Question

Air is expanded in an adiabatic turbine of 85 percent isentropic efficiency from an inlet state of \(2200 \mathrm{kPa}\) and \(300^{\circ} \mathrm{C}\) to an outlet pressure of \(200 \mathrm{kPa}\). Calculate the outlet temperature of air and the work produced by this turbine per unit mass of air.

Short answer

Question: Determine the final temperature and work produced per unit mass of air for an adiabatic turbine with an initial pressure of 300 kPa, a final pressure of 100 kPa, an initial temperature of 500 K, and isentropic efficiency of 80%. Answer: To find the final temperature and work produced per unit mass of air for the adiabatic turbine, follow these steps: 1. Use the isentropic efficiency equation to determine the isentropic final temperature, with given values and γ ≈ 1.4 for air: $$ T_{2isentropic} = 500 \times \left(\frac{100}{300}\right)^\frac{1.4-1}{1.4} \approx 361.1 K $$ 2. Find the actual enthalpy change and outlet temperature using the isentropic efficiency (80%) and specific heat at constant pressure for air (Cp ≈ 1005 J/kg·K): $$ h_{2actual} = h_{1} - 0.8(h_{1} - h_{2isentropic}) $$ $$ T_{2actual} = 500 + \frac{h_{2actual} - h_{1}}{1005} \approx 378.4 K $$ 3. Calculate the work produced per unit mass of air using the determined outlet temperature: $$ W_{turbine} = 1005(500 - 378.4) \approx 122196 J/kg $$ So, the final temperature is approximately 378.4 K, and the work produced per unit mass of air is approximately 122,196 J/kg.

Step-by-step solution

Use isentropic efficiency equation to determine the isentropic final temperature

The isentropic efficiency of the turbine process is given by: $$ \eta_{\text{isentropic}} = \frac{h_{1} - h_{2actual}}{h_{1} - h_{2isentropic}} $$ We'll first determine the isentropic enthalpy change, then find the actual enthalpy change using the given isentropic efficiency. For an isentropic process, we can use the following relation: $$ \frac{P_{2isentropic}}{P_{1}} = \left(\frac{T_{2isentropic}}{T_{1}}\right)^\frac{\gamma}{\gamma-1} $$ where \(\gamma\) is the specific heat ratio (\(\approx 1.4\) for air). Now, isolate \(T_{2isentropic}\) and plug in the given numbers: $$ T_{2isentropic} = T_{1} \left(\frac{P_{2isentropic}}{P_{1}}\right)^\frac{\gamma-1}{\gamma} $$

Find the actual enthalpy change and outlet temperature

Using the isentropic efficiency equation, solve for \(h_{2 actual}\): $$ h_{2actual} = h_{1} - \eta_{\text{isentropic}}(h_{1} - h_{2isentropic}) $$ Now, since we have an adiabatic process, the internal energy change is only due to changes in temperature. Therefore: $$ h_{2actual} - h_{1} = C_{p}(T_{2actual} - T_{1}) $$ where \(C_{p}\) is the specific heat of air at constant pressure (\(\approx 1005 \mathrm{J/kg\cdot K}\) for air). Solve for \(T_{2actual}\): $$ T_{2actual} = T_{1} + \frac{h_{2actual} - h_{1}}{C_{p}} $$

Calculate the work produced per unit mass of air

Applying the first law of thermodynamics for the adiabatic process: $$ W_{turbine} = h_{1} - h_{2actual} $$ Now, since the process is adiabatic and the only energy change is due to changes in temperature, we can rewrite the work in terms of temperature: $$ W_{turbine} = C_{p}(T_{1} - T_{2actual}) $$ Now, we have all the necessary equations to find the outlet temperature and the work produced per unit mass of air. Plug in known values into the equations from Steps 1-3 to find \(T_{2isentropic}\), \(T_{2actual}\), and \(W_{turbine}\).

Key concepts

Isentropic Efficiency

Understanding isentropic efficiency is essential when studying turbines and other thermodynamic cycles. It describes the performance of a turbine by comparing the actual work output with the ideal or maximum possible work output in an isentropic process—a process that is both adiabatic and reversible.

Isentropic efficiency, often expressed as a percentage, is calculated using the formula:
\[\begin{equation}\eta_{\text{isentropic}} = \frac{h_{1} - h_{2\text{actual}}}{h_{1} - h_{2\text{isentropic}}}\end{equation}\]
where:

• \(h_1\) represents the initial enthalpy of the gas entering the turbine,
• \(h_2\text{actual}\) is the actual enthalpy of the gas exiting the turbine, and
• \(h_2\text{isentropic}\) is the enthalpy the gas would have if it expanded isentropically (without any entropy change).

The closer the isentropic efficiency is to 100%, the more efficiently the turbine is performing relative to an ideal process without any heat loss or friction.

Enthalpy Change

Enthalpy change in the context of turbines refers to the difference in the enthalpy (total heat content) of a fluid between the inlet and the outlet of the turbine. It's a measure of the energy transferred in or out of the system, excluding any work done by the system or changes in kinetic or potential energy.

Specifically, for adiabatic turbines, the enthalpy change can be directly related to the work done by the system since the process is adiabatic and therefore no heat is transferred:
\[\begin{equation}W_{\text{turbine}} = h_{1} - h_{2\text{actual}}\end{equation}\]
where \( W_{\text{turbine}} \) is the work done by the turbine. When gases or vapors expand through a turbine, their pressure and temperature drop, which is reflected in the enthalpy change. The exact relationship depends on the properties of the fluid and the conditions of the process.

Specific Heat Ratio

The specific heat ratio, denoted as \( \gamma \), is a dimensionless number representing the ratio of the specific heat at constant pressure \( C_p \) to the specific heat at constant volume \( C_v \). It is a crucial property for gases, especially in thermodynamic calculations for processes such as the expansion or compression in turbines or compressors. For an ideal gas, the specific heat ratio has a significant impact on the temperature and pressure changes during an adiabatic process.

The specific heat ratio is used in equations describing isentropic processes:
\[\begin{equation}\left( \frac{P_{2\text{isentropic}}}{P_{1}} \right) = \left( \frac{T_{2\text{isentropic}}}{T_{1}} \right)^\frac{\gamma}{\gamma-1}\end{equation}\]
The value of \( \gamma \) varies with the type of gas and its conditions. For air, a common approximation is \( \gamma \approx 1.4 \), which is used in thermodynamic calculations related to air compression and expansion.

First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system; it can only change forms. In terms of turbines and other thermodynamic devices, the first law is often utilized to relate the heat transfer and the work done with the change in internal energy (or enthalpy in constant-pressure processes) of the working fluid.

For an adiabatic turbine, the first law simplifies since there is no heat transfer (\( Q = 0 \)), and the work done by the turbine (\( W_{\text{turbine}} \)) is related to the enthalpy change of the fluid:
\[\begin{equation}W_{\text{turbine}} = h_{1} - h_{2\text{actual}}\end{equation}\]
This relationship allows engineers to predict the performance and efficiency of turbines by analyzing how the enthalpy of the working fluid changes as it moves through the system.