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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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 kinematic viscosity mu
V, D, and dynamic 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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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)[source]¶
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