Question

Air enters the compressor of an ideal air-standard Brayton cycle at \(100 \mathrm{kPa}, 300 \mathrm{~K}\), with a volumetric flow rate of \(5 \mathrm{~m}^{3} / \mathrm{s}\). The compressor pressure ratio is 10 . For turbine inlet temperatures ranging from 1000 to \(1600 \mathrm{~K}\), plot (a) the thermal efficiency of the cycle. (b) the back work ratio. (c) the net power developed, in \(\mathrm{kW}\).

Short answer

Calculate the cycle efficiency, BWR, and net power developed by the Brayton cycle as a function of the turbine inlet temp, ranging from 1000K to 1600K. Use thermodynamic equations and plots for each.

Step-by-step solution

Understand the Brayton Cycle Basics

The Brayton cycle consists of four processes: isentropic compression, constant-pressure heat addition, isentropic expansion, and constant-pressure heat rejection. The cycle efficiency, back work ratio, and net power are dependent on the turbine inlet temperature and pressure ratio.

Calculate the Compressor Exit State

Given: Initial pressure, \(P_1 = 100 \, kPa\) Initial temperature, \(T_1 = 300 \, K\) Volumetric flow rate, \( \dot{V} = 5 \, m^3/s\) Pressure ratio, \( rp = 10 \). Use the isentropic relation for the ideal gas (air): \[ T_2 = T_1 \times rp^{(\frac{\frac{u - 1}{1}})}. \]

Determine the Turbine Exit State

For a turbine inlet temperature range from 1000 K to 1600 K, calculate the turbine exit temperature using the isentropic relation: \[ T_4 = T_3 / rp^{(\frac{u - 1}{1})}. \]

Calculate the Thermal Efficiency

The thermal efficiency of the Brayton cycle is given by: \[ \eta = 1 - ( 1 / rp^{(\frac{u - 1}{1})}). \] Plot the thermal efficiency against the turbine inlet temperature.

Calculate the Back Work Ratio

The back work ratio (BWR) is the ratio of compressor work to turbine work: \[ BWR = W_c / W_t. \] Plot BWR against the turbine inlet temperature.

Calculate the Net Power Developed

The net power developed is the difference between the work done by the turbine and the work required by the compressor: \[ W_(net) = W_t - W_c. \] The mass flow rate can be calculated using: \[ \dot{m} = \frac{ \dot{V} \times \rho_1}{V_1.} \] Net power developed in \(kW \), \([P = \dot{m} \times c_p \times ( T_3 - T_4 - T_2 + T_1,)\right].\)... \Plot the net power vs. turbine inlet temperature.

Key concepts

Isentropic Compression

Isentropic compression is a vital process in the Brayton cycle. During this phase, air is compressed in the compressor with no heat transfer. For an ideal gas like air, the relation between temperature and pressure during isentropic compression is given by the following formula: \[ T_2 = T_1 \times rp^{(\frac{\frac{u - 1}{1}})} \]. Here, \(T_1\) is the initial temperature, \(rp\) is the pressure ratio, and \(u\) is the specific heat ratio (typically 1.4 for air). This raised temperature, \(T_2\), at the end of the compression process is crucial for determining the efficiency and performance of the cycle. The pressure ratio affects how much the temperature increases. A higher pressure ratio leads to a higher temperature after compression, increasing the overall efficiency of the Brayton cycle.

Thermal Efficiency

Thermal efficiency indicates how effectively the Brayton cycle converts heat into work. It’s a measure of performance, and for the Brayton cycle, it is calculated using: \[ \eta = 1 - ( 1 / rp^{(\frac{u - 1}{1})} ) \]. Here, \(rp\) represents the pressure ratio and \(u\) is the specific heat ratio. This equation shows that thermal efficiency depends directly on the pressure ratio. A higher pressure ratio results in a higher efficiency since it decreases the fraction \(1 / rp^{(\frac{u - 1}{1})}\). Thermal efficiency also depends on the temperatures at turbine inlet and outlet. Increasing the turbine inlet temperature while maintaining high-pressure ratios can enhance the cycle's thermal efficiency significantly.

Back Work Ratio

The back work ratio (BWR) is an important measure in the Brayton cycle. It represents the fraction of the turbine work that is used to drive the compressor. Mathematically, it is defined as: \[ BWR = W_c / W_t \]. Here, \(W_c\) is the work required by the compressor and \(W_t\) is the work produced by the turbine. Typically, in a Brayton cycle, the compressor and turbine work are substantial, so the BWR is usually less than 1 but significant. A lower BWR means that less of the turbine's work is needed to drive the compressor, leaving more net power output. It’s beneficial for overall efficiency and power generation.

Net Power Calculation

Calculating the net power developed in a Brayton cycle is essential for determining how much usable power can be generated. Net power is obtained by subtracting the work done by the compressor from the work produced by the turbine: \[ {W_{net}} = {W_t} - {W_c} \]. To find the actual net power in kW, you need the mass flow rate \(\dot{m}\), which can be calculated using: \[ \dot{m} = \frac{ \dot{V} \times \rho_1 }{V_1} \]. \(\dot{V}\) is the volumetric flow rate, and \( \rho_1\) is the density of air at the initial state. Once you have the mass flow rate, the power equation incorporates specific heat \(c_p\) and temperature changes: \[ {P} = \dot{m} \times c_p \times ( T_3 - T_4 - T_2 + T_1 ) \]. This incorporation helps in plotting the net power against the turbine inlet temperature, revealing how the cycle's performance varies with temperature.