3260
Ind. Eng. Chem. Res. 1998, 37, 3260-3267
Evaluated Equation Forms for Correlating Thermodynamic and Transport Properties with Temperature Thomas E. Daubert† Department of Chemical Engineering, 165 Fenske Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802
Equation forms developed for use in correlating thermodynamic and transport properties with temperature have been developed and tested over almost 2 decades using a large pool of critically evaluated data over a wide temperature range. Usually only one equation form has been used for any single property in a given phase; however, alternate forms are sometimes recommended. Properties treated are solid and liquid vapor pressure, heat of vaporization, liquid and solid density, solid, liquid, and ideal gas heat capacity, second virial coefficient, liquid and vapor viscosity, liquid and vapor thermal conductivity, and surface tension. Equation forms vary from empirical polynomials to theoretically based forms, including polynomials, exponentials, inverse power, and trigonometric functions in temperature. Introduction During the past 18 years, our research group has been responsible for evaluating and compiling the pure component physical, thermodynamic, and transport property data published as the Data Compilation of the Design Institute for Physical Property Data (Daubert et al., extant 1997) of the AIChE. Fifteen temperature dependent properties are included, each of which requires a correlating equation for presentation of coefficients and use for each of the over 1500 chemicals in the compilation. General criteria were established for selection of equation forms. A limitation of no more than five adjustable constants for computer format reasons was made; however, relaxation of this criterion showed no significant change in property fit. For liquid properties an equation form that would fit the entire range between the triple point and the critical point was desired. All equation forms were required to interpolate and extrapolate over the entire region of applicability. Studies of numerous correlating equations have been made, and usually a single equation form was selected per property to be used for every compound. The correlating equations range from straight lines and polynomials to reduced forms and exponentials. This paper details the correlating equation selection process including the quantitative evaluation of candidate equations and discusses the various equation forms by property. Besides the general criteria discussed in the previous paragraph, specific criteria for certain properties are applied and discussed. For example, equations for liquid properties are usually valid at atmospheric pressure up to the normal boiling temperature and at the saturation pressure at higher temperatures. Finally we summarize our recommendations, including alternates to the selected equation forms for use by the reader. Correlating Equations Vapor Pressure. Vapor pressure correlations are both the most familiar and the most important of all as † Phone: 814-863-4816. Fax:
[email protected].
814-865-7846. E-mail:
accuracy in the calculation of vapor pressure is essential to its use as a base property for calculation of the acentric factor, thermal properties, and equilibrium properties. The equation form must be interpolatable and extrapolatable between the triple point and the critical point. The ability to calculate hypothetical vapor pressures above the critical point would also be desirable for use in vapor-liquid equilibrium calculations involving low molecular weight organics. The fact that the Clapeyron (1834) equation dP/dT ) ∆Hv/(T∆V) presents an exact theoretical thermodynamic relationship for the dependence of vapor pressure on temperature gives a starting point to a correlation by predicting a linear relationship between the logarithm of vapor pressure and the reciprocal absolute temperature.
ln P ) A +
B T
Antoine (1888) determined that modification of the equation to
ln P ) A +
B T+C
(1)
correlates vapor pressure data below the normal boiling point quite accurately. Riedel (1954) following Frost and Kalkwarf (1953) modified the equation by adding two terms to take into account nonlinearities in behavior, leading to the equation
ln P ) A +
B + C ln T + DTE T
(2)
This equation can be fit nonlinearly by allowing E to vary unconstrained. However, if E is set to a relatively small positive integer, usually 1, 2, or 6, accuracy can be maintained and multiple linear regression can be used for fitting over the entire temperature range ranging from the triple point temperature, usually not measured but taken as the melting point temperature, to the critical temperature. Studies of the effect of the value of E on the accuracy of the correlating equation show that the sum of squared error may increase
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Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3261
monatomically with E, may decrease and then become level, or may reach a minimum and then increase with increasing E. However, the higher E becomes the smaller D becomes. Thus, the use of E as a small positive integer is a practical one with almost negligible effect on the correlation accuracy. Since a plot of ln P vs reciprocal temperature should be slightly S-shaped, the values of A and D should be positive with B and C negative given E a positive integer. Unless this is true, the Clapeyron equation, dP/dT ) ∆Hv/(T∆V), will not predict a reasonable heat of vaporization using this correlation for ln P. In fact, unless the vapor pressure data range correlated covers a large portion of the possible range, the strict Clapeyron test may not work well even with the correct signs on the correlator coefficients. Hicks (1978) advanced a set of rules for constrained fitting of vapor pressure data such that the “sign rule” of +, -, -, + applicable to the A-D coefficients will automatically be met. These rules postulate the Sshaped ln P vs 1/T curve with limits on the value of ∆H/ ∆Z as calculated by the Clapeyron relation as RT2(d(ln P)/dT) in the low and high reduced temperature ranges and require a minimum in ∆H/∆Z in the Tr range of 0.8-0.9. Although useful, constrained fits to attain the correct signs did not reduce and sometimes increased errors over an approach where no data points were privileged, as were the critical and normal boiling points for Hicks, and where all experimental data were utilized in regression. For cases where most data must be predicted, the methods were essentially equivalent. Using all available experimental data and adding predicted data where necessary to complete the range and curve shape in general are the preferred method of correlation. A second widely used method for vapor pressure data correlation is the method of Wagner (1973).
ln Pr )
1 (At + Bt1.5 + Ct3 + Dt6) Tr
(3)
were sparse. Equation 4 may be slightly better at extrapolation above Tc but overall is not as accurate a correlator as eq 2. A sign rule for eq 4 was derived with A and B being + and -, respectively. C and D must be - and +, respectively, while E can be + or -, although - predominates. In summary, eq 2 is the most accurate and flexible equation for correlating vapor pressure with the added advantage of extrapolatability above the critical temperature when necessary. Equation 3 is almost equivalent with the advantage of exact prediction of the critical pressure but with the disadvantage of its inability to be extrapolated. Solid vapor pressure is also correlated by eq 2 using values of E of 1, 2, or 6 according to which gives the most accurate fit. Heat of Vaporization. Correlation of heat of vaporization with temperature must be consistent with the exact Clapeyron equation dP/dT ) ∆Hv/(T∆V). This exact equation also relates vapor pressure and vapor and liquid densities. In addition, by definition the heat of vaporization must be zero at the critical point. This latter requirement leads the correlator to an equation form which uses (1 - Tr) as the independent variable. Previous workers have used the following equation forms C
∆Hv ) A(B(1-Tr) - 1) ∆Hv ) A(1 - Tr)B ∆Hv ) B(1 - Tr)A + C(1 - Tr)2A + D(1 - Tr)3A + ... (5) Evaluation of these forms showed the third eq 5 truncated after the second term (three constants) to be the most accurate as a correlator. Later two additional forms were compared to eq 5
∆Hv ) A(1 - Tr)(B+CTr+DTr +ETr ) 2
where Pr ) P/Pc, Tr ) T/Tc, and t ) 1 - Tr. This equation is only applicable to the critical point and cannot be extrapolated. Constrained fits using the same general criteria as those for the Riedel equation and forcing the data through the critical and normal boiling points gave slightly more accurate results using Wagner, a 3.4% average error vs a 5.5% average error for Riedel for 13 representative compounds. However, the normal unconstrained Riedel fit for the same data which yields the correct signs with no privileged points resulted in the most accurate result with a 2.7% average error. Thus, the Riedel equation again was deemed superior. Later in the work a modified Riedel equation used by API was tested.
ln P ) A +
E B + C ln T + DT2 + 2 T T
(4)
This equation adds a fifth term, while the exponent on the fourth term is set at 2. When 34 compounds distributed almost equally among alcohols, ethers, acids, and ketones were used to compare eqs 2 and 4, the former equation yielded a lower deviation for the normal boiling point to the critical point. The lack of a required sign rule for eq 4 sometimes allowed the shape of the curve in the low-temperature range to be odd if data
(
)
1 - Tr ∆Hv ) A 1 - Tbr
3
(6)
(B+CTr+DTr2+ETr3)
(7)
Note that eq 7 simply expands the common Watson (1931) equation by adding the terms above B to the exponent for predicting heats of vaporization at any temperature from the heat of vaporization at the normal boiling point. When eqs 5-7 were compared using five very different compound types, all three were quite accurate and extrapolated very well and could have been chosen. However, since eq 6 has linear coefficients which can be regressed more easily as well as does not require the normal boiling point as an input, it was chosen as the preferred correlating equation. It should be noted that correlation over a short temperature range may be more accurate using T rather than Tr; however, no such form will predict zero heat of vaporization at the critical temperature. Solid Density. Data on solid density as a function of temperature are scarce. Most data are only reported at room temperature. As the variation with temperature is not strong, a linear equation is normally adequate with a data range of the lowest data point to the triple point of the compound. In a few cases with a larger amount of data a simple quadratic polynomial
3262 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
was recommended. For most situations use of a singleexponential value over the entire temperature range is sufficient. Liquid Density. Equation forms successful for both predicting and correlating liquid density as a function of only temperature are limited to those containing the temperature as a reduced parameter. Thus, equations chosen for study use either reduced or pseudoreduced parameters. Equation 8 is derived from work of Yen and Woods (1966), while eq 9 follows Rackett’s (1970) work. B is either a regression constant or could be set to Tc. C is either a regression constant or could be set to Tc.
( )
4
FL ) A +
T
Kj 1 ∑ B j)1
FL ) AB-[1+(1-T/C)
j/3
(8)
D]
(9)
Various forms of eq 8 were tested, the two most complete of which are given as eqs 8-1 and 8-2. Equation 9 was also evaluated. Nine polar compounds of
(
T 1/3 T 2/3 +D1B B
(
T Tc
FL ) A + C 1 FL ) A + C 1 -
)
)
E
(
)
(
T Tc
+D 1-
)
F
(8-1) (8-2)
various chemical families which had extended range, high-quality liquid density data were tested over the entire range from the triple point to the critical point. Equations 8-1, 8-2, and 9 are the only equations which interpolated or extrapolated data well. Average percent errors were 0.28, 0.28, and 0.37 for eqs 8-1, 8-2, and 9, respectively. Equation 8-2, however, was very difficult to regress and required excess computation time. When plots of data vs regressions for eqs 8-1 and 9 were compared, the former equation did not behave smoothly as the density drops rapidly near the critical point, while the latter equation was quite smooth in its behavior. When the two equations were further tested for small data sets, the polynomial form of (8-1) was not reliable outside the range when compared to the exponential form of (9). This is in line with the Spencer and Danner (1972) evaluation of Rackett’s predictor, where A ) Pc/ RTc, C ) Tc, and D ) 2/7 with B being a regression parameter called ZRA allowing an equation to be built with as little as only one point of data. Because of all of these advantages, eq 9 is recommended for correlation of liquid density. In practice the C constant of eq 9 has always been set to Tc, the critical temperature. The options of setting D ) 2/7 or A ) Pc/RTc are also allowed. The option of setting C and D is often desirable when the data set is sparse or over a limited range and when agreement between the calculated and experimental (or estimated) critical volume do not closely agree. Equation 8 can be expanded to eq 10. Equation 10 has been found to be useful for correlating liquid density very accurately over the entire range where large amounts of data exist, such as is the case for water which shows an average percent deviation of 0.5% from the triple point to the critical point with no bias.
FL ) A + B(1 - Tr)1/3 + C(1 - Tr)2/3 + D(1 - Tr) + E(1 - Tr)4/3 (10) Heat Capacity. Solid, liquid, and ideal gas heat capacities are very important in many thermodynamic calculations. Solid heat capacity is a weak function of temperature, and little past correlational work is available. The problem that the heat capacity of most crystalline substances contains discontinuities because of solid transitions at low temperatures must be acknowledged. Obviously no equation form can bridge such discontinuities. If discontinuities occur within the useful temperature range, data on each side of the discontinuity are treated separately with separate equations. As data are usually not plentiful and cover only a short temperature range, a simple polynomial in temperature is used.
CPs ) A + BT + CT2 + DT3 + ET4
(11)
Most data can be fit with a linear equation with a quadratic necessary for a few systems. Liquid heat capacity data as a function of temperature are more plentiful than are solid data. Again polynomials such as eq 11 are useful for correlation if sufficient data are available. However, the degree of the polynomial must be restricted to the smallest number which will adequately account for all real inflections in the data. Use of additional terms may reduce the magnitude of the error deviations but will often add unwarranted inflections resulting in poor interpolation or extrapolation. For third and fourth degree polynomial use, extreme care must be taken to force curves to behave reasonably. An alternate equation form (12) has been proposed by Zabransky et al. (1990), where t ) 1 - Tr. Nine representative compounds were chosen for testing of the equation for various situations. Results showed that
CPL )
A2 C2t3 CDt4 D2t5 + B - 2ACt - ADt2 t 3 2 5 (12)
eq 12 is superior to eq 11 with wide range data that approach the critical temperature. No significant difference exists if data are available only up to approximately the normal boiling point. Equation 11 will correlate limited data with sharp minima more accurately. As eq 12 extrapolates more accurately than eq 11, it should be used as a default. Ideal gas heat capacity data have been correlated in many ways. Most are of the polynomial form, as listed by composite eq 13. Forms using various combinations
CP° ) A + BT + CT2 + DT3 +
E F H + 1/2 + FT4 + 2 T T T (13)
of three to five of the terms have been recommended. No matter which equation is selected, any correlation must exhibit (1) high accuracy over the wide temperature range usually desired, (2) accurate interpolation of all data with smooth curves and no unwarranted inflections, (3) acceptable extrapolation of data within and slightly above and below the desired range, and (4)
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3263
characteristics which are integrable for use in enthalpy and entropy calculations. Another set of equations for testing was based on the basic exponential form of eq 14. Certain modifications adding other reciprocal temperature terms were also considered. Polynomials below level 4 were discarded
CP° ) A + Be-C/T + E/T D
(14)
as low-temperature minima could not be fit. Polynomials with added reciprocal terms generally behaved strangely at the upper and lower temperature limits. Forms of eq 14 tended to model the experimental data more closely between 100 and 1500 K, the chosen reasonable limits of applicability. Equation 14 as written tended to extrapolate poorly at the low-temperature limit, while eq 14-1 using only the first two terms of eq 14 fit both ends reasonably well.
CP° ) A + B exp(-C/TD)
(14-1)
The only disadvantage of eq 14-1 is that it must be integrated numerically rather than analytically to obtain enthalpy or entropy. While tedious, the calculations can be programmed quite easily to solve the resulting series. For the calculation of enthalpy, the integral DH ) ∫TT12 must be evaluated over the desired temperature range. Equation 14-1 for CP° may be rewritten as
CP° ) A + B(1 - C/TD + C2/2T2D - C3/6T3D + ...) or
∑ m
CP° ) A + B +
[
m
(-1)
BCm
]
∫TT CP° dT ) (A + B)T + ∑
(m!)(1 - mD)
|
T1
T
A T
+
B T
n
+
[
]
BCm
∑ (-1)mm!T(mD + 1) m)1
∫TT
1
2
CP T
|
(-1)mBCmT(-mD) T2 m!(-mD)
T1
C D B + + Tn1 Tn2 TE
(17)
D E B C + + + T T3 T8 T9
(16-1)
Similarly eq 17 yielded eq 17-1, where -1 e E e +9, E * 0, 3, or 8.
dT ) (A + B) ln T +
∑ m
(16)
where n1 * n2 * E but all are integers between -1 and +9 (E would vary by compound). When nine representative compounds were evaluated for all 210 possible combinations of eq 16, the most accurate overall correlator was identified as eq 16-1.
β)A+
Thus,
∆S )
C D E B + n + n + n n1 2 3 T T T T 4
β)A+
For entropy calculations, the integral ∆S ) dT for constant pressure. Using the series expansion again,
)
When the same 13 compounds were correlated by eqs 14-1 and 15, results were essentially equal as were the integrations to enthalpy. Note that the sign of C or E is immaterial to the calculation. After extensive discussions by the experts on the steering committee, the Aly and Lee eq (15) was chosen to replace the earlier eq 14-1 for ease of use. Results of this change give equivalent accuracy except for extrapolation below about 200 K for compounds where data only exist above this point; in this region eq 14-1 extrapolates better than eq 15. Second Virial Coefficient. Second virial coefficients vary from negative values at the low-temperature limit through zero into positive numbers at higher temperatures with a hyperbolic shape. Equations proposed varied from polynomials to reciprocal polynomials to combined polynomial-exponential curves. Forms resulting from Kreglewski’s potential function (Kreglewski, 1974) are
(14-2)
∫TT12(CP/T)
CP°
S° ) A ln T + B((C/T) coth(C/T) - ln sinh(C/T)) D((E/T) tanh(E/T) - ln cosh(E/T)) + SCON (15-2)
where n1 * n2 * n3 * n4 and n1-n4 integers between -1 and +9 (exponents of T would be fixed for all compounds) and
(-1)mBCmT(1-mD) T2
m)1
H° ) AT + BC coth(C/T) - DE tanh(E/T) + HCON (15-1)
β)A+
2
1
n
CP° ) A + B((C/T)/sinh(C/T))2 + D((E/T)/cosh(E/T))2 (15)
m!TmD
Thus,
∆H )
careful selection of regression coefficients A-D for heat capacity curves with wide low-temperature ranges of essentially constant heat capacity. As an alternative the eq form (15) of Aly and Lee (1981) was considered as it is directly integrable to enthalpy or entropy. This leads to eqs 15-1 and 15-2 for enthalpy and entropy, where HCON and SCON are integration constants.
(14-3)
This equation was found to be satisfactory for hundreds of compounds; the only exception was hydrogen where the odd shape of the data curve required a split into two ranges. Problems with series convergence were later noted for nitrogen, oxygen, and air and required
β)A+
B C D + 8+ E 3 T T T
(17-1)
Equation 17-1 regressions for the representative compounds yielded values of 1 or 9 for E; this observation is consistent with eq 16-1. Compound by compound comparisons showed eq 16-1 to be the most accurate correlator for the second virial
3264 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
coefficient. The equation extrapolates well except below the usual 100 K low-temperature limit. Extrapolation above the 1500 K high-temperature limit is very good. Viscosity. Any appropriate liquid viscosity-temperature correlating equation must be able to fit lowpressure experimental data from the triple point to the critical point with the ability to smoothly interpolate between data points without inflections and to extrapolate actual data to the entire range. Liquid viscosity decreases with temperature. The simple Andrade (1934) eq (18) and another similar eq (19) have been used to
ln(µL) ) A + B/T
(18)
ln(µL) ) A + B ln T
(19)
correlate viscosity data. In an attempt to improve the correlation and noting that liquid viscosity behavior somewhat follows vapor pressure behavior, eq 18 was modified to eq 20.
ln(µL) ) A + B/T + C ln T
(20)
Preliminary evaluations using common compounds such as methanol consistently ranked eq 19 as giving the poorest fit with equation 20 being superior to eq 18. All equations yielded reasonable fits in the medium temperature range, but only eq 20 was accurate near the triple and critical points. An evaluation with two dozen disparate compounds with data available over most of the range for most compounds confirmed the accuracy of eq 20 in correlating, interpolating, and extrapolating liquid viscosity data. Except for inorganic acids, correlation errors were less than 1%. An even more accurate form which improves the fit near the critical point is the fully expanded Riedel-type eq (21) which adds an additional term to eq (20). The vast majority of systems where the fourth term was necessary could be accurately correlated with the exponent E set to a value of 10.
ln(µL) ) A + B/T + C ln T + DTE
(21)
As some perceived that eq 21 sometimes was not accurate enough, two other equations were evaluateds temperature-dependent eq 22 attributable to Walther and an equation in reduced temperature (23), where t ) 1 - Tr. When a large number of compounds were ATB
µL ) 1010
-C
ln(µL) ) A + Bt1/3 + Ct2/3 + Dt + Et4/3
(22) (23)
correlated, eq 21 was again determined to be the most accurate with eq 23 the poorest and eq 22 with C set to 0.9 almost equivalent. For compounds such as propane and butane, which were especially difficult, only eq 21 gave reasonable results. Further studies were carried out on the use of eq 21 without setting the exponent E to a value of 10. A study using 20 widely different compounds with values of E from 1 to 10 showed that half of the compounds were most accurately correlated with E ) 10, while data for other compounds correlated better with E less than 10. However, the average percent deviation between the optimum E and E ) 10 for the 10 compounds was only 0.5% with a maximum of 1.6% for a completely fluorin-
ated compound. When a further study tested values of E from -12 to +12, a few compounds were better correlated using negative values for E. As a result of these studies it is recommended that eq 21 be used for correlating liquid viscosity with temperature. In some cases the fourth term (DTE) is statistically insignificant and can be dropped; if the term is retained, values of E other than 10 should be evaluated if high accuracy is required. Whatever final equation form and value for E is chosen should be tested near the melting point if this region is important as the E value most affects correlation at low temperatures. Vapor viscosity as a function of temperature at low pressures well below saturation, where the pressure effect can be neglected, has previously been correlated by many different equation forms. Vapor viscosity increases with temperature and has a magnitude much smaller than liquid viscosity. Often vapor viscosities at temperatures as high as 1500 K are required. The viscosity increase with temperature has widely varying slopessome curving up, some curving down, and some almost straight lines. Inflections sometimes occur. The most simple expression is the exponential form of eq 24. Sutherland’s (Sutherland and Maass, 1932) equa-
µv ) ATB
(24)
tion (25) modifies the exponential eq 24. Testing of eq 25 for 26 different gases yielded average errors of 1.3% compared to 1.8% for eq 24 but was inadequate for curves containing an inflection. Other two constant variations with an exponential denominator and various polynomials were all inferior.
µv )
AT3/2 T+B
(25)
Equation 26, which could incorporate three or four constants, was selected for testing on the basis of several past studies. This equation can be used with either C or D zero or both C and D zero, reducing eq 26 to eq 24.
[
µv ) ATB/ 1 +
]
C D + T T2
(26)
If D ) 0 and B ) 1/2, the Sutherland equation results. The equation has much flexibility and was determined to be most useful using forms of all four constants for wide range data, with three constants (D ) 0) for most situations, and with C and D both set to zero, where either very narrow range data were available or where statistical analysis did not warrant three constants. For a data set of 24 widely different compounds with wide range data, 10 compounds required 4 constants, 11 compounds required 3 constants, and 3 compounds (benzene, toluene, and 1,3-butadiene) only required 2 constants. Average deviations were only about 0.5%. Correlators should check whether all constants of eq 26 are significant statistically or necessary for curve shape when fitting data for any compound. Thermal Conductivity. Liquid thermal conductivity data are sparse and notoriously inaccurate. Any correlation must be able to show that the property decreases with increasing temperature for essentially all fluids except a few such as water, which first increase and then decrease, and some inorganic acids, which increase. No theoretical or empirical equation form is satisfactory to describe liquid thermal conductivity.
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3265 Table 1. Key for the Equation Formsa equation
properties
Y ) A + BT + + + Y ) exp[A + B/T + C ln T + DTE] Y ) ATB/[1 + C/T + D/T2] Y ) A + B/T + C/T3 + D/T8 + E/T9 Y ) A/B(1+(1-T/C)D) Y ) A(1 - Tr)(B+CTr+DTr2+ETr3) Y ) A + B[(C/T)/sinh(C/T)]2 + D[(E/T)/cosh(E/T)]2 Y ) A2/t + B - 2ACt - ADt2 - C2t3/3 - CDt4/2 - D2t5/5, where t ) (1 - Tr) ln Y ) A + B/T + C ln T + DT2 + E/T2 Y ) A + B(1 - Tr)1/3 + C(1 - Tr)2/3 + D(1 - Tr) + E(1 - Tr)4/3 CT2
DT3
Fs, CPL, CPs, kL P, µL µv, kv β FL ∆Hv, σ CP° CPL P FL(H2O)
ET4
CP° ) A + B[(C/T)/sinh(C/T)]2 + D[(E/T)/cosh(E/T)]2 integrates to H° and S° H° ) AT + BC coth(C/T) - DE tanh(E/T) + HCON S° ) A ln T + B[(C/T) coth(C/T) - ln sinh(C/T)] - D[(E/T) tanh(E/T) - ln cosh(E/T)] + SCON a Y ) the property; T ) temperature in kelvin; T ) reduced temperature; A-E ) specific regression constants for compound and r property.
Thus, the polynomial form of eq 27 was selected.
kL ) A + BT + CT2 + DT3 + ET4
(27)
Compounds are fit according to the data availability using from linear to quartic forms. Average fitting errors were about 5%. Most compounds only require a cubic equation; however, compounds where data are available at temperatures approaching the critical one require the quartic form. If data are sparse and very narrow in range, a linear form suffices. Water requires a quartic form. Equation 28 is an alternate form which
kL ) A + Bt1/3 + Ct2/3 + Dt + Et4/3
(28)
can be useful in correlating liquid thermal conductivity, where t ) 1 - Tr. This equation was touted to be superior to the polynomial form above the normal boiling point where extrapolation is often required. However, extensive evaluations of the equation show that the equation is equivalent in correlating available data, show a minimum above the normal boiling point where a third or fourth order polynomial shows a minimum, and sometimes give a negative value of A which is physically impossible as the value should approximate the liquid thermal conductivity at the critical point. Both eqs 27 and 28 give average errors of about 10%. Equation 29 was used by Jamieson (1979) to correlate liquid thermal conductivity in either a two or four constant form, where t ) 1 - Tr
kL ) A(1 + t2/3 + B(t1/3 - 3t2/3 + 3t)) kL ) A(1 + Bt1/3 + Ct2/3 + Dt)
(29)
A study of 12 compounds with adequate experimental data varying from simple hydrocarbons to alcohols and esters shows that the simple polynomial eq 27 with only two constants fit better than the two constant eq (29) for nine compounds and that the three or four constant polynomial fit equally well or better than the four constant eq (29) for all 12 compounds. Thus, eq 29 is not recommended. Correlation of low-pressure vapor thermal conductivity data is similar to correlation of low-pressure vapor viscosity data. Equations such as (24), a modified (25), and (26) are normally recommended. Equation 26 with three constants was found to correlate vapor data for a wide variety of compounds typified by oxygen and
Figure 1. Vapor pressure for methanol using eq 2.
Figure 2. Heat of vaporization for methanol using eq 6.
cumene. Compounds with large amounts of experimental data such as water, carbon dioxide, and methane are better fit and extrapolated to as high as 1500 K using eq 26 with four constants. No better correlating equation is apparent. Surface Tension. Selection of a correlating equation for surface tension requires accurate interpolation and extrapolation as well as prediction of a zero value at the critical point. An equation form with such characteristics is given as eq 30. Guggenheim (1945) modified
σ ) A(1 - T/B)C
(30)
the equation to eq 31 to apply corresponding states principles. Equation 31, which is a simplification of eq 6, meets the criterion of a zero surface tension at the critical point not necessarily met by eq 30. Neverthe-
σ ) A(1 - (T/Tc))C
(31)
3266 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Figure 3. Second virial coefficient for methanol using eq 16-1. Figure 7. Liquid heat capacity for methanol using eq 11.
Figure 4. Surface tension for methanol using eq 31. Figure 8. Ideal gas heat capacity for methanol using eq 15.
Figure 5. Liquid density for methanol using eq 9. Figure 9. Liquid viscosity of methanol using eq 21.
Figure 6. Solid heat capacity for methanol using eq 11.
less, both equations were evaluated. For a selected group of six compounds of varying types with complete data available, eq 30 as expected fits the data better overall but loses its accuracy totally near the critical point. Average errors for eqs 30 and 31 were almost
Figure 10. Vapor viscosity for methanol using eq 26.
the same. On the basis of the estimated data accuracy, use of the more theoretically correct eq 31 does not compromise accuracy of correlation to any meaningful extent and is recommended. Use of eq 6, an expanded version of eq 31, is justified only when sufficient data
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3267 T ) absolute temperature t ) 1 - Tr V ) volume Greek Letters β ) second virial coefficient µ ) absolute viscosity F ) density σ ) surface tension Subscripts
Figure 11. Liquid thermal conductivity for methanol using eq 27.
b ) normal boiling c ) critical L ) liquid r ) reduced s ) solid v ) vapor Superscripts ° ) ideal gas Units of all properties are variable but must be consistent.
Literature Cited
Figure 12. Vapor thermal conductivity for methanol using eq 26.
are available and the simpler equation does not fit the data accurately. Summary Table 1 summarizes the recommended equation forms for correlating temperature dependent property data for pure chemicals throughout their full range of applicability within the indicated phase. Sample plots for 12 of the 13 properties discussed in this paper are shown in Figures 1-12 for methanol as an example over the entire applicable temperature range for each property. Solid density is not plotted as the property is essentially constant. These plots show the shape of each curve with either experimental data limits or extrapolation to theoretical limits such as the critical temperature or triple point temperature. Acknowledgment The work described in this paper is a composite of many studies carried out by the Data Compilation Project financially sponsored by about two dozen chemical companies through the Design Institute for Physical Property Data (DIPPR) administered through the American Institute of Chemical Engineers from 1980-1997 in the Department of Chemical Engineering at The Pennsylvania State University. The author and principal investigator deeply appreciates the past and present work of professional staff and graduate/undergraduate students who contributed to these studies. Nomenclature A-E ) regression constants CP ) heat capacity ∆Hv ) heat of vaporization k ) thermal conductivity P ) vapor pressure
Aly, F. A.; Lee, L. L. Self-Consistent Equations for Calculating the Ideal Gas Heat Capacity Enthalpy, and Entropy. Fluid Phase Equilib. 1981, 6, 169. Andrade, E. N. Philos. Mag. 1934, 17, 497, 698. Antoine, C. Thermodynamic Vapor Pressures: New Relation between the Pressures and the Temperatures (Thermodynamique, Tensions des Vapeurs: Novelle Relation entre les Tensions et les Temperatures). C. R. Hebd. Seances Acad. Sci. 1888, 107, 681, 836, 1146. Clapeyron, E. J. Memoirs on the motive power of heat. L’ecole Polytechnique 1834, 14 (23), 153. Daubert, T. E.; Danner, R. P.; Sibul, H. M.; Stebbins, C. C. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Taylor & Francis: Bristol, PA, extant 1997. Frost, A. A.; Kalkwarf, D. R. A Semiempirical Equation for the Vapor Pressure of Liquids as a Function of Temperature. J. Chem. Phys. 1953, 21, 264. Guggenheim, E. A. The Principle of Corresponding States. J. Chem. Phys. 1945, 13, 253. Hicks, C. P. The Constrained Fitting of Vapor Pressures. National Physical Laboratory Report CHEM 77; National Physical Laboratory: Teddington, U.K., 1978. Jamieson, D. T. Thermal Conductivity of Liquids. J. Chem. Eng. Data 1979, 24, 244. Kreglewski, A. Internal Report, API Research Project 44 & TRC; Texas A&M: College Station, TX, 1974. Rackett, H. G. Equation of State for Saturated Liquids. J. Chem. Eng. Data 1970, 15, 514. Riedel, L. A New Universal Vapor Pressure Formula (in German). Chem.-Ing.-Tech. 1954, 26, 83. Spencer, C. F.; Danner, R. P. Improved Equation for Prediction of Saturated Liquid Density. J. Chem. Eng. Data 1972, 17, 236. Sutherland, B. P.; Maass, O. Measurement of the Viscosity of Gases Over a Large Temperature Range. Can. J. Res. 1932, 6, 428. Wagner, W. New Vapor Pressure Measurements for Argon and Nitrogen and a New Method for Establishing Rational Vapor Pressure Equations. Cryogenics 1973, 13, 470. Watson, K. M. Prediction of Critical Temperatures and Heats of Vaporization. Ind. Eng. Chem. 1931, 23, 360. Yen, L. C.; Woods, S. S. A Generalized Equation for Computer Calculation of Liquid Densities. AIChE J. 1966, 12, 95. Zabransky, M.; Ruzicka, V., Jr.; Majer, V. Heat Capacities of Organic Compounds in the Liquid State. I. C1 to C18 1-Alkanols. J. Phys. Chem. Ref. Data 1990, 19, 719.
Received for review December 1, 1997 Revised manuscript received May 11, 1998 Accepted May 13, 1998 IE9708687
