Silanes and Siloxanes Thermal Conductivity in the Liquid Phase: A Critical Review and an Improved Prediction Method

As well known the Fourier’s law [4], for example in the mono-dimensional case, contains the thermal conductivity λ:

$\mathrm{dq}=-\lambda \mathrm{d} \mathrm{A} \frac{\mathrm{dT}}{\mathrm{dx}}$    (1)

where, dq is the heat flow along the x axis, dA is an infinitesimal area normal to the direction x and T is the temperature. The SI units are used, unless differently stated.

The thermal conductivity λ of the substances in the liquid phase is closely related to temperature, pressure and chemical structure, but rigorous theoretical investigations, based on classical or quantum mechanics, lead to general results usually not acceptable for engineering purposes because of large errors; the theory of the liquid state is far away from producing acceptable results when, for example, the saturation line is investigated. In the liquid phase reasonable approximations are available only for some compounds characterized by simple chemical structure and in any case when several ad hoc constants are introduced. For gases and solids on the contrary models based on the similitaries with the characteristics of the perfect gas and on the crystalline simmetries respectively are valid instruments for general theories. Moreover, accurate experimental thermal conductivity data for compounds in the liquid state are very often not available at all, and due to the lack of a general theory, empirical or semiempirical equations can provide reasonably accurate thermal conductivity data valid for engineering purposes. In this work, on the basis of investigations presented in previous papers [1, 2] the families of silanes and siloxanes (21 chlorosilanes, 5 cyclosiloxanes, 10 linear siloxanes, 11 silanes and 20 other silanes) are taken into consideration using a larger database [3] in order to extend the use of a general formula valid for organic compounds (alcohols, alkanes, ketones,….) and to improve preceeding results obtained in the case of the cited silanes and siloxanes, for which experimental thermal conductivity data at atmospheric pressure or along the saturation line in the liquid phase are available in very few cases. Substantially two parameters characterize the thermal conductivity of the investigated compounds, the first is an exponent a, the same for the all the compounds of the family, and the second is λ0.618, that is the thermal conductity estimated at the value of the reduced temperature equal to (Φ – 1) the mantissa of the “golden ratio” Φ = (1 + √5)/2 = 1.618033… The proposed method is acceptable for engineering purposes and comparable with the method proposed by DIPPR [3] (a linear correlation with various and different parameters). An extensive and critical comparison between very different methods points out that the method proposed in this work can be considered valid with absolute errors not greater than 5%.

Because the dependence of the thermal conductivity λ upon the pressure P for the compounds in the liquid phase is negligible up to 40-50 bar, only the temperature is taken into account in this work concerning λ at atmospheric pressure or along the saturation line. On the other hand, as above pointed-out, very few accurate experimental λ values for liquids as function of temperature are available in technical and scientific literature. Moreover, with very limited exceptions, from the normal melting point to the normal boiling point and over, the liquid thermal conductivity decreases almost linearly, and the critical point represents a singularity.

Some interesting models [5] present the liquid thermal conductivity as function of temperature and density by the sum of three contributions (“dilute gas contribution”, “excess contribution” and “critical enhancement contribution”), but in this work, because the explored range is near or along the saturation line, the density dependence, as for the pressure dependence, is neglected.

In this work the Eq. (3) is examined again with respect to A which is given by expression (5); this procedure changes the problem from A to h and, while A has a precise meaning (it is the thermal conductivity at the reduced temperature Tr=0.618…), h derives from a process of optimization and has units of measure without any particular meaning. Substantially the factor h and the exponent a are coupled to give the minimum mean absolute deviation between experimental (when available) and predicted λ data. Finally, at the end of the mathematical examination, the value of the factor h determines the value of the exponent a and vice versa; moreover, the exponent a is adimensional and is linked to the slope of the function (3) at every value of the reduced temperature. What is proposed is to return to the physical meaning of A, that is:

$\mathrm{A}=\lambda 0.618033989 \ldots$    (6)

At this point the value of A has to be found in the scientific and technical literature, rendering the choice of the exponent a entirely independent. In any case the liquid thermal conductivity, which is quite linear as function of the temperature, depends upon two parameters and what is proposed is to accept expression (6) in order to search for the value of a. The procedure adopted in this work is based on the following points:

- the value of (6) is taken from the Database DIPPR (3);

- the families taken into account are Silanes and Siloxanes.

The choice of the Database DIPPR has the following reason:

- the available liquid experimental or predicted λ data in technical and scientific literature are by different apparatuses and mathematical procedures with very different accuracies;

- the test of Eq. (5) where A is given by expression (6) has to be developed by taking into account the large, updated and reliable collection of thermal conductivity data contained in the DIPPR database: the λ data are clearly reported as experimental, predicted and smoothed; in any case the accuracy of the thermal conductivity data is clearly indicated.

In the discussion about the results, it is very important to outline the need of a large and accurate database containing reliable data with a clear claimed accuracy in large temperature ranges for a number of compounds as large as possible.

The DIPPR database contains in a common reduced temperature range (usually from 0.3 to 0.8) for each compound whether experimental thermal conductivity data with their sources or λ data obtained from a linear regression obtained by the available experimental data and some other predicted data (when this is possible).

Finally, without a reliable rigorous theory of the liquid phase, it has to be again pointed out that an acceptable matching has to be reached between accuracy and generality of the equations minimizing the number of parameters characterizing the single compound.

Substantially, given the very different and complicate structures of the investigated silanes and siloxanes, the obtained results are generally satisfying as follows:

- on the basis of the data collected from the DIADEM database, the use of the Eq. (5), already given for different organic families, is generalized, taking into account the value of the thermal conductivity at a specific value of the reduced temperature as shown by the expression (6) to cover silanes and siloxanes;

- the families differ from each other for the value of the exponent a (the same for the compounds of each family, with the only exception of silane characterized by small value of the molar mass M);

- factor A has a clear physical meaning expressed by Eq. (6);

- the five investigated families are characterized by very different and complex structures;

- the method can be easily extended to other families.

- the method proposed in the present paper generally leads to low average absolute deviations, being less than 5%.