Q.26 Prandtl number is the ratio of
(A) thermal diffusivity to momentum diffusivity
(B) mass diffusivity to momentum diffusivity
(C) momentum diffusivity to thermal diffusivity
(D) thermal diffusivity to mass diffusivity
Correct Answer: (C) momentum diffusivity to thermal diffusivity
The Prandtl number (Pr) precisely measures how momentum spreads through a fluid compared to heat, making option C the standard definition used across heat transfer and fluid mechanics.
Option Analysis
(A) Thermal diffusivity to momentum diffusivity
This inverts the Prandtl number (1/Pr), which applies to fluids where heat diffuses faster than momentum, like liquid metals (Pr < 1), but does not define Pr itself.
(B) Mass diffusivity to momentum diffusivity
This describes the Schmidt number (Sc = ν/D), relevant for mass transfer processes, not heat transfer.
(C) Momentum diffusivity to thermal diffusivity
Correct. Pr = ν/α = (μ/ρ) / (k/(ρ Cp)), where ν is kinematic viscosity (momentum diffusivity) and α is thermal diffusivity. This ratio governs boundary layer thickness ratios in convection.
(D) Thermal diffusivity to mass diffusivity
This defines the Lewis number (Le = α/D), used in combined heat and mass transfer, such as evaporation or combustion.
Prandtl Number Fundamentals
The Prandtl number, a key dimensionless group in fluid mechanics, represents the ratio of momentum diffusivity to thermal diffusivity. Named after Ludwig Prandtl, it helps engineers predict how heat and momentum interact in flowing fluids during convection processes.
Formula and Calculation
Prandtl number formula: Pr = ν/α = cpμ/k
- ν (momentum diffusivity) = kinematic viscosity = μ/ρ
- α (thermal diffusivity) = k/(ρcp)
- μ: dynamic viscosity, k: thermal conductivity, cp: specific heat, ρ: density
Typical Values
| Fluid | Prandtl Number (Pr) | Interpretation |
|---|---|---|
| Air | ~0.7 | Momentum and thermal diffusion similar |
| Water | ~7 | Momentum diffuses faster than heat |
| Liquid Mercury | ~0.025 | Heat diffuses much faster (Pr << 1) |
| Engine Oil | ~1000 | Very viscous, momentum dominates (Pr >> 1) |
Physical Significance
When Pr > 1, the velocity boundary layer grows thicker than the thermal one, common in oils; Pr < 1 means the opposite, as in liquid metals where conduction dominates convection. This ratio directly influences Nusselt number correlations for convective heat transfer design in heat exchangers and cooling systems.
Applications in Engineering
Prandtl number guides boundary layer analysis in aerodynamics, boiler design, and biochemical reactors—fields where molecular biology and fermentation interests overlap with scaled-up microbial processes. For air (Pr ≈ 0.71), it ensures balanced heat transfer modeling in HVAC or bioprocess ventilation.
