Dimensionless Numbers in Process Engineering: Definitions, Formulas, and Applications

Dimensionless numbers are very important in Process Engineering, when needing a refresher about one of them, this page is giving the necessary details to understand calculate key dimensionless numbers :

Eu : Euleur number
Fo : Fourier number
Gr : Grashof number
Ma : Mach number
Nu : Nusselt number
Pe : Peclet number
Re : Reynolds number
Sc : Schmidt number
St : Stanton number
We : Weber number

1. Dimensionless numbers : symbols, formula, meaning

Euler Number (Eu)

Symbol: Eu
Formula: Eu = (ρ * v^2 * L) / P

ρ: Fluid density (kg/m³)
v: Flow velocity (m/s)
L: Characteristic length (m)
P: Pressure (Pa)

Meaning: The Euler number is a measure of the ratio of kinetic energy to the potential energy of a fluid flow. It's used to determine whether pressure forces or kinetic forces dominate in a flow.
Applications: Essential in the analysis of compressible flows in fluid dynamics and aerodynamics. It's crucial in designing nozzles, diffusers, and other flow control devices.

Fourier Number (Fo)

Symbol: Fo
Formula: Fo = α * t / L^2

α: Thermal diffusivity (m²/s)
t: Time (s)
L: Characteristic length (m)

Meaning: The Fourier number represents the ratio of thermal diffusivity (α) to the square of a characteristic length (L) and time (t). It indicates how quickly heat conduction occurs in a solid or fluid.
Applications: Critical in heat conduction analysis in materials and process systems. Used in the design of heat exchangers and predicting temperature profiles during heating or cooling.

Froude Number (Fr)

Symbol: Fr
Formula: Fr = v / √(g * L)

v: Flow velocity (m/s)
g: Acceleration due to gravity (m/s²)
L: Characteristic length (m)

Meaning: The Froude number characterizes the relative significance of inertial forces to gravitational forces in a fluid flow. It distinguishes between different flow types, such as subcritical and supercritical.
Applications: Used in open-channel flow analysis, dam design, and ship stability calculations. Determines if waves will form in flowing water.

You can access examples of calculations using the Froude number here :

Grashof Number (Gr)

Symbol: Gr
Formula: Gr = (g * β * ΔT * L^3) / ν^2

g: Acceleration due to gravity (m/s²)
β: Coefficient of thermal expansion (1/K)
ΔT: Temperature difference (K)
L: Characteristic length (m)
ν: Kinematic viscosity (m²/s)

Meaning: The Grashof number characterizes the importance of buoyancy forces compared to viscous forces in natural convection flows. It helps determine whether natural convection occurs.
Applications: Used in the analysis of natural convection heat transfer in various systems, including buildings, electronics cooling, and solar collectors.

You can access examples of calculations using the Grashof number here :

Heat transfer coefficient in natural convection

Mach Number (Ma)

Symbol: Ma
Formula: Ma = v / c

v: Flow velocity (m/s)
c: Speed of sound in the fluid (m/s)

Meaning: The Mach number quantifies the flow velocity of a fluid compared to the speed of sound in that fluid. It determines whether the flow is subsonic, sonic, or supersonic.
Applications: Essential in aerodynamics, especially in the design of aircraft and rockets. Also used in high-speed flows like those in nozzles and compressors.

Nusselt Number (Nu)

Symbol: Nu
Formula: Nu = (h * L) / k

h: Convective heat transfer coefficient (W/(m²·K))
L: Characteristic length (m)
k: Thermal conductivity (W/(m·K))

Meaning: The Nusselt number characterizes the convective heat transfer in fluid flows. It relates the rate of heat transfer to the thermal conductivity and length scale of the system.
Applications: Essential in designing heat exchangers, condensers, and cooling systems. It governs the efficiency of heat transfer processes.

You can access examples of calculations using the Nusselt number here :

Peclet Number (Pe)

Symbol: Pe
Formula: Pe = (v * L) / α

v: Flow velocity (m/s)
L: Characteristic length (m)
α: Thermal diffusivity (m²/s)

Meaning: The Peclet number represents the ratio of convective heat transfer to conductive heat transfer within a fluid flow. It indicates whether conduction or convection dominates in heat transfer.
Applications: Used in heat and mass transfer analysis. In chemical engineering, it's vital for modeling transport processes in porous media, such as catalytic reactions in packed beds.

Prandtl Number (Pr)

Symbol: Pr
Formula: Pr = μ * cp / k

μ: Dynamic viscosity (Pa·s)
cp: Specific heat capacity at constant pressure (J/(kg·K))
k: Thermal conductivity (W/(m·K))

Meaning: The Prandtl number describes the relative importance of momentum diffusivity (viscosity) to thermal diffusivity. It indicates whether temperature or velocity gradients dominate in a flow.
Applications: Crucial for analyzing heat transfer in liquids and gases. Used in designing processes involving convection and diffusion, like combustion and melting.

You can access examples of calculation using the Prandlt number here :

Rayleigh Number (Ra)

Symbol: Ra
Formula: Ra = (g * β * ΔT * L^3) / (ν * α)

g: Acceleration due to gravity (m/s²)
β: Coefficient of thermal expansion (1/K)
ΔT: Temperature difference (K)
L: Characteristic length (m)
ν: Kinematic viscosity (m²/s)
α: Thermal diffusivity (m²/s)

Meaning: The Rayleigh number combines the effects of buoyancy (Grashof number) and thermal diffusivity (Peclet number) in natural convection flows. It characterizes the onset of convection.
Applications: Used to predict the transition from laminar to turbulent natural convection in various systems, including geophysics, metallurgy, and astrophysics.

Reynolds Number (Re)

Symbol: Re
Formula: Re = (ρ * v * L) / μ

ρ: Fluid density (kg/m³)
v: Fluid velocity (m/s)
L: Characteristic length (m)
μ: Dynamic viscosity (Pa·s)

Meaning: The Reynolds number represents the ratio of inertial forces to viscous forces within a fluid flow. It helps classify flow regimes as laminar or turbulent.
Applications: Used in designing pipelines, heat exchangers, and pumps. It helps determine whether flow is turbulent or laminar, impacting heat transfer, friction, and mixing efficiency.

You can access examples of calculation using the Rey number here :

Schmidt Number (Sc)

Symbol: Sc
Formula: Sc = μ / (ρ * D)

μ: Dynamic viscosity (Pa·s)
ρ: Fluid density (kg/m³)
D: Mass diffusivity (m²/s)

Meaning: The Schmidt number relates the momentum diffusivity (viscosity) to mass diffusivity (diffusion coefficient). It is important in mass transfer processes involving molecular diffusion.
Applications: Used in chemical engineering for modeling mass transfer in gas-liquid and liquid-liquid systems, such as absorption and extraction processes.

Stanton Number (St)

Symbol: St
Formula: St = (h / (ρ * v * cp))

h: Convective heat transfer coefficient (W/(m²·K))
ρ: Fluid density (kg/m³)
v: Fluid velocity (m/s)
cp: Specific heat capacity at constant pressure (J/(kg·K))

Meaning: The Stanton number represents the ratio of heat transfer at the surface to the thermal capacity of the fluid. It quantifies the effectiveness of heat transfer.
Applications: Used to assess heat exchanger performance and efficiency. It's crucial in designing and optimizing heat exchangers for various industrial processes.

Weber Number (We)

Symbol: We
Formula: We = (ρ * v^2 * L) / σ

ρ: Fluid density (kg/m³)
v: Flow velocity (m/s)
L: Characteristic length (m)
σ: Surface tension (N/m)

Meaning: The Weber number describes the relative significance of inertial forces to surface tension forces in a fluid flow. It determines whether droplets or jets form.
Applications: Important in fluid dynamics and multiphase flow studies, especially in predicting the breakup of droplets or bubbles in processes like spraying and atomization.

Common Dimensionless Numbers in Process Engineering

Including Reynolds number, Prandtl number, Fourier number