Text 3-9: Problem 4-27 deals with the optimal thickness L * of insulation for minimizing the cost of maintaining a tank at elevated temperature 7; located in a building at temperature Ta. There are Y hours per year when heat is not required for the building and X hours per year when cooling is required. H is annual hours, A is the tank heat transfer area, C is the annual cost of unit volume of insulation of thermal conductivity k, R is the thermal resistance without insulation between tank contents and the building air, C3 is the cost of the tank heating energy, C4 is the cost of the building heating energy, C5 is the cost of the building cooling energy, and COP is the coefficient of performance of the cooling equipment. When C = HC3 - YC4+ C5X/COP, the optimal insulation thickness L* and cost C* is given by L*/Rk = max {[kC(Ti Ta)/C]/2/Rk - 1,0} and C */A = - CL*2/Rk Cost coefficients are to be evaluated for: A = 100 ft, R = 0.5 h ft F/Btu, k = 0.05 Btu/h ft F, T- Ta=100 F, H = 8766 h/yr, X = 2100 h/yr, Y = 4766 h/yr, C3 = C4 = $5/106 Btu for natural gas, C5 $20/106 Btu, and COP = 3 for vapor-compression refrigeration. The insulation lifetime n = 20 yr, the annual interest rate i = 12%, and the annual inflation rate j = 2%. (a) Obtain a numerical value for C as the capital-recovery factor, CRF, multiplied by an insulation purchase cost Cp obtained from Means Mechanical Cost Data. (b) Determine the optimal insulation thickness. (c) Show that the simple payback period tp for installation the optimal insulation thickness determined in part (b), with the purchase cost of the insulation divided by the annual energy cost saving, is tp R kC CRF C(Ti-Ta) 1/2 Provide numerical values for the purchase cost and tp. [Answers: L * = 4 in., $400, tp = 0.6 yr]