# Expressing thermal efficiency for a closed Brayton cycle in terms of static enthalpies

I have a Brayton cycle made of a compressor (stage 2 to 3), a heat exchanger (stage 3 to 4) and a turbine (stage 4 to 5). The turbine and compressor are connected through a shaft. The cycle is closed via a heat exchanger, where the hot recirculating air is cooled. Here is an image:

The book from which this image was taken (Fundamentals of Propulsion, by Ronald D. Flack) defines an important coefficient, called the thermodynamic efficiency, given by: $$\eta_{th}=\dfrac{\dot{W}_{net}}{\dot{Q}_{in}}$$ Right after defining this coefficient, the author says that for the ideal case this is equal to: $$\eta_{th}=1-\dfrac{h^{\prime}_5-h_2}{h_4-h^{\prime}_3}$$ where, as he says, "the primes (') serve as reminders that the processes are ideal".

How did he get to that last expression, using enthalpies?

I tried using the general expression for the conservation of energy for a (stationary) control volume (CV): $$0=\dot{Q}_\text{CV}-\dot{W}_\text{CV}+\dot{m}_\text{in}\left(h_\text{in}+\frac{v_\text{in}^2}{2}\right)-\dot{m}_\text{out}\left(h_\text{out}+\frac{v_\text{out}^2}{2}\right)$$ where:

• $\dot{Q}_\text{CV}$ is the heat transfer rate over the boundary of the CV.
• $\dot{W}_\text{cv}$ is the energy transfer by work across the boundary of the control volume.

I can apply this formula to the compressor, for example, and I'd get the following: $$\dot{W}_{c}=\dot{m}\left(h_{02}-h_{03}\right)$$

But all in all, I don't know how to express the thermal efficiency in terms of static enthalpies, as using my approach all I'd get would be stagnation enthalpies.

• Hi Jose - we tend to prefer that questions ask something specific. If all you can say is "I don't know how to continue", that's not really asking anything. Could you consider whether you can focus on a more specific aspect? That being said, this question is not too bad, relatively speaking. – David Z Mar 11 '16 at 13:22
• Hey @DavidZ thanks for your insights. Inside the "Question" block I formulated the question, which is: "how did he get to that last expression, using enthalpies?". Anyway I made it clearer at the end of the "My Attempt" block and deleted that "I don't know how to continue" part. I know there are some mathematical operations behind, and that's my specific question. Does this work or not yet? Thanks again. – Jose Lopez Garcia Mar 11 '16 at 13:35
• Yeah, that's somewhat better. Like I said, it's not really a bad question. I think your formatting (e.g. the section headings) might be getting in the way of readability a bit. If you like, I could make some slight edits to improve that. – David Z Mar 11 '16 at 13:45
• Hey @DavidZ, please go ahead and help me with the formatting. Would be great! – Jose Lopez Garcia Mar 11 '16 at 13:53
• All done, take a look and make sure everything looks OK. The changes are minor but I think they help focus on the real question. – David Z Mar 11 '16 at 14:27

You should check first law for compressor, turbine and high temperature heat exchanger.

If compressor and turbine are ideal, then we have:

First law for compressor: $w_c=h_2-h'_3$

First law for high temperature heat exchanger: $q_{in}=h_4-h'_3$

First law for turbine: $w_t=h_4-h'_5$

$$\large{\eta_{th}}=\large{\frac{w_{net}}{q_{in}}}$$

$w_{net}=w_t+w_c=h_4-h'_5+h_2-h'_3=(h_4-h'_3)-(h'_5-h_2)$

$$\Longrightarrow\;\large{\eta_{th}}=\large{\frac{(h_4-h'_3)-(h'_5-h_2)}{h_4-h'_3}}=1-\large{\frac{h'_5-h_2}{h_4-h'_3}}$$