Question 9.8: A Gas Turbine with Reheating and Intercooling An ideal gas-t...

An ideal gas-turbine cycle with two stages of compression and two stages of expansion is considered. The back work ratio and the thermal efficiency of the cycle are to be determined for the cases of no regeneration and maximum regeneration.
Assumptions 1 Steady operating conditions exist. 2 The air-standard assumptions are applicable. 3 Kinetic and potential energy changes are negligible.
Analysis The T-s diagram of the ideal gas-turbine cycle described is shown in Fig. 9–46. We note that the cycle involves two stages of expansion, two stages of compression, and regeneration.
For two-stage compression and expansion, the work input is minimized and the work output is maximized when both stages of the compressor and the turbine have the same pressure ratio. Thus,

\frac{P_{2}}{P_{1}}=\frac{P_{4}}{P_{3}}=\sqrt{8}=2.83 \quad \text { and } \quad \frac{P_{6}}{P_{7}}=\frac{P_{8}}{P_{0}}=\sqrt{8}=2.83

Air enters each stage of the compressor at the same temperature, and each stage has the same isentropic efficiency (100 percent in this case). Therefore, the temperature (and enthalpy) of the air at the exit of each compression stage will be the same. A similar argument can be given for the turbine.
Thus,

At inlets:    T_{1}=T_{3}, \quad h_{1}=h_{3} \quad \text { and } \quad T_{6}=T_{8}, \quad h_{6}=h_{8}

At exits:  T_{2}=T_{4}, \quad h_{2}=h_{4} \quad \text { and } \quad T_{7}=T_{9}, \quad h_{7}=h_{9}

Under these conditions, the work input to each stage of the compressor will be the same, and so will the work output from each stage of the turbine.
(a) In the absence of any regeneration, the back work ratio and the thermal efficiency are determined by using data from Table A–17 as follows:

\begin{gathered} T_{1}=300 K \rightarrow h_{1}=300.19 kJ / kg \\ P_{r 1}=1.386 \\ P_{r 2}=\frac{P_{2}}{P_{1}} P_{r 1}=\sqrt{8}(1.386)=3.92 \rightarrow T_{2}=403.3 K \\ h_{2}=404.31 kJ / kg \end{gathered}

\begin{gathered} T_{6}=1300 K \rightarrow h_{6}=1395.97 kJ / kg \\ P_{r 6}=330.9 \\ P_{r 7}=\frac{P_{7}}{P_{6}} P_{r 6}=\frac{1}{\sqrt{8}}(330.9)=117.0 \rightarrow T_{7}=1006.4 K \\ h_{7}=1053.33 kJ / kg \end{gathered}

Then

\begin{aligned}w_{\text {comp,in }} &=2\left(w_{\text {comp,in,I }}\right)=2\left(h_{2}-h_{1}\right)=2(404.31-300.19)=208.24 kJ / kg \\w_{\text {turb,out }} &=2\left(w_{\text {turb,out, } I }\right)=2\left(h_{6}-h_{7}\right)=2(1395.97-1053.33)=685.28 kJ / kg \\w_{\text {net }} &=w_{\text {turb,out }}-w_{\text {comp,in }}=685.28-208.24=477.04 kJ / kg \\q_{\text {in }} &=q_{\text {primary }}+q_{\text {releat }}=\left(h_{6}-h_{4}\right)+\left(h_{8}-h_{7}\right) \\&=(1395.97-404.31)+(1395.97-1053.33)=1334.30 kJ / kg\end{aligned}

Thus,

r_{ bw }=\frac{w_{\text {comp,in }}}{w_{\text {turb,out }}}=\frac{208.24 kJ / kg }{685.28 kJ / kg }=0.304 \text { or } 30.4 \%

and

\eta_{\text {th }}=\frac{w_{\text {net }}}{q_{\text {in }}}=\frac{477.04 kJ / kg }{1334.30 kJ / kg }=0.358 \text { or } 35.8 \%

A comparison of these results with those obtained in Example 9–5 (singlestage compression and expansion) reveals that multistage compression with intercooling and multistage expansion with reheating improve the back work ratio (it drops from 40.3 to 30.4 percent) but hurt the thermal efficiency (it drops from 42.6 to 35.8 percent). Therefore, intercooling and reheating are not recommended in gas-turbine power plants unless they are accompanied by regeneration.

(b) The addition of an ideal regenerator (no pressure drops, 100 percent effectiveness) does not affect the compressor work and the turbine work. Therefore, the net work output and the back work ratio of an ideal gas-turbine cycle are identical whether there is a regenerator or not. A regenerator, however, reduces the heat input requirements by preheating the air leaving the compressor, using the hot exhaust gases. In an ideal regenerator, the compressed air is heated to the turbine exit temperature T_{9} before it enters the combustion chamber. Thus, under the air-standard assumptions, h_{5}=h_{7}=h_{9} .
The heat input and the thermal efficiency in this case are

\begin{aligned} q_{\text {in }} &=q_{\text {primary }}+q_{\text {reheat }}=\left(h_{6}-h_{5}\right)+\left(h_{8}-h_{7}\right) \\ &=(1395.97-1053.33)+(1395.97-1053.33)=685.28 kJ / kg \end{aligned}

and

\eta_{\text {th }}=\frac{w_{\text {net }}}{q_{\text {in }}}=\frac{477.04 kJ / kg }{685.28 kJ / kg }=0.696 \text { or } 6 9 . 6 \%

Discussion Note that the thermal efficiency almost doubles as a result of regeneration compared to the no-regeneration case. The overall effect of twostage compression and expansion with intercooling, reheating, and regeneration on the thermal efficiency is an increase of 63 percent. As the number of compression and expansion stages is increased, the cycle will approach the Ericsson cycle, and the thermal efficiency will approach

\eta_{ th , Ericsson }=\eta_{ th , Carnot }=1-\frac{T_{L}}{T_{H}}=1-\frac{300 K }{1300 K }=0.769

Adding a second stage increases the thermal efficiency from 42.6 to 69.6 percent, an increase of 27 percentage points. This is a significant increase in efficiency, and usually it is well worth the extra cost associated with the second stage. Adding more stages, however (no matter how many), can increase the efficiency an additional 7.3 percentage points at most, and usually cannot be justified economically.