A long rod of 60-mm diameter and thermophysical properties rho = 8000 kg/m3 , c = 500 J/kg·K, and k = 50 W/m·K is initially at a uniform temperature and is heated in a forced convection furnace maintained at 750 K. The convection coefficient is estimated to be 1000 W/m2 ·K. (a) What is the centerline temperature of the rod when the surface temperature is 550 K? (b) In a heat-treating process, the centerline temperature of the rod must be increased from Ti = 300 K to T = 500 K. Compute and plot the centerline temperature histories for h = 100, 500, and 1000 W/m2 ·K. In each case the calculation may be terminated when T = 500 K.

An external consultant recommends that a plant installs a bank of capacitors for power factor correction. this will reduce the peak electrical demand charges by an average of 93 kw every month. the plant current pays $13 per kw in peak demand charges. the capacitor bank will include 223 kw of fixed capacitors, and 183 of variable capacitors. the fixed capacitors cost $59 per kw, and the variable capacitors will cost $65 per kw. the consultant charges 21% of the equipment costs to install the capacitors. because this project will reduce the demand for the electric utility, they are prepared to provide a one-time rebate of $42 per kw of reduced demand. what is the simple payback period for this project (in years)?