The Cold Air-Standard Assumption

Ch 9, Lesson E, Page 11 - The Cold Air-Standard Assumption

• We can use the 1st Law from the two heat exchangers to eliminate QC and QH from the equation for the thermal efficiency of the cycle, but that leaves with all four specific enthalpies to determine.
• This seems like a step backwards until we make the COLD Air-Standard Assumption.
• Here, we assume that the heat capacities. CP and CV, of the air are constants.
• This makes life easier because we can replace the ΔH terms in the numerator and denominator of the equation for the efficency with CP time ΔT.
• Then, good things start happening.
• First, the heat capacity cancels.
• The next simplification is easier to see if we factor the numerator and denominator, as shown.
• This helps because we can find a relationship between the ratio of T3 to T4 and the ratio of T2 to T1.
• There are two key ideas here.
• One is that the compressor and turbine are isentropic.
• This lets us use the relationships we derived in lesson 7E for isentropic processes for ideal gases.
• The other key is that because the heat exchangers are isobaric, P1 = P2 and P3 = P4.
• When we use the relationship shown here for the isentropic turbine and the isentropic compressor, we find that the ratio of T1 to T4 is equal to the ratio of T2 to T3.
• A little algebraic manipulation shows that the ratio of T3 to T4 is equal to the ratio of T2 to T1.
• Yes !  This gives us crucial cancellation back in our equation for the thermal efficiency of the cycle.
• In fact the cycle efficiency depends only on the temperature change across the compressor.
• Alternatively, we can use the same isentropic relationship shown here to express the thermal efficiency in terms of the pressure ratio at the compressor, rp.  The pressure ratio is the ratio of the compressor outlet pressure to the inlet pressure.
• The thermal efficiency of the Air-Standard Brayton Cycle with constant heat capacities is 1 minus one over the pressure ratio to the gamma minus 1 over γ power.
• Remember that the thermal efficiency depends on the pressure ratio and the heat capacity ratio.
• Now, let’s see what this relationship looks like.