Question

Consider an ideal air-standard Brayton cycle with minimum and maximum temperatures of \(300 \mathrm{~K}\) and \(1500 \mathrm{~K}\), respectively. The pressure ratio is that which maximizes the net work developed by the cycle per unit mass of air flow. On a cold air-standard basis, calculate (a) the compressor and turbine work per unit mass of air flow, each in \(\mathrm{kJ} / \mathrm{kg}\). (b) the thermal efficiency of the cycle. (c) Plot the thermal efficiency versus the maximum cycle temperature ranging from 1200 to \(1800 \mathrm{~K}\).

Short answer

Calculate optimal P_ratio = (T_{max}/T_{min})^{γ/(2(γ-1))}. Use this P_ratio to find W_c and W_t, then calculate η = 1 - 1/(P_ratio)^{(γ-1)/γ}. Finally, plot η for T_{max} ranging from 1200 K to 1800 K.

Step-by-step solution

Identify key parameters and assumptions

Given an ideal air-standard Brayton cycle with minimum temperature (T_{min}) = 300 K and maximum temperature (T_{max}) = 1500 K. Identify that the pressure ratio maximizes the net work of the cycle per unit mass of air. Assume specific heat capacity at constant pressure (c_p) and constant volume (c_v) are constant, with γ = c_p/c_v.

Calculate the optimal pressure ratio

The optimal pressure ratio (P_ratio) for maximizing net work output of the Brayton cycle is given by: P_ratio = (T_{max}/T_{min})^{γ/(2(γ-1))}. Substituting the values, find the optimal P_ratio. For air, γ ≈ 1.4.

Calculate the turbine work (W_t)

Use the relation: W_t = c_p(T_{max} - T_{min} * (P_ratio)^{(γ-1)/γ}). Substituting the values of T_{max}, T_{min} and c_p = 1.005 kJ/kg*K (specific heat capacity of air at constant pressure) and calculated P_ratio from Step 2, find W_t.

Calculate the compressor work (W_c)

Use the relation W_c = c_p(T_{min} * (P_ratio)^(γ-1/γ) - T_{min}). Substituting c_p, T_{min} and optimal P_ratio, calculate W_c.

Calculate the thermal efficiency (η)

The thermal efficiency of the Brayton cycle is given by: η = 1 - 1/(P_ratio)^{(γ-1)/γ}. Using the optimal P_ratio calculated in Step 2, find η.

Plot thermal efficiency vs. maximum cycle temperature

For the range of maximum temperatures from 1200 K to 1800 K, repeat Steps 2 and 5 to calculate η for each value of T_{max}. Plot the thermal efficiency as a function of T_{max}.

Key concepts

ideal air-standard Brayton cycle

The Brayton cycle is a thermodynamic cycle that describes the workings of a gas turbine engine. The 'ideal air-standard Brayton cycle' assumes the working fluid is air, which behaves as an ideal gas.
This cycle consists of four main processes: two isentropic (adiabatic and reversible) and two isobaric (constant pressure) processes.
The cycle has the following sequence:

• Isentropic compression
• Isobaric heat addition
• Isentropic expansion
• Isobaric heat rejection

Understanding this cycle helps in analyzing the efficiency and work output of practical gas turbine engines.

thermal efficiency

In thermodynamics, 'thermal efficiency' is a measure of how well a heat engine converts heat into work. For the Brayton cycle, thermal efficiency \( \eta\) is defined as the ratio of the net work output to the heat input.
Mathematically, it is expressed as: \[\eta = 1 - \frac{1}{(P_{ratio})^{(\gamma-1)/\gamma}}\] Here, \( P_{ratio} \) is the pressure ratio, and \( \gamma \) is the ratio of specific heats ( \( \gamma = c_p / c_v \) ).
A higher thermal efficiency indicates a more effective cycle in converting heat into useful work, reducing energy losses.

pressure ratio

The 'pressure ratio' in a Brayton cycle is the ratio of the pressure after compression to the initial pressure of the air before compression. It is a critical factor that influences the efficiency and work output of the gas turbine.
The optimal pressure ratio, especially for maximum net work output, can be calculated using: \[ P_{ratio} = \left( \frac{T_{max}}{T_{min}} \right)^{\frac{\gamma}{2(\gamma-1)}} \]
Where \( T_{max} \) and \( T_{min} \) refer to the maximum and minimum temperatures of the cycle, respectively. Using this ratio helps maximize the work produced by the turbine while minimizing the work required by the compressor.

compressor work

'Compressor work' refers to the energy required to compress the air in the Brayton cycle, which is one of the main energy inputs.
It can be expressed with the equation: \[ W_c = c_p \left( T_{min} \left( P_{ratio} \right)^{\frac{\gamma-1}{\gamma}} - T_{min} \right) \]
Here, \( c_p \) is the specific heat capacity at constant pressure, and \( T_{min} \) is the minimum temperature.
Efficient compressors are designed to minimize this work to improve the overall cycle efficiency.

turbine work

'Turbine work' is the work generated by expanding the high-pressure, high-temperature air within the turbine. This expansion process produces useful mechanical energy that can be converted into electricity or used for propulsion.
The formula to calculate this work is: \[ W_t = c_p (T_{max} - T_{min} \cdot (P_{ratio})^{(\gamma-1)/\gamma}) \]
Where \( T_{max} \) is the maximum temperature of the cycle. Using these values helps determine the useful work generated by the turbine. Efficient turbines maximize this work, contributing to higher overall efficiency for the cycle.