Consider a pan placed on top of electric range to cook. The bottom section of the pan is 0.4 cm thick and has a diameter of 18 cm. The electric heating unit on the range top consumes 800 W of power during cooking, and 80 percent of the heat generated in the heating element is transferred uniformly to the pan. Assuming constant thermal conductivity and steady state operation. The heat transfer can be approximated as being one-dimensional through the bottom surface to the top. Take the direction normal to the bottom surface to be the x-axis. a)Perform an energy balance on a thin element of the pan during a small-time interval △t , and indicate which terms may be neglected or ignored and why? b)Assuming constant thermal conductivity, obtain the differential equation that describes the variation of the temperature in the bottom section of the pan during steady state operation.

A soldering iron has a cylindrical tip of 2.5 mm in diameter and 20 mm in length. With age and usage, the tip has oxidized and has an emissivity of 0.80. Assuming that the average convection heat transfer coefficient over the soldering iron tip is 25 W/m2·K, and the surrounding air temperature is 20°C, determine the power required to maintain the tip at 400°C.

A counter-flow double-pipe heat exchanger is to heat water from 20°C to 80°C at a rate of 1.2 kg/s. The heating is to be accomplished by geothermal water available at 160°C at a mass flow rate of 2 kg/s. The inner tube is thin-walled and has a diameter of 1.5 cm. If the overall heat transfer coefficient of the heat exchanger is 640 W/m2?K, determine the length of the heat exchanger required to achieve the desired heating.