The thermal efficiency of a gas turbine cycle with...

The thermal efficiency of a gas turbine cycle with regeneration in terms of $\mathrm{T}_{3}$ (maximum temperature), $\mathrm{T}_{1}$ (minimum temperature), $r_{p}$ (pressure ratio) and $\gamma\left(=\frac{c_{p}}{c_{v}}\right)$ is given by (a) $1-\frac{\mathrm{T}_{1}}{\mathrm{~T}_{3}} r_{p}^{\left(\frac{\gamma}{\gamma-1}\right)}$ (b) $1-\frac{\mathrm{T}_{3}}{\mathrm{~T}_{1}} r_{p}^{\left(\frac{\gamma}{\gamma-1}\right)}$ (c) $1-\frac{\mathrm{T}_{3}}{\mathrm{~T}_{1}} r_{p}^{\left(\frac{\gamma-1}{\gamma}\right)}$ (d) $1-\frac{\mathrm{T}_{1}}{\mathrm{~T}_{3}} r_{p}^{\left(\frac{\gamma-1}{\gamma}\right)}$

The thermal efficiency of a gas turbine cycle with regeneration in terms of $\mathrm{T}_{3}$ (maximum temperature), $\mathrm{T}_{1}$ (minimum temperature), $r_{p}$ (pressure ratio) and $\gamma\left(=\frac{c_{p}}{c_{v}}\right)$ is given by
(a) $1-\frac{\mathrm{T}_{1}}{\mathrm{~T}_{3}} r_{p}^{\left(\frac{\gamma}{\gamma-1}\right)}$
(b) $1-\frac{\mathrm{T}_{3}}{\mathrm{~T}_{1}} r_{p}^{\left(\frac{\gamma}{\gamma-1}\right)}$
(c) $1-\frac{\mathrm{T}_{3}}{\mathrm{~T}_{1}} r_{p}^{\left(\frac{\gamma-1}{\gamma}\right)}$
(d) $1-\frac{\mathrm{T}_{1}}{\mathrm{~T}_{3}} r_{p}^{\left(\frac{\gamma-1}{\gamma}\right)}$

Key Concepts

Gas Turbine Cycle

A gas turbine cycle is a thermodynamic process that converts the energy stored in fuel into mechanical work by compressing air, heating it through combustion, and then expanding the high-temperature gases in a turbine. This cycle is the basis for many power generation and propulsion systems, and its performance is evaluated through various efficiency metrics.

Thermal Efficiency

Thermal efficiency is a measure of how well a thermodynamic cycle converts heat energy into useful work. In cycles such as the gas turbine cycle, it is expressed as the ratio of the net work output to the heat input. Enhancing thermal efficiency means better fuel utilization and reduced waste heat, which is pivotal in power generation and engine design.

Regeneration

Regeneration in a thermodynamic cycle refers to the process of recovering waste heat from the exhaust to preheat the incoming air or fuel mixture before combustion. This process improves the overall thermal efficiency of the cycle by reducing the amount of fuel needed to reach the desired combustion temperature, thereby lowering energy losses.

Pressure Ratio

Pressure ratio is defined as the ratio of the pressure after compression to the pressure before compression in the cycle. In a gas turbine cycle, a higher pressure ratio generally leads to greater cycle efficiency up to a point, as it increases the temperature rise during compression, which in turn affects the expansion process in the turbine.

Specific Heat Ratio (?)

The specific heat ratio, denoted by ? and defined as the ratio of specific heats at constant pressure (cp) to constant volume (cv), is a key parameter in determining the properties of working gases. It plays an essential role in the mathematical relationships describing the thermodynamic processes in the cycle, influencing the efficiency and performance of devices like turbines and compressors.

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