FPROPS/Thermal conductivity

Calculation of thermal conductivity in FPROPS is in development. This development is driven by an application requiring transport properties of carbon dioxide, so the first correlations being implemented will be for that. Although textbooks such as Incropera and DeWitt, Holman and Cengel use <math>k</math> for thermal conductivity, most publications on thermophysical properties use the convention of <math>\lambda</math>, and that convention will be used on this page.

Correlations typically specify conductivity in terms of ideal (zero-density limit) <math>\lambda_0</math>, residual <math>\lambda_r</math> and critical region enhancement function <math>\lambda_c</math>, as follows:

<math>\lambda \left(\rho,T \right) = \lambda_0 \left(T \right) + \lambda_r \left(\rho, T\right) + \lambda_c \left(\rho, T\right)</math>

Ideal component

The ideal (zero-density limit) component has been expressed in several places as:

<math>\lambda_0\left(T\right) = \frac{0.177568 \sqrt{T} \left(c_p^0/R\right)}{\sigma^2 \sqrt{M} \; \mathcal{C}_\lambda^\ast\left(T^\ast\right)}</math>.

Here, <math>c_p^0/R</math> is the isobaric heat capacity divided by the gas constant (either both specific or both molar), <math>\sigma</math> is length scaling parameter (in units of nm, which is messy), a constant particular to the fluid in question, and <math>M</math> is relative molecular mass of the fluid.

The reduced temperature <math>T_\ast</math> is calculated as

<math>T_\ast = \frac{T}{\epsilon / k}</math>

where <math>\epsilon / k</math> is the 'energy scaling parameter', specific to the fluid in question. The reduced effective cross-section for thermal conductivity, <math>\mathcal{C}_\lambda^\ast \left(T^\ast\right)</math> is usually provided in the form of a power series like

<math>\mathcal{C}_\lambda^\ast \left(T^\ast\right) = \sum_{i=0}^{n}{\frac{b_i}{\left(T^\ast\right)^i}}</math>

where the constants <math>b_i</math> would be tabulated for each fluid separately.

Essentially, the ideal part of the thermal conductivity, then, is a ratio of two power series in terms of <math>T</math>: one derived from experimentally-determined <math>c_p^0\left(T\right)</math>, and the other a reduced effective cross-section derived experimentally from low-density conductivity data, <math>\mathcal{C}_\lambda^\ast \left(T^\ast\right)</math>. To accommodate as many different ways that authors may publish their results, we can allow arbitrary power series on both numerator and denominator, and we can allow the already-defined <math>c_p^0</math> function to be referenced/used as the denominator if desired.

Note that one publication provides the function <math>\lambda_0</math> in the form

<math>\lambda_0</math> = N_1 \frac{\eta_0\left(T\right)}{1 \mbox{\mu Pa \cdot s}} + N_2 \tau^{t_2} + N_3 \tau^{t_3}</math>

and this form does not comply with the above ratio of power series, since zero-density viscosity is, in turn, defined as a ratio of power series (see FPROPS/Viscosity).