Question

A gas-turbine power plant operates on the simple Brayton cycle with air as the working fluid and delivers \(32 \mathrm{MW}\) of power. The minimum and maximum temperatures in the cycle are 310 and \(900 \mathrm{K},\) and the pressure of air at the compressorexit is 8 times the value at the compressor inlet. Assuming an isentropic efficiency of 80 percent for the compressor and 86 percent for the turbine, determine the mass flow rate of air through the cycle. Account for the variation of specific heats with temperature.

Short answer

Answer: Following the provided step-by-step solution and using the given information, determine the actual temperature after compression (T_{2a}), the actual temperature after expansion (T_{4a}), the work done by the compressor (W_c), the work done by the turbine (W_t), and the heat input to the cycle (Q_{in}). Then, calculate the mass flow rate (m) using the equation: m = \frac{W_{net}}{\dot{W}_t - \dot{W}_c}

Step-by-step solution

Actual Temperature after Compression

Let's find the actual temperature after compression using isentropic efficiency: $$\eta_c = \frac{T_{2s}-T_1}{T_{2a}-T_1}$$ Solve for \(T_{2a}\): $$ T_{2a} = T_1 + \frac{T_{2s}-T_1}{\eta_c}$$ Since air has varying specific heats, we need to use the relationship: $$\frac{T_{2s}}{T_1} = \left(\frac{P_2}{P_1}\right)^{(\gamma - 1)/\gamma}$$ Inserting the given pressure ratio, specific heat values for air, and the isentropic temperature ratio, we can find the actual temperature after compression, \(T_{2a}\).

Actual Temperature after Expansion

Now find the actual temperature after expansion using the isentropic efficiency for the turbine: $$\eta_t = \frac{T_3-T_{4s}}{T_3-T_{4a}}$$ Solve for \(T_{4a}\): $$T_{4a} = T_3 - \eta_t (T_3-T_{4s})$$ Again, we need to find the isentropic temperature ratio: $$\frac{T_3}{T_{4s}} = \left(\frac{P_2}{P_1}\right)^{(\gamma - 1)/\gamma}$$ Using the given pressure ratio and specific heat values for air, we can find the actual temperature after expansion, \(T_{4a}\).

Calculate Work and Heat Interaction Parameters

We can now find the work done by the compressor, the work done by the turbine, and the heat input to the cycle: $$W_c = c_p (T_{2a} - T_1)$$ $$W_t = c_p (T_3 - T_{4a})$$ $$Q_{in} = c_p (T_3 - T_{2a})$$

Calculate Mass Flow Rate

We know the power output of the cycle (\(W_{net}\)), and we can express it as: $$W_{net} = m\dot{W}_{net}$$ Where \(m\) denotes the mass flow rate (kg/s) and \(\dot{W}_{net}\) refers to the net work per unit mass. We can express the net work per unit mass using the turbine and compressor works: $$\dot{W}_{net} = \dot{W}_t - \dot{W}_c$$ So, we can rewrite the power output as: $$W_{net} = m(\dot{W}_t - \dot{W}_c)$$ Finally, rearrange the equation to solve for the mass flow rate \(m\): $$m = \frac{W_{net}}{\dot{W}_t - \dot{W}_c}$$ By inserting the values for the works and the given power output, we can determine the mass flow rate of air through the cycle.

Key concepts

Isentropic Efficiency

Isentropic efficiency forms a cornerstone concept in thermodynamics, especially when discussing engines and compressors. Specifically, it's a metric used to gauge how closely a real process approximates an ideal isentropic process—which is a reversible adiabatic process. For a compressor, the isentropic efficiency (\text{\text{{\(\text{\text{\text{{\)\text{\text{\text{{\(\text{\text{{\text{\text{{\text{\text{\text{{\)\text{\text{{\text{{{compressor_efficiency}}} is defined by the temperature rise in the compressor compared to what the temperature rise would be in an ideal isentropic compressor. Mathematically, it's expressed as:
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Gas Turbine Power Plant

A gas turbine power plant is a type of power station that uses a gas (typical air) as the working fluid to convert fuel into electricity. At the heart of this system is the Brayton cycle, which consists of four basic processes: isentropic compression in a compressor, constant pressure heat addition in a combustion chamber, isentropic expansion in a turbine, and constant pressure heat rejection in the exhaust. These power plants are valued for their high power-to-weight ratio and relatively quick start-up times. Understanding the thermodynamic cycle that governs this system is key to analyzing and optimizing its performance. Importantly, the mass flow rate of air through the cycle is a crucial parameter. It influences the amount of power generated since it determines how much air moves through the system—and consequently, how much work is done by the turbine and required by the compressor.

Specific Heats of Air

The specific heats of air—represented as cp (specific heat at constant pressure) and cv (specific heat at constant volume)—are significant because they influence the thermodynamic processes occurring in a gas turbine engine. For instance, the work done on or by the fluid during compression or expansion is directly related to the specific heat and temperature change.
In the context of the Brayton cycle, variations in specific heats with temperature can't be ignored. Air, as an ideal gas, has specific heats that vary with temperature; hence, it's important to use correct values for cp when calculating the energy transfer during the isentropic processes. Accurate values ensure precise calculations of work and heat transfer, leading to an accurate determination of the thermal efficiency and performance of the power plant.

Thermodynamics of Brayton Cycle

The Brayton cycle, also known as the gas-turbine cycle, is the ideal cycle for gas-turbine engines that operate primarily on the principles of thermodynamics. In this cycle, two isentropic processes and two constant pressure processes form a closed loop that describes the continuous conversion of thermal energy into mechanical work and then into electricity.
The thermodynamic efficiency of the Brayton cycle depends on the temperatures and pressures at various points in the cycle, as well as on the isentropic efficiencies of the turbine and compressor. When solving for the mass flow rate, it's essential to calculate the actual work interactions for both the compressor and the turbine, considering the real (not ideal) process efficiencies. Armed with the temperature terms and specific heats, engineers use these calculated dynamics to determine the performance of the cycle—reflecting the critical importance of thermodynamics in the design and operation of modern gas turbine power plants.