Question

A gas turbine unit receives air at 1 bar and 300 k and compresses it adiabatically to 6.2 bar. The compressor efficiency is 88%. The fuel has a heating valve of 44186 kj/kg and the fuel-air ratio is 0.017 kj/kg of air. The turbine internal efficiency is 90%. Calculate the work of turbine and compressor per kg of air compressed and thermal efficiency

Answer 1

Answer:

Explanation:

ANSWERS:

Work of Turbine=473.91 KJ/Kg

Work of Turbine=238.39 KJ/Kg

Thermal Efficiency= 31.36%

Answer 2

Answer:

Work of turbine and compressor per kg of air compressed and thermal efficiency is $=180.29, \mathrm{~kJ} / \mathrm{kg}$,$24 \%$

Explanation:

Gas turbine cycle is shown by $1-2-3-4$on T-S diagram,

Given:

$\begin{aligned}&P_{1}=1 \text { bar }, P_{2}=P_{3}=6.2 \text { bar }, F / \text { A ratio }=0.017 \\&T_{1}=300 K, \eta_{\text {compr. }}=88 \%, \eta_{\text {turb }}=90 \%\end{aligned}$

Heating value of fuel $=44186 \mathrm{~kJ} / \mathrm{kg}$

For process $1-2$being isentropic,

$\begin{aligned}&\frac{T_{2}}{T_{1}}=\left(\frac{P_{2}}{P_{1}}\right)^{\frac{\gamma}{\gamma-1}} \\&T_{2}=505.26 K\end{aligned}$

Considering compressor efficiency,

$\eta_{\text {compr }}=\frac{T_{2}-T_{1}}{T_{2^{\prime}}-T_{1}}, 0.88= \frac{(505.26-300)}{\left(T_{2^{\prime}}-300\right)}$

Actual temperature after compression, $T_{2}^{\prime}=533.25 \mathrm{~K}$

During process $2-3$ due to combustion of unit mass of air compressed the energy balance shall be a Heat added $=m_{f} \times Heating value$

$&=\left(\left(m_{a}+m_{f}\right) \cdot c_{\mathrm{p}, \mathrm{comb}} \cdot T_{3}\right)-\left(m_{a} \cdot c_{\mathrm{p}, \text { air }} \cdot T_{2^{\prime}}\right) \\ \text { or }\\ \quad\left(\frac{m_{f}}{m_{a}}\right) \times 44186 &=\left(\left(1+\frac{m_{f}}{m_{a}}\right) \cdot c_{\mathrm{p}, \text { comb }} \cdot T_{3}\right)-\left(c_{\mathrm{p}, \text { air }} \times 533.25\right)$

$\text { Here, } & \frac{m_{f}}{m_{a}} &=0.017, c_{\mathrm{p}, \text { comb }}=1.147 k J / k g \cdot K, c_{\mathrm{p}, \text { air }}=1.005 k J / k g \cdot K \end{aligned}$

Upon substitution

$\begin{aligned}(0.017 \times 44186) &=\left((1+0.017) \times 1.147 \times T_{3}\right)-(1.005 \times 533.25) \\T_{3} &=1103.37 \mathrm{~K}\end{aligned}$

For expansion $3-4$ being

$\begin{aligned}\frac{T_{4}}{T_{3}} &=\left(\frac{P_{4}}{P_{3}}\right)^{\frac{n-1}{n}} \\T_{4} &=1103.37 \times\left(\frac{1}{6.2}\right)^{\frac{0.33}{1.33}} \\T_{4} &=701.64 K\end{aligned}$

Actual temperature at turbine inlet considering internal efficiency of turbine,

$\begin{aligned}\eta_{t u r b} &=\frac{T_{3}-T_{4^{\prime}}}{T_{3}-T_{4}} ; 0.90=\frac{\left(1103.37-T_{4^{\prime}}\right)}{(1103.37-701.64)} \\T_{4^{\prime}} &=741.81 K\end{aligned}$

$\text { Compressor work, per kg of air compressed }=c_{\mathrm{p}, \text { air }} \cdot\left(T_{2^{\prime}}-T_{1}\right) \\W_{C} &=1.005 \times(533.25-300) \\W_{C} &=234.42 \mathrm{~kJ} / \mathrm{kg} \text { or air } \\\text { Compressor work } &=\mathbf{2 3 4 . 4 2} \mathbf{~ k J} / \mathbf{k g} \text { of air } \\\text { Turbine work, per kg of air compressed }=c_{\mathrm{p}, \text { comb }} \cdot\left(T_{3}-T_{4^{\prime}}\right) \\&=1.147 \times(1103.37-741.81)$

$\text { Turbine work } &=\mathbf{4 1 4 . 7 1} \mathbf{k J} / \mathbf{k g} \text { of air } \\W_{T} &=414.71, \mathrm{~kJ} / \mathrm{kg} \text { of air } \\\text { Net work } &=W_{T}-W_{C}=(414.71-234.42)$

$W_{n e t} &=180.29, \mathrm{~kJ} / \mathrm{kg} \text { of air } \\\text { Heat supplied } &=0.017 \times 44186=751.162 \mathrm{~kJ} / \mathrm{kg} \text { of air } \\\text { Thermal efficiency } &=\frac{W_{n e t}}{\text { Heat supplied }}=\frac{180.29}{751.162}=24 \% \\\text { Thermal efficiency } &=\mathbf{2 4 \%}\end{aligned}$