Question 12.PS.74: Repeat Problem 12.67, but assume variable specific heat. The......
Repeat Problem 12.67, but assume variable specific heat. The ideal gas air tables, Table A.7, are recommended for this calculation (and the specific heat from Fig. 5.10 at high temperature).
Table A.7 is used with interpolation.
T _1=283.2 \,K , \quad u _1=202.3 \,kJ / kg , \quad v _{ r 1}=210.44
Compression 1 to 2: s _2= s _1 \Rightarrow \text { From definition of the } v _{ r } function
v _{ r 2}= v _{ r 1}\left( v _2 / v _1\right)=210.4(1 / 7)=30.063
Interpolate to get:
T _2=603.9 \,K , \quad u _2=438.1 \,kJ / kg
\begin{gathered}\Rightarrow-{ }_1 w _2= u _2- u _1=235.8 \,kJ / kg , \\u _3=438.1+1800=2238.1 \Rightarrow T _3= 2 5 7 3 . 4 \,K , \quad v _{ r 3}=0.34118 \\P _3=90 \times 7 \times 2573.4 / 283.2= 5 7 2 5 \,k P a\end{gathered}
Expansion 3 to 4: s _4= s _3 \Rightarrow \text { From the } v _{ r } \text { function as before }
v _{ r 4}= v _{ r 3}\left( v _4 / v _3\right)=0.34118(7)=2.3883
Interpolation \Rightarrow T _4=1435.4 \,K , \quad u _4=1145.8 \,kJ / kg
{ }_3 w _4= u _3- u _4=2238.1-1145.8=1092.3 \,kJ / kg
Net work, efficiency and mep
\begin{aligned}➔& w _{ net }={ }_3 w _4+{ }_1 w _2=1092.3-235.8=856.5 \,kJ / kg \\& \eta_{ TH }= w _{ net } / q _{ H }=856.5 / 1800= 0 . 4 7 6\end{aligned}
\begin{aligned}& v _1= RT _1 / P _1=(0.287 \times 283.2) / 90=0.9029 \,m ^3 / kg \\& v _2=(1 / 7) v _1=0.1290 \,m ^3 / kg \\& P _{\text {meff }}=\frac{ w _{\text {net }}}{ v _1- v _2}=856.5 /(0.9029-0.129)= 1 1 0 7 \,k P a \end{aligned}