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Ind. Eng. Chem. Res. 2007, 46, 5810-5814
Critical Modification to the Vogel-Tammann-Fulcher Equation for Temperature Effect on the Density of Water Faruk Civan* Mewbourne School of Petroleum and Geological Engineering, The UniVersity of Oklahoma, T 301 Sarkeys Energy Center, 100 East Boyd Street, Norman, Oklahoma 73019-1003
A Vogel-Tammann-Fulcher equation-type asymptotic exponential function is modified by including a secondorder term to correlate the temperature dependence of the anomalous pure water density behavior. This approach yields an accurate correlation of the liquid water density in the temperature range of -30 °C to 89 °C at 1 atm, with an average relative deviation from the data that is on the order of 10-4. 1. Introduction Since its inception (see the literature of Vogel,1 Tammann and Hesse,2 and Fulcher3), the Vogel-Tammann-Fulcher (VTF) equation has been applied successfully for correlating the temperature dependency of various physical and chemical parameters of practical importance. Numerous applications reported in the literature indicate that the VTF equation performs well, in comparison to other approaches and empirical correlations of various parameters that involve a material properties description.4-7 Garcı´a-Colı´n et al.8,9 explained the theoretical basis for the VTF equation. Because it is theoretically based, it may allow safer extrapolation beyond the range of experimental data, and much more reliably, than empirical correlations that have been derived by curvefitting without any theoretical foundation. However, the VTF equation is not a panacea for all problems. For example, a critical modification may be required for applications that involve liquid water, as demonstrated in this paper. The conventional VTF equation cannot describe the anomalous temperature-density behavior of liquid water. Countereffects of the structural collapsing and thermal expansion of liquid water have been identified as being the primary processes responsible for this anomaly.10-12 However, rigorous theoretical description and correlation of this peculiar property of water has been of a very challenging issue.13 Yet, it is paramount to use an accurate correlation of water density in the modeling of processes that involve liquid water, such as freezing and thawing of moist soil.14-16 Ironically, the anomalous behavior of liquid water is one of its outstanding characteristic properties, providing unique advantages in many natural and engineered processes. This peculiar feature of liquid water influences the course of such processes in complicated ways. For example, the unfrozen water content and wettability of porous materials and soil are inherently dependent on the temperature-density behavior of water, which makes the modeling of relevant processes a considerable challenge.17,18 In this paper, first, the previously developed outstanding correlations are reviewed, justifying the motivation for the present study, and then a critical modification in the VTF equationsby adding a second-order termsis proposed and validated by accurately correlating typical water density data. To the author’s knowledge, this is the first attempt at modifying this equation in this manner. This modification enables the * Tel.: (405) 325-6778. Fax: (405) 325-7477. E-mail:
[email protected].
adequate representation of the aforementioned countereffects on the temperature-density behavior of liquid water and leads to an accurate correlation of the experimental data in the temperature range of -30 °C to 89 °C at 1 atm. This is demonstrated by correlating the temperature-density data of Weast and Astle.19 This is the primary purpose of the present study, because an accurate and mathematically simple correlation of water density at atmospheric pressure frequently is required for the modeling and simulation of many natural processes. Hence, the extension of the applicability of the present approach to correlating the water density under other conditions is beyond the scope of this short paper. The latter is left for future studies. 2. Previous Attempts at Correlating the Density of Water Only the outstanding attempts that have been made at correlating the ordinary liquid water density will be reviewed here, to justify the motivation for the present study. It should be noted a priori, however, that none are completely and truly theoretically based and they all involve many empirically estimated coefficients, including the most publicized IAPWS95 formulation.20,21 Such equations can be best referred to as being semiempirical. The IAPWS-95 formulation of the pure water equation of state is based on the Helmholtz free energy, which is expressed by a mathematically complicated empirical equation that contains many coefficients determined by means of tedious nonlinear regression of experimental data.20 Although the IAPWS-95 formulation incorporates theoretically rigorous and thermodynamically consistent principles, it is by no means a convenient approach that would be the preferred choice of an equation for applications of practical importance. Frequently, natural and engineering processes that involve the common substance of ordinary water are investigated by means of highly computationally intensive numerical simulation approaches. In such exercises, a mathematically complicated series sum of 56 terms, inherently accompanied by more than 56 empirically determined coefficients that must be provided in a tabular form of the IAPWS-95 formulation, will certainly introduce an excessive amount of computational burden. (See, for example, eqs 6.4-6.6 on page 429 in Wagner and Pruss,20 which are not repeated here because of the large space requirement.) Because most simulations require simultaneous iterative solutions of many nonlinear and highly coupled sets of equations, it is apparent that a mathematically compact, simple, and reasonably
10.1021/ie070714j CCC: $37.00 © 2007 American Chemical Society Published on Web 07/20/2007
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accurate equation that contained a few fitting coefficients, as proposed in this paper, would be preferable for computationally intensive simulations. In fact, the development of a practical correlation by a small modification to the well-known VTF equation was the primary motivation of the present author, which, nevertheless, yielded a correlation with accuracy comparable to that of the IAPWS-95 formulation. Hence, the approach of Wagner and Pruss20 leads to unnecessarily complicated equations for the purposes of practical importance. In the literature, this problem has been circumvented conveniently by resorting to simpler approaches, as described in the following discussion. Ahmed23 reports that the Hewlett-Packard HP41C Petroleum Fluids PAC manual24 offers an empirical correlation for the pure water density, with respect to pressure, by means of the following quadratic equation:
Fsw ) A1 + A2p + A3p2 Fw
(1)
where p denotes the absolute pressure (in units of psia), Fsw represents the density of water under standard conditions, and Fw is the density of water under actual conditions. The empirical parameters of eq 1 (Al, A2, and A3) are correlated, with respect to temperature, by means of the following quadratic equation:
Ai ) ai + biT + ciT2
(2)
where T denotes the temperature (in degrees Fahrenheit). The empirical correlations of water density given by eqs 1 and 2 contain nine empirical fitting constants (ai, bi, and ci, for i ) 1, 2, and 3), the values of which are reported by Ahmed.23 McCain22 presented a water density correlation, which is given by
Fsw ) (1 + ∆VT)(1 + ∆Vp) Fw
(3)
where the following empirical polynomial correlations are used:
∆VT ) - 1.0001 × 10-2 + 1.33391 × 10-4T + 5.50654 × 10-7T2 (4) ∆Vp ) -1.95301 × 10-9pT - 1.72834 × 10-13p2T - 3.58922 × 10-7p - 2.25341 × 10-10p2 (5) where p denotes the absolute pressure (in units of psia) and T denotes the temperature (in degrees Fahrenheit). Stopa and Wojnarowski25 presented the following empirical correlation:
Fw ) 1043.196 - 42.966623 exp(0.0068950122T) (6) where Fw is the pure water density (expressed in units of kg/ m3) and T denotes the temperature (in degrees Celsius). Comparing eq 6 with the correlations given by McCain22 and Ahmed,23 it is obvious that the empirical correlation of Stopa and Wojnarowski25 is advantageous, because it is a compact and simple three-parameter mathematical expression. Therefore, it would be preferred in complicated simulation studies, as mentioned previously. However, the application conditions of
Figure 1. Correlation of the pure water density using the modified VogelTammann-Fulcher (VTF) equation.
this correlation does not include liquid water at
