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

Δ

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

Δ

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( $δ = ρ / ρ_{c}$ )。 The normalized density ( $δ = ρ / ρ_{c}$ ).
tau	規格化した温度の逆数( $τ = T_{c} / T$ )。 Inverse of the normalized temperature ( $τ = T_{c} / T$ ).
Ndiff_delta	$δ$ での微分回数。「戻り値」の項目参照。 The number of differentiation by $δ$ . See “Return value” section for more detail.
Ndiff_tau	$τ$ での微分回数。「戻り値」の項目参照。 The number of differentiation by $τ$ . 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

Δ

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

Δ

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	$Δ$	$θ^{2} + B_{i} {\tilde{δ}}^{a_{i}}$
1	0	$\frac{\partial Δ}{\partial δ}$	$(δ - 1) [\frac{2 A_{i} θ}{β_{i}} {\tilde{δ}}^{1 / (2 β_{i}) - 1} + 2 B_{i} a_{i} {\tilde{δ}}^{a_{i} - 1}]$
2	0	$\frac{\partial^{2} Δ}{\partial δ^{2}}$	$\begin{array}{rcl} \frac{2 A_{i} θ}{β_{i}} {\tilde{δ}}^{1 / (2 β_{i}) - 1} + 2 B_{i} a_{i} {\tilde{δ}}^{a_{i} - 1} \\ + 4 B_{i} a_{i} (a_{i} - 1) {\tilde{δ}}^{a_{i} - 1} + \frac{2 A_{i}^{2}}{β_{i}^{2}} {\tilde{δ}}^{1 / β_{i} - 1} \\ + \frac{4 A_{i} θ}{β_{i}} (\frac{1}{2 β_{i}} - 1) {\tilde{δ}}^{1 / (2 β_{i}) - 1} \end{array}$
0	1	$\frac{\partial Δ}{\partial τ}$	$- 2 θ$
0	2	$\frac{\partial^{2} Δ}{\partial τ^{2}}$	$2$
1	1	$\frac{\partial^{2} Δ}{\partial δ \partial τ}$	$- \frac{2 A_{i}}{β_{i}} {\tilde{δ}}^{1 / (2 β_{i}) - 1} (δ - 1)$

ここで

\begin{matrix} (1) & θ \equiv (1 - τ) + A_{i} {\tilde{δ}}^{1 / (2 β_{i})} \end{matrix}

\begin{matrix} (2) & \tilde{δ} \equiv (δ - 1)^{2} \end{matrix}

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

β_{i} = 0.3

である。
Here,

θ

and

\tilde{δ}

are defined as Eqs. (

1

) and (

2

), respectively. The constants are given as

A_{i} = 0.32

B_{i} = 0.2

a_{i} = 3.5

, and

β_{i} = 0.3

, regardless of whether

i = 55

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);