A steel frame (K = 45 W m^–1 °C^–1) of total length 60 cm and cross-sectional area 0.20 cm², forms three sides of a square. The free ends are maintained at 20°C and 40°C. Find the rate of heat flow through a cross-section of the frame.

An icebox almost completely filled with ice at 0°C is dipped into a large volume of water at 20°C. The box has walls of surface area 2400 cm², thickness 2.0 mm and thermal conductivity 0.06 W m^–1 °C^–1. Calculate the rate at which the ice melts in the box. Latent heat of fusion of ice = 3.4 × 10⁵ J kg^–1.

A pitcher with 1 mm thick porous walls contains 10 kg of water. Water comes to its outer surface and evaporates at the rate of 0.1 g s^–1. The surface area of the pitcher (one side) = 200 cm². The room temperature = 45°C, latent heat of vaporization = 2.27 × 10⁶ J kg^–1, and the thermal conductivity of the porous walls = 0.80 J s^–1m^–1 °C^–1. Calculate the temperature of water in the pitcher when it attains a constant value.

Water at 50°C is filled in a closed cylindrical vessel of height 10 cm and cross-sectional area 10 cm². The walls of the vessel are adiabatic but the flat parts are made of 1 mm thick aluminium (K = 200 J s^–1 m^–1 °C^–1). Assume that the outside temperature is 20°C. The density of water is 1000 kg m^–3, and the specific heat capacity of water = 4200 J k^–1 m^–1 °C^–1. Estimate the time taken for the temperature to fall by 1.0°C. Make any simplifying assumptions you need but specify them.

The left end of a copper rod (length = 20 cm, area of cross-section = 0.20 cm²) is maintained at 20°C and the right end is maintained at 80°C. Neglecting any loss of heat through radiation, find

(a) the temperature at a point 11 cm from the left end and

(b) the heat current through the rod. Thermal conductivity of copper = 385 W m^–1 °C^–1.