# thermo.dippr module¶

thermo.dippr.EQ100(T, A=0, B=0, C=0, D=0, E=0, F=0, G=0, order=0)[source]

DIPPR Equation # 100. Used in calculating the molar heat capacities of liquids and solids, liquid thermal conductivity, and solid density. All parameters default to zero. As this is a straightforward polynomial, no restrictions on parameters apply. Note that high-order polynomials like this may need large numbers of decimal places to avoid unnecessary error.

$Y = A + BT + CT^2 + DT^3 + ET^4 + FT^5 + GT^6$
Parameters: T : float Temperature, [K] A-G : float Parameter for the equation; chemical and property specific [-] order : int, optional Order of the calculation. 0 for the calculation of the result itself; for 1, the first derivative of the property is returned, for -1, the indefinite integral of the property with respect to temperature is returned; and for -1j, the indefinite integral of the property divided by temperature with respect to temperature is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given. Y : float Property [constant-specific; if order == 1, property/K; if order == -1, property*K; if order == -1j, unchanged from default]

Notes

The derivative with respect to T, integral with respect to T, and integral over T with respect to T are computed as follows. All derivatives and integrals are easily computed with SymPy.

$\frac{d Y}{dT} = B + 2 C T + 3 D T^{2} + 4 E T^{3} + 5 F T^{4} + 6 G T^{5}$
$\int Y dT = A T + \frac{B T^{2}}{2} + \frac{C T^{3}}{3} + \frac{D T^{4}}{4} + \frac{E T^{5}}{5} + \frac{F T^{6}}{6} + \frac{G T^{7}}{7}$
$\int \frac{Y}{T} dT = A \log{\left (T \right )} + B T + \frac{C T^{2}} {2} + \frac{D T^{3}}{3} + \frac{E T^{4}}{4} + \frac{F T^{5}}{5} + \frac{G T^{6}}{6}$

References

Examples

Water liquid heat capacity; DIPPR coefficients normally listed in J/kmol/K.

>>> EQ100(300, 276370., -2090.1, 8.125, -0.014116, 0.0000093701)
75355.81000000003

thermo.dippr.EQ101(T, A, B, C, D, E)[source]

DIPPR Equation # 101. Used in calculating vapor pressure, sublimation pressure, and liquid viscosity. All 5 parameters are required. E is often an integer. As the model is exponential, a sufficiently high temperature will cause an OverflowError. A negative temperature (or just low, if fit poorly) may cause a math domain error.

$Y = \exp\left(A + \frac{B}{T} + C\cdot \ln T + D \cdot T^E\right)$
Parameters: T : float Temperature, [K] A-E : float Parameter for the equation; chemical and property specific [-] Y : float Property [constant-specific]

Notes

This function is not integrable for either dT or Y/T dT.

References

Examples

Water vapor pressure; DIPPR coefficients normally listed in Pa.

>>> EQ101(300, 73.649, -7258.2, -7.3037, 4.1653E-6, 2)
3537.44834545549

thermo.dippr.EQ102(T, A, B, C, D, order=0)[source]

DIPPR Equation # 102. Used in calculating vapor viscosity, vapor thermal conductivity, and sometimes solid heat capacity. High values of B raise an OverflowError. All 4 parameters are required. C and D are often 0.

$Y = \frac{A\cdot T^B}{1 + \frac{C}{T} + \frac{D}{T^2}}$
Parameters: T : float Temperature, [K] A-D : float Parameter for the equation; chemical and property specific [-] order : int, optional Order of the calculation. 0 for the calculation of the result itself; for 1, the first derivative of the property is returned, for -1, the indefinite integral of the property with respect to temperature is returned; and for -1j, the indefinite integral of the property divided by temperature with respect to temperature is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given. Y : float Property [constant-specific; if order == 1, property/K; if order == -1, property*K; if order == -1j, unchanged from default]

Notes

The derivative with respect to T, integral with respect to T, and integral over T with respect to T are computed as follows. The first derivative is easily computed; the two integrals required Rubi to perform the integration.

$\frac{d Y}{dT} = \frac{A B T^{B}}{T \left(\frac{C}{T} + \frac{D}{T^{2}} + 1\right)} + \frac{A T^{B} \left(\frac{C}{T^{2}} + \frac{2 D}{T^{3}} \right)}{\left(\frac{C}{T} + \frac{D}{T^{2}} + 1\right)^{2}}$
$\int Y dT = - \frac{2 A T^{B + 3} \operatorname{hyp2f1}{\left (1,B + 3, B + 4,- \frac{2 T}{C - \sqrt{C^{2} - 4 D}} \right )}}{\left(B + 3\right) \left(C + \sqrt{C^{2} - 4 D}\right) \sqrt{C^{2} - 4 D}} + \frac{2 A T^{B + 3} \operatorname{hyp2f1}{\left (1,B + 3,B + 4,- \frac{2 T}{C + \sqrt{C^{2} - 4 D}} \right )}}{\left(B + 3\right) \left(C - \sqrt{C^{2} - 4 D}\right) \sqrt{C^{2} - 4 D}}$
$\int \frac{Y}{T} dT = - \frac{2 A T^{B + 2} \operatorname{hyp2f1}{\left (1,B + 2,B + 3,- \frac{2 T}{C + \sqrt{C^{2} - 4 D}} \right )}}{\left(B + 2\right) \left(C + \sqrt{C^{2} - 4 D}\right) \sqrt{C^{2} - 4 D}} + \frac{2 A T^{B + 2} \operatorname{hyp2f1}{\left (1,B + 2,B + 3, - \frac{2 T}{C - \sqrt{C^{2} - 4 D}} \right )}}{\left(B + 2\right) \left(C - \sqrt{C^{2} - 4 D}\right) \sqrt{C^{2} - 4 D}}$

References

Examples

Water vapor viscosity; DIPPR coefficients normally listed in Pa*s.

>>> EQ102(300, 1.7096E-8, 1.1146, 0, 0)
9.860384711890639e-06

thermo.dippr.EQ104(T, A, B, C, D, E, order=0)[source]

DIPPR Equation #104. Often used in calculating second virial coefficients of gases. All 5 parameters are required. C, D, and E are normally large values.

$Y = A + \frac{B}{T} + \frac{C}{T^3} + \frac{D}{T^8} + \frac{E}{T^9}$
Parameters: T : float Temperature, [K] A-E : float Parameter for the equation; chemical and property specific [-] order : int, optional Order of the calculation. 0 for the calculation of the result itself; for 1, the first derivative of the property is returned, for -1, the indefinite integral of the property with respect to temperature is returned; and for -1j, the indefinite integral of the property divided by temperature with respect to temperature is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given. Y : float Property [constant-specific; if order == 1, property/K; if order == -1, property*K; if order == -1j, unchanged from default]

Notes

The derivative with respect to T, integral with respect to T, and integral over T with respect to T are computed as follows. All expressions can be obtained with SymPy readily.

$\frac{d Y}{dT} = - \frac{B}{T^{2}} - \frac{3 C}{T^{4}} - \frac{8 D}{T^{9}} - \frac{9 E}{T^{10}}$
$\int Y dT = A T + B \log{\left (T \right )} - \frac{1}{56 T^{8}} \left(28 C T^{6} + 8 D T + 7 E\right)$
$\int \frac{Y}{T} dT = A \log{\left (T \right )} - \frac{1}{72 T^{9}} \left(72 B T^{8} + 24 C T^{6} + 9 D T + 8 E\right)$

References

Examples

Water second virial coefficient; DIPPR coefficients normally dimensionless.

>>> EQ104(300, 0.02222, -26.38, -16750000, -3.894E19, 3.133E21)
-1.1204179007265156

thermo.dippr.EQ105(T, A, B, C, D)[source]

DIPPR Equation #105. Often used in calculating liquid molar density. All 4 parameters are required. C is sometimes the fluid’s critical temperature.

$Y = \frac{A}{B^{1 + (1-\frac{T}{C})^D}}$
Parameters: T : float Temperature, [K] A-D : float Parameter for the equation; chemical and property specific [-] Y : float Property [constant-specific]

Notes

This expression can be integrated in terms of the incomplete gamma function for dT, but for Y/T dT no integral could be found.

References

Examples

Hexane molar density; DIPPR coefficients normally in kmol/m^3.

>>> EQ105(300., 0.70824, 0.26411, 507.6, 0.27537)
7.593170096339236

thermo.dippr.EQ106(T, Tc, A, B, C=0, D=0, E=0)[source]

DIPPR Equation #106. Often used in calculating liquid surface tension, and heat of vaporization. Only parameters A and B parameters are required; many fits include no further parameters. Critical temperature is also required.

\begin{align}\begin{aligned}Y = A(1-T_r)^{B + C T_r + D T_r^2 + E T_r^3}\\Tr = \frac{T}{Tc}\end{aligned}\end{align}
Parameters: T : float Temperature, [K] Tc : float Critical temperature, [K] A-D : float Parameter for the equation; chemical and property specific [-] Y : float Property [constant-specific]

Notes

The integral could not be found, but the integral over T actually could, again in terms of hypergeometric functions.

References

Examples

Water surface tension; DIPPR coefficients normally in Pa*s.

>>> EQ106(300, 647.096, 0.17766, 2.567, -3.3377, 1.9699)
0.07231499373541

thermo.dippr.EQ107(T, A=0, B=0, C=0, D=0, E=0, order=0)[source]

DIPPR Equation #107. Often used in calculating ideal-gas heat capacity. All 5 parameters are required. Also called the Aly-Lee equation.

$Y = A + B\left[\frac{C/T}{\sinh(C/T)}\right]^2 + D\left[\frac{E/T}{ \cosh(E/T)}\right]^2$
Parameters: T : float Temperature, [K] A-E : float Parameter for the equation; chemical and property specific [-] order : int, optional Order of the calculation. 0 for the calculation of the result itself; for 1, the first derivative of the property is returned, for -1, the indefinite integral of the property with respect to temperature is returned; and for -1j, the indefinite integral of the property divided by temperature with respect to temperature is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given. Y : float Property [constant-specific; if order == 1, property/K; if order == -1, property*K; if order == -1j, unchanged from default]

Notes

The derivative with respect to T, integral with respect to T, and integral over T with respect to T are computed as follows. The derivative is obtained via SymPy; the integrals from Wolfram Alpha.

$\frac{d Y}{dT} = \frac{2 B C^{3} \cosh{\left (\frac{C}{T} \right )}} {T^{4} \sinh^{3}{\left (\frac{C}{T} \right )}} - \frac{2 B C^{2}}{T^{3} \sinh^{2}{\left (\frac{C}{T} \right )}} + \frac{2 D E^{3} \sinh{\left (\frac{E}{T} \right )}}{T^{4} \cosh^{3}{\left (\frac{E}{T} \right )}} - \frac{2 D E^{2}}{T^{3} \cosh^{2}{\left (\frac{E}{T} \right )}}$
$\int Y dT = A T + \frac{B C}{\tanh{\left (\frac{C}{T} \right )}} - D E \tanh{\left (\frac{E}{T} \right )}$
$\int \frac{Y}{T} dT = A \log{\left (T \right )} + \frac{B C}{T \tanh{ \left (\frac{C}{T} \right )}} - B \log{\left (\sinh{\left (\frac{C}{T} \right )} \right )} - \frac{D E}{T} \tanh{\left (\frac{E}{T} \right )} + D \log{\left (\cosh{\left (\frac{E}{T} \right )} \right )}$

References

 [2] Aly, Fouad A., and Lloyd L. Lee. “Self-Consistent Equations for Calculating the Ideal Gas Heat Capacity, Enthalpy, and Entropy.” Fluid Phase Equilibria 6, no. 3 (January 1, 1981): 169-79. doi:10.1016/0378-3812(81)85002-9.

Examples

Water ideal gas molar heat capacity; DIPPR coefficients normally in J/kmol/K

>>> EQ107(300., 33363., 26790., 2610.5, 8896., 1169.)
33585.90452768923

thermo.dippr.EQ114(T, Tc, A, B, C, D, order=0)[source]

DIPPR Equation #114. Rarely used, normally as an alternate liquid heat capacity expression. All 4 parameters are required, as well as critical temperature.

\begin{align}\begin{aligned}Y = \frac{A^2}{\tau} + B - 2AC\tau - AD\tau^2 - \frac{1}{3}C^2\tau^3 - \frac{1}{2}CD\tau^4 - \frac{1}{5}D^2\tau^5\\\tau = 1 - \frac{T}{Tc}\end{aligned}\end{align}
Parameters: T : float Temperature, [K] Tc : float Critical temperature, [K] A-D : float Parameter for the equation; chemical and property specific [-] order : int, optional Order of the calculation. 0 for the calculation of the result itself; for 1, the first derivative of the property is returned, for -1, the indefinite integral of the property with respect to temperature is returned; and for -1j, the indefinite integral of the property divided by temperature with respect to temperature is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given. Y : float Property [constant-specific; if order == 1, property/K; if order == -1, property*K; if order == -1j, unchanged from default]

Notes

The derivative with respect to T, integral with respect to T, and integral over T with respect to T are computed as follows. All expressions can be obtained with SymPy readily.

$\frac{d Y}{dT} = \frac{A^{2}}{T_{c} \left(- \frac{T}{T_{c}} + 1\right)^{2}} + \frac{2 A}{T_{c}} C + \frac{2 A}{T_{c}} D \left( - \frac{T}{T_{c}} + 1\right) + \frac{C^{2}}{T_{c}} \left( - \frac{T}{T_{c}} + 1\right)^{2} + \frac{2 C}{T_{c}} D \left( - \frac{T}{T_{c}} + 1\right)^{3} + \frac{D^{2}}{T_{c}} \left( - \frac{T}{T_{c}} + 1\right)^{4}$
$\int Y dT = - A^{2} T_{c} \log{\left (T - T_{c} \right )} + \frac{D^{2} T^{6}}{30 T_{c}^{5}} - \frac{T^{5}}{10 T_{c}^{4}} \left(C D + 2 D^{2} \right) + \frac{T^{4}}{12 T_{c}^{3}} \left(C^{2} + 6 C D + 6 D^{2} \right) - \frac{T^{3}}{3 T_{c}^{2}} \left(A D + C^{2} + 3 C D + 2 D^{2}\right) + \frac{T^{2}}{2 T_{c}} \left(2 A C + 2 A D + C^{2} + 2 C D + D^{2}\right) + T \left(- 2 A C - A D + B - \frac{C^{2}}{3} - \frac{C D}{2} - \frac{D^{2}}{5}\right)$
$\int \frac{Y}{T} dT = - A^{2} \log{\left (T + \frac{- 60 A^{2} T_{c} + 60 A C T_{c} + 30 A D T_{c} - 30 B T_{c} + 10 C^{2} T_{c} + 15 C D T_{c} + 6 D^{2} T_{c}}{60 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D - 6 D^{2}} \right )} + \frac{D^{2} T^{5}} {25 T_{c}^{5}} - \frac{T^{4}}{8 T_{c}^{4}} \left(C D + 2 D^{2} \right) + \frac{T^{3}}{9 T_{c}^{3}} \left(C^{2} + 6 C D + 6 D^{2} \right) - \frac{T^{2}}{2 T_{c}^{2}} \left(A D + C^{2} + 3 C D + 2 D^{2}\right) + \frac{T}{T_{c}} \left(2 A C + 2 A D + C^{2} + 2 C D + D^{2}\right) + \frac{1}{30} \left(30 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D - 6 D^{2}\right) \log{\left (T + \frac{1}{60 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D - 6 D^{2}} \left(- 30 A^{2} T_{c} + 60 A C T_{c} + 30 A D T_{c} - 30 B T_{c} + 10 C^{2} T_{c} + 15 C D T_{c} + 6 D^{2} T_{c} + T_{c} \left(30 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D - 6 D^{2}\right)\right) \right )}$

Strictly speaking, the integral over T has an imaginary component, but only the real component is relevant and the complex part discarded.

References

Examples

Hydrogen liquid heat capacity; DIPPR coefficients normally in J/kmol/K.

>>> EQ114(20, 33.19, 66.653, 6765.9, -123.63, 478.27)
19423.948911676463

thermo.dippr.EQ115(T, A, B, C=0, D=0, E=0)[source]

DIPPR Equation #115. No major uses; has been used as an alternate liquid viscosity expression, and as a model for vapor pressure. Only parameters A and B are required.

$Y = \exp\left(A + \frac{B}{T} + C\log T + D T^2 + \frac{E}{T^2}\right)$
Parameters: T : float Temperature, [K] A-E : float Parameter for the equation; chemical and property specific [-] Y : float Property [constant-specific]

Notes

No coefficients found for this expression. This function is not integrable for either dT or Y/T dT.

References

thermo.dippr.EQ116(T, Tc, A, B, C, D, E, order=0)[source]

DIPPR Equation #116. Used to describe the molar density of water fairly precisely; no other uses listed. All 5 parameters are needed, as well as the critical temperature.

\begin{align}\begin{aligned}Y = A + B\tau^{0.35} + C\tau^{2/3} + D\tau + E\tau^{4/3}\\\tau = 1 - \frac{T}{T_c}\end{aligned}\end{align}
Parameters: T : float Temperature, [K] Tc : float Critical temperature, [K] A-E : float Parameter for the equation; chemical and property specific [-] order : int, optional Order of the calculation. 0 for the calculation of the result itself; for 1, the first derivative of the property is returned, for -1, the indefinite integral of the property with respect to temperature is returned; and for -1j, the indefinite integral of the property divided by temperature with respect to temperature is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given. Y : float Property [constant-specific; if order == 1, property/K; if order == -1, property*K; if order == -1j, unchanged from default]

Notes

The derivative with respect to T and integral with respect to T are computed as follows. The integral divided by T with respect to T has an extremely complicated (but still elementary) integral which can be read from the source. It was computed with Rubi; the other expressions can readily be obtained with SymPy.

$\frac{d Y}{dT} = - \frac{7 B}{20 T_c \left(- \frac{T}{T_c} + 1\right)^{ \frac{13}{20}}} - \frac{2 C}{3 T_c \sqrt[3]{- \frac{T}{T_c} + 1}} - \frac{D}{T_c} - \frac{4 E}{3 T_c} \sqrt[3]{- \frac{T}{T_c} + 1}$
$\int Y dT = A T - \frac{20 B}{27} T_c \left(- \frac{T}{T_c} + 1\right)^{ \frac{27}{20}} - \frac{3 C}{5} T_c \left(- \frac{T}{T_c} + 1\right)^{ \frac{5}{3}} + D \left(- \frac{T^{2}}{2 T_c} + T\right) - \frac{3 E}{7} T_c \left(- \frac{T}{T_c} + 1\right)^{\frac{7}{3}}$

References

Examples

Water liquid molar density; DIPPR coefficients normally in kmol/m^3.

>>> EQ116(300., 647.096, 17.863, 58.606, -95.396, 213.89, -141.26)
55.17615446406527

thermo.dippr.EQ127(T, A, B, C, D, E, F, G, order=0)[source]

DIPPR Equation #127. Rarely used, and then only in calculating ideal-gas heat capacity. All 7 parameters are required.

$Y = A+B\left[\frac{\left(\frac{C}{T}\right)^2\exp\left(\frac{C}{T} \right)}{\left(\exp\frac{C}{T}-1 \right)^2}\right] +D\left[\frac{\left(\frac{E}{T}\right)^2\exp\left(\frac{E}{T}\right)} {\left(\exp\frac{E}{T}-1 \right)^2}\right] +F\left[\frac{\left(\frac{G}{T}\right)^2\exp\left(\frac{G}{T}\right)} {\left(\exp\frac{G}{T}-1 \right)^2}\right]$
Parameters: T : float Temperature, [K] A-G : float Parameter for the equation; chemical and property specific [-] order : int, optional Order of the calculation. 0 for the calculation of the result itself; for 1, the first derivative of the property is returned, for -1, the indefinite integral of the property with respect to temperature is returned; and for -1j, the indefinite integral of the property divided by temperature with respect to temperature is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given. Y : float Property [constant-specific; if order == 1, property/K; if order == -1, property*K; if order == -1j, unchanged from default]

Notes

The derivative with respect to T, integral with respect to T, and integral over T with respect to T are computed as follows. All expressions can be obtained with SymPy readily.

$\frac{d Y}{dT} = - \frac{B C^{3} e^{\frac{C}{T}}}{T^{4} \left(e^{\frac{C}{T}} - 1\right)^{2}} + \frac{2 B C^{3} e^{\frac{2 C}{T}}}{T^{4} \left(e^{\frac{C}{T}} - 1\right)^{3}} - \frac{2 B C^{2} e^{\frac{C}{T}}}{T^{3} \left(e^{\frac{C}{T}} - 1\right)^{2}} - \frac{D E^{3} e^{\frac{E}{T}}}{T^{4} \left(e^{\frac{E}{T}} - 1\right)^{2}} + \frac{2 D E^{3} e^{\frac{2 E}{T}}}{T^{4} \left(e^{\frac{E}{T}} - 1\right)^{3}} - \frac{2 D E^{2} e^{\frac{E}{T}}}{T^{3} \left(e^{\frac{E}{T}} - 1\right)^{2}} - \frac{F G^{3} e^{\frac{G}{T}}}{T^{4} \left(e^{\frac{G}{T}} - 1\right)^{2}} + \frac{2 F G^{3} e^{\frac{2 G}{T}}}{T^{4} \left(e^{\frac{G}{T}} - 1\right)^{3}} - \frac{2 F G^{2} e^{\frac{G}{T}}}{T^{3} \left(e^{\frac{G}{T}} - 1\right)^{2}}$
$\int Y dT = A T + \frac{B C^{2}}{C e^{\frac{C}{T}} - C} + \frac{D E^{2}}{E e^{\frac{E}{T}} - E} + \frac{F G^{2}}{G e^{\frac{G}{T}} - G}$
$\int \frac{Y}{T} dT = A \log{\left (T \right )} + B C^{2} \left( \frac{1}{C T e^{\frac{C}{T}} - C T} + \frac{1}{C T} - \frac{1}{C^{2}} \log{\left (e^{\frac{C}{T}} - 1 \right )}\right) + D E^{2} \left( \frac{1}{E T e^{\frac{E}{T}} - E T} + \frac{1}{E T} - \frac{1}{E^{2}} \log{\left (e^{\frac{E}{T}} - 1 \right )}\right) + F G^{2} \left( \frac{1}{G T e^{\frac{G}{T}} - G T} + \frac{1}{G T} - \frac{1}{G^{2}} \log{\left (e^{\frac{G}{T}} - 1 \right )}\right)$

References

>>> EQ127(20., 3.3258E4, 3.6199E4, 1.2057E3, 1.5373E7, 3.2122E3, -1.5318E7, 3.2122E3)