Eleborate on how to use these formulas in exam

 

Key Formulae

GAS TURBINES

Compressor Isentropic Efficiency: ${}_c=\frac{Isentropic\text{P}\text{o}\text{w}\text{e}\text{r}\text{I}\text{n}\text{p}\text{u}\text{t}}{\text{A}\text{c}\text{t}\text{u}\text{a}\text{l}\text{P}\text{o}\text{w}\text{e}\text{r}\text{I}\text{n}\text{p}\text{u}\text{t}}=\frac{T_{isen}}{T_{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}}$

Turbine Isentropic Efficiency: $,{}_T=\frac{\text{A}\text{c}\text{t}\text{u}\text{a}\text{l}\text{P}\text{o}\text{w}\text{e}\text{r}\text{O}\text{u}\text{t}\text{p}\text{u}\text{t}}{Isentropic\text{P}\text{o}\text{w}\text{e}\text{r}\text{O}\text{u}\text{t}\text{p}\text{u}\text{t}}=\frac{T_{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}}{T_{isen}}$

For isentropic compression$/$expansion:

$\frac{T_{\text{f}\text{i}\text{n}\text{a}\text{l}}}{T_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}}=(\frac{p_{\text{f}\text{i}\text{n}\text{a}\text{l}}}{p_{\text{i}\text{n}\text{i}\text{t}\text{i}\text{a}\text{l}}})\frac{-1}{}$

For a simple single shaft Gas Turbine with constant $C_p$ and $$ throughout:

Specific work:

$\frac{W}{C_p{T}_1}=\frac{T_3}{T_1}{}_T(1-r_{p_t}^{\frac{1-}{}})-\frac{1}{{}_c}(r_{p_c}^{\frac{-1}{}}-1)$

maximum specific work occurs when: $,r_p=({}_C{}_T\frac{T_3}{T_1})\frac{}{2(-1)}$

Thermal efficiency:

${}_{th}=1-\frac{\frac{T_3}{T_1}\{1-{}_T(1-r_{p_t}^{\frac{1-}{}})\}-1}{\frac{T_3}{T_1}-\{1+\frac{1}{{}_C}(r_{p_c}^{\frac{-1}{}}-1)\}}$

Regenerator thermal ratio $=\frac{T_{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}}{T_{\text{m}\text{a}\text{x}\text{i}\text{m}\text{u}\text{m}}}$

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RECIPROCATING COMPRESSORS

Volumetric efficiency:

${}_{\text{v}\text{o}\text{l}}=1-\frac{V_{\text{c}\text{l}\text{e}\text{a}\text{r}\text{a}\text{n}\text{c}\text{e}}}{V_{\text{s}\text{w}\text{e}\text{p}\text{t}}}(r_p^{\frac{1}{n}}-1)$

Net Indicated Work per cycle:

$W_{\text{n}\text{e}\text{t}}=\frac{n}{n-1}{p}_{\text{i}\text{n}\text{l}\text{e}\text{t}}{V}_{\text{i}\text{n}\text{d}\text{u}\text{c}\text{e}\text{d}}(r_p^{n-1}-1)$

Stage pressure ratio for minimum work: $r_p=R_p\frac{1}{N}$

where $R_p=$ overall pressure ratio, and $N$ is the number of stages

Isothermal efficiency:

${}_{\text{i}\text{s}\text{o}}=\frac{ln{r}_p}{\frac{n}{n-1}\{r_p^{\frac{n-1}{n}}-1\}}$

REFRIGERATION AND HEAT PUMPS

Refrigeration Effect $=m{h}_{\text{e}\text{v}\text{a}\text{p}\text{o}\text{r}\text{a}\text{t}\text{o}\text{r}}$

Heating Effect $=m{h}_{\text{c}\text{o}\text{n}\text{d}\text{e}\text{n}\text{s}\text{e}\text{r}}$

$CO{P}_{\text{c}\text{h}\text{a}\text{r}\text{t}}=\frac{h_{\text{e}\text{v}\text{a}\text{p}\text{o}\text{r}\text{a}\text{t}\text{o}\text{r}}}{h_{\text{c}\text{o}\text{m}\text{p}\text{r}\text{e}\text{s}\text{s}\text{o}\text{r}}},($refrigeration$)$

$CO{P}_{\text{c}\text{h}\text{a}\text{r}\text{t}}=\frac{h_{\text{c}\text{o}\text{n}\text{d}\text{e}\text{n}\text{s}\text{e}\text{r}}}{h_{\text{c}\text{o}\text{m}\text{p}\text{r}\text{e}\text{s}\text{s}\text{o}\text{r}}}($heat pump$)$

Compressor Isentropic efficiency $=\frac{Isentropic\text{P}\text{o}\text{w}\text{e}\text{r}\text{I}\text{n}\text{p}\text{u}\text{t}}{\text{A}\text{c}\text{t}\text{u}\text{a}\text{l}\text{P}\text{o}\text{w}\text{e}\text{r}\text{I}\text{n}\text{p}\text{u}\text{t}}=\frac{h_{isen}}{h_{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}}$

Where: $m$ is refrigerant mass flow rate

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IC ENGINES

Air Standard thermal efficiencies:

For Dual Combustion Cycle:

${}_{th}=1-\frac{1}{r_c^{-1}}[\frac{-1}{(-1)+(-1)}]$

$\frac{\text{I}\text{M}\text{E}\text{P}}{p_1}=\frac{r_c}{(r_c-1)}\frac{r_c^{-1}[(-1)+(-1)]-(-1)}{(-1)}$

For Otto Cycle:

${}_{th}=1-\frac{1}{r_c^{y-1}}$

$\frac{\text{I}\text{M}\text{E}\text{P}}{p_1}=\frac{Q_{in}}{p_1{V}_1}(\frac{r_c}{r_c-1}){}_{\text{o}\text{t}\text{t}\text{o}}$

For Diesel Cycle:

${}_{th}=1-\frac{1}{r_c^{-1}}[\frac{-1}{(-1)}]$

$\frac{\text{I}\text{M}\text{E}\text{P}}{p_1}=\frac{r_c^{}(-1)-r_c(-1)}{(-1)(r_c-1)}$

Volume at BDC: $V_{BDC}=\frac{r_c}{r_c-1}{V}_s,($where $V_s$ is the stroke volume$)$

Mean effective pressure $($MEP$)=\frac{W_{\text{n}\text{e}\text{t}}}{V_s}$

Brake or Indicated Power $=$ PLAN

$P=$ BMEP $or$ IMEP

$L=$ stroke Length

$A=$ piston face Area

$N=No.of$ power strokes per $sec.$

BSFC $=\frac{\text{F}\text{u}\text{e}\text{l}\text{C}\text{o}\text{n}\text{s}\text{u}\text{m}\text{p}\text{t}\text{i}\text{o}n}{\text{B}\text{r}\text{a}\text{k}\text{e}\text{P}\text{o}\text{w}\text{e}\text{r}}$

Volumetric efficiency $($for $4-$stroke naturally aspirated engines$)$

${}_v=\frac{\text{v}\text{o}\text{l}\text{.}of\text{f}\text{r}\text{e}\text{e}\text{a}\text{i}\text{r}\text{t}\text{r}\text{a}\text{p}\text{p}\text{e}\text{d}\text{p}\text{e}\text{r}\text{c}\text{y}\text{c}\text{l}\text{e}}{\text{s}\text{w}\text{e}\text{p}\text{t}\text{v}\text{o}\text{l}\text{.}of\text{c}\text{y}\text{l}\text{i}\text{n}\text{d}\text{e}\text{r}}=\frac{\text{t}\text{r}\text{a}\text{p}\text{p}\text{e}\text{d}\text{m}\text{a}\text{s}\text{s}}{\text{t}\text{h}\text{e}\text{o}\text{r}\text{e}\text{t}\text{i}\text{c}\text{a}\text{l}\text{m}\text{a}\text{s}\text{s}}$

i$.$e$.,{}_v=\frac{m_a^{}}{\frac{1}{2}{}_a{V}_s{N}_e}$

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STEADY STATE CONDUCTION

$1-$D steady: $Q?\text{'}\text{'}=-\frac{T}{x},$ plane

$Q$