This is a long problem. Apologies and thanks in advance. 1. Given: In the GTHTR300 plant, a fraction b of the coolant mass flow rate is bled off at the exit of the compressor to cool the turbine disks, shaft bearings and the electrical generator. Due to temperature limitations of the electrical insulation materials in the generator, the temperature of the helium bleed flow at the inlet of the mixing chamber must not exceed 550 K, and T8 is taken equal to this temperature (T8 = 550 K) in the power plant. The bleed flow fraction b is then calculated using an energy balance of the bleed lines, assuming that the bleed flow picks up the shaft mechanical losses and the generator electrical losses. Problem: For a given compressor pressure ratio, pC, develop analytical expressions for the bleed flow fraction, b, the net electrical power delivered to the Grid, Pe, and the plant’s thermal efficiency, h, as functions of the 15 parameters above. The bleed flow fraction to cool the Reactor Pressure Vessel (RPV) (Figure 1) can be neglected in the plant’s performance analysis. Use the expression for h to obtain the plant’s thermal efficiencies of unrecuperated (e = 0) and recuperated (e = 1) ideal Brayton cycles, and compare your results with those given in the textbook on pages 191 and 193. Calculate the state point temperatures, bleed flow fraction and compressor mass flow rate, turbine and compressor works, generator and Grid electrical output, and thermal powers exchanged in the recuperator and precooler for the reference values of the 15 parameters given above and for a compressor pressure ratio of 2.0. You may compare your answers with the reported GTHTR300 plant performance [1 – 4]. Given Values: (1) Working fluid (helium) molecular weight: M = 4.003 g/mole (2) Stagnation temperature at the compressor inlet: T1 = 301 K (3) Stagnation pressure at the compressor exit: P2 = 7.11 MPa (4) Stagnation temperature at the reactor exit: T4 = 1123 K (5) Reactor thermal power: Qin = 600 MW (6) Recuperator effectiveness: e = 0.95 (7) Six-stage, axial flow turbine polytropic efficiency: hT = 92.8% (8) Twenty-stage, axial flow compressor polytropic efficiency: hC = 90.5% (9) Rotating shaft mechanical efficiency: hM = 99.0% (10) Electrical generator efficiency: he = 98.7% (11) Relative pressure losses in recuperator’s hot leg: DP76 / P7 = 1.9% (12) Relative pressure losses in recuperator’s cold leg: DP23 / P2 = 1.5% (13) Relative pressure losses in nuclear reactor: DP34 / P3 = 1.7% (14) Relative pressure losses in pre-cooler: DP61 / P6 = 1.5% (15) Stagnation temperature of bleed flow at mixing chamber inlet: T8 = 550 K 3. The attempt at a solution I know that the bleed flow fraction is a ratio of the bleed flow mass rate divided by the total mass flow. And I have found the bleed flow mass flow rate to be m=-[T2(he-1)+(hm+he-2)(T5-T4)-T1(he-1)]Qin/(Cp(T4-T3)[T2-T1)(he-1)-(T8-T2)] and the total mass flow is the mass flow rate of the turbine (mt=Qin/[Cp(T4-T3)] plus the bleed flow mass flow rate. Now because I am using Helium, I am assuming an isentropic with y=Cp/Cv = 1.66. Solving for P3, I get P3=P2(1-DP23), like wise for P4. These values are P3=7.00335, P4=6288429 MPa. From here, I solved for T2, and T3 using T4/T3=(P4/P3)^(y-1/y) and T3/T2=(P3/P2)^(y-1/y). T2=1137.6, and T3=1130.68. I think these numbers are way, way too high, and I want to know where I am going wrong.