The documentation of “IAPWS95/viscosity.h”

IAPWS95/viscosity.h マニュアル

(The documentation of IAPWS95/viscosity.h)

Last Update: 2023/9/22

IAPWS95/viscosity.hでは粘性係数を計算する関数が定義されている。
Functions to compute the viscosity are defined in IAPWS95/viscosity.h.

粘性係数の計算式はWagner and Pruss (2002)には含まれておらず、 Huber et al. (2009)において別途定義されている。 2023年9月21日現在、 NIST Chemistry WebBook において同論文の式が用いられている。
The equation of the viscosity is not included in Wagner and Pruss (2002) but is defined separately in Huber et al. (2009). The equation in that paper is used in NIST Chemistry WebBook as of 21 September 2023.

Huber et al. (2009)において粘性係数

μ

は

\begin{matrix} (1) & \frac{μ}{μ_{r e f}} = {\bar{μ}}_{0} (τ) {\bar{μ}}_{1} (δ, τ) {\bar{μ}}_{2} (δ, τ) \end{matrix}

の形で与えられる(同論文(2)式)。ここで

μ_{r e f} \equiv 1 \times 10^{- 6}

[Pa s]は基準となる粘性係数を表す。また

δ \equiv ρ / ρ_{c}

は無次元化した密度、

τ \equiv T_{c} / T

は無次元化した温度の逆数であり、これらはIAPWS-95全体の表記に合わせてある (そのためHuber et al. (2009)の表記とは異なる)。
Huber et al. (2009) gives the viscosity

μ

in the form of Eq. (

1

), which is from Eq. (2) of the paper, where

μ_{r e f} \equiv 1 \times 10^{- 6}

[Pa s] is a reference viscosity;

δ \equiv ρ / ρ_{c}

is a nondimensional density and

τ \equiv T_{c} / T

is the inverse of a nondimensional temperature, defined consistently with the notation of IAPWS-95. Note that the notation is different from that of Huber et al. (2009).

{\bar{μ}}_{0}

は低密度極限での粘性係数を表し、

\begin{matrix} (2) & {\bar{μ}}_{0} (τ) = \frac{100}{\sqrt{τ} \sum_{i = 0}^{3} H_{i} τ^{i}} \end{matrix}

で与えられる(論文(11)式)。ここで

H_{i}

は表1のように与えられる。
The factor

{\bar{μ}}_{0}

represents the viscosity in the limit of zero density and is given as Eq. (

2

), which is from Eq. (11) of the paper. The values of

H_{i}

are given in Table 1.

表1. (

2

)式の係数

H_{i}

。 Huber et al. (2009)の表2に基づく。
Table 1. The coefficients

H_{i}

of Eq. (

2

) from Table 2 of Huber et al. (2009).

$i$	$H_{i}$
0	1.67752
1	2.20462
2	0.6366564
3	-0.241605

{\bar{μ}}_{1}

は密度増加の粘性係数への寄与を表し、

\begin{matrix} (3) & {\bar{μ}}_{1} (δ, τ) = \exp [δ \sum_{i = 0}^{5} (τ - 1)^{i} \sum_{j = 0}^{6} H_{i j} (δ - 1)^{j}] \end{matrix}

で与えられる(同論文(12)式)。ここで

H_{i j}

は表2のように与えられる。
The factor

{\bar{μ}}_{1}

represents the contribution to the viscosity due to increasing density and is given as Eq. (

3

) which is from Eq. (12) of the paper. The values of

H_{i j}

are given in Table 2.

表2. (

3

)式の係数

H_{i j}

。 Huber et al. (2009)の表3に基づく。
Table 2. The coefficients

H_{i j}

of Eq. (

3

) from Table 3 of Huber et al. (2009).

	$i = 0$	$i = 1$	$i = 2$	$i = 3$	$i = 4$	$i = 5$
$j = 0$	5.20094e-01	8.50895e-02	-1.08374	-2.89555e-01	0	0
$j = 1$	2.22531e-01	9.99115e-01	1.88797	1.26613	0	1.20573e-01
$j = 2$	-2.81378e-01	-9.06851e-01	-7.72479e-01	-4.89837e-01	-2.57040e-01	0
$j = 3$	1.61913e-01	2.57399e-01	0	0	0	0
$j = 4$	-3.25372e-02	0	0	6.98452e-02	0	0
$j = 5$	0	0	0	0	8.72102e-03	0
$j = 6$	0	0	0	-4.35673e-03	0	-5.93264e-04

{\bar{μ}}_{2}

は臨界点近傍での振る舞いを表し、論文(33)式, (20)-(23)式, (26)-(28)式, および(34)式の直後の注釈により

\begin{matrix} (4) & {\bar{μ}}_{2} (δ, τ) = \exp [χ_{μ} Y (δ, τ)] \end{matrix}

\begin{matrix} (5) & Y (δ, τ) = {\begin{cases} Y_{1} (δ, τ) & (ξ \leq 0.3817016416 \times 10^{- 9} [m]) \\ Y_{2} (δ, τ) & (ξ > 0.3817016416 \times 10^{- 9} [m]) \end{cases} \end{matrix}

\begin{matrix} (6) & Y_{1} (δ, τ) = \frac{1}{5} q_{C} q_{D}^{5} ξ (δ, τ)^{6} [1 - q_{C} ξ (δ, τ) + q_{C}^{2} ξ (δ, τ)^{2} - \frac{765}{504} q_{D}^{2} ξ (δ, τ)^{2}] \end{matrix}

\begin{array}{rcl} Y_{2} (δ, τ) & = & \frac{1}{12} \sin [3 ψ_{D} (δ, τ)] - \frac{1}{4 q_{C} ξ (δ, τ)} \sin [2 ψ_{D} (δ, τ)] \\ + \frac{1}{q_{C}^{2} ξ (δ, τ)^{2}} [1 - \frac{5}{4} q_{C}^{2} ξ (δ, τ)^{2}] \sin [ψ_{D} (δ, τ)] \\ (7) & - \frac{1}{q_{C}^{3} ξ (δ, τ)^{3}} {[1 - \frac{3}{2} q_{C}^{2} ξ (δ, τ)^{2}] ψ_{D} (δ, τ) - | q_{C}^{2} ξ (δ, τ)^{2} - 1 |^{3 / 2} l (δ, τ)} \end{array}

\begin{matrix} (8) & l (δ, τ) = {\begin{cases} \ln \frac{1 + w (δ, τ)}{1 - w (δ, τ)} & (q_{C} ξ (δ, τ) > 1) \\ 2 \tan^{- 1} | w (δ, τ) | & (q_{C} ξ (δ, τ) \leq 1) \end{cases} \end{matrix}

\begin{matrix} (9) & w (δ, τ) = \sqrt{| \frac{q_{C} ξ (δ, τ) - 1}{q_{C} ξ (δ, τ) + 1} |} \tan \frac{ψ_{D} (δ, τ)}{2} \end{matrix}

\begin{matrix} (10) & ψ_{D} (δ, τ) = \cos^{- 1} \frac{1}{\sqrt{1 + q_{D}^{2} ξ (δ, τ)^{2}}} \end{matrix}

\begin{matrix} (11) & ξ (δ, τ) = ξ_{0} {[\frac{Δ \bar{χ} (δ, τ)}{Γ_{0}}]}^{ν / γ} \end{matrix}

\begin{matrix} (12) & Δ \bar{χ} (δ, τ) = max {0, δ \frac{P_{c}}{ρ_{c}} [\frac{1}{{(\frac{\partial P}{\partial ρ})}_{T}} - \frac{1}{{{(\frac{\partial P}{\partial ρ})}_{T} |}_{τ = τ_{R}}} \frac{τ}{τ_{R}}]} \end{matrix}

と書ける。ここで

P

は圧力、

P_{c}

は臨界点での圧力であり、他の定数は表3のように与えられる。
The factor

{\bar{μ}}_{2}

represents the behaviour of the viscosity near the critical point, given as Eqs. (

4

)-(

12

) according to Eqs. (33), (20)-(23), (26)-(28), and the note immediately after Eq. (34) of the paper. In these equations,

P

is a pressure and

P_{c}

is the pressure at the critical point. The other constants in these equations are given in Table 3.

表3. (

4

)-(

12

)式の定数。 Huber et al. (2009)の表5に基づく。
Table 5. The constants in Eqs. (

4

)-(

12

) from Table 5 of Huber et al. (2009).

定数 Constant	値 Value
$χ_{μ}$	0.068
$q_{C}$	$\frac{1}{1.9 \times 10^{- 9}}$ [m $^{- 1}$ ]
$q_{D}$	$\frac{1}{1.1 \times 10^{- 9}}$ [m $^{- 1}$ ]
$ν$	0.630
$γ$	1.239
$ξ_{0}$	$0.13 \times 10^{- 9}$ [m]
$Γ_{0}$	0.06
$τ_{R}$	$\frac{1}{1.5}$

このヘッダファイル内で定義されている関数を以下に示す。各関数の詳細は関数名をクリックしてリンク先を参照のこと。
Functions defined in this header file are listed below. For details of individual functions, click the links.

関数名 Function name	機能・用途 Purpose
IAPWS95_calculate_viscosity_mu0	${\bar{μ}}_{0} (τ)$ を計算する。 Compute ${\bar{μ}}_{0} (τ)$ .
IAPWS95_calculate_viscosity_mu1	${\bar{μ}}_{1} (δ, τ)$ を計算する。 Compute ${\bar{μ}}_{1} (δ, τ)$ .
IAPWS95_calculate_viscosity_mu2	${\bar{μ}}_{2} (δ, τ)$ を計算する。 Compute ${\bar{μ}}_{2} (δ, τ)$ .
IAPWS95_calculate_viscosity	Huber et al. (2009)の計算式を用いて指定された密度・温度における粘性係数を計算する。 Compute the viscosity at given density and temperature using the equations of Huber et al. (2009).

◆検証 (Validation)

関数IAPWS95_calculate_viscosity を用いて粘性係数を計算し、 Huber et al. (2009)の表6, 表7と厳密に一致する結果が得られることを確認した。
The viscosities computed by function IAPWS95_calculate_viscosity were exactly identical to the numerical values shown in Tables 6 and 7 of Huber et al. (2009).

◆引用文献 (References)

Huber ML, Perkins RA, Laesecke A, Friend DG, Sengers JV, Assael MJ, Metaxa IN, Vogel E, Mares R, Miyagawa K (2009) New international formulation for the viscosity of H2O, J Phys Chem Ref Data 38(2), 101-125. https://doi.org/10.1063/1.3088050
Wagner W, Pruss A (2002) The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, J Phys Chem Ref Data 31(2), 387-535. https://doi.org/10.1063/1.1461829