ARTICLE pubs.acs.org/EF
This paper was withdrawn on August 23, 2012 (Energy Fuels 2012, DOI: 10.1021/ef301392b)
Estimation of Surface Tensions of Paraffin Hydrocarbons Using a Novel Predictive Tool Approach and Vandermonde Matrix Alireza Bahadori* Environmental Innovations Research Centre, School of Environmental Science and Management, Southern Cross University, PO Box 157, Lismore, New South Wales 2480, Australia ABSTRACT: Surface tension is an important property used in the design of fractionators, absorbers, two-phase pipelines, and petroleum reservoir engineering calculations, and it is an important property where foaming, wetting, emulsification, and droplet formation are likely to occur. The petroleum industry is especially interested in the surface tensions of paraffin hydrocarbons to improve production and increase oil yields. In this work, a simple Arrhenius-type function combined with Vandermonde matrix is presented for the estimation of the surface tensions of paraffin hydrocarbons as a function of molecular weight and temperature. The surface tension is calculated for temperatures in the range 250420 K and molecular weights between 30 and 240. The proposed tool is superior, owing to its accuracy and clear numerical background, wherein the relevant coefficients can be retuned quickly if more data become available. Estimations are found to be in excellent agreement with the reliable data in the literature with average absolute deviation being less than 1.1%. The tool developed in this study can be of immense practical value for the engineers and scientists to have a quick check of the surface tension of the paraffin hydrocarbons at various conditions without opting for any experimental measurements. In particular, chemical and process engineers would find the approach to be user-friendly with transparent calculations involving no complex expressions.
1. INTRODUCTION One of the most striking demonstrations of intermolecular forces is the tension at the surface of a liquid.1,2 Numerous methods have been proposed to estimate the surface tension of pure liquids and liquid mixtures.3,4 One of the simplest is the empirical formula proposed by Macleod5 that expresses the surface tension of a liquid in equilibrium with its own vapor as a function of the liquid and vapor phase densities. However, Sugden6,7 modified this expression as a function of a temperature-independent parameter, parachor, and indicated a way to estimate it from molecular structure. Quayle8 used experimental surface tension and density data for numerous compounds to calculate the parachors of hydrocarbons. It has been shown that the parachor is a weak function of temperature for a variety of fluids within wide range of temperatures,58 and thus, it is generally assumed to be a constant. The good performance and extreme simplicity of its analytical form have made Macleod equation5 a very popular method for surface tension calculation.913 Nevertheless, there are various shortcomings for the application of this equation: (1) the parachor is actually a temperature-dependent parameter whose functional form with temperature was not known, (2) the empirical nature of parachor poses difficulty in deriving a more accurate expression for it, and (3) the absolute average percent deviation in surface tension prediction increases with increasing complexity of the molecular structure of fluid under consideration.14 Macleod’s empirical expression for surface tension calculation has proven to work very well for many substances over a wide range of temperatures. Nonetheless, deviations with respect to temperature are generally observed. Thus, efforts should be made to derive the functionality of surface tension with respect to temperature. The bottom-line is that correlations of physical properties should be sought only in terms of independent r 2011 American Chemical Society
variables such as temperature, pressure, molecular weight, and concentration.15 In view of the above-mentioned issues, it is necessary to develop an accurate and simple correlation that is easier than the existing approaches and less complicated with fewer computations for predicting the surface tension of paraffin hydrocarbons as a function of molecular weight and temperature. The paper discusses the formulation of such equations in a systematic manner along with a sample example to show the simplicity of the model and usefulness of such tools. The proposed method is exponential function, which leads to well-behaved (i.e., smooth and non-oscillatory) equations, enabling more accurate and nonoscillatory predictions, and this is the distinct advantage of the proposed method in comparison with previously developed methods.16
2. METHODOLOGY FOR THE DEVELOPMENT OF NOVEL CORRELATION The primary purpose of the present study is to accurately correlate the surface tensions of paraffin hydrocarbons as a function of molecular weight and temperature. This is done by a simple correlation, using an Arrhenius-type asymptotic exponential function, with a small modification of the VogelTammannFulcher (VTF) equation (Vogel,17 Tammann and Hesse,18 Fulcher19). This is important, because such an accurate and mathematically simple correlation of the surface tensions of paraffin hydrocarbons as a function of molecular weight and temperature is required frequently for the quick engineering calculations to avoid the additional computational burden of complicated calculations. The VTF equation is an asymptotic Received: September 2, 2011 Revised: November 7, 2011 Published: November 07, 2011 5695
dx.doi.org/10.1021/ef201317r | Energy Fuels 2011, 25, 5695–5699
Energy & Fuels
ARTICLE
exponential function that is given in the following general form:15 ln f ¼ ln fc
E RðT Tc Þ
Table 1. Tuned Coefficients Used in eqs 1013 ð1Þ
In eq 2, f is a properly defined temperature-dependent parameter, the units for which are determined individually for a certain property; fc is a preexponential coefficient, having the same unit as the property of interest; T and Tc are the actual temperature and the characteristic-limit temperature, respectively (both given in degrees Kelvin); E is referenced as the activation energy of the process causing parameter variation (given in units of J/kmol); and R is the universal gas constant (R = 8.314 J/kmol K). A special case of the VFT equation for Tc = 0 is the well-known Arrhenius equation.20 For the purpose of the present application, which involves the correlation of the surface tensions of paraffin hydrocarbons as a function of molecular weight and temperature, the VTF equation has been modified by adding second-order and third order terms, as follows:15
coefficient
value
A1
1.4533851511 101
B1
3.566784747 103
C1
1.612520992 105
D1
8.9266105865 106
A2
9.7583129667 103
B2
2.7812788879 106
C2
1.0679264826 108
D2 A3
6.0896474940 109 2.9276256778 106
B3
7.6092394098 108
C3
2.4336871495 1010
D3
1.3903544950 1012
ð2Þ
A4
2.9482231725 108
In eq 2, Tc has been considered zero to convert eq 2 to the well-known Arrhenius equation20 type.
C4
1.9662175686 1012
D4
1.0689869316 1014
b c d ln f ¼ ln fc þ þ þ T Tc ðT Tc Þ2 ðT Tc Þ3
ln f ¼ ln fc þ
b c d þ 2 þ 3 T T T
ð3Þ
The required data to develop this correlation includes the reported data21,22 for the loss surface tension of paraffin hydrocarbons as a function of molecular weight and temperature. The following methodology has been applied to develop this correlation. 2.1. Vandermonde matrix. Vandermonde matrix is a matrix with the terms of a geometric progression in each row (i.e., an m n matrix).23,24 2 3 1 α1 α21 3 3 3 αn1 1 6 7 6 1 α2 α22 3 3 3 αn2 1 7 6 7 2 n 1 7 V ¼6 ð4Þ 6 1 α3 α3 3 3 3 α3 7 6l 7 l l ⋱ l 4 5 1 αm α2m 3 3 3 αnm 1 or j1
V i, j ¼ αi
ð5Þ
for all indices i and j. The determinant of a square Vandermonde matrix (where m = n) can be expressed as23 det V ¼
Y
ðαj αi Þ
1ei
