A large container is separated into two halves by a 2.0 cm thick glass divider. The area of the divider is 0.25 m^2 and the thermal conductivity of glass is 0.90W/m•K. Each side is filled with 30 kg of water, specific heat 4180 J/kg•K, at different temperatures. Initially, the temperature difference is 20 K. How long will it take before the temperature difference is 1.0 K?
To solve this problem, we need to calculate the rate of heat transfer through the glass divider, and then determine how long it takes for the temperature difference to change from 20 K to 1.0 K.
1. Calculate the rate of heat transfer (Q) through the glass divider.
- The rate of heat transfer is given by the formula: Q = k * A * ΔT / d, where k is the thermal conductivity, A is the area of the divider, ΔT is the temperature difference, and d is the thickness of the divider.
- Substitute the given values: Q = 0.90 W/m•K * 0.25 m^2 * (20 K - 1.0 K) / 0.02 m
- Calculate Q: Q = 0.90 * 0.25 * 19 / 0.02 = 8.55 W
2. Calculate the amount of heat required to reduce the temperature difference from 20 K to 1.0 K.
- The amount of heat (Q) required to change the temperature is given by the formula: Q = m * c * ΔT, where m is the mass of water, c is the specific heat capacity of water, and ΔT is the temperature difference.
- Substitute the given values: Q = 30 kg * 4180 J/kg•K * (20 K - 1.0 K)
- Calculate Q: Q = 30 * 4180 * 19 = 239,220 J
3. Determine the time required for the temperature difference to change by dividing the amount of heat required by the rate of heat transfer.
- The time (t) is given by the formula: t = Q / P, where Q is the amount of heat required and P is the rate of heat transfer.
- Substitute the calculated values: t = 239,220 J / 8.55 W
- Calculate t: t ≈ 27,974 seconds or approximately 7.8 hours
Therefore, it will take approximately 7.8 hours for the temperature difference to change from 20 K to 1.0 K in the container.