Analytical Expressions of Thermodynamic and Transport Properties of the Martian Atmosphere in a Wide Temperature and Pressure Range

Abstract

Calculation of thermodynamic and transport properties of CO₂/N₂/O₂/Ar system (Martian atmosphere) have been performed in a wide pressure (0.01–100 bar) and temperature range (50–50,000 K). A self-consistent approach for the thermodynamic properties and higher order approximation of the Chapman–Enskog method for the transport coefficients have been used. Debye–Hückel corrections have been included in the calculation of thermodynamic properties while collision integrals derived following a phenomenological approach and accounting also for resonant processes contributions have been used. Moreover, charge–charge interactions have been obtained by using a screened Coulomb potential. Calculated values have been fitted by closed forms ready to be inserted in fluid dynamic codes in order to simulate plasma conditions for different technological applications. Comparison with data present in literature is also reported.

Appendix: Analytical Expressions of Thermodynamic and Transport Properties

In this appendix we report analytical expressions of thermodynamic and transport properties in the temperature range 50–50,000 K and in the pressure range 0.01–100 bar.

The following set of functions has been used:

Gaussian

$$ \begin{aligned} \gamma (T;c,\Updelta ) & = & e^{{ - q^{2} }} ,\quad q = \frac{T - c}{\Updelta } \\ \gamma_{i} (T) & = & \gamma (T;c_{i} ,\Updelta_{i} ) \\ \end{aligned} $$

(12)

Sigmoid

$$ \begin{aligned} \sigma (T;c,\Updelta ) & = & \frac{{e^{q} }}{{e^{q} + e^{ - q} }},\quad q = \frac{T - c}{\Updelta } \\ \sigma_{i} (T) & = & \sigma (T;c_{i} ,\Updelta_{i} ) \\ \end{aligned} $$

(13)

Special functions

$$ \begin{aligned} \xi (T,a,c,\Updelta ,w) & = & a - ce^{ - (T/\Updelta )w} \\ \xi_{i} (T) & = & \xi (T;a_{i} ,c_{i} ,\Updelta_{i} ,w_{i} ) \\ \end{aligned} $$

(14)

$$ \begin{aligned} \phi (T;a,w) & = & aT^{w} \\ \phi_{i} (T) & = & \phi (T;a_{i} ,w_{i} ) \\ \end{aligned} $$

(15)

The dependence of the fitted data on the pressure has been calculated by fitting the parameters a _i , c _i , Δ _i , w _i as a function of the pressure logarithm. In particular this expression has been used

$$ C = \sum\limits_{j = 0}^{n} {\alpha_{j} x^{j} } $$

(16)

where x = log(P) and C represents any of the parameters (or its natural logarithm) a _i , c _i , Δ _i , w _i of listed functions. α _j coefficients calculated by Eq. (6) are reported in Tables 2–10. In this way all the quantities are expressed as a function of two variables, P and T. Extrapolation out of the pressure range is not recommended, while the temperature dependence is very accurate up to 80,000 K. Relative errors of analytical expressions are always less than 5 %. Figure 14 shows the comparison of specific heats at constant pressure at P = 1, 100 bar and the percentage relative errors.

Fig. 14

Comparison of specific heats at constant pressure and percentage relative errors at P = 1 bar (data: square, analytical values: full line) and P = 100 bar (data: circle, analytical values: dashed line) obtained by using the analytical expression given by Eq. (22)

To calculate electron molar fractions, mean molar mass, specific enthalpy, specific heat, specific entropy thermal and electric conductivities and viscosity, the following analytical expressions can be used

Electron molar fraction, T < 20,000 K

$$ \begin{gathered} \chi_{e} (P,T) = (1 - \sigma_{0} (T))\exp [\sum\limits_{i = 0}^{6} {\delta_{i} (P)T^{i} } ] \hfill \\ + a_{1} \sigma_{0} (T)\sigma_{1} (T)\sigma_{m} (T) + \sigma_{0} (T)[a_{2} \sigma_{2} (T) + a_{3} \sigma_{3} (T) - a_{4} \gamma_{4} (T)] \hfill \\ \end{gathered} $$

(17)

where

$$ \delta_{i} (P) = \sum\limits_{k = 0}^{6} {\beta_{k,i} } [\log (P)]^{k} $$

Electron molar fraction, 20,000 K < T < 50,000 K

$$ \chi_{e} = \sum\limits_{j = 1}^{N} {a_{j} \sigma_{j} } (T) $$

(18)

Mean Molar Mass [kg/mol] and Gas Density [kg/m³]

$$ \bar{M} = c_{0} - \sum\limits_{j = 1}^{6} {a_{j} \sigma_{j} } (T)+ a_{7} \gamma_{7} (T) $$

(19)

$$ \rho = \frac{P}{RT}\bar{M} $$

(20)

Specific Enthalpy [cal/g]

$$ H = \sum\limits_{j = 0}^{4} {c_{j} T^{j} + } \sum\limits_{j = 1}^{7} {a_{j} \sigma_{j} (T)} $$

(21)

Specific heat [cal/g/K]

$$ c_{p} = \sum\limits_{j = 0}^{1} {c_{j} T^{j} + } \sum\limits_{j = 1}^{5} {a_{j} \sigma_{j} (T)} + \sum\limits_{j = 6}^{19} {a_{j} \gamma_{j} (T)} $$

(22)

Specific Entropy [cal/g/K]

$$ S = \sum\limits_{j = 1}^{7} {a_{j} \sigma_{j} } (T) + \phi_{0} (T) $$

(23)

Electric Conductivity [S/m]

$$ \log \sigma = \xi_{0} (\log (T)) + \sum\limits_{j = 1}^{7} {a_{j} \sigma_{j} } (T) $$

(24)

Thermal Conductivity [W/K/m]

$$ \log \lambda = a_{0} + \sum\limits_{j = 1}^{6} {a_{j} \sigma_{j} } (\log (T)) + \sum\limits_{j = 7}^{11} {a_{j} \gamma_{j} } (\log (T)) $$

(25)

Viscosity [Kg/m/s]

$$ \log \eta = \log [\xi_{0} (T) + \sum\limits_{j = 1}^{5} {a_{j} \sigma_{j} } (T)] + \sum\limits_{j = 6}^{10} {a_{j} \sigma_{j} } (T) $$

(26)