Question

For an ideal gas with constant specific heats show that the compressor and turbine isentropic efficiencies may be written as $$\eta_{C}=\frac{\left(P_{2} / P_{1}\right)^{(k-1) / k}}{\left(T_{2} / T_{1}\right)-1} \text { and } \eta_{T}=\frac{\left(T_{4} / T_{3}\right)-1}{\left(P_{4} / P_{3}\right)^{(k-1) / k}-1}$$ The states 1 and 2 represent the compressor inlet and exit states and the states 3 and 4 represent the turbine inlet and exit states.

Short answer

Question: Show that the isentropic efficiencies of a compressor and a turbine can be written in the given expressions, involving the ratios of pressure and temperature at the inlet and exit states. Answer: The isentropic efficiency for a compressor can be expressed as: $$\eta_C = \frac{\left(\frac{P_2}{P_1}\right)^{(k-1)/k}-1}{\frac{T_2}{T_1}-1}$$ And the isentropic efficiency for a turbine can be expressed as: $$\eta_T = \frac{\frac{T_4}{T_3}-1}{\left(\frac{P_4}{P_3}\right)^{(k-1)/k}-1}$$ Where k is the specific heat ratio and the subscripts denote the inlet (1,3) and exit (2,4) states for the compressor and turbine, respectively.

Step-by-step solution

Understanding the concept of isentropic efficiency

Isentropic efficiency is a measure of the performance of an ideal thermodynamic process compared to the same process executed in an isentropic (i.e., reversible and adiabatic) manner. For a compressor and turbine, the isentropic efficiency relates the actual work done to the isentropic work done.

Write down the isentropic efficiency formulas for compressor and turbine

The isentropic efficiency for the compressor (η_C) is given by: $$\eta_C = \frac{h_2^{is} - h_1}{h_2 - h_1}$$ And the isentropic efficiency for the turbine (η_T) is given by: $$\eta_T = \frac{h_3 - h_4}{h_3 - h_4^{is}}$$ Where the superscript "is" denotes the isentropic (ideal) conditions.

Express enthalpy change in terms of temperature change and specific heat

For an ideal gas with constant specific heats, the change in enthalpy can be related to the change in temperature and specific heat constant (C_p) as follows: $$\Delta h = C_p \Delta T$$ Substitute this relation into the isentropic efficiency formulas, we get: $$\eta_C = \frac{C_p(T_2^{is} - T_1)}{C_p(T_2 - T_1)}$$ $$\eta_T = \frac{C_p(T_3 - T_4)}{C_p(T_3 - T_4^{is})}$$

Use the relation between temperature and pressure ratio for ideal gas with constant specific heats

The relationship between temperature ratio and pressure ratio for an ideal gas with constant specific heats (k) undergoing an isentropic process can be written as follows: $$\frac{T_2^{is}}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k-1)/k}$$ $$\frac{T_4^{is}}{T_3} = \left(\frac{P_4}{P_3}\right)^{(k-1)/k}$$

Solve the isentropic efficiency formulas with temperature and pressure ratio

For the compressor, divide both sides of the formula by \(T_1\). Then replace the temperature ratio with its corresponding pressure ratio, we get: $$\eta_C = \frac{\left(\frac{P_2}{P_1}\right)^{(k-1)/k}-1}{\frac{T_2}{T_1}-1}$$ Similarly, for the turbine, divide both sides of the formula by \(T_3\). Then replace the temperature ratio with its corresponding pressure ratio, we obtain: $$\eta_T = \frac{\frac{T_4}{T_3}-1}{\left(\frac{P_4}{P_3}\right)^{(k-1)/k}-1}$$ These are the desired expressions for compressor and turbine isentropic efficiencies in terms of pressure and temperature ratios.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of our exercise, it provides a framework for understanding how energy is transferred in the form of work and heat during the operation of compressors and turbines, which are fundamental components in many industrial processes such as power generation and refrigeration.

At the heart of thermodynamics are the laws that govern these energy transformations, and isentropic processes are idealized thermodynamic processes that are both reversible and adiabatic—meaning no heat is exchanged with the surroundings. Understanding isentropic efficiency helps students gauge how close real-world compressors and turbines come to this ideal performance. Remarkably, it quantitatively measures the deviation from the ideal, isentropic process, giving us a tool for maximizing the efficiency of these machines.

Compressor and Turbine Performance

Evaluating compressor and turbine performance finds great use in engineering fields, especially when dealing with engines and various HVAC systems. Isentropic efficiency is an essential metric for assessing these machines' performance, reflecting how effectively they convert input energy into useful output. A compressor's purpose is to increase the pressure of a gas, whereas a turbine's goal is to extract energy from a fluid.

Higher isentropic efficiencies indicate better performance where the device approaches the ideal behavior dictated by isentropic processes. For a student or engineer, understanding how to calculate these efficiencies can be crucial for designing more efficient systems, or for diagnosing and improving current systems. The formulas provided in the exercise reflect the relationship between the actual work and the ideal, isentropic work, which is vital for analyzing real-world scenarios.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics which relates the pressure, volume, and temperature of an ideal gas with its amount. Expressed as \( PV = nRT \), it tells us that for a fixed amount of gas, the product of pressure and volume is directly proportional to its temperature. Here, 'P' represents the pressure, 'V' is the volume, 'n' is the number of moles, 'R' is the ideal gas constant, and 'T' is the temperature.

In the context of our exercise, understanding the Ideal Gas Law is critical as it underpins the relationship we use between temperature and pressure during isentropic processes of an ideal gas. The compressors and turbines are assumed to be dealing with ideal gases, which simplifies the analysis and helps develop the mathematical expressions for isentropic efficiencies based on real and ideal changes in temperature and pressure.

Specific Heats

Specific heats are a measure of how much heat energy is required to raise the temperature of a substance by a certain amount, typically one degree. For gases, specific heats can be defined at constant pressure (\( C_p \) ) or at constant volume (\( C_v \) ), and they are key properties in thermodynamic processes. The ratio of specific heats (\( k = C_p/C_v \) ) for an ideal gas is used to relate changes in temperature to changes in pressure during isentropic processes, as shown in the provided exercise.

For students tackling thermodynamics, specific heats are critically important for solving a wide array of problems, not only because they measure energy change but also because they allow for the simplification of complex relationships, such as the one between enthalpy and temperature. Grasping the concept of specific heats and their application is thus fundamental for understanding and calculating the isentropic efficiencies of compressors and turbines.