$(100$ points$)$ Assume that the pressure ratio $R_p=\frac{P_2}{P_1}$ across the compressor of an ideal thermodynamic Brayton cycle is used to maximum the net$-$work output per unit mass flow rate. Also assume that the states at the isentropic compressor inlet and the temperature at the isentropic turbine inlet are fixed. Use cold air$-$standard analysis and ignore potential and kinetic energy effects. Each component can be analyzed as a control volume at steady$-$state conditions and assume that there are no pressure drop for flow through the heat exchangers. The working fluid is air modeled as an ideal gas. Assume that the specific heat $C_p=1\mathrm{.}005k\frac{J}{K}g.K$ and the specific heat ratio $k=1\mathrm{.}4$ are constants.

$3\mathrm{.}1$ Using an energy balance yields the net work of the entire cycle defined as

a$.w_{\text{c}\text{y}\text{c}\text{l}\text{e}}=C_p(T_3-T_4+T_1-T_2)$

b$.w_{\text{c}\text{y}\text{c}\text{l}\text{e}}=-C_p(T_3-T_4+T_1-T_2)$

c$.w_{\text{c}\text{y}\text{c}\text{l}\text{e}}=C_p(T_3-T_4+T_1+T_2)$

d$.$ None of the above

$3\mathrm{.}2$ The temperature ratios as functions of compression ratio $R_p$ are

$\frac{T_2}{T_1}=(\frac{P_2}{P_1}{)}^{\frac{k-1}{k}}=(R_p{)}^{\frac{k-1}{k}}$

$\frac{T_4}{T_3}=(\frac{P_4}{P_3}{)}^{\frac{k-1}{k}}=(\frac{P_1}{P_2}{)}^{\frac{k-1}{k}}=(\frac{1}{R_p}{)}^{\frac{k-1}{k}}$

True False

$3\mathrm{.}3$ Converting $w_{\text{c}\text{y}\text{c}\text{l}\text{e}}$ as a function of compression ratio $R_p$ yields

$w_{\text{c}\text{y}\text{c}\text{l}\text{e}}=C_p{T}_1(\frac{T_3}{T_1}-\frac{T_3}{T_1}(\frac{1}{R_p}{)}^{\frac{k-1}{k}}-(R_p{)}^{\frac{k-1}{k}}+1)$

True

False

$3$

$3\mathrm{.}4$ The derivative of $w_{\text{c}\text{y}\text{c}\text{l}\text{e}}$ as a function of compression ratio $R_p$ is

$\frac{del{w}_{\text{c}\text{y}\text{c}\text{l}\text{e}}}{del{R}_p}=C_p{T}_1[-\frac{T_3}{T_1}\frac{1}{k}(k-1)(R_p{)}^{[\frac{1-k1}{k}]-1}-\frac{1}{k}(k-1)(R_p{)}^{[\frac{k-1}{k}]-1}]$

True

False

$3\mathrm{.}5$ The compression ratio is

a$.R_p=15$

b$.R_p=28$

c$.R_p=7$

d$.$ None of the above

$3\mathrm{.}6$ The pressure at state $2$ is

a$.P_2=1,500$kPa

b$.P_2=2,500$kPa

c$.P_2=980$kPa

d$.$ None of the above

$3\mathrm{.}7$ The pressure at state $3$ is

a$.P_3=1,500$kPa

b$.P_3=2,500$kPa

c$.P_3=980$kPa

d$.$ None of the above

$3\mathrm{.}8$ The temperature at state $2$ is

a$.T_2=54\mathrm{.}20C$

b$.T_2=66\mathrm{.}31C$

c$.T_2=74\mathrm{.}93C$

d$.$ None of the above

$3\mathrm{.}9$ The temperature at state $4$ is

a$.T_4=521\mathrm{.}26C$

b$.T_4=732\mathrm{.}66C$

c$.T_4=614\mathrm{.}82C$

d$.$ None of the above

$3\mathrm{.}10$ The value of the cycle net$-$work output is

a$.w_{\text{c}\text{y}\text{c}\text{l}\text{e}}=409\mathrm{.}83k\frac{J}{K}g$

b$.w_{\text{c}\text{y}\text{c}\text{l}\text{e}}=667\mathrm{.}54k\frac{J}{K}g$

c$.w_{\text{c}\text{y}\text{c}\text{l}\text{e}}=556\mathrm{.}77k\frac{J}{K}g$

d$.$ None of the above

$3\mathrm{.}11$ The cycle thermal efficiency is

a$.n_{th}=38\%$

b$.n_{th}=50\%$

c$.n_{th}=44\%$

d$.$ None of the above

$3\mathrm{.}12$ For a constant pressure and quasi$-$static process, the specific enthalpy of state $1$ is

a$.h_1=299\mathrm{.}49k\frac{J}{K}g$

c$.h_1=278\mathrm{.}98k\frac{J}{K}g$

b$.h_1=345\mathrm{.}76k\frac{J}{K}g$

d$.$ None of the above

$3\mathrm{.}13$ The temperature of the state $2$ is

a$.T_2=373C$

c$.T_2=264C$

b$.T_2=471C$

d$.$ None of the above

$3\mathrm{.}14$ For a constant pressure and quasi$-$static process, the specific enthalpy of state $2$ is

a$.h_2=649\mathrm{.}23k\frac{J}{K}g$

c$.h_2=456\mathrm{.}12k\frac{J}{K}g$

b$.h_2=299\mathrm{.}56k\frac{J}{K}g$

d$.$ None of the above

$3\mathrm{.}15$ For a constant pressure and quasi$-$static process, the specific enthalpy of the state $3$ is

a$.h_3=1410\mathrm{.}02k\frac{J}{K}g$

b$.h_3=1432\mathrm{.}46k\frac{J}{K}g$

c$.h_2=1365\mathrm{.}78k\frac{J}{K}g$

d$.$ None of the above

$3\mathrm{.}16$ The temperature of the state $4$ is

a$.T_4=374\mathrm{.}19C$

b$.T_4=471\mathrm{.}50C$

c$.T_4=264\mathrm{.}67C$

d$.$ None of the above

$3\mathrm{.}17$ For a constant pressure and quasi$-$static process, the specific enthalpy of the state $4$ is

a$.h_4=650\mathrm{.}43k\frac{J}{K}g$

b$.h_4=912\mathrm{.}34k\frac{J}{K}g$

c$.h_4=765\mathrm{.}82k\frac{J}{K}g$

ALL QUESTIONS PLIZ