturated state variables applied to nitrogen
Using nitrogen as an example, a new method o f establishing equations o f state for pure fluid substances is demonstrated. With the aid of these equations exact phase equilibrium data can be calculated. This method together with a new equation o f state permits the vapour pressure and the densities on the phase boundary curves to be calculated directly from the equation of state (application of the Maxwell criterion). From the critical point to the triple point values are obtained which are of high accuracy and which lie within the deviation range of the results measured throughout the whole o f the range.
A method to establish equations of state exactly representing all saturated state variables applied to nitrogen W. Wagner A knowledge of liquid-vapour phase equilibria and state variables is of decisive importance for the technical systems involved in processes of gas liquefaction, gas separation, and refrigeration. For this reason a great deal of attention has been given in recent years to methods of calculating liquid-vapour phase equilibrium data for pure substances and multicomponent mixtures. The mixtures mainly of concern are binary or ternary systems whose pure components have low boiling points. These include combinations of various pure and halogen-substitutes hydrocarbons [for example, n-pentanepropane-methane, difluoromonochloromethane (R 22)pentafluoromonochloroethane (R 115)] and systems involving cryogenic substances (for example helium-nitrogen, helium-hydrogen-carbon dioxide). Exact prediction of the required phase equilibrium data for the economic operation of such equipment is very difficult if conventional methods are employed. The values determined are often so inexact that the equipment must be inordinately over-dimensioned if the quantity and quality of the product are to be guaranteed. This leads to an increase in investment and operation costs.
Traditional methods of determining phase equilibrium data
The author is with the Institute of Thermodynamics, Technical University, Brunswick, Germany. Received 25 September 1971.
214
Ps = Ps(P", T)
(1)
(2)
The value of the saturated liquid density p' is calculated as a function of the temperature from an empirical relationship
p' = p'(T)
(3)
Since the phase equilibrium data Ps, P', and p" are related to one another for a fixed temperature (see equation 7) but as (1) and (3) are independent of one another, inconsistent phase equilibrium values may result from calculations. 1 In contrast to this the phase equilibrium values, in addition to the state variables of the homogeneous phases, could also be calculated consistently from one suitable equation of state.
Mixtures. The basis for calculating phase equilibrium data for mixtures at a fixed temperature and a known total pressure is provided by the conditions T '= T";
Pure substances. The data for the liquid vapour equilibria of a pure substance is normally calculated with the aid of three equations which are independent of one another. The density of the saturated vapour p" can be determined from an equation of state applying only to the gas phase if the saturation pressure is known at each temperature, for example, from a vapour pressure equation Ps = Ps(T)
The variable p" is then arrived at by iteration from the equation of state
P' =P";
3~' =3~"
(4)
The temperature T, the pressure p, and the fugacity f of the component i of the mixture in the gaseous and liquid phase must be equal respectively. Phase equilibrium data determined experimentally can be extrapolated and interpolated with the aid of empirical graphs. 2,3 More accurate values are obtained by application of analytical methods which are based on exact thermodynamic relationships.4,5, 6 The difficulties encountered in calculating phase equilibrium data for mixtures thus do not result from a lack of exact equations. The main source of inaccuracies lies in the difficulty in determining the required thermodynamic values in the specific molar volume v', and in the fact that
C R Y O G E N I C S . J U N E 1972
the partial molar volume ?[ of the component i of the liquid mixture are very unreliable particularly at high equilibrium pressures. As a basic prerequisite for determining the above-mentioned variables it is also necessary to measure certain phase equilibrium values directly for each mixture. Moreover the methods are rather complicated and usually have to be modified to an extreme extent to suit various types of mixtures.
Description of the problem. The uncertainties described above in the determination of the fugacity f ' of the liquid phase of pure substances and mixtures could be avoided and the number of required equations could be reduced to one if the fugacity f ' could be calculated directly from an equation of state. For this purpose the following conditions would have to be fulfilled: 1. An equation of state for pure fluid substances must be available which permits the whole fluid range to be reproduced with a single set of constants approximately up to the three-fold critical density. 2. This equation of state must be constructed in such a manner that for the pure components prediction of the vapour pressure and the saturation densities on the phase boundary curves is possible with the aid of this equation alone. 3. Rules for calculating the combination of constants of the equation of state of the pure substances must be available. Combination rules of this type already exist. 7,8,9 When taking into consideration the ~reconditions t to 3 listed above, the fugacity equation 1"
The curve of an isotherm in the p, v diagram is shown in Fig.1. The equation of state describes the real behaviour in the two homogeneous states (gas and liquid) and prorides the curve of the 'theoretical' isotherm in the twophase range (dotted curve in Fig. 1). The equilibrium existing between the saturated liquid and the saturated vapour demands that the conditions in (4) be fulfilled. If the usual thermodynamic relationships for f ' a n d f " are substituted into (4), 13,14,15 then the relationship known under the name 'Maxwell criterion' is obtained 16 V st
Ps(V" - v') = I p(v, T) r dv
(6)
i 12
By substitution of the density p for specific volume v we obtain the equation p'
Ps
-
p,
-
=
,
,,
p~,r)
r
(7)
02
P Geometrically this condition implies that the area under the isotherm T = T' = T" between the limits v" and v' must be equal to an area formed by the rectangle with the corners l, 2, 3, 4 (see Fig.l). The equation to be developed should also fulfil the 'normal' demands which can be expected from an equation of state. The individual conditions are as follows:
\YiP/
= RTl°ge ~-P --
+
1. The p, p, T data measured in the range (subscript tr = triple point)
P -
+
o
~ni J
T, V, n/~i
R T
f
- -
Ttr ~< T ~< 400 K
(5)
P
Ptr ~
0 <<.p<.3Pk for determining the variables fi and fi' need only be solved within the limits p = 0 and p = p" or p = p'. In the above equation R is the universal gas constant, y the molar concentration, V the total volume, n the number of moles, and i a serial index. This method used together with the traditional equations of state produces incorrect values for the fugacity f ' even for the pure substances, l, 11,12 Thus using the example of nitrogen a new method will be given here for establishing equations of state, initially for pure fluid substances, taking into consideration preconditions 1 and 2. Using such equations alone it will then be possible to calculate phase equilibrium values firstly for pure substances and secondly utilizing suitable combination rules for mixtures consistent with the data of the homogeneous phases.
(8)
must be reproduced to an optimum extenl. Critical point
Liquid
~ "
~
Area A I
Area [FT[Y~ Area AI2'
A .~ 1 2 ~
['--3 Area A2*
Q.
ff (3-
;3 Demands placed on the equation of state If an equation of state p = p(v, T) applies to the whole fluid range, that is, for the gas and liquid states then an equilibrium relationship for calculating the thermal variables of state on the phase boundary curve can be derived.
C R Y O G E N I C S . JUNE 1972
F"
g"
Specific volume, V Fig.1 Representation of the Maxwell criterion. Dashed isotherm: qualitative curve in accordance with (19). Dot--dash isotherm: mathematically possible curve of isotherm in two-phase region. On fulfilment of the Maxwell criterion A1 = A2 or AI" = A2"
215
2. By taking the limit p ~ 0 the equation of state for ideal gases can be derived directly from the equation of state for real pure substances.
p = R Tp
(9)
3. The following should apply at the critical point (subscript c):
The subsidiary conditions at the critical point are not included in the fitting procedure but are exactly fulfilled with the aid of Lagrange multiplicators. In order that the coefficients of the equation of state fit the measured p, P, T values of the homogeneous phase (number: n l ) the first partial sum of(11) is required n1
p(rc, pc) pc
(lOa)
[(Op/OT)p]c = °tc = (TIPsx dPs/dT)c
(lOb)
[(DP/aP)T] c = 0 = [ ( a 2 p / a p 2 ) T ] c
(lOc)
lim(a2p/aT2)t) = 0
(lOd)
=
T-.).=
Glr2= E 1
G 1 [p(T, P)calc - p(T, P)meas] 2
Condition 4 ensures that extrapolation to regions of low density and temperature will produce reliable results. In addition the usual combination rules for the second virial coefficient will apply with the corresponding constants in an equation of state for the mixture. Determination of the constants The coefficients of an equation of state are on the other hand determined in such a manner that the system simultaneously fits the measured values of the homogenous phase including the phase boundary curve and the equilibrium condition (equation 7). The individual conditions are here taken into account as partial sums in the least square relations LSR.
G1r2 +
n,)
G2r2 + E
G3r2
(12)
1
The second partial sum n2
n2
,-
4. Measured results for the second virial coefficient should be described as exactly as possible.
LSR =
E
n1
1
G 2 [p(T, O'smoothed) -
Ps(T)] 2 +
G2 [p(T, Osmoothed)-
Ps(T)] 2
1
n2 +E
(13)
1
serves to reproduce n2, p, p, T values on the saturated liquid and saturated vapour line consistent with the measured data of the homogeneous phases. The Maxwell rule (equation 7) is fulfilled with the aid of the third partial sum:
1
L# (14)
-+Min (11)
1
Auxiliary equations G = weight factor, suitable chosen; r
=
residue
Table 1. Selected critical data and important constants for nitrogen Critical t e m p e r a t u r e Critical pressure Critical d e n s i t y Universal gas c o n s t a n t M o l a r mass Conversion f a c t o r
Tc Pc Pc R
= = = =
126.26 K 33.655 atm 11.25 mole 1-1 0 . 0 8 2 0 5 7 4 I arm mole 1 K 1
M = 2 8 . 0 1 6 g mole -1 11 = 1 0 0 0 . 0 2 8 cm 3
Table 2. Reduced variables employed Reduced Reduced Reduced Reduced Reduced Reduced Reduced
216
temperature pressure density second virial c o e f f i c i e n t gas c o n s t a n t specific e n t h a l p y specific v a p o r i z a t i o n e n t h a l p y
0 = T/T c 7r ~ Br o hr z~hr
= P/Pc =P/Pc = BPc = RTcPc/Pc = h / R Tc = Ah/RT c
The existing, highly dispersed results 15 measured for the phase boundary curves are not sufficient for effective determination of the constants in the equation of state. The measured data for the saturated liquid and saturated vapour line only contain pairs of values for T and p' or p". It is therefore necessary to allocate a pressure to each measured temperature. This cannot be determined directly from the data measured for the vapour pressure since the temperature values of vapour pressure measurements do not coincide with those of the density measurements. For this reason interpolation equations for the saturated liquid and vapour densities and a vapour pressure equation must be established. In addition an analytical relationship for the second virial coefficient is developed which is required when the fourth condition is taken into account. Auxiliary equations need not be developed if reliable information on the relevant substance for calculating the abovementioned values is available in the literature. In the sections below, these auxiliary equations are given for nitrogen (for detailed treatment see WagnerlS). The individual values were in each case reduced to the critical data (see Tables 1 and 2 as well as by WagnerlS).
C R Y O G E N I C S . J U N E 1972
The second virial coefficient. The analytical relationship for the second virial coefficient B(T) should fulfill the
O.I <1
following conditions: 1. Close reproduction of the reduced measured values B r as listed by Wagner 15 must be obtained.
O
L80°
o
S
g o
o o
'~x x
o
o
o~
oO
o
o
~'
I10 = 1.73
d~
<
1
1.7
I
t.5
I.'25
--
1.0
0175
0150
R¢ciprocol reduced temperature, I ] 0
0.906788 (15)
fulfilled the above-mentioned conditions best and was employed to describe the curve of the second virial coefficient for nitrogen.
mits the measured values in the critical range (T > 0.9 Tc) to be calculated with high accuracy.
The constants N 1 to N 4 were determined by the method of least square errors. The accuracy of the values measured for Br(O ) decreases pronouncedly in the direction of low temperatures. In order to take this factor into account the squared sum of the relative errors was minimized. For this reason the function Br(O ) had first to be shifted in such a manner that it did not pass through zero.
The following boundary and auxiliary conditions were tested for various approaches:
The deviation of the individual measured values from (15) is shown in Fig.2. The analysed derivatives dBr/d(1/O ) and d2Br/d(1/O) 2 exhibited a curve completely without oscillations. The calculated Boyle temperature is TB = 325.26 K and thus lies only 0.4 K below the value theoretically obtained by Morsy. 18 The inversion temperature has the value T1 = 620 K and thus deviates by 2.1% from the data obtained by Straub et al. 19
3. Inclusion of the Plank criterion 37. (da/dO)c = O.
A strict test of the quality of an equation for the second virial coefficient is provided by a comparison o f measured and calculated specific isochoric and isobaric residual heat capacities. To carry out this test, measured values obtained by van Itterbeek 20 and Lestz 21 for Aev/R and Acp/R were used. The agreement for reduced pressures ir up to approximately n -~ 1 is excellent.
Vapour pressure curve. The commonest known equations for nitrogen 23-35 were tested in a preliminary investigation 22 by a comparison with 154 measured values. 15 Here it turned out that equations of simple structure either only adequately reproduce the individual authors' own measured results or are only applicable to a small part of the whole temperature range. The vapour pressure formulas put forward by Riedel, 33 Stein, 34 and Strobridge 35 on the other hand permit exact calculation of the vapour pressure. At the extremes of the temperature range the s-sharped curve of the vapour pressure curve is not completely described by a logen s, 1/0 graph. As a result of this the relative error in the vicinity of the critical point lies above the average mean error of the whole range. On the basis of a proposal put forward by Martin, 36 a new vapour pressure equation was established which also per-
C R Y O G E N I C S . JUNE 1972
0.35
Fig.2 Absolute deviation AB r = Br, meas -- Br, calc of the reduced measured values Br, meas for the second virial coefficient 1 5 in relationship to the data calculated in accordance w i t h (19) for nitrogen
B r = N 1 + N 2 0 - I +N30"tlogeO +N40-2
N 4 = -0.724141
O
-O~1
The equation
N3=-0.021492
I0 =
o
3. The specific residual heat capacities at constant pressure Acp and constant volume Ac v should be qualitatively reproduced with correct values in the range of low densities (slightly real behaviour).
N2 =
°'~'1
O
2. The sqcond derivative d2Br/d(1/O) 2 should exhibit a curve which is thermodynamically tenable.
N 1 = 0.469015
o
E
1. Exact calculation of the critical pressure Pc = 33.54 atm at Tc given in the literature 35 : (logens) 0 = ] = 0. 2. Reproduction of the most commonly used value for
ec = [d(l°geTrs)/d(l°geO)] c = 6.0. 4. Prediction of the normal boiling point at the temperature T b = 77.352 K: (logens) 0 = 0b = log e (1 atm/Pe). 5. Exact reproduction of the triple point (Ttr = 63.145 K; Ptr = 94.6 tort = 0.1245 atm): (logeTrs)0 = 0 tr = (0.1245 atm/Pc) Prediction of the named boundary and auxiliary conditions (with the exception of the Plank criterion) exerts an advantageous effect on the quality of the new vapour pressure equation. Fixing of the gradient of the a-curve at the critical point influences the reproduction of the measured values and the curve of the derivatives dns/dO and d2ns/dO 2 in an unfavourable manner and was therefore not taken into account. As a result of the slight change in the prediction for the critical pressure (Pc, new = 33.655 atm) and the value of a c ( = 6.09) a furthe~ improvement was achieved. Thus the new vapour pressure equation is as follows: 4
logeTrs = E N i O i -
2 + N508 + N609 + N71oge(N8_ 0) (15)
i=1 with the following values for the constants: N] = - 7 . 0 4 4 6 2 2 N4=4.750931 N7=-0.104115
N 2 = 12.167604 N 5 = -0.504249 N 8 = 1.069222
N 3 = -9.862838 N 6 =0.215142
The values o f N 1 to N 8 were determined by the least squares fit taking into account constraints. 17
217
The average relative error APs' rel = (1 - Ps, meas/Ps, calc) amounts to 0.14%. Measured results in the vicinity of the critical point were reproduced with particular accuracy (Fig.3). The vapour pressure formula over the whole temperature range provides a thermodynamically tenable, oscillation-free curve for the derivatives t~, dot/dO, d(logeZrs)/ d(1/O) and d2(logelrs)/d(1/O ) 2.
O.50 o .
~O.25
_~
1. Optimum reproduction of the smoothed men~ured values 15 for the reduced temperature range 0tr ~< 0 ~< 0 c and tenable reproduction of the derivatives in the 0, 6 1 graph.
The following equation provides good reproduction of the smoothed values in relationship to the transformed temperature 0 t = (1 - 0) 0.42 (see Fig.4): 3
(17)
NiOi + N409 + N5 l°ge0
i=1 N 2 =-1.503284 N5 = 4.313671
o
o
o° o °°
ooo
o°o
/ o°°
__o 0.50
o.s
g
o
o
o
;.7
I
I
0.8
o
0.9
1.o
Reduced temperoture,9
0.8
~'-
0.5
__8
0
Q.
"~ - 0 . 5 I
I
I
0.6
0.7
0.8
I
0.9
1.0
Reduced temperature, O
4
NiO ti + N50 t9 + N 6 loge0
Oo
OOooa
ooo¢O
o.2s
o
o
o
o
g
o
o
oo
o "--,,-.~o °
•
o
°
co
oo
N 3 = 6.417808
Hardly any simple analytic means of determining the relationship p" = p"(T) are available in the literature. If however (17) is expanded by one term and if the constants are made to fit the data on the saturated vapour line, then the following expression is obtained:
l°ge6" = E
9
-o.a 0.5
N 1 = 2.494026 N4 = 4.109196
o
o
o
Fig.4 Relative deviation Ap're I = (1 -- P'calctP'meas) x 100 of the values measured for the saturated liquid density 15 and relative deviation of the values calculated solely from the equation of state (19) (extended curve) from those calculated from equation 17 for nitrogen
2. Exact reproduction of the reduced critical density at 0 = 1:6'(0 = 1)=6"(0 = 1)= I.
l°ge6' = E
o
o
t-~
Equations for the saturated liquid and vapour density. The equations to be developed for calculating the reduced saturated liquid density 6' = 6'(0) = p'(T)/Oc and the saturated vapour density 6" = 6 "(0) = p"(T)/Oc only had to fulfill two conditions:
o
(18)
Fig.5 R e l a t i v e d e v i a t i o n A p , re I = (1 - - P " c a l c / P " s m o o t h e d ) x 100 of the data calculated solely from the equation of state (19) from the values obtained from equation 18
the temperature. The relative deviation as compared with the smoothed data (see Table 9.715) represented in Fig.5 lies below 0.01% over the whole temperature range.
i=1 N 1 =-0.297090 N 4 = 21.263018
N 2 =-7.281406 N 5 = 25.748240
Equation
N 3 = 27.164961 N 6 =42.746767
On account of the fact that doubts exist in the literature regarding the applicability of the Maxwell rule to any equation of state, test calculations were carried out with several known equations. Here it was found that only with the Benedict-Webb-Rubin equation 38 and its extensions, 35,39-41 is success achieved in reproducing phase equilibrium data for light, saturated hydrocarbons in the temperature range of T > 0.7 Tc with an accuracy of approximately 3%.
with 0 t = (1 - 0) 0.29. With the aid of (18), the saturated vapour density can readily be calculated as a function of -
0.8 0.5 o
.%*o~
% aa o
o
a
o°
o oa o °
o
8 °_ o
o o ~
o
o
-0.5 I
-O.8 ce
03
0.6
I
I
0.7
0.8
I
0.9
Reduced ternperoture, O Fig.3 R e l a t i v e d e v i a t i o n Aps " rel = (1 of the measured vapour pressure values of the values calculated solely from the (extended curve) from those calculated equation (16) for nitrogen
218
~-~Ps calc/Ps meas) x 1 0 0 i o a~d relative deviation equation of state (19) from the vapour pressure
of state
1.0
In addition attempts were made to make the BWR equation in the form put forward by Strobridge, 35 fit for one isotherm the predictions 1 and 2 stated in the section 'Description of the problem'. Here it was found that prediction of phase equilibrium data was possible for a single isotherm. If however the method was applied to the determination of constants for several sub-critical isotherms at the same time, pronounced dispersions were observed. (Parallel to this work Bender 42 succeeded in determining the constants in an exten ded Strobridge equation taking into consideration the Maxwell rule.) Moreover reproduction of the second virial coefficient was influenced unfavourably. The experiments
C R Y O G E N I C S . J U N E 1972
were used to serve as a basis for the new development o f an equation o f state. For the new equation of state the constants were selected from a general virial statement up to the 13th power in/5 and an isochoric representation consisting of seven terms, in such a manner that an optimum fit with the individual conditions was obtained. In accordance with the above the equation of state in the reduced form (see Table 2) is as follows: lr = 00/5 + Br(O)oO/52
NIO + N 2 +N31oge0 +
NiO-i+3
E
/53
i=4 + (N80 + N 9 +N101Oge0 + N l 1 0 " 4 ) 6 4 13.
+ (N120"2 +N130"3 + N 1 4 0 - 4 ~ 5 + (NlslOge0 +N160"1)~6 +N17/57 + N180-168 +N190-1/59 +N200-1/511
(19)
with the following values for the constants: N 1 = 2.898009804 x 101 N 3 = - 1 . 1 0 3 6 1 6 7 3 3 x 102 N 5 = 3.730999120 x 101 N T = 4.013569816 x 10 -2 N g = 5.337164652 x 10 ° N i l = 9.574551968 x 10 -2 N13 = - 1 . 4 0 6 3 4 6 4 9 8 x 10 ° N15 = 5.826957437 x 10 -1 NI7 = 6.980906270 x 10 -2 N19 = - 1 . 0 0 9 9 3 6 6 7 1 x 10 1
N 2 = 8.009920081 x 101 N 4 = - 1 . 3 9 7 5 5 1 4 0 0 x 102 N 6 = - 5 . 1 7 3 7 5 1 5 3 4 x 10 ° N 8 = 4.731825202 x 10 ° NIO = 4.210224653 x 10 ° N12 = 2.953938544 x 10 ° NI4 = 2.083912318 x 10 "1 N16 = 1.497232942 x 10 ° N18 = 3.397909092 x 10 -1 N20 = 2.438999293 x 10 -3
I
2
5
IO
20
40
Density p, tool I -t
Fig.6 Measured isotherms of nitrogen in a p, p-diagram. Representation of the range of validity of the equation of state (19)
T h e e x p r e s s i o n B r ( O ) i s t h e r e d u c e d second virialcoefficient (compare equation 15).
the experimentally obtained enthalpy 44 in the gas and liquid ranges can also be satisfactorily reproduced with the proposed equation of state.
Discussion
The new equation of state (19) covers the range o f state within the boundaries quoted in (8). This range is shown in Fig.6. The average relative error of all measured values 15 amounts to 0.45% for the pressure in the gas range (p <~Pc) and to 0.86% for the density in the liquid range (p > Pc)" On application of a relationship for the specific isobaric heat capacity c~ in the state of the ideal gas for nitrogen, for example, equation 20 43 o Cp
T N3 =N[ +N 2 -+ (cal mole -1 K "l) 1K (T/1 K) 2
(20) [exp(N 5)
N 1 = 6.9550789 N4=2.006387
112 N 2 = 8.0448 x 10 -6 N 3 = 0.3786 U 5 = 3 3 5 3 . 4 0 6 1 / ( T / 1 K)
C R Y O G E N I C S . J U N E 1972
As an exainple, the saturation values calculated in accordance with the scheme in Fig.7 and the vaporization enthalpy z2dt are listed for nitrogen in Table 3 for certain temperatures. Ah = A h r R T c was determined with
N 2 exp(N5) + N4
Phase equilibria. The state variables on the liquid-gaseous phase boundary curve can be determined with the proposed equation of state alone, from the triple point to the critical point. In addition to the vapour pressure and the saturated liquid and vapour densities extremely accurate values are also obtained for the vaporization enthalpy.
Ah r = o
6"
6'
+o
7r - 0 ,
;
(21)
6
The relative deviations in the values ofPs, p', p", and Ah in relation to the measured data or in relation to the values
219
Table 3. Variables of state calculated solely with the aid of the equation of state (19): vapour pressure, density of saturated liquid and vapour as well as vaporization enthalpy for nitrogen Ah =h"-h',
T,K
, Ps, atm p',moler 1 p",molel "1 Jrnole "1
65 70 75 80 85 90 95 100 105 110 115 120 125 126.26
0.1721 0.3817 0.7509 1.348 2.254 3.556 5.345 7.714 10.755 14.557 19.217 24.855 31.682 33.655
30.7265 29.9918 29.1875 28.3458 27.4704 26.5539 25.5851 24.5485 23.4209 22.1614 20.6870 18.7803 15.2662 11.25
0.03265 0.06788 0.1264 0.2169 0.3498 0.5378 0.7967 1.147 1.618 2.255 3.147 4.511 7.480 11.25
lished permitting phase equilibrium calculations to be applied to mixtures too without additional relationships being required. In addition the new equation of state for nitrogen allows the whole fluid range to be described in accordance with the measured values in the limits Ttr(63 K) ~< T~< 3Tc (about 400 K)
6 019.9 5817.8 5 638.1 5 470.6 5 292.3 5 083.9 4 832.7 4 530.8 4 170.9 3741.9 3218.4 2 531.8 1 308.3 0
Ptr(0.12 atm) <. p <<.6Pc (about 200 atm) 0 ~
I
i
I
I
I Ps (estimated),AP, et =I¢~,Sz-- I I
t
p'(r,p )--equation of state-~'(r,p ) I
I f'(T~p~ ~equation H.,
1 lA,new-.-,.'-,'l
calculated from the auxiliary equations are shown as a function of the temperature in Figs 3 - 5 and 8. Fig.9 shows how the quotient of the fugacity of the liquid and gaseous phase varies in relation to the reduced temperature. For reliable calculations of the values Ps, P', and p" the following must apply:
f"(0,8") ['(0,8')
I
Yes Yes
,,.o
)
n',o
C'A'new/,/A'o,d) y~s
(22)
I Ap---Ap
If the condition in (7) is not fulfilled by the equation of state, then the expression f " / f ' is particularly dependent on the temperature:
f"(o, ~") - ;'(0,a')
,,/
n,o
( No
- constant = 1
I
of stote~f"(T,p") I
•. ~ S i g n
I
(A:new)* sign ( A / o l d ) )
y ,s
I p--Ap/2 I 1 F(O ) =~constant
1
I A,old--A,ne.;p --p,-Ap I
(23)
j Curve o f the isotherms in the two phase region. If the
Result
I
T,p s , p" p'
Maxwell criterion is taken into account during determination of the constants of the equation of state (19), then for certain sub-critical isotherms the dot-dashed curve shown in Fig.1 may arise. The area fit remains valid but in the region of the sub-critical isotherm between points A and B, a stable phase could be possible since the expression 45 applies: (24) T If the constants for (19) are determined in the manner described then for all sub-critical isotherms only three points of intersection with the corresponding isobars are obtained in each case (compare with Fig.l, dashed curve).
Fig.7 Scheme f o r calculating the phase equilibrium values ps,p', and p" for a selected temperature T
o-8 I
-~o.s °
0 -~. 0 g
i
;
,~-O.S 0'.6
017 018 Reduced temperolure O
019
I.O
Summary If an equation of state fulf'dls the Maxwell rule in the region Ttr < T < T c, an important prerequisite is estab-
220
Fig.8 Relative deviation Ahre I = (1 -- AhcalclAhroeas) x 1OO of the values measured for the vaporization enthalpy ,o~ f r o m the values calculated solely f r o m the equation of state (19)
CRYOGENICS. JUNE 1972
8
1.2
Benedict, M., Webb, G. B., and Rubin, L. C. Chem Engng Progress 47 (1951) 419
9
o.6 °1
0.8
"without
-I / 0
J
10 consideration of the condition equation (7) ti
11 12
13 14
0.4
15 16 17 18 19
0.2
01 O.S
i
i
l
i
0.6
0.7
0.8
0.9
1.0
Reduced tcml>znatur¢ Fig.9 Curve of the quotient f"/f" in relation to the reduced temperature (calculated in accordance with equation 19)
20 21 22 23 24 25 26 27
The e q u a t i o n o f state (19) also fulfills the auxiliary conditions at the critical point. It undergoes a transition to the e q u a t i o n o f state o f the ideal gas on f o r m a t i o n o f the limiting value p -+ 0 and contains directly an expression for exact r e p r o d u c t i o n o f the second virial coefficient. In addition new relationships for the vapour pressure and the densities on the phase b o u n d a r y curves as well as an e q u a t i o n for the second virial coefficient was established and c o m p a r e d with existing relationships.
30 31 32 33 34 35 36 37 38
References Rombusch, U. K., and Giesen, H. Kdltetechnik 16 (1964) 66 Hadden, S. T. Chem Engng Prog Symp Series 49 (1953) 53 Cajander, B. C., Hipkin, H. G., and Lenoir, J. M. J Chem EngngData 5 (1960) 251 Sood, S. K., and Haselden, G. G. Cryogenics 10 (1970) 199 Loftier, H. J. Thermodynamik, Bd II: Gemische und Chemische Reaktionen (Springer-Verlag, Berlin, New York, 1969) Prausnitz, J. M. Molecular Thermodynamics of Fluid-PhaseEquilibria (Prentice-HaU, 1969) Beattie, J. A., and lkehara, S. Proc A m A cad Arts and Sci 64 (1930) 127
CRYOGENICS.
28 29
J U N E 1972
39 40 41 42 43 44 45
Wagner, W. Eine neue Berechnungsformel zur Ermittlung der Konstanten einer the mischen Zustandsgleichung ftir binffrc Gemische (unpublished) Rossini, F. D. (ed), and Beattie, J. A. Thermodynamics and Physics of Matter (Princeton University Press, New Jersey, 1955) Stein, W. A. Forsch Ing Wesen 34 (1968) 193 Baehr, H. D. Zur Thermodynamik der Zweiphasengleichgewichte (Verlag der Akademie der Wiss u d Lit in Mainz in Kommission bei Franz Steiner Verlag Gmbtl Wiesbaden, 1952) Lewis, G. N. Z P h y s C h e m 3 8 ( 1 9 0 1 ) 2 0 5 L/Jffler, H. J. Thermodynamik, Bd I Grundlagen und Anwendung auf reine Stoffe (Springer-Verlag, Berlin, 1969) Wagner, W. Dissertation TU Braunschweig (1970) Maxwell, J. C. Nature 35 (1875) 357 Hust, J. G., and McCarty, R. D. Cryogenics 7 (1967) 200 Morsy, T. E. Dissertation TIt Karlsruhe (1963) Straub, D., Schaber, A., and Morsy, T. E. Kd/tetechnik 17 (1965) 212 Van ltterbeek, A., De Roop, W., and Forrez, G. A ppl Sci Res Section A 6 (1957) 421 Lestz, S. S. J Chem Phys 38 (1961 ) 2830 Wiistner, R. Report of Institut fiir Thermodynamik der TU Braunschweig (1970) Crommelin, C. A. Corn Phys Lab Leiden 145 d (1914) 29 Cath, P.C. ComPhysLabLeiden 152d(1918) 45 Porter, F., and Perry, J. H. J A m e r Chem Soc 48 (1926) 2059 Dodge, B. F., and Davis, H. N. J C h e m A m e r S o c 4 9 ( 1 9 2 7 ) 610 Giauque, W. F., and Clayton, J. O. J A mer Chem Soc 55 (1953) 4875 Henning, F., and Otto, J. Phys Z 37 (1936) 633 Friedman, A. S., and White, D. J A mer Chem Soc 72 (1950) 3931 White, D., Friedman, A. S., and Johnston, H. L. J A mer Chem Soc 73 (1951) 5713 Michels, A., Waasenaar, T., De Graaf, W., and Prins, C. Physica 19 (1953) 26 Armstrong, G. T. J Res NBS 53 (1954) 263 Riedel, L. Chem Ing Techn 26 (1954) 263 Stein, W. A. Kidtetechnik-Klirnatisierung 22 (1970) 7 Strobridge, T. R. NBS Technical Note 129 A (1963) Martin, J. J., Kapoor, R. M., and Shinn, R. D. Dechema Monograph 32 (1958) 46 Hank, R., and Riedel, L. IngArch 16 (1948) 255 Benedict, M., Webb, G. B., and Rubin, L. C. J Chem Phys 8 (1940) 334 Bloomer, O. T., and Rao, K. N. Inst (;as Technology Res Bull 18 (1951) Morsy, T. E. Kiiltetechnik 17 (1965) 273 Stewart, R. B., Prydz, P., and Timmerhaus, K. D. Advances in Cryogenic Engng 13 (1968) 384 Bender, E. KhTtetechnik-Klimatisierung (forthcoming) Barieau, R. E. J Phys Chern 69 (1965) 495 Mage, D. T., Jones, M. L. Jr, Katz, D. L , and Roebuck, J. R. Chem Engng Progr Symp Series 59 (1963) 61 Haase, R. Thermodynamik der Mischphasen (SpringerVerlag, Berlin, 1956)
221
A method to establish equations of state exactly representing all saturated state variables applied to nitrogen W. Wagner A knowledge of liquid-vapour phase equilibria and state variables is of decisive importance for the technical systems involved in processes of gas liquefaction, gas separation, and refrigeration. For this reason a great deal of attention has been given in recent years to methods of calculating liquid-vapour phase equilibrium data for pure substances and multicomponent mixtures. The mixtures mainly of concern are binary or ternary systems whose pure components have low boiling points. These include combinations of various pure and halogen-substitutes hydrocarbons [for example, n-pentanepropane-methane, difluoromonochloromethane (R 22)pentafluoromonochloroethane (R 115)] and systems involving cryogenic substances (for example helium-nitrogen, helium-hydrogen-carbon dioxide). Exact prediction of the required phase equilibrium data for the economic operation of such equipment is very difficult if conventional methods are employed. The values determined are often so inexact that the equipment must be inordinately over-dimensioned if the quantity and quality of the product are to be guaranteed. This leads to an increase in investment and operation costs.
Traditional methods of determining phase equilibrium data
The author is with the Institute of Thermodynamics, Technical University, Brunswick, Germany. Received 25 September 1971.
214
Ps = Ps(P", T)
(1)
(2)
The value of the saturated liquid density p' is calculated as a function of the temperature from an empirical relationship
p' = p'(T)
(3)
Since the phase equilibrium data Ps, P', and p" are related to one another for a fixed temperature (see equation 7) but as (1) and (3) are independent of one another, inconsistent phase equilibrium values may result from calculations. 1 In contrast to this the phase equilibrium values, in addition to the state variables of the homogeneous phases, could also be calculated consistently from one suitable equation of state.
Mixtures. The basis for calculating phase equilibrium data for mixtures at a fixed temperature and a known total pressure is provided by the conditions T '= T";
Pure substances. The data for the liquid vapour equilibria of a pure substance is normally calculated with the aid of three equations which are independent of one another. The density of the saturated vapour p" can be determined from an equation of state applying only to the gas phase if the saturation pressure is known at each temperature, for example, from a vapour pressure equation Ps = Ps(T)
The variable p" is then arrived at by iteration from the equation of state
P' =P";
3~' =3~"
(4)
The temperature T, the pressure p, and the fugacity f of the component i of the mixture in the gaseous and liquid phase must be equal respectively. Phase equilibrium data determined experimentally can be extrapolated and interpolated with the aid of empirical graphs. 2,3 More accurate values are obtained by application of analytical methods which are based on exact thermodynamic relationships.4,5, 6 The difficulties encountered in calculating phase equilibrium data for mixtures thus do not result from a lack of exact equations. The main source of inaccuracies lies in the difficulty in determining the required thermodynamic values in the specific molar volume v', and in the fact that
C R Y O G E N I C S . J U N E 1972
the partial molar volume ?[ of the component i of the liquid mixture are very unreliable particularly at high equilibrium pressures. As a basic prerequisite for determining the above-mentioned variables it is also necessary to measure certain phase equilibrium values directly for each mixture. Moreover the methods are rather complicated and usually have to be modified to an extreme extent to suit various types of mixtures.
Description of the problem. The uncertainties described above in the determination of the fugacity f ' of the liquid phase of pure substances and mixtures could be avoided and the number of required equations could be reduced to one if the fugacity f ' could be calculated directly from an equation of state. For this purpose the following conditions would have to be fulfilled: 1. An equation of state for pure fluid substances must be available which permits the whole fluid range to be reproduced with a single set of constants approximately up to the three-fold critical density. 2. This equation of state must be constructed in such a manner that for the pure components prediction of the vapour pressure and the saturation densities on the phase boundary curves is possible with the aid of this equation alone. 3. Rules for calculating the combination of constants of the equation of state of the pure substances must be available. Combination rules of this type already exist. 7,8,9 When taking into consideration the ~reconditions t to 3 listed above, the fugacity equation 1"
The curve of an isotherm in the p, v diagram is shown in Fig.1. The equation of state describes the real behaviour in the two homogeneous states (gas and liquid) and prorides the curve of the 'theoretical' isotherm in the twophase range (dotted curve in Fig. 1). The equilibrium existing between the saturated liquid and the saturated vapour demands that the conditions in (4) be fulfilled. If the usual thermodynamic relationships for f ' a n d f " are substituted into (4), 13,14,15 then the relationship known under the name 'Maxwell criterion' is obtained 16 V st
Ps(V" - v') = I p(v, T) r dv
(6)
i 12
By substitution of the density p for specific volume v we obtain the equation p'
Ps
-
p,
-
=
,
,,
p~,r)
r
(7)
02
P Geometrically this condition implies that the area under the isotherm T = T' = T" between the limits v" and v' must be equal to an area formed by the rectangle with the corners l, 2, 3, 4 (see Fig.l). The equation to be developed should also fulfil the 'normal' demands which can be expected from an equation of state. The individual conditions are as follows:
\YiP/
= RTl°ge ~-P --
+
1. The p, p, T data measured in the range (subscript tr = triple point)
P -
+
o
~ni J
T, V, n/~i
R T
f
- -
Ttr ~< T ~< 400 K
(5)
P
Ptr ~
0 <<.p<.3Pk for determining the variables fi and fi' need only be solved within the limits p = 0 and p = p" or p = p'. In the above equation R is the universal gas constant, y the molar concentration, V the total volume, n the number of moles, and i a serial index. This method used together with the traditional equations of state produces incorrect values for the fugacity f ' even for the pure substances, l, 11,12 Thus using the example of nitrogen a new method will be given here for establishing equations of state, initially for pure fluid substances, taking into consideration preconditions 1 and 2. Using such equations alone it will then be possible to calculate phase equilibrium values firstly for pure substances and secondly utilizing suitable combination rules for mixtures consistent with the data of the homogeneous phases.
(8)
must be reproduced to an optimum extenl. Critical point
Liquid
~ "
~
Area A I
Area [FT[Y~ Area AI2'
A .~ 1 2 ~
['--3 Area A2*
Q.
ff (3-
;3 Demands placed on the equation of state If an equation of state p = p(v, T) applies to the whole fluid range, that is, for the gas and liquid states then an equilibrium relationship for calculating the thermal variables of state on the phase boundary curve can be derived.
C R Y O G E N I C S . JUNE 1972
F"
g"
Specific volume, V Fig.1 Representation of the Maxwell criterion. Dashed isotherm: qualitative curve in accordance with (19). Dot--dash isotherm: mathematically possible curve of isotherm in two-phase region. On fulfilment of the Maxwell criterion A1 = A2 or AI" = A2"
215
2. By taking the limit p ~ 0 the equation of state for ideal gases can be derived directly from the equation of state for real pure substances.
p = R Tp
(9)
3. The following should apply at the critical point (subscript c):
The subsidiary conditions at the critical point are not included in the fitting procedure but are exactly fulfilled with the aid of Lagrange multiplicators. In order that the coefficients of the equation of state fit the measured p, P, T values of the homogeneous phase (number: n l ) the first partial sum of(11) is required n1
p(rc, pc) pc
(lOa)
[(Op/OT)p]c = °tc = (TIPsx dPs/dT)c
(lOb)
[(DP/aP)T] c = 0 = [ ( a 2 p / a p 2 ) T ] c
(lOc)
lim(a2p/aT2)t) = 0
(lOd)
=
T-.).=
Glr2= E 1
G 1 [p(T, P)calc - p(T, P)meas] 2
Condition 4 ensures that extrapolation to regions of low density and temperature will produce reliable results. In addition the usual combination rules for the second virial coefficient will apply with the corresponding constants in an equation of state for the mixture. Determination of the constants The coefficients of an equation of state are on the other hand determined in such a manner that the system simultaneously fits the measured values of the homogenous phase including the phase boundary curve and the equilibrium condition (equation 7). The individual conditions are here taken into account as partial sums in the least square relations LSR.
G1r2 +
n,)
G2r2 + E
G3r2
(12)
1
The second partial sum n2
n2
,-
4. Measured results for the second virial coefficient should be described as exactly as possible.
LSR =
E
n1
1
G 2 [p(T, O'smoothed) -
Ps(T)] 2 +
G2 [p(T, Osmoothed)-
Ps(T)] 2
1
n2 +E
(13)
1
serves to reproduce n2, p, p, T values on the saturated liquid and saturated vapour line consistent with the measured data of the homogeneous phases. The Maxwell rule (equation 7) is fulfilled with the aid of the third partial sum:
1
L# (14)
-+Min (11)
1
Auxiliary equations G = weight factor, suitable chosen; r
=
residue
Table 1. Selected critical data and important constants for nitrogen Critical t e m p e r a t u r e Critical pressure Critical d e n s i t y Universal gas c o n s t a n t M o l a r mass Conversion f a c t o r
Tc Pc Pc R
= = = =
126.26 K 33.655 atm 11.25 mole 1-1 0 . 0 8 2 0 5 7 4 I arm mole 1 K 1
M = 2 8 . 0 1 6 g mole -1 11 = 1 0 0 0 . 0 2 8 cm 3
Table 2. Reduced variables employed Reduced Reduced Reduced Reduced Reduced Reduced Reduced
216
temperature pressure density second virial c o e f f i c i e n t gas c o n s t a n t specific e n t h a l p y specific v a p o r i z a t i o n e n t h a l p y
0 = T/T c 7r ~ Br o hr z~hr
= P/Pc =P/Pc = BPc = RTcPc/Pc = h / R Tc = Ah/RT c
The existing, highly dispersed results 15 measured for the phase boundary curves are not sufficient for effective determination of the constants in the equation of state. The measured data for the saturated liquid and saturated vapour line only contain pairs of values for T and p' or p". It is therefore necessary to allocate a pressure to each measured temperature. This cannot be determined directly from the data measured for the vapour pressure since the temperature values of vapour pressure measurements do not coincide with those of the density measurements. For this reason interpolation equations for the saturated liquid and vapour densities and a vapour pressure equation must be established. In addition an analytical relationship for the second virial coefficient is developed which is required when the fourth condition is taken into account. Auxiliary equations need not be developed if reliable information on the relevant substance for calculating the abovementioned values is available in the literature. In the sections below, these auxiliary equations are given for nitrogen (for detailed treatment see WagnerlS). The individual values were in each case reduced to the critical data (see Tables 1 and 2 as well as by WagnerlS).
C R Y O G E N I C S . J U N E 1972
The second virial coefficient. The analytical relationship for the second virial coefficient B(T) should fulfill the
O.I <1
following conditions: 1. Close reproduction of the reduced measured values B r as listed by Wagner 15 must be obtained.
O
L80°
o
S
g o
o o
'~x x
o
o
o~
oO
o
o
~'
I10 = 1.73
d~
<
1
1.7
I
t.5
I.'25
--
1.0
0175
0150
R¢ciprocol reduced temperature, I ] 0
0.906788 (15)
fulfilled the above-mentioned conditions best and was employed to describe the curve of the second virial coefficient for nitrogen.
mits the measured values in the critical range (T > 0.9 Tc) to be calculated with high accuracy.
The constants N 1 to N 4 were determined by the method of least square errors. The accuracy of the values measured for Br(O ) decreases pronouncedly in the direction of low temperatures. In order to take this factor into account the squared sum of the relative errors was minimized. For this reason the function Br(O ) had first to be shifted in such a manner that it did not pass through zero.
The following boundary and auxiliary conditions were tested for various approaches:
The deviation of the individual measured values from (15) is shown in Fig.2. The analysed derivatives dBr/d(1/O ) and d2Br/d(1/O) 2 exhibited a curve completely without oscillations. The calculated Boyle temperature is TB = 325.26 K and thus lies only 0.4 K below the value theoretically obtained by Morsy. 18 The inversion temperature has the value T1 = 620 K and thus deviates by 2.1% from the data obtained by Straub et al. 19
3. Inclusion of the Plank criterion 37. (da/dO)c = O.
A strict test of the quality of an equation for the second virial coefficient is provided by a comparison o f measured and calculated specific isochoric and isobaric residual heat capacities. To carry out this test, measured values obtained by van Itterbeek 20 and Lestz 21 for Aev/R and Acp/R were used. The agreement for reduced pressures ir up to approximately n -~ 1 is excellent.
Vapour pressure curve. The commonest known equations for nitrogen 23-35 were tested in a preliminary investigation 22 by a comparison with 154 measured values. 15 Here it turned out that equations of simple structure either only adequately reproduce the individual authors' own measured results or are only applicable to a small part of the whole temperature range. The vapour pressure formulas put forward by Riedel, 33 Stein, 34 and Strobridge 35 on the other hand permit exact calculation of the vapour pressure. At the extremes of the temperature range the s-sharped curve of the vapour pressure curve is not completely described by a logen s, 1/0 graph. As a result of this the relative error in the vicinity of the critical point lies above the average mean error of the whole range. On the basis of a proposal put forward by Martin, 36 a new vapour pressure equation was established which also per-
C R Y O G E N I C S . JUNE 1972
0.35
Fig.2 Absolute deviation AB r = Br, meas -- Br, calc of the reduced measured values Br, meas for the second virial coefficient 1 5 in relationship to the data calculated in accordance w i t h (19) for nitrogen
B r = N 1 + N 2 0 - I +N30"tlogeO +N40-2
N 4 = -0.724141
O
-O~1
The equation
N3=-0.021492
I0 =
o
3. The specific residual heat capacities at constant pressure Acp and constant volume Ac v should be qualitatively reproduced with correct values in the range of low densities (slightly real behaviour).
N2 =
°'~'1
O
2. The sqcond derivative d2Br/d(1/O) 2 should exhibit a curve which is thermodynamically tenable.
N 1 = 0.469015
o
E
1. Exact calculation of the critical pressure Pc = 33.54 atm at Tc given in the literature 35 : (logens) 0 = ] = 0. 2. Reproduction of the most commonly used value for
ec = [d(l°geTrs)/d(l°geO)] c = 6.0. 4. Prediction of the normal boiling point at the temperature T b = 77.352 K: (logens) 0 = 0b = log e (1 atm/Pe). 5. Exact reproduction of the triple point (Ttr = 63.145 K; Ptr = 94.6 tort = 0.1245 atm): (logeTrs)0 = 0 tr = (0.1245 atm/Pc) Prediction of the named boundary and auxiliary conditions (with the exception of the Plank criterion) exerts an advantageous effect on the quality of the new vapour pressure equation. Fixing of the gradient of the a-curve at the critical point influences the reproduction of the measured values and the curve of the derivatives dns/dO and d2ns/dO 2 in an unfavourable manner and was therefore not taken into account. As a result of the slight change in the prediction for the critical pressure (Pc, new = 33.655 atm) and the value of a c ( = 6.09) a furthe~ improvement was achieved. Thus the new vapour pressure equation is as follows: 4
logeTrs = E N i O i -
2 + N508 + N609 + N71oge(N8_ 0) (15)
i=1 with the following values for the constants: N] = - 7 . 0 4 4 6 2 2 N4=4.750931 N7=-0.104115
N 2 = 12.167604 N 5 = -0.504249 N 8 = 1.069222
N 3 = -9.862838 N 6 =0.215142
The values o f N 1 to N 8 were determined by the least squares fit taking into account constraints. 17
217
The average relative error APs' rel = (1 - Ps, meas/Ps, calc) amounts to 0.14%. Measured results in the vicinity of the critical point were reproduced with particular accuracy (Fig.3). The vapour pressure formula over the whole temperature range provides a thermodynamically tenable, oscillation-free curve for the derivatives t~, dot/dO, d(logeZrs)/ d(1/O) and d2(logelrs)/d(1/O ) 2.
O.50 o .
~O.25
_~
1. Optimum reproduction of the smoothed men~ured values 15 for the reduced temperature range 0tr ~< 0 ~< 0 c and tenable reproduction of the derivatives in the 0, 6 1 graph.
The following equation provides good reproduction of the smoothed values in relationship to the transformed temperature 0 t = (1 - 0) 0.42 (see Fig.4): 3
(17)
NiOi + N409 + N5 l°ge0
i=1 N 2 =-1.503284 N5 = 4.313671
o
o
o° o °°
ooo
o°o
/ o°°
__o 0.50
o.s
g
o
o
o
;.7
I
I
0.8
o
0.9
1.o
Reduced temperoture,9
0.8
~'-
0.5
__8
0
Q.
"~ - 0 . 5 I
I
I
0.6
0.7
0.8
I
0.9
1.0
Reduced temperature, O
4
NiO ti + N50 t9 + N 6 loge0
Oo
OOooa
ooo¢O
o.2s
o
o
o
o
g
o
o
oo
o "--,,-.~o °
•
o
°
co
oo
N 3 = 6.417808
Hardly any simple analytic means of determining the relationship p" = p"(T) are available in the literature. If however (17) is expanded by one term and if the constants are made to fit the data on the saturated vapour line, then the following expression is obtained:
l°ge6" = E
9
-o.a 0.5
N 1 = 2.494026 N4 = 4.109196
o
o
o
Fig.4 Relative deviation Ap're I = (1 -- P'calctP'meas) x 100 of the values measured for the saturated liquid density 15 and relative deviation of the values calculated solely from the equation of state (19) (extended curve) from those calculated from equation 17 for nitrogen
2. Exact reproduction of the reduced critical density at 0 = 1:6'(0 = 1)=6"(0 = 1)= I.
l°ge6' = E
o
o
t-~
Equations for the saturated liquid and vapour density. The equations to be developed for calculating the reduced saturated liquid density 6' = 6'(0) = p'(T)/Oc and the saturated vapour density 6" = 6 "(0) = p"(T)/Oc only had to fulfill two conditions:
o
(18)
Fig.5 R e l a t i v e d e v i a t i o n A p , re I = (1 - - P " c a l c / P " s m o o t h e d ) x 100 of the data calculated solely from the equation of state (19) from the values obtained from equation 18
the temperature. The relative deviation as compared with the smoothed data (see Table 9.715) represented in Fig.5 lies below 0.01% over the whole temperature range.
i=1 N 1 =-0.297090 N 4 = 21.263018
N 2 =-7.281406 N 5 = 25.748240
Equation
N 3 = 27.164961 N 6 =42.746767
On account of the fact that doubts exist in the literature regarding the applicability of the Maxwell rule to any equation of state, test calculations were carried out with several known equations. Here it was found that only with the Benedict-Webb-Rubin equation 38 and its extensions, 35,39-41 is success achieved in reproducing phase equilibrium data for light, saturated hydrocarbons in the temperature range of T > 0.7 Tc with an accuracy of approximately 3%.
with 0 t = (1 - 0) 0.29. With the aid of (18), the saturated vapour density can readily be calculated as a function of -
0.8 0.5 o
.%*o~
% aa o
o
a
o°
o oa o °
o
8 °_ o
o o ~
o
o
-0.5 I
-O.8 ce
03
0.6
I
I
0.7
0.8
I
0.9
Reduced ternperoture, O Fig.3 R e l a t i v e d e v i a t i o n Aps " rel = (1 of the measured vapour pressure values of the values calculated solely from the (extended curve) from those calculated equation (16) for nitrogen
218
~-~Ps calc/Ps meas) x 1 0 0 i o a~d relative deviation equation of state (19) from the vapour pressure
of state
1.0
In addition attempts were made to make the BWR equation in the form put forward by Strobridge, 35 fit for one isotherm the predictions 1 and 2 stated in the section 'Description of the problem'. Here it was found that prediction of phase equilibrium data was possible for a single isotherm. If however the method was applied to the determination of constants for several sub-critical isotherms at the same time, pronounced dispersions were observed. (Parallel to this work Bender 42 succeeded in determining the constants in an exten ded Strobridge equation taking into consideration the Maxwell rule.) Moreover reproduction of the second virial coefficient was influenced unfavourably. The experiments
C R Y O G E N I C S . J U N E 1972
were used to serve as a basis for the new development o f an equation o f state. For the new equation of state the constants were selected from a general virial statement up to the 13th power in/5 and an isochoric representation consisting of seven terms, in such a manner that an optimum fit with the individual conditions was obtained. In accordance with the above the equation of state in the reduced form (see Table 2) is as follows: lr = 00/5 + Br(O)oO/52
NIO + N 2 +N31oge0 +
NiO-i+3
E
/53
i=4 + (N80 + N 9 +N101Oge0 + N l 1 0 " 4 ) 6 4 13.
+ (N120"2 +N130"3 + N 1 4 0 - 4 ~ 5 + (NlslOge0 +N160"1)~6 +N17/57 + N180-168 +N190-1/59 +N200-1/511
(19)
with the following values for the constants: N 1 = 2.898009804 x 101 N 3 = - 1 . 1 0 3 6 1 6 7 3 3 x 102 N 5 = 3.730999120 x 101 N T = 4.013569816 x 10 -2 N g = 5.337164652 x 10 ° N i l = 9.574551968 x 10 -2 N13 = - 1 . 4 0 6 3 4 6 4 9 8 x 10 ° N15 = 5.826957437 x 10 -1 NI7 = 6.980906270 x 10 -2 N19 = - 1 . 0 0 9 9 3 6 6 7 1 x 10 1
N 2 = 8.009920081 x 101 N 4 = - 1 . 3 9 7 5 5 1 4 0 0 x 102 N 6 = - 5 . 1 7 3 7 5 1 5 3 4 x 10 ° N 8 = 4.731825202 x 10 ° NIO = 4.210224653 x 10 ° N12 = 2.953938544 x 10 ° NI4 = 2.083912318 x 10 "1 N16 = 1.497232942 x 10 ° N18 = 3.397909092 x 10 -1 N20 = 2.438999293 x 10 -3
I
2
5
IO
20
40
Density p, tool I -t
Fig.6 Measured isotherms of nitrogen in a p, p-diagram. Representation of the range of validity of the equation of state (19)
T h e e x p r e s s i o n B r ( O ) i s t h e r e d u c e d second virialcoefficient (compare equation 15).
the experimentally obtained enthalpy 44 in the gas and liquid ranges can also be satisfactorily reproduced with the proposed equation of state.
Discussion
The new equation of state (19) covers the range o f state within the boundaries quoted in (8). This range is shown in Fig.6. The average relative error of all measured values 15 amounts to 0.45% for the pressure in the gas range (p <~Pc) and to 0.86% for the density in the liquid range (p > Pc)" On application of a relationship for the specific isobaric heat capacity c~ in the state of the ideal gas for nitrogen, for example, equation 20 43 o Cp
T N3 =N[ +N 2 -+ (cal mole -1 K "l) 1K (T/1 K) 2
(20) [exp(N 5)
N 1 = 6.9550789 N4=2.006387
112 N 2 = 8.0448 x 10 -6 N 3 = 0.3786 U 5 = 3 3 5 3 . 4 0 6 1 / ( T / 1 K)
C R Y O G E N I C S . J U N E 1972
As an exainple, the saturation values calculated in accordance with the scheme in Fig.7 and the vaporization enthalpy z2dt are listed for nitrogen in Table 3 for certain temperatures. Ah = A h r R T c was determined with
N 2 exp(N5) + N4
Phase equilibria. The state variables on the liquid-gaseous phase boundary curve can be determined with the proposed equation of state alone, from the triple point to the critical point. In addition to the vapour pressure and the saturated liquid and vapour densities extremely accurate values are also obtained for the vaporization enthalpy.
Ah r = o
6"
6'
+o
7r - 0 ,
;
(21)
6
The relative deviations in the values ofPs, p', p", and Ah in relation to the measured data or in relation to the values
219
Table 3. Variables of state calculated solely with the aid of the equation of state (19): vapour pressure, density of saturated liquid and vapour as well as vaporization enthalpy for nitrogen Ah =h"-h',
T,K
, Ps, atm p',moler 1 p",molel "1 Jrnole "1
65 70 75 80 85 90 95 100 105 110 115 120 125 126.26
0.1721 0.3817 0.7509 1.348 2.254 3.556 5.345 7.714 10.755 14.557 19.217 24.855 31.682 33.655
30.7265 29.9918 29.1875 28.3458 27.4704 26.5539 25.5851 24.5485 23.4209 22.1614 20.6870 18.7803 15.2662 11.25
0.03265 0.06788 0.1264 0.2169 0.3498 0.5378 0.7967 1.147 1.618 2.255 3.147 4.511 7.480 11.25
lished permitting phase equilibrium calculations to be applied to mixtures too without additional relationships being required. In addition the new equation of state for nitrogen allows the whole fluid range to be described in accordance with the measured values in the limits Ttr(63 K) ~< T~< 3Tc (about 400 K)
6 019.9 5817.8 5 638.1 5 470.6 5 292.3 5 083.9 4 832.7 4 530.8 4 170.9 3741.9 3218.4 2 531.8 1 308.3 0
Ptr(0.12 atm) <. p <<.6Pc (about 200 atm) 0 ~
I
i
I
I
I Ps (estimated),AP, et =I¢~,Sz-- I I
t
p'(r,p )--equation of state-~'(r,p ) I
I f'(T~p~ ~equation H.,
1 lA,new-.-,.'-,'l
calculated from the auxiliary equations are shown as a function of the temperature in Figs 3 - 5 and 8. Fig.9 shows how the quotient of the fugacity of the liquid and gaseous phase varies in relation to the reduced temperature. For reliable calculations of the values Ps, P', and p" the following must apply:
f"(0,8") ['(0,8')
I
Yes Yes
,,.o
)
n',o
C'A'new/,/A'o,d) y~s
(22)
I Ap---Ap
If the condition in (7) is not fulfilled by the equation of state, then the expression f " / f ' is particularly dependent on the temperature:
f"(o, ~") - ;'(0,a')
,,/
n,o
( No
- constant = 1
I
of stote~f"(T,p") I
•. ~ S i g n
I
(A:new)* sign ( A / o l d ) )
y ,s
I p--Ap/2 I 1 F(O ) =~constant
1
I A,old--A,ne.;p --p,-Ap I
(23)
j Curve o f the isotherms in the two phase region. If the
Result
I
T,p s , p" p'
Maxwell criterion is taken into account during determination of the constants of the equation of state (19), then for certain sub-critical isotherms the dot-dashed curve shown in Fig.1 may arise. The area fit remains valid but in the region of the sub-critical isotherm between points A and B, a stable phase could be possible since the expression 45 applies: (24) T If the constants for (19) are determined in the manner described then for all sub-critical isotherms only three points of intersection with the corresponding isobars are obtained in each case (compare with Fig.l, dashed curve).
Fig.7 Scheme f o r calculating the phase equilibrium values ps,p', and p" for a selected temperature T
o-8 I
-~o.s °
0 -~. 0 g
i
;
,~-O.S 0'.6
017 018 Reduced temperolure O
019
I.O
Summary If an equation of state fulf'dls the Maxwell rule in the region Ttr < T < T c, an important prerequisite is estab-
220
Fig.8 Relative deviation Ahre I = (1 -- AhcalclAhroeas) x 1OO of the values measured for the vaporization enthalpy ,o~ f r o m the values calculated solely f r o m the equation of state (19)
CRYOGENICS. JUNE 1972
8
1.2
Benedict, M., Webb, G. B., and Rubin, L. C. Chem Engng Progress 47 (1951) 419
9
o.6 °1
0.8
"without
-I / 0
J
10 consideration of the condition equation (7) ti
11 12
13 14
0.4
15 16 17 18 19
0.2
01 O.S
i
i
l
i
0.6
0.7
0.8
0.9
1.0
Reduced tcml>znatur¢ Fig.9 Curve of the quotient f"/f" in relation to the reduced temperature (calculated in accordance with equation 19)
20 21 22 23 24 25 26 27
The e q u a t i o n o f state (19) also fulfills the auxiliary conditions at the critical point. It undergoes a transition to the e q u a t i o n o f state o f the ideal gas on f o r m a t i o n o f the limiting value p -+ 0 and contains directly an expression for exact r e p r o d u c t i o n o f the second virial coefficient. In addition new relationships for the vapour pressure and the densities on the phase b o u n d a r y curves as well as an e q u a t i o n for the second virial coefficient was established and c o m p a r e d with existing relationships.
30 31 32 33 34 35 36 37 38
References Rombusch, U. K., and Giesen, H. Kdltetechnik 16 (1964) 66 Hadden, S. T. Chem Engng Prog Symp Series 49 (1953) 53 Cajander, B. C., Hipkin, H. G., and Lenoir, J. M. J Chem EngngData 5 (1960) 251 Sood, S. K., and Haselden, G. G. Cryogenics 10 (1970) 199 Loftier, H. J. Thermodynamik, Bd II: Gemische und Chemische Reaktionen (Springer-Verlag, Berlin, New York, 1969) Prausnitz, J. M. Molecular Thermodynamics of Fluid-PhaseEquilibria (Prentice-HaU, 1969) Beattie, J. A., and lkehara, S. Proc A m A cad Arts and Sci 64 (1930) 127
CRYOGENICS.
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J U N E 1972
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Wagner, W. Eine neue Berechnungsformel zur Ermittlung der Konstanten einer the mischen Zustandsgleichung ftir binffrc Gemische (unpublished) Rossini, F. D. (ed), and Beattie, J. A. Thermodynamics and Physics of Matter (Princeton University Press, New Jersey, 1955) Stein, W. A. Forsch Ing Wesen 34 (1968) 193 Baehr, H. D. Zur Thermodynamik der Zweiphasengleichgewichte (Verlag der Akademie der Wiss u d Lit in Mainz in Kommission bei Franz Steiner Verlag Gmbtl Wiesbaden, 1952) Lewis, G. N. Z P h y s C h e m 3 8 ( 1 9 0 1 ) 2 0 5 L/Jffler, H. J. Thermodynamik, Bd I Grundlagen und Anwendung auf reine Stoffe (Springer-Verlag, Berlin, 1969) Wagner, W. Dissertation TU Braunschweig (1970) Maxwell, J. C. Nature 35 (1875) 357 Hust, J. G., and McCarty, R. D. Cryogenics 7 (1967) 200 Morsy, T. E. Dissertation TIt Karlsruhe (1963) Straub, D., Schaber, A., and Morsy, T. E. Kd/tetechnik 17 (1965) 212 Van ltterbeek, A., De Roop, W., and Forrez, G. A ppl Sci Res Section A 6 (1957) 421 Lestz, S. S. J Chem Phys 38 (1961 ) 2830 Wiistner, R. Report of Institut fiir Thermodynamik der TU Braunschweig (1970) Crommelin, C. A. Corn Phys Lab Leiden 145 d (1914) 29 Cath, P.C. ComPhysLabLeiden 152d(1918) 45 Porter, F., and Perry, J. H. J A m e r Chem Soc 48 (1926) 2059 Dodge, B. F., and Davis, H. N. J C h e m A m e r S o c 4 9 ( 1 9 2 7 ) 610 Giauque, W. F., and Clayton, J. O. J A mer Chem Soc 55 (1953) 4875 Henning, F., and Otto, J. Phys Z 37 (1936) 633 Friedman, A. S., and White, D. J A mer Chem Soc 72 (1950) 3931 White, D., Friedman, A. S., and Johnston, H. L. J A mer Chem Soc 73 (1951) 5713 Michels, A., Waasenaar, T., De Graaf, W., and Prins, C. Physica 19 (1953) 26 Armstrong, G. T. J Res NBS 53 (1954) 263 Riedel, L. Chem Ing Techn 26 (1954) 263 Stein, W. A. Kidtetechnik-Klirnatisierung 22 (1970) 7 Strobridge, T. R. NBS Technical Note 129 A (1963) Martin, J. J., Kapoor, R. M., and Shinn, R. D. Dechema Monograph 32 (1958) 46 Hank, R., and Riedel, L. IngArch 16 (1948) 255 Benedict, M., Webb, G. B., and Rubin, L. C. J Chem Phys 8 (1940) 334 Bloomer, O. T., and Rao, K. N. Inst (;as Technology Res Bull 18 (1951) Morsy, T. E. Kiiltetechnik 17 (1965) 273 Stewart, R. B., Prydz, P., and Timmerhaus, K. D. Advances in Cryogenic Engng 13 (1968) 384 Bender, E. KhTtetechnik-Klimatisierung (forthcoming) Barieau, R. E. J Phys Chern 69 (1965) 495 Mage, D. T., Jones, M. L. Jr, Katz, D. L , and Roebuck, J. R. Chem Engng Progr Symp Series 59 (1963) 61 Haase, R. Thermodynamik der Mischphasen (SpringerVerlag, Berlin, 1956)
221