DIPPR Fit Equations (chemicals.dippr)

This module contains implementations of various numered property equations used by the DIPPR, the Design Institude for Physical Property Research.

No actual data is included in this module; it is just functional implementations of the formulas and some of their derivatives/integrals.

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

Equations

chemicals.dippr.EQ100(T, A=0, B=0, C=0, D=0, E=0, F=0, G=0, order=0)

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

Tfloat

Temperature, [K]

A-Gfloat

Parameter for the equation; chemical and property specific [-]

orderint, 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.

Returns

Yfloat

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.

$$\frac{d Y}{dT} = B + 2 C T + 3 D T^{2} + 4 E T^{3} + 5 F T^{4} + 6 G T^{5}$$

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

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

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

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

chemicals.dippr.EQ101(T, A, B, C=0.0, D=0.0, E=0.0, order=0)

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

Tfloat

Temperature, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

orderint, optional

Order of the calculation. 0 for the calculation of the result itself; for n, the nth derivative of the property is returned. No other integrals or derivatives are implemented, and an exception will be raised if any other order is given.

Returns

Yfloat

Property [constant-specific]

Notes

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

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

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

$$\frac{d^3 Y}{dT^3} = \frac{\left(- \frac{6 B}{T} + 2 C + D E^{3} T^{E} - 3 D E^{2} T^{E} + 2 D E T^{E} + \left(- \frac{B}{T} + C + D E T^{E}\right)^{3} + 3 \left(- \frac{B}{T} + C + D E T^{E}\right) \left(\frac{2 B}{T} - C + D E^{2} T^{E} - D E T^{E}\right)\right) e^{A + \frac{B}{T} + C \log{\left(T \right)} + D T^{E}}}{T^{3}}$$

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

Examples

Water vapor pressure; DIPPR coefficients normally listed in Pa.

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

chemicals.dippr.EQ102(T, A, B, C=0.0, D=0.0, order=0)

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

Tfloat

Temperature, [K]

A-Dfloat

Parameter for the equation; chemical and property specific [-]

orderint, 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.

Returns

Yfloat

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

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

Examples

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

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

chemicals.dippr.EQ104(T, A, B, C=0.0, D=0.0, E=0.0, order=0)

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

Tfloat

Temperature, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

orderint, 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.

Returns

Yfloat

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.

$$\frac{d Y}{dT} = - \frac{B}{T^{2}} - \frac{3 C}{T^{4}} - \frac{8 D}{T^{9}} - \frac{9 E}{T^{10}}$$

$$\int Y dT = A T + B \ln{\left (T \right )} - \frac{1}{56 T^{8}} \left(28 C T^{6} + 8 D T + 7 E\right)$$

$$\int \frac{Y}{T} dT = A \ln{\left (T \right )} - \frac{1}{72 T^{9}} \left(72 B T^{8} + 24 C T^{6} + 9 D T + 8 E\right)$$

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

Examples

Water second virial coefficient; DIPPR coefficients normally dimensionless.

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

chemicals.dippr.EQ105(T, A, B, C, D, order=0)

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

$$Y = \frac{A}{B^{1 + \left(1-\frac{T}{C}\right)^D}}$$

Parameters

Tfloat

Temperature, [K]

A-Dfloat

Parameter for the equation; chemical and property specific [-]

orderint, optional

Order of the calculation. 0 for the calculation of the result itself; for 1, 2, and 3, that derivative of the property is returned; No other integrals or derivatives are implemented, and an exception will be raised if any other order is given.

Returns

Yfloat

Property [constant-specific]

Notes

This expression can be integrated in terms of the incomplete gamma function for dT, however nans are the only output from that function. For Y/T dT no integral could be found.

$$\frac{d Y}{dT} = \frac{A B^{- \left(1 - \frac{T}{C}\right)^{D} - 1} D \left(1 - \frac{T}{C}\right)^{D} \log{\left(B \right)}}{C \left(1 - \frac{T}{C}\right)}$$

$$\frac{d^2 Y}{dT^2} = \frac{A B^{- \left(1 - \frac{T}{C}\right)^{D} - 1} D \left(1 - \frac{T}{C}\right)^{D} \left(D \left(1 - \frac{T}{C} \right)^{D} \log{\left(B \right)} - D + 1\right) \log{\left(B \right)}} {C^{2} \left(1 - \frac{T}{C}\right)^{2}}$$

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

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

Examples

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

>>> EQ105(300., 0.70824, 0.26411, 507.6, 0.27537)
7.593170096339237

chemicals.dippr.EQ106(T, Tc, A, B, C=0.0, D=0.0, E=0.0, order=0)

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.

$$Y = A(1-T_r)^{B + C T_r + D T_r^2 + E T_r^3}$$

$$Tr = \frac{T}{Tc}$$

Parameters

Tfloat

Temperature, [K]

Tcfloat

Critical temperature, [K]

A-Dfloat

Parameter for the equation; chemical and property specific [-]

orderint, optional

Order of the calculation. 0 for the calculation of the result itself; for 1, 2, and 3, that derivative of the property is returned; No other integrals or derivatives are implemented, and an exception will be raised if any other order is given.

Returns

Yfloat

Property [constant-specific]

Notes

This form is used by Yaws with only the parameters A and B.

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

$$\frac{d Y}{dT} = A \left(- \frac{T}{T_{c}} + 1\right)^{B + \frac{C T} {T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}} \left( \left(\frac{C}{T_{c}} + \frac{2 D T}{T_{c}^{2}} + \frac{3 e T^{2}} {T_{c}^{3}}\right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} - \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}} {T_{c}^{3}}}{T_{c} \left(- \frac{T}{T_{c}} + 1\right)}\right)$$

$$\frac{d^2 Y}{dT^2} = \frac{A \left(- \frac{T}{T_{c}} + 1\right)^{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}} {T_{c}^{3}}} \left(2 \left(D + \frac{3 e T}{T_{c}}\right) \log{\left( - \frac{T}{T_{c}} + 1 \right)} + \left(\left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}}\right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} + \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}}{\frac{T}{T_{c}} - 1}\right)^{2} + \frac{2 \left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}} \right)}{\frac{T}{T_{c}} - 1} - \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}}{\left(\frac{T}{T_{c}} - 1\right)^{2}}\right)}{T_{c}^{2}}$$

$$\frac{d^3 Y}{dT^3} = \frac{A \left(- \frac{T}{T_{c}} + 1\right)^{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}} {T_{c}^{3}}} \left(\frac{6 \left(D + \frac{3 e T}{T_{c}}\right)} {\frac{T}{T_{c}} - 1} + \left(\left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}}\right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} + \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}}{\frac{T}{T_{c}} - 1}\right)^{3} + 3 \left(\left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}} \right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} + \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}} {T_{c}^{3}}}{\frac{T}{T_{c}} - 1}\right) \left(2 \left(D + \frac{3 e T} {T_{c}}\right) \log{\left(- \frac{T}{T_{c}} + 1 \right)} + \frac{2 \left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}}\right)} {\frac{T}{T_{c}} - 1} - \frac{B + \frac{C T}{T_{c}} + \frac{D T^{2}} {T_{c}^{2}} + \frac{e T^{3}}{T_{c}^{3}}}{\left(\frac{T}{T_{c}} - 1\right)^{2}}\right) + 6 e \log{\left(- \frac{T}{T_{c}} + 1 \right)} - \frac{3 \left(C + \frac{2 D T}{T_{c}} + \frac{3 e T^{2}}{T_{c}^{2}} \right)}{\left(\frac{T}{T_{c}} - 1\right)^{2}} + \frac{2 \left(B + \frac{C T}{T_{c}} + \frac{D T^{2}}{T_{c}^{2}} + \frac{e T^{3}} {T_{c}^{3}}\right)}{\left(\frac{T}{T_{c}} - 1\right)^{3}}\right)} {T_{c}^{3}}$$

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

Examples

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

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

chemicals.dippr.EQ107(T, A=0, B=0, C=0, D=0, E=0, order=0)

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

Tfloat

Temperature, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

orderint, 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.

Returns

Yfloat

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.

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

$$\int Y dT = A T + \frac{B C}{\tanh{\left (\frac{C}{T} \right )}} - D E \tanh{\left (\frac{E}{T} \right )}$$

$$\int \frac{Y}{T} dT = A \ln{\left (T \right )} + \frac{B C}{T \tanh{ \left (\frac{C}{T} \right )}} - B \ln{\left (\sinh{\left (\frac{C}{T} \right )} \right )} - \frac{D E}{T} \tanh{\left (\frac{E}{T} \right )} + D \ln{\left (\cosh{\left (\frac{E}{T} \right )} \right )}$$

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

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

chemicals.dippr.EQ114(T, Tc, A, B, C, D, order=0)

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

$$Y = \frac{A^2}{\tau} + B - 2AC\tau - AD\tau^2 - \frac{1}{3}C^2\tau^3 - \frac{1}{2}CD\tau^4 - \frac{1}{5}D^2\tau^5$$

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

Parameters

Tfloat

Temperature, [K]

Tcfloat

Critical temperature, [K]

A-Dfloat

Parameter for the equation; chemical and property specific [-]

orderint, 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.

Returns

Yfloat

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.

$$\frac{d Y}{dT} = \frac{A^{2}}{T_{c} \left(- \frac{T}{T_{c}} + 1\right)^{2}} + \frac{2 A}{T_{c}} C + \frac{2 A}{T_{c}} D \left( - \frac{T}{T_{c}} + 1\right) + \frac{C^{2}}{T_{c}} \left( - \frac{T}{T_{c}} + 1\right)^{2} + \frac{2 C}{T_{c}} D \left( - \frac{T}{T_{c}} + 1\right)^{3} + \frac{D^{2}}{T_{c}} \left( - \frac{T}{T_{c}} + 1\right)^{4}$$

$$\int Y dT = - A^{2} T_{c} \ln{\left (T - T_{c} \right )} + \frac{D^{2} T^{6}}{30 T_{c}^{5}} - \frac{T^{5}}{10 T_{c}^{4}} \left(C D + 2 D^{2} \right) + \frac{T^{4}}{12 T_{c}^{3}} \left(C^{2} + 6 C D + 6 D^{2} \right) - \frac{T^{3}}{3 T_{c}^{2}} \left(A D + C^{2} + 3 C D + 2 D^{2}\right) + \frac{T^{2}}{2 T_{c}} \left(2 A C + 2 A D + C^{2} + 2 C D + D^{2}\right) + T \left(- 2 A C - A D + B - \frac{C^{2}}{3} - \frac{C D}{2} - \frac{D^{2}}{5}\right)$$

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

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

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

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

chemicals.dippr.EQ115(T, A, B, C=0, D=0, E=0, order=0)

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.

$$Y = \exp\left(A + \frac{B}{T} + C\ln T + D T^2 + \frac{E}{T^2}\right)$$

Parameters

Tfloat

Temperature, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

orderint, optional

Order of the calculation. 0 for the calculation of the result itself; for 1, 2, and 3, that derivative of the property is returned; No other integrals or derivatives are implemented, and an exception will be raised if any other order is given.

Returns

Yfloat

Property [constant-specific]

Notes

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

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

$$\frac{d^2 Y}{dT^2} = \left(\frac{2 B}{T^{3}} - \frac{C}{T^{2}} + 2 D + \frac{6 E}{T^{4}} + \left(\frac{B}{T^{2}} - \frac{C}{T} - 2 D T + \frac{2 E}{T^{3}}\right)^{2}\right) e^{A + \frac{B}{T} + C \log{\left(T \right)} + D T^{2} + \frac{E}{T^{2}}}$$

$$\frac{d^3 Y}{dT^3} =- \left(3 \left(\frac{2 B}{T^{3}} - \frac{C}{T^{2}} + 2 D + \frac{6 E}{T^{4}}\right) \left(\frac{B}{T^{2}} - \frac{C}{T} - 2 D T + \frac{2 E}{T^{3}}\right) + \left(\frac{B}{T^{2}} - \frac{C}{T} - 2 D T + \frac{2 E}{T^{3}}\right)^{3} + \frac{2 \left( \frac{3 B}{T} - C + \frac{12 E}{T^{2}}\right)}{T^{3}}\right) e^{A + \frac{B}{T} + C \log{\left(T \right)} + D T^{2} + \frac{E}{T^{2}}}$$

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

chemicals.dippr.EQ116(T, Tc, A, B, C, D, E, order=0)

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.

$$Y = A + B\tau^{0.35} + C\tau^{2/3} + D\tau + E\tau^{4/3}$$

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

Parameters

Tfloat

Temperature, [K]

Tcfloat

Critical temperature, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

orderint, 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.

Returns

Yfloat

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

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

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

chemicals.dippr.EQ127(T, A, B, C, D, E, F, G, order=0)

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

Tfloat

Temperature, [K]

A-Gfloat

Parameter for the equation; chemical and property specific [-]

orderint, 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.

Returns

Yfloat

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.

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

$$\int Y dT = A T + \frac{B C^{2}}{C e^{\frac{C}{T}} - C} + \frac{D E^{2}}{E e^{\frac{E}{T}} - E} + \frac{F G^{2}}{G e^{\frac{G}{T}} - G}$$

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

References

1

Design Institute for Physical Properties, 1996. DIPPR Project 801 DIPPR/AIChE

Examples

Ideal gas heat capacity of methanol; DIPPR coefficients normally in J/kmol/K

>>> EQ127(20., 3.3258E4, 3.6199E4, 1.2057E3, 1.5373E7, 3.2122E3, -1.5318E7, 3.2122E3)
33258.0

Jacobians (for fitting)

chemicals.dippr.EQ101_fitting_jacobian(Ts, A, B, C, D, E)

Compute and return the Jacobian of the property predicted by DIPPR Equation # 101 with respect to all the coefficients. This is used in fitting parameters for chemicals.

Parameters

Tslist[float]

Temperatures of the experimental data points, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

Returns

jaclist[list[float, 5], len(Ts)]

Matrix of derivatives of the equation with respect to the fitting parameters, [various]

chemicals.dippr.EQ102_fitting_jacobian(Ts, A, B, C, D)

Compute and return the Jacobian of the property predicted by DIPPR Equation # 102 with respect to all the coefficients. This is used in fitting parameters for chemicals.

Parameters

Tslist[float]

Temperatures of the experimental data points, [K]

A-Dfloat

Parameter for the equation; chemical and property specific [-]

Returns

jaclist[list[float, 4], len(Ts)]

Matrix of derivatives of the equation with respect to the fitting parameters, [various]

chemicals.dippr.EQ105_fitting_jacobian(Ts, A, B, C, D)

Compute and return the Jacobian of the property predicted by DIPPR Equation # 105 with respect to all the coefficients. This is used in fitting parameters for chemicals.

Parameters

Tslist[float]

Temperatures of the experimental data points, [K]

A-Dfloat

Parameter for the equation; chemical and property specific [-]

Returns

jaclist[list[float, 4], len(Ts)]

Matrix of derivatives of the equation with respect to the fitting parameters, [various]

chemicals.dippr.EQ106_fitting_jacobian(Ts, Tc, A, B, C, D, E)

Compute and return the Jacobian of the property predicted by DIPPR Equation # 106 with respect to all the coefficients. This is used in fitting parameters for chemicals.

Parameters

Tslist[float]

Temperatures of the experimental data points, [K]

Tcfloat

Critical temperature, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

Returns

jaclist[list[float, 5], len(Ts)]

Matrix of derivatives of the equation with respect to the fitting parameters, [various]

chemicals.dippr.EQ107_fitting_jacobian(Ts, A, B, C, D, E)

Compute and return the Jacobian of the property predicted by DIPPR Equation # 107 with respect to all the coefficients. This is used in fitting parameters for chemicals.

Parameters

Tslist[float]

Temperatures of the experimental data points, [K]

A-Efloat

Parameter for the equation; chemical and property specific [-]

Returns

jaclist[list[float, 5], len(Ts)]

Matrix of derivatives of the equation with respect to the fitting parameters, [various]