Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
81
REVIEW Cubic Equations of State-Which? Joseph J. Martin Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigart 48 109
An analysis is made of volume-cubic equations of state, starting with the most general generating equation from which all specific forms can easily be derived. One equation is shown to be both the simplest and the best in performance. A new quantity, the t-chart sum, is presented, which is a useful way to compare all equations of
state, whether cubic or otherwise, while at the same time permitting excellent predictions of the second virial coefficient at the critical temperature from the critical compressibility factor. Extensive comparisons of reliable experimental PVTdata with a number of equations of state are given for a wide range of substances from nonpolar to polar, and these demonstrate the superiority of the simplest two-term cubic equation.
Introduction Since the time of van der Waals, a century ago, there has been a steady outpouring of equations of state to represent the pressure-volume-temperature behavior of fluids. These have ranged from simple expressions with one or two constants to complicated forms with up to more than 50 constants. The longer equations have been utilized for high-precision work and one can find in the literature many interesting and useful ones such as the BenedictWebb-Rubin ( 3 ) ,the Strobridge (30),the virial form of Onnes ( 2 4 ) to the 17th power in volume, the MartinStanford (21),and some recent semitheoretical formulations with two or three dozen universal constants and a handful of molecular-parameter constants characteristic of the substance being represented. Although these long and complex equations are desirable for the precise representation of PVT data and calculation of simple thermodynamic properties, they are not generally preferred for involved thermodynamic calculations such as vapor pressure and latent heat of vaporization, mixture behavior and activity coefficients of mixture components, or multicomponent vapor-liquid equilibrium ratios because they require tedious manipulation and excessive computer storage in lengthy iterative calculations which tax even the most modern electronic machines. The attractiveness, then, of the shorter equations lies in their simplicity of calculation. The bulk of these short equations may be shown to be cubic in volume and these include such well-known forms as van der Waals (32), Clausius (6),Berthelot ( 4 ) ,Onnes third degree virial ( 2 4 ) , Redlich-Kwong ( 2 7 ) , Wilson (33), Barner-PigfordSchreiner (2),Martin (201, Lee-Edmister (15),Soave (29), Dingran-Thodos (9),Usdin-McAuliffe (31),Redlich (28), Peng-Robinson (25),Fuller (12),and Won (34). It is these short cubic equations that are to be examined here. If one is searching for a suitable equation of state and has ruled out the extended equations as being too cumbersome and wishes to use a simple cubic, he might think his problems are over, but may be in a quandary when he faces the forelisted array of available cubic equations. How does he decide which one to use? A previous study of cubic equations by Abbott ( I ) analyzed some of their characteristics and behavior, but did not address the central question, similar to the queen in Snow White who asked, 0019-787417911018-0081$01,0010
“Mirror, mirror on the wall, who’s the fairest of them all?” Thus, it behooves this study to present a set of arguments which will prove beyond any doubt the superiority of one form over the rest, a formidable objective considering the vast amount of effort that has gone into this subject previously and the necessity to run down every conceivable alley of approach no matter how blind. In the pursuit of this objective it is hoped to clear up some of the myths about the behavior of cubic equations and the supposed advantages of one favorite form over another. The principles developed here are fundamental and applicable to all equations of state, be they cubic or not. It has been common practice in recent years to compare the simpler equations of state by noting how well they predict saturated liquid volumes and vapor pressures through the equality of liquid and gas fugacities. This has merit because of the way such short equations are used; however, in this work the comparisons will be made directly to the PVT behavior in the liquid and gas states because this is more fundamental and through exact thermodynamic relations precludes all equilibrium comparisons. Also the comparisons will be for computed pressures rather than computed volumes which is a much more severe test in the liquid phase. It is only by making comparisons with actual PVT data that the subtleties in behavior of the various cubic equations can be detected. A. General Source of all Equations of State That Give Pressure as a Cubic Function of Volume The analysis will be initiated by setting down the most general form of the volume-cubic equation of state, with pressure, P, a function of specific volume, V , and temperature, T
which is somewhat different and more inclusive than one given earlier ( I ) . Here R is the universal gas constant, a and 6 are functions of temperature, and P and y are constants. The latter could also be taken as temperature functions, but usually the improvement by doing so is marginal. By specializing the constants and by straightforward algebraic rearrangement including simple translation in volume, all forms of cubic equations can 0 1979 American Chemical Society
82
Ind. Eng. Chern. Fundarn., Vol. 18, No. 2, 1979
easily be obtained from eq 1. Following are a number of examples. I. Let P = y = 0 to get
which is the virial equation (third degree) of Onnes (24). 11. Rearrange eq 1 to P = RT(V + p) - aV + V(V + 0) V ( V + P)(V + y)
+
6 V(V + P)(V +y) (3)
or
RT a - PRT ay + 6 p=-(4) V + P V ( V + P ) V ( V + P ) ( V +y) NOWlet p = -b = -7, CY - PRT = a , a y + 6 = c to give a C p = - -RT + (5) V - b V ( V - b) V ( V - b ) ( V +b ) which is the Lee-Edmister equation (15),that as a surprise to some is just a rearranged form of P = RT/ V - a/V(V + P ) + 6/V(V + P)(V - PI. 111. Let 6 = 0 in eq 1 to yield +
Now translate by t in volume so that a p = - -RT V - t (V-t+P)(V-t+y)
or
a p = - -RT V - b V+2bV-b2 The denominator of the last term may be factored to give a p = - -RT (15) V - b V ( V + b) + b ( V - b) which is the Peng-Robinson equation (25). VIII. A form which is not in the literature but has interesting properties is obtained from eq 7 by letting t = b = (P + y)/2, c = - [ ( p - y)/2I2, and a = a, giving p = - -RT U V-b V + C A specialized case of this occurs if y = -P, then a p = -RT -(17) v v+c Examples other than the eight presented can be given, but they add nothing to the techniques of rearrangement of eq 1. Clearly, if all cubic equations are merely specialized cases of eq 1,attention should be focused here to determine the characteristics of the different forms. This may be done by first solving for the constants in eq 1 by the classical method of van der Waals (32) that the first two pressure-volume derivatives vanish at the critical point
(7)
Next let t = P = b, y = 2t = 2P, and a = a to get a p = - -RT V - b V ( V +b) which is the Redlich-Kwong equation (27). IV. If in eq 7 we let P = y = t = b, and a = a, we have
which is the van der Waals equation (32). V. If eq 6 is translated by P t in volume instead of just t as in eq 7 RT a P= (10) v-p-t ( V - P - t + P ) ( V - P - t + y)
+
Letting P = b, y = P, and a = a, we have
or by the alternate equivalent technique of using three equal volume roots at the critical point, as explained by Martin and Hou (16). The latter is simpler and so will be employed here. If eq 1 is multiplied out in descending powers of volume, it may be written as
vPyRT a P P A t the critical point with three equal roots (V - VC)3= 0 which may be expanded to v3 - 3VCV+ 3v:v-
v,3 = 0
-
0 (19)
(20) (21)
Comparing eq 19 and 21 for each power of Vat the critical point, where a and 6 are taken as their critical temperature values, for v2 which is the equation developed by Martin (20). It can also be obtained by translating the van der Waals eq 9 by 1
L.
+
VI. If in eq 7 weset t = /3 = b, y = t c and a = a, we get a p = - -RT (12) V - b V(V+C) which is the form given by Usdin and McAuliffe (31) and by Fuller (12). VII. Referring again to eq 7 , let t = b, P = (2 + &)b, y = ( 2 - h ) b , and a = a to yield a p = - -RT
v-
[ V - b + (2
+ & ) b ] [ V - b + (2 - &)b]
(13)
for V'
for
V"
It is useful to put these into dimensionless forms with PcV,/RT, = Z,, so that eq 22 becomes (P + YIP, = 1 - 32, R TC
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
variables PR = PIP,, VR = V/V,, T R = TIT,, and Z, = (PCVclR Tc)exp
eq 23 becomes
-- - 3 prp: R2T:
CUP, (P + r)Pc 2 2 - p R2TC2 RTC
83
+
(26)
and eq 24 becomes
The last three equations are independent, but there are four unknowns, a , P, y, and 6, or their dimensionless equivalents, if P,, V,, and T , are known. Thus, a unique solution can be obtained only if one of the unknowns is specified-usually as zero. To exploit this approach eq 25 and 27 are inserted into eq 26 to get
CUP, R2T,2
- - - --
6P,2 2: R3TC3
+ 32:
-
32,+ 1
(28)
This may be translated by t by replacing V in eq 35 with V - t or V Rin eq 36 or 37 with VR - tP,/Z&T,. If we then let t = P or y so that from eq 33 CD
.
and (39)
which will be examined for its various possibilities.
B. Consideration of Equations for the Critical Isotherm Suppose our interest first is only along the critical temperature line and we want to study two-term cubic equations (which the majority of them are), so 6 is set equal to 0 and eq 28 becomes
-PC .-
- -2: + 32:
-32,
+1
R2TC2 The value of 2, can be set as we please, so after some pretrial let us take it to be 0.25. This makes
-aPc R2T:
64
-
t , thus,
Here the superscripts, bt and at, mean before and after translation. For eq 37 translated according to eq 38 it is seen from eq 41 that 2, = 0.25 + 1/8 = 0.375 (42)
Equation 25 then becomes
RTC + -rpc =64yPc RT,
1 4
or
4RTc + -64= o
(32)
Solving gives
YPC = -1 RT,
8
(33)
Inserting this into eq 31 yields PPC = -1 -
8
(34)
Here, the unknowns, a , P, and y, have been solved for as dimensionless quantities to be used in the equation of state in reduced or dimensionless form which is obtained by writing eq 1, with 6 = 0, as
R T V ,T , - (aP,/ R2T:) (R2T,2VC2) (35) pc vpcvcTc PC2VC2(V + P)(V + y) or with the definitions of the reduced experimental
_p --
which follows from V being replaced with V giving
- -27
at the critical temperature, and
RT,
which is a well-known reduced form of the van der Waals equation. It is instructive to note the effect of translation on a calculated or selected 2, (not the experimental Z,). We first note that
As emphasized earlier (20,221, eq 39 gives results consistent with its development only if 1, = 0.375 for which value it gives PR = 1 at the critical point where V R= 1 at TR = 1. If, however, the actual V , or Zc is used for a given substance, the representation along the TR = 1 isotherm up to the critical density will be vastly improved even though the prediction right at VR = 1 is slightly off. Table I gives a comparison with the National Bureau of Standards data (13) of the predictions for argon whose Z, = 0.29121. The earlier discussions (20,21)showed that this principle must hold for the reduced form of any equation of state; Le., wherever Z, enters, the experimental value should be employed and not the value used to get the equation or predicted by the equation. The reason is simply that the reduced volume is based on the actual critical volume so the value of Z, must also be calculated with the actual critical volume. It is of interest next to look at the second virial coefficient of translated equations. If eq 7, where 6 = 0, is rearranged to solve for z CY z = 1 + - -tP RT RT[1 + P / ( V - t ) ] [ V -t + y] (43) Now in general as P
-
0
RT P
V - - + €(T)
(44)
84
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
Table I. Pressures on the Critical Isotherm for Argon Using Values of 2, = 0.375 and 0.29121 in Eq 39 VR P, NBS % dev pR o ' 2 I ' 100 20 5 2.5 1.25 1.0
- -
0.033 95 0.162 09 0.540 73 0.834 2 1 0.995 97 1.000 00
0.026 46 0.128 09 0.451 43 0.750 77 0.989 09 1.000 00
-22.1 -21.0 -16.5 - 10.0 - 0.7 0
where t is the residual volume. Inserting this into eq 43 and letting P 0 or V a
% dev
-0.1 + 0.6 t 2.1 + 3.3 + 0.4 + 4.2
0.033 9 0 0.163 03 0.552 30 0.862 35 1.000 0 4 1.041 7 4
Substance 1
2 /
Differentiating with respect to pressure a t constant temperature
This may be put into reduced form
-
It has been shown earlier (18) that this slope of a constant TR line as PR 0 on the usual compressibility plot of 2 vs. P R is just the generalized second virial coefficient, BPJRT, where B is the second virial coefficient in the expression, i = 1 + B / V + C / P .... If we now combine eq 41 and 47 with 2 = Z and Zcat being Z, for any equation with or without translation
+
ZC-T,(dz) dPR
ffPC as PR R2T,2TR
TR
-+
0
Figure 1. Generalized compressibility or i chart. Table 11. 5Chart Sum, E , for a Number of Substances substance Z C a E Ne
0.309 0.305 0.295 0.294 0.291 0.291 0.290 0.289 0.288 0.288 0.28 3 0.281 0.279 0.279 0.278 0.278 0.278 0.274 0.274 0.274 0.274 0.271 0.270 0.270 0.268 0.267 0.264 0.261 0.260 0.259 0.259 0.248 0.248 0.242 0.238 0.234
(48)
At the critical temperature this becomes
where BG1is the generalized second virial coefficient at a reduced temperature of 1. The interesting characteristic of this quantity is that it is independent of translation because t has been eliminated. As an example for eq 37 zc-
(
= 0.25
g)TR=l,pR=O
27 += 0.671875 64
(50)
Thus, for van der Waals eq 39 it follows automatically that
P,=o = 0.671875
(51)
This can be checked by combining eq 30, 38, 42, and 47 at TR = 1 = 0.375 -
1 8
-
27 += 0.671875 64
(52)
The constancy of eq 50,51, and 52 leads one to speculate the whether this quantity, which will be designated i-chart sum
x,
0.634 0.630 0.625 0.618 0.621 0.623 0.623 0.630 0.621 0.621 0.624 0.618 0.619 0.619 0.623 0.614 0.617 0.619 0.624 0.615 0.622 0.616 0.622 0.633 0.633 0.61 3 0.624 0.629 0.620 0.61 4 0.611 0.604 0.606 0.612 0.603 0.574
average (exclusive of H,O) = 0.62 - ( dz/dPR )TR =
is the same for all actual substances. One might guess this to be so by noting the shape of the TR = 1 line on the .Z chart for two specific cases as shown in Figure 1. If a substance has a small negative slope at PR = 0, it will have
0.325 0.325 0.330 0.324 0.330 0.332 0.333 0.341 0.339 0.333 0.341 0.337 0.340 0.340 0.345 0.336 0.339 0.345 0.350 0.341 0.348 0.345 0.352 0.363 0.365 0.346 0.368 0.368 0.360 0.355 0.352 0.356 0.358 0.37 0 0.365 0.340
,,p
= = - BPc 1R Tc,
a high t,,and vice versa. The verification of this idea is shown by Table 11, where it will be seen that the i-chart sum is remarkably constant. The only notable exception is water, which is highly polar, but so also is ammonia which behaves like the other substances. In fact, what variability there is in may be due to lack of precision
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
in the experimental determination of 2 , and B P J R T , , or (dZ/dPR)TR=l=o, though it is possible that better data will show ten&g to be slightly lower for some compounds = such as alcohols and ketones. The average value of 0.62 allows one to predict with considerable confidence the second virial coefficient at the critical temperature from the critical compressibility factor for any substance (54)
Returning to eq 37, it may be translated in VR by (0+ t)P,/Z$T, instead of just tP,/@T,. Taking @P,/RT, = 118 as before 27/64 TR (55) pR= z,VR - t P c / R T c - 1/8 (z,VR - tP,/RT,)' which is the reduced form of eq 11. The development leading to eq 37 was on the assumption of a value for 2, of 0.25, but the procedure could have followed Example V of the several cubic equations by setting p = y. This would have made eq 25 become
85
Now three arbitrary values of 2, (0.23, 0.25, 0.27) are inserted into eq 65 and each equation is translated by t P , / R T c to give a good fit at about twice the critical density. As shown in the pioneer paper (20) on volume translation, a cubic equation cannot be adjusted to fit the critical isotherm at reduced densities of both 1.5 and 2.0. The translation selected in most cases in this paper will give deviations that are slightly negative a t twice the critical density and of the order of 25% positive at about 1.5 times the critical density. If interest is in reduced densities only to 1.5, the translation can be reduced and a good fit obtained at 1.5, but large negative deviations would occur at twice the critical density and would be far worse if the equation were to be used at even higher densities. Table I11 gives the comparison with NBS data for argon for the three values of 2,. For all densities up to the critical the deviations are well below 1% for 2, = 0.25. The results are not so good for 2, = 0.23, where relatively large negative deviations occur, and for 2, = 0.27, where large positive deviations occur for densities below the critical. Above the critical the three equations give similarly large deviations. The best equation overall is for 2, = 0.25, translated by 0.082, and can be written
and eq 27 become (6 = 0)
-P'P-,' R2T$
- 2:
(57)
Combining eq 56 and 57 to eliminate P P J R T , yields 4ZC3- 2,'
+ 62, - 1 = 0
(58)
This may be solved by Cardan's method for the roots, 2, = 114, 1, and 1. Since 2, = 1 implies ideal gas at the critical point, only the first root is meaningful in this application and it is identical with the assumption which gave eq 30, 33, and 34. Because eq 55 differs from eq 39 only by translation, it follows that = 0.671875. As pointed out earlier (20), however, the translation characteristic makes eq 55 much more powerful than eq 39 because 2, is not constrained to 0.375 and can be made more realistic. If in eq 29 2, is taken to be less than 0.25, the following results 2, = 0.24; aP,/R2T,2 = 0.4390; = 0.6790 (59) 2, = 0.23; aP,/R2T: = 0.4565;
= 0.6865
(60)
Since the desired value of 2 is 0.62, it is clear that to make 2, < 0.25 is going in the wrong direction. For values greater than 0.25, we get 2, = 0.26; aP,/R2T: = 0.4052; 2, = 0.27; aP,/R2TC2= 0.3890;
E= 0.6652
X
2, = 0.3; aP,/R2Tc2 = 0.343; 2, = 1 / 3 ; aP,/R2T,' = 0.2963;
= 0.6590 = 0.643
(61) (62)
l ) / [ ( Z , V R - tP,/RT,)'
-
+ (1
3PPc - - 1 - 32,
RTC and eq 27 with 6 = 0 gives
-2p2pc2 - - 2: R2TC2
=0.6296 (64)
(-ZC3+ 32,' - 3 2 , + 3 2 , ) ( Z C V-~ tP,/RT,) + Z23 (65)
-
which is the equivalent of eq 55 or 37 with translation. The problem of large deviations for p R above 1.2 is characteristic of all cubic equations of state, for the actual data tend to follow a fourth-degree equation as discussed previously (21). The major difference between the many different cubic equations that have been proposed is whether they fit better at about two-thirds or one-half the critical volume. One of the major advantages of eq 67 is that a small change in translation will permit fitting exactly where one chooses. In obtaining eq 65 from eq 25,26, and 27, it was not necessary to solve explicitly for /3 and y, but if one wishes to do this, he would find that p and y are real numbers for 2,s 0.25, but imaginary or complex numbers for 2, > 0.25. We can now show how some of the well-known cubic equations fit into the general pattern. As shown in example 111, if we let y = 2P and t = p, we get the Redlich-Kwong equation. Using dimensionless quantities, eq 25 becomes
(63)
Here the trend is toward 0.62, but it must be noted that is indicative of the behavior only up to the critical density, so the behavior above the critical density (or volumes less than the critical volume) must be studied. Therefore, eq 36 will first be combined with eq 25, 26, 27 to obtain after translation
PR = TR/(Z,VR - tP,/RT,)
or in its reduced form before a specific translation is made 27/64 TR (67) PR = Z,VR- tP,/RTc (B,VR + 1 / 8 - t P , / R T J 2
Eliminating PP,/RT, between eq 68 and 69 yields an expression for 2, (similar to eq 58) 92:
-
182,'
+ 122,
-
2 =0
(70)
Again this may be solved by Cardan's method to give two roots that are complex and the third which is real and useful 2, = 0.246693
The translation for eq 67 is, therefore, from eq 7 1
(71)
86
Ind. Eng. Chem. Fundam., Vol. 18, No. 2 , 1979
1 - 32, -tP,- ---PP, - -bP, =-RT, RT, RT, 3
- 0.08664035
(72)
and using eq 71 in eq 65 a t the critical temperature
TR
P -
- Z,VR - 0.08664035
"m I
3mC.lmt-mmwo
$23
3oo*t-wmom"woomo
5 0 0 0 0 0 0 0 0 A i i ~ ~ eo im 1
I
1
+ + + + + + + + A * + W
++
I
00 Icoa,
-
0.4274802 Z,VR(Z,VR 0.08664035) (73)
+
which is the reduced form of the Redlich-Kwong equation before temperatures other than the critical are considered. Table I11 gives the results predicted by this equation and it will be seen that the fit is similiar to that of eq 66. This is expected because the two equations have a similar mathematical form as discussed before (20) and since the Redlich-Kwong equation gives Zcat = 0.246693 + 0.08664035 = 113 according to eq 41, while the example of eq 66 gives 2cat = 0.25 + 0.082 = 0.332, which values are almost identical. The Redlich-Kwong, however, must be slightly inferior because its Z-chart sum is = 0.246693 0.42748 = 0.674173, which is to be compared with 0.671875 of eq 66 and the desired value of about 0.62. Furthermore, the Redlich-Kwong will not fit other substances nearly as well as for argon whereas eq 66 can easily be made to do so by translation. Naturally, if one wishes to fix the translation for many substances and never allow it to vary (thus, fixed 2cat),this can be done by using an average value that is best for all the substances involved. Of course, the Redlich-Kwong equation itself could be translated to give a better fit of the experimental data than given by eq 73 but it would be no better than eq 67 and would be more complex in form. From example VI1 if we let
+
P = (2 + f i ) b and y
2-d2 or y = -
= (2 - &)b
we obtain the Peng-Robinson equation. Now eq 25 becomes 4PPc
= 1 - 32,
(74)
and eq 27 becomes
Eliminating pPC2/R2Tc2 yields the cubic equation in 2,
As with eq 70 two roots are complex and the third and useful root is 2, = 0.229605235
(77)
This may be inserted into eq 75 to give according to Example VI1
bP, -tP,- - --
RT,
RT,- (2
PPC
+ &)RT,
= 0.077796074
(78)
Using eq 77 and 78 in eq 65 gives P R = TR/(Z,VR - 0.077796074) 0.457235529/[z,vR(zcvR + 0.077796074) +0.077796O74(ZcV~- 0.077796074)l (79) which is the reduced form of the Peng-Robinson equation
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 87
for the critical isotherm. Table I11 gives the predictions of this equation, and it is seen that these are somewhat lower than those of the equation based on 2, = 0.23 and translated by 0.09. If the latter had been translated by 0.0774 so that its Zcat = 0.3074 which is almost the same as 2, = 0.229605 + 0.077796 = 0.307401 in the PengRobinson, the two would predict almost the same. Looking at the Z-chart sum for the Peng-Robinson equation, we find 1 = 0.229605 0.457236 = 0.68684. Recalling again that the desired value is 0.62, it is seen that eq 79 is somewhat worse than eq 55 or 67 or even the RedlichKwong equation. From example VI if we let y = P + c and t = 0,we get the Usdin-McAuliffe or Fuller equation (12). We note that if c = 0, we obtain 2, = 0.25 from eq 56, 57, and 58, which with the indicated translation gives the van der Waals equation (39). If c = P, eq 68,60, and 70 give 2, = 0.246693 which with translation gives the Redlich-Kwong equation (73). If we take y = m@,which is equivalent to c = (m 1)P or P + y = (m + 1)P, m may be set at will. In this procedure eq 25 becomes -PP, - --1 - 32, (80) RT, m + l
+
and eq 27 with 6 = 0 becomes P2PC2 2:
--
R2T,2
--
m
(81)
Once m is set, P can be eliminated between eq 80 and 81 to give a cubic in 2, whose roots can be determined as in eq 58 or 70. Alternatively, 2, may be set and the two equations solved for m. This is easier because m occurs in a quadratic rather than a cubic. As an example, take 2, = 0.245 which determines cyPc/R2T? by eq 29. Then eq 80 and 81 give m = 2.34962058, and either eq 80 or 81 gives PPJRT, = 0.079113438. Noting that P = t , eq 65 yields
TR
0.430368875 ZcVR(ZcV~ + 0.106773124) (82) The results of this are tabulated in Table 111. Had eq 67 been translated by 0.075 instead of 0.082, it would have given almost the same results. This is because of similarity of form and Z,st. For eq 82 it is ZFt = 0.245 + 0.079113438 = 0.324113438 and for eq 67 it would be 0.25 + 0.075 = 0.325. The two equations are of about equivalent simplicity, but it is easier to fix the translation in eq 67 than to assume 2, and calculate m, P, and cy in eq 65 to get eq 82. From example VI11 it is seen through eq 25 that if 2, < 113, b turns out to be a positive number and the results are similar to those from eq 65 for 0.25 C 2, < 113. If 2, > 113, however, the value of the 2-chart sum can be made close to 0.62 and the predictions are fantastically good below the critical density. This is shown in Table I11 for 2, = 0.42, which gives the equation TR 0.195112 PR = ZCVR -t 0.13 (Z,VR)~ + (0.239140)2 (83)
PR =
Z,VR
- 0.079113438-
whose 2-chart sum is = 0.42 + 0.195112 = 0.615112. It might be expected that this would be a good equation to use for temperatures away from the critical and densities up to the critical. Unfortunately, it does not have the ability to inflect properly at the critical as can be seen by the low pressures above the critical density. This effect is carried back to lower densities for temperatures above
the critical if the numerator of the second term is taken as a temperature function (Le., a ( T ) which is 0.195112 a t T,). For temperatures below the critical and densities up to the critical, the equation does an excellent job. For the three-term cubic it is instructive to start with the conditions of example 11, backtrack to I, and then generalize. In eq 1 let P = -7, as done to get eq 5. From eq 25 it follows that 2, = 113. Combining eq 26 and 27 then gives a relation between cy and 6
Now let cyPc/R2T,2= 113, which makes 6P:/R3T: = 1/27. Putting this into eq 27 makes = y = 0, which gives the simple virial eq 2 that in reduced form is
This equation can be obtained directly as in example I by letting = y = 0. The predictions of eq 85 are given in Table 111. A t low densities the results are good, but a t higher densities it gives significant deviations so that the average overall densities to twice the critical are not so good as for the two-term eq 55 or 66. Equation 85 can be translated to improve the predictions at high density, but this will worsen those below the critical density and the average predictions will be no better than eq 66. If interest is only in densities less than the critical, eq 85 can be translated by just the right amount to do a slightly better job than any of the other cubics. This is because its 2-chart sum, E, of 113 + 113 = 213, is less than and closer to 0.62 than any of the others. The Lee-Edmister equation 5 permits setting b at something other than 0 which is the unique value giving the virial equation. An equivalent way to do this in eq 84 is to set aP,/R2T,2 at a value other than 113, since eq 5 has been shown to be a form of eq 1 with P = -7. If cyP,/RT, is taken as 8/27, then 6PC2/R3Tc3 = 0 and PyPc2/R2Tc2 = 1/27 from eq 27 so that in reduced form eq 1 becomes m
n
In-
This is just a reduced form of eq 17 previously considered and it predicts high for densities below the critical and low for densities well above the critical. By trial it is found that the best three-term equation for argon, with P = -y is obtained by letting aP,/R2T,2 = 0.34 which makes 6P,2/R3T2= 0.437037037 and /3-yP,2/R2T,2 = -11150. This gives
pR=---TR 0.34 ~CVR (Z,VR)~ - 1/150
0.0437037037 Z,VR[(Z,VR)~ - 1/150] (87) which is the reduced form of eq 1 that is equivalent to eq 5. Table I11 gives the results for this equation and it is seen that it is about as good, but no better than eq 66. Varying cyP,/R2T,2 does not improve the situation. Its third term, therefore, does not give it any advantage at the critical temperature over two-term cubic equations. For temperatures away from the critical, it does offer the possibility of two temperature functions (i.e., a(?")and G(T))compared to just one for two-term equations, but this does not give it any better overall behavior because of the overpowering defect of all cubic equations at reduced densities in the range of 1.5 to 2.0. +
88
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 PI
IO
12
14
16
,-R
' 8
20
22
24
F i g u r e 3. Generalized vapor pressures with critical-point slopes.
+I
~
-1.0
Figure 2. Typical pressure-temperature graph.
C. Consideration for Temperatures Other Than the Critical As mentioned for eq 1, the volume constants, /3 and y could be dependent upon temperature; however, here they will be assumed to be temperature independent as is customarily done, and also any translation in volume will be constant, while a and 6 will be considered to be temperature dependent. Since eq 66 is as good or better than all of the other two-term cubic equations and at the same time is the simplest in form, it will be examined first. To make its a a function of temperature, the least complicated expression that can be employed is TR", and it may be introduced into eq 66 as
This is just a slight modification of what Berthelot ( 4 ) did 80 years ago when he put T in the denominator of the second term of the van der Waals equation. To determine the value of n, use is made of the slope of the critical volume line on a pressure-temperature diagram or the slope at the high end of a vapor-pressure plot. As developed and emphasized in earlier papers (16,19,20,21),the critical isometric is essentially, though not exactly, straight for all substances and its slope is the slope of the vapor-pressure curve just as it approaches the critical point. Figure 2 shows this characteristic and also 2, shows that for rather high temperatures, where T R the critical isometric (and all the other isometrics) tends to curve down very slightly; however, this effect is minor compared to others and is outside the range of temperatures encountered most often and considered in this discussion. Figure 3 shows the generalized vapor-pressure plot of log PRvs. l/TR, presented several times in the literature (16,19,21). The parameter on each curve is the slope M = dPR/dTR as T R 1. All that is needed to establish which curve a substance follows, and thus what will be the slope of the critical isometric, is the critical point and one vapor-pressure point such as the normal boiling point. The curves for different substances never cross each other on this plot so that a single vapor-pressure point determines a curve all the way to the critical point with a unique slope at the critical point. (This is often useful for getting intermediate vapor pressures.) In the case of argon which is the example used in this discussion, the normal boiling point is 87.28 K, the critical temperature is 180.86 K, and
-
-
Figure 4. Behavior of the second derivative of pressure with respect to temperature a t lower temperatures.
the critical pressure is 48.34 atm, so 1/TR = 1.7285 and PR= 0.020687. Placing this point in Figure 3 shows that the slope of the critical isometric is about 6.1. (Because of the inherent lack of precision in a derivative compared to a primitive, this should not be taken to be necessarily better than f 5 % .) To utilize this, differentiate eq 88 to get
One quickly notes that for all isometrics (dPR/dTR)Vas given by eq 89 decreases with temperature if n is a positive constant. This is in accordance with the experimental facts for volumes greater than the critical volume, as shown in Figure 2, but it is contrary to the facts for volumes less than the critical volume, and of course, the critical volume line has little appreciable curvature as mentioned above. Also, as the volume decreases to about half the critical volume, (dPR/dTR)Vtends to become constant again. In another way (dzpR/dTR2)V is negative for large volumes, zero at about the critical volume, positive for volumes less than the critical, and negative again for volumes less than half the critical volume. This is shown in Figure 4 where the solid line is the experimental data. Despite the inability of eq 88 or any similar two-term cubic, to give the correct curvature to the isometrics by making CY a function of temperature, it is common practice to use such equations and fit the data as best as possible. The dashed line in Figure 4 shows how this approximation works. The results, although not really precise, are good enough to make useful calculations. The principal reason the approximation works is that the curvature of the isometrics is small and other deviations are of much greater significance. To approximate the critical isometric with eq 88, one can use eq 89 to calculate its slope at T R = 1 and T R = 1.5 and note the downward curvature. The reason for selecting this temperature range is not only that most situations involve temperatures below T R = 1.5, but that the critical isometric is linear between 1 and 1.5 and only starts to
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
89
Table IV. Comparison of Six Equations of State with Data ( I 3 ) for Argon Martin-with Yartin-wlth ... ~
pR Data
Redlich-
io0
0.5
FC =
20
1.2 1.5 0.7
,02001 .03395 ,04088 .05127 ,10704 .16209 ,19808 ,25159 ,22813 30565 ,38079 ,49183 ,43008 ,54073 ,70432 ,94212 .73591 . 83421 1.2134 1.7430 .92682 *.9632 I.5861 2.4836 . 99560 1.9C02 3.2217 . 99946 2.0343.6167 .99999 2.2174 4.0471 1.0006 2. 3983 4 . 5252 1.0058 2.6105 5.0738 ,96551 1.3635 3.1966 6.4759 . 79003 1. 3426 4.1993 8. 5241 .64 I 32 2.1534 5.9763 11.619
i
48.34atm C
!0
1.2 1.5 0.8
3
1.2 1.5 .87
13.41 ymole,
.
1
R =
,082jjjjlatrr1.2
z
q r r o l e . K1.5 2.3 .95
=
c. 29121
1 1.2 1. 5 1 U 3 . 6 .99 1 1.2 1.5 ./0.8 1 1.2 1.5 d 3 . 9 1 1.2 1.5 i.2 1.3 1,’l.l 1 1.2
1. 5 1. 2 1 1.2 1.5 1./1.4 .99 1 i. 2 1.5 1/1.6 .96 j.;
1.2 1.5 lVi.8 .92 1.2 1.5 .86
;/2
,45894 4 . 0320
1.2 1.5
9.0682 16,318
_ -
-0.08664
?_VR
Peng-
,02002 .a3394 ,04087 ,05125 ,10716
!Bl
Dev.
’R
smaller translation (E) g
(D)
oil..
Dev.
‘2
1.01 ,02005 + . 1 7 -.03 ,03334 -.03 -.03 ,04087 -.03 -.03 .05126 -.01
+.11 ,10756 +.49 , 1 6 1 8 7 -.14 -.14 .19784 -.12 -.11 ,25155 -.02 t.12 ,22910 t.13 t . 2 2 ,30497 - . 2 2 -.20 ,38014 -.17 -.19 ,49193 7 . 0 2 +.28 ,43251 - . 5 7 -.15 , 5 3 9 9 3 - . 1 3 -.25 ,70302 -.18 -.23 ,94390 1.19 1.19 ,74278 t.93
. 1 6 1 8 7 -.14
,19781 ,25130 ,22841 ,30497 ,38002 ,49092 .43128 ,53993 ,70256 ,93998 ,74169 ,83827 1.2130 1.7502 ,93542 ,96826 1.6040 2.5049 ,99931 1.9475 3.2802 1.0000 2.1248 3.7314 1.0033 2.3195 4.1604 1.0195 2.5429 4.6702 1.0598 2.8076 3.2457 1.1471 1.2646 3.5202 6.6643 i.1435 1.7432 4.6021 5.5905 1.1542 2.6717 6.2600 11.277 .95351 4.3180 3.8113 15.117
Martin
Robinson
(A)
’R
TC = 150.86X
Soave
Kwong
,02003 ,03390 ,04082 ,05120 ,10703 ,16089 ,19562 .25002 ,22690 ,30160
+.07 -.16
.02001 ,03394 ,04088 ,05126 .lo699 ,16189 ,19789 ,25142 ,22806 ,30503 ,38031 ,49136 ,43048 .59004 ,70351 .94i45 ,74057 .83327 1.2156 1.7541 ,93480 ,96797 1.6080 2.5097 ,99933 1.9515 3.2813 1.0300 2.1275 1.6974 1.0029 2.3193 4.1478 1.0174 2.3371
-.15
-.i4 .OO -.74 -.74
-.62 -.54 -1.3 , 3 7 5 7 7 ->.3 ,48623 -1.1 ,42590 - . 9 7 ,53023 -1.9 ,68905 -2.2 . 9 2 4 5 2 -1.9 ,72381 -.83 t.50 ,83837 +.50 ,82040 -1.7 +.2l 1.2147 +.36 1.1784 -2.1 c.41 1.7651 1 1 . 3 1.7052 -2.2 r.93 ,93576 +.96 ,92450 -.25 +.53 ,96826 +.53 ,95463 - . a 9 +1.1 1.6077 tl..’, 1.3508 -2.2 t.56 2.3366 +2.1 2.4235 -2.4 t.38 ,99937 - . 3 d .99654 t.09 +2.5 1.9537 12.8 ;.8709 -1.5 +l.8 3 . 3 3 3 9 +3.5 3.1467 -2.3 t.05 1.0000 - . 3 5 ,99988 + . a 4 t 3 . 4 2.1325 +3.8 2.0291 -1.2 t2.3 3.7678 - 4 . 2 3.1298 -2.4 t.33 1.0033 + . 3 3 1.3002 +.02 14.6 2.3288 1 5 . 0 2.1964 -.95 +2.8 4.2405 -4.8 3.9381 -2.7 +:.9 1.3195 t ? . ? L.0039 +.33 -6.0 2.5539 +6.3 2.3793 -.79 - 3 . 2 4.7650 +5.3 4.3802 -3.2 +5.4 1.0598 -5.4 1.0182 +?.2 + 7 . 5 2.8204 + a , ? 2.5863 - . 9 4 7 3 . 4 5.3560 t5.6 4.8655 -4.1 t19. 1.1486 +19. 1.0069 + 4 . 3 +18. 1.2616 +;8. 1.1084 +3.7 + l o . 3.5370 +ll. 3.1063 -2.9 12.9 6.8081 + 5 . l 6 . C 0 9 3 -7.2 -45. 1.1533 +46. ,83217 - 5 . 3 +30. 1.7432 -30. 1.3416 -.07 +lo. 4.6231 - 1 0 , 3.8411 -8.5 +.78 8.7707 -2.9 7.4723 -12. +78. 1.:860 -83. ,54764 -15. -24. 1.6717 - 2 4 . 1.8117 -15. +4.8 6.2855 -5.2 4.9027 -18. -2.9 11.497 -1.: 9.3949 -19. +lo8 1.0468 +128 - . @ E 0 2 -118 +7.1 4.3180 -7.1 2.6523 -34. -2.8 8.8416 -2.5 6.4521 -29. -7.4 15.358 -5.8 11.979 -27.
‘R
,01994 ,03392 .04087 ,05126 ,10594 ,16150 ,19770 ,25136 ,22528 ,30366 .37968 ,49122 ,42345 .33588 ,70191 ,94131 ,72937 ,82997 1.2143 L.7551 ,92738 ,96137 1.6107 2.5096 ,99804 1.9552 3.2568 .99999 2.1267 3.6648 1.0006 2.3072 4.0855 1.0069 2.5037 4.5314 1.0255 2.7234 5.0251 1.0167 1.1298 3.2625 6.1489 ,81123 1.3713 3.9879 7.5287 ,44780 1.8166 4.9733 9.2500 -.3691 2.5451 6.3088 11.421
-.04
-.03 -.02 -.02 -.05 -.12 -.lo -.07
-.03 -.20 -.13
-.lo +.a9 -.l3 -.12 -.07 +.63
-.11 +.43 +.64
+.86 +.50 tl.9 +1.1 +.37
+2.7 C1.9
+.Os +3.6 +2.2
+.29 +4.6
12.5 +1.7 +5.8 4 . 6 4 4 2 +2.7 1 , 0 5 4 4 ‘+4.8 2.7923 +7.0 5.1995 -2.5 :.1235 +16. 1.2401 -16. 3.4662 +8.4 6.5466 +1.1 i.0773 +36. 1.6655 t 2 4 . 4.4614 76.2 a.3318 -2.3 ,99922 + 5 4 . 2.4740 +15. 5.9407 -.60 18.752 -7.5 ,63539 +38. 3.8622 -4.2 8.1382 -10. 14.094 - 1 4 .
3.42748 TR0‘5f V !?cVRf0.08564;
(D) P
Linear Isometrics
(F)
Dev.
-.35 -.OX -.04
-.02 -1.0 -.35 -.19 -.09 -1.3 -.65 -.29 -.12 -1.5 -.90 -.34 -.09 -.89
-.05 -.32
+.69 1.06 -.19 -1.6 -1.1 -.25 +2.9 +i.4 +.05 13.3
t1.3 +.06 -4.:
+.95 7.63 +4.4
C.14 +2.0 +4.3
-.96 +5.3 t5.7 +2.0 -5.1 t2.7 +2.l -5.0 -12. -3:. -16. -17. -20. -180 -37. -30. -30.
‘R
~
Dev.
,02008 +.32 ,03393 -.05 .04086 -.06 ,51251 -.03 ,10795 +.88 ,16171 -.23 ,19753 -.28 ,25126 - . 1 3 ,22984 + . 7 5 ,30439 - . 4 1 ,37893 - . 4 8 ,49078 -.21 ,43375 +.85 ,53809 - . 4 9 ,69861 -.Xi ,93940 - . 2 9 ,74311 t.98 ,83439 +.02 1.1995 - . 9 3 1.7472 i . 2 4 .93436 +.8l ,96494 +.18 1.5766 - . 5 3 2.4942 + . 4 3 .99880 t.32 1.8983 -.OB 3.2484 + . d 3 1.0000 +.05 2.0599 +.2a 3.6497 +.91 1.00l5 + . l 5 2.2326 +.68 4.0791 +.79 1.0116 +1.1 2.4254 tl.1 4.3461: + . 4 6 1.0388 +:.3 2.5473 - 1 . 4 5,0605 - . i o 1.0789 ti2. 1.1809 +11. 3.2i97 -.66 6 . 2 7 ~ 3-3.1 1.0014 +27. 1.5076 +12. 4 . 0 3 a i -3.d 7.8343 -8.1 ,87751 +36. 2.1131 -1.7 5.2159 -13. 9.8642 -15. ,50252 +9.3 3.1372 -22. 6.9039 - 2 4 . 12.5‘.7 -23.
? 7 ! 64Tno’j5 n l~cV,+0.043)
R - -zcVR-O. 082
c R T
R IBj Pa = =z C v R - 3 . 0 8 5 5 4
(c)
-
‘a
V -0.0778 2 2 = 5-___-
c R
0.42748 [1+0.48(1-?R0‘5) 1 Z c V R ( Z c V R + O . 08664)
dTR v
= 6.857 when TR = 1
which is high, and
(“) dTR
v
= 5.888 when T R = 1.5
which is low. The slope at an average temperature of TR = 1.25 is 6.25 which compares favorably with the 6.1 from the reduced vapor-pressure charts, so p R =
27 / 64TR0’55 TR ZCVR - 0.082 ( i c V + ~ 0.043)2
(90)
TR
-
__ 27/(64~,~”)
ZcVR-0.067
0.457236 [1+0.375(1-TRo”j l 2 - FC V R ~ Z c V R + 0 . 0 7 7 8 ) + 0 . 0 7 7 8 ( T c v R - o o 7 7 8 ) IF)
droop a little as TR 2. In the case of eq 88 for argon whose 2, = 0.29121 one quickly finds that for n = 0.55 at VR = 1, eq 89 gives
(”)
( E ) PR =
‘R
=
TR
i:c V R - 0 . 0 7 5
-
0 . 5 -0.178125TR
(;cvR+0.0j)2
should be a good equation for argon. This is shown to be true in Table IV where eq 90 is presented as eq (D) and compared with the NBS data on argon. (In making these comparisons one can emphasize either the absolute deviations or percentage deviations depending on the ultimate use of the equation of state. Here a little more weight is given to the percentage deviation.) The selection of the amount of translation depends in part upon the critical compressibility factor and the two are simply related in a linear fashion. To fit data a t about twice the critical density, but have fairly high positive deviations (20-2570) a t about two-thirds the critical volume, along the critical isotherm, use tP,/RT, = 0.8572,
-
0.1674
(91)
To fit data well around two-thirds the critical volume, but have high negative deviations a t twice the critical and higher densities, use
90
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
tP,/RT, = 0.7522, - 0.152
(92)
These equations give for argon the values of 0.082 and 0.067, respectively. Clearly anything in between is acceptable depending upon where the fit is to be maximized above the critical density. A variation of (D) is the temperature function. In (E) it has been made linear with the slope of the critical isometric taken as 6.155, and the translation made a little less than (D) and (E). The performance is exactly what would be expected. Below the critical density it predicts too high at the lowest temperatures because it does not allow the isometrics to curve down as they really do. In the region of the critical density it is superb because it does not force downward curvature on isometrics that are essentially straight, as does eq (D). At the higher densities the effect of curvature of the isometrics is completely dominated by the large deviations in the critical isotherm a t reduced densities around 1.5 to 2.0, and its smaller translation makes it better at 1.5 and worse at 2.0. On an overall basis (E) performs as well as any other equation, so one must conclude that in the two-term cubic linear isometrics are as good as isometrics that always curve down. A number of years ago there were claims that the Redlich-Kwong equation was the best of the two-parameter equations of state so it has been selected for comparison as eq (A) in Table IV. For densities up to 1/4 the critical it is not as good as (D). For densities around the critical the two are similar. At 2/3 the critical volume (A) is not as good as (D), but at twice the critical density it is a little better though a slight change in translation would make (D) just as good. Overall the two equations are about a standoff for argon, but other substances will reveal appreciable differences. Also the Redlich-Kwong can never be quite as good as eq (D) or (E) or eq 55 because its 2-chart sum is = 0.33333 + 0.42748 - 0.08664 = 0.67417, which is slightly greater than that of (D). There have been several attempts to improve the Redlich-Kwong equation either by changing the temperature function or by changing the volume-function constants. Soave chose the former procedure, replacing He made m a the simple l/'Td.5 with [l + m(1 - TRo.5)]2. function of the acentric factor (26) which is just an empirical parameter somewhat equivalent to the fundamental quantity, M , but which does not constitute a basic fluid characteristic as does M which is the slope of the critical volume line on P-T coordinates. Soave's eq (B) is shown in Table IV and the results indicate no improvement over the Redlich-Kwong as the two predict almost the same since they are identical at the critical temperature and since it is for substances with lower experimental 2, that a variable temperature function, such as Soave's or l/TRn, is needed in place of 1/TRo,5. The Peng-Robinson eq ( C ) has been shown to be a specific case of the all-inclusive eq 1 presented in this study, but it represented a new development because it was not based on previous well-known equations. It uses the same temperature function as Soave, [l+ ~ ( 1TR',~)]', with K a function of the acentric factor. Its principal difference from the Soave equation is that it predicts the critical compressibility factor to be 0.3074 instead of 1/3. This is not beneficial for argon, but it does help for substances whose z, is around 0.26 to 0.27. Table IV shows that below the critical density it is inferior to (A), (B), or (D) because its .%chart sum, 2 = 0.3074 + 0.4572 - 0.0778 = 0.6868, is higher than for (A), (B), or (D). The main reason for the large negative deviations in this range is that the generalized second virial coefficient at the critical
'Pable V. substance Ar
c,H4 C4F8 i-C CHF, 3 "
-(dZ/dP&R=l,
-
ZC
0.291 0.281 0.278 0.270 0.259 0.242
M
a
6.1 6.5 7.8 7.2 7.4 7.4
0.332 0.337 0.336 0.352 0.352 0.37 0
p R = o = -BG,.
temperature for (C)is -0.379 whereas the true value for argon is -0.332. At 213 the critical volume the PengRobinson is better than (D) while at twice the critical density, it is not so good. The conclusion from the data on argon is that no one equation stands clearly above the others, even though no other equation is better than (D), and (D) is as simple as any other in form. In order to test the versatility of any of the equations, it is necessary to look at other substances carefully chosen over a range of parameters that determine the PVT behavior. These have been shown to be z,,M , and (dz/dPR),,=,, as PR 0. Substances have been selected that have dependable wide-range PVT data available and are listed in Table V. The last two substances in Table V are polar, but that does not affect the principles involved unless there is a high degree of chemical association of vapor-phase molecules. It would have been more desirable to have selected a nonpolar substance with a low z,, such as octane or nonane, but the experimental PVT data for these are so extremely limited that no reliable comparisons can be made. For ethylene the temperature function to be used in eq 67 is TR0.@while the translation is 0.074 and the resulting eq (D) up to the critical density is clearly superior to the Redlich-Kwong, the Soave, the Peng-Robinson, and the Chueh-Prausnitz (5). (See Table VI.) As expected, the latter equation is better than the Redlich-Kwong, Soave, and Peng-Robinson below the critical because the constants of the Redlich-Kwong were arbitrarily adjusted to get a better fit for saturated vapor volumes which, of course, are densities below the critical. Even in this range, however, the Chueh-Prausnitz is inferior to eq (D), partly because it has the same temperature function as the Redlich-Kwong. Above the critical density its deviations are so very large it must be dismissed as a serious contender for the throne of cubic equations. At reduced densities of about 1.5 the Peng-Robinson is better than (D), but this is simply because the translation for (D) has been selected by eq 91 so that it will do a better job near twice the critical and higher densities. As with argon, the translation could have been set by eq 92 and the fit at pR = 1.5 would have been equal to that of the Peng-Robinson, while still being far better than the Peng-Robinson below the critical density. Equation (F)has been introduced here to show how another simple temperature function, ek(l-Td, behaves. It gives about the same fit as the temperature power function, TRn. The more complex temperature function of Soave, or the linear temperature function (20) could just as easily have been chosen and the results would have been as indicated for argon. The overall conclusion for ethylene is that eq (D) is both the best and the simplest. Table VI1 gives the comparisons for perfluorocyclobutane. Up to a reduced density of 0.8 eq (D)is better than all the others, particularly if reliance is placed on the higher precision data. From the critical density to about 1.8 times the critical the Peng-Robinson is very slightly better. In general, the comparisons are about the same as for argon.
-
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
91
Table VI. Comparison of Six Equations of State with Data ( I 0 ) for Ethylene
-
C2H4
ETHYLENE
Red 1 ichKwonc (A) P , bar
Tc = 282.35K
0.8
Pc = 1.5
5 0 . 4 1 9 7 bar
-25 10 75 150 -25
10 7.635 gmo 1e / d r n 3 -C =
3.5
R = ,0831433
bar.dn3/qrnole.F
Z
75 150 0 75 150
10
5.0
75 150
= 0.281304
10
6.5
75
150 10
7.5 8.0 10.0
75 150 10 75 150 10 15 150
10
11.0
75
150 12.0
10 75 125
0
13.0 14.0
15.0
15.5
50 100 0 75 0 50 -25 0 30
Tu 0.08664
TR C
R
-0.0778
€
P
Dev. t.46
-.08 -.37 -.47 t1.4 t.01 +.57 -.74 t1.4
-.71 -1.1 +.47
-.18 -.74 t.07 t1.2
-.os
C.68 t2.8 t.49
+1.9 +3.7 t.76 tL8. +7.8 t1.3 +33. t9.3 t.99 +45. t9.5 t2.4 +79. t16. +4.0
+67. +6.1 t47. +8.7
+199 t38.
+13.
13.932 16,570 21.413 26.931 22.251 27.692 37.604 48.804 41.380 74.393 106.10 50.415 96.849 148.11 51.278 119.14 193.71 51.731 135.97 228.36 52.393 145.50 247.54 62.243 195.93 342.25 74.623 232.19 404.67 94.994 279.39 415.12 92.277 259.96 421.00 133.36 422.62 194.37 420.61
Dev. +.21 -.07 t.06 +.33 t.89
+.03 s.27 t.77
+.97 +1.5 t2.5 t.53
+3.1 t4.3 t.17
+5.7 +6.3
+.E1 t7.9
+7.5 t2.0 t9.1 t8.0
+18. +14. +9.2
+33. +15. +8.7 +46.
+15. +11.
t75. +20. t10. +64. t11. t46. +12. 108.58 t 1 7 8 234.66 +36. 382.25 +15.
Martin (3)
P 13.779 16.374 21.150 26.605 21.905 27.187 36.845 47.804 40.732 71.902 102.09 49.895 92.929 140.87 51.182 113.06 181.74 51.396 127.48 211.78 51.619 135.33 228.03 55.890 174.12 304.84 61.859 200.10 352.89 72.048 232.50
Dev. -.E9 -1.3 -1.2 -.E9 -.67
-1.8 -1.8 -1.3 -.62 -1.9 -1.4
-.51 -1.1 -.EO -.02 +.25
-.31 +.15 tl.l -.32
+.50 >1.5 -.49 t6.3 tl.3 -2.8
+lo. -.50 -5.2
+11. -3.9
351.51 -7.3 58.779 203.03 342.48 78.304 325.02 107.83 298.2: 22.456 127.39 250.60
112. -6.2 -10. -3.4
-15. -19. -21. -42. -26. -24.
P
Dev.
13.894 -.06
16.530 -.31 2 1 . 3 1 7 -.39 26.732 22.155 27.581 37.280 48.122 41.237 72.919 102.55 50.261 93.964 140.85 51.233 114.05 180.91 51.531 128.62 210.39
-.42 t.46 -.37
-.55 -.64 t.62
-.55
-.91 t.22 t.03 t.81 t.06 tl.l -.76 t.42 +2.0 -.98 51.955 t 1 . 2 136.66 +2.5 226.38 -1.2 5 8 . 5 6 9 +11. 1 7 7 . 1 7 +3.1 302.44 -3.6 66.848 t 1 9 . 204.69 t 1 . 8 350.24 -5.9 80.180 t23. 239.03 -1.2 352.27 -7.1 70.308 +34. 214.16 -1.0 347.76 -9.0 94.601 t17. 335.82 -12. 129.22 -3.3 315.27 -16. 45.066 t16. 151.26 -12. 272.26 -18.
%
3
%
%
Martin with Exponential Temperature Function (Fi
P 13.939 16.544 21.301 26.703 22.278 27.610 37.227 48.028 41.226 72.568 102.27 49.885 93.447 140.83 50.648 113.92 182.28 51.138 129.49 213.90 51.864 138.39 231.52 62.430 186.60 320.00 75.578 222.09 379.53 97.140 268.94 393.35 97.904 255.72 404.61 141.81 413.78 206.91 421.61 127.57 249.94 390.95
Dev. +.26 -.23 -.47 -.52 +l.O -.27 -.73 -.E4 1.59 -1.0 -1.2
-.53 -.52 -.E3
-1.1 tl.0
-.01 -.35 t2.7 +.68 +.98 t3.8 +1.0 t19. 18.6 t2.1 t35. +11. t2.0 +49. +11. t3.8
+86. t18.
+5.9 t75. +8.6 t55. t12. t227 +45. t18.
P
13.943 16.530 21.302 26.764 22.319 27.578 37.236 48.222 41.397 72.649 103.06 50.238 93.447 141.81 51.196 113.23 182.43 51.484 127.57 212.33 51.902 135.49 228.55 58.493 175.48 305.57 66.759 202.72 353.89
Dev. t.29 -.32 -.46 -.30 t1.2 -.38 -.71 -.44
+1.0 -.92 -.42 +.18 -.52
-.14
+.01 t.41 +.07 +.33 +1.2
-.06 tl.l
+1.6
-.27 +11. +2.1 -2.6 +19. 1.90 -4.9 80.073 t 2 3 . 236.77 -2.2 353.66 -6.7 71.824 t 3 7 . 211.18 -2.4 346.83 -9.2 96.297 t19. 332.97 -13. 131.10 -1.9 311.58 -17. 55.152 +41. 1 5 3 . 2 3 -11. 2 6 9 . 1 8 -19.
T"
TR0'5?cVR(ZcVR+ 0 . 0 8 6 6 4 ) 0 . 4 2 7 4 6 1 1 + 0.615(1 - T q 0 ' 5 ) 1 2
TR ZcVR- il.08664
f V
13.966 16.569 21.321 26.718 22.367 27.688 37.289 48.074 41.561 72.800 102.44 50.383 93.761 140.95 51.227 114.17 182.20 51.665 129.56 213.50 452.32 138.32 230.90 62.134 185.36 317.75 74.495 219.75 375.84 94.845 264.99 388.18 94.145 250.24 397.48 135.47 404.02 196.73 408.32 116.65 237.15 375.97
Robinson (Ci
(a)
0.42748
-
~~
ZcVR-
13.903 16.582 21.401 26.843 22.054 27.684 37.502 48.434 40.984 73.323 103.49 50.149 93.932 142.00 51.193 112.77 182.30 51.317 126.05 212.47 51.359 133.33 229.16 52.579 171.87 313.57 56.102 201.12 372.14 65.202 241.97 379.14 52.597 216.39 382.13 81.075 380.91 133.57 375.49 39.002 172..31 331.68
Soave
%
P
data
ChuehPrausnitz (El
Peng-
Z
-
C
v R ( ZC v R
t 0-.98664)
0 . 4 5 7 2 3 6 [1+0.504(1-TR0'5) 1' ZCVR(i V tO.0778) +0.0778(5cVR-0.0778) C
R
Table VI11 gives the comparisons for isopentane. Up to the critical temperature eq (D) is clearly the best. Above the critical temperature the Peng-Robinson and eq (D) give about the same average deviation, both of which are better than the Redlich-Kwong, the Soave, and the Chueh-Prausnitz. At the higher densities the PengRobinson and eq (D) are similar. The conclusion is that eq (D) is the simplest and best for an overall fit for isopentane. Table IX gives the comparisons for trifluoromethane. At all densities eq (D) is superior to the other equations while retaining the simplest form. Table X gives the results for ammonia. Again for all densities eq (D) is superior to all the others. Considering the performance for the wide range of substances the inescapable conclusion is that eq (D), or eq 67 with the temperature function, TRn,(or A + BTR) is not only the simplest but the best of the two-term cubics. The three-term virial does have a slightly lower value of E, but it is one term more complicated. Certainly, there is no reason to go to other more complex two-term cubics, such as the Peng-Robinson, for they do not do as well as (D). One of the reasons for the superiority of (D) is found in Table V, which lists the PVT behavior parameters, e,, M , and B G ~one , of which is not independent because it has
been shown that = f , - B G is ~ practically constant a t 0.62. Using a single empirical parameter, such as the acentric factor, in a correlating equation of state cannot possibly do as well as (D) unless there is a consistent relation between M and 2, or BG1,which there is not for the six substances chosen. The Soave and Peng-Robinson equations, for example, are constrained to fixed values of 2, (1/3 and 0.3074, respectively) with the temperature function allowed to vary with a single parameter, but no allowance is made for different values of e,. Equation (D) does not have this pitfall as it is not locked into a fixed 2, for all substances. Since in (D) 2, = 0.25 tP,/RT,, 2, varies from 0.25 + 0.082 = 0.332 for argon down to 0.25 + 0.04 = 0.29 for ammonia. At the same time the temperature function, TRn,is varying simultaneously with M and 2, as shown by eq 89 where M and the translation, tP,lRT,, enter into the slope of the critical isometric. Thus, accounting for differences in e,, having the smallest value of E, better performance, and inherent simplicity are the compelling features that make (D) superior to all others. Clearly, other temperature functions besides TRn can be used in eq 67 and they will perform as described for argon and ethylene, these being the linear and exponential functions. The temperature function of Soave could easily be used but nothing is gained by its more complex form.
+
92
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
Table VII. Comparison of Four Equations of State with Data ( I 7 ) for Perfluorocyclobutane
TC
=
1.37518 i.37758
699.27R
i.38062 1.38351
Pc =
.66C78:
,682863 ,684335 ,465549
403.6gsi. Fc
=
38. 71b/ft3. MW =
,411692
200.044. R = 10.131F
&
1bnole.R zc =
C.27801
'
,406226 .407?67 ,438148 ,218651 ,218911 ,219291 ,126033 ,125177 ,126257 ,126582 *.la6771
191.78 545.24 617.24 682.92 527.69 618.14 688.25 572.63 641.52 7i1.18 785.07 862.42 571.32 710.19 761.52 851.83 559.62 616.10 689.15 596. 02 636.28 689.96 634.03 670.19 689.60 651.44 768.55 849.84 699.08 761.69 851.69 941.69 1031.7
*.08C078
,066534
*.053385
,044823 *.040339
*.032031
699.06 761.69 851.69 941.69 iG31.7 685.89 761.67 820.86 875.49 699.08 761.69 851.69 896.69 986.69 694.68 769.11 864.33 699.08 761.69 851.69 941.69 1031.7 699.08 761.69 851.69 941.69 1031.7
.030030
'.026593
'.022879
,022497 *.020020
,019670
,017931
*.017795
+*.014356
**.012920
*
** (A) PR ~
699.52 738.79 776.94 852.59 716.69 806.69 696.69 986.69 619.08 761.69 851.69 941.69 1031.7 707.28 782.89 857.52 699.08 761.69 851.69 941.69 1331.7 698.8 752.14 796.81 838.86 862.04 695.95 763.76 801.60 861.36 699.08 761.69 851.69 941.69 1031.7 661.07 699.27 761.69 851.69 941.69 63i.29 699.27 761.69 851.69
17.R3
18.363 + 3 . 2
20.11 23.39
23.473 i l . 8 23.297 1.90 25.852 1 . 4 7 38.421 l 4 . l 4 5 . 7 8 1 11.6 51.426 +. 62 59.676 +2.2 6 8 . 1 6 7 -1.5 76.689 -.66 8 5 . 6 7 7 -.95 95,040 -1.7 66.389 11.6 85.829 -2.4 95.709 -1.8 105,40 - 2 . 1 65.497 + 4 . 6 73.700 12.5 83.565 il.0 119.26 i 4 . 8 130.96 -2.9 145.12 -1.4 195.67 i4.2 214.47 12.9 224.46 12.3 204.23 i2.4 264.42 -.a6 305.30 -2.2 258.31 i . 1 5 297.43 -1. 6 352.67 -2.8 406.97 -3.5 460.56 -3.9 309.46 1 . 3 8 364.96 -2.0 442.99 -3.7 519.40 -4.5 594.55 -5.0 320.22 i2.2 410.92 -1.9 475.78 - 4 . 0 534.68 -4.3 312.79 -.76 465.96 -3.1 596.11 -5.2 659.83 -5.9 785.C) - 6 . 7 382.55 11.7 522.45 - 4 . c 695.37 -6.3 398.31 + . 7 1 535.81 -4.0 727.04 -6.5 912.34 -7.9 1093.0 -8.9 403.00 c . 2c 591.81 - 4 . 4 8 5 3 . 5 8 -7.2 1106.5 -9.0 1352.4 - 1 0 . 404.46 + . I 4 535.42 -3.6 660.06 -5.0 900.85 -7.5 475.23 -.85 825.42 -5.5 1161.5 - 8 . 5 1486.8 -11. 423.01 15.0 738.05 -.49 1173.3 -6.4 1592.4 -10. 1999.0 -13. 470.50 13.9 858.08 -1.8 1226.5 -6.6 481.45 -18. 873.48 1 4 . 5 1415.0 -5.5 1936.1 -11. 2441.6 - 1 5 , 492.75 123. 837.81 1 7 . 2 1119.0 +.69 1378.2 -3.7 i519.1 - 5 . 8 572.90 i l l . 1066.8 110. 1365.5 12.0 1795.5 - 5 . 3 608.64 - 4 0 . 1089.5 19.9 1753.7 -5.1 2393.1 -12. 3013.2 -17. 861.44 -226 1318.6 l65. 2043.8 122. 3051.0 t4.2 4022.7 -3.9 1169.0 +528 2206.9 +52. 3125.1 119. 4453.3 r2.4
25.73 36.92 45.05 51.11 58.40 69.20 77.20 86.50 96.70 65.37 87.95 97.47 107.70 62.62 71.92 82.73 113.79 127.3 143.09 187.75 208.37 219.33 199.5 266.7 312.0 257.91 302.15 362.81 421.53 479.11 308.29 372.55 459.77 543.76 625.9 313.40 419.00 495.50 558. 90 369.97 480.62 629.04 700.83 041.84 376.00 544.00 742.0 395.48 558.11 777.54 990.26 1199.3 102.18
619.08 919.72 1215.5 1508.3 403.90 555.1 695.4 973.4 479.26 873.64 1269.8 1665.5 403.00 741.70 1253.8 1773.3 2295.6 453.00 a73.30 13-3.4 436.9;
835.39 1497.C
2171.8 2854.8 c02.00 781.40 1111.4
1430.4 16L2.4 405.90 987.40 1338.4 1895.4 434.15 991.76 1847.6 2:20.a 3607.0 264.68 798.6 1671.0 2929.3 4186.9 186.21 1455.6 2621.2 4301.7
-2.5 11.4 1.69
16.195 2C.331 23.191 25.778 37.827 45.367 51.150 58.632 67.406 76.215 85.493 95.145 65.066 85.232 95.469 105.59 64.077 72.601 82.84d 115.50 127.97 143.04 188.15 208.76 219.73 197.62 263.43 308.01 253.07 296.27 357.10 416.65 4 7 5. 819 302.71 365.04 452.46 537.67 620.99 310.13
1.44 13.3 4 5 . 5 9 2 11.2 5 1 . 4 0 2 1.57
59.040 i1.l 67.882 - 1 . 9 76.747 -.58 55.569 i . 3 0 85.896 - 2 . 3 96.231 - 1 . 3
-06.28 -1.3 64.575 *3.1 73.174 - 1 . 7 63.50: -.93 +2.8
+1.8 r1.3
+2.0 t2.O 11.8
+,E5 1.73 +.70
1.15 1.19
365.14
t.64
426.05 485,66 309.44 374.17 464.65 552.53 638.16 316.25 424.03 500.69 569.95 372.73
11.1 il.4 - , 37 1.44 11.1
485.77
+1.1
11.6
12.0 1.91 il.2 11.1 +2.0 +,75
642.74 -2.9 7 1 9 . 0 7 12.6 867.91 13.1 380.48 11.2 553.03 11.7 7 6 4 . 8 6 13.1 398.20 1.69 569.56 12.; 896.46 13.7 1033.8 14.4 1252.9 +4.5 402.82 + . 1 7 642.43 13.8 972.69 i5.8 1288.6 16.0 1592.2 -5.6 404.69 *.20 571.71 i2.9 730.34 i 5 . 0 1035.3 16.4 495.12 +3.3 943.84 18.0 1371.0 i8.0 1779.8 16.1 422.80 14.9 829.87 tl2. 1389.4 -11. 1922.9 + a . 4 2433.9 - 5 . 0 482.88 16.6 983.68 113. 1156.9 rll. 481.08 +la. 989.10 t l 8 . 1687.1 1 1 3 . 2352.3 2989.3 16.3 14.1 491.83 939.12 1302.7 1636.5 1817.3 565.29 1233.8 1590.5 2143.2 608.20 1230.7 2086.2 2901.5 3682.4 733.20 1318.6 2245.8 3526.4 4749.7 895.51 2206.9 3364.9 4967.7
11.9 -1.1 +.56 37 -2.2 4 5 . 4 7 5 *.94 51.330 1 . 4 3
-2.5
18.140 20.323 23.220 25.824 37.726
*.lo 1.98 1.43
5s.65a 1 . 4 4
+.44 r.19
-.
-2.6 -1.3 -1.2 -1.6
67.683 -2.2 75.638 - . 77 6 5 . 5 3 2 -1.2 95.532 -1.2 -.46 65.079 - . 4 4 85.722 -2.5 -3.1 96.000 -1.5 -2.1 -2.1 1 0 5 . 9 9 -1.6 12.3 64.010 12.2 1.95 72.859 + 1 . 3 i . 1 4 63.312 r . 70 + I .5 115.73 11.2 -.52 128.82 1.96 -.c4 144.36 1.88 189.40 t.88 r.22 211.12 i 1 . 3 1.18 1.17 222.49 il. 4 -.94 199.45 -.03 267.08 + . I 4 -1.2 -1.3 3 1 1 . 4 4 -.la -1.9 256.65 - . 4 9 -2.0 300.78 -.45 361.36 -.lo -1.6 419.47 - . 4 8 -1.2 -.a4 475.77 -.70 -1.8 307.20 -.35 -2.0 371.21 1 . 3 6 458.29 -.32 -1.6 -i.l 541.10 -.49 620.80 -.82 -.a0 -1.0 113.60 1.64 420.23 1.29 413.05 -1.4 486.55 -1.8 4 9 3 . 8 8 -.33 553.14 - 1 . 0 559.25 1.06 370.72 1.07 365.85 -1.1 472.78 -1.6 480.85 + . 0 5 621.90 -1.1 629.42 C.36 694.68 - . a 7 700.14 -.39 837.10 -.56 836.09 -.68 375.77 -. 06 3 7 8 . 5 4 1.68 537.53 -1.2 546.18 1.40 737.16 -.65 744.07 1.28 3 9 4 . 4 0 -.27 396.76 1.32 554.02 - . 7 3 562.46 + . 7 R 775.80 -.22 783.08 1.71 989.81 - . 0 5 988.48 -.la 1182.7 -1.4 1197.1 -.la 402.45 C.07 402.76 i . 1 4 622.73 + . 5 9 631.33 i2.0 928.02 1.90 933.93 +1.5 1214.0 -.12 1221.8 1.51 1505.6 -.I8 1477.6 -2.0 402.56 - . 3 3 404.03 1.03 542.98 12.2 564.35 11.6 702.96 +I. 1 712.42 +2.5 983.96 il. 1 989.23 i l . 6 489.79 +2.2 485.14 A1.2 894.01 i2.3 903.77 t3.5 1285.9 +1.3 1281.3 +.91 .~ 1662.9 - . l 5 1632.7 -2.0 413.13 +2.5 408.40 +1.3 775.05 14.5 789.65 16.5 1285.1 c2.5 1281.8 i2.2 1740.6 -1.8 1767.8 -.31 2167.5 -5.6 2236.0 -2.6 460.10 c1.6 4 6 8 . 0 7 C3.3 91i.18 + 4 . 3 926.01 16.0 1339.7 C2.0 1341.4 C2.1 433.14 t6.4 447.51 +lo. 886.87 C6.1 909.10 C8.8 1515.8 c1.3 L 5 1 3 . 5 +1.1 2114.2 -2.7 2073.1 -4.5 2692.5 - 5 , l 2594.4 -9.1 437.01 t8.7 453.11 113. 835.91 17.0 659.75 110. 1161.2 i 4 . 5 1180.1 16.2 i467.6 12.6 1460.6 +2.; 1621.1 1 . 5 4 1623.4 +.68 487.32 t20. 460.16 1 i 3 . i050.2 16.4 1081.3 t10. 1391.7 i 4 . 0 1 3 7 C . l 12.4 1863.3 - 1 . 7 1903.2 c.41 493.54 il4. 522.74 +20. 1077.3 +8.6 1045.2 i5.9 1806.8 -2.2 1805.9 -2.3 2536.7 -6.6 2475.1 -9.0 3100.9 - 1 4 . 3239.3 - 1 0 . 346.16 i31. 4 0 3 . 8 4 +53. 653.96 -6.9 917.84 1 1 5 . 1701.3 cl.8 1660.3 -.E4 2777.7 -5.2 2736.1 -6.6 3849.3 -8.1 3688.6 -12. 185.21 -.54 242.78 1 3 0 . 1305.9 -10. 1 3 6 4 . 0 -6.3 2297.6 -12. 2304.1 -12. 3673.4 - 1 5 . 3548.9 -18.
86.278 -.49 95.779 - . 9 5
i16.96 129.63 144.92 191.47 212.63 223.43 201.19 268.64 314.18 258.29 302.73
i2.2 11.1
i22. i20. 117. -14. +13.
-39. 125. +19. 113. +lo.
124. 113. 16.6 i2.1 1177 C65. +34.
+20. +13. 1381 +52. C28. 115.
Higher p r e c ~ s ~ odna t a Extension o f h i g h e r p r e c i s i o n data t o twice the c r i t i c a l density IC)
R
0.42748
p=L 0.457236 Ilr0.89!l-TR0'ji -
TC vR -0.08664 181
TR i
18.240 20.383 23.250 25.843 38.312
=
TCVR-0.O8664
I
.
0.42746 Iltl 0 ll-TRo'5 , 1
TcvR(IcvR+O. 08664)
v
C R
-0.~778
z
'1 v ( tC v R + ~ . o ~ ~ ~ ~ ~ ~ . o ~ ~ ~ I z ~ v ~ - o . o ~ ~ ~ I
c H
ID)
27/64TR1"
TR
px=-
Z C V R -0.071
-
(TcVR+2.05412
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
93
Table VIII. Comparison of Five Equations of State with Data ( 8 ) for Isopentane ISOPEN TANE
RedlichKwong
Soave
(Ai
(Bl 3
%
T, K 310
TC = 460.39K
330
p C = 33.37atm350
V = .306 cm3 /gmol
370
R = 82.05606 c m atm grnole .K
390
z
410
-
=
0.270297
430
450
455
Z
,9445 ,9395 ,9149 .9671 ,8771 .E832 .9729 ,8301 .9045 .9775 ,7726 ,8286 .9207 .9811 .7014 ,7715 ,8589 ,9335 .9840 ,6087 ,6290 .7325 .E132 .E822 ,9437 ,9864 ,4649 .4875 ,6219 .7104 ,7822 .a445 ,9006 ,9521 .9883 ,4204 .4547 .5391 .6460 ,7256 ,7923 ,8512 ,9046 ,9539
470
,9887 ,6046 .4386 ,3680
500
600
-
(A) P
R
18)
.2949 .2585 .2295 .5891 .7641 ,8926 ,9899 .7651 .6106 .4532 .3932 ,3671 .4363 .5771 .7097 .a200 .9152 .9919 1.0606 .9309 .a059 ,6996 .6625 ,6937 ,7539 .7919 .E327 .a747 ,9170 .9588 .9959
-zcVR-0.08664 T
P R = ZcVR-0.08664
1.35 1 2.47 1 4.19 4
1 6.66
4
1 10.08 8 4 1 14.64 12 8 4 1 20.63 20 16 12. 8 4
1 28.41 28 24 20 16 12 8 4 1 30.71 30
28 24 20 16
12 8 4 1 150
100 80 60 50
40 30 20 10 1 200 150 100 80 60 50 40 30 20 10 1 300 250 200 150 100 80 60 50
40 30 20 10 1
%
%
%
Dev.
P
Dev.
P
Dev.
P
Dev.
P
Dev.
1.3673 1.0092 2.5138 1.0065 4.2855 4.0866 1.0047 6.8484 4.0598 1.0033 10.416 8.1932 4.0419 1.0024 15.196 17.332 8.1285 4.0208 1.0017 21.466 20.765 16.420 12.209 8.0839 4.0194 1.0011 29.377 28.957 24.679 20.427 16.248 12.128 8.0521 4.0119 1.0007 31.448 30.865 28.829 24.580 20.369 16.217 12.112 8.0456 4.0106 1.0007 213.19 142.38 115.98 89.762 75.630 57.007 30.570 20.227 10.047 1.0004 258.68 199.00 134.81 105.90 68.007 50.407 40.127 30.110 20.042 10.007 1.0000 309.95 267.15 218.57 161.46 100.12 78.270 58.462 48.828 39.206 29.538 19.789 9.9470 ,99948
t1.3 t.92 tl.8 +.65 t2.3 t2.2 t.47 t2.8 t1.5 t.33 t3.3 t2.4 +1.1 t.24 t3.8 +2.8 t1.6 t.72 t.17 t4.1 +3.8 t2.4 tl.7 +1.1 t.49 t.ll t3.4 t3.4 t2.8 +2.1 +1.6 +1.1 +.65 t.30 +.07 +2.4 t2.9 t3.0 +2.4 +1.9 t1.4 +.93 t.57 t.27 +.07 +42. +42. 145. t50. t51. +43. t1.9 tl.l t.47 +.04 t29. +33. +35. +32. t13. t.81 t.32 t.37 +.21 +.07 00 t3.3 t6.9 +9.3 f7.6 +.12 -2.2 -2.6 -2.3 -2.0 -1.5 -1.1 -.53 -.05
1.3617 1.0062 2.4983 1.0043 4.2486 4.0534 1.0030 6.7719 4.0363 1.0021 10.274 8.1151 4.0258 1.0015 14.965 12.202 8.0811 4.0186 1.0011 21.141 20.477 16.281 12.145 8.0591 4.0139 1.0008 29.066 28.680 24.548 20.356 16.210 12.109 8.0447 4.0102 1.0006 31.210 30.683 28.712 24.518 20.334 16.198 12.102 8.0419 4.0097 1.0006 216.29 145.08 118.47 92.002 77.689 58.735 30.759 20.280 10.057 1.0005 271.23 210.34 144.32 114.21 73.808 53.483 41.357 30.594 20.210 10.042 1.0003 346.50 300.71 248.00 184.47 112.64 85.995 62.367 51.358 40.711 30.325 20.115 10.023 1.0002
t.87 t.62 t1.2 t.43 t1.4 tl.3 t.30 t1.7 t.91 t.21 +1.9 11.4 1.65
1.3587 1.0046 2.4888 1.0027 4.2224 4.0295 1.0015 6.7098 4.0136 1.0006 10.144 8.0306 4.0041 1.0001 14.720 12.029 8.0004 3.9980 ,99977 20.749 20.093 16.000 11.977 7.9823 3.9942 .99952 28.690 28.251 24.041 19.955 15.931 11.949 7.9715 3.9916 .99944 31.016 30.295 28.167 23.998 19.934 15.027 11.944 7.9696 3.9913 .99945 132.39 95.256 81.169 67.056 59.384 49.232 30.069 19.886 9.9496 ,99938 182.32 148.38 110.29 92.286 66.779 51.691 40.248 29.840 19.836 9.9416 .99929 275.19 244.77 208.56 162.59 105.42 81.939 60.103 49.730 39.619 29.678 19.812 9.9433 .99937
t.64 +.46 +.76 t.27 +.77 c.74 t.15 t.75 +.34 t.06 t.63 t.38 1.10 t.01 +.55 +.23 t.01 -.05 -.02 t.58 t.47 00 +.19 t.22 +.15 -.05 +.99 t.90 t.17 -.23 -.40 -.43 -.36 -.21 -.06 t1.0 t.98 1.60 -.01 -.33 -.46 -.47 -.38 -.22 -.06 -12. -4.7 t1.5 +12. tl9. t23. +.23 -.57 -.50 -.06 -8.8 -1.0 t10. t15. t11. +3.4 +.62 -.53 -.82 -.58 -.07
1.3478 .99878 7.4642 .99914 4.1757 3.9877 .99944 6.6342 3.9923 .99959 10.039 7.9791 3.9960 .99975 14.601 11.979 7.9923 3.9981 .99986 20.645 20.019 16.013 12.004 8.0000 3.9998 ,99991 28.629 28.230 24.173 20.093 16.044 12.017 8.0045 4.0004 1.0000 30.939 30.329 28.317 24.193 20.102 16.049 12.019 8.0053 4.0007 1.0001 114.76 88.008 77.113 65.629 59.081 49.907 30.485 20.115 10.014 1.0001 162.98 138.23 108.13 92.722 68.934 53.337 41.267 30.437 20.115 10.015 1.0000 256.45 232.39 202.59 162.12 107.74 84.087 61.634 50.894 40.425 30.164 20.042
-.16 -.12 -.23 -.09 -.34 -.31 -.06 -.39 -.19 -.04 -.41 -.26 -.lo -.03 -.27 -.17 -.lo -.05 -.01 t.07 t.09 +.OB +.03 00 00 -.01 +.75 t.82 t.72 t.47 +.27 t.14 t.06 t.01 00 +.75 tl.l tl.l t.80
1.3648 1.0078 2.5065 1.0054 4.2662 4.0693 1.0038 6.8032 4.0456 1.0025 10.317 8.1375 4.0300 1.0017 14.993 12.213 8.0833 4.0187 1.0011 21.056 20.396 16.228 12.115 8.0464 4.0108 1.0006 28.530 28.172 24.234 20.163 16.098 12.051 8.0205 4.0045 1.0003 30.436 29.973 28.160 24.174 20.122 16.074 12.038 8.0152 4.0034 1.0002 259.67 169.39 136.10 103.19 85.467 62.025 29.977 20.021 10.004 1.0000 306.04 231.02 151.68
tl.l +.78 t1.5 t.54 t1.8 t1.7 1.38 +2.2 +1.1 +.25 +2.4 +1.7 +.75 t.17 t2.4 +1.8 t1.0 +.47 +.11 t2.1 t2.0 t1.4 +.96 +.58 t.27 t.06 +.42 t.61 1.98 t.81 t.61 +.43 +.26 t.ll t.03 -.E9 -.09 t.57 +.72 +.61 t.46 +.32 t.19
0.42748[1+0.82(l-T:'5)
-
ChuehPrausnitz (El
Martin (Di
P
atm
0.42748 TR0'5ZcVR(ZcVR+0.08664)
I
=
P,
PengRob 1n son (Cl
f.15
+2.2 t1.7 t1.0 t.47
+.11 +2.5 t2.4 t1.8 t1.2 +.74 t.35 t.08 +2.3 t2.4 t2.3 t1.8 t1.3 t.91 1.56 t.26 t.06 t1.6 +2.3 t2.5 t2.2 t1.7 t1.2 +.e5 t.52 t.24 t.06 t44. t45. +48. t53. 155. +47. +2.5 +1.4 t.57 t.05 +36. +40. 144. t43. +23. t7.0 t3.4 +2.0 +1.1 t.42 t.03 t16. +20. +24. 123. t13. +7.5 t4.0 t2.7 tl.8 +1.1 t.58 t.23 t.02
-8.3
TR
'1
TR
ZcVR(ZcVR+0.08664)
TcVR-0.0906
(CI T, 0.457236[1+0.71(1-T-0'51 l 2 m - zcVR(z V +0.07781t0.0778(ZcVR-0.07781 K PR = zcVR-0.0778 C R
The differences of all temperature functions are not significant at high densities because of the inherent defect of all cubics for reduced densities of 1.5 to 2.0.
D. The Best Two-Term Cubic Equation of State In summary, the best two-term cubic equation is also
t.30 +.16 +.07 +.02 +.01 -23. -12. -3.6
t9.4 tl8. t25. t1.6 t.58 +.14 +.01 -19. -7.8 +8.1 t16. +15. +6.7 +3.2 +1.5 +.58 +.15 00 -15. -7.0 +1.2 18.1 +7.7 t5.1 +2.7 +1.8 tl.l +.55 t.21 1 0 . 0 0 5 t.05 1.0000 00
-2.1 t4.3 +8.4 t5.4 t2.4 t.17 -.54 -.95 -1.1 -.94 -.57 -.06
ZcVR-0.064
+.51
-
116.38
70.615 49.862 39.416 29.733 19.891 9.9726 .9997 342.43 291.56 234.59 168.94 100.97 78.041 58.037 48.471 38.952 29.386 19.715 9.9291 ,99930
+.OS +.02 t73. t69. t70. t72. t71. t55. -.OS -.01 t.04 00 t53. +54. +52. +45. tl8. -.28 -1.5 -.E9 -.55 -.27 -.03 +14. +17. 117. t13. t.97 -2.5 -3.3 -3.1 -2.6 -2.1 -1.4 -.71 -.07
27/64TR1' (?cVR+0.06112 0.4450 TRO'% V
c R
(?cVRt0.0906)
the simplest and it can be written either as eq 67 with a temperature function such as TR" in the second term or alternatively as
94
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
Table IX. Comparison of Four Equations of State with Data (18) for Trifluoromethane TRIFLUOROMETHANE
v TC =
p,
55.61 61.20 16.30 90.25 104.17 79.45 92.46 103.13 123.92 148.08 179.78 203.82 236.26 276.15 313.09 270.8 298.0 330.7 332.2 315.2 430.3 484.3 409.0 416.9 548.0 612.2 672.5 725.5 525.8 589.9 699.2 802.0 906.1 1036.5 654.3 742.7 910.0 1162.6 1392.5 684.3 834.1 1082.2 764.1 1014.7 1268.9 1496.4 1148.6 1913.1 715.0 1019.0 1224.4 1504.1 1749.9 1995.4 705.8 1024.3 1254.5 1555.7 1974.3 693.2 115.7 1014.3 1313.6 1649.1 1978.2 106.6
57.253 67.884 76.056 89.532 103.01 87.101 93.944 103.48 123.16 146.29 187.06 208.02 237.20 214.23 209.58 281.36 304.56 332.68 333.91 373.06 425.53 418.41 422.99 480.18 543.96 604.54
.31066 ft3/ltmole. R MW = 70.019
.20158
.12836
,082677
,055465
.047352
563.06
,033604
,028644
,023589
606.44 545.6! 574.22 602.73 628.97 657.22 683.05 545.69 568.67 581.21 613.06 634.82 656.98 538.33 561.00 576.86
,023646
,021531
,020316
596.89 624.23 536.11 538.33 558.08 517.12 591.56 617.85 536.54 538.33 559.85 575.60 593.64 610.16 523.91 538.33
.016045
548.51 561.74 512.13 598.68 500.02 505.25 514.58 522.15 538.33 544.27
736.6
1097.5 1314.6 1698.1 1991.0 680.8 781.1 974.8 1229.1 1435.8 1971.5 461.8 645.3 965.4 1226.6 1192.4 1967.9
660.73
111.85 541.28 600.11 691.28 190.06 884.08 1006.4 667.11 145.29 911.11 1134.3 1349.3 694.08 833.49 1066.6 772.08 1029.5 1281.4 1509.4 1151.0 1968.8 828.09 1094.6 1306.3 1597.1 1838.3 2080.8 951.85 1317.5 1569.5 1883.8 2306.0 1017.2 1045.6 1388.2
1113.1 2058.2 2395.5 1167.5 1201.8 1610.6 1905.5 2239.3 2541.4 1339.3 1455.3 1611.8 1950.7 2167.7 2715.2 2855.1 3056.1 3393.0 3664.5 4239.7 4449.2
0.42748 iR ZcVR-0.08664
(B) PR =
-
IC)
400.1 466.4 517.66 602.60 687.91 412.49 463.35 504.59 590.22 691.63 455.44 493.01 546.04 614.02 679.54 419.89 505.55 536.95 538.33 502.52 642.41 103.64 510.78 548.42 591.05 632.09 670.58 705.91 519.08 539.42 576.68 613.11 650.58 100.06 534.86 552.38 587.93 643.00 694.8 531.68
psi'
=
IB)
,99819
R = 10.7315
p
(A1 PS1
P
.70008
= .259344
Robinson
data
= 32.78 lbift’
z
Soave
T, R
Pc =
701.42 p s i
Kwong
ft3/lb
538.33 R
F,
Penq-
Redlich-
CHF3
TR
zcVR-0.08664
(C) P =
TR0’5EcVR(ZcVRt0. 08664) 0.42748 ilt0.86 u - T ~ ” ~ )1’ ~cVRl~cVR+0.08664)
where A(TR)= 2 7 / 6 4 T ~B~ = , tPc/RTc,the translation, and C = 1/8 - B. The translation is set to fit best at a reduced density of 1.5 or 2.0, depending upon what behavior is sought, and n is selected to give a reasonable
% Dev.
t3.0 t1.0 -.32
-.EO -1.1 t3.3 t1.6 t.34 -.62 -1.2 t4.1 +2.l +.40 -.69 -1.1 +3.9 t2.2 +.60 t.52 -.57 -1.1 -1.2 +3.4 t.69 -.74 -1.3 -1.8 -1.9 +4.1 +1.8 -.21 -1.5 -2.4 -3.0
t2.1 +.35 -1.1 -2.4 -3.1 t1.4 -.07
-1.4 t1.0 +1.5 t.99 +.E7 +.14 -.22 t6.9 tl.4 t6.7 t6.2 +5.1 i4.3 t35. t29. +25. +21. 117. t47. t46. t37. t30. t25. +21. t65. t63. t41. t39. t32. +28. t97. t86. +72. 159. t51. t38. t512 +373 t251 +199 t136 t126
% Dev.
P
56.818 61.686 75.999 89.707 103.30 81.410 93.525 103.29 123.44 141.11 184.15 206.75 237.42 276.31 313.38 217.55 302.42 332.59 333.91 315.91 432.15 488.17 418.68 481.15 552.11 618.88 680.72 736.88 570.31 601.16 111.05 816.68 923.63 1062.5 665.06
755.98 931.63 1212.3 1463.9 693.42 858.51 1135.1 785.90 1096.7 1401.4 1671.0 1969.1 2232.1 846.43 1169.9 1421.2 1780.8 2014.0 2368.7 951.85 1396.1 1102.8 2085.5 2599.8 1011.2 1045.6 1461.7 1857.5
2276.8 2681.5 1160.2 1201.8 1697.7 2055.9 2461.7 2829.1 1315.0 1455.3 1711.3 2055.1 3218.2 2982.4 2612.8 2831.4 3236.1 3551.7 4239.7 4488.3
TR ZcVR-O. 0778
ID)
t2.3 t.12 -.39 -.60
-.74 +2.5 +1.2 +.16 -.39 -.65 t2.8 11.4 +.49 t.06 +.09 t2.5 tl.5 t.57 t.52 t.19 t.43 t.92 t2.4 t1.0 +.75 tl.l tl.2 +1.6 t2.8 t1.9 tl.1 t1.8 +1.9 t2.5 t1.7 t1.8 +3.0 t4.3 t5.1 +1.3 +2.9 +4.9 +2.9 +8.l t10. t12. t13. +13.
+9.2 t15. +11. tl8. t19. t19. t35. +36. t36. +34. +32. t46. +46. +44. +41. t38. t36. t64. t63. +55. t50. t45. +42. +93.
+86. t16. +61. +61. +51. +458 t339 t235 +19C +136 +128
Martin ID1
%
e
P
Dev.
P
Dev.
56.645 67.398 15.673 89.328 102.98 80.948 92.915 102.68 122.71 146.27 182.67 204.34 234.57 212.98 309.65 293.34 291.65 321.11 328.46 369.62 424.83 480.53 410.63 471.62 539.81 604.65 664.82 119.54 530.31 588.32 693.00 193.81 896.24 1029.5 651.38 142.15 911.93 1169.5 1406.5 588.97 841.58 1098.1 774.88 1056.5 1333.3 1584.3 1850.9 2091.5 807.27 1091.3 1328.4 1646.1 1911.2 2117.5 808.11 1200.9 1472.6 1812.0 2269.2 823.18 853.57 1220.2 1569.7 1940.4 2304.3 894.11 930.59 1366.0 1681.0 2038.3 2362.3 935.49 1057.9 1286.1 1581.9 1812.1 2394.0 1098.2 1287.9 1624.7 1896.5 2413.1 2683.3
+1.9 +.29 -.E2 -1.0 -1.1 t1.9 t.56 -.44 -.97 -1.2 t1.6 t.25 -.71 -1.2 -1.1 t.94 -.12 -1.1 -1.1 -1.5 -1.3 -.78 +4.0 -1.1 -1.5 -1.2 -1.1 -.E2 t.89 -.27 -.E9 -1.0 -1.1 -.68 t.47 -.07 +.21 t.60 t1.0 +.68 +.91 +1.5 t1.4 t4.1 +5.1 +5.9 t5.9 t6.0 +4.2 tl.7 +8.5 t9.5 19.2 t9.1
56.364 61.316 15.155 89.466 103.09 80.496 92.904 102.19 122.98 146.48 181.88 204.47 235.43 214.07 310.49 272.63 298.09 328.54 329.81 311.64 426.75 481.57 410.97 414.18 543.32 601.90 667.02 120.26 530.18 590.46 697.34 798.08 898.50 1021.0 651.40 144.52 915.19 1166.6 1391.6 688.99 843.94 1097.2 114.52 1050.5 1315.3 1550.7 1196.3 2014.6 801.70 1080.6 1298.7 1593.7 1834.7 2074.0 175.48 1142.8 1391.5 1697.4 2101.9 118.43 806.74 1145.8 1462.9 1193.5 2112.8 822.98 856.62 1253.3 1535.1 1850.0 2131.6 824.51 935.59 1144.5 1404.1 1606.5 2110.0 525.33 686.45 969.97 1196.5 1670.7 1841.6
+1.4 t.26 -.71 -.E7 -1.0 t1.3 t.48 -.33 -.76 -1.1 t1.2 +.32 -.35 -.75 -.E3 t.68 1.03 -.65 -.lo -.95 -.E3 -.56 t.48 -.57
+15. t17. tl7. 1-16. 115. +19. +19. +20. t19. t18. +16. t21. t26. t24. t22. t20. t19. t37. t35. +32. t29. t26. +21. t134 t100 t68. t54. +38. +36.
-.E5 -.70 -.E1 -.72 +.E3 +.09 -.03
-.49 -.E4 -.92 t.49 t.24 +.57 +.34 -.06 +.69 +1.2 +1.4 +1.4 t3.5 t3.1 t3.6 t2.7 12.1 t3.4 +6.0
t6.1 +6.0
+4.9 +3.9 tl0. +12. t11. t9.1 +6.5 +12. +13. t13. t11. +8.8 +6.8 +16. t16. +14. t12. t9.0 +7.1 t21. +20. +17. t14. 112. t2.0 +12. t6.4 +.47 -2.5 -6.8
-6.4
0.451236 1 1 + 0 . 1 5 ( 1 - T ~ ’ 51’ )
-
Z V (fcVRtO.077 8)to. 017 8 ( ;cVR-O. 077 8 ) C R
21/64TR zcVR-0.055
l z c V R t0.07)2
average slope M of the criticcl isometric based on the generalized vapor-pressure diagram. It has been shown that no other two-term cubic can be developed to be superior to eq 93 because they all come from the single
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
95
Table X. Comparison of Four Equations of State with Data (23) for Ammonia tlH3 AMMONIA
T,OC T
-
= C
405.44K P
45 30 10 20
=
V cm 3/gm
P IkPa data
2190 1140 620.3 699.6 341.5 161.7 776.7 303.2 145.0 64.77 802.1 395.4 169.2 89.49 49.17 852.5 277.7 133.8 61.93 31.07 902.6 599.0 295.4 143.4 67.10 38.53 18.04 927.6 615.8 259.5 148.1 69.83 35.84 14.34 952.5 632.6 267.0 129.8 66.28 30.21 10.32 977.4 649.4 274.5 133.8 68.72 35.42 16.9 8.524 6.549 989.8 657.7 278.2 135.8 69.91 34.41 16.97 12.67 8.590 6.557 4.706 3.620 2.588 2.263 2.127 1200 598.1 297.2 130 66.98 33.69 17.03 8.396 4.681 2.801 1470 587.2 292.7 132.1 68.32 34.03 17.62 8.490 4.553
101.33 200 200 400 800 200 500 1000 2000 200 400 900 1600 2600 200 600 1200 2200 4000 200 300 600 1200 2400 3800 6200 200 300 700 1200 2400 4200 7400 200 300 700 1400 2600 5000 9000 200 300 700 1400 2600 4600 7800 10400 10800 200 300 700 1400 2600 4800 8000 9400 10900 11500 11800 12000 16000 26000 36000 200 400 800 1800 3400 6400 11400 19000 28000 50000 200 500 1000 2200 4200 8200 15000 28000 50000
~~
11278 kPa
50
v = 4.247 cm3/gm z
60
=
.242004 80 R = 8 3 14.4
cm3. kPa/gmole. K 100
110
120
130
135
220
330
(A!
PR = (B) PR =
TR zcVR-0.08664 TR zCVR-.O8664
-
50
0.42748
RedlichKwong
soave
(A)
(B)
P
+.96 t1.4 +1.4 +.72 11.5 t3.4 r.38
+.98 +2.1 t5.3 t.30 +.62 +1.5 1-3.0 +6.0 +.20 t.62 t1.3 +2.7 +6.6 C.13 t.19 t.37 +.79 +1.7 +3.1 +7.1 +.lo +.16 +.35 1.59 ~ 1 . 3 t2.6 +6.4 +.08 t.12 t.27 t.55 +1.1 t2.3 t5.2 t.06 t.08
+.18 +.40 +.78 +1.5 +2.7 t3.0 +1.5 +.05 +.DE t.16 +.32 +.65 t1.3 +2.6 t2.6 +2.2 tl.O +2.1 t28. +197 +267 t283 -.03 -.06 -.12 -.28 -.46 -.E0 -1.2 -.82 +6.4 t54. -.03 -.16 -.30 -.64 -1.1 -1.8 -2.1 -.28 +6.1
P
PR =
(D)
TR zcVR-0.0778
-
P
Dev.
50.435 102.63 202.36 201.14 404.77 821.73 200.57 503.73 1016.1 2081.6 200.45 401.85 909.89 1635.9 2717.7 200.30 602.80 1211.6 2244.1 4194.7 200.21 300.46 601.71 1207.4 2432.3 3891.3 6519.8 200.16 300.39 702.03 1205.8 2425.6 4287.3 7745.1 200.14 300.32 701.67 1406.7 2624.4 5099.7 9338.0 200.10 300.24 701.25 1405.4 2619.5 4665.2 6003.7 10674. 10904. 200.10 300.24 701.15 1404.6 2617.7 4863.9 8191.6 9659.1 11185. 11686. 12164. 15565. 47798. 95910. 138214 200.01 400.09 800.29 1801.6 3408.8 6443.4 11617. 20196. 33672. 86079. 200.05 499.90 999.84 2199.5 4203.0 8249.5 15392. 30754. 61771.
(C)
TR0'5Z C V R (ZcVRt0.00664) 0.42748 [1+0.86(1-TR0'5) 1
Dev.
+.87 t1.2 +l.2 +.57 tl.2 +2.7 +.29 +.75 +1.6 +4.1 t.23 +.46 +1.1 +2.3 +4.5 +.15 +.47 t.96 +2.0 t4.9 t.10 1.15 +.28 +.61 +1.3 +2.4 +5.2 +.08 t.13 t.29 c.48 tl.l +2.1 +4.7 t.07
+.11 t.24 +.48 +2.9 t2.0 13.8 +.05 t.08 +.18 1.38 +.75 t1.4 +2.6 t2.6 t.97 t.05 +.08 t.16 t.33 1.68 +1.3 t2.4 +2.8 +2.6 +1.6 t3.1 130. +199 t269 t284 t.01 +.02 +.04 +.09 +.26 t.68 C1.9 +6.3
+20. +72. +.03 -.02 -.02 -.02 +.07 +.60 +2.6 C9.8 t24.
Martin (D! -5
4
%
%
50.479 102.79 202.82 201.82 406.02 827.24 200.75 504.91 1021.2 2106.8 200.60 402.46 913.19 1647.6 2755.9 200.40 603.70 1215.4 2258.7 4263.0 200.26 300.58 602.19 1209.4 2441.5 3919.0 6640.8 200.20 300.47 702.47 1207.1 2431.6 4309.3 7874.7 200.16 300.36 701.89 1407.7 2628.1 5116.7 9470.8 200.11 300.24 701.29 1405.5 2620.2 4667.5 8013.3 10710. 10962. 200.10 300.23 701.11 1404.4 2617.0 4861.0 8180.3 9639.8 11144. 11619. 12043. 15374. 47462. 95493. 137753 199.94 399.78 799.02 1795.0 3384.2 6348.5 11260. 18844. 29793. 76776. 199.94 499.20 997.05 2185.9 4152.6 8050.6 14679. 27922. 53069.
PengRobinson (C)
50.421 102.58 202.16 200.95 404.02 818.64 200.40 502.65 1011.7 2063.8 200.29 401.18 906.47 1625.0 2688.8 200.15 601.38 1205.9 2224.9 4123.3 200.06 300.12 600.36 1202.0 2411.0 3839.3 6400.9 200.01 300.06 700.26 1200.6 2404.1 4226.4 7601.9 200.00 300.00 699.94 1399.9 2601.2 5017.9 9199.5 200.00 299.93 699.58 1398.7 2596.9 4597.1 7834.0 10537. 10891. 200.00 299.93 699.50 1398.0 2595.5 4791.3 8015.0 9448.2 11010. 11610. 11947. 13564. 30163. 54482. 74957. 199.90 399.66 798.58 1793.1 3379.2 6343.5 11327. 19468. 31406. 68443. 200.00 499.39 997.83 2190.0 4169.2 8126.9 15008. 29464. 56966.
Dev. +.E4 +1.2 +1.0 t.48 e1.0 t2.3 +.20 +.53 +1.2 +3.2 t.14 +.30
+.72 +1.6 t3.4 t.07 +.23 +.49 tl.l +3.3 +.03
+.04 +.06 1.17
+.46 +1.0 +3.2 +.01 +.02 t.04 +.05 +.21 +.63 t2.7 00 00 -.01 -.01 +.05
t.36 +2.2 -.01 -.02 -.06 -.09 -.12 -.06 +.44 t1.3 +.84 -.02 -.02 -.07 -.13 -.17 -.18 t.19 +.52 c1.0 +.95 +1.2 +13. +89. t110 +lo8 -.05
-.08 -.18 -.38 -.61 -.E8 -.64
t2.5 t12. t37. -.01 -.12 -.22 -.46 -.73 -.E9 t.05 +5.2 +14.
P
Dev.
50.298 102.21 201.23 200.50 402.12 810.26 200.18 501.23 1005.6 2033.9 200.12 400.50 902.76 1612.0 2646.9 200.05 600.49 1202.1 2212.4 4068.1 200.00 300.00 599.91 1200.1 2402.2 3813.1 6298.0 199.99 299.99 699.83 1199.2 2399.2 4204.i 7493.; 199.98 299.95 699.68 1398.7 2596.6 4995.8 9089.7 199.96 299.90 699.43 1398.1 2594.3 4586.4 7790.7 1046.1 10852. 199.96 299.92 699.41 1397.7 2593.6 4782.3 7979.7 9394.9 10947. 11575. 11891. 12610. 18245. 24454. 28734. 199.91 399.69 798.70 1793.6 3380.3 6343.3 11300. 19206. 29497. 50581. 199.96 499.29 997.41 2187.8 4160.2 8084.2 14812. 28305. 50548.
t.60 t.87 +.62 t.25 +.53 +1.3 +.09 1.25 +.56 t1.7 t.06 +.12 +.31 +.75 +1.8 +.G3 t.08 +.17 +.47 t1.7 GO -.02 -.02 00 +.09 +.35 +1.6 -.0l 00 -.02 -.06 -.04 +.lo +1.3 -.01 -.02 -.05 -.09 -.13 -.OB +1.3 -.02 -.03 -.G8 -.14 -.22 -.30 -.12 +.59 +.48 -.G2 -.03
-.08 -.17 -.25 -.37 -.25 -.05 t.44 +.65 c.77 t5.1 t14. -6.0 -20. -.04 -.08 -.16 -.36 -.5B -.E9 -.E8 +1.1 +5.4 t1.2 -.02 -.14 -.26 -.56 -.95 -1.4 -1.3 il.1
tl.l
0.457236 rlC0.75 (l-TR0") 1 Z V
c R ( Z CV R+0.0778!t0.0778(ZcVR-0.0778)
2 7/64 TR1'
IR
z C V R (icVR+0.08664)
generating eq 6, and eq 93 has the smallest value of 1 which makes it the best. If eq 93 is applied to mixtures, it is a simple procedure
'R=--
2 V -0.04 C R
(ZcVR+0.085)2
to combine constants just as with van der Waals or any other equation. To show how the procedure works, eq 93 can be transformed from its generalized form back to its
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
96
Table XI. Compressibility Factor for n-C,H, at 470 K
specific form
RT
P=
-
A R2T: / Pc
--.-RT -
For a given substance, i, this is
A 1R2Tct2/ pci RT (95) V - BcRTc,/Pc, (V + CLRT,,/PJ2 where A, = 27/64TRlniaccording to the definition of A. Now van der Waals procedure of combining the volume constants, b and c, linearly, and the a by geometric mean gives good results, though linear cube root may be better for a. Thus, the constants for a mixture of two substances, 1 and 2, are P=
BmTc,/Pc, = XlBlTcl/Pcl+ ~2B2Tc2/PcZ
(96) (97)
and
CmTc,/Pc, = XICITcl/Pcl + x2C27'c2/pcz (98) For a single molar gas constant, R, the concentrations, x l and x2, are to be mole fractions, and it is to be noted that R has been divided out of both sides of eq 96,97, and 98. These will give good results in the expression
for mixtures, particularly if the magnitudes of the a, b, and c terms being combined do not differ greatly from substance to substance. Equation 99 can be differentiated with respect to the moles of 1 and the moles of 2 to give the individual component fugacities in the usual expression
1 10 20 30 40 50 60 80 100 150 200 250
0.9894 0.8859 0.7453 0.5292 0.2047 0.2440 0.2840 0.3634 0.4414 0.6304 0.8128 0.9901
125.518 11.2388 4.72755 2.23786 0.64922 0.61909 0.60048 0.57628 0.55997 0.53316 0.51557 0.50243
0.9996 9.9694 19.935 30.112 46.863 51.663 55.441 61.566 66.649 77.186 85.976 93.171
-0.04 -0.31 -0.33 0.37 17.16 3.33 -7.60 -23.04 -33.35 -48.54 - 57.01 -62.49
will predict the PVT behavior of C 0 2as well or better than any other two-term cubic equation. One further point concerning the selection of 0.068 .for the translation is worth noting. To fit the critical point exactly, according to eq 41, one would set tP,/RT, = 0.2745 - 0.25 = 0.0245. To fit the generalized second virial coefficient exactly at T R = 1 in eq 47 one would take tP,/RT, = 27/64 - 0.345 = 0.0769. The actual selection of 0.068 leans in the direction of the second virial coefficient, since this gives a better fit below the critical density where the predictions are less sensitive to variations in the critical compressibility factor. As a second example take hexafluorobenzene, C6F,. Its constants (11) are T, = 516.74 K, Pc = 32.304 atm, pc = 0.542 g/cm3, mol wt = 186.057 2, = 0.2612, BG1= -0.368, and the NBT = 353.41 K. From the latter, PR and 1/TR establish M = 8.0 on the generalized vapor-pressure chart. Equations 91 and 92 give the range of translation from 0.0445 to 0.0565, and 0.056 will be chosen to give a better fit near twice the critical density. If n is taken as 1.3 in TRn,dPR/dTR= 7.9 for the critical isometric at TR = 1.25, and this compares favorably with M = 8.0. Thus, the best two-term cubic equation for hexafluorobenzene is
fi
In - = x ;P J*"'
[ -vt!- ( + -
RT
dni
T,V,n,
]dVt - In p v t (100) nRT
where xi = ni/n = ni/(nl n2)and V , = nV. Equation 99 is useful in equilibrium calculations. The discussion will be concluded with three examples that show the simplicity and power of eq 93 and the ease of obtaining appropriate constants. Suppose a PVT equation is desired for carbon dioxide. The input constants (22)are T , = 304.19 K, P, = 72.85 atm, pc = 0.468 g/cm3, 2, = 0.2745, BG1 = -0.345, 217.59 K for a boiling point at 5.3477 atm (there is no normal boiling point at 1 atm because this is below the triple point), and mol wt = 44.01. From these data the PR and 1/TR at the given boiling point establish a point on the generalized vaporpressure diagram which shows M = 7.15. According to eq 91 and 92 the translation, tP,/RT,, may range from 0.054 to 0.068, depending upon where the best fit at high density is to be obtained. A value of 0.068 will be chosen to give good results near a reduced density of 2.0 rather than 1.5. In the temperature function TRn it is quickly found from eq 89 that n = 0.9 gives a slope of 7.10 for the critical isometric at TR = 1.25, which agrees well with M = 7.15. Thus, one can say with confidence that
The last example is normal pentane, whose constants (7) are T , = 469.65, P, = 33.25 atm, V , = 304 cm3/g-mol, 2, = 0.2623, the NBT = 309.19 K, BG = -0.352, and mol wt = 72.151. PR and l / T R at the boiling point give = 7.35 from the generalized vapor-pressure diagram. Equations 91 and 92 give tPc/RT,ranging from 0.045 to 0.057, and 0.057 will be selected to favor the highest densities. If n is taken as 1in TRn,dPR/dTR for the criticd isometric at TR = 1.25 is 7.35, which is equal to M . Thus, an excellent equation for n-C5H12 is
Now a recent publication (7) has reviewed the PVT data
on n-C5H12 and given a table of compressibility factors for
a temperature of 470 K which is almost the critical temperature. Thus, a check was made to determine how well eq 103 would predict. The results are given in Table XI. Up to the critical density eq 103 does an excellent job, as expected. Above the critical density at 50 atm, eq 103 should predict about 20% high instead of predicting within 3 70. As twice the critical density is approached eq 103 should predict within 5 to 2090, as all cubic equations with the same 2, = 0.25 + 0.057 = 0.307 tend to do. The fact that eq 103 predicts some 60% low indicates that the correlation is in error a t high densities. This is especially interesting in that a cubic equation, which is an obvious
Ind. Eng.
approximation, is able to reveal an error in a correlation in the literature to give a n unexpected additional advantage to equations of the form of eq 103. Nomenclature a = constant in equation of state; may be function of temperature A = constant in equation of state; may be function of temperature b = constant in equation of state B = constant in equation of state; second virial coefficient c = constant in equation of state C = constant in equation of state d = differentiation sign i = refers to single component m = refers to a mixture; variable in generalized equation n = exponent on temperature P = pressure R = gas constant t = translation in volume T = temperatures; absolute K or R V = volume per unit mass 2 = compressibility factor, experimental Z = compressibility factor, calculated or fixed = constant in equation of state; may be function of temperature p = constant in equation of state y = constant in equation of state 6 = constant in equation of state; may be function of temperature BG = generalized second virial coefficient BGi = generalized second virial coefficient at reduced temperature of 1 P,,V,, T , = refgr to critical point values PR, VR,TR = reduced values, Le., PIP,, VI V,, T I T ,
Chem. Fundam., Vol. 18, No. 2,
1979
97
Literature Cited (1) ' Abbott, M. M., AIChE J . , 19 (3),596 (1973). (2) Barner, H. E., Pigford, R. L., Schreiner, W. C., Proc. API, 46,244 (1966). (3) Benedict, M., Webb, G. B., Rubin, L. C., J . Chem. Phys., 8,334 (1940). (4) Berthelot, D. J., J . Phys., 8,263 (1899). (5) Chueh, P. L., Prausnitz, J. M., Znd. Eng. Chem. Fundam., 6,492 (1967). (6) Clausius, R.. Ann. Phys. Chem., IX, 337 (1881). (7) Das, T. R., Reed, C. O., Jr., Eubank, P. T., J . Chem. Eng. Data, 22, 3 (1977). (8) Das, T. R., Reed, C. O., Jr., Eubank, P. T., J . Chem. Eng. Data. 22, 9 (1977). (9) Dingrani, J. G., Thodos, G., Can. J . Chem. Eng., 53, 317 (1975). (10) Douslin, D. R., Harrison, R. H., Moore, R. T., J . Chem. Thermodyn., 1, 305 (1969). (1 1) Douslin, D. R., Harrison, R. H., J. Chem. Thermodyn., 8,301-330 (1976). (12)Fuller, G. G., Ind. Eng. Chem. Fundam., 15, 254 (1976). (13) Gosman, A. L., McCarty, R. D., Hust, J. G., NSRDS-NBS,27 (1969). (14)Hou, Y . C., Martin, J. J., AIChE J . , 5 ( I ) , 125. (15) Lee, B. I., Edmister, W. C.. Ind. Eng. Chem. Fundam., 10, 32 (1971). (16)Martin, J. J., Hou, Y. C., AIChE J . , 1, 142 (1955). (17) Martin, J. J., J , Chem. Eng. Data, 7, 66 (1962). (18) Martin, J. J., Chem. Eng. Prog. Symp. Ser., 59, No. 44, 120 (1963). (19)Martin, J. J., Edwards, J. B., AIChE J . , 11 (2),331 (1965). (20)Martin, J. J., Ind. Eng. Chem., 59 (12),34 (1967). (21)Martin, J. J., Stanford, T. G., AIChE Symp. Ser., 70, 1 (1974). (22) Michels, C., Michels, A., Doctoral Dissertation, University of Amsterdam, 1937. (23) Milora, S . L., Combs, S . K., Oak Ridge Natl. Lab., TM-547 (1977). (24)Onnes, ti. K., Commun. Phys. Lab., Leiden, Holland, 71 (1901). (25)Peng, D. Y., Robinson, D. B., Ind. Eng. Chem. Fundam., 15,59 (1976). (26) Pitzer, K. S.,Lippman, D. Z.,Curl, R. F., Jr., Huggins, C. M., Petersen, D. E., J . Am. Chem. SOC.,77, 3427,3433 (1955). (27)Redlich, O.,Kwong, J. N. S.,Chem. Rev., 44, 233 (1949). (28)Redlich, O.,Ind. Eng. Chem. Fundam., 14, 257 (1975). (29) Soave, G., Chem. Eng. Sci., 27, 1197 (1972). (30) Strobridge, T. R., NaN. Bur. Stand. Tech. Note, 129 (1962). (31) Usdin, E., McAuliffe, J. C., Chem. Eng. Sci., 31 (ll),1077 (1976). (32)van der Waals, J. D.. Doctoral Dissertation, Leiden, Holland, 1873. (33) Wilson, G. M., Adv. Cryog. Eng., 9, 168 (1964). (34) Won, K. W., 69th Annual AIChE Meeting, Chicago, Ill., Nov 1976.
(Y
Received f o r review June 13, 1978 Accepted December 18, 1978
ARTICLES Interferometric Study of Forced Convection Mass Transfer Boundary Layers in Laminar Channel Flow F. R. McLarnon,* R. H. Muller, and C. W. Tobias Materials and Molecular Research Division, Lawrence Berkeley Laboratory, and Depatfment of Chemical Engineering, University of California, Berkeley, California 94720
Double beam interferometry has been used to determine the development of local mass-transfer boundary layers under laminar flow conditions. A traveling, dual-emission laser interferometer has been employed in the study of a model transfer process, the electrodeposition of copper from copper sulfate solution in a flow channel of rectangular cross section. Concentration profiles in the boundary layer have been derived from experimental interferograms which were corrected for optical aberrations. Mass balance considerations were used to select the correct concentration contour from those associated with practically indistinguishable interference fringes. Asymptotic solutions to the convectwe diffusion equation have been found to closely describe the transient growth and steady-state behavior of laminar, forced convection boundary layers.
Introduction Mass transport often limits the rate of chemical processes. Many industrially important electrochemical re*Department of Chemical Engineering, University of California, Davis, Calif. 95616. 0019-7874/79/1018-0097$01.00/0
actions fall in this category; some are also used to model transfer processes. The limiting current technique (Selman, 1971) is an established method for the experimental analysis of problems involving ionic transport to and from electrodes. However, this method gives no direct information about the nature of mass-transfer processes 0 1979 American Chemical Society
