NBPhO 2025

Difficulty medium

Prerequisites

• Ideal gas law $pV = nRT$ and $p = \rho RT/\mu$
• Concept of saturated vapour pressure and phase equilibrium
• Fourier's law of heat conduction, $q = -\kappa\,\nabla T$
• Fick's law of diffusion, $J = -D\,\nabla n$
• Latent heat $L$ as an enthalpy (already includes $p\,\Delta V$ work)
• Steady-state 1D transport → linear profiles

Learning objectives

• Identify when a saturated-vapour pressure (not atmospheric) sets the mechanical equilibrium of a liquid–gas interface
• Write a latent-heat balance for evaporative cooling and justify dropping the vapour mass on the liquid side
• Recognise that, in steady state, conduction and diffusion fluxes in a laminar layer share the same $1/d$ scaling so the layer thickness cancels
• Reduce coupled heat-and-mass transport to a single implicit equation for the wet-bulb temperature
• Solve a transcendental equation graphically by overlaying a straight line on the saturation curve
• Explain why a dry sauna at $110\ °\text{C}$ is tolerable but the same temperature at 100% humidity is not

Watch out for

• Treating $L$ as an internal-energy change. $L$ is the enthalpy of vaporisation and already bakes in the $p\,\Delta V$ work the vapour does on the piston — adding a separate $p\,\Delta V$ term double-counts.
• Setting the vapour pressure equal to atmospheric. Saturated vapour pressure at $90\ °\text{C}$ is only ~$70\ \text{kPa}$, well below atmospheric; water is liquid in the initial state, and the force calculation is driven by the $30\ \text{kPa}$ deficit.
• Forgetting that at the wet skin surface the relative humidity is 1 ($r_s = 1$), regardless of the bulk $r$. The density at $x = 0$ is $\rho_{\text{sat}}(T_s)$, not $r \cdot \rho_{\text{sat}}(T_s)$.
• Keeping $d$ in the final balance. Both fluxes scale as $1/d$ in steady state, so $d$ cancels — this is the whole reason the wet-bulb temperature is a well-defined material property of the air state.
• Mixing up flux signs. Heat flows from hot air down to cool skin; water vapour flows up from the wet skin to drier air. The two fluxes are anti-parallel and the energy balance uses their magnitudes.