Problem 14.2 (Stoecker 3rd. ed.): Pure water recovery (distillation) in boiler operation. (Use Lagrange multiplier method) One of the methods by which some of the water from the blowdown of a boiler can be recovered is to throttle it to a flash tank, generating vapor and concentrated liquid with unwanted metal ions for discharge. Then, the vapor from the flash tank moves to a condenser, where all of the vapor is condensed to the steam, which is recovered as a condensate from the bottom of the condenser as shown below. The condensate can be reused as a boiler feedwater, if desired. Given: The enthalpy of water: h [kJ/kg] = 4.19t The enthalpy of water vapor: hg [kJ/kg] = 2502 + 1.815 t Condenser heat transfer: UA = 335/(wc)0.2 Throttle valve Blowdown, 2.8 kg/s Saturated vapor h = 1185 kJ/kg 1 C Flash tank Condenser UA 25C 70 kg/s wd kg/s t C wc kg/s Discharged Recovered condensate (a) (b) From Condenser heat transfer, we get f = UA - 335/(w0.2) Construct the heat balance equation at the flash tank. The thermal energy into the flash tank is equal to the sum of the thermal energy out of the flash tank. From the heat balance equation at the flash tank, construct f (c) Construct the heat balance equation at the condenser. The thermal energy into the condenser is equal to the sum of the thermal energy out of the condenser. From the heat balance equation at the flash tank, construct f (d) Determine the flow rate of condensed water recovered, wc, through the system simulation using Lagrange multiplier method. (d-1) Construct the matrix of partial derivatives t UA W af af2 af2 1 8x3 af3 af 3 af3 1 ax3 af1 af1 af1 1 8x2 ax3 t UA Wc (d-2) Construct the matrix of partial derivatives with numbers using the following values near the solutions: t=40C, wc = 1.2 kg/s, and UA = 320 kW/K, t UA Wc f (d-3) Multiply the lower-triangular terms by the constant (d-4) Solve the matrix to determine the values of the constant (d-5) Determine whether the calculation shall converge or diverge.