Heat Loss

An immersion heater of power J = 500 W is used to heat water in a bowl. After 2 minutes, the temperature increases from T₁ = 85^oC to T₂ = 90°C. The heater is then switched off for an additional minute, and the temperature drops by ∆T = 1^oC. Estimate the mass of the water in the bowl. The thermal capacity of water c = 4.2 × 10³ J/kg K.

SOLUTION

Let τ₁ = 2 min be the time the heater is operating. The energy added to the water and bowl will heat the water as well as the environment. We will assume that the heat loss Q₁ to the surroundings is proportional to the elapsed time τ₁ and that the changing temperature difference between the water and room temperature T ~ 23^oC. During this phase, we may write

(1)

The heat loss Q₁ is actually a time integral of some proportionality constant λ times the temperature difference as the water heats up. However,  only varies by 5^oC out of an average 65^oC, so we will ignore the variation. The heat loss during the second phase is given by

(2)

Just as in the heating phase, the heat loss is proportional to the elapsed time τ₂ = τ₁/2. Since  again only changes a little, we have

(3)

We may now eliminate Q₁ from(1), yielding