Dimensionless Numbers
Dimensionless numbers are mathematical ratios that are used in physics and engineering to describe physical systems or phenomena. They are typically used to compare physical quantities that have different units, by eliminating the effects of units and focusing on the underlying physical relationships. These are nondimensional quantity of dimension one and can not be measured on a scale of physical units. All quantities of dimension one are said to have the SI coherent derived unit "one" with the symbol "1". It is a pure number, thus always having a dimension of 1. The number does not change even if the number system you are working in does. Dimensionless numbers or quantities are used in many disiplines such as chemistry, economics, engineering, mathematics, and physics.
These dimensionless numbers provide important insights into the behavior of physical systems, and they are widely used in engineering and physics to analyze and optimize the performance of various systems.
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Dimensionless numbers Glossary
A
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 Abbe Number  The measure of material's dispersion (variation of refractive index versus wavelength) with high values of V indicating dispersion. It is used to classify glass and other optically transparent materials.
 Aeration Number  Is used for the agitated mixing of gas and liquids.
 Air–fuel Ratio  The mass ratio of air to fuel mixture present in an internal combustion engines.
 Alfven Number  The steady flow past a fixed object of a conduction fluid or characteristic number for the relation between plasma speed and Alfven wave speed.
 Algebric Difference in Grade  The difference in grades.
 Archimedes Number  Analyzes flow as it relates to a system of density differences. It is used when dealing with gravitational settling of particles in fluid.
 Arrhenius Equation  Where the temperature dependance of the reaction rateconstant which is the rate of chemical reaction.
 Atomic Number  Equal to the number of protons in an atom's nucleus. The atomic number determines which element an atom is.
 Atwood Number  Describes density difference between two adjacent fluids with a common interface.
B
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 Bagnold Number  The ratio of grain collision stresses to various fluid stresses in a granular flow with interstitial Newtonian fluid.
 Bejan Number  Fluid Mechanics  The pressure drop along a length of channel.
 Bejan Number  Heat Transfer  The pressure drop along a channel of length.

Bejan Number  Mass Transfer  The pressure drop along a channel of length.

Bingham Number  The ratio of yield stress to visvous stress.

Biot Number  The ratio of internal thermal resistance of solid to fluid thermal resistance. This is used for heat transfer between fluids and solids.
 Blake Number  Is proportional to inertial force and viscous force and is used in momentum transfer and flow through beds of solids.
 Bodenstein Number  Used in mass transfer and diffusion in reactors calculations in particular.
 Bond Number  Represents the relationship of gravitational force to surface tension force.
 Brinkman Number  The heat conduction from a wall to a flowing viscous fluid, comminly used in polymer processes.
 BrownellKatz number  A combination of Capillary number and Bond number.
 Buckling Coefficient  Is an instability that leads to a failure mode.
C
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 Capillary Number  Representing the relative effect of viscous forces against the surface tension between a liquid/gas or liquid/liquid interface.
 Carnot Efficiency  The theroetical maximum efficiency of any heat engine depending only on the temperatures it operates between.
 Cauchy Number  The ratio of inertial force to compressibility force in a flow. When the compressibility is important the elastic forces must be considered along with inertial forces.

Cavitation Number  Expresses the relationship between the difference of a local absolute pressure from the vapor pressure and the kinetic energy per volume.
 Chandrasekhar Number  Used in magnetic convection to represent ratio of the Lorentz force to the viscosity.

Chezy Coefficient  Dependent on surface roughness and the hydraulic mean depth.
 Coefficient of Discharge  See Discharge Coefficient
 Coefficient of Friction  See Friction Coefficient
 Coefficient of Kinetic Friction  See Kinetic Friction Coefficient
 Cohesion Number  Used in partical tchnology by which the cohesivity of different powders can be compared.
 Colburn J Factors  Heat transfer equation to calculate the natural convection moment of heat from vertical surfaces or horizontal cylinders of fluid (gasses or liquids) flowing past these surfaces.
 Compound Machines Efficiency  A simple machine is a mechanical device that changes the direction or magnitude of a force. A compound machine is machines connected in series.
 Compressibility  Measures the change in volume under external forces for any liquid.
 Compressibility Factor  Corrects for deviation from the ideal gas law to account for the real gases behavior.
 Compression Ratio  The ratio of the maximum volume to the minimum volume in a cylinder.
 Consistancy Index  The range of water content to the firmness of the soil.
 Constrictivity  A parameter used to describe transportation processes in porous media.
 Correction Factor  A factor multiplied with the result of an equation to correct for a known amount of systemic error.
 Curvature Coefficient  Classifies a soil as well graded or poorly graded.

Courant Number  The measure of how much information traverses a computational grid cell in a given time interval.
D
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Damping Ratio  Describes how oscillations in a system decay after a disturbance.
 Darcy Friction Factor  Helps calculate the friction losses in pipe flow and open channel flow. This coefficient is used in the DarcyWeisbach equation.
 Darcy Friction Factor  Brkić  Equations that allow you to calculate the friction losses in pipe flow and open channel flow.
 Darcy Friction Factor  Free Surface Flow  Equations that allow you to calculate the friction losses in pipe flow and open channel flow.
 Darcy Friction Factor  Laminar Flow  Equations that allow you to calculate the friction losses in pipe flow and open channel flow.
 DarcyWeisbach Friction Factor  See Darcy Friction Factor

DarcyWeisbach Equation  The most common way of expressing the pressure drop of a piped fluid. The equation is valid for fully developed, steady state and incompressible flow.
 Dean Number  Used in momentum transfer for the flow in curved pipes and channels.

Deborah Number  The ratio of fluidity of materials under specific flow conditions, primarly used in rheology
 Degree of Saturation  The ratio of volume of water to the volume of voids.
 Determination Coefficient  The porportion of the variation in the dependent variable that is predicated from the independent variables.
 Discharge Coefficient  The ratio of actual discharge to the theoretical discharge.
 Drag Coefficient  Used to quantify the amount of force or resistance an object experienced as it moves through the fluid stream.
 Dukhin Number  The surface conductivity to various electrokinetic and electroacoustic effects, as well as the electrical conductivity and permittivity of fluid heterogeneous systems.
E
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Eckert Number  Used to calculate the ratio of the kinetic energy to enthalpy.
 Efficiency  Expressed in percentage and always less than 100%.
 Ekman Number  In fluid dynamics it describes the ratio of various forces to Coriolis forces.
 Elasticity Number  The ratio of elastic force to internal force in viscoelastic flow.
 Elongation Percentage  The percentage of elongation at the fracture.
 Emissivity  A measure of a material's ability to emit thermal radiation. It is defined as the ratio of the thermal radiation emitted by a material to the thermal radiation emitted by a blackbody at the same temperature.

Energy Efficiency Rating  Measure the efficiency with which a product uses energy to function. It is calculated by dividing a product's BTU output by its wattage.
 Eotvos Number  Measures the importance of gravitational forces compared to surface tension forces for the movement of liquid front.

Ericksen Number  The ratio of the viscous to elastic forces.
 Euler Number  Used for analyzing flow where the differential pressure between two points is important.
F
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Factor of Safety  The ability of a system's structural capacity to be usable beyond it's expected or acrual loads.
 Fan Efficiency  The ratio of input and output power.
 Fanning Friction Factor  The ratio of shear stress acting on the surface of a solid.

Flow Coefficient  Described as the volume (in US gallons) of water at 60°F that will flow per minute through a valve with a pressure drop of 1 psi across the valve.
 Fourier Modulus  See Fourier Number
 Fourier Number  The ratio of heat conduction rate to the rate of thermal energy storage in a solid.
 Fresnel Number  Occuring in optics, in paticular in scalar diffraction theory.
 Friction Coefficient  The ratio between two contacting surfaces and the frictional force that resists the normal force of the object.
 Friction Factor  Used in internal flow calculations with the DarcyWeisbach equation. Depending on the Reynolds Number, the friction factor may be calculated one of several ways.
 Froude Number  The ratio of inertial force to gravitational force. It is used for wave and surface behavior for mixed natural and forced convection.
G
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 Galilei Number  See Galileo Number
 Galileo Number  Used in fluid dynamics to describe fluid film flow over walls. It is a ratio of forces present in the flow of viscous fluids.
 Gas Compressibility Factor  A factor independent of the quantity of gas and determined by the character of the gas, the temperature, and pressure.
 Glide Ratio  Heavierthanair flight without the use of thrust.
 Golden Ratio (Divine Porportion, Golden Mean, Golden Section)  A mathematical ratio commonly found in nature and design.
 Gortler Number  The secondary flows that appear in the boundary layer flow along a concave wall.
 Graetz Number  Characterizes laminar flow with heat transfer in a conduit. This number is used to determine the thermal development on the entrance to ducts.

Grashof Number  The ratio of buoyant to viscous forces.
 Gravimetric Factor  The number by which the weight of a gravimetric determination has to be multiplied in order to obtain the mass of the analyte it contains.
H
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Hagen Number  Used in forced flow calculations. The equation is similar to the Grashof Number but for forced flow rather than natural convection.

Hartmann Number  The ratio of electromagnetic force to the viscous.
 Hatta Number  Compares the rate of reaction in a liquid film to the rate of diffusion through the film.
 HazenWilliams Coefficient  Used in the HazenWilliams Equation. The lower the coefficient, the smoother the pipe is. The higher the coefficient, the less fluid flow is restricted.
 HazenWilliams friction coefficient  See HazenWilliams Coefficient
 Heat Capacity Ratio  See Specific Heat Ratio

Hedstrom Number  A dimensionless quantity used in fluid dynamics.
 Helmholtz Number  The ratio of a characteristic length in acoustics.
 Hydraulic Efficiency  Is the ratio of hydrodynamic energy in the form of fluid to the amount of mechanical energy delivered to the rotor.
 Hydraulic Gradient  The change in height (pressure) to length between any two points.
I
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 Iribarren Number  Used to model several effects of surface gravity waves on beaches and coastal surfaces.
J
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Jakob Number  The ratio of sensible latent heat absorbed or released during liquid vapor phase change.
K
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 Karlovitz Number  The ratio of chemical time scale to Kolmogorov time scale.
 Keulegan–Carpenter Number  Describes the relative importance of the drag forces over inertia forces for bluff objects in an oscillatory fluid flow.
 Kinetic Friction Coefficient  The amount of force that resists motion at a constant velocity.
 Knudsen Number  Used in fluid dynamics to determine what type of analysis should be used to model an object moving through a fluid. It is defined as the ratio of the molecular mean free path length to a representative physical length scale.
 Kutateladze Number  Is the reciprocal of Jakok Number. Note that there is a second Kutateladze Number that is related to electric arcs.
L
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Laplace Number  Used in the charcaterization of free surface fluid dynamics. It is a ratio of surface tension to the momentum transport inside a fluid.

Lewis Number  The ratio of thermal diffusivity to mass diffusivity. It is used to characterize fluid flows where there is simultaneous heat and mass transfer.
 Lift Coefficient  Represents the lift generated by a lifting body to the fluid density around the body, the density of the fluid around the body, velocity of the fulid and an associated reference area.
 Liquid Critical Pressure Ratio Factor FF  The ratio between pressure in vena contracta and the pressure at the maximum effective choke pressure drop across the valve.

Liquid Index  Scaling the natural moisture content of a soil sample to the liquid limit and plastic limit.
 Liquid Limit  The minimum water content at which soil just begins to flow.
 Liquid Pressure Recovery Factor  The ratio of pressure drop that occures between the vena contracta and the upstream pressure.
 LockhartMartinelli Parameter  Used in twophase flow calculations. It expresses the liquid fraction of a flowing fluid.
 Lorentz Factor  The factor by which time, length, and relativistic mass change for an object while that object is motion.
 Loss Coefficient  Measures the minor loss to the change in velocity due to friction thru pipes, fittings, and valves.
 Love Number  Parameters that measure the rigidity of a planetary body and the susceptibility of its shape to change in responce to a tidal potential.
 Lundquist Number  A ratio that compares the timescale of an Alfven wave crossing to the timescale of resistive diffusion.
M
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Mach Number  The ratio of the velocity of flow to the velocity of sound. The speed of sound in this equation is dependent on the density of the medium that the sound is traveling through.
 Machinability Index  Used to compare the machinability of different materials in the various cutting process.
 Manning's Roughness Coefficient  Measures the frictional resistance exerted by a channel, culvert or pipe on the flow.
 Marangoni Number  Compares the rate of transport due to Marangoni flows, with a rate of transport of diffusion.
 Margin of Safety  The factor of safety minus 1.
 Markstein Number  Characterizes the effect of local heat release of a propagating flame on variations in the surface topology along the flame and the associated local flame front curvature.
 Mass Transfer Coefficient  A porportional constant to the difference in the concentrations and the rate of mass transfer.
 Mechanical Efficiency  The ratio of the work output to work inpit.
 Moisture Content  The quantity of water contained in a material.
 Mole Fraction  The number of molecules (moles) of a solution in a mixture divided by the total number of moles present in the solution.

Morton Number  The shape of bubbles or drops moving in a surrounding fluid or continuous phase.
 Motor Efficiency  The ratio of shaft power out and electric power input of a motor.
N
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 Normalized Water Content  The water content for which the gradient becomes zero.
 Number Density  The number of items per unit volume.

Nusselt Number  Describes the ratio of convective heat transfer to conductive heat across a boundary.
O
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 Octane Number  The antiknock properties of a liquid motor fuel with a higher number indicating a smaller likelihood of knocking.

Ohnesorge Number  Relates the viscous force to inertial forces and surface tension forces.
 Overall Efficiency  Measures the ratio of the output of actual power to the input of actual power. It determines the amount of energy lost overall.
P
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Péclet Number  Defined as a ratio of heat transport by convection to heat transport by conduction.
 Pi  Archimedes' constant number that never ends and never repeats. The ratio of the circumference of a circle to the diameter.
 Pigging Efficiency  Expressed in percentage and always less than 100%.
 Pipeline Parameter  Proportional to maximum water hammer pressure rise and static pressure.
 Piping Geometry Factor  The pressure and velocity changes caused by fittings such as bends, expanders, reducers, tees, and Y's if directly conected to the valve.
 Pitting Resistance Equivalent Number (PREN)  A way to measure the pitting corrosion resistance of various types of stainless steel in a chlorine based environment.
 Plastic Index  The range of water content over which the soil remains in the plastic state.
 Plastic Limit  The water content at which the soil changes from semisolid state to solid state.
 Poisson's Ratio  The elastic ratio between lateral strain and longitudinal strain.
 Porosity  The ratio between the pore volume of a substance and its total volume. Porosity of a measure of a rocks capacity to store fluids.
 Power Number  Represents the proportional drag force and inertial force. It is used in momentum transfer and power consumption by fans, mixers and pump impellers at a specified rotational speed.
 Prandtl Number  In fluid dynamics is used to calculate force by the ratio of momentum diffusivity (kinematic viscosity) and thermal diffusivities.
 Pressure Coefficient  The relative pressures throughout a flow field in fluid dynamics.
 Pump Efficiency  The ratio of total horsepower output to shaft horsepower input of the pump or power output to power input.
Q
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 Quality Factor  The underdamped condition of an oscillator or resonator.
R
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 Ratio of Specific Heat  See Specific Heat Ratio
 Rayleigh Number  Associated with free or natural convection. It is a modified Grashof number used for natural convection calculations.
 Relative Density  The density or ratio of any substance relative to another substance.
 Relative Humidity  A measure of the amount of moisture in the air with respect to the temperature.
 Relative Roughness  Of a pipe is a ratio of the surface roughness to the diameter of the pipe.

Resistance Coefficient  How much resistance to the flow an obstacle has.
 Reynolds Number  Measures the ratio of inertial forces to viscosity forces.
 Reynolds Number for Liquid 

Richardson Number  The ratio of the buoyancy term to the flow shear term.

Roshko Number  Describing oscillating flow mechanisms.

Rossby Number  Used in describing fluid flow. It is the ratio of internal to Coriolis force.

Rouse Number  Used in fluid dynamics showing how a concentration of suspended sediment will be transported in a flowing fluid.
 Runoff Curve Number  The water flow that occurs when the soil is infiltrated to full capacity and excess water or rain.
S
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 Safety Factor  See Factor of Safety

Schmidt Number  Used in analyzing fluid flows where there is an interface between two different fluids.

Shape Factor  Used in structural it is the strength of the shape no matter the scale for a given load, bending, torsion, or twisting.
 Shear Strain  Is opposing forces acting parrallel to the crosssection of a body.

Sherwood Number  The ratio between convective mass transfer and diffusive mass transfer.
 Shrinkage Limit  The maximum water content at which further reduction in the water content will not cause decrease in volume of soil.
 Skin Friction Coefficient  The ratio of shear stress and dynamic pressure of a free stream.
 Soil Erodibility Factor  A quantitative description of soil detatchment by runoff and raindrop impact of a particular soil.

Sommerfield Number  Used extensively in hydrodynamic lubrication analysis.
 Specific Gravity  The density or ratio of any substance to another substance.
 Specific Gravity of Gas  The ratio of the density of the gas to the density of air at a standard pressure and temperature.
 Specific Gravity of Soil  The mass of solids in the soil compared to the mass of water at the same volume.
 Specific Heat Ratio  The ratio of two specific heats or the ratio of the heat capacity at constant pressure to heat capacity at constant volume.
 Spring Index  The ratio between the spring and wire diameter for a helical spring.

Stanton Number  The heat transfer into a fluid to the thermal capacity of fluid.
 Static Efficiency  A measure of an air mover's efficiency based on its air horsepower in terms of flow and static pressure vs. required shaft input power.
 Static Friction Coefficient  The amount of force that resists motion that is on the verge of motion.

Stefan Number  The ratio of sensible heat to latent heat.

Stokes Number  The behavior of particles suspended in a fluid flow.

Stuart Number  The ratio of electromagnetic to inertial forces.
 Strain  A linear strain or longitudinal strain, is the deformation, stretched or compressed, of a material compared to its original length.
 Strouhal Number  Used to describe oscillating flow. It is used in fluid dynamics and is the ratio of inertial forces due to the unsteadiness of the flow or acceleration of inertial forces due to the changes in velocity between points in a flow field.
 Suction Head Coefficient  The angular velocity of the turbomachine shaft and the turbomachine impeller diameter.
 Support Practice Factor  Control particles that reduce the erosion potential of runoff by their influence on drainage paterns.
T
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Tayor Number  The quantity that characterizes the importance of centrifugal forces due to the rotation of a fluid about an axis, relative to viscous forces.
 Thermal Efficiency  The fraction of heat that is converted to work or desired output divided by required input.
 Tortuosity  The dispersion of fluid flow in porous media.
 Total Pump Efficency  The conversion of one form of energy into another form of energy.

Turbulent Prandtl Number  The ratio between the momentum eddy diffusivity and the heat transfer eddy diffusivity.
U
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 Uniformity Coefficient  Classifies a soil as well graded or poorly graded.

Ursell Number  Indicates the nonlinearity of long surface gravity waves on a fluid layer.
V
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Vadasz Number  Governs the effect of porosity on flow in a porous media.
 Valve Coefficient  See Flow Coefficient
 Void Ratio  The ratio of the volume of voids to the volume of solids.
 Volumetric Efficiency  The more air inside the combustion chamber, the more fuel that can be burned and the higher the output engine torque and power.

Von Karman Constant  The logarithmic law describing the distribution of the longitudinal velocity in the wallnormal direction of a turbulent fluid flow near a boundary with a noslip condition.
W
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 Water Content  The ratio of weight of soil to the weight of water or the quantity of water contained in a material.
 Wahl Correction Factor  A method to find out the effect of direct sheat and change in coil curvature in a spring.
 Weber Number  Used in fluid mechanics is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces.

Weissenberg Number  Compares the elastic forces to the viscous forces.

Womersley Number  The pulsatile flow frequency in relation to viscous effects.
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Tags: Glossaries