Dimensionless numbers (fluids.core)

This module contains basic fluid mechanics and engineering calculations which have been found useful by the author. The main functionality is calculating dimensionless numbers, interconverting different forms of loss coefficients, and converting temperature units.

For reporting bugs, adding feature requests, or submitting pull requests, please use the GitHub issue tracker or contact the author at Caleb.Andrew.Bell@gmail.com.

Dimensionless Numbers

fluids.core.Archimedes(L, rhof, rhop, mu, g=9.80665)

Calculates Archimedes number, Ar, for a fluid and particle with the given densities, characteristic length, viscosity, and gravity (usually diameter of particle).

$$Ar = \frac{L^3 \rho_f(\rho_p-\rho_f)g}{\mu^2}$$

Parameters

Lfloat

Characteristic length, typically particle diameter [m]

rhoffloat

Density of fluid, [kg/m^3]

rhopfloat

Density of particle, [kg/m^3]

mufloat

Viscosity of fluid, [N/m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

Arfloat

Archimedes number []

Notes

Used in fluid-particle interaction calculations.

$$Ar = \frac{\text{Gravitational force}}{\text{Viscous force}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Archimedes(0.002, 2., 3000, 1E-3)
470.4053872

fluids.core.Bejan_L(dP, L, mu, alpha)

Calculates Bejan number of a length or Be_L for a fluid with the given parameters flowing over a characteristic length L and experiencing a pressure drop dP.

$$Be_L = \frac{\Delta P L^2}{\mu \alpha}$$

Parameters

dPfloat

Pressure drop, [Pa]

Lfloat

Characteristic length, [m]

mufloat, optional

Dynamic viscosity, [Pa*s]

alphafloat

Thermal diffusivity, [m^2/s]

Returns

Be_Lfloat

Bejan number with respect to length []

Notes

Termed a dimensionless number by someone in 1988.

References

1

Awad, M. M. “The Science and the History of the Two Bejan Numbers.” International Journal of Heat and Mass Transfer 94 (March 2016): 101-3. doi:10.1016/j.ijheatmasstransfer.2015.11.073.

2

Bejan, Adrian. Convection Heat Transfer. 4E. Hoboken, New Jersey: Wiley, 2013.

Examples

>>> Bejan_L(1E4, 1, 1E-3, 1E-6)
10000000000000.0

fluids.core.Bejan_p(dP, K, mu, alpha)

Calculates Bejan number of a permeability or Be_p for a fluid with the given parameters and a permeability K experiencing a pressure drop dP.

$$Be_p = \frac{\Delta P K}{\mu \alpha}$$

Parameters

dPfloat

Pressure drop, [Pa]

Kfloat

Permeability, [m^2]

mufloat, optional

Dynamic viscosity, [Pa*s]

alphafloat

Thermal diffusivity, [m^2/s]

Returns

Be_pfloat

Bejan number with respect to pore characteristics []

Notes

Termed a dimensionless number by someone in 1988.

References

1

Awad, M. M. “The Science and the History of the Two Bejan Numbers.” International Journal of Heat and Mass Transfer 94 (March 2016): 101-3. doi:10.1016/j.ijheatmasstransfer.2015.11.073.

2

Bejan, Adrian. Convection Heat Transfer. 4E. Hoboken, New Jersey: Wiley, 2013.

Examples

>>> Bejan_p(1E4, 1, 1E-3, 1E-6)
10000000000000.0

fluids.core.Biot(h, L, k)

Calculates Biot number Br for heat transfer coefficient h, geometric length L, and thermal conductivity k.

$$Bi=\frac{hL}{k}$$

Parameters

hfloat

Heat transfer coefficient, [W/m^2/K]

Lfloat

Characteristic length, no typical definition [m]

kfloat

Thermal conductivity, within the object [W/m/K]

Returns

Bifloat

Biot number, [-]

Notes

Do not confuse k, the thermal conductivity within the object, with that of the medium h is calculated with!

$$Bi = \frac{\text{Surface thermal resistance}} {\text{Internal thermal resistance}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Biot(1000., 1.2, 300.)
4.0
>>> Biot(10000., .01, 4000.)
0.025

fluids.core.Boiling(G, q, Hvap)

Calculates Boiling number or Bg using heat flux, two-phase mass flux, and heat of vaporization of the fluid flowing. Used in two-phase heat transfer calculations.

$$\text{Bg} = \frac{q}{G_{tp} \Delta H_{vap}}$$

Parameters

Gfloat

Two-phase mass flux in a channel (combined liquid and vapor) [kg/m^2/s]

qfloat

Heat flux [W/m^2]

Hvapfloat

Heat of vaporization of the fluid [J/kg]

Returns

Bgfloat

Boiling number [-]

Notes

Most often uses the symbol Bo instead of Bg, but this conflicts with Bond number.

$$\text{Bg} = \frac{\text{mass liquid evaporated / area heat transfer surface}}{\text{mass flow rate fluid / flow cross sectional area}}$$

First defined in [4], though not named.

References

1

Winterton, Richard H.S. BOILING NUMBER. Thermopedia. Hemisphere, 2011. 10.1615/AtoZ.b.boiling_number

2

Collier, John G., and John R. Thome. Convective Boiling and Condensation. 3rd edition. Clarendon Press, 1996.

3

Stephan, Karl. Heat Transfer in Condensation and Boiling. Translated by C. V. Green.. 1992 edition. Berlin; New York: Springer, 2013.

4

W. F. Davidson, P. H. Hardie, C. G. R. Humphreys, A. A. Markson, A. R. Mumford and T. Ravese “Studies of heat transmission through boiler tubing at pressures from 500 to 3300 pounds” Trans. ASME, Vol. 65, 9, February 1943, pp. 553-591.

Examples

>>> Boiling(300, 3000, 800000)
1.25e-05

fluids.core.Bond(rhol, rhog, sigma, L)

Calculates Bond number, Bo also known as Eotvos number, for a fluid with the given liquid and gas densities, surface tension, and geometric parameter (usually length).

$$Bo = \frac{g(\rho_l-\rho_g)L^2}{\sigma}$$

Parameters

rholfloat

Density of liquid, [kg/m^3]

rhogfloat

Density of gas, [kg/m^3]

sigmafloat

Surface tension, [N/m]

Lfloat

Characteristic length, [m]

Returns

Bofloat

Bond number []

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

Examples

>>> Bond(1000., 1.2, .0589, 2)
665187.2339558573

fluids.core.Capillary(V, mu, sigma)

Calculates Capillary number Ca for a characteristic velocity V, viscosity mu, and surface tension sigma.

$$Ca = \frac{V \mu}{\sigma}$$

Parameters

Vfloat

Characteristic velocity, [m/s]

mufloat

Dynamic viscosity, [Pa*s]

sigmafloat

Surface tension, [N/m]

Returns

Cafloat

Capillary number, [-]

Notes

Used in porous media calculations and film flow calculations. Surface tension may gas-liquid, or liquid-liquid.

$$Ca = \frac{\text{Viscous forces}} {\text{Surface forces}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Kundu, Pijush K., Ira M. Cohen, and David R. Dowling. Fluid Mechanics. Academic Press, 2012.

Examples

>>> Capillary(1.2, 0.01, .1)
0.12

fluids.core.Cavitation(P, Psat, rho, V)

Calculates Cavitation number or Ca for a fluid of velocity V with a pressure P, vapor pressure Psat, and density rho.

$$Ca = \sigma_c = \sigma = \frac{P-P_{sat}}{\frac{1}{2}\rho V^2}$$

Parameters

Pfloat

Internal pressure of the fluid, [Pa]

Psatfloat

Vapor pressure of the fluid, [Pa]

rhofloat

Density of the fluid, [kg/m^3]

Vfloat

Velocity of fluid, [m/s]

Returns

Cafloat

Cavitation number []

Notes

Used in determining if a flow through a restriction will cavitate. Sometimes, the multiplication by 2 will be omitted;

$$Ca = \frac{\text{Pressure - Vapor pressure}} {\text{Inertial pressure}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Cavitation(2E5, 1E4, 1000, 10)
3.8

fluids.core.Confinement(D, rhol, rhog, sigma, g=9.80665)

Calculates Confinement number or Co for a fluid in a channel of diameter D with liquid and gas densities rhol and rhog and surface tension sigma, under the influence of gravitational force g.

$$\text{Co}=\frac{\left[\frac{\sigma}{g(\rho_l-\rho_g)}\right]^{0.5}}{D}$$

Parameters

Dfloat

Diameter of channel, [m]

rholfloat

Density of liquid phase, [kg/m^3]

rhogfloat

Density of gas phase, [kg/m^3]

sigmafloat

Surface tension between liquid-gas phase, [N/m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

Cofloat

Confinement number [-]

Notes

Used in two-phase pressure drop and heat transfer correlations. First used in [1] according to [3].

$$\text{Co} = \frac{\frac{\text{surface tension force}} {\text{buoyancy force}}}{\text{Channel area}}$$

References

1

Cornwell, Keith, and Peter A. Kew. “Boiling in Small Parallel Channels.” In Energy Efficiency in Process Technology, edited by Dr P. A. Pilavachi, 624-638. Springer Netherlands, 1993. doi:10.1007/978-94-011-1454-7_56.

2

Kandlikar, Satish G. Heat Transfer and Fluid Flow in Minichannels and Microchannels. Elsevier, 2006.

3

Tran, T. N, M. -C Chyu, M. W Wambsganss, and D. M France. Two-Phase Pressure Drop of Refrigerants during Flow Boiling in Small Channels: An Experimental Investigation and Correlation Development.” International Journal of Multiphase Flow 26, no. 11 (November 1, 2000): 1739-54. doi:10.1016/S0301-9322(99)00119-6.

Examples

>>> Confinement(0.001, 1077, 76.5, 4.27E-3)
0.6596978265315191

fluids.core.Dean(Re, Di, D)

Calculates Dean number, De, for a fluid with the Reynolds number Re, inner diameter Di, and a secondary diameter D. D may be the diameter of curvature, the diameter of a spiral, or some other dimension.

$$\text{De} = \sqrt{\frac{D_i}{D}} \text{Re} = \sqrt{\frac{D_i}{D}} \frac{\rho v D}{\mu}$$

Parameters

Refloat

Reynolds number []

Difloat

Inner diameter []

Dfloat

Diameter of curvature or outer spiral or other dimension []

Returns

Defloat

Dean number [-]

Notes

Used in flow in curved geometry.

$$\text{De} = \frac{\sqrt{\text{centripetal forces}\cdot \text{inertial forces}}}{\text{viscous forces}}$$

References

1

Catchpole, John P., and George. Fulford. “DIMENSIONLESS GROUPS.” Industrial & Engineering Chemistry 58, no. 3 (March 1, 1966): 46-60. doi:10.1021/ie50675a012.

Examples

>>> Dean(10000, 0.1, 0.4)
5000.0

fluids.core.Drag(F, A, V, rho)

Calculates drag coefficient Cd for a given drag force F, projected area A, characteristic velocity V, and density rho.

$$C_D = \frac{F_d}{A\cdot\frac{1}{2}\rho V^2}$$

Parameters

Ffloat

Drag force, [N]

Afloat

Projected area, [m^2]

Vfloat

Characteristic velocity, [m/s]

rhofloat

Density, [kg/m^3]

Returns

Cdfloat

Drag coefficient, [-]

Notes

Used in flow around objects, or objects flowing within a fluid.

$$C_D = \frac{\text{Drag forces}}{\text{Projected area}\cdot \text{Velocity head}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Drag(1000, 0.0001, 5, 2000)
400.0

fluids.core.Eckert(V, Cp, dT)

Calculates Eckert number or Ec for a fluid of velocity V with a heat capacity Cp, between two temperature given as dT.

$$Ec = \frac{V^2}{C_p \Delta T}$$

Parameters

Vfloat

Velocity of fluid, [m/s]

Cpfloat

Heat capacity of the fluid, [J/kg/K]

dTfloat

Temperature difference, [K]

Returns

Ecfloat

Eckert number []

Notes

Used in certain heat transfer calculations. Fairly rare.

$$Ec = \frac{\text{Kinetic energy} }{ \text{Enthalpy difference}}$$

References

1

Goldstein, Richard J. ECKERT NUMBER. Thermopedia. Hemisphere, 2011. 10.1615/AtoZ.e.eckert_number

Examples

>>> Eckert(10, 2000., 25.)
0.002

fluids.core.Euler(dP, rho, V)

Calculates Euler number or Eu for a fluid of velocity V and density rho experiencing a pressure drop dP.

$$Eu = \frac{\Delta P}{\rho V^2}$$

Parameters

dPfloat

Pressure drop experience by the fluid, [Pa]

rhofloat

Density of the fluid, [kg/m^3]

Vfloat

Velocity of fluid, [m/s]

Returns

Eufloat

Euler number []

Notes

Used in pressure drop calculations. Rarely, this number is divided by two. Named after Leonhard Euler applied calculus to fluid dynamics.

$$Eu = \frac{\text{Pressure drop}}{2\cdot \text{velocity head}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Euler(1E5, 1000., 4)
6.25

fluids.core.Fourier_heat(t, L, rho=None, Cp=None, k=None, alpha=None)

Calculates heat transfer Fourier number or Fo for a specified time t, characteristic length L, and specified properties for the given fluid.

$$Fo = \frac{k t}{C_p \rho L^2} = \frac{\alpha t}{L^2}$$

Inputs either of any of the following sets:

• t, L, density rho, heat capacity Cp, and thermal conductivity k

• t, L, and thermal diffusivity alpha

Parameters

tfloat

time [s]

Lfloat

Characteristic length [m]

rhofloat, optional

Density, [kg/m^3]

Cpfloat, optional

Heat capacity, [J/kg/K]

kfloat, optional

Thermal conductivity, [W/m/K]

alphafloat, optional

Thermal diffusivity, [m^2/s]

Returns

Fofloat

Fourier number (heat) []

Notes

$$Fo = \frac{\text{Heat conduction rate}} {\text{Rate of thermal energy storage in a solid}}$$

An error is raised if none of the required input sets are provided.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Fourier_heat(t=1.5, L=2, rho=1000., Cp=4000., k=0.6)
5.625e-08
>>> Fourier_heat(1.5, 2, alpha=1E-7)
3.75e-08

fluids.core.Fourier_mass(t, L, D)

Calculates mass transfer Fourier number or Fo for a specified time t, characteristic length L, and diffusion coefficient D.

$$Fo = \frac{D t}{L^2}$$

Parameters

tfloat

time [s]

Lfloat

Characteristic length [m]

Dfloat

Diffusivity of a species, [m^2/s]

Returns

Fofloat

Fourier number (mass) []

Notes

$$Fo = \frac{\text{Diffusive transport rate}}{\text{Storage rate}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

Examples

>>> Fourier_mass(t=1.5, L=2, D=1E-9)
3.7500000000000005e-10

fluids.core.Froude(V, L, g=9.80665, squared=False)

Calculates Froude number Fr for velocity V and geometric length L. If desired, gravity can be specified as well. Normally the function returns the result of the equation below; Froude number is also often said to be defined as the square of the equation below.

$$Fr = \frac{V}{\sqrt{gL}}$$

Parameters

Vfloat

Velocity of the particle or fluid, [m/s]

Lfloat

Characteristic length, no typical definition [m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

squaredbool, optional

Whether to return the squared form of Froude number

Returns

Frfloat

Froude number, [-]

Notes

Many alternate definitions including density ratios have been used.

$$Fr = \frac{\text{Inertial Force}}{\text{Gravity Force}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Froude(1.83, L=2., g=1.63)
1.0135432593877318
>>> Froude(1.83, L=2., squared=True)
0.17074638128208924

fluids.core.Froude_densimetric(V, L, rho1, rho2, heavy=True, g=9.80665)

Calculates the densimetric Froude number $Fr_{den}$ for velocity V geometric length L, heavier fluid density rho1, and lighter fluid density rho2. If desired, gravity can be specified as well. Depending on the application, this dimensionless number may be defined with the heavy phase or the light phase density in the numerator of the square root. For some applications, both need to be calculated. The default is to calculate with the heavy liquid ensity on top; set heavy to False to reverse this.

$$Fr = \frac{V}{\sqrt{gL}} \sqrt{\frac{\rho_\text{(1 or 2)}} {\rho_1 - \rho_2}}$$

Parameters

Vfloat

Velocity of the specified phase, [m/s]

Lfloat

Characteristic length, no typical definition [m]

rho1float

Density of the heavier phase, [kg/m^3]

rho2float

Density of the lighter phase, [kg/m^3]

heavybool, optional

Whether or not the density used in the numerator is the heavy phase or the light phase, [-]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

Fr_denfloat

Densimetric Froude number, [-]

Notes

Many alternate definitions including density ratios have been used.

$$Fr = \frac{\text{Inertial Force}}{\text{Gravity Force}}$$

Where the gravity force is reduced by the relative densities of one fluid in another.

Note that an Exception will be raised if rho1 > rho2, as the square root becomes negative.

References

1

Hall, A, G Stobie, and R Steven. “Further Evaluation of the Performance of Horizontally Installed Orifice Plate and Cone Differential Pressure Meters with Wet Gas Flows.” In International SouthEast Asia Hydrocarbon Flow Measurement Workshop, KualaLumpur, Malaysia, 2008.

Examples

>>> Froude_densimetric(1.83, L=2., rho1=800, rho2=1.2, g=9.81)
0.4134543386272418
>>> Froude_densimetric(1.83, L=2., rho1=800, rho2=1.2, g=9.81, heavy=False)
0.016013017679205096

fluids.core.Graetz_heat(V, D, x, rho=None, Cp=None, k=None, alpha=None)

Calculates Graetz number or Gz for a specified velocity V, diameter D, axial distance x, and specified properties for the given fluid.

$$Gz = \frac{VD^2\cdot C_p \rho}{x\cdot k} = \frac{VD^2}{x \alpha}$$

Inputs either of any of the following sets:

• V, D, x, density rho, heat capacity Cp, and thermal conductivity k

• V, D, x, and thermal diffusivity alpha

Parameters

Vfloat

Velocity, [m/s]

Dfloat

Diameter [m]

xfloat

Axial distance [m]

rhofloat, optional

Density, [kg/m^3]

Cpfloat, optional

Heat capacity, [J/kg/K]

kfloat, optional

Thermal conductivity, [W/m/K]

alphafloat, optional

Thermal diffusivity, [m^2/s]

Returns

Gzfloat

Graetz number []

Notes

$$Gz = \frac{\text{Time for radial heat diffusion in a fluid by conduction}} {\text{Time taken by fluid to reach distance x}}$$

$$Gz = \frac{D}{x}RePr$$

An error is raised if none of the required input sets are provided.

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> Graetz_heat(1.5, 0.25, 5, 800., 2200., 0.6)
55000.0
>>> Graetz_heat(1.5, 0.25, 5, alpha=1E-7)
187500.0

fluids.core.Grashof(L, beta, T1, T2=0, rho=None, mu=None, nu=None, g=9.80665)

Calculates Grashof number or Gr for a fluid with the given properties, temperature difference, and characteristic length.

$$Gr = \frac{g\beta (T_s-T_\infty)L^3}{\nu^2} = \frac{g\beta (T_s-T_\infty)L^3\rho^2}{\mu^2}$$

Inputs either of any of the following sets:

• L, beta, T1 and T2, and density rho and kinematic viscosity mu

• L, beta, T1 and T2, and dynamic viscosity nu

Parameters

Lfloat

Characteristic length [m]

betafloat

Volumetric thermal expansion coefficient [1/K]

T1float

Temperature 1, usually a film temperature [K]

T2float, optional

Temperature 2, usually a bulk temperature (or 0 if only a difference is provided to the function) [K]

rhofloat, optional

Density, [kg/m^3]

mufloat, optional

Dynamic viscosity, [Pa*s]

nufloat, optional

Kinematic viscosity, [m^2/s]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

Grfloat

Grashof number []

Notes

$$Gr = \frac{\text{Buoyancy forces}}{\text{Viscous forces}}$$

An error is raised if none of the required input sets are provided. Used in free convection problems only.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

Example 4 of [1], p. 1-21 (matches):

>>> Grashof(L=0.9144, beta=0.000933, T1=178.2, rho=1.1613, mu=1.9E-5)
4656936556.178915
>>> Grashof(L=0.9144, beta=0.000933, T1=378.2, T2=200, nu=1.636e-05)
4657491516.530312

fluids.core.Hagen(Re, fd)

Calculates Hagen number, Hg, for a fluid with the given Reynolds number and friction factor.

$$\text{Hg} = \frac{f_d}{2} Re^2 = \frac{1}{\rho} \frac{\Delta P}{\Delta z} \frac{D^3}{\nu^2} = \frac{\rho\Delta P D^3}{\mu^2 \Delta z}$$

Parameters

Refloat

Reynolds number [-]

fdfloat, optional

Darcy friction factor, [-]

Returns

Hgfloat

Hagen number, [-]

Notes

Introduced in [1]; further use of it is mostly of the correlations introduced in [1].

Notable for use use in correlations, because it does not have any dependence on velocity.

This expression is useful when designing backwards with a pressure drop spec already known.

References

1(1,2)

Martin, Holger. “The Generalized Lévêque Equation and Its Practical Use for the Prediction of Heat and Mass Transfer Rates from Pressure Drop.” Chemical Engineering Science, Jean-Claude Charpentier Festschrift Issue, 57, no. 16 (August 1, 2002): 3217-23. https://doi.org/10.1016/S0009-2509(02)00194-X.

2

Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.

3

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

Example from [3]:

>>> Hagen(Re=2610, fd=1.935235)
6591507.17175

fluids.core.Jakob(Cp, Hvap, Te)

Calculates Jakob number or Ja for a boiling fluid with sensible heat capacity Cp, enthalpy of vaporization Hvap, and boiling at Te degrees above its saturation boiling point.

$$Ja = \frac{C_{P}\Delta T_e}{\Delta H_{vap}}$$

Parameters

Cpfloat

Heat capacity of the fluid, [J/kg/K]

Hvapfloat

Enthalpy of vaporization of the fluid at its saturation temperature [J/kg]

Tefloat

Temperature difference above the fluid’s saturation boiling temperature, [K]

Returns

Jafloat

Jakob number []

Notes

Used in boiling heat transfer analysis. Fairly rare.

$$Ja = \frac{\Delta \text{Sensible heat}}{\Delta \text{Latent heat}}$$

References

1

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Jakob(4000., 2E6, 10.)
0.02

fluids.core.Knudsen(path, L)

Calculates Knudsen number or Kn for a fluid with mean free path path and for a characteristic length L.

$$Kn = \frac{\lambda}{L}$$

Parameters

pathfloat

Mean free path between molecular collisions, [m]

Lfloat

Characteristic length, [m]

Returns

Knfloat

Knudsen number []

Notes

Used in mass transfer calculations.

$$Kn = \frac{\text{Mean free path length}}{\text{Characteristic length}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Knudsen(1e-10, .001)
1e-07

fluids.core.Lewis(D=None, alpha=None, Cp=None, k=None, rho=None)

Calculates Lewis number or Le for a fluid with the given parameters.

$$Le = \frac{k}{\rho C_p D} = \frac{\alpha}{D}$$

Inputs can be either of the following sets:

• Diffusivity and Thermal diffusivity

• Diffusivity, heat capacity, thermal conductivity, and density

Parameters

Dfloat

Diffusivity of a species, [m^2/s]

alphafloat, optional

Thermal diffusivity, [m^2/s]

Cpfloat, optional

Heat capacity, [J/kg/K]

kfloat, optional

Thermal conductivity, [W/m/K]

rhofloat, optional

Density, [kg/m^3]

Returns

Lefloat

Lewis number []

Notes

$$Le=\frac{\text{Thermal diffusivity}}{\text{Mass diffusivity}} = \frac{Sc}{Pr}$$

An error is raised if none of the required input sets are provided.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

3

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> Lewis(D=22.6E-6, alpha=19.1E-6)
0.8451327433628318
>>> Lewis(D=22.6E-6, rho=800., k=.2, Cp=2200)
0.00502815768302494

fluids.core.Mach(V, c)

Calculates Mach number or Ma for a fluid of velocity V with speed of sound c.

$$Ma = \frac{V}{c}$$

Parameters

Vfloat

Velocity of fluid, [m/s]

cfloat

Speed of sound in fluid, [m/s]

Returns

Mafloat

Mach number []

Notes

Used in compressible flow calculations.

$$Ma = \frac{\text{fluid velocity}}{\text{sonic velocity}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Mach(33., 330)
0.1

fluids.core.Morton(rhol, rhog, mul, sigma, g=9.80665)

Calculates Morton number or Mo for a liquid and vapor with the specified properties, under the influence of gravitational force g.

$$Mo = \frac{g \mu_l^4(\rho_l - \rho_g)}{\rho_l^2 \sigma^3}$$

Parameters

rholfloat

Density of liquid phase, [kg/m^3]

rhogfloat

Density of gas phase, [kg/m^3]

mulfloat

Viscosity of liquid phase, [Pa*s]

sigmafloat

Surface tension between liquid-gas phase, [N/m]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

Mofloat

Morton number, [-]

Notes

Used in modeling bubbles in liquid.

References

1

Kunes, Josef. Dimensionless Physical Quantities in Science and Engineering. Elsevier, 2012.

2

Yan, Xiaokang, Kaixin Zheng, Yan Jia, Zhenyong Miao, Lijun Wang, Yijun Cao, and Jiongtian Liu. “Drag Coefficient Prediction of a Single Bubble Rising in Liquids.” Industrial & Engineering Chemistry Research, April 2, 2018. https://doi.org/10.1021/acs.iecr.7b04743.

Examples

>>> Morton(1077.0, 76.5, 4.27E-3, 0.023)
2.311183104430743e-07

fluids.core.Nusselt(h, L, k)

Calculates Nusselt number Nu for a heat transfer coefficient h, characteristic length L, and thermal conductivity k.

$$Nu = \frac{hL}{k}$$

Parameters

hfloat

Heat transfer coefficient, [W/m^2/K]

Lfloat

Characteristic length, no typical definition [m]

kfloat

Thermal conductivity of fluid [W/m/K]

Returns

Nufloat

Nusselt number, [-]

Notes

Do not confuse k, the thermal conductivity of the fluid, with that of within a solid object associated with!

$$Nu = \frac{\text{Convective heat transfer}} {\text{Conductive heat transfer}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> Nusselt(1000., 1.2, 300.)
4.0
>>> Nusselt(10000., .01, 4000.)
0.025

fluids.core.Ohnesorge(L, rho, mu, sigma)

Calculates Ohnesorge number, Oh, for a fluid with the given characteristic length, density, viscosity, and surface tension.

$$\text{Oh} = \frac{\mu}{\sqrt{\rho \sigma L }}$$

Parameters

Lfloat

Characteristic length [m]

rhofloat

Density of fluid, [kg/m^3]

mufloat

Viscosity of fluid, [Pa*s]

sigmafloat

Surface tension, [N/m]

Returns

Ohfloat

Ohnesorge number []

Notes

Often used in spray calculations. Sometimes given the symbol Z.

$$Oh = \frac{\sqrt{\text{We}}}{\text{Re}}= \frac{\text{viscous forces}} {\sqrt{\text{Inertia}\cdot\text{Surface tension}} }$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

Examples

>>> Ohnesorge(1E-4, 1000., 1E-3, 1E-1)
0.01

fluids.core.Peclet_heat(V, L, rho=None, Cp=None, k=None, alpha=None)

Calculates heat transfer Peclet number or Pe for a specified velocity V, characteristic length L, and specified properties for the given fluid.

$$Pe = \frac{VL\rho C_p}{k} = \frac{LV}{\alpha}$$

Inputs either of any of the following sets:

• V, L, density rho, heat capacity Cp, and thermal conductivity k

• V, L, and thermal diffusivity alpha

Parameters

Vfloat

Velocity [m/s]

Lfloat

Characteristic length [m]

rhofloat, optional

Density, [kg/m^3]

Cpfloat, optional

Heat capacity, [J/kg/K]

kfloat, optional

Thermal conductivity, [W/m/K]

alphafloat, optional

Thermal diffusivity, [m^2/s]

Returns

Pefloat

Peclet number (heat) []

Notes

$$Pe = \frac{\text{Bulk heat transfer}}{\text{Conduction heat transfer}}$$

An error is raised if none of the required input sets are provided.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Peclet_heat(1.5, 2, 1000., 4000., 0.6)
20000000.0
>>> Peclet_heat(1.5, 2, alpha=1E-7)
30000000.0

fluids.core.Peclet_mass(V, L, D)

Calculates mass transfer Peclet number or Pe for a specified velocity V, characteristic length L, and diffusion coefficient D.

$$Pe = \frac{L V}{D}$$

Parameters

Vfloat

Velocity [m/s]

Lfloat

Characteristic length [m]

Dfloat

Diffusivity of a species, [m^2/s]

Returns

Pefloat

Peclet number (mass) []

Notes

$$Pe = \frac{\text{Advective transport rate}}{\text{Diffusive transport rate}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

Examples

>>> Peclet_mass(1.5, 2, 1E-9)
3000000000.0

fluids.core.Power_number(P, L, N, rho)

Calculates power number, Po, for an agitator applying a specified power P with a characteristic length L, rotational speed N, to a fluid with a specified density rho.

$$Po = \frac{P}{\rho N^3 D^5}$$

Parameters

Pfloat

Power applied, [W]

Lfloat

Characteristic length, typically agitator diameter [m]

Nfloat

Speed [revolutions/second]

rhofloat

Density of fluid, [kg/m^3]

Returns

Pofloat

Power number []

Notes

Used in mixing calculations.

$$Po = \frac{\text{Power}}{\text{Rotational inertia}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Power_number(P=180, L=0.01, N=2.5, rho=800.)
144000000.0

fluids.core.Prandtl(Cp=None, k=None, mu=None, nu=None, rho=None, alpha=None)

Calculates Prandtl number or Pr for a fluid with the given parameters.

$$Pr = \frac{C_p \mu}{k} = \frac{\nu}{\alpha} = \frac{C_p \rho \nu}{k}$$

Inputs can be any of the following sets:

• Heat capacity, dynamic viscosity, and thermal conductivity

• Thermal diffusivity and kinematic viscosity

• Heat capacity, kinematic viscosity, thermal conductivity, and density

Parameters

Cpfloat

Heat capacity, [J/kg/K]

kfloat

Thermal conductivity, [W/m/K]

mufloat, optional

Dynamic viscosity, [Pa*s]

nufloat, optional

Kinematic viscosity, [m^2/s]

rhofloat

Density, [kg/m^3]

alphafloat

Thermal diffusivity, [m^2/s]

Returns

Prfloat

Prandtl number []

Notes

$$Pr=\frac{\text{kinematic viscosity}}{\text{thermal diffusivity}} = \frac{\text{momentum diffusivity}}{\text{thermal diffusivity}}$$

An error is raised if none of the required input sets are provided.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

3

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> Prandtl(Cp=1637., k=0.010, mu=4.61E-6)
0.754657
>>> Prandtl(Cp=1637., k=0.010, nu=6.4E-7, rho=7.1)
0.7438528
>>> Prandtl(nu=6.3E-7, alpha=9E-7)
0.7000000000000001

fluids.core.Rayleigh(Pr, Gr)

Calculates Rayleigh number or Ra using Prandtl number Pr and Grashof number Gr for a fluid with the given properties, temperature difference, and characteristic length used to calculate Gr and Pr.

$$Ra = PrGr$$

Parameters

Prfloat

Prandtl number []

Grfloat

Grashof number []

Returns

Rafloat

Rayleigh number []

Notes

Used in free convection problems only.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Rayleigh(1.2, 4.6E9)
5520000000.0

fluids.core.relative_roughness(D, roughness=1.52e-06)

Calculates relative roughness eD using a diameter and the roughness of the material of the wall. Default roughness is that of steel.

$$eD=\frac{\epsilon}{D}$$

Parameters

Dfloat

Diameter of pipe, [m]

roughnessfloat, optional

Roughness of pipe wall [m]

Returns

eDfloat

Relative Roughness, [-]

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> relative_roughness(0.5, 1E-4)
0.0002

fluids.core.Reynolds(V, D, rho=None, mu=None, nu=None)

Calculates Reynolds number or Re for a fluid with the given properties for the specified velocity and diameter.

$$Re = \frac{D \cdot V}{\nu} = \frac{\rho V D}{\mu}$$

Inputs either of any of the following sets:

• V, D, density rho and dynamic viscosity mu

• V, D, and kinematic viscosity nu

Parameters

Vfloat

Velocity [m/s]

Dfloat

Diameter [m]

rhofloat, optional

Density, [kg/m^3]

mufloat, optional

Dynamic viscosity, [Pa*s]

nufloat, optional

Kinematic viscosity, [m^2/s]

Returns

Refloat

Reynolds number []

Notes

$$Re = \frac{\text{Momentum}}{\text{Viscosity}}$$

An error is raised if none of the required input sets are provided.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Reynolds(2.5, 0.25, 1.1613, 1.9E-5)
38200.65789473684
>>> Reynolds(2.5, 0.25, nu=1.636e-05)
38202.93398533008

fluids.core.Schmidt(D, mu=None, nu=None, rho=None)

Calculates Schmidt number or Sc for a fluid with the given parameters.

$$Sc = \frac{\mu}{D\rho} = \frac{\nu}{D}$$

Inputs can be any of the following sets:

• Diffusivity, dynamic viscosity, and density

• Diffusivity and kinematic viscosity

Parameters

Dfloat

Diffusivity of a species, [m^2/s]

mufloat, optional

Dynamic viscosity, [Pa*s]

nufloat, optional

Kinematic viscosity, [m^2/s]

rhofloat, optional

Density, [kg/m^3]

Returns

Scfloat

Schmidt number []

Notes

$$Sc =\frac{\text{kinematic viscosity}}{\text{molecular diffusivity}} = \frac{\text{viscous diffusivity}}{\text{species diffusivity}}$$

An error is raised if none of the required input sets are provided.

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Schmidt(D=2E-6, mu=4.61E-6, rho=800)
0.00288125
>>> Schmidt(D=1E-9, nu=6E-7)
599.9999999999999

fluids.core.Sherwood(K, L, D)

Calculates Sherwood number Sh for a mass transfer coefficient K, characteristic length L, and diffusivity D.

$$Sh = \frac{KL}{D}$$

Parameters

Kfloat

Mass transfer coefficient, [m/s]

Lfloat

Characteristic length, no typical definition [m]

Dfloat

Diffusivity of a species [m/s^2]

Returns

Shfloat

Sherwood number, [-]

Notes

$$Sh = \frac{\text{Mass transfer by convection}} {\text{Mass transfer by diffusion}} = \frac{K}{D/L}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

Examples

>>> Sherwood(1000., 1.2, 300.)
4.0

fluids.core.Stanton(h, V, rho, Cp)

Calculates Stanton number or St for a specified heat transfer coefficient h, velocity V, density rho, and heat capacity Cp [1] [2].

$$St = \frac{h}{V\rho Cp}$$

Parameters

hfloat

Heat transfer coefficient, [W/m^2/K]

Vfloat

Velocity, [m/s]

rhofloat

Density, [kg/m^3]

Cpfloat

Heat capacity, [J/kg/K]

Returns

Stfloat

Stanton number []

Notes

$$St = \frac{\text{Heat transfer coefficient}}{\text{Thermal capacity}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ: Wiley, 2011.

Examples

>>> Stanton(5000, 5, 800, 2000.)
0.000625

fluids.core.Stokes_number(V, Dp, D, rhop, mu)

Calculates Stokes Number for a given characteristic velocity V, particle diameter Dp, characteristic diameter D, particle density rhop, and fluid viscosity mu.

$$\text{Stk} = \frac{\rho_p V D_p^2}{18\mu_f D}$$

Parameters

Vfloat

Characteristic velocity (often superficial), [m/s]

Dpfloat

Particle diameter, [m]

Dfloat

Characteristic diameter (ex demister wire diameter or cyclone diameter), [m]

rhopfloat

Particle density, [kg/m^3]

mufloat

Fluid viscosity, [Pa*s]

Returns

Stkfloat

Stokes numer, [-]

Notes

Used in droplet impaction or collection studies.

References

1

Rhodes, Martin J. Introduction to Particle Technology. Wiley, 2013.

2

Al-Dughaither, Abdullah S., Ahmed A. Ibrahim, and Waheed A. Al-Masry. “Investigating Droplet Separation Efficiency in Wire-Mesh Mist Eliminators in Bubble Column.” Journal of Saudi Chemical Society 14, no. 4 (October 1, 2010): 331-39. https://doi.org/10.1016/j.jscs.2010.04.001.

Examples

>>> Stokes_number(V=0.9, Dp=1E-5, D=1E-3, rhop=1000, mu=1E-5)
0.5

fluids.core.Strouhal(f, L, V)

Calculates Strouhal number St for a characteristic frequency f, characteristic length L, and velocity V.

$$St = \frac{fL}{V}$$

Parameters

ffloat

Characteristic frequency, usually that of vortex shedding, [Hz]

Lfloat

Characteristic length, [m]

Vfloat

Velocity of the fluid, [m/s]

Returns

Stfloat

Strouhal number, [-]

Notes

Sometimes abbreviated to S or Sr.

$$St = \frac{\text{Characteristic flow time}} {\text{Period of oscillation}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> Strouhal(8, 2., 4.)
4.0

fluids.core.Suratman(L, rho, mu, sigma)

Calculates Suratman number, Su, for a fluid with the given characteristic length, density, viscosity, and surface tension.

$$\text{Su} = \frac{\rho\sigma L}{\mu^2}$$

Parameters

Lfloat

Characteristic length [m]

rhofloat

Density of fluid, [kg/m^3]

mufloat

Viscosity of fluid, [Pa*s]

sigmafloat

Surface tension, [N/m]

Returns

Sufloat

Suratman number []

Notes

Also known as Laplace number. Used in two-phase flow, especially the bubbly-slug regime. No confusion regarding the definition of this group has been observed.

$$\text{Su} = \frac{\text{Re}^2}{\text{We}} =\frac{\text{Inertia}\cdot \text{Surface tension} }{\text{(viscous forces)}^2}$$

The oldest reference to this group found by the author is in 1963, from [2].

References

1

Sen, Nilava. “Suratman Number in Bubble-to-Slug Flow Pattern Transition under Microgravity.” Acta Astronautica 65, no. 3-4 (August 2009): 423-28. doi:10.1016/j.actaastro.2009.02.013.

2

Catchpole, John P., and George. Fulford. “DIMENSIONLESS GROUPS.” Industrial & Engineering Chemistry 58, no. 3 (March 1, 1966): 46-60. doi:10.1021/ie50675a012.

Examples

>>> Suratman(1E-4, 1000., 1E-3, 1E-1)
10000.0

fluids.core.Weber(V, L, rho, sigma)

Calculates Weber number, We, for a fluid with the given density, surface tension, velocity, and geometric parameter (usually diameter of bubble).

$$We = \frac{V^2 L\rho}{\sigma}$$

Parameters

Vfloat

Velocity of fluid, [m/s]

Lfloat

Characteristic length, typically bubble diameter [m]

rhofloat

Density of fluid, [kg/m^3]

sigmafloat

Surface tension, [N/m]

Returns

Wefloat

Weber number []

Notes

Used in bubble calculations.

$$We = \frac{\text{inertial force}}{\text{surface tension force}}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

3

Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010.

Examples

>>> Weber(V=0.18, L=0.001, rho=900., sigma=0.01)
2.916

Loss Coefficient Converters

fluids.core.K_from_f(fd, L, D)

Calculates loss coefficient, K, for a given section of pipe at a specified friction factor.

$$K = f_dL/D$$

Parameters

fdfloat

friction factor of pipe, []

Lfloat

Length of pipe, [m]

Dfloat

Inner diameter of pipe, [m]

Returns

Kfloat

Loss coefficient, []

Notes

For fittings with a specified L/D ratio, use D = 1 and set L to specified L/D ratio.

Examples

>>> K_from_f(fd=0.018, L=100., D=.3)
6.0

fluids.core.K_from_L_equiv(L_D, fd=0.015)

Calculates loss coefficient, for a given equivalent length (L/D).

$$K = f_d \frac{L}{D}$$

Parameters

L_Dfloat

Length over diameter, []

fdfloat, optional

Darcy friction factor, [-]

Returns

Kfloat

Loss coefficient, []

Notes

Almost identical to K_from_f, but with a default friction factor for fully turbulent flow in steel pipes.

Examples

>>> K_from_L_equiv(240)
3.5999999999999996

fluids.core.L_equiv_from_K(K, fd=0.015)

Calculates equivalent length of pipe (L/D), for a given loss coefficient.

$$\frac{L}{D} = \frac{K}{f_d}$$

Parameters

Kfloat

Loss coefficient, [-]

fdfloat, optional

Darcy friction factor, [-]

Returns

L_Dfloat

Length over diameter, [-]

Notes

Assumes a default friction factor for fully turbulent flow in steel pipes.

Examples

>>> L_equiv_from_K(3.6)
240.00000000000003

fluids.core.L_from_K(K, D, fd=0.015)

Calculates the length of straight pipe at a specified friction factor required to produce a given loss coefficient K.

$$L = \frac{K D}{f_d}$$

Parameters

Kfloat

Loss coefficient, []

Dfloat

Inner diameter of pipe, [m]

fdfloat

friction factor of pipe, []

Returns

Lfloat

Length of pipe, [m]

Examples

>>> L_from_K(K=6, D=.3, fd=0.018)
100.0

fluids.core.dP_from_K(K, rho, V)

Calculates pressure drop, for a given loss coefficient, at a specified density and velocity.

$$dP = 0.5K\rho V^2$$

Parameters

Kfloat

Loss coefficient, []

rhofloat

Density of fluid, [kg/m^3]

Vfloat

Velocity of fluid in pipe, [m/s]

Returns

dPfloat

Pressure drop, [Pa]

Notes

Loss coefficient K is usually the sum of several factors, including the friction factor.

Examples

>>> dP_from_K(K=10, rho=1000, V=3)
45000.0

fluids.core.head_from_K(K, V, g=9.80665)

Calculates head loss, for a given loss coefficient, at a specified velocity.

$$\text{head} = \frac{K V^2}{2g}$$

Parameters

Kfloat

Loss coefficient, []

Vfloat

Velocity of fluid in pipe, [m/s]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

headfloat

Head loss, [m]

Notes

Loss coefficient K is usually the sum of several factors, including the friction factor.

Examples

>>> head_from_K(K=10, V=1.5)
1.1471807396001694

fluids.core.head_from_P(P, rho, g=9.80665)

Calculates head for a fluid of specified density at specified pressure.

$$\text{head} = {P\over{\rho g}}$$

Parameters

Pfloat

Pressure fluid in pipe, [Pa]

rhofloat

Density of fluid, [kg/m^3]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

headfloat

Head, [m]

Notes

By definition. Head varies with location, inversely proportional to the increase in gravitational constant.

Examples

>>> head_from_P(P=98066.5, rho=1000)
10.000000000000002

fluids.core.f_from_K(K, L, D)

Calculates friction factor, fd, from a loss coefficient, K, for a given section of pipe.

$$f_d = \frac{K D}{L}$$

Parameters

Kfloat

Loss coefficient, []

Lfloat

Length of pipe, [m]

Dfloat

Inner diameter of pipe, [m]

Returns

fdfloat

Darcy friction factor of pipe, [-]

Notes

This can be useful to blend fittings at specific locations in a pipe into a pressure drop which is evenly distributed along a pipe.

Examples

>>> f_from_K(K=0.6, L=100., D=.3)
0.0018

fluids.core.P_from_head(head, rho, g=9.80665)

Calculates head for a fluid of specified density at specified pressure.

$$P = \rho g \cdot \text{head}$$

Parameters

headfloat

Head, [m]

rhofloat

Density of fluid, [kg/m^3]

gfloat, optional

Acceleration due to gravity, [m/s^2]

Returns

Pfloat

Pressure fluid in pipe, [Pa]

Examples

>>> P_from_head(head=5., rho=800.)
39226.6

Temperature Conversions

These functions used to be part of SciPy, but were removed in favor of a slower function convert_temperature which removes code duplication but doesn’t have the same convenience or easy to remember signature.

fluids.core.C2K(C)

Convert Celsius to Kelvin.

Parameters

Cfloat

Celsius temperature to be converted, [degC]

Returns

Kfloat

Equivalent Kelvin temperature, [K]

Notes

Computes K = C + zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> C2K(-40)
233.14999999999998

fluids.core.K2C(K)

Convert Kelvin to Celsius.

Parameters

Kfloat

Kelvin temperature to be converted.

Returns

Cfloat

Equivalent Celsius temperature.

Notes

Computes C = K - zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> K2C(233.15)
-39.99999999999997

fluids.core.F2C(F)

Convert Fahrenheit to Celsius.

Parameters

Ffloat

Fahrenheit temperature to be converted.

Returns

Cfloat

Equivalent Celsius temperature.

Notes

Computes C = (F - 32) / 1.8.

Examples

>>> F2C(-40.0)
-40.0

fluids.core.C2F(C)

Convert Celsius to Fahrenheit.

Parameters

Cfloat

Celsius temperature to be converted.

Returns

Ffloat

Equivalent Fahrenheit temperature.

Notes

Computes F = 1.8 * C + 32.

Examples

>>> C2F(-40.0)
-40.0

fluids.core.F2K(F)

Convert Fahrenheit to Kelvin.

Parameters

Ffloat

Fahrenheit temperature to be converted.

Returns

Kfloat

Equivalent Kelvin temperature.

Notes

Computes K = (F - 32)/1.8 + zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> F2K(-40)
233.14999999999998

fluids.core.K2F(K)

Convert Kelvin to Fahrenheit.

Parameters

Kfloat

Kelvin temperature to be converted.

Returns

Ffloat

Equivalent Fahrenheit temperature.

Notes

Computes F = 1.8 * (K - zero_Celsius) + 32 where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> K2F(233.15)
-39.99999999999996

fluids.core.C2R(C)

Convert Celsius to Rankine.

Parameters

Cfloat

Celsius temperature to be converted.

Returns

Rafloat

Equivalent Rankine temperature.

Notes

Computes Ra = 1.8 * (C + zero_Celsius) where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> C2R(-40)
419.66999999999996

fluids.core.K2R(K)

Convert Kelvin to Rankine.

Parameters

Kfloat

Kelvin temperature to be converted.

Returns

Rafloat

Equivalent Rankine temperature.

Notes

Computes Ra = 1.8 * K.

Examples

>>> K2R(273.15)
491.66999999999996

fluids.core.F2R(F)

Convert Fahrenheit to Rankine.

Parameters

Ffloat

Fahrenheit temperature to be converted.

Returns

Rafloat

Equivalent Rankine temperature.

Notes

Computes Ra = F - 32 + 1.8 * zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> F2R(100)
559.67

fluids.core.R2C(Ra)

Convert Rankine to Celsius.

Parameters

Rafloat

Rankine temperature to be converted.

Returns

Cfloat

Equivalent Celsius temperature.

Notes

Computes C = Ra / 1.8 - zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> R2C(459.67)
-17.777777777777743

fluids.core.R2K(Ra)

Convert Rankine to Kelvin.

Parameters

Rafloat

Rankine temperature to be converted.

Returns

Kfloat

Equivalent Kelvin temperature.

Notes

Computes K = Ra / 1.8.

Examples

>>> R2K(491.67)
273.15

fluids.core.R2F(Ra)

Convert Rankine to Fahrenheit.

Parameters

Rafloat

Rankine temperature to be converted.

Returns

Ffloat

Equivalent Fahrenheit temperature.

Notes

Computes F = Ra + 32 - 1.8 * zero_Celsius where zero_Celsius = 273.15, i.e., (the absolute value of) temperature “absolute zero” as measured in Celsius.

Examples

>>> R2F(491.67)
32.00000000000006

Miscellaneous Functions

fluids.core.thermal_diffusivity(k, rho, Cp)

Calculates thermal diffusivity or alpha for a fluid with the given parameters.

$$\alpha = \frac{k}{\rho Cp}$$

Parameters

kfloat

Thermal conductivity, [W/m/K]

rhofloat

Density, [kg/m^3]

Cpfloat

Heat capacity, [J/kg/K]

Returns

alphafloat

Thermal diffusivity, [m^2/s]

References

1

Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984.

Examples

>>> thermal_diffusivity(k=0.02, rho=1., Cp=1000.)
2e-05

fluids.core.c_ideal_gas(T, k, MW)

Calculates speed of sound c in an ideal gas at temperature T.

$$c = \sqrt{kR_{specific}T}$$

Parameters

Tfloat

Temperature of fluid, [K]

kfloat

Isentropic exponent of fluid, [-]

MWfloat

Molecular weight of fluid, [g/mol]

Returns

cfloat

Speed of sound in fluid, [m/s]

Notes

Used in compressible flow calculations. Note that the gas constant used is the specific gas constant:

$$R_{specific} = R\frac{1000}{MW}$$

References

1

Green, Don, and Robert Perry. Perry’s Chemical Engineers’ Handbook, Eighth Edition. McGraw-Hill Professional, 2007.

2

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> c_ideal_gas(T=303, k=1.4, MW=28.96)
348.9820953185441

fluids.core.nu_mu_converter(rho, mu=None, nu=None)

Calculates either kinematic or dynamic viscosity, depending on inputs. Used when one type of viscosity is known as well as density, to obtain the other type. Raises an error if both types of viscosity or neither type of viscosity is provided.

$$\nu = \frac{\mu}{\rho}$$

$$\mu = \nu\rho$$

Parameters

rhofloat

Density, [kg/m^3]

mufloat, optional

Dynamic viscosity, [Pa*s]

nufloat, optional

Kinematic viscosity, [m^2/s]

Returns

mu or nufloat

Dynamic viscosity, Pa*s or Kinematic viscosity, m^2/s

References

1

Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006.

Examples

>>> nu_mu_converter(998., nu=1.0E-6)
0.000998

fluids.core.gravity(latitude, H)

Calculates local acceleration due to gravity g according to [1]. Uses latitude and height to calculate g.

$$g = 9.780356(1 + 0.0052885\sin^2\phi - 0.0000059^22\phi) - 3.086\times 10^{-6} H$$

Parameters

latitudefloat

Degrees, [degrees]

Hfloat

Height above earth’s surface [m]

Returns

gfloat

Acceleration due to gravity, [m/s^2]

Notes

Better models, such as EGM2008 exist.

References

1

Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.

Examples

>>> gravity(55, 1E4)
9.784151976863571