It has been claimed that “smart,” hydrogel-filled double-pane windows can be used to shift energy usage in buildings to off-peak times by delaying the heat transfer as a result of the heat capacity of the hydrogel.

Let’s consider a double-pane window with a 5.0 mm gap filled with a polymer-based hydrogel having a thermal conductivity of 0.30 W/m K, and density of 1,050 kg/m³, and a heat capacity of 4,000 J/kg K. The room temperature is kept at 25°C while the external temperature varies between 25°C at night to 35°C in the morning.

(a) Assuming that the heat transfer resistance through the hydrogel is dominant, predict the heat transfer flux into the room as a function of time if, starting at 25°C, the external temperature suddenly reaches 35°C. Determine the cumulative heat transfer flux into the room over a 1-h period and compare your answer with the cumulative flux that would be obtained during the same time if energy accumulation in the hydrogel were neglected.