EuPhO 2026 — Задача 3. Dry ice hockey (10 pts)

Theoretical Problems · 10 т.

Условие

At pressure $p_0 = 100\ \mathrm{kPa}$, solid $\mathrm{CO_2}$ (dry ice) sublimates (goes from solid to gaseous state) at $T_\mathrm{s} = -78.5\,{}^{\circ}\mathrm{C}$. Its saturated vapour pressure follows the Clausius–Clapeyron relation:

$\frac{\mathrm{d}p_{\mathrm{sat}}}{\mathrm{d}T} = \frac{\mu\lambda p_{\mathrm{sat}}}{RT^2},$

where the latent heat of sublimation is $\lambda = 600\ \mathrm{kJ/kg}$, the molar mass is $\mu = 0.044\ \mathrm{kg/mol}$, and the gas constant is $R = 8.3\ \mathrm{J/(mol \cdot K)}$. The thermal conductivity of the $\mathrm{CO_2}$ gas is $\kappa = 10\ \mathrm{mW/(m \cdot K)}$ and its dynamic viscosity is $\eta = 10\ \mu\mathrm{Pa \cdot s}$. The density of dry ice is $\rho = 1500\ \mathrm{kg/m^3}$ and the gravitational acceleration is $g = 10\ \mathrm{m/s^2}$.

A puck of radius $r = 10\ \mathrm{mm}$ consists of a disc of dry ice of thickness $h = 1\ \mathrm{mm}$, and a metal disc of mass $M = 0.01\ \mathrm{kg}$ on top of it. The initial temperature of the puck is $T_\mathrm{s}$ and the ambient pressure is $p_0$.

The puck is placed on a horizontal metal plate which is held at a constant temperature $T = T_\mathrm{s} + \Delta T$, and given an initial horizontal velocity $v = 10\ \mathrm{mm/s}$. After a very long time, the horizontal displacement $L$ of the metal disc is measured.

Treat the $\mathrm{CO_2}$ gas as ideal. Assume no tilting of the puck at any moment, all surfaces are perfectly smooth, and the thermal conductivities of the metal, dry ice, and gas satisfy $\kappa_\mathrm{metal} \gg \kappa_\mathrm{ice} \gg \kappa$.

a) When $\Delta T$ is sufficiently small, $L$ is negligible and independent of $\Delta T$. However, when $\Delta T$ reaches a critical value $\Delta T_c$, the function $L(\Delta T)$ starts to grow. Estimate $\Delta T_c$. [2 т.]

b) Estimate the maximal value $L_{\max}$ of the function $L(\Delta T)$. [8 т.]

Решение

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Оригинал в Архива: eupho2026_theory_problems.pdf · официални решения: eupho2026_theory_solutions.pdf