Thermodynamic properties of ideal gases

Calculating $c_p, h, s$

The $c_p, h, s$ of a particular species is calculated using the following NASA-9 polynomials:

\[\begin{aligned} \cphatR &= %\frac{\hat{c}_p^\circ(T)}{\overline{R}} &= && a_1T^{-2} &+ &a_2 T^{-1} &+ &a_3 &+ &a_4 T &+ &a_5T^2 &+ &a_6 T^3 &+ &a_7 T^4 &&&&\\ \hhatRT &= &-&a_1T^{-2} &+ &a_2\frac{\ln(T)}{T} &+ &a_3 &+ &a_4\frac{T}{2} &+ &a_5\frac{T^2}{3} &+ &a_6\frac{T^3}{4} &+ &a_7\frac{T^4}{5} &+ &\frac{b_1}{T}&&\\ \frac{\hat{\phi}^\circ(T)}{\Ru} &= &-&a_1\frac{T^{-2}}{2} &- &a_2T^{-1} &+ &a_3 \ln(T) &+ &a_4 T &+ &a_5 \frac{T^2}{2} &+ &a_6 \frac{T^3}{3} &+ &a_7 \frac{T^4}{4} && &+ &b_2 , \end{aligned}\]

where the $^\circ$ and $\hat{\ }$ denote standard pressure ($P_\mathrm{std}=101.325\, \mathrm{kPa}$) and molar basis respectively and $\Ru = 8.3144\, \mathrm{J\, mol^{-1}\, K^{-1}}$ is the universal gas constant.

$\hat{\phi}^\circ$is the entropy complement function (we use $\phi$ instead of $s$ to emphasize that the entropy complement is only a function of temperature and not, in general, equal to the entropy) and is equal to $\hat{s}^\circ$ only at standard pressure. At any pressure other than $\Pstd$ the entropy $\hat{s}$ is given by,

\[\frac{\hat{s}(T)}{\Ru} = \phihatR - \ln{\frac{P}{\Pstd}}.\]

Cp(Tarray::AbstractVector{T}, a::AbstractVector{T}) where T

Cp0/R = a₁T⁻² + a₂T⁻¹ + a₃ + a₄T + a₅T² + a₆T³ + a₇T⁴

Cp(T, sp::AbstractSpecies)

h(TT::AbstractVector{type}, a::AbstractVector{type}) where type

H0/RT = -a1*T^-2 + a2*T^-1*ln(T) + a3 + a4*T/2 + a5*T^2/3 + a6*T^3/4 + a7*T^4/5 + b1/T
      = -a1*T₁   + a2*T₂*T₈      + a3 + a4*T₄/2 + a5*T₅/3  + a6*T₆/4  + a7*T₇/5  + a₈*T₂

h(T, sp::AbstractSpecies)

𝜙(TT::AbstractVector{type},a::AbstractVector{type}) where type

S0/R = -a1*T^-2/2 - a2*T^-1 + a3*ln(T) + a4*T + a5*T^2/2 + a6*T^3/3.0 + a7*T^4/4 + b2 
     = -a1*T₁/2   - a2*T₂   + a3*T₈    + a4*T₄+ a5*T₅/2  + a6*T₆/3.0  + a7*T₇/4  + a₉   

Representing mixtures

The molar specific heat, enthalpy, and entropy of a mixture of ideal gases at standard pressure, $\Pstd$, is given by,

\[\begin{aligned} \cpbarR &= \sum_{i=1}^n \Xi\left.\cphatR\right|_i\\ \hbarRT &= \sum_{i=1}^n \Xi\left.\hhatRT\right|_i\\ \phibarR &= \sum_{i=1}^n \Xi\left.\phihatR\right|_i + \Xi \ln{\Xi}. \\ \end{aligned}\]

The term $\Xi \ln{\Xi}$ in the entropy complement function represents the entropy of mixing when multiple species are present in the gas mixture.

When $P \neq \Pstd$, $\hat{c}_p$, $\hat{h}$, and $\phi$ are as above but $s$ has a pressure term.

\[\begin{aligned} \overline{\frac{\hat{c}_p (T)}{\Ru}} = \cpbarR &= \sum_{i=1}^n \Xi\left.\cphatR\right|_i\\ \overline{\frac{\hat{h} (T)}{\Ru T}} = \hbarRT &= \sum_{i=1}^n \Xi\left.\hhatRT\right|_i\\ \overline{\frac{\hat{\phi}(T)}{\Ru}} = \phibarR &= \sum_{i=1}^n \Xi\left.\phihatR\right|_i + \Xi \ln{\Xi} \\ \overline{\frac{\hat{s}(T)}{\Ru}} &= \sum_{i=1}^n \Xi\left.\phihatR\right|_i + \Xi \ln{\Xi} - \Xi \ln{\frac{P}{\Pstd}}. \\ \end{aligned}\]

Representing mixtures with fixed composition

For mixtures with fixed composition these calculations can be considerably sped up (see performance) by defining an equivalent set of coefficients $\left(\bar{a}_1\; \mathrm{to}\;\bar{a}_7\; \mathrm{and}\; \bar{b}_1,\bar{b}_2\right)$ as follows:

\[\begin{aligned} \cpbarR &= %\frac{\hat{c}_p^\circ(T)}{\overline{R}} &= && \bar{a}_1T^{-2} &+ &\bar{a}_2 T^{-1} &+ &\bar{a}_3 &+ &\bar{a}_4 T &+ &\bar{a}_5T^2 &+ &\bar{a}_6 T^3 &+ &\bar{a}_7 T^4 &&&&\\ \hbarRT &= &- &\bar{a}_1T^{-2} &+ &\bar{a}_2\frac{\ln(T)}{T} &+ &\bar{a}_3 &+ &\bar{a}_4\frac{T}{2} &+ &\bar{a}_5\frac{T^2}{3} &+ &\bar{a}_6\frac{T^3}{4} &+ &\bar{a}_7\frac{T^4}{5} &+ &\frac{\bar{b}_1}{T}&&\\ \phibarR &= &- &\bar{a}_1\frac{T^{-2}}{2} &- &\bar{a}_2T^{-1} &+ &\bar{a}_3 \ln(T) &+ &\bar{a}_4 T &+ &\bar{a}_5 \frac{T^2}{2} &+ &\bar{a}_6 \frac{T^3}{3} &+ &\bar{a}_7 \frac{T^4}{4} && &+ &\bar{b}_2 , \end{aligned}\]

where for each species $i$ in the mixture (containing $n$ individual species),

\[\begin{aligned} \bar{a}_j &= \sum_{i=1}^n a_{j,i} &\forall i \in n \\ \bar{b}_1 &= \sum_{i=1}^n b_{1,i} &\forall i \in n \\ \bar{b}_2 &= \sum_{i=1}^n \left(b_{2,i} + \Xi \ln{\Xi} \right)&\forall i \in n. \end{aligned} \]

The coefficients on the right-hand side are the original NASA-9 polynomial coefficients for each component species.

Thermodynamic derivatives

Additionally it is also useful to calculate the following derivatives,

\[\begin{aligned} \frac{d h}{d T} & = c_p\\ \frac{ds}{dT} = \frac{d\phi}{dT} & = \frac{c_p}{T}\\ \frac{d c_p}{d T} & = R_\mathrm{univ}\left(-2a_1T^{-3} - a_2 T^{-2} + a_4 + 2a_5T + 3a_6T^2 + 4a_7T^3\right)\\ \end{aligned}\]

where the last derivative $\displaystyle {\frac{dc_p}{dT}}$ is obtained by differentiating the polynomial representation of $c_p$.