Planning a School Water Tank Under Winter Constraints
A girls' secondary school in Bamyan has a rooftop water tank of 2.0 m height and 1.5 m diameter, filled from a spring-fed line. In winter the tank is exposed and the caretaker reports that a full tank froze solid over three consecutive nights when the minimum air temperature was -12 °C, -15 °C and -9 °C. Water has a specific heat capacity of about 4200 J/(kg·°C), a latent heat of fusion of about 334000 J/kg, and density 1000 kg/m³. The tank is uninsulated steel, and the caretaker says the surface felt as cold as the air within an hour of sunset. The school wants to know whether insulation alone, or draining the tank nightly, is the better response.
1. Estimate the mass of water in the full tank and explain why the freezing event is plausible given the reported temperatures.
Read the worked answer
The tank is a cylinder: V = πr²h with r = 0.75 m, h = 2.0 m. V = π × (0.75)² × 2.0 ≈ π × 0.5625 × 2.0 ≈ 3.53 m³. Mass m = ρV ≈ 1000 × 3.53 ≈ 3530 kg.
Freezing that mass requires removing the latent heat Q = mL ≈ 3530 × 334000 ≈ 1.18 × 10⁹ J, plus sensible cooling from, say, 10 °C to 0 °C: 3530 × 4200 × 10 ≈ 1.48 × 10⁸ J. The latent term dominates.
The plausibility argument: an uninsulated steel tank has a very small thermal resistance, so its surface tracks air temperature closely. Over three nights at -9 to -15 °C there is a sustained temperature difference of roughly 10-15 °C between water and air, and a large exposed steel surface area (side plus top, roughly 2πrh + πr² ≈ 9.4 + 1.8 ≈ 11 m²). Heat loss is continuous and the water is not being replaced at night, so the total heat that must be removed is achievable across the three nights. The judgement is that the event is physically consistent, not an exaggeration.
2. The school can afford either (a) wrapping the tank in insulation or (b) a valve and drain routine each evening. Using the physics, recommend one and justify it.
Read the worked answer
Insulation slows heat loss but does not stop it; it only lengthens the time to freeze. With a large thermal mass (≈3530 kg) and only three cold nights, insulation might buy a few extra hours per night, but the tank would still be at risk on an extended cold spell, and a partially full tank freezes faster per unit mass.
Draining removes the water entirely, so there is no mass to freeze. The residual film is small and cannot burst the tank. Draining also protects the pipework if the valve is at the low point.
The judgement: draining is the more reliable response because it addresses the cause (standing water) rather than slowing the rate. Insulation is a useful supplement for daytime and for the inlet line, but on its own it leaves a residual risk that a single very cold night could convert into a burst tank. Recommend drain-and-refill, with insulation as a secondary measure.