Dimensionless numbers are ratios that compare competing transport effects such as inertia, viscosity, diffusion, conduction, convection, and buoyancy. This cheat sheet helps engineering students identify which physical effects control a flow, heat transfer, or mass transfer problem. It is especially useful for choosing correlations, checking assumptions, and scaling lab results to real systems.
These groups appear throughout fluid mechanics, heat transfer, and mass transfer because they make complex systems easier to compare.
The most common numbers include Re for flow regime, Pr and Sc for property ratios, Nu and Sh for convective transfer strength, and Pe for advection versus diffusion. Bi and Fo are central in transient conduction, while Gr and Ra measure buoyancy driven motion in natural convection. Each number is dimensionless, so units must cancel completely when the formula is written correctly.
A good first step in any transport problem is to identify the dominant mechanism, then choose the dimensionless group that compares it to the competing mechanism.
Key Facts
- Reynolds number is Re = rho V L / mu = V L / nu, and it compares inertial forces to viscous forces.
- Prandtl number is Pr = nu / alpha = mu Cp / k, and it compares momentum diffusivity to thermal diffusivity.
- Schmidt number is Sc = nu / D_AB, and it compares momentum diffusivity to mass diffusivity.
- Nusselt number is Nu = h L / k, and it compares convective heat transfer to conductive heat transfer across a length L.
- Sherwood number is Sh = k_c L / D_AB, and it compares convective mass transfer to molecular diffusion.
- Peclet number is Pe = Re Pr for heat transfer and Pe_m = Re Sc for mass transfer, comparing advection to diffusion.
- Biot number is Bi = h L_c / k_s, and Bi < 0.1 usually supports the lumped capacitance approximation for transient conduction.
- Rayleigh number is Ra = Gr Pr, where Gr = g beta DeltaT L^3 / nu^2, and it measures buoyancy driven natural convection relative to damping effects.
Vocabulary
- Characteristic length
- A representative length scale L used in a dimensionless number, such as pipe diameter, plate length, or volume divided by surface area.
- Dynamic viscosity
- Dynamic viscosity mu measures a fluid's resistance to shear deformation and has units of Pa s.
- Kinematic viscosity
- Kinematic viscosity nu is dynamic viscosity divided by density, so nu = mu / rho.
- Thermal diffusivity
- Thermal diffusivity alpha = k / (rho Cp) measures how quickly temperature changes spread through a material.
- Mass diffusivity
- Mass diffusivity D_AB measures how quickly species A spreads through species B due to concentration gradients.
- Transport correlation
- A transport correlation is an empirical or semiempirical equation that relates dimensionless numbers, such as Nu = C Re^m Pr^n.
Common Mistakes to Avoid
- Using the wrong characteristic length is a common error because Re, Nu, Sh, Bi, Gr, and Ra all depend on L in ways tied to the geometry and physical process.
- Mixing dynamic viscosity and kinematic viscosity is wrong because Re = rho V L / mu uses mu, while Re = V L / nu uses nu, and substituting one for the other changes the units.
- Applying a correlation outside its valid range gives unreliable results because formulas for Nu, Sh, or friction often require specific ranges of Re, Pr, Sc, geometry, and boundary conditions.
- Assuming Bi < 0.1 without computing it is unsafe because the lumped capacitance model only works when internal conduction resistance is much smaller than surface convection resistance.
- Forgetting that properties may depend on temperature or concentration can distort dimensionless numbers because rho, mu, k, Cp, and D_AB often change significantly across a transport process.
Practice Questions
- 1 Water flows through a pipe with rho = 998 kg/m^3, mu = 0.001 Pa s, V = 2.0 m/s, and D = 0.05 m. Calculate Re and state whether inertia or viscosity is more dominant.
- 2 Air has nu = 1.6e-5 m^2/s and alpha = 2.3e-5 m^2/s. Calculate Pr and explain what it says about momentum diffusion compared with thermal diffusion.
- 3 A solid sphere has h = 40 W/m^2 K, k_s = 15 W/m K, and characteristic length L_c = 0.004 m. Calculate Bi and decide whether lumped capacitance is likely reasonable.
- 4 Two flows have the same Reynolds number but different Prandtl numbers. Explain why their velocity boundary layers may be dynamically similar while their thermal boundary layers are not.