Question

Air enters the compressor of a gas turbine at \(100 \mathrm{kPa}\), \(300 \mathrm{~K}\). The air is compressed in two stages to \(900 \mathrm{kPa}\), with intercooling to \(300 \mathrm{~K}\) between the stages at a pressure of \(300 \mathrm{kPa}\). The turbine inlet temperature is \(1480 \mathrm{~K}\) and the expansion occurs in two stages, with reheat to \(1420 \mathrm{~K}\) between the stages at a pressure of \(300 \mathrm{kPa}\). The compressor and turbine stage efficiencies are 84 and \(82 \%\), respectively. The net power developed is \(1.8 \mathrm{MW}\). Determine (a) the volumetric flow rate, in \(\mathrm{m}^{3} / \mathrm{s}\), at the inlet of each compressor stage. (b) the thermal efficiency of the cycle. (c) the back work ratio.

Short answer

a) Use Ideal Gas Law for volumetric flow rate at inlet stages. b) Thermal efficiency from \(\eta = 1- \frac{Q_{out}}{Q_{in}}\). c) Back work ratio from \(\frac{W_{compressor}}{W_{turbine}}\).

Step-by-step solution

Define State Points

Identify and define the states at each point in the cycle. Assume air behaves as an ideal gas.

State 1 (Inlet of the First Compressor Stage)

State 1: Pressure \(P_1 = 100 \ \mathrm{kPa}\), Temperature \(T_1 = 300 \ \mathrm{K}\).

State 2 (First Stage Compression)

Use isentropic relations and efficiencies to determine the properties at State 2. The pressure ratio \(\frac{P_2}{P_1} = 3\). The isentropic temperature relation is \(T_2' = T_1 \left(\frac{P_2}{P_1}\right)^{\frac{\gamma-1}{\gamma}}\), where \(T_2'\) is the temperature after ideal compression and \ \gamma = 1.4\ is the specific heat ratio for air. Find \(T_2'\) and then use the efficiency to find \(T_2\).

Intermediate Cooling (State 3)

State 3: After intercooling, the pressure is dropped to 300 kPa, and the temperature is reset to 300 K.

State 4 (Second Stage Compression)

Similar to the first stage compression, but from State 3. Pressure ratio \(\frac{P_4}{P_3} = 3\). Apply isentropic relations and then efficiency to find \(T_4\).

Volumetric Flow Rate Calculation at Compressor Inlet

Use the Ideal Gas Law to find volumetric flow rates: \(\dot{V} = \frac{\dot{m}RT}{P}\) for States 1, 3, and 4.

Thermal Efficiency

Thermal efficiency \(\eta = 1- \frac{Q_{out}}{Q_{in}}\). Calculate heat added in terms of specific heat at constant pressure and temperature differences at each stage.

Back Work Ratio

Back work ratio is \(\frac{W_{compressor}}{W_{turbine}}\). Calculate using \(W = \dot{m} \cdot \(h_2 - h_1\)\) for each stage.

Key concepts

Volumetric Flow Rate Calculation

In gas turbine cycles, determining the volumetric flow rate is essential to understand the amount of air entering the system.
The Ideal Gas Law provides a straightforward way to find this. The law states that for an ideal gas, the pressure, volume, and temperature are related as follows: \[ PV = nRT \]
Here, we rearrange the formula to find the volumetric flow rate: \[ \frac{\text{V}}{t} = \frac{\text{nRT}}{P} \]
Utilizing the given data—pressure, temperature, and the gas constant for air—we can determine the volumetric flow rates at various stages. In this example, you'll calculate the flow rate at each compressor stage inlet. Ensure you use the correct units to match the desired output, typically in cubic meters per second (m³/s). This step-by-step calculation helps understand how the properties of air change at each stage of compression and cooling in the cycle.

Thermal Efficiency

Thermal efficiency is a measure of how well a system converts heat into work.
For gas turbine cycles, it’s crucial because it shows the percentage of energy input that is useful output. The efficiency (\(\text{η}\)) can be calculated with: \[ \text{η} = 1 - \frac{Q_{out}}{Q_{in}} \]
Here, Q_out represents the heat rejected, and Q_in is the heat added.
Specific heats at constant pressure (Cp) and temperature differences across various stages of the cycle provide the needed values. Make sure to meticulously account for heat exchanges at each stage of the turbine and any intercoolers or reheats to get accurate results. Higher thermal efficiency indicates a more effective cycle at converting heat into work, which is the goal for many industrial applications.

Back Work Ratio

The back work ratio is a critical concept in gas turbine cycles.
It tells us how much of the turbine's work output is used to drive the compressor. It's represented by: \[ \text{Back Work Ratio} = \frac{W_{compressor}}{W_{turbine}} \]
The lower the ratio, the more efficient the system. Here, W_compressor and W_turbine are the works done by the compressor and turbine respectively.
To find these values, use the mass flow rates (\(\text{ṁ}\)) and enthalpy changes (\(h_2 - h_1\)) at each stage. The compressor and turbine stages are analyzed individually, and losses due to inefficiencies must be considered. This ratio helps in understanding the energetic costs within the cycle and is a key indicator of the system's performance.

Isentropic Relations

Isentropic processes are idealized processes where entropy remains constant.
They’re crucial in analyzing gas turbines, simplifying the calculations. For an isentropic process involving an ideal gas, the relationship between temperature and pressure is given by: \[ T_2' = T_1 \times \bigg(\frac{P_2}{P_1}\bigg)^{\frac{\text{γ}-1}{\text{γ}}} \]
Here, \(T_2'\) and \(T_1\) are temperatures, \(P_2\) and \(P_1\) are pressures, and \(\text{γ}\) is the specific heat ratio (1.4 for air). These relations help predict the final state after an ideal compression or expansion. Efficiencies must be applied subsequently for real scenarios. Following these steps for each stage and comparing to actual conditions reveals inefficiencies and areas for optimization.

Ideal Gas Law

The Ideal Gas Law is fundamental in the analysis of gas turbine cycles.
It establishes a relationship between pressure (P), volume (V), and temperature (T) for a given quantity of gas (n), represented as: \[ PV = nRT \]
Where R is the universal gas constant. In applications where mass flow rates (\(\text{ṁ}\)) and specific gas constant (R_specific) are used instead, it modifies to: \[ PV = \text{ṁ} R_{specific} T \]
This equation allows us to derive properties like volumetric flow rate and specific volume at different stages. By applying this law, you simplify complex dynamics to manageable calculations, enabling accurate predictions and performance assessments of the cycle under various operating conditions.