関数IAPWS95_calculate_Delta マニュアル

(The documentation of function IAPWS95_calculate_Delta)
Last Update: 2023/9/22


◆機能・用途(Purpose)

残差項の計算式に登場する\(\Delta\)とその導関数を計算する。 関数IAPWS95_calculate_residual の内部で用いる補助関数である。
Compute \(\Delta\) and its derivatives that appear in the equations of the residual part. This is a supplementary function used internally in function IAPWS95_calculate_residual.

◆形式(Format)

#include <IAPWS95/forward.h>
inline double IAPWS95_calculate_Delta
(const double delta,const double tau,
 const int Ndiff_delta,const int Ndiff_tau)


◆引数(Arguments)

delta 規格化した密度(\(\delta=\rho/\rho_c\))。
The normalized density (\(\delta=\rho/\rho_c\)).
tau 規格化した温度の逆数(\(\tau=T_c/T\))。
Inverse of the normalized temperature (\(\tau=T_c/T\)).
Ndiff_delta \(\delta\)での微分回数。「戻り値」の項目参照。
The number of differentiation by \(\delta\). See “Return value” section for more detail.
Ndiff_tau \(\tau\)での微分回数。「戻り値」の項目参照。
The number of differentiation by \(\tau\). See “Return value” section for more detail.


◆戻り値(Return value)

指定した密度・温度に対応する\(\Delta\)またはその導関数の値。 引数Ndiff_delta, Ndiff_tauの値に応じて以下の量が計算される。
The value of \(\Delta\) or its derivative for the given density and temperature. The following quantities are computed depending on the values of arguments Ndiff_delta and Ndiff_tau.
Ndiff_delta Ndiff_tau 計算する量
Quantity to compute 		計算式(Wagner and Pruss (2002)の表6.5に基づく)
The equation (based on Table 6.5 of Wagner and Pruss (2002))
0 0 \(\Delta\) \[\begin{equation*} \theta^2+B_i\tilde{\delta}^{a_i} \end{equation*}\]
1 0 \(\PartialDiff{\Delta}{\delta}\) \[\begin{equation*} (\delta-1)\left[ \frac{2A_i\theta}{\beta_i}\tilde{\delta}^{1/(2\beta_i)-1} +2B_ia_i\tilde{\delta}^{a_i-1} \right] \end{equation*}\]
2 0 \(\PartialDDiff{\Delta}{\delta}\) \[\begin{eqnarray*} & & \frac{2A_i\theta}{\beta_i}\tilde{\delta}^{1/(2\beta_i)-1} +2B_ia_i\tilde{\delta}^{a_i-1} \nonumber \\ & & +4B_ia_i(a_i-1)\tilde{\delta}^{a_i-1} +\frac{2A_i^2}{\beta_i^2} \tilde{\delta}^{1/\beta_i-1} \nonumber \\ & & +\frac{4A_i\theta}{\beta_i}\left(\frac{1}{2\beta_i}-1\right) \tilde{\delta}^{1/(2\beta_i)-1} \end{eqnarray*}\]
0 1 \(\PartialDiff{\Delta}{\tau}\) \[\begin{equation*} -2\theta \end{equation*}\]
0 2 \(\PartialDDiff{\Delta}{\tau}\) \[\begin{equation*} 2 \end{equation*}\]
1 1 \(\PartialCrossDiff{\Delta}{\delta}{\tau}\) \[\begin{equation*} -\frac{2A_i}{\beta_i}\tilde{\delta}^{1/(2\beta_i)-1}(\delta-1) \end{equation*}\]

ここで \[\begin{equation} \theta\equiv (1-\tau)+A_i\tilde{\delta}^{1/(2\beta_i)} \label{eq.theta} \end{equation}\] \[\begin{equation} \tilde{\delta}\equiv (\delta-1)^2 \label{eq.delta_tilde} \end{equation}\] である。またWagner and Pruss (2002)の表6.2によれば \(i=55\), \(i=56\)のいずれの場合にも \(A_i=0.32\), \(B_i=0.2\), \(a_i=3.5\), \(\beta_i=0.3\)である。
Here, \(\theta\) and \(\tilde{\delta}\) are defined as Eqs. (\ref{eq.theta}) and (\ref{eq.delta_tilde}), respectively. The constants are given as \(A_i=0.32\), \(B_i=0.2\), \(a_i=3.5\), and \(\beta_i=0.3\), regardless of whether \(i=55\) or \(i=56\), according to Table 6.2 of Wagner and Pruss (2002).

◆使用例(Example)

const double rho=20.0;
const double T=600.0;
double dDelta_ddelta=IAPWS95_calculate_Delta (rho/IAPWS95_rhoc,IAPWS95_Tc/T,1);