1) (60) Make the usual assumptions for an ideal air-standard Brayton cycle, i.e Steady state Fluid is ideal gas with constant specific heats Combustion is replaced by constant-pressure heat addition Inlet/exhaust is replaced with constant-pressure heat removal. Fuel mass flowrate is neglected Neglect kinetic and potential energy changes. Turbine and compressor are adiabatic; There are no internal irreversibilities Now comes the interesting part. Assume that state 1 and T3 are constant, and you are investigating the effect of different choices of T, i.e. you are investigating the pressure ratio as a design parameter. Notes: For fixed inlet state, there is a one-to-one correspondence between values of pressure ratio and values of T. Keeping T3 constant is a reasonable choice for a design study, if you select T3 to be the highest temperature that can be tolerate by turbine blade materials. For each choice of T2, it is possible to select a value of Qin/m that will result in the required value of T3. Only positive values of Qin/m are physically realistic (a) Sketch T-s diagrams for three different ideal Brayton cycles that have the same state 1 and the same value of T3. Then use the diagram and energy balances to help you fill out the following table: Quantity As T increases, does the quantity increase, decrease, or stay the same? Ta Pressure ratio Turbine power output per mass flowrate Compressor power input per mass flowrate Combustor heat transfer rate per mass flowrate (b) Use energy balances to write an expression for W/m as a function of T2, T1, T3, T4, K, Cp, and/or c. Then express T4 in terms of a subset of the T2, T1, T3, k, Cp, and/or Cv. Substitute in to obtain Wnet/m as a function of T2, T1, T3, K, Cp, and/or Cv. W net vs. T. This (e) For T1-300 K, P=1 bar, T3=1500 K, k=1.4, Cp=1004 J/(kgK), plot should be a quantitative, properly formatted plot. Your plot should have at least 10 values of T2, including one that produces a value close to the maximum of Vnet/m What are the values of T2 and of the pressure ratio corresponding to the highest value of Wnet/m? Also plot efficiency versus T2. You may derive your own expression for efficiency or use an already derived expression. (d) Now get the same result with calculus. Perform the appropriate derivative (and set equal to zero) to find an expression for the choice of T2 that gives the maximum net power output per mass flow rate working fluid, for fixed T1, P1, T3, and gas properties. Plug in numerical values from c) to find whether your result is consistent with (c). (e) Refer to your plot from part c and also consider how efficiency varies with the choice of T2, as seen in the following figure: Efficiency, dimensionless Ideal Brayton cycle efficiency vs post-compressor temperature, for T =300 K and k=1.4 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 1400 1600 Post-compressor temperature T2, K How can you combine these two pieces of information to make recommendations for a choice of T2? Are some choices of T2 clearly poor choices?