IAPWS: International Association for the Properties of Water and Steam (chemicals.iapws)

This module contains the core of the IAPWS-95 and IAPWS-97 standards. The objective of this module is to contain extremely fast functions to calculate several basic properties of water.

The simplest interfaces are iapws95_rho for density calculation only and iapws95_properties for some basic properties.

For reporting bugs, adding feature requests, or submitting pull requests, please use the GitHub issue tracker.

IAPWS-95 Basic Solvers

chemicals.iapws.iapws95_rho(T, P)

Calculate the density of water according to the IAPWS-95 standard given a temperature T and pressure P. The phase is determined in this calculation.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water, [kg/m^3]

See also

iapws95_rhol_sat

iapws95_rhog_sat

Notes

There is a sudden transition at the saturation pressure between liquid and vapor density, by design.

This solution is iterative due to the nature of the equation. The solution procedure begins with IAPWS-97’s explicit equations as an initial guess, extrapolating when out of range. If the temperature is under the critical temperature, the saturation density is calculated, and used to ensure the solver begins in the feasible region. Newton’s method converges extremely, normally after 2 or 3 iterations.

Temperatures under 273.15 K are not officially supported by [1], but a solution is still attempted down to 235 K.

References

1(1,2)

Wagner, Wolfgang, and Andreas Pruß. “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.” Journal of Physical and Chemical Reference Data 31, no. 2 (2002): 387-535.

Examples

>>> iapws95_rho(T=300.0, P=1e6)
996.96002269499

1 GPa and 5000 K are suggested as upper limits of [1] although there are no hardcoded limits for temperature and pressure.

>>> iapws95_rho(T=5000.0, P=1e9)
326.79451662743

chemicals.iapws.iapws95_P(T, rho)

Calculate the pressure of water according to the IAPWS-95 standard given a temperature T and mass density rho.

Parameters

Tfloat

Temperature, [K]

rhofloat

Mass density of water, [kg/m^3]

Returns

Pfloat

Pressure, [Pa]

Notes

The IAPWS-95 model is explicit with inputs of temperature and density, so this is a direct calculation with no iteration required.

References

1

Wagner, Wolfgang, and Andreas Pruß. “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.” Journal of Physical and Chemical Reference Data 31, no. 2 (2002): 387-535.

Examples

>>> iapws95_P(330.0, iapws95_rho(T=330.0, P=8e5))
8e5
>>> iapws95_P(823.0, 40.393893559703734)
14e6

Not all temperature and density inputs provide a stable solution; for example anything between the vapor and gas saturation curves. In some but not all of these cases a negative pressure is returned:

>>> iapws95_P(T=300, rho=300)
-1.526394720e+23

chemicals.iapws.iapws95_T(P, rho)

Calculate the temperature of water according to the IAPWS-95 standard given a density rho and pressure P.

Parameters

Pfloat

Pressure, [Pa]

rhofloat

Mass density of water, [kg/m^3]

Returns

Tfloat

Temperature, [K]

Notes

This solution is iterative due to the nature of the equation. The solution procedure begins with IAPWS-97’s equations as an initial guess, extrapolating when out of range. Newton’s method converges extremely, normally after 2 or 3 iterations.

Due to water’s unique density curve, there is a temperature region spanning 273.15 K to 280.005 K where there are two solutions. No guarantee is made as to which solution will be returned.

References

1

Wagner, Wolfgang, and Andreas Pruß. “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.” Journal of Physical and Chemical Reference Data 31, no. 2 (2002): 387-535.

Examples

>>> iapws95_T(P=1e6, rho=995.0)
306.461547194

IAPWS-97 Basic Solvers

chemicals.iapws.iapws97_rho(T, P, use_95_boundary=False)

Calculate the density of water in kg/m^3 according to the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

use_95_boundarybool, optional

If True, respect the IAPWS-95 vapor pressure curve instead of the IF-97 one, [-]

Returns

rhofloat

Mass density of water, [kg/m^3]

Notes

The range of validity of this formulation is as follows:

For $P \le 100 \text{ MPa}$:

$$273.15 \text{ K} \le T \le 1073.15 \text{ K}$$

For $P \le 50 \text{ MPa}$:

$$1073.15 \text{ K} \le T \le 2273.15 \text{ K}$$

A ValueError is raised if the temperature or the pressure is out of bounds.

IAPWS is implemented in four regions in the T-P domain: Region 1 (liquid), region 2 (gas and supercritical gas), region 5 (high temperature gas), and region 3 (near-critical). Significant discontinuities exist between the transitions of each regions. In region 3, there are 26 sub-regions and the correlation has the least accuracy.

For many applications, the discontinuities in IF-97 can be problematic and the slower IAPWS-95 must be used. IAPWS-95 also has a wider range of applicability.

References

1

Cooper, JR, and RB Dooley. “Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam.” The International Association for the Properties of Water and Steam 1 (2007): 48.

Examples

>>> iapws97_rho(648.6, 22.5e6)
353.06081088726
>>> iapws97_rho(330.0, 8e5)
985.10498080770
>>> iapws97_rho(823.0, 14e6)
40.39293607288123
>>> iapws97_rho(2000.0, 3e7)
32.11456228328856

chemicals.iapws.iapws97_P(T, rho)

Calculate the pressure of water according to the IAPWS-97 standard given a temperature T and mass density rho.

Parameters

Tfloat

Temperature, [K]

rhofloat

Mass density of water, [kg/m^3]

Returns

Pfloat

Pressure, [Pa]

Notes

The range of validity of this formulation is as follows:

For $P \le 100 \text{ MPa}$:

$$273.15 \text{ K} \le T \le 1073.15 \text{ K}$$

For $P \le 50 \text{ MPa}$:

$$1073.15 \text{ K} \le T \le 2273.15 \text{ K}$$

A ValueError is raised if the temperature or density is out of bounds.

Newton’s method with analytical derivatives is used here to solve these equations. The solver tolerance is as tight as it can be without causing wasted iterations that do not improve the result at all. Pressure changes quickly with density however, and some discrepancy between solvers is to be expected.

For region 3, there are really two formulations present in IAPWS-97. There is a Helmholtz energy equation (Temperature and density dependent), and also 26 separate backwards equations for rho which depend on T and P. The Helmholtz energy equation is much more accurate and does not have discontinuities. The two sets of equations agree closely not not perfectly. By design, iapws97_rho implements the 26 T-P equations and this implements the Helmholtz energy equation. This means that in region 3 solutions will not be consistent. For consistency requirements, IAPWS-95 is recommended.

This solver does not have any issues with multiple solutions. The solvers have been checked to achieve a relative solution tolerance of 5e-9 on 100 million points.

References

1

Cooper, JR, and RB Dooley. “Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam.” The International Association for the Properties of Water and Steam 1 (2007): 48.

Examples

>>> iapws97_P(330.0, iapws97_rho(T=330.0, P=8e5))
8e5
>>> iapws97_P(823.0, 40.39293607288123)
14e6
>>> iapws97_P(T=2000.0, rho=32.11456228328856)
3e7

Region 3 point - does not implement the same equations as iapws97_rho!

>>> iapws97_P(648.6, iapws97_rho(T=648.6, P=22.5e6))
22499974.093936257

chemicals.iapws.iapws97_T(P, rho)

Calculate the temperature of water according to the IAPWS-97 standard given a pressure P and mass density rho.

Parameters

Pfloat

Pressure, [Pa]

rhofloat

Mass density of water, [kg/m^3]

Returns

Tfloat

Temperature, [K]

Notes

The range of validity of this formulation is as follows:

For $P \le 100 \text{ MPa}$:

$$273.15 \text{ K} \le T \le 1073.15 \text{ K}$$

For $P \le 50 \text{ MPa}$:

$$1073.15 \text{ K} \le T \le 2273.15 \text{ K}$$

A ValueError is raised if the pressure or density is out of bounds.

Newton’s method with analytical derivatives is used here to solve these equations. The solver tolerance is as tight as it can be without causing wasted iterations that do not improve the result at all.

Due to water’s unique density curve, there is a temperature region spanning 273.15 K to 280.005 K where there are two solutions. No guarantee is made as to which solution will be returned.

References

1

Cooper, JR, and RB Dooley. “Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam.” The International Association for the Properties of Water and Steam 1 (2007): 48.

Examples

>>> iapws97_T(8e5, iapws97_rho(T=330.0, P=8e5))
330.0
>>> iapws97_T(14e6, 40.39293607288123)
823.0
>>> iapws97_T(P=3e7, rho=32.11456228328856)
2000.0

IAPWS-95 Properties

chemicals.iapws.iapws95_properties(T, P)

Calculate some basic properties of water according to the IAPWS-95 standard given a temperature T and pressure P.

The properties are density rho, internal energy U, entropy S, enthalpy H, isochoric heat capacity Cv, isobaric heat capacity Cp, speed of sound w, Joule-Thomson coefficient JT, isothermal throttling coefficient delta_T, isentropic temperature-pressure coefficient beta_s, and the derivative of mass density with respect to pressure at constant temperature drho_dP.

This function is intended as a demonstration of how to use the IAPWS-95 equations. For that reason, mass-units are used in all returned variables.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water, [kg/m^3]

Ufloat

Internal energy of water, [J/(kg)]

Sfloat

Entropy of water, [J/(kg*K)]

Hfloat

Enthalpy of water, [J/(kg)]

Cvfloat

Isochoric heat capacity, [J/(kg*K)]

Cpfloat

Isobaric heat capacity, [J/(kg*K)]

wfloat

Speed of sound, [m/s]

JTfloat

Joule-Thomson coefficient, [K/Pa]

delta_Tfloat

Isothermal throttling coefficient, [J/(kg*Pa)]

beta_sfloat

Isentropic temperature-pressure coefficient, [K/Pa]

drho_dPfloat

Derivative of mass density with respect to pressure at constant temperature, [kg/(m^3*Pa)]

Notes

Hundreds of useful properties can be obtained from the IAPWS-95 model. It is intended for this function to serve as a useful starting point to those. Calculating every property with every set of units is beyond the scope of chemicals. The functions like iapws95_dAr_ddelta can be used directly in your own implementation - where you can calculate only those properties which are necessary, for maximum speed.

The formulas are as follows:

$$\frac{u(\delta, \tau)}{R T}=\tau\left(\phi_{\tau}^{\mathrm{o}} +\phi_{\tau}^{\mathrm{r}}\right)$$

$$\frac{s(\delta, \tau)}{R}=\tau\left(\phi_{\tau}^{\mathrm{o}} +\phi_{\tau}^{\mathrm{r}}\right)-\phi^{\mathrm{o}}-\phi^{\mathrm{r}}$$

$$\frac{h(\delta, \tau)}{R T}=1+\tau\left(\phi_{\tau}^{\mathrm{o}} +\phi_{\tau}^{\mathrm{r}}\right)+\delta \phi_{\delta}^{\mathrm{r}}$$

$$\frac{c_{v}(\delta, \tau)}{R}=-\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}} +\phi_{\tau \tau}^{\mathrm{r}}\right)$$

$$\frac{c_{p}(\delta, \tau)}{R}=-\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}} +\phi_{\tau \tau}^{\mathrm{r}}\right)+\frac{\left(1+\delta \phi_{\delta}^{\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}} \right)^{2}}{1+2 \delta \phi_{\delta}^{\mathrm{r}}+\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}}$$

$$\frac{w^{2}(\delta, \tau)}{R T}=1+2 \delta \phi_{\delta}^{\mathrm{r}} +\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}-\frac{\left(1+\delta \phi_{\delta}^{\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}} \right)^{2}}{\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}}+\phi_{\tau \tau}^{\mathrm{r}}\right)}$$

$$\mu R \rho=\frac{-\left(\delta \phi_{\delta}^{\mathrm{r}}+\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}+\delta \tau \phi_{\delta \tau}^{ \mathrm{r}}\right)}{\left(1+\delta \phi_{\delta}^{\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}}\right)^{2}-\tau^{2}\left( \phi_{\tau \tau}^{\mathrm{o}}+\phi_{\tau \tau}^{\mathrm{r}}\right) \left(1+2 \delta \phi_{\delta}^{\mathrm{r}}+\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}\right)}$$

$$\delta_{T} \rho=1-\frac{1+\delta \phi_{\delta}^{\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}}}{1+2 \delta \phi_{\delta}^{\mathrm{r}} +\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}}$$

$$\beta_{S} \rho R=\frac{1+\delta \phi_{\delta}^{\mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}}}{\left(1+\delta \phi_{\delta}^{ \mathrm{r}}-\delta \tau \phi_{\delta \tau}^{\mathrm{r}}\right)^{2} -\tau^{2}\left(\phi_{\tau \tau}^{\mathrm{o}}+\phi_{\tau \tau}^{ \mathrm{r}}\right)\left(1+2 \delta \phi_{\delta}^{\mathrm{r}} +\delta^{2} \phi_{\delta \delta}^{\mathrm{r}}\right)}$$

This derivative isn’t part of the same table of properties, but it is needed by the transport calculation routines:

$$\left(\frac{\partial \rho}{\partial P}\right)_{T} = \frac{1}{ R T\left(1+2 \delta \alpha_{\delta}^{\mathrm{r}}+\delta^{2} \alpha_{\delta \delta}^{\mathrm{r}}\right)}$$

References

1

Wagner, Wolfgang, and Andreas Pruß. “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use.” Journal of Physical and Chemical Reference Data 31, no. 2 (2002): 387-535.

Examples

>>> iapws95_properties(T=300.0, P=1e6)
(996.96002269, 112478.998245, 392.813902893, 113482.047492, 4127.21730497, 4178.103605593, 1503.035983829, -2.202166728257e-07, 0.000920088074745, 1.985617879134e-08, 4.48108429028e-07)

>>> rho, U, S, H, Cv, Cp, w, JT, delta_T, beta_s, drho_dP = iapws95_properties(T=500.0, P=1e5)
>>> w
548.3138393244

IAPWS Saturation Pressure/Temperature

chemicals.iapws.iapws95_Psat(T)

Compute the saturation pressure of the IAPWS-95 equation using high- fidelity polynomial fits. These have a relative accuracy of under 1e-12, and are generated by solving the saturation equations under the high-precision environment of mpmath. The range of the fit is 235 K to 647.096 K, the critical point.

$$P_{sat} = P_c \exp(\text{polynomial}(a (T - b)))$$

Parameters

Tfloat

Temperature at which to calculate the saturation condition, [K]

Returns

Psatfloat

Saturation vapor pressure, [Pa]

See also

iapws95_saturation

Notes

This method should be used in preference to iapws95_saturation. Although using mpmath generates slightly different results than using plain floating point numbers, the requirement for the saturation curve is to be smooth, and continuous; mpmath makes this easy and the saturation equations were solved extremely high precision, well under a floating point’s error.

The polynomial coefficients have been carefully chosen to be able to be evaluated accurately with horner’s method, although they are derived as a Chebyshev approximation originally.

Examples

>>> iapws95_Psat(400.0)
245769.3455

chemicals.iapws.iapws95_dPsat_dT(T)

Compute the temperature derivative of saturation pressure of the IAPWS-95 equation using high- fidelity polynomial fits. The range of the fit is 235 K to 647.096 K, the critical point.

$$P_{sat} = P_c \exp(\text{polynomial}(a (T - b)))$$

$$\frac{\partial P_{sat}}{\partial T} = a P_c \exp(\text{polynomial}(a (T - b))) \exp\left(\frac{\partial \text{polynomial}(a (T - b))}{\partial T}\right)$$

Parameters

Tfloat

Temperature at which to calculate the saturation condition and its temperature derivative, [K]

Returns

dPsat_dTfloat

First temperature derivative of Saturation vapor pressure, [Pa/K]

Psatfloat

Saturation vapor pressure, [Pa]

Notes

Psat must be calculated in the calculation of the derivative, so it is returned as well which may be useful in some applications.

Examples

>>> iapws95_dPsat_dT(400.0)
(7483.62075827, 245769.3455657)

chemicals.iapws.iapws92_Psat(T)

Compute the saturation pressure of the IAPWS-92 equation.

$$P_{sat} = P_c \exp\left(\frac{T_c}{T}[a_1\tau + a_2\tau^{1.5} + a_3\tau^3 + a_4\tau^{3.5} a_5\tau^4 + a_6\tau^{7.5}]\right)$$

Parameters

Tfloat

Temperature at which to calculate the saturation condition and its temperature derivative, [K]

Returns

Psatfloat

Saturation vapor pressure, [Pa]

Notes

The coefficients are [-7.85951783, 1.84408259, -11.7866497, 22.6807411, -15.9618719, 1.80122502]

Examples

>>> iapws92_Psat(400.0)
245765.2635418

chemicals.iapws.iapws92_dPsat_dT(T)

Compute the temperature derivative of saturation pressure of the IAPWS-92 equation.

$$P_{sat} = P_c \exp\left(\frac{T_c}{T}[a_1\tau + a_2\tau^{1.5} + a_3\tau^3 + a_4\tau^{3.5} a_5\tau^4 + a_6\tau^{7.5}]\right)$$

Parameters

Tfloat

Temperature at which to calculate the saturation condition and its temperature derivative, [K]

Returns

dPsat_dTfloat

First temperature derivative of saturation vapor pressure, [Pa/K]

Psatfloat

Saturation vapor pressure, [Pa]

Notes

The coefficients are [-7.85951783, 1.84408259, -11.7866497, 22.6807411, -15.9618719, 1.80122502]

Examples

>>> iapws92_dPsat_dT(400.0)
(7483.47094105, 245765.263541)

chemicals.iapws.iapws95_Tsat(P)

Compute the saturation temperature of the IAPWS-95 equation. The range of the fit is 235 K to 647.096 K, the critical point.

Parameters

Psatfloat

Saturation vapor pressure specified, [Pa]

Returns

Tfloat

Temperature at which the saturation pressure occurs, [K]

See also

iapws95_Psat

Tsat_IAPWS

Notes

This method is quite fast and precise because it starts with great initial guesses and the equation is well-bounded. The precision of this calculation should be the same as iapws95_Psat.

Examples

>>> iapws95_Tsat(iapws95_Psat(400.0))
400.0

chemicals.iapws.iapws95_saturation(T, xtol=1e-05, rhol_guess=None, rhog_guess=None)

Solve the vapor-liquid saturation equations of IAPWS-95 given a specified temperature. With floating point numbers, the achievable tolerance is somewhat low so xtol is exposed as a setting - it can be adjusted somewhat. Density guesses may be provided, otherwise they will be estimated.

$$G_{liq}(T, \rho_l) = G_{vap}(T, \rho_g)$$

$$P_{liq}(T, \rho_l) = P_{vap}(T, \rho_g)$$

Parameters

Tfloat

Temperature at which to solve for saturation condition, [K]

xtolfloat

Tolerance for solver, [-]

rhol_guessfloat, optional

Liquid density of water at saturation (guess), [kg/m^3]

rhog_guessfloat, optional

Vapor density of water at saturation (guess), [kg/m^3]

Returns

Psatfloat

Saturation vapor pressure, 3[Pa]

rholfloat

Saturation liquid water density, [kg/m^3]

rhogfloat

Saturation vapor water density, [kg/m^3]

Notes

This is not a perfect function.

With mpmath multiple precision, the equation can be solved down to 233.6 K and up to 647.095999995 K - within 10 parts in a billion of the critical point exactly.

Reasons for non-convergence include floating point issues as delta becomes 1, and zero division errors in the matrix inverse.

Examples

>>> iapws95_saturation(400.0, xtol=1e-6)
(245769.345, 937.4860, 1.3694075)
>>> iapws95_saturation(647.0955, xtol=1e-7)
(22063866.35, 325.70, 318.277)

chemicals.iapws.iapws11_Psub(T)

Compute the sublimation pressure of the frozen water using the IAPWS-11 equation from the Revised Release on the Pressure along the Melting and Sublimation Curves of Ordinary Water Substance.

$$P_{sub} = P_t\exp\left( \theta^{-1} \sum_{i=1}^3 a_i \theta^{b_i} \right)$$

$$\theta = \frac{T}{T_t}$$

Parameters

Tfloat

Temperature at which to calculate the sublimation condition [K]

Returns

Psubfloat

Sublimation vapor pressure, [Pa]

Notes

The triple temperature is 273.16 K, and triple pressure 611.657 Pa.

The coefficients are as follows:

ais = [-0.212144006E2, 0.273203819E2, -0.610598130E1]

bis = [0.333333333E-2, 0.120666667E1, 0.170333333E1]

The equation is valid from 50 K to the triple temperature.

Examples

>>> iapws11_Psub(230.0)
8.947352740189151

IAPWS Saturation Density

chemicals.iapws.iapws95_rhol_sat(T)

Compute the saturation liquid density of the IAPWS-95 equation using high- fidelity polynomial fits. These have a relative accuracy of under 1e-13, except near the critical point where it rises to 1e-10, and are generated by solving the saturation equations under the high-precision environment of mpmath. The range of the fit is 235 K to 647.096 K, the critical point.

Parameters

Tfloat

Temperature at which to calculate the saturation condition, [K]

Returns

rholfloat

Saturation liquid density, [kg/m^3]

See also

iapws92_rhol_sat

Notes

This method should be used in preference to iapws92_rhol_sat.

Examples

>>> iapws95_rhol_sat(400.0)
937.48603939

chemicals.iapws.iapws95_rhog_sat(T)

Compute the saturation vapor density of the IAPWS-95 equation using high- fidelity polynomial fits. These have a relative accuracy of under 1e-13, except near the critical point where it rises to 1e-10, and are generated by solving the saturation equations under the high-precision environment of mpmath. The range of the fit is 235 K to 647.096 K, the critical point.

Parameters

Tfloat

Temperature at which to calculate the saturation condition, [K]

Returns

rholfloat

Saturation vapor density, [kg/m^3]

See also

iapws92_rhog_sat

Notes

This method should be used in preference to iapws92_rhog_sat.

Examples

>>> iapws95_rhog_sat(400.0)
1.3694075410

chemicals.iapws.iapws95_drhol_sat_dT(T)

Compute the first temperature derivative of saturation liquid density of the IAPWS-95 equation using high-fidelity polynomial fits. The actual saturated liquid density is returned as well.

The range of the fit is 235 K to 647.096 K, the critical point.

Parameters

Tfloat

Temperature at which to calculate the saturation condition and its derivative, [K]

Returns

drhol_dTfloat

First temperature derivative of saturation liquid density, [kg/(m^3*K)]

rholfloat

Saturation liquid density, [kg/m^3]

Examples

>>> iapws95_drhol_sat_dT(400.0)
(-0.835194603380, 937.486039392)

chemicals.iapws.iapws92_rhol_sat(T)

Calculates saturation liquid mass density of water using the IAPWS SR1-86(1992) [1] [2] explicit equation.

$$\frac{\rho^{sat}_l}{\rho_c} = 1 + b_1\tau^{1/3} + b_2\tau^{2/3} + b_3 \tau^{5/3} + b_4\tau^{16/3} + b_5\tau^{43/3} + b_6\tau^{110/3}$$

$$\tau = 1 - \frac{T}{T_c}$$

Parameters

Tfloat

Temperature of water, [K]

Returns

rhol_satfloat

Saturation liquid mass density of water [kg/m^3]

See also

iapws95_rhol_sat

Notes

This equation is fit to experimental data to within its accuracy. It does not satisfy the equilibrium conditions for the IAPWS-95 or IAPWS-97 formulations.

The values of the constants are as follows:

b1 = 1.99274064; b2 = 1.09965342; b3 = -0.510839303; b4 = -1.75493479; b5 = -45.5170352; b6 = -6.74694450e5

References

1

IAPWS, Secretariat, B Dooley, and EPRI. “Revised Supplementary Release on Saturation Properties of Ordinary Water Substance”, 1992.

2

Wagner, Wolfgang, and A. Pruss. “International Equations for the Saturation Properties of Ordinary Water Substance. Revised According to the International Temperature Scale of 1990. Addendum to J. Phys. Chem. Ref. Data 16, 893 (1987).” Journal of Physical and Chemical Reference Data 22, no. 3 (May 1, 1993): 783-87. https://doi.org/10.1063/1.555926.

Examples

>>> iapws92_rhol_sat(300.)
996.5089712803

chemicals.iapws.iapws92_rhog_sat(T)

Calculates saturation vapor mass density of water using the IAPWS SR1-86(1992) [1] [2] explicit equation.

$$\ln \left(\frac{\rho^{sat}_g}{\rho_c}\right) = 1 + c_1\tau^{2/6} + c_2\tau^{4/6} + c_3 \tau^{8/6} + c_4\tau^{18/6} + c_5\tau^{37/6} + c_6\tau^{71/6}$$

$$\tau = 1 - \frac{T}{T_c}$$

Parameters

Tfloat

Temperature of water, [K]

Returns

rhog_satfloat

Saturation vapor mass density of water [kg/m^3]

See also

iapws95_rhog_sat

Notes

This equation is fit to experimental data to within its accuracy. It does not satisfy the equilibrium conditions for the IAPWS-95 or IAPWS-97 formulations.

The values of the constants are as follows:

c1 = -2.03150240; c2 = -2.68302940; c3 = -5.38626492; c4 = -17.2991605; c5 = -44.7586581; c6 = -63.9201063

References

1

IAPWS, Secretariat, B Dooley, and EPRI. “Revised Supplementary Release on Saturation Properties of Ordinary Water Substance”, 1992.

2

Wagner, Wolfgang, and A. Pruss. “International Equations for the Saturation Properties of Ordinary Water Substance. Revised According to the International Temperature Scale of 1990. Addendum to J. Phys. Chem. Ref. Data 16, 893 (1987).” Journal of Physical and Chemical Reference Data 22, no. 3 (May 1, 1993): 783-87. https://doi.org/10.1063/1.555926.

Examples

>>> iapws92_rhog_sat(300.)
0.0255887212886

IAPWS Constants

chemicals.iapws.iapws95_Tc = 647.096

Critical temperature of water in K according to IAPWS-95, also used in IAPWS-97

chemicals.iapws.iapws95_Pc = 22064000.0

Critical pressure of water in Pa according to IAPWS-95, also used in IAPWS-97

chemicals.iapws.iapws95_rhoc = 322.0

Critical density of water in kg/m^3 according to IAPWS-95, also used in IAPWS-97

chemicals.iapws.iapws95_MW = 18.015268

Molecular weight of water in g/mol according to IAPWS-95, also used in IAPWS-97

chemicals.iapws.iapws95_R = 461.51805

Specific gas constant in J/(kg*K) according to IAPWS-95

chemicals.iapws.iapws97_R = 461.526

Specific gas constant in J/(kg*K) according to IAPWS-97

chemicals.iapws.iapws95_Tt = 273.16

Triple temperature of water in K according to IAPWS

IAPWS-97 Region 1

chemicals.iapws.iapws97_G_region1(tau, pi)

Calculates the dimensionless Gibbs free energy for water according to the IAPWS-97 standard (for region 1).

$$\gamma = \sum_{i=1}^{34} I_i(7.1-\pi)^{I_i}(\tau - 1.222)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (1386 K)/T [-]

pifloat

Dimensionless pressure, P/(16.53 MPa), [-]

Returns

Gfloat

Dimensionless Gibbs energy G/(RT), [-]

Examples

>>> iapws97_G_region1(1386/277.15, 101325/16.53E6)
-0.00016341033954414

chemicals.iapws.iapws97_dG_dpi_region1(tau, pi)

Calculates the derivative of dimensionless Gibbs free energy with respect to pi for water according to the IAPWS-97 standard (for region 1).

$$\frac{\partial \gamma}{\partial \pi} = \sum_{i=1}^{34} -n_i I_i(7.1-\pi)^{I_i-1}(\tau - 1.222)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (1386 K)/T [-]

pifloat

Dimensionless pressure, P/(16.53 MPa), [-]

Returns

dG_dpifloat

Derivative of dimensionless Gibbs energy G/(RT) with respect to pi, [-]

Notes

Used in density solution. This contains a hand-optimized implementation with a single division, no power operations, 65 multiplications, 16 local variables, and a minimum number of additions.

Examples

>>> iapws97_dG_dpi_region1(1386/277.15, 101325/16.53E6)
0.1292327182544

chemicals.iapws.iapws97_d2G_dpi2_region1(tau, pi)

Calculates the second derivative of dimensionless Gibbs free energy with respect to pi for water according to the IAPWS-97 standard (for region 1).

$$\frac{\partial^2 \gamma}{\partial \pi^2} = \sum_{i=1}^{34} n_i I_i(I_i-1)(7.1-\pi)^{I_i-2}(\tau - 1.222)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (1386 K)/T [-]

pifloat

Dimensionless pressure, P/(16.53 MPa), [-]

Returns

d2G_dpi2float

Second Derivative of dimensionless Gibbs energy G/(RT) with respect to pi, [-]

Examples

>>> iapws97_d2G_dpi2_region1(1386/277.15, 101325/16.53E6)
 -0.0010570100274769

chemicals.iapws.iapws97_dG_dtau_region1(tau, pi)

Calculates the derivative of dimensionless Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 1).

$$\frac{\partial \gamma}{\partial \tau} = \sum_{i=1}^{34} n_i(7.1-\pi)^{I_i}J_i(\tau - 1.222)^{J_i-1}$$

Parameters

taufloat

Dimensionless temperature, (1386 K)/T [-]

pifloat

Dimensionless pressure, P/(16.53 MPa), [-]

Returns

dG_dtaufloat

Derivative of dimensionless Gibbs energy G/(RT) with respect to tau, [-]

Examples

>>> iapws97_dG_dtau_region1(1386/277.15, 101325/16.53E6)
0.026440334282967

chemicals.iapws.iapws97_d2G_dtau2_region1(tau, pi)

Calculates the second derivative of dimensionless Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 1).

$$\frac{\partial^2 \gamma}{\partial \tau^2} = \sum_{i=1}^{34} n_i(7.1-\pi)^{I_i}J_i(J_i-1)(\tau - 1.222)^{J_i-2}$$

Parameters

taufloat

Dimensionless temperature, (1386 K)/T [-]

pifloat

Dimensionless pressure, P/(16.53 MPa), [-]

Returns

d2G_dtau2float

Second Derivative of dimensionless Gibbs energy G/(RT) with respect to tau, [-]

Examples

>>> iapws97_d2G_dtau2_region1(1386/277.15, 101325/16.53E6)
-0.3645169808573

chemicals.iapws.iapws97_d2G_dpidtau_region1(tau, pi)

Calculates the second derivative of dimensionless Gibbs free energy with respect to tau and pi for water according to the IAPWS-97 standard (for region 1).

$$\frac{\partial^2 \gamma}{\partial \tau \partial \pi} = \sum_{i=1}^{34} -n_iI_i(7.1-\pi)^{I_i}J_i(\tau - 1.222)^{J_i-1}$$

Parameters

taufloat

Dimensionless temperature, (1386 K)/T [-]

pifloat

Dimensionless pressure, P/(16.53 MPa), [-]

Returns

d2G_dpidtaufloat

Second Derivative of dimensionless Gibbs energy G/(RT) with respect to tau and pi, [-]

Examples

>>> iapws97_d2G_dpidtau_region1(1386/277.15, 101325/16.53E6)
0.025837659858819

IAPWS-97 Region 2

chemicals.iapws.iapws97_G0_region2(tau, pi)

Calculates the dimensionless ideal gas Gibbs free energy for water according to the IAPWS-97 standard (for region 2).

$$\gamma^\circ = \ln \pi + \sum_{i=1}^9 n_i^\circ \tau^{J_i^\circ}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

G0float

Dimensionless ideal gas Gibbs energy G0/(RT), [-]

Examples

>>> iapws97_G0_region2(540/300.0, 101325/1e6)
3.3180953922351

chemicals.iapws.iapws97_dG0_dtau_region2(tau, pi)

Calculates the first derivative of dimensionless ideal gas Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 2).

$$\frac{\partial \gamma^\circ}{\partial \tau} =\sum_{i=1}^9 n_i^\circ J_i^\circ\tau^{J_i^\circ-1}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

dG0_dtaufloat

First derivative of dimensionless ideal gas Gibbs energy G0/(RT) with respect to tau, [-]

Notes

This function does not depend on pi but it is accepted for consistency.

Examples

>>> iapws97_dG0_dtau_region2(540/300.0, 101325/1e6)
10.2374188173906

chemicals.iapws.iapws97_d2G0_dtau2_region2(tau, pi)

Calculates the second derivative of dimensionless ideal gas Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 2).

$$\frac{\partial^2 \gamma^\circ}{\partial \tau^2} =\sum_{i=1}^9 n_i^\circ J_i^\circ( J_i^\circ-1)\tau^{J_i^\circ-2}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2G0_dtau2float

Second derivative of dimensionless ideal gas Gibbs energy G0/(RT) with respect to tau, [-]

Notes

This function does not depend on pi but it is accepted for consistency.

Examples

>>> iapws97_d2G0_dtau2_region2(540/300.0, 101325/1e6)
-1.2472096479372

chemicals.iapws.iapws97_Gr_region2(tau, pi)

Calculates the dimensionless residual Gibbs free energy for water according to the IAPWS-97 standard (for region 2).

$$\gamma^r = \sum_{i=1}^{43} n_i \pi^{I_i} (\tau - 0.5)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

Grfloat

Dimensionless residual Gibbs energy Gr/(RT), [-]

Examples

>>> iapws97_Gr_region2(540/300.0, 101325/1e6)
-0.71851548053980

chemicals.iapws.iapws97_dGr_dpi_region2(tau, pi)

Calculates the first derivative of dimensionless residual Gibbs free energy with respect to pi for water according to the IAPWS-97 standard (for region 2).

$$\frac{\partial \gamma^r}{\partial \pi} = \sum_{i=1}^{43} n_i I_i \pi^{I_i-1} (\tau - 0.5)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

dGr_dpifloat

Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to pi, [-]

Notes

Used in density solution.

Examples

>>> iapws97_dGr_dpi_region2(540/300.0, 101325/1e6)
-27.7714056629532

chemicals.iapws.iapws97_d2Gr_dpi2_region2(tau, pi)

Calculates the second derivative of dimensionless residual Gibbs free energy with respect to pi for water according to the IAPWS-97 standard (for region 2).

$$\frac{\partial^2 \gamma^r}{\partial \pi^2} = \sum_{i=1}^{43} n_i I_i (I_i-1)\pi^{I_i-2} (\tau - 0.5)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2Gr_dpi2float

Second Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to pi, [-]

Examples

>>> iapws97_d2Gr_dpi2_region2(540/300.0, 101325/1e6)
-983.15187604898

chemicals.iapws.iapws97_dGr_dtau_region2(tau, pi)

Calculates the first derivative of dimensionless residual Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 2).

$$\frac{\partial \gamma^r}{\partial \tau} = \sum_{i=1}^{43} n_i \pi^{I_i} J_i (\tau - 0.5)^{J_i-1}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

dGr_dtaufloat

Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to tau, [-]

Examples

>>> iapws97_dGr_dtau_region2(540/300.0, 101325/1e6)
-18.1535856049444

chemicals.iapws.iapws97_d2Gr_dtau2_region2(tau, pi)

Calculates the second derivative of dimensionless residual Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 2).

$$\frac{\partial^2 \gamma^r}{\partial \tau^2} = \sum_{i=1}^{43} n_i \pi^{I_i} J_i (J_i-1) (\tau - 0.5)^{J_i-2}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2Gr_dtau2float

Second derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to tau, [-]

Examples

>>> iapws97_d2Gr_dtau2_region2(540/300.0, 101325/1e6)
-470.9302933324787

chemicals.iapws.iapws97_d2Gr_dpidtau_region2(tau, pi)

Calculates the second derivative of dimensionless residual Gibbs free energy with respect to tau and pi for water according to the IAPWS-97 standard (for region 2).

$$\frac{\partial^2 \gamma^r}{\partial \tau \partial \pi} = \sum_{i=1}^{43} n_i I_i \pi^{I_i-1} J_i (\tau - 0.5)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2Gr_dpidtau_float

Second derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to tau and pi, [-]

Examples

>>> iapws97_d2Gr_dpidtau_region2(540/300.0, 101325/1e6)
-735.391845360247

IAPWS-97 Region 3

chemicals.iapws.iapws97_A_region3(tau, delta)

Calculates the dimensionless Helmholtz free energy for water according to the IAPWS-97 standard (for region 3).

$$\frac{f(\rho, T)}{RT} = \phi(\delta, \tau) = n_1\ln\delta + \sum_{i=2}^{40} n_i \delta^{I_i}\tau^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

Afloat

Helmholtz free energy A/(RT), [-]

Examples

>>> iapws97_A_region3(647.096/500.0, 400.0/322.0)
-3.0336402168865

chemicals.iapws.iapws97_dA_ddelta_region3(tau, delta)

Calculates the derivative of dimensionless Helmholtz free energy with respect to delta for water according to the IAPWS-97 standard (for region 3).

$$\frac{\partial \phi(\delta, \tau)}{\partial \delta} = \frac{n_1}{\delta} + \sum_{i=2}^{40} n_i I_i \delta^{I_i-1}\tau^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

dA_ddeltafloat

Derivative of dimensionless Helmholtz free energy with respect to delta, [-]

Examples

>>> iapws97_dA_ddelta_region3(647.096/500.0, 400.0/322.0)
7.35562435092

chemicals.iapws.iapws97_d2A_ddelta2_region3(tau, delta)

Calculates the second derivative of dimensionless Helmholtz free energy with respect to delta for water according to the IAPWS-97 standard (for region 3).

$$\frac{\partial^2 \phi(\delta, \tau)}{\partial \delta^2} = \frac{-n_1}{\delta^2} + \sum_{i=2}^{40} n_i I_i (I_i-1)\delta^{I_i-2}\tau^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d2A_ddelta2float

Second derivative of dimensionless Helmholtz free energy with respect to delta, [-]

Examples

>>> iapws97_d2A_ddelta2_region3(647.096/500.0, 400.0/322.0)
-2.2858869882497

chemicals.iapws.iapws97_dA_dtau_region3(tau, delta)

Calculates the derivative of dimensionless Helmholtz free energy with respect to tau for water according to the IAPWS-97 standard (for region 3).

$$\frac{\partial \phi(\delta, \tau)}{\partial \tau} = + \sum_{i=2}^{40} n_i J_i \delta^{I_i}\tau^{J_i-1}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

dA_dtaufloat

Derivative of dimensionless Helmholtz free energy with respect to tau, [-]

Examples

>>> iapws97_dA_dtau_region3(647.096/500.0, 400.0/322.0)
-24.9687028688

chemicals.iapws.iapws97_d2A_dtau2_region3(tau, delta)

Calculates the second derivative of dimensionless Helmholtz free energy with respect to tau for water according to the IAPWS-97 standard (for region 3).

$$\frac{\partial^2 \phi(\delta, \tau)}{\partial \tau^2} = + \sum_{i=2}^{40} n_i J_i (J_i-1)\delta^{I_i}\tau^{J_i-2}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d2A_dtau2float

Second derivative of dimensionless Helmholtz free energy with respect to tau, [-]

Examples

>>> iapws97_d2A_dtau2_region3(647.096/500.0, 400.0/322.0)
-373.6565823701

chemicals.iapws.iapws97_d2A_ddeltadtau_region3(tau, delta)

Calculates the second derivative of dimensionless Helmholtz free energy with respect to tau and delta for water according to the IAPWS-97 standard (for region 3).

$$\frac{\partial^2 \phi(\delta, \tau)}{\partial \tau \partial \delta} = + \sum_{i=2}^{40} n_i J_i \delta^{I_i-1}\tau^{J_i-1}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d2A_ddeltadtaufloat

Second derivative of dimensionless Helmholtz free energy with respect to tau and delta, [-]

Examples

>>> iapws97_d2A_ddeltadtau_region3(647.096/500.0, 400.0/322.0)
145.85190014717

IAPWS-97 Region 3 PT Backwards Equation Boundaries

chemicals.iapws.iapws97_boundary_3uv(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3uv(22.3E6)
647.7996121480069

chemicals.iapws.iapws97_boundary_3ef(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3ef(40E6)
713.959399239744

chemicals.iapws.iapws97_boundary_3cd(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3cd(25E6)
649.3659208321279

chemicals.iapws.iapws97_boundary_3gh(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3gh(25E6)
656.69805722612

chemicals.iapws.iapws97_boundary_3ij(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3ij(25E6)
660.7865756716819

chemicals.iapws.iapws97_boundary_3jk(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3jk(25E6)
668.1915358826951

chemicals.iapws.iapws97_boundary_3mn(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3mn(22.8E6)
649.6054132953997

chemicals.iapws.iapws97_boundary_3qu(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3qu(22E6)
645.6355027340121

chemicals.iapws.iapws97_boundary_3rx(P)

Calculates the transition temperature for a region 3 PT backwards equation transition.

Parameters

Pfloat

Pressure [Pa]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3rx(22E6)
648.26227536701

chemicals.iapws.iapws97_boundary_3wx(logP_MPa, logP_MPa_inv)

Calculates the transition temperature for a region 3 PT backwards equation transition (for one of “wx”, “ab”, or “op”; the others do not use a log fit). The parameters are provided in the specific units for speed savings only.

Parameters

logP_MPafloat

Natural logarithm of pressure in units of MPa [log(MPa)]

logP_MPa_invfloat

Inverse of Natural logarithm of pressure in units of MPa [1/log(MPa)]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3wx(log(22.3), 1/log(22.3))
648.204947950734

chemicals.iapws.iapws97_boundary_3ab(logP_MPa, logP_MPa_inv)

Calculates the transition temperature for a region 3 PT backwards equation transition (for one of “wx”, “ab”, or “op”; the others do not use a log fit). The parameters are provided in the specific units for speed savings only.

Parameters

logP_MPafloat

Natural logarithm of pressure in units of MPa [log(MPa)]

logP_MPa_invfloat

Inverse of Natural logarithm of pressure in units of MPa [1/log(MPa)]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3ab(log(40), 1/log(40))
693.0341408296053

chemicals.iapws.iapws97_boundary_3op(logP_MPa, logP_MPa_inv)

Calculates the transition temperature for a region 3 PT backwards equation transition (for one of “wx”, “ab”, or “op”; the others do not use a log fit). The parameters are provided in the specific units for speed savings only.

Parameters

logP_MPafloat

Natural logarithm of pressure in units of MPa [log(MPa)]

logP_MPa_invfloat

Inverse of Natural logarithm of pressure in units of MPa [1/log(MPa)]

Returns

T_transfloat

Transition temperature [K]

Examples

>>> iapws97_boundary_3op(log(22.8), 1/log(22.8))
650.010694314133

IAPWS-97 Region 3 PT Backwards Equations

chemicals.iapws.iapws97_region3_a(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_b(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_c(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_d(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_e(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_f(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_g(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_h(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_i(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_j(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_k(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_l(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_m(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_n(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_o(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_p(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_q(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_r(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_s(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_t(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_u(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_v(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_w(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_x(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_y(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

chemicals.iapws.iapws97_region3_z(T, P)

Calculate the mass density water in one of the 26 region 3 backwards regions of the IAPWS-97 standard.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Returns

rhofloat

Mass density of water in region 3, [kg/m^3]

Notes

Significant discontinuities exist between each region. These functions are automatically generated and are not to be edited directly.

IAPWS-97 Region 5

chemicals.iapws.iapws97_G0_region5(tau, pi)

Calculates the dimensionless ideal gas Gibbs free energy for water according to the IAPWS-97 standard (for region 5).

$$\gamma^\circ = \ln \pi + \sum_{i=1}^6 n_i^\circ \tau^{J_i^\circ}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

G0float

Dimensionless ideal gas Gibbs energy G/(RT), [-]

Examples

>>> iapws97_G0_region5(1000.0/1500, 101325/1e6)
-14.9741430290056

chemicals.iapws.iapws97_dG0_dtau_region5(tau, pi)

Calculates the first derivative of dimensionless ideal gas Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 5).

$$\frac{\partial \gamma^\circ}{\partial \tau} =\sum_{i=1}^6 n_i^\circ J_i^\circ\tau^{J_i^\circ-1}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

dG0_dtaufloat

First derivative of dimensionless ideal gas Gibbs energy G/(RT) with respect to tau, [-]

Notes

This function does not depend on pi but it is accepted for consistency.

Examples

>>> iapws97_dG0_dtau_region5(1000.0/1500, 101325/1e6)
11.311766995978

chemicals.iapws.iapws97_d2G0_dtau2_region5(tau, pi)

Calculates the second derivative of dimensionless ideal gas Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 5).

$$\frac{\partial^2 \gamma^\circ}{\partial \tau^2} =\sum_{i=1}^6 n_i^\circ J_i^\circ( J_i^\circ-1)\tau^{J_i^\circ-2}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2G0_dtau2float

Second derivative of dimensionless ideal gas Gibbs energy G/(RT) with respect to tau, [-]

Notes

This function does not depend on pi but it is accepted for consistency.

Examples

>>> iapws97_d2G0_dtau2_region5(1000.0/1500, 101325/1e6)
-12.744650271463655

chemicals.iapws.iapws97_Gr_region5(tau, pi)

Calculates the dimensionless residual Gibbs free energy for water according to the IAPWS-97 standard (for region 5).

$$\gamma^r = \sum_{i=1}^{6} n_i \pi^{I_i} (\tau)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

Grfloat

Dimensionless residual Gibbs energy Gr/(RT), [-]

Examples

>>> iapws97_Gr_region5(1000/300.0, 101325/1e6)
-0.0194648291645718

chemicals.iapws.iapws97_dGr_dpi_region5(tau, pi)

Calculates the first derivative of dimensionless residual Gibbs free energy with respect to pi for water according to the IAPWS-97 standard (for region 5).

$$\frac{\partial \gamma^r}{\partial \pi} = \sum_{i=1}^{6} n_i I_i \pi^{I_i-1} (\tau)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

dGr_dpifloat

Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to pi, [-]

Notes

Used in density solution.

Examples

>>> iapws97_dGr_dpi_region5(1000/300.0, 101325/1e6)
-0.213281155629998

chemicals.iapws.iapws97_d2Gr_dpi2_region5(tau, pi)

Calculates the second derivative of dimensionless residual Gibbs free energy with respect to pi for water according to the IAPWS-97 standard (for region 5).

$$\frac{\partial^2 \gamma^r}{\partial \pi^2} = \sum_{i=1}^{6} n_i I_i (I_i-1) \pi^{I_i-2} (\tau)^{J_i}$$

Parameters

taufloat

Dimensionless temperature, (540 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2Gr_dpi2float

Second derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to pi, [-]

Examples

>>> iapws97_d2Gr_dpi2_region5(1000/300.0, 101325/1e6)
-0.4179905782304291

chemicals.iapws.iapws97_dGr_dtau_region5(tau, pi)

Calculates the first derivative of dimensionless residual Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 5).

$$\frac{\partial \gamma^r}{\partial \tau} = \sum_{i=1}^{6} n_i \pi^{I_i} J_i(\tau)^{J_i-1}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

dGr_dtaufloat

Derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to tau, [-]

Examples

>>> iapws97_dGr_dtau_region5(1000/300.0, 101325/1e6)
-0.02200629869194

chemicals.iapws.iapws97_d2Gr_dtau2_region5(tau, pi)

Calculates the second derivative of dimensionless residual Gibbs free energy with respect to tau for water according to the IAPWS-97 standard (for region 5).

$$\frac{\partial^2 \gamma^r}{\partial \tau^2} = \sum_{i=1}^{6} n_i \pi^{I_i} J_i(J_i-1)(\tau)^{J_i-2}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2Gr_dtau2float

Second derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to tau, [-]

Examples

>>> iapws97_d2Gr_dtau2_region5(1000/300.0, 101325/1e6)
-0.0239165867999155

chemicals.iapws.iapws97_d2Gr_dpidtau_region5(tau, pi)

Calculates the second derivative of dimensionless residual Gibbs free energy with respect to tau and pi for water according to the IAPWS-97 standard (for region 5).

$$\frac{\partial^2 \gamma^r}{\partial \tau \partial \pi} = \sum_{i=1}^{6} n_i I_i \pi^{I_i-1} J_i(\tau)^{J_i-1}$$

Parameters

taufloat

Dimensionless temperature, (1000 K)/T [-]

pifloat

Dimensionless pressure, P/(1 MPa), [-]

Returns

d2Gr_dpidtaufloat

Second derivative of dimensionless residual Gibbs energy Gr/(RT) with respect to tau and pi, [-]

Examples

>>> iapws97_d2Gr_dpidtau_region5(1000/300.0, 101325/1e6)
-0.27438379131103097

IAPWS-95 Ideal Gas Terms

chemicals.iapws.iapws95_A0(tau, delta)

Calculates the ideal gas Helmholtz energy of water according to the IAPWS-95 standard.

$$\phi^\circ = \ln \delta + n_1 + n_2\tau + n_3\ln \tau + \sum_{i=4}^8 n_i \ln \left[1 - \exp(-\gamma_i \tau) \right]$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

A0float

Ideal gas dimensionless Helmholtz energy A/(RT) [-]

Notes

This implementation is checked to have a relative error always under 1e-15.

Examples

>>> iapws95_A0(647.096/300.0, 999.0/322)
9.537075529761053

chemicals.iapws.iapws95_dA0_dtau(tau, delta)

Calculates the first derivative of ideal gas Helmholtz energy of water with respect to tau according to the IAPWS-95 standard.

$$\frac{\partial \phi^\circ}{\partial \tau} = n_2 + \frac{n_3}{\tau} + \sum_{i=4}^8 n_i\gamma_i \left[\left(1-\exp(-\gamma_i \tau) \right)^{-1} - 1 \right]$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

dA0_dtaufloat

First derivative of ideal gas dimensionless Helmholtz energy A/(RT) with respect to tau [-]

Notes

This implementation is checked to have a relative error always under 1e-15.

Examples

>>> iapws95_dA0_dtau(647.096/300.0, 999.0/322)
8.079705548882

chemicals.iapws.iapws95_d2A0_dtau2(tau, delta)

Calculates the second derivative of ideal gas Helmholtz energy of water with respect to tau according to the IAPWS-95 standard.

$$\frac{\partial^2 \phi^\circ}{\partial \tau^2} = \frac{n_3}{\tau^2} + \sum_{i=4}^8 n_i\gamma_i ^2 \exp(-\gamma_i \tau) \left[\left(1-\exp(-\gamma_i \tau) \right)^{-2}\right]$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d2A0_dtau2float

Second derivative of ideal gas dimensionless Helmholtz energy A/(RT) with respect to tau [-]

Notes

This implementation is checked to have a relative error always under 1e-15.

Examples

>>> iapws95_d2A0_dtau2(647.096/300.0, 999.0/322)
-0.653543047751809

chemicals.iapws.iapws95_d3A0_dtau3(tau, delta)

Calculates the third derivative of ideal gas Helmholtz energy of water with respect to tau according to the IAPWS-95 standard.

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d3A0_dtau3float

Third derivative of ideal gas dimensionless Helmholtz energy A/(RT) with respect to tau [-]

Notes

This implementation is checked to have a relative error always under 1e-15. This equation is not explicitly in IAPWS-95, but is needed to compute some second derivatives.

Examples

>>> iapws95_d3A0_dtau3(647.096/300.0, 999.0/322)
0.6222542507278

chemicals.iapws.iapws95_A0_tau_derivatives(tau, delta)

Calculates the ideal gas Helmholtz energy of water and its first three derivatives with respect to tau according to the IAPWS-95 standard. As each of those calls spends most of their time computing exponentials which are the same for each function, function offers a time saving.

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

A0float

Ideal gas dimensionless Helmholtz energy A/(RT) [-]

dA0_dtaufloat

First derivative of ideal gas dimensionless Helmholtz energy A/(RT) with respect to tau [-]

d2A0_dtau2float

Second derivative of ideal gas dimensionless Helmholtz energy A/(RT) with respect to tau [-]

d3A0_dtau3float

Third derivative of ideal gas dimensionless Helmholtz energy A/(RT) with respect to tau [-]

Notes

The extra cost of calling this function vs iapws95_A0 alone is ~15% with numba, ~40% with PyPy, and 120% with CPython.

Examples

>>> iapws95_A0_tau_derivatives(647.096/300.0, 999.0/322)
(9.53707552976, 8.0797055488, -0.65354304775, 0.62225425072)

IAPWS-95 Residual Terms

chemicals.iapws.iapws95_Ar(tau, delta)

Calculates the residual Helmholtz energy of water according to the IAPWS-95 standard.

$$\phi^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} \delta^{d_{i}} \tau^{t_{i}}+\sum_{i=8}^{51} n_{i} \delta^{d_{i}} \tau^{t_{i}} \mathrm{e}^{-\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}} \tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i} \right)^{2}-\beta_{i}\left(\tau-\gamma_{i}\right)^{2}}+\sum_{i=55}^{56} n_{i} \Delta^{b_{i}} \delta \psi$$

$$\Delta=\theta^{2}+B_{i}\left[(\delta-1)^{2}\right]^{a_{i}}$$

$$\theta=(1-\tau)+A_{i}\left[(\delta-1)^{2}\right]^{\frac{1}{2 \beta_{i}}}$$

$$\psi=e^{-C_{i}(\delta-1)^{2}-D_{i}(\tau-1)^{2}}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

Arfloat

Residual Helmholtz energy A/(RT) [-]

Notes

This is an optimized implementatation taking 9 exp calls, 4 sqrts, and 3 powers. It was generated using SymPy’s CSE functionality, with select polynomial optimizations by hand as well. It is over 10x faster than a naive implementation.

This implementation has been tested against a straightforward implementation with the equations given in IAPWS-95.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative error was 5.0416E-15, with a maximum relative error of 1.118E-9 and a standard deviation of 5.773e-13.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 90000 points, the mean relative error was 3.14E-15, with a maximum relative error of 3.54e-12 and a standard deviation of 3.017E-14.

This comparison indicates that this implementation is more accurate than the straightforward implementation.

Examples

>>> iapws95_Ar(647.096/300.0, 999.0/322)
-9.57577716026768

chemicals.iapws.iapws95_dAr_ddelta(tau, delta)

Calculates the first derivative of residual Helmholtz energy of water with respect to delta according to the IAPWS-95 standard.

$$\phi_{\delta}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} d_{i} \delta^{d_{i}-1} \tau^{t_{i}}+\sum_{i=8}^{51} n_{i} \mathrm{e}^{-\delta^{c_{i}}}\left[ \delta^{d_{i}-1} \tau^{t_{i}}\left(d_{i}-c_{i} \delta^{c_{i}}\right) \right]+\sum_{i=52}^{54} n_{i} \delta^{d_{i}} \tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i}\right)^{2}-\beta_{i} \left(\tau-\gamma_{i}\right)^{2}}\left[\frac{d_{i}}{\delta}-2 \alpha_{i} \left(\delta-\varepsilon_{i}\right)\right]+\sum_{i=55}^{56} n_{i} \left[\Delta^{b_{i}}\left(\psi+\delta \frac{\partial \psi}{\partial \delta}\right)+\frac{\partial \Delta^{b_{i}}}{\partial \delta} \delta \psi\right]$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

dAr_ddeltafloat

First derivative of residual Helmholtz energy A/(RT) with respect to delta, [-]

Notes

This is an optimized implementatation taking 8 exp calls, 4 sqrts, and 2 powers. It was generated using SymPy’s CSE functionality, with select polynomial optimizations by hand as well. It is over 10x faster than a naive implementation.

This implementation has been tested against a straightforward implementation with the equations given in IAPWS-95.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative error was 4.033E-15, with a maximum relative error of 3.8765e-10 and a standard deviation of 3.189e-13.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 90000 points, the mean relative error was 6.046E-15, with a maximum relative error of 3.39E-10 and a standard deviation of 7.056E-13.

There was a singularity at tau = delta = 1, but the limit is correctly returned.

Examples

>>> iapws95_dAr_ddelta(647.096/300.0, 999.0/322)
-0.3093321202374

chemicals.iapws.iapws95_d2Ar_ddelta2(tau, delta)

Calculates the second derivative of residual Helmholtz energy of water with respect to delta according to the IAPWS-95 standard.

$$\begin{aligned} \phi_{\delta \delta}^{\mathrm{r}}=& \sum_{i=1}^{7} n_{i} d_{i}\left(d_{i}-1\right) \delta^{d_{i}-2} \tau^{t_{i}} +\sum_{i=8}^{51} n_{i} \mathrm{e}^{-\delta^{6}}\left[\delta^{d_{i}-2} \tau^{t_{i}}\left(\left(d_{i}-c_{i} \delta^{c_{i}}\right)\left(d_{i} -1-c_{i} \delta^{c_{i}}\right)-c_{i}^{2} \delta^{c_{i}}\right)\right] +\sum_{i=52}^{54} n_{i} \tau^{t_{i}} \mathrm{e}^{-\alpha_{i} \left(\delta-\varepsilon_{i}\right)^{2}-\beta_{i}\left(\tau-\gamma_{i} \right)^{2}} \\ & \cdot\left[-2 \alpha_{i} \delta^{d_{i}}+4 \alpha_{i}^{2} \delta^{d_{i}}\left(\delta-\varepsilon_{i}\right)^{2} -4 d_{i} \alpha_{i} \delta^{d_{i}-1}\left(\delta-\varepsilon_{i}\right) +d_{i}\left(d_{i}-1\right) \delta^{d_{i}-2}\right]+\sum_{i=55}^{56} n_{i}\left[\Delta^{b_{i}}\left(2 \frac{\partial \psi}{\partial \delta} +\delta \frac{\partial^{2} \psi}{\partial \delta^{2}}\right)+2 \frac{\partial \Delta^{b_{i}}}{\partial \delta}\left(\psi+\delta \frac{\partial \psi}{\partial \delta}\right)+\frac{\partial^{2} \Delta^{b_{i}}}{\partial \delta^{2}} \delta \psi\right]\end{aligned}$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d2Ar_ddelta2float

Second derivative of residual Helmholtz energy A/(RT) with respect to delta, [-]

Notes

This is an optimized implementatation taking 4 exp calls, 4 sqrts, and 2 powers. It was generated using SymPy’s CSE functionality, with select polynomial optimizations by hand as well. It is over 10x faster than a naive implementation.

This implementation has been tested against a straightforward implementation with the equations given in IAPWS-95.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative error was 9.566e-16, with a maximum relative error of 1.0518E-10 and a standard deviation of 6.20265E-14.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 250000 points, the mean relative error was 1.039E-15, with a maximum relative error of 2.431E-11 and a standard deviation of 5.31708E-14.

Examples

>>> iapws95_d2Ar_ddelta2(647.096/300.0, 999.0/322)
1.7862535141735987

chemicals.iapws.iapws95_d3Ar_ddelta3(tau, delta)

Calculates the third derivative of residual Helmholtz energy of water with respect to delta according to the IAPWS-95 standard.

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d3Ar_ddelta3float

Third derivative of residual Helmholtz energy A/(RT) with respect to delta, [-]

Notes

No equation is given for this in IAPWS-95, and the derivative was symbolically computed with SymPy.

This is an optimized implementatation. It was generated using SymPy’s CSE functionality.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-4 kg/m^3 to 5000 kg/m^3, 90000 points were evaluated. The mean relative error was 5.41E-13, with a maximum relative error of 6.3957e-11 and a standard deviation of 3.346e-12.

90000 points were also evaluated with mpmath. The mean relative error was 1.41959E-14, with a maximum relative error of 5.8878E-10 and a standard deviation of 1.978E-12.

Evaluating 10000 points in the 1e-10 to 1e-4 range, the mean relative error was 1.2E-16, maximum relative error 1.2e-16, and standard deviation 6.66e-16.

Examples

>>> iapws95_d3Ar_ddelta3(647.096/300.0, 999.0/322)
0.33621190578

chemicals.iapws.iapws95_dAr_dtau(tau, delta)

Calculates the first derivative of residual Helmholtz energy of water with respect to tau according to the IAPWS-95 standard.

$$\phi_{\tau}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} t_{i} \delta^{d_{i}} \tau^{t_{i}-1}+\sum_{i=8}^{51} n_{i} t_{i} \delta^{d_{i}} \tau^{t_{i}-1} \mathrm{e}^{-\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}} \tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i} \right)^{2}-\beta_{i}\left(\tau-\gamma_{i}\right)^{2}} \left[\frac{t_{i}}{\tau}-2 \beta_{i}\left(\tau-\gamma_{i}\right) \right]+\sum_{i=55}^{56} n_{i} \delta\left[\frac{\partial \Delta^{b_{i}}}{\partial \tau} \psi+\Delta^{b_{i}} \frac{\partial \psi}{\partial \tau}\right]$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

dAr_dtaufloat

Derivative of residual Helmholtz energy A/(RT) with respect to tau, [-]

Notes

This is an optimized implementatation taking 9 exp calls, 4 sqrts, and 2 powers. It was generated using SymPy’s CSE functionality, with a limited amount of horner polynomial optimizations as well. It is over 10x faster than a naive implementation.

This implementation has been tested against a straightforward implementation with the equations given in IAPWS-95.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 250000 points were evaluated. The mean relative error was 5.68E-14, with a maximum relative error of 6.73E-9 and a standard deviation of 1.35E-11.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 90000 points, the mean relative error was 4.66E-14, with a maximum relative error of 4.25E-10 and a standard deviation of 1.77E-12.

The maximum error ocurs in the extremely low density regime, $\rho < 1e-6$.

Examples

>>> iapws95_dAr_dtau(647.096/300.0, 999.0/322)
-7.7043336309570

chemicals.iapws.iapws95_d2Ar_dtau2(tau, delta)

Calculates the second derivative of residual Helmholtz energy of water with respect to tau according to the IAPWS-95 standard.

$$\phi_{\tau \tau}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} t_{i}\left(t_{i} -1\right) \delta^{d_{i}} \tau^{t_{i}-2}+\sum_{i=8}^{51} n_{i} t_{i} \left(t_{i}-1\right) \delta^{d_{i}} \tau^{t_{i}-2} \mathrm{e}^{ -\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}} \tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i}\right)^{2} -\beta_{i}\left(\tau-\gamma_{i}\right)^{2}}\left[\left(\frac{t_{i}} {\tau}-2 \beta_{i}\left(\tau-\gamma_{i}\right)\right)^{2}-\frac{t_{i}} {\tau^{2}}-2 \beta_{i}\right] +\sum_{i=55}^{56} n_{i} \delta\left[\frac{\partial^{2} \Delta^{b_{i}}} {\partial \tau^{2}} \psi+2 \frac{\partial \Delta^{b_{i}}}{\partial \tau} \frac{\partial \psi}{\partial \tau}+\Delta^{b_{i}} \frac{\partial^{2} \psi}{\partial \tau^{2}}\right]$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d2Ar_dtau2float

Second derivative of residual Helmholtz energy A/(RT) with respect to tau, [-]

Notes

This is an optimized implementatation taking 9 exp calls, 4 sqrts, and 2 powers. It was generated using SymPy’s CSE functionality, with a limited amount of horner polynomial optimizations as well. It is over 10x faster than a naive implementation.

This implementation has been tested against a straightforward implementation with the equations given in IAPWS-95.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative error was 4.595E-16, with a maximum relative error of 1.835e-10 and a standard deviation of 1.209E-13.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 250000 points, the mean relative error was 2.6026E-16, with a maximum relative error of 2.36E-12 and a standard deviation of 8.055E-15.

This comparison indicates this implementation is more accurate than the straightforward implementation.

Examples

>>> iapws95_d2Ar_dtau2(647.096/300.0, 999.0/322)
-1.2616419775539

chemicals.iapws.iapws95_d2Ar_ddeltadtau(tau, delta)

Calculates the second derivative of residual Helmholtz energy of water with respect to tau and also delta according to the IAPWS-95 standard.

$$\phi_{\delta \tau}^{\mathrm{r}}=\sum_{i=1}^{7} n_{i} d_{i} t_{i} \delta^{d_{i}-1} \tau^{t_{i}-1}+\sum_{i=8}^{51} n_{i} t_{i} \delta^{d_{i}-1} \tau^{t_{i}-1}\left(d_{i}-c_{i} \delta^{c_{i}}\right) \mathrm{e}^{-\delta^{c_{i}}}+\sum_{i=52}^{54} n_{i} \delta^{d_{i}} \tau^{t_{i}} \mathrm{e}^{-\alpha_{i}\left(\delta-\varepsilon_{i} \right)^{2}-\beta_{i}\left(\tau-\gamma_{i}\right)^{2}}\left[\frac{d_{i}} {\delta}-2 \alpha_{i}\left(\delta-\varepsilon_{i}\right)\right]\left[ \frac{t_{i}}{\tau}-2 \beta_{i}\left(\tau-\gamma_{i}\right)\right] \sum_{i=55}^{56} n_{i}\left[\Delta^{b_{i}}\left(\frac{\partial \psi} {\partial \tau}+\delta \frac{\partial^{2} \psi}{\partial \delta \partial \tau}\right)+\delta \frac{\partial \Delta^{b_{i}}}{\partial \delta} \frac{\partial \psi}{\partial \tau}+\frac{\partial \Delta^{b_{i}}}{\partial \tau}\left(\psi+\delta \frac{\partial \psi} {\partial \delta}\right)+\frac{\partial^{2} \Delta^{b_{i}}}{\partial \delta \partial \tau} \delta \psi\right]$$

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d2Ar_ddeltadtaufloat

Second derivative of residual Helmholtz energy A/(RT) with respect to tau and delta, [-]

Notes

This is an optimized implementatation taking 11 exp calls, 4 sqrts, and 3 powers. It was generated using SymPy’s CSE functionality, with select polynomial optimizations by hand as well. It is over 10x faster than a naive implementation.

This implementation has been tested against a straightforward implementation with the equations given in IAPWS-95.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 4E6 points were evaluated. The mean relative error was 2.82E-14, with a maximum relative error of 8.404E-9 and a standard deviation of 5.166e-12.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 90000 points, the mean relative error was 6.974E-14, with a maximum relative error of 4.286E-9 and a standard deviation of 4.286E-12.

Examples

>>> iapws95_d2Ar_ddeltadtau(647.096/300.0, 999.0/322)
-0.198403562385

chemicals.iapws.iapws95_d3Ar_ddeltadtau2(tau, delta)

Calculates the third derivative of residual Helmholtz energy of water with respect to delta once and tau twice according to the IAPWS-95 standard.

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d3Ar_ddeltadtau2float

Third derivative of residual Helmholtz energy A/(RT) with respect to delta once and tau twice, [-]

Notes

This is an optimized implementatation. It was generated using SymPy’s CSE functionality.

No equation is given for this in IAPWS-95, and the derivative was symbolically computed with SymPy.

Like many higher-order derivatives of functions with exponentials, this one balloons to use many, many terms.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 250000 points were evaluated. The mean relative error was 7.936e-16, with a maximum relative error of 1.965E-11 and a standard deviation of 4.7938E-14.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 90000 points, the mean relative error was 6.08E-16, with a maximum relative error of 3.537E-12 and a standard deviation of 1.85197E-14.

Examples

>>> iapws95_d3Ar_ddeltadtau2(647.096/300.0, 999.0/322)
1.081479970332

chemicals.iapws.iapws95_d3Ar_ddelta2dtau(tau, delta)

Calculates the third derivative of residual Helmholtz energy of water with respect to delta twice and tau one according to the IAPWS-95 standard.

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d3Ar_ddeltadtau2float

Third derivative of residual Helmholtz energy A/(RT) with respect to delta twice and tau once, [-]

Notes

This is an optimized implementatation. It was generated using SymPy’s CSE functionality.

No equation is given for this in IAPWS-95, and the derivative was symbolically computed with SymPy.

Like many higher-order derivatives of functions with exponentials, this one balloons to use many, many terms.

Over a linear temperature range of 200 K to 5000 K and a logarithmic density range of 1E-10 kg/m^3 to 5000 kg/m^3, 250000 points were evaluated. The mean relative error was 3.629e-15, with a maximum relative error of 8.38E-11 and a standard deviation of 2.1214E-13.

Over the same range, the model was evaluated to a precision of 50 decimal places with mpmath, and on 10000 points, the mean relative error was 2.4e-15, with a maximum relative error of 7.62E-12 and a standard deviation of 7.818E-14.

Examples

>>> iapws95_d3Ar_ddelta2dtau(647.096/300.0, 999.0/322)
0.015646982949077

chemicals.iapws.iapws95_d4Ar_ddelta2dtau2(tau, delta)

Calculates the fourth derivative of residual Helmholtz energy of water with respect to tau twice and delta twice according to the IAPWS-95 standard.

Parameters

taufloat

Dimensionless temperature, (647.096 K)/T [-]

deltafloat

Dimensionless density, rho/(322 kg/m^3), [-]

Returns

d4Ar_ddelta2dtau2float

Fourth derivative of residual Helmholtz energy A/(RT) with respect to tau and delta, [-]

Examples

>>> iapws95_d4Ar_ddelta2dtau2(647.096/300.0, 999.0/322)
-2.656422915480