Ind. Eng. Chem. Res. 1988, 27, 874-882
874
The enthalpy of dissociation and heat capacity of the hydrate could not be determined precisely because of uncertainty in the composition of the hydrate. Since this sample was used without grinding to a fine powder, it dissociated in the two-step process discussed above. Most of the dissociation occurred between 220 and 260 K. The specific enthalpy of dissociation of hydrate into ice and gas was found to be 27.8 J g-l based on all the water present as hydrate and 33.1 J g-l based on 20% water present as ice. The specific heat of the hydrate in the range 100-210 K was found to be about 10% lower than that of ice if all the water is assumed to be present as hydrate and about 10% higher than that of ice if 20% water is assumed to be present as ice. Conclusion The small sample sizes and relatively large ice content or uncertainty about the ice content made it difficult to make a thorough study of the samples. A sample size of about 10 g of an intact, naturally occurring gas hydrate is highly desirable. Then a payt of the sample can first be dissociated to determine the composition of the enclathrated gas, and the hydrate can then be scanned in the calorimeter under the pressure of gas of the appropriate composition. Moreover, a larger sample size would allow determinations of the effect of sedimentary material on the thermal properties and the effect of grain size of the hydrate on its dissociation characteristics. The present work, however, does indicate that the properties of the naturally occurring hydrates are broadly the same as those of the laboratory-synthesizedhydrates, and likewise it may be possible that the thermodynamic properties of the sediment-consolidated hydrates are similar to those of sediment-consolidated ice.
Registry No. CH3CH3, 74-84-0; H3CCH2CH3,74-98-6; (H3C)*CHCH,, 106-97-8.
Literature Cited Brooks, J. M.; Kennicutt, M. C.; Fay, R. R.; McDonald, T. J. Science (Washington,D.C.) 1984,225, 409. Cherskii, N. V.; Groisman, A. G.; Nikitina, L. M.; Tsarev, V. P. Dokl. Akad. Nauk SSSR 1982,265, 185. Cox, J. L., Ed. Natural Gas Hydrates: Properties, Occurrence and Recouery; Butterworth, Boston, 1983. Davidson, D. W. In Water: A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1973; Vol. 2, Chapter 3. Davidson, D. W.; Desando, M. A.; Gough, S. R.; Handa, Y. P.; Ratcliffe, C. I.; Ripmeester, J. A.; Tse, J. s. J. Zncl. Phenom. 1987,5, 219. Davidson, D. W.; Garg, S. K.; Gough, S. R.; Handa, Y. P.; Ratcliffe, C. I.; Ripmeester, J. A.; Tee, J. S. Geochim. Cosmochim. Acta 1986, 50, 619. Elliot, G. R. B.; Vanderborgh, N. E.; Barraclough, B. L. U.K. Patent Appl. 2 093 503, 1982. Groisman, A. G.; Sawin, A. Z.; Barachov, S. P.; Tsarev, V. P. Izu. Sib. Otd. Akad. Nauk SSSR, Ser. Khim. Nauk 1985,1, 44. Hammerschmidt, E. G. Ind. Eng. Chem. 1934, 26, 851. Handa, Y. P. J . Chem. Thermodyn. 19868, 18, 891. Handa, Y. P. J . Chem. Thermodyn. 1986b, 18,915. Handa, Y. P.; Hawkins, R. E.; Murray, J. J. J . Chem. Thermodyn. 1984, 16, 623. Hiza, M. J.; Kidnay, A. J.; Miller, R. C. Equilibrium Properties of Fluid Mixtures; IFI/Plenum: New York, 1982; Vol. 2. Katz, M. L. U S . Patent 3916993, 1975. Kvenvolden, K. A.; McMenamin, M. A. US'. Geol. Suru. Circ. 1980, 825, 1. Kvenvolden, K. A.; Claypool, G. E.; Threlkeld, C. N.; Sloan, E. D. Org. Geochem. 1984,6, 703. McGuire, P. L. In Alternate Energy Sources;Veziro, T. N., Ed.; Ann Arbor Science: Ann Arbor, MI, 1982; Vol. 4; p 203.
Received for review August 24, 1987 Accepted December 11, 1987
Virial Equations for Light and Heavy Water Philip G. Hill* and R. D. Chris MacMillan Department of Mechanical Engineering, University of British Columbia, Vancouver, British Columbia, Canada V 6 T 1W5
New fundamental equations have been developed for the vapor states of H 2 0 and D20. For H20, the PVT data, supplemented by numerous accurate throttling data (isenthalpic and isothermal), and the Osborne determinations of saturation vapor enthalpy and density permit formulation with statistically significant third and fourth virial coefficients and a representation of all available data u p t o a temperature of 300 O C . Based on more limited experimental data, and in recognition of the similarity of H 2 0 and D20 states, a corresponding formulation has been developed for D20 which may be used t o determine the equilibrium vapor states of D20 u p to 300 "C. 1. Introduction
The subcritical temperature vapor states of H 2 0 are defined by abundant experimental data, Besides PVT data (which are difficult to use alone in establishing virial coefficients), there are numerous isenthalpic and isothermal throttling coefficients, some specific heat and speed of sound data, and accurate determinations by Osborne et al. (1937, 1939a,b) of the saturated vapor density and enthalpy. The use of these data in a selective regression analysis which retains only statistically significant terms makes it possible to establish the second, third, and fourth coefficients of a virial expansion in the form
-P-
- 1 + B2p
+ B3p2 + B*p3
PRT 0888-5885/88/2627-0874$01.50/0
and to represent thereby all experimental data within what is believed to be the accuracy of those data. For DzO, experimental data on the vapor states are few in addition to PUTdata. However, the close similarity of the DzO and HzO vapor states, coupled with the direct experimental determinations of the small differences in the second virial coefficient of DzO and HzO, makes it possible to derive a corresponding virial equation for DzO from the HzO virial equation. As was shown by Hill and MacMillan (1980), such an equation is required (in conjunction with the quite accurately known vapor pressures of DzO) to establish the saturation vapor densities and enthalpies of DzO. The Hill and MacMillan (1980) derivation of a virial equation for H 2 0 provided values within the estimated 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 875 accuracy of the correlations for saturated vapor volume ug and saturated vapor enthalpy h of Osborne et al. (1939a,b), from 0 to 325 "C. For D26,it was possible to demonstrate that the calculated values of the saturated vapor enthalpy (h,) were within f 2 J of those calculated from the Clausius-Clapeyron relationship
h, = hf + (ug - uf)TdP,,,/dT
In p
2 Bi+2 + iC-pi+l i0C + 1
1
+ #o(T)
(3)
and using the identity P = p2(t3,b/ap)T, we obtain the virial equation 2
P = pRT[1 + eBi+2pi+1]
(2)
is0
in conjunction with hf values deduced from the liquid D20 C, data of Rivkin and Egorov (1963). The calculation of these vapor saturation states was a necessary step in the derivation of the fundamental equation of state for heavy water (Hill et al., 1982). Kell et al. (1968a) determined an equation of state for H 2 0 vapor with correlations for B2 and B3 for H 2 0 (from 150 to 450 "C) by using a virial functional form truncated to two coefficients. From their measurements, Kell et al. (196813) estimated correspondingB2and B, values for D20. Although the two sets of values for B2 are not very different from each other (on a molar basis), a modified analysis did provide estimates for the quantity BHzO - BDZDHowever, below about 200 "C, there are indications of increasing uncertainty in volume measurements approaching the saturation pressure, perhaps hs a result of adsorption. Thus, the B2 correlation, set out in Keenan et al. (1969), which itself was a modification of Keyes (1962), seemed preferable to the Kell et al. (1968a) correlation. The Keyes (1962) correlation has a range of validity which extends to 0 "C and is based on (ah/ap)Tmeasurements (which are unaffected by adsorption) in the low-temperature range, and this B2 correlation was used in conjunction with the B, correlation of Kell et al. (1986a) to form the basis of the Hill and MacMillan (1980) values. Above 200 "C, the addition of a correlation for the term B4 was required to achieve satisfactory agreement with the Osborne et al. (1939b) values of u, and h,. The correlation of B2 values of LeFevre et al. (1975) in the low-temperature region is based on the steady-flow data of Wormald (1964) and Nightingale (1970) in addition to the data of Collins and Keyes (1938) and employs a theoretically based functional form. The scatter of the Wormald (1964) data is less than that of the Nightingale (1970) data where they overlap, although the Wormald (1964) data below about 70 "C differed systematicallyfrom the correlation. Young (1987) has recently developed an empirical correlation for B3 (to be used in conjunction with the Le Fevre et al. (1975) B2 correlation), for the purpose of calculating the properties of steam in finite difference calculations, where computational speed is an important consideration. The virial correlation of Hill and MacMillan (1980) satisfactorily represented the Osborne et al. (1939b) saturation values, but the degree of agreement was surprising given that the B2 and B3 correlations used were derived independently of each other. It was therefore decided to perform a simultaneous correlation of the data using a virial functional form which includes virial coefficients up to the fourth to provide a consistent set of coefficients and in order to provide a more accurate basis for determining the vapor saturation states of D20, as well as providing a convenient correlation to represent H20 vapor states. Such a correlation is sufficiently flexible to allow the inclusion of the raw data of Osborne et al. (1937, 1939a) and thus provides data for determining the values of B2 in addition to the steady-flow data used by LeFevre et al. (1975). 2. The Virial Correlation Functional Form Expressing the Helmholtz free energy of the vapor as
(4)
The enthalpy may be obtained from eq 3 by using the identity
h = a(#T)/aT + p l P in which T = 1/T. The result is
From this it follows that
From (2) we obtain 2
(aP/ap)T = RT[1 + CBi+2(i+ ~ ) P ' + ~ I i=O
(7)
Combining (6) and (7) gives
i=O
The Joule-Thompson coefficient
may be evaluated by using eq (8) and (from eq 5)
in which
Specific internal energy is evaluated as
u=h-P/p
(11)
the specific heat a t constant volume is
c, = (au/aT), and the speed of sound is then
= The specific entropy is
(2J52!
(12)
876 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988
-(d$/dT),
s
d(&r2) 2 -- C bi(i + 9)Ti+'
d7 i=O The form chosen for the fourth virial coefficient was
2
In
s =
p
+ i=o CBi+2i+1
2
(2
i+l
(14)
~Ci(lOOO/T)l-k+ C7 In T
i=O
2 d dr i=O The value of the gas constant R for H 2 0 is 0.46151 kJ(kgK) and for D20 is 0.415 147 kJ/(kg.K).
- (B473)= CCi(i + 1 0 ) ~ ' + ~
+ C8 In (T)(T/lOOO) (15)
(where the subscript 0 refers to zero pressure). The coefficients for &, kJ/kg, are C1 = 1892.70718, C z = 694.12683, C3 = -260.046196, C4 = -40.670098, C5 = 5.168428, c6 = 1.027395, c7 = 33.922127, and c8 = -1113.706. The values of il0were obtained from a correlation of the C, values of Woolley (1979),with C1 ihitialized so that hf N 0 at 0.01 "C (using eq 2 in conjunction with the quantity h, - ho calculated from eq 24) and Cz initialized so that sf = 0, using the relation sfg= hf / T . The form chosen for the second virial coefficient is an extension of the three-term expression of LeFevre et al. (1975) to 6
Bz = C a i f i ( T )
(16)
i=l
in which
f l ( T ) = (1 + T/a)-' f 2 ( T )= (1 - e-Bir)5/2(T/p)1/2ea/T f 3 ( T )= P/T f 4 ( T )= ( P / T ) 2 f s ( T )= ( P / T I 3
f 6 ( T )= ( P / T ) * and a = 10000 K and /3 = 1500 K. (The explanation for the choice of the first three terms of this functional form is set out in LeFevre et al. (1975).) The appropriate temperature derivative of the second virial coefficient appearing in eq 5, 6, 8, and 10 is
3. The H20Correlations The correlation data base included the heat data of Osborne et al. (1937,1939a) and the PUTmeasurements of Kell and McLaurin (1977) and Kell et al. (1985). The raw data of Osborne (1937, 1939a) are measurements of the quantities
P = v ~ dP,/dT T a = hf - hf,(Uf/Uf,)
Saturation values for h, were obtained by an integration of a correlation of the a values combined with the y values, while ug values were calculated by using the vapor pressure equation of Saul and Wagner (1984) to evaluate dP,/dT. Saul and Wagner (1984) discuss the procedure in more detail and provide correlations along saturation of the Osborne et al. (1937,1939a)data. (See also Wagner and Sengers (1986)J This data base, with the zero pressure enthalpies, was used to determine the coefficients a,, b,, and c, for the virial coefficient functions Bz,B,, and B,. The procedure used was least-squares minimization of the objective function
D=E(
Pobs
6-pp c d c
) +
in which eq 4 was used for the calculated values Pdc and the observed values of Pobsat saturation are calculated from Saul and Wagner (1987). The calculated enthalpy differences were obtained by using eq 5 to write hg - ho = R
P
++ T
5P - -(e@/T2T
g3
= 2(P/T)
g4
= 3(P/Tl2
g5
= 4(P/T)3
g6
= 5(P/T)4
1
The form chosen for the third virial coefficient was 2
B3 = x b i ~ ~ + ~ i=O
and the appropriate temperature derivative is
(18)
(22)
y = v,T dP,/dT
in which (and following from eq 16)
1 2
(20)
for which the appropriate temperature derivative is
The values of ilo are evaluated by $om = i=l
B4 = C C ~ T ~ + ~
2
T
i=o(L
p,'+l
d
(Bi+2ri+') +C l ) ryi d7
(24)
in which p, = l/u, was calculated from the Osborne (1937, 1939a) y values. The values 6P and 6h are estimates of the experimental error in P and h, respectively. While a "preliminary virial correlation" of these data was able to generate calculated values which for the most part were within the error estimates of the data base, Figure 1 indicates a systematic discrepancy between the lowtemperature h, values from Osborne et al. (1939a) and the correlation. A second deficiency of this preliminary correlation involved a significant discrepancy between the correlation and the steady-flow data of Wormald (1964). In order to correct these deficiencies, the data of Wormald (1964) were added to the correlation data base, using an additional set of coefficients, and truncating terms of order P.The Wormald (1964) data present an enthalpy
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 877 + 0sborne.Stinson.Ginnings 119371
Table I. Coefficients of Virial Equation a, = 0.160946868952524 X m3/kg a2 = -0.1270588577231565 X lo-* m3/kg u3 = -0.1947567798197269 X m3/kg as = 0.2960495174903245 X lo4 m3/kg bo = 0.960 620 298 296 245 5 X (m3/kg)2 b2 = -0.327 058 191623 207 6 X (m3/kg)2 c2 = 0.464 002 359 159 200 1 X (m3/kg)3 d3 = -0.3153631770586872 X (m3/kg)(m2/N)K3
x Osborne.Srinson.Ginn1ng~ll9391
-
--
Cn
Uagner.Sengers(19861 Virial I P r e l ~ m i n a r y l
X 01 1
1
0
I
50
1
I /c
I
100
I50
I
200
I
250
Figure 1. Enthalpy of the saturated vapor a t 0 OC d t d 250 OC. Comparison of experimental values with the preliminary virial correlation aa the base line (H20).
increment CP at pressures PI and P2along an isotherm T,. Thus,
where 3
A 3 ( T )= Cdiri i=l T
= 1/T
Pm = (PI + P2)/2 (A more detailed description of this procedure is outlined in LeFevre et al. (1975).) The A, function does not comprise a part of the final set of virial coefficients, as it is a function of P, but the steady-flow data do assist in the determination of the B2(7') coefficients a t low temperatures. The discrepancy between the preliminary virial correlation and the Wormald data was dealt with by multiplying the h values obtained from the Osborne et al. (1937, 1939aj values by a factor of 0.9997 (i.e., reducing h, by 0.8 J a t 0 "C). This repositioning of the Osborne et al. (1937, 1939a) values is within their stated experimental uncertainty and analogous to the adjustments made to 0.1-MPa PUT values in Kell (1975) to ensure consistency with the accepted value of the maximum absolute density of standard mean ocean water. In addition, a similar comparison of PUT measurements in the range 0-150 "C with speed of sound measurements resulted in a repositioning of the PUT values to be more consistent with the speed of sound values (see Kell and Whalley (1975)). The discrepancy between the Wormald (1964) values and the Osborne (1939a) values is suggestive of some systematic error in one or both sets of measurements. The agreement between the older Collins and Keyes (1938) and the Wormald (1964) values tends to confirm their reliability (see Le Fevre et al. (1975)). Ordinarily, it is appropriate to include only the most accurate data in a least-squares regression and particularly to exclude any data which show evidence of systematic error. The Osborne et al. (1937) measurements are still the most extensive and reliable data available above 150 OC along the saturation line. However, without adjustment of the Osborne et al. (1937, 1939a) values, a correlation including both the Wormald (1964) data (in the temperature range 70-140 "C) and the raw Osborne et al. (1937) values (in the temperature range 100-300 "C) incorporates an unsatisfactory wrinkle where the two inconsistent data sets merge, and the resulting calculated derivative quantities are inconsistent with experimental results.
Table 11. Calculated Virial Coefficient Values for HzO T,K Bz, m3 ka-' Ea, (m3 kg-1)2 E,, (m3 k ~ - ' ) ~ 273.16 -0.103 -0.030 2 0.000 055 -0.021 4 -0.085 1 0.000 040 283.15 293.15 0.000029 -0.071 4 -0.015 3 303.15 0.000021 -0.060 I -0.011 0 313.15 -0.052 2 -0.008 04 0.000016 0.000 012 323.15 -0.005 90 -0.045 3 333.15 -0.004 36 -0.039 I 0.000 009 2 343.15 -0.035 1 0.000 007 0 -0.003 24 0.000 005 4 -0.002 43 -0.031 2 353.15 0.000 004 2 -0.001 82 -0.028 0 363.15 0.000 003 3 -0.025 2 -0.001 38 373.15 0.000 002 6 -0.001 05 -0.022 9 383.15 -0.000 796 0.000 002 1 -0.020 8 393.15 -0.000608 0.000 001 6 -0.019 1 403.15 -0.000 465 0.000 001 3 -0.017 5 413.15 0.000001 1 -0.000 356 -0.016 2 423.15 -0.015 0 -0.000 274 0.000 000 86 433.15 -0.000 210 0.00000070 -0.013 9 443.15 0.000 000 58 -0.000 161 -0.013 0 453.15 0.000 000 47 -0.012 1 -0.000 123 463.15 -0.011 4 0.000 000 39 -0.000 094 2 473.15 -0.000 071 7 0.000 000 32 -0.010 7 483.15 -0.000 054 2 0.000 000 27 493.15 -0.010 0 0.000 000 22 503.15 -0.009 44 -0.000 040 6 0.000 000 19 513.15 -0.008 92 -0.0000300 -0.000 021 9 0.000 000 16 523.15 -0.008 43 0.000 000 13 -0.000 015 5 533.15 -0.007 98 -0.0000106 0.000 000 11 -0.007 57 543.15 -0.000 006 8 0.000 000 10 -0.007 18 553.15 0.000 000 081 -0.000 003 9 -0.006 83 563.15 -0.000 001 7 0.000 000 070 -0.006 50 573.15
The extensive Osborne et al. (1937, 1939a) measurements are internationally recognized for their comprehensiveness and accuracy (see Haar et al. (1984)), and it must be conceded that there remains the alternative possibility of achieving consistency between the Osborne et al. (1937, 1939a) values and the Wormald (1964) values by means of an adjustment to the Saul and Wagner (1987) saturation pressure correlation. However, the Saul and Wagner (1987) correlation is also internationally accepted as an accurate and convenient correlation of the best available H 2 0 saturation pressure values. The virial correlations developed in this work are sufficiently flexible that, in the event further experimental data leads to a revision of the Saul and Wagner (1987) P, correlation, a corresponding revised set of saturation values may be calculated without difficulty. Table I sets out the coefficients selected from the "term bank" of possible coefficients (eq 16, 18, 20, and 25) using a stepwise regression procedure designed to ensure that only coefficients statistically significantly different from zero are included in the final correlation. Tables I1 and I11 set out the values calculated at saturation from the coefficients of Table I and the corresponding values of B z , B3, and B,.
H20Correlations with Experimental Data The values of B2 and B3 calculated from the correlations are compared with the values of other researchers (including values calculated from the two recent global HzO 4. Comparisons of t h e
878 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 Haar.Gallapher.Kc11119841
Table 111. Calculated Saturation Values for HzO
X KcII,~cLaurin.Uhallcy119681 - - Young119871
sgrkJ (kg
T, K 273.16 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15 363.15 373.15 383.15 393.15 403.15 413.15 423.15 433.15 443.15 453.15 463.15 473.15 483.15 493.15 503.15 513.15 523.15 533.15 543.15 553.15 563.15 573.15
kg m-3 0.004 854 3 0.009 404 2 0.017 306 0.030 399 0.051 211 0.083 090 0.130 33 0.198 27 0.293 43 0.423 55 0.597 68 0.826 28 1.1212 1.4960 1.965 6 2.546 8 3.2584 4.121 0 5.157 9 6.394 8 7.8610 9.589 1 11.617 13.987 16.750 19.967 23.709 28.065 33.148 39.104 46.132
PB,kPa
pg,
0.611 659 1.227 92 2.33849 4.245 10 7.381 12 12.344 6 19.9331 31.1777 47.3759 70.120 7 101.325 143.243 198.483 270.019 361.191 475.712 617.659 791.468 1001.9 1254.2 1553.7 1906.2 2317.8 2795.0 3344.6 3973.5 4689.2 5499.6 6412.7 7437.5 8583.1
K)-'
h,, kJ kg-' 2499.9 2518.3 2536.5 2554.7 2572.7 2590.5 2608.0 2625.3 2642.2 2658.7 2674.7 2690.2 2705.0 2719.1 2732.4 2774.9 2756.4 2766.8 2776.1 2784.1 2790.8 2796.0 2799.6 2801.5 2801.5 2799.5 2795.2 2788.3 2778.6 2765.6 2748.5
- - H1ll.tlacil1llanll988) - Hi11119871
0 in
9.1519 8.8964 8.6628 8.4489 8.2526 8.0719 7.9053 7.7512 7.6082 7.4752 7.3512 7.2351 7.1261 7.0234 6.9262 6.8339 6.7460 6.6618 6.5808 6.5026 6.4268 6.3529 6.2806 6.2093 6.1388 6.0686 5.9982 5.9272 5.8548 5.7805 5.7031
260
230
200
290
320
350
t/C
Figure 3. B3 a t 200 "C 6 t 6 350 "C. Comparison of experimental values and calculated correlation values (H,O). +
0sborne.et al.il93711Rdjusledl X 0sborne.el a1.11939)(Rdjusiedl - - Uagner.Sengersl19861
-
Hill.flactli llani19881
Yaar ,Gal 1 agher,Kel!! 13841 - - LeFeure.N~ght1ngale.Rosclls751 - H~ll.naclllian119881
-
x Kell.iicLaur1n.Uhallcy119681 + ~e.l.nctaw1nll977~ - Hi1!113871
n
X 1
1
1
100
50
1
150
1
200
1
250
l/C
Figure 4. Enthalpy of the saturated vapor a t 0 "C < t 6 250 "C. Comparison of experimental values and calculated correlation values, with Hill, MacMillan (1988) as the base line.
trend would be consistent with the adsorption of water molecules on the walls of the vessel. This tendency seems to be considerably stronger a t 150 and 175 " C . Quite apart from adsorption, however, the relative uncertainty in the volume measurements will decrease as the specific volume increases. Given that the absolute uncertainty 6m in the mass charged into the vessel is independent of pressure or temperature, one may write 3
t I56
I
I
200
I75
I
I
225
25C
I
2'5
1
300
l/C
Figure 2. B2 at 150 "C d t < 300 "C. Comparison of experimental values and calculated correlation values (H,O).
correlations of Haar et al. (1984) and Hill (1987))in Figures 2 and 3, in which the results of this work are labeled "Hill, MacMillan (1988)". The agreement between the virial correlation and the adjusted Osborne (1937, 1939a) h, values is set out in Figure 4. For temperatures above 200 "C, it appears that the relative differences between the equation and the Kell and McLaurin (1977) and Kell et al. (1985) PUTdata are of the order of the relative uncertainty which can be ascribed to the volumes deduced from the Osborne measurements, i.e., 6~,/.ug N 0.0005. However, it may be noted that the deviations are all negative close to saturation and that this
in which V is the volume of the vessel. If the precision of measurement of the low-temperature liquid (u 1cm3/g) is of the order of 6u/v = 3 X this would imply that in general 6u/u 3 x 10-5u
-
and that near saturation 6u/u would be 0.0015 (at 250 "C), 0.0038 (at 200 "C), and 0.0118 (at 150 "C).In light of this argument, the degree of agreement between the virial equations, the Osborne et al. (1937, 1939a) volumes, and the Kell and McLaurin (1977) and Kell et al. (1985) volumes is considerably better than one might expect. A comparison with the Wormald (1964) steady-flowdata is set out in Figure 5. Comparisons of the correlation with
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 879 --
Haar.Gallaghcr.Kcllll3841 Hill,nacflillan119881 V i r i a l IPrelininaryI Uornald119641
'
Hill.~acRillan(19881 Haar.Callagher.Kcllll9841
x Sirota.Tinro1I19561 - Hi11119871
x - H~11119871
'1
1
p .................................................................. I J4pn4 ........................................................................ ------.-__ x x x x x x x x
IC
UL' I Y Y :.i.Y U
I
Lo
2
....... 89.94.. 0
Lo
Lo
-
Saruraiion
I
I
I
220
20 0
240
I
260
I
280
I
300
l/C
Figure 7. Specific heat capacity at constant pressure at P = 1.96 and 5.88 MPa. Comparison of experimental values and calculated correlation values (H,O).
0
ru 0 ,
I
0
I
0 02
I
PInPa
0 .04
I
0 .06
I
I
0 .08
0. I '
Figure 5. Isothermal Joule-Thomson plot a t 60 "C 6 t 6 140 "C. Comparison of experimental values and calculated correlation values (H20).
-.--
Haar.Gallaghcr.Kcllll9841 Hili.~acnillan119881 Uagner.Scnpcrs(19861
+ Osborne.Slinson.Cinningsll9391 - Hi1111987)
9
- ' - Hill.llac~lllan119881 '
+
-
Haar.Callaghcr.Kclll13~1
+\
Novikou.Rvdoninl19681
CPO
Hi11119871
+
i
P U
:
:
273 15
-+ Lo
'\\
x
283 15
293 15
303 15
313 15
323 15
T/K
Figure 8. Enthalpy difference increments (&,AT') at 273 K C T Q 323 K. Comparison of experimental values and calculated correlation values (H,O).
important advantage of the virial functional form approach to representing combined ug and h vapor saturation values is that consistency with the well-known C,, values of Woolley (1979) is ensured (see Figure 8).
5. The D20Virial Correlation The values of Kell et al. (1968b) of B2 for DzO were not very different from the HzO values, on a molar basis, but a modified analysis was able to suggest values for the function B2 - BZm, and these values are set out in Figure 9, along w i g the correlation of those values used in Hill and MacMillan (1980). With the availability of the more recent D 2 0 PUT measurements of Kell and McLaurin
880 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 '
+ x
-
0 Ln
9
Hlll.flacflillanll3BOl
Table IV. Calculated Virial Coefficient Values for D,O
Kell.flcLaur1n.Uhalleyll968bl
Kell.ncLaurin!l9771 E o u a t i a n (271
"1
50
100
I50
200
250
30 0
t/C
Figure 9. (BZDz0 - BZHzO) at 50 "C < t 6 300 OC. Comparison of experimental values and calculated correlation values (H,O).
(1977), it is possible to use the H 2 0 virial correlation (adjusted for the difference in molecular weights by a factor of 18.016/20.028, and assuming B3 and B4 are identical for H 2 0 and DzO) to calculate the AB2 value associated with each data point. Those values are also plotted in Figure 9. While the AE correlation of Hill and MacMillan (1980) closely followed the Kell and McLaurin (1977) values, the discrepancy between the H 2 0B2values associated with the Kell and McLaurin (1977) PUT data and the LeFevre et al. (1975) correlation values (particularly below 200 "C) indicated that those values should be given much less weight. The resulting correlation for AB2 is as follows:
AB2 = (0.0063701361 - 0.1019727 X 10-4T)10-3m3/mol (27)
Equation 15 is used to calculate $o for D20, using coefficients Ci (kJ/kg) from a correlation of the ideal gas specific heats of Friedman and Haw (1954), with C1= 1866.7679, CQ= 1793.9404, C3 = 64.605, C4 = -284.8833, C5 = 100.1333, C, = -13.135, C7 = 0.32684, and C8 = -1211.253. The values of C, and C2are initialized as for HzO, at the triple point temperature of 3.8 "C. The resulting saturation values and virial coefficient values are set out in Tables IV and V, using the D 2 0 saturation pressure correlation of Hill and MacMillan (1979). 6. Comparison of the D,O Virial Correlation with
Experimental Data Figures 10 and 11set out the B2 and B3 values obtained from the D 2 0 virial correlation and compare those values with Kell et al. (196813) and Kell and McLaurin (1977). At present, the Joule-Thomson values of Juza et al. (1966) are the only derivative measurements available to evaluate the D20 correlation, and those values are set out in Figure 12. Application of the correction suggested by Ertle (1979) would lead to the Juza et al. (1966) values being increased by about 3%, which is in good agreement with the correlation. The character of the slope of the calculated D 2 0 values a t 128 "C of this correlation is comparable to that of H 2 0 and of the data and suggests
T,K
Bz, m3 kg-'
B,, (m3 kg-')*
B4,(m3 kg-1)3
276.95 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15 363.15 373.15 383.15 393.15 403.15 413.15 423.15 433.15 443.15 453.15 463.15 473.15 483.15 493.15 503.15 513.15 523.15 533.15 543.15 553.15 563.15 573.15
-0.086 1 -0.076 7 -0.064 4 -0.054 8 -0.047 1 -0.040 9 -0.035 9 -0.031 7 -0.028 2 -0.025 3 -0.022 8 -0.020 7 -0.018 9 -0.017 3 -0.015 9 -0.014 7 -0.013 6 -0.012 6 -0.011 8 -0.011 0 -0.010 3 -0.009 66 -0.00908 -0.008 56 -0.008 08 -0.007 63 -0.007 23 -0.006 85 -0.006 50 -0.006 17 -0.005 87
-0.021 4 -0.017 3 -0.0124 -0.008 94 -0.006 5 1 -0.004 77 -0.003 53 -0.002 62 -0.001 96 -0.001 48 -0.001 12 -0.000 846 -0.000 644 -0.000 492 -0.000 376 -0.000 288 -0.000 221 -0.000 170 -0.000 130 -0.000 100 -0.000 076 3 -0.000 058 0 -0.000 043 8 -0.000 032 8 -0.000024 3 -0.000017 7 -0.0000126 -0.000 008 6 -0.000005 5 -0.000 003 2 -0.000 0014
0.000 035 0.000 029 0.000021 0.000016 0.000012 0.000 008 8 0.000006 7 0.000 005 1 0.000 004 0 0.OOO 003 1 0.000002 4 0.000 001 9 0.000 001 5 0.000 001 2 0.000001 0 0.000 000 78 0.000 000 63 0.000 000 51 0.000 000 42 0.000 000 34 0.000 000 28 0.000 000 24 0.000 000 20 0.000 000 16 0.000 000 14 0.000 000 12 0.000 000 10 0.000 000 082 0.000 000 070 0.000 000 059 0.000 000 051
Table V. Calculated Saturation Values for D,O s.,
T,K
Ps,kPa
276.95 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15 363.15 373.15 383.15 393.15 403.15 413.5 423.15 433.15 443.15 453.15 463.15 473.15 483.15 493.15 503.15 513.15 523.15 533.15 543.15 553.15 563.15 573.15
0.660 135 1.026 36 1.999 12 3.701 51 6.548 70 11.1208 18.1996 28.804 7 44.228 6 66.068 3 96.252 137.062 191.149 261.545 351.669 465.323 606.694 780.344 991.199 1244.5 1546.0 1901.5 2317.5 2800.4 3357.2 3995.3 4722.3 5546.3 6475.8 7519.9 8688.5
p,,
kg m-,
0.005 7444 0.008737 2 0.016 444 0.029 459 0.050 494 0.083 181 0.132 22 0.203 53 0.304 35 0.443 33 0.630 69 0.878 22 1.1994 1.609 5 2.125 7 2.767 3 3.555 5 4.514 3 5.670 4 7.053 7 8.697 8 10.641 12.927 15.606 18.738 22.394 26.661 31.646 37.487 44.368 52.541
h,, kJ kg-' 2323.8 2334.2 2351.0 2367.7 2384.3 2400.8 2417.1 2433.2 2448.9 2464.4 2479.4 2494.0 2508.0 2521.4 2534.1 2546.0 2557.0 2567.1 2576.1 2584.0 2590.6 2595.8 2599.5 2601.6 2601.9 2600.3 2596.4 2590.0 2580.8 2568.2 2551.5
kJ (kg
K)-' 8.3908 8.2448 8.0265 7.8271 7.6447 7.4773 7.3233 7.1812 7.0499 6.9281 6.8148 6.7091 6.6100 6.5169 6.4290 6.3458 6.2666 6.1908 6.1182 6.0480 5.9801 5.9138 5.8489 5.7850 5.7216 5.6583 5.5947 5.5302 5.4642 5.3957 5.3239
a significant improvement over that in Hill and MacMillan (1980). The D,O virial correlation has a recommended range of 0 "C < t < 300 "C. 7. Conclusions
A virial equation has been derived for H 2 0 which represents the saturated vapor volumes and enthalpies derived
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 881 - --
x
Ln
9
- ' - Hill.nacnillanll3801 - - Hill.nacflillan11988) x Juza e1 al.119661
Hill.nacllillanll5BO~ Hill.tlacnillan119881 Kcll.llcLaurin.Uhallcy11968)
+ Krll.tlcLaurinll9771
'1 Saturation
-"?
/ /
--/
0
/
0
0
1
1
I
I
I
I
I
0
0.5
1
I
.5
2
2.5
P/nPa
Figure 12. Isenthalpic JouleThomson plot at 128 "C d t d 250 "C. Comparison of experimental values and calculated correlation values.
measurements in deriving the virial coefficients. Such data are relatively abundant for H 2 0 but for DzO only a few Joule-Thomson data are available. More throttling data would assist in more accurately determining the DzO saturation states and associated virial coefficients. Acknowledgment
The National Sciences and Engineering Research Council of Canada is gratefully acknowledged. The authors are very much indebted to the late Dr. G. S. Kell for the use of his unpublished data on HzO and DzO volumes and to Professor Edwin LeFevre for critical comments. Nomenclature
"! 0
200
I
I
240
220
I
260
I
280
1
300
t/c
Figure 11. Third virial coefficient B3 at 200 O C Q t Q 350 "C (DzO).
from the Osborne et al. (1937,1939a) measurements, the Joule-Thomson data, the Kell and McLaurin (1977) and Kell et al. (1985) PUT data, and the steady-flow data of Wormald (1964) within what are believed to be the experimental uncertainties. The range of the equation is from 0 to 300 "C and from zero pressure to saturation. By a simple transformation, a virial equation has been derived for DzO. This equation provided the same degree of agreement with the Kell and McLaurin (1977) DzOPVT data as it does for the Kell and McLaurin (1977) and Kell et al. (1985) HzO PUT data, up to 300 OC. Because of the difficulty of making accurate volume measurements a t low density and, additionally, the evidence of absorption a t temperatures below 200 O C in even the best PUT data, it is vital to use throttling or other
Bz = second virial coefficient, m3 kg-' B3 = third virial coefficient, (m3kg-1)2 B4 = fourth virial coefficient, (m3 kg-')3 C, = isopiestic specific heat, kJ (kg K)-l C, = isochoric specific heat, kJ (kg K)-' h = specific enthalpy, kJ kg-' P = pressure, kPa Paat, P, = saturation vapor pressure, kPa R = gas constant kJ (kg K)-I s = entropy, kJ (kg K)-' T = thermodynamic temperature, K u = specific internal energy, k J kg-' u = specific volume, m3kg-' w = speed of sound, ms-' Greek Symbols 6 = uncertainty A = change in value of property p
= density, kg m-3
)I
= Helmholtz free energy, J. = u
T
= 1/T, K-'
Subscripts calc = calculated values crit, c = critical point f = saturated liquid g = saturated vapour obs = experimental values 0 = zero pressure
- T,,kJ
kg-'
Ind. Eng. Chem. Res. 1988,27, 882-892
882
Registry No. HzO, 7732-18-5; D20, 7789-20-0.
Literature Cited Collins, S. C.; Keyes, F. G. Proc. Am. Acad. Arts Sci. 1938,72,283. Ertle, S. Ph. D. Thesis, Technische Universitat Miinchen, Germany, 1979. Friedman, A. S.; Haar, L. J . Chem. Phys. 1954,22,2051. Haar, L.; Gallagher, J. S.; Kell, G. S. NBSINRC Steam Tables; Hemisphere: Washington, D.C., 1984. Hill, P. G. "A Unified Equation of State for H,O". Paper submitted to the International Association for the Properties of Steam, Sept 1987. Hill, P. G.; MacMillan, R. D. C. Znd. Eng. Chem. Fundam. 1979,18, 412. Hill, P. G.; MacMillan, R. D. C. J . Phys. Chem. Ref. Data 1980,9, 735. Hill, P. G.; MacMillan, R. D. C.; Lee, J. J . Phys. Chem. Ref. Data 1982,11, 1. Juza, J.; Kmonicek, V.; Sifner, 0.;Schovanec, K. Physica 1966,32, 362. Keenan, J. H.; Keyes, F. G.; Hill, P. G.; Moore, J. G. Steam Tables; Wiley: New York, 1969. Kell, G. S. J . Chem. Eng. Data 1975,20,97. Kell, G. S.; McLaurin, G. E., National Research Council, Ottawa, 1977, unpublished data. Kell, G. S.; Whalley, E. J . Chem. Phys. 1975,62,3496. Kell, G. S.; McLaurin, G. E.; Whalley, E. J . Chem. Phys. 1968a,48, 3805. Kell, G. S.; McLaurin, G. E.; Whalley, E. J. Chem. Phys. 1968b,49, 2839. Kell, G. S.; McLaurin, G. E.; Whalley, E. Philos. Trans. R. Soc. London, A 1985,315,235.
Keyes, F. G. Int. J . Heat Mass Transfer 1962,5,137. LeFevre, E. J.; Nightingale, M. R.; Rose, J. W. J . Mech. Eng. Sei. 1975,17, 243. Nightingale, M. R. Ph.D. Thesis, University of London, Great Britain, 1970. Novikov, I. I.; Avdonin, V. I. "Velocity of Sound in Saturated and Superheated Steam". Report to 7th International Conference Properties of Water and Steam, Tokyo, 1968. Osborne, N. S.; Stimson, H. F.; Ginnings, D. C. J . Res. Nat. Bur. Stand. 1937,18, 389. Osborne, N. S.; Stimson, H. F.; Ginnings, D. C. J . Res. Nat. Bur. Stand. 1939a, 23, 197. Osborne, N.S.; Stimson, H. F.; Ginnings, D. C. J . Res. Nat. Bur. Stand. 1939b,23, 261. Rivkin, S. L.; Egorov, B. N. At. Energ. 1963,14, 416. Saul, A.; Wagner, W. "Correlation Equations for the Vapour Pressure and for the Orthobaric Densities of Water Substance". Proceedings of the 10th International Conference on the Properties of Steam, Sept 2-7, 1984, Moscow, USSR. Saul, A,; Wagner, W. J. Phys. Chem. Ref. Data 1987,in press. Sirota, A. M.; Timrot, T. L. Teploenergetika 1956,3, 16. Wagner, W.; Sengers, J. V. "Draft Release on Saturation Properties of Ordinary Water Substance". Technical Report BN1051, IAPS, June 1986. Woolley, H. W. "Thermodynamic Properties for H 2 0 in the Ideal Gas State". Proceedings of the 9th International Conference on the Properties of Steam, Sept 10-14, 1979, Munchen, Germany. Wormald, C. J. "Thermodynamics of Vapours". Ph.D. Thesis, University of Reading, Great Britain, 1964. Young, J. B. J . Eng. Gas Turbines Power 1987,in press.
Received for review February 24, 1987 Revised manuscript received December 2, 1987 Accepted December 16, 1987
A Correlation for Thermodynamic Properties of Heavy Fossil-Fuel Fractions Barry J. Schwarz and John M. Prausnitz* Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory, and Chemical Engineering Department, University of California, Berkeley, California 94720
A correlation is developed for calculation of thermodynamic properties of heavy fossil fuels, based on a recent version of the perturbed-hard-chain equation of state. Since the correlation does not require experimental vapor pressures or densities as input data, i t is useful for heavy fossil fuels where vapor pressures and densities are difficult or impossible to measure. For fossil-fuel fractions, equation-of-state parameters are found from approximate molecular-structure (characterization) data coupled with a calibration based on pure-component (model-compound) data. Interaction parameters are given for mixtures of fossil fuels with methane, ethane, carbon dioxide, hydrogen, and hydrogen sulfide. While the correlation presented here is useful for calculating vapor-liquid equilibria, it is not intended for conventional fossil fuels, where other correlations are satisfactory. Instead, i t is intended for heavy fossil fuels where conventional correlations are often not reliable and where the usual input data are often unavailable.
As the world continues to deplete its petroleum reserves, high-quality crude oil can remain neither plentiful nor inexpensive. When reserves of light crude oil run low, heavy crude oil, coal liquids, and other heavy fossil fuels may be required to meet the world's energy needs. Heavy fossil fuels contain molecules that are larger and more aromatic and that contain more heteroatoms (nitrogen, oxygen, and sulfur) than those found in light crudes. To process heavy fossil fuels, their thermodynamic properties must be known. Most current correlations for thermodynamic properties of fossil fuels use normal boiling point and liquid density at 60 O F as input parameters (American Petroleum Institute, 1983; Tsonopoulos et al., 1986; Twu, 1984; Lin and Chao, 1984; Watanasiri et al., 1985). While these corre0888-5885/88/2627-0882$01.50/0
lations are useful in many cases, they have some serious disadvantages. Many of the boiling-point and density-based correlations are not applicable to fuels containing a significant number of heteroatoms. Further, for heavy fossil fuels, the needed liquid densities and boiling points may be experimentally inaccessible. In some cases, accuracy may be lost upon using a correlation to extrapolate available densities and vapor pressures to 60 OF and the normal boiling point. For very heavy fossil fuels, reliable vapor pressures and liquid densities may be unavailable, thereby preventing use of the common correlations. Several attempts have been made to construct correlations based on group-contribution methods or on molecular-structure data [e.g., Allen (1986), Smith et al. (1976), 0 1988 American Chemical Society
