Question

the rate at which heat is transferred between two bodies is typically proportional to the temperature difference between them. Under these conditions, show that the cost of running an air conditioner is proportional to the temperature difference between inside and outside squared.

Answer 1

As we know that the rate of heat loss between a hot and cold body is given by:

Where A is the area of interface between hot and cold body, T_h is the temperature of hot body and T_c is the temperature of the cold body and 'k' is the heat resistivity of the material.

Hence we need to supply exactly the same amount of power to maintain such a temperature difference. hence, we need to provide electricity at power:

As the cost is given by total energy consumed, we get total energy consumption as:

If the ac is run for time 't'.

Hence, for a price of per unit "r"heating cost will be:

Which is proportional to temperature difference.

(Pl note that cost will be proportional to temperature difference not its square)