Thermal conductivity

All thermal conductivity derivatives are determined by perturbation.

Pure compound thermal conductivities are only available for use with internal mixture thermal conductivity routines. They cannot be used if no internal mixture routine is selected. For CAPE-OPEN version 1.1 external calculations, pure compound thermal conductivities are evaluated for each compound as mixture thermal conductivities with composition set to unity for that compound and zero for all other compounds.

Liquid Mixture Thermal Conductivity

Ideal

The ideal liquid thermal conductivity mixing:

$\begin{displaymath}\lambda^L_m = \sum^{c}_{i=1} x_i \lambda^L_i\end{displaymath}$

DIPPR 9I

The DIPPR procedure 9I for liquid thermal conductivity mixing:

$\displaystyle F_{v,i}$	$\textstyle =$	$\displaystyle x_i / \sum^{c}_{i=1} x_i / \rho^L_i$
$\displaystyle \lambda_{ij}$	$\textstyle =$	$\displaystyle 2 / ( 1 / \lambda_i + 1 / \lambda_j )$
$\displaystyle \lambda^L_m$	$\textstyle =$	$\displaystyle \sum^{c}_{i=1}\sum^{c}_{j=1} F_{v,i} F_{v,j} \lambda_{ij}$

DIPPR 9H

The DIPPR procedure 9H for liquid thermal conductivity mixing:

$\begin{displaymath}{1 \over \sqrt{\lambda^L_m}} =\sum^{c}_{i=1} {w_i \over (\lambda^L_i)^2}\end{displaymath}$

where w_i is the weight fraction of compound i.

High pressure correction

For all liquid thermal conductivity mixing models, a correction is applied when the pressure is larger than 35 bar:

$\begin{displaymath}\lambda_{hp} = \left(0.63 T^{1.2}_r P_r / (30 + P_r) +0.98 + 0.0079 P_r T^{1.4}_r \right) \lambda\end{displaymath}$

This is DIPPR procedure 9G-1 where the mixture parameters are computed by the ideal mixing rules.

Pure compound liquid thermal conductivity

Temperature Correlation

The parameters for the temperature correlation for liquid thermal conductivity of pure compounds are available through TEA's PCD data files. If the reduced temperature exceeds 0.99, a value of 0.99 is used.

Pachaiyappan et al.

$\displaystyle f$	$\textstyle =$	$\displaystyle 3 + 20 (1 - T_r)^{2/3}$
$\displaystyle b$	$\textstyle =$	$\displaystyle 3 + 20 (1 - 273.15 / T_{c,i})^{2/3}$
$\displaystyle \lambda_i$	$\textstyle =$	$\displaystyle c 10^{-4} M^x_i \rho^L_i (f/b)$

for straight chain hydrocarbons:

$c=1.811$

$x=1.001$

otherwise:

$c=4.407$

$x=0.7717$

If the reduced temperature exceeds 0.99, a value of 0.99 is used. If the critical temperature is below 273.15 K, a value of 3 is used for b.

Sato-Riedel

$\displaystyle f$	$\textstyle =$	$\displaystyle 3 + 20 (1 - T_r)^{2/3}$
$\displaystyle b$	$\textstyle =$	$\displaystyle 3 + 20 (1 - T_b)^{2/3}$
$\displaystyle \lambda_i$	$\textstyle =$

Here, the reduced boiling temperature T_b,r is substituted by a default value of 0.65 if the normal boiling temperature is not available. If the reduced temperature exceeds 1, a value of 1 is used.

DIPPR procedure 9E

The Latini et al. procedure (DIPPR procedure 9E) for liquid thermal conductivity of pure compounds:

$\displaystyle \lambda^L_i$	$\textstyle =$	$\displaystyle {A (1-T_r)^{0.38} \over T^{1/6}_r}$
$\displaystyle A$	$\textstyle =$	$\displaystyle {A^* T^{\alpha}_b \over M^{\beta}_i T^{\gamma}_c}$

where parameters A^*, α, β and γ depend on the class of the compound:

Family	A^*	α	β	γ
Saturated hydrocarbons	0.0035	1.2	0.5	0.167
Olefins	0.0361	1.2	1.0	0.167
Cycloparaffins	0.0310	1.2	1.0	0.167
Aromatics	0.0346	1.2	1.0	0.167
Alcohols, phenols	0.00339	1.2	0.5	0.167
Acids (organic)	0.00319	1.2	0.5	0.167
Ketones	0.00383	1.2	0.5	0.167
Esters	0.0415	1.2	1.0	0.167
Ethers	0.0385	1.2	1.0	0.167
Refrigerants:
R20, R21, R22, R23	0.562	0.0	0.5	-0.167
Others	0.494	0.0	0.5	-0.167

If the reduced temperature exceeds 0.99, a value of 0.99 is used.

Per compound

When selecting the Per Compound routine, the above methods can be selected on a per-compound basis in the compounds tab.

Vapor Mixture Thermal Conductivity

Ideal

The mixture vapour thermal conductivity is computed from the pure compound thermal conductivities as follows:

$\begin{displaymath}\lambda^V_m = \sum^{c}_{i=1} x_i \lambda^V_i\end{displaymath}$

Kinetic theory

This is DIPPR procedure 9D:

$\begin{displaymath}\lambda^V_m = \sum^{c}_{i=1} {x_i \lambda^V_i \over\sum^{c}_{j=1} x_j \phi_{ij}}\end{displaymath}$

where interaction parameters φ_ij are computed from:

$\begin{displaymath}\phi_{ij} = 0.25 (1+\sqrt{{\eta_i \over \eta_j}{M_j \ov......}})^2{T+\sqrt{1.5^2 T_{b,i} T_{b,j}} \over T+1.5 T_{b,i}}\end{displaymath}$

Note that the compound viscosities are required for this evaluation.

High pressure correction

If the system pressure is larger than 1 atmosphere a correction is applied according to DIPPR procedure 9C-1. Mixture parameters are computed using the ideal mixing rules. Critical and reduced densities are computed from:

$\displaystyle \rho_c$	$\textstyle =$	$\displaystyle {1 \over V_{c,m}}$
$\displaystyle \rho_r$	$\textstyle =$	$\displaystyle {\rho \over \rho_c}$

If the reduced density is below 0.5, then

$a=2.702$

$b=0.535$

$c=-1$

else, if the reduced density is below 2, then

$a=2.528$

$b=0.67$

$c=-1.069$

otherwise

$a=0.574$

$b=1.155$

$c=2.016$

The high pressure thermal conductivity correction is then calculated from:

$\begin{displaymath}\Delta \lambda = {a 10^{-8} (\exp (b \rho_r) + c) \over......_m} T^{1/6}_{c,m} \over P^{2/3}_{c,m} }\right) Z^5_{c,m} }\end{displaymath}$

which must be added to the calculated thermal conductivity for low pressure.

Pure compound vapor thermal conductivity

Temperature Correlation

The parameters for the temperature correlation for vapor thermal conductivity of pure compounds are available through TEA's PCD data files.

DIPPR procedure 9B-3

DIPPR procedure 9B-3 for pure compound thermal conductivity:

$\begin{displaymath}\lambda^V_i = (1.15 (C_p - R) + 16903.36) \eta^V_i / M_i\end{displaymath}$

DIPPR procedure 9B-2

DIPPR procedure 9B-2 for pure compound thermal conductivity:

$\begin{displaymath}\lambda^V_i = (1.3 (C_p - R) +14644 -2928.8/T_r) \eta^V_i / M_i\end{displaymath}$

This method is recommended for linear molecules.

DIPPR procedure 9B-1

DIPPR procedure 9B-1 for pure compound thermal conductivity:

$\begin{displaymath}\lambda^V_i = 2.5 (C_p - R) \eta^V_i / M_i\end{displaymath}$

This method is suitable for mono-atomic gases only.

Misic-Thodos 2

This method is used for methane and cyclic compounds.

$\displaystyle \xi$	$\textstyle =$	$\displaystyle 2173.424 T^{1/6}_{c,i} \sqrt{M_i} \over P^{2/3}_{c,i}$
$\displaystyle \lambda_i$	$\textstyle =$	$\displaystyle 4.91~10^{-7} T_r C_p / \xi$

This method should be used when T_r < 1. A warning is produced if this is not the case.

Misic-Thodos 1

This method is used for all compounds but methane and cyclic compounds.

$\displaystyle \xi$	$\textstyle =$	$\displaystyle 2173.424 T^{1/6}_{c,i} \sqrt{M_i} \over P^{2/3}_{c,i}$
$\displaystyle \lambda_i$	$\textstyle =$	$\displaystyle 11.05~10^{-8} (14.52 T_r -5.14)^{1/6} C_p / \xi$