Mathematically relate the heat transfer vector for conduction to fourier's law of heat conduction. how does isotropy relate to idea of the heat transfer rate?

Along 8-cm diameter steam pipe whose external surface temperature is 900c connects two buildings. the pipe is exposed to ambient air at 70c with a wind speed of 50 km/hr blowing across the pipe. determine the heat loss from the pipe per unit length. (b) air at 500c enters a section of a rectangular duct (15 cm x 20 cm) at an average velocity of 7 m/s. if the walls of the duct are maintained at 100c. a) the length of the tube for an exit temperature of the air to be 40 0c. b)the rate of heat transfer from the air. c) the fan power needed to overcome the pressure drop in this section of the duct.

Starting wih an energy balance on a rectangular volume element, derive the one- dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation.

An air-conditioning system consists of a heating section and an evaporative cooler. air enters the heating section at 10°c and 70 percent relative humidity at a rate of 30m^3/min, and it leaves the evaporative cooler at 20°c and 60 percent relatively humidity. if the air conditioning system operates at a total pressure of 98 kpa, determine (a) the temperature and relative humidity of the air when it leaves the heating section, (b) the rate of heat transfer in the heating section, and (c) the rate of water added to air in the evaporative cooler