Eq. (6) is simpler than equations 1-5. Since parameters ofEq. (2)-(4) at the critical point for most of the liquids are not available readily, the simpler equation like ...
Indian Journal of Chemical Technology Vol. 7, September 2000 pp.264-265
Note
A simple two-parameter equation for predicting the densities of pure liquids and a case study involving six esters Shekhar Kumar* & SB Koganti Reprocessing R&D Division , Reprocessing Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India
Received 17 March 1999; accepted 25 July 2000 A simple two-parameter empirical equation with one reference density value at reference temperature is proposed for predicting saturated liquid densities. The proposed equation is simpler than the other correlations reported in the literature. For the case study invol vi ng six liquids, standard deviations for the proposed equation are lower than those for the DIPPR equation.
Experimental data on saturated liquid density are abundant in the literature and traditionall y correlated 1 2 by various empirical correlations. Francis • correlated experimental liquid density data with the following equation with a good accuracy:
c
p = A-Bt---
... (1)
E-t
The main drawbac k of Eq.(l) is its failure at the critical point and usage of non-linear regression for estimation of the coefficients. Martin 3 suggested the following empirical equation for reduced density of pure liquids.
~ = I+A(l-T,.)t t3 + B(I-T,)2t3 Pc + C(1- T,.) + D(l- T,.)
... (2)
Yen and Woods 4 observed that Eq.(2), when used with three terms as shown in Eq.(3), could predict experimental data for 62 liquids with an average deviation of 0.3%.
... (3)
+ D(l-T,)4t 3 An extended form of Eq. (2) was proposed by Wagner and Pruss 5 for predicting saturated liquid
* For correspondence
~ = 1+ A(l-T,.)t/3 + B(l-T,.)2t3 Pc + C( 1 - T,)st3 + D(l- T,.)t6t3
... (4)
+ E(l-T,.)43/3 + F(l-T,.)II 0/3
Another popular correlation was DIPPR equation 6 as reported by Daubert and Danner : a
... (5)
Though the DIPPR equation has four adjustable parameters, predicted values from DIPPR equation are often not likely to be better than 0.1 % over any range of temperature and should not be used if better accuracy is desired (Marsh K N, 1998, Personal Communication). The present authors observed that the reported experimental data on saturated liquid density for most of the liquids could be correlated with a sufficient accuracy by the following simple empirical equation, _P'-'1' - - - = 1 + a(t- 298.15) + b(t- 298.15) 2
413
E.!_= 1+ A(l-T,)t t 3 + B(l-T,)2 t3 Pc
density of water on ITS-90 scale. ITS-90 (the international temperature scale-1990) is the latest universal standard for the temperature. It replaces IPTS-68 standard (International Practical Temperature Scale of 1968), used earlier. The ITS-90 equation is listed as
... (6)
P298 . 15K
This equation was first reported by the authors in a recent paper7 for conelating the den sity of n-dodecane. This eq uation ha s only two adjustable parameters as aga inst four in the DIPPR equation. A simple multiple linear regression is sufficient for estimating the coefficients of Eq. (6) and an accurate experimental value is needed at the reference temperature of 298.15 K or any other suitable temperature. Eq. (6) is simpler than equations 1-5. Since parameters ofEq. (2)-(4) at the critical point for most of the liquids are not available readily, the simpler equation like Eq.6 may be used with much ease. To explore the applicability of Eq. (6) in predicting the liquid densities, reported precise experimental
NOTES
265
Table !-Parameters of Eq.(6) Compound Tri-n-Butyl Phosphate
B
A K-1
P298.1 5 K g/mL
7
-8 .96Ix10-4
0.97249
No. of points
K-2 4.539xl0·
7
15
a (Eq.(6)
a (DIPPR Eq.)
g/mL
g/mL
1.643xl0·
4
1.80x 10·4
4
1.87x Io· 4
Diethyl azelate
0.96670
-9.117 xl0-4
4.703 xl0·
15
1.655x 10'
Dibutyl suberate
0.94436
-8 .713 xl0-4
4.699 x10· 7
14
1.402xl0.4
1.92x 10-4
-8 .945 xi0-4
4.805 x10·
7
15
1.528x10'
4
1.86x 10' 4
7
15
1.787xl0-4
2.19xl0' 4
15
1.465xl0' 4
2.53x I 0' 4
Diethyl debacate
0.95913
Diethyl phthalate
1.11381
-8.111 x i0-4
5.836 x10·
Dioctyl phthalate
0.98000
-7 .863 xi0-4
6.599 xl0' 7
(
Note: Std. Deviation calculated as a =
I lP calc
- P meas
)2
(N -(p + 1))
data 8 for six esters were used. These esters are important industrial chemicals. The reference density values along with the coefficients of Eq.(6) and standard deviations are listed in Table 1. It may be observed that for all the six liquids studied here, standard deviations for Eq.(6) are lesser than that reported for the DIPPR equation. On the average, the overall standard deviation for Eq.(6), L58xl0· 4 g/mL, is 22% lower than that reported for the DIPPR equation (2.03xl0' 4 g/mL). I
Conclusion A simple two-parameter empirical equation requiring one reference density value at the reference temperature, already proposed for predicting saturated liquid densities, has been tested. The proposed equation is simpler than other correlations. For the case study involving six esters, standard deviations for the proposed equation are lower than those for the DIPPR equation. Acknowledgements Authors sincerely thank Dr. Placid Rodriguez, Director, IGCAR and Dr. Baldev Raj, Director, Reprocessing Group, IGCAR for the keen interest and encouragement.
where N is no. of points and p is no. of parameters in the equation
Nomenclature a,b A, B C,D F,F N
=adjustable parameters and coefficients of Eq.(6) =adjustable parameters =adjustable parameters =adjustable parameters =number of points = temperature (K) = reduced temperature (T/Tc) =critical temperature (K)
T, T, Greek =density (g/mL) p Pc =density at critical temperature (g/mL) a = standard deviation (g/mL), Defined at the bottom of Table I.
References I Francis A, lnd Eng Clum, 49(1957) 1779. 2 Francis A, Chem Eng Sci, I0( 1959) 37. 3 Martin J J, Thermodynamic and Transport Properties of Gases, Liquids and Solids (A merican Society of Mechanical Engineers and McGraw Hill Publishing Company, New York), 1959, 110. 4 Yen L C & Woods S S, A/ChE J, 12(1966) 95. 5 Wagner W & Pruss A, J Phys Chem Ref Data , 22( 1993) 783. 6 Daubert T E & Danner R P, Physical Properties and Thermodynamic Properties of Pure Chemicals: Data Compilation (Hemisphere Publishing Corporation, New York), 1989. 7 Kumar S & Koganti S B, J Nucl Sci Eng, 35(1998) 309. 8 De Lorenzi L, Fermegelia M & Torriano G, J Chem Eng Data , 42(1997) 919.
