Boundary-layer thickness ratio and Prandtl number The ratio of thermal boundary-layer thickness to hydrodynamic boundary-layer thickness over a surface scales approximately as (Prandtl number)^n. What is the typical value of the exponent n (engineering correlation)?

Introduction / Context:
When a fluid flows over a surface, two boundary layers develop: a hydrodynamic boundary layer where velocity transitions from zero at the wall to free-stream, and a thermal boundary layer where temperature transitions from wall temperature to free-stream. The relative growth of these layers depends on the Prandtl number, Pr = ν/α, where ν is kinematic viscosity and α is thermal diffusivity.

Given Data / Assumptions:

• External flow over a flat plate (laminar regime reference) or correlations adapted for engineering estimates.
• Prandtl number greater than about 0.6 for common liquids and gases in many applications.
• Interest in scaling of δ_t / δ.

Concept / Approach:
Analyses and empirical correlations show that for Pr > 0.6, the thermal boundary layer is thinner than the velocity boundary layer when Pr > 1, and thicker when Pr < 1. A commonly used engineering correlation is δ_t / δ ≈ Pr^(−1/3). This captures the trend that increasing Pr (lower thermal diffusivity) reduces thermal boundary-layer thickness relative to the hydrodynamic layer.

Step-by-Step Solution:

Verification / Alternative check:
More detailed similarity solutions (Pohlhausen, Leveque approximations) yield exponents around −1/3 for many practical ranges. Turbulent-flow analogies also retain a −1/3 exponent in simplified forms for Prandtl numbers not far from unity.

Why Other Options Are Wrong:

−2/3 or −1: exaggerate the sensitivity to Pr and do not match standard correlations.1 or 0: would imply the thermal layer is equal to or independent of velocity layer thickness, which is not generally correct.

Common Pitfalls:
Assuming the correlation is exact for all flows. The exponent can change for extreme Pr or different geometries, but −1/3 is a widely taught engineering rule-of-thumb for external laminar flow and often used in turbulent analogies.

Final Answer:

-1/3