The objective of this problem is to optimize a hot-oil loop in a petrochemical plant with consideration of the total worth of the costs for economic life of the facility utilizing parameters given. In refineries and petrochemical plants one of the methods of distributing heat is to heat the oil in a central furnace and pump the heated oil into heat exchangers placed into spots where heating is necessary. For the hot oilloop in the petrochemical plant under discussion a representative figure to describe the hot oil-loop is provided in Figure 1. In the plant there is a furnace, three reboilers and a pump. Given data: Oil data: Oil chemically stable up to 370C Specific heat, 2.1KJ/KgK Densitv 850Kr/m Thus T1=200C0,T2=150C0,T3=175C0 Temperature of fluid heated in each boiler is essentially constant. Cost of the reboiler is $90 per square meter of heat exchanger area a) Find W1,W2 and W3 then find the total flow rate assume that Tb=370C b) Calculate the total area of the reboilers assume the area of the reboilers is constant then calculate the cost. Furnace data: Treat the furnace as a combination of a combustion chamber and a counter flow heat exchanger Combustion air is heated to temperature Th in the combustion chamber and then flows overheatexchanger surface. U value of furnace heat exchanger, 30W/mK. First cost of furnace, $30 per square meter of heat -exchanger area. Cost of heating by natural gas, $1.20 per gigajoule Flow rate of combustion gas, 6.0Kg/s Specific heat of combustion gas, 1.1KJ/Kg. K Pump, motor, and electric data: Efficiency of pump (P),85 percent Efficiency of electric motor for the pump, 90 percent First cost of pump. $160 for each Kilogram per second of oil flow. First cost of electric motor, $50 per Kilowatt shaft power output Cost of electricity, 3.2 cents per kilowatt-hour. An approximate measure of the pressure drop throughout the loop is given by p.KPa=8.2w1.8 where w is the system flow rate in Kilograms per second. Economic data: Plan on full-time operation for the 5-year economic life of the facility Interest rate, 10 percent c) System Oil-Pressure Drop d) Size Of the Electrical Motor e) Find the optimum values of leaving temperature of oil ta,tb,th,tg Use Lagrange Multipliers and find the cost function and constraint function assume Tb=370C use oil total flow rate you found in part (a) Note: Plan on full-time operation for the 5-year economic life of the facility Interest rate, 10 percent Heat loss= Heat gain wgCpT=UAlnTbTaTaTb For constraint function Furnace Heat transfer= Oil Heat transfer QF=wgcp(tgth)=Qo=wocp(tatb)