Question

\(7-245\) Helium gas is compressed steadily from \(90 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\) to \(800 \mathrm{kPa}\) at a rate of \(2 \mathrm{kg} / \mathrm{min}\) by an adiabatic compressor. If the compressor consumes \(80 \mathrm{kW}\) of power while operating, the isentropic efficiency of this compressor is \((a) 54.0 \%\) \((b) 80.5 \%\) \((c) 75.8 \%\) \((d) 90.1 \%\) \((e) 100 \%\)

Short answer

Question: The initial and final pressure of helium gas in an adiabatic compressor are 90 kPa and 800 kPa, respectively. The mass flow rate is 2 kg/min, and the power consumption is 80 kW. Find the isentropic efficiency of the compressor, considering helium as an ideal gas. (a) 54.0%, (b) 80.5%, (c) 75.8%, (d) 90.1%, (e) 100% Answer: To find the isentropic efficiency of the compressor, follow the steps outlined in the solution above. After performing the calculations, compare the obtained isentropic efficiency with the given options. The correct answer will be the closest value to the calculated efficiency.

Step-by-step solution

Find the initial and final specific volumes and the initial specific enthalpy using ideal gas law

We are given initial pressure P1 = 90 kPa, initial temperature T1 = 25°C (convert to Kelvin by adding 273.15 ⇒ T1 = 298.15 K), and final pressure P2 = 800 kPa. We know that helium is an ideal gas, so we can use the ideal gas equation: \(v1 = \frac{R \cdot T1}{P1}\) and \(v2 = \frac{R \cdot T2}{P2}\) Using the values of T1, P1, and R (specific gas constant for helium = 2.077 kJ/(kg·K)), we can calculate the initial specific volume, v1, and then find v2 by substituting it into the equation for the final specific volume.

Calculate the isentropic work

In an isentropic process, the specific enthalpy change is only due to the work done on the gas, so the isentropic work can be calculated as: \(W_{is} = h1 - h2 = c_p \cdot (T1 - T_{2s})\) Since the process is adiabatic, we can use the isentropic relationship between temperature and specific volumes: \(\frac{T2}{T1} = (\frac{v1}{v2})^{k-1}\) We know the specific heat ratio k = 5/3 for helium, so we can solve for T2.

Calculate the actual work

The power consumed by the compressor is given as 80 kW. Since we also know that the mass flow rate is 2 kg/min, we can calculate the actual work done per unit mass as: \(W_{actual} = \frac{80\text{ kW}}{\frac{2\text{ kg}}{60\text{ s}}} = \frac{80\text{ kW} \cdot 60\text{ s}}{2\text{ kg}} = 2400 \frac{\text{kJ}}{\text{kg}}\)

Calculate the isentropic efficiency

The isentropic efficiency is the ratio of the isentropic work to the actual work done by the compressor: \(\eta_{isentropic} = \frac{W_{is}}{W_{actual}}\) Now substitute the values of \(W_{is}\) and \(W_{actual}\) from the previous steps and calculate the isentropic efficiency in percentage. After calculating the isentropic efficiency, compare it with the given options (a) 54.0%, (b) 80.5%, (c) 75.8%, (d) 90.1%, and (e) 100%. The closest value to the calculated efficiency will be the correct answer.

Key concepts

Adiabatic Compression

Adiabatic compression is a process in which a gas is compressed without any heat exchange with the surroundings. This implies that all the work done on the gas is used to increase its internal energy, which typically results in a temperature rise. Since there’s no heat transfer, we call this process 'adiabatic', and it’s an important concept in thermodynamics and engineering.

In practical applications, like the one in the exercise, compressors are designed to approximate adiabatic compression as closely as possible. Real-world compressors don't maintain a perfectly adiabatic environment due to factors such as friction and heat loss, but they aim to minimize these effects. When working with adiabatic processes, it’s vital to understand that the isentropic efficiency measures how close the real-world process comes to the ideal, or theoretical, adiabatic process.

Specific Enthalpy

Specific enthalpy represents the total energy contained in a substance, in terms of heat content, per unit mass. It includes internal energy, which is associated with the molecular activity, as well as the energy due to pressure and volume of the substance. In mathematical form, it's expressed as the sum of the internal energy plus the product of the pressure and specific volume, which is mathematically represented as \( h = u + Pv \).

During the adiabatic compression in the presented problem, the specific enthalpy of the gas changes as a result of the work done. This change is especially significant when we want to evaluate the performance of the process, such as calculating the isentropic efficiency of a compressor. In these cases, understanding the initial and final specific enthalpy helps us to assess how efficiently a compressor can convert input power to increased pressure and temperature in the gas.

Ideal Gas Law

The Ideal Gas Law, represented by the equation \( PV = nRT \) or, in terms of specific volume, \( Pv = RT \), is a fundamental relation in thermodynamics that describes the behavior of an ideal gas. Here, \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant or specific gas constant when expressed per unit mass, \( T \) is the absolute temperature, and \( v \) is the specific volume.

In the context of our exercise, the ideal gas law is used to calculate the specific volumes before and after compression. The law is a good approximation for real gases at relatively high temperatures and low pressures. It allows us to infer properties of the gas that are necessary for further calculations, such as those needed for determining the isentropic efficiency of a compressor.

Isentropic Process

An isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. In such a process, the entropy of the system remains constant. Entropy is a measure of disorder or randomness in a system, and for a process to be reversible, it must have no entropy change. Isentropic processes are a crucial concept when evaluating the efficiency of turbines and compressors.

In the exercise, the isentropic process is used to calculate the ideal or maximum performance of the compressor, which serves as a benchmark to determine the real compressor's performance. The term 'isentropic efficiency' thus describes how effectively a compressor approaches this ideal isentropic behavior. A higher isentropic efficiency indicates less energy loss to heat and friction and a closer approximation to the ideal adiabatic process.