Question

An ideal gas expands in an adiabatic turbine from \(1200 \mathrm{K}\) and \(900 \mathrm{kPa}\) to \(800 \mathrm{K}\). Determine the turbine inlet volume flow rate of the gas, in \(\mathrm{m}^{3} / \mathrm{s}\), required to produce turbine work output at the rate of \(650 \mathrm{kW}\). The average values of the specific heats for this gas over the temperature range and the gas constant are \(c_{p}=1.13 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}, c_{v}=\) \(0.83 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K},\) and \(R=0.30 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\).

Short answer

The required turbine inlet volume flow rate for the gas to produce a turbine work output of 650 kW is approximately 0.7814 \(m^3/s\).

Step-by-step solution

Find the adiabatic index

Using the given specific heat values \(c_p = 1.13 \, \mathrm{kJ/kgK} \;\) and \(\,c_v = 0.83 \, \mathrm{kJ/kgK}\), we can find the adiabatic index (also known as the heat capacity ratio) using the formula: \(\gamma = \displaystyle\frac{c_p}{c_v}\) Plugging in the values: \(\gamma = \displaystyle\frac{1.13}{0.83} = 1.3614\)

Compute the turbine work per unit mass

We can use the adiabatic index to find the work done by the gas in the turbine, per unit mass, with the following equation: \(w = c_v (T_1 - T_2)\) Where \(w\) is the work done per unit mass, \(T_1\) is the initial temperature, and \(T_2\) is the final temperature. Plugging in the given temperatures and specific heat value: \(w = 0.83 (1200 - 800) \, \mathrm{kJ/kg} = 332 \, \mathrm{kJ/kg}\)

Determine the mass flow rate

Since we know the desired power output and the work done per unit mass, we can now find the mass flow rate (\(\dot{m}\)): \(\dot{P} = w \cdot \dot{m}\) Where \(\dot{P}\) is the power output. Rearranging for the mass flow rate: \(\dot{m} = \displaystyle\frac{\dot{P}}{w} = \frac{650}{332} = 1.957 \, \mathrm{kg/s}\)

Calculate the volume flow rate

Finally, we can determine the volume flow rate (\(\dot{V}_1\)) of the gas entering the turbine using the ideal gas law: \(\dot{V}_1 = \displaystyle\frac{\dot{m} \cdot R \cdot T_1}{P_1}\) Where \(R\) is the gas constant, \(T_1\) is the initial temperature, and \(P_1\) is the initial pressure. Plugging in the given values and the mass flow rate determined earlier: \(\dot{V}_1 = \displaystyle\frac{1.957 \cdot 0.3 \cdot 1200}{900} = 0.7814 \, \mathrm{m^3/s}\)

Final Answer

The required turbine inlet volume flow rate for the gas to produce a turbine work output of 650 kW is \(\approx 0.7814 \, \mathrm{m^3/s}\).

Key concepts

Adiabatic Process

An adiabatic process is a thermodynamic change in which a system does not exchange heat with its surroundings. This concept is crucial when analyzing systems like adiabatic turbines, where the gas within is compressed or expanded without heat transfer. Specifically, during an adiabatic process, all the system's work results in changes to the internal energy, which in turn affects the system's temperature and pressure.

In the context of a turbine, as gas expands adiabatically, its temperature decreases because it does work on the surrounding blades, converting internal energy into mechanical energy. It's also important to note that adiabatic processes are reversible only when they occur without any entropy change or friction, which is an idealization in real-world applications.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the amount of substance present (n) through the gas constant (R). The law is often expressed as:
\[PV = nRT\]
It serves as an excellent approximation for the behavior of many gases under various conditions, though it assumes no interactions between gas molecules and that the volume occupied by the gas particles themselves is negligible.

For practical uses, such as in the problem of calculating the turbine inlet volume flow rate, the ideal gas law helps to relate the physical properties of the gas at different states, which allows the determination of the flow rate given the mass flow rate, temperature, and pressure.

Specific Heats

Specific heats are properties that represent the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius or Kelvin. There are two specific heats important in thermodynamics: specific heat at constant volume (\(c_v\)) and at constant pressure (\(c_p\)).

The values of \(c_v\) and \(c_p\) differ for various substances, and their ratio (\(\gamma = c_p/c_v\)) is significant in analyzing adiabatic processes for an ideal gas. This ratio is also known as the adiabatic index or heat capacity ratio, which plays a key role in the determination of the work done by the gas in processes like adiabatic expansion or compression.

Volume Flow Rate Calculation

Volume flow rate (dot)V) measures how much volume of fluid travels past a point in a given unit of time, indicating the throughput of fluid in systems such as pipes, channels, or in this case, a turbine. To calculate the volume flow rate of a gas, we typically use the ideal gas law in combination with the mass flow rate (dot)m), temperature, and pressure of the gas:
\[dot)V = dot)m \frac{RT}{P}\]
This equation connects the mass flow rate and the thermodynamic state of the gas to its volume flow. For the adiabatic turbine problem specifically, the volume flow rate needed to produce a certain power output is key to designing and evaluating the performance of the turbine system.