The Whistling Tea Kettle · AI Generator #GEN-FAF3AA49

Physics · thermodynamics

Problem

A tea kettle whistles when steam escapes through a small circular hole in its spout. The whistle occurs because the escaping steam creates vortices that oscillate at a specific frequency. For a particular kettle, the whistle frequency is found to be proportional to the square root of the absolute temperature of the steam. When the kettle is at sea level (atmospheric pressure P0=1.01×105  PaP_0 = 1.01 \times 10^5\;\text{Pa}), it whistles at f1=2400  Hzf_1 = 2400\;\text{Hz} when the water just begins to boil. The steam escapes through a circular hole of diameter d=3.0  mmd = 3.0\;\text{mm}. The kettle is then taken to a mountain cabin at an altitude where the atmospheric pressure is P2=0.85×105  PaP_2 = 0.85 \times 10^5\;\text{Pa}. (a) At what temperature does water boil at the mountain cabin? (b) What is the whistle frequency when the water just begins to boil at the mountain cabin? (c) If the hole diameter were increased to d′=4.5  mmd' = 4.5\;\text{mm} at sea level, by what factor would the mass flow rate of steam increase? Assume steam behaves as an ideal gas with molar mass M=18.0  g/molM = 18.0\;\text{g/mol}, and use the Clausius-Clapeyron relation: dPdT=LTΔV\frac{dP}{dT} = \frac{L}{T \Delta V} where L=2.26×106  J/kgL = 2.26 \times 10^6\;\text{J/kg} is the latent heat of vaporization.

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