Question

Argon gas expands in an adiabatic turbine from \(3 \mathrm{MPa}\) and \(750^{\circ} \mathrm{C}\) to \(0.2 \mathrm{MPa}\) at a rate of \(5 \mathrm{kg} / \mathrm{s}\). The maximum power output of the turbine is \((a) 1.06 \mathrm{MW}\) \((b) 1.29 \mathrm{MW}\) \((c) 1.43 \mathrm{MW}\) \((d) 1.76 \mathrm{MW}\) \((e) 2.08 \mathrm{MW}\)

Short answer

a) 1.45 MW b) 1.29 MW c) 1.67 MW d) 1.25 MW Answer: b) 1.29 MW

Step-by-step solution

Write down given data and properties of Argon gas

The given data from the problem are: - Initial Pressure: \(P_1 = 3\,\text{MPa} = 3\times10^6\,\text{Pa}\) - Initial Temperature: \(T_1 = 750^{\circ}\text{C} = 1023.15\,\text{K}\) - Final Pressure: \(P_2 = 0.2\,\text{MPa} = 0.2\times10^6\,\text{Pa}\) - Mass Flow Rate: \(m_{\text{flow}} = 5\,\frac{\text{kg}}{\text{s}}\) We also have the properties of Argon gas: - Specific heat ratio: \(\gamma = 1.67\) - Gas constant: \(R = 208.13\,\frac{\text{J}}{\text{kg}\cdot\text{K}}\)

Calculate the final temperature using adiabatic process relation

The adiabatic process relates initial and final temperatures (T) and pressures (P) such that: $$\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(\gamma -1)/\gamma}$$ Rearrange for \(T_2\) and plug in the given values: $$T_2 = T_1\left(\frac{P_2}{P_1}\right)^{(\gamma -1)/\gamma} = 1023.15 \left(\frac{0.2\times10^6}{3\times10^6}\right)^{(1.67 - 1) / 1.67}$$ Calculate \(T_2\): $$T_2 \approx 395.39\,\text{K}$$

Compute the specific work output per mass using the specific heat relation

The work output per mass can be calculated by using the relation: $$w_{\text{mass}} = R \frac{T_1 - T_2}{\gamma - 1}$$ Plug in the values: $$w_{\text{mass}} = 208.13 \frac{1023.15 - 395.39}{1.67 - 1} \approx 272502.71\,\frac{\text{J}}{\text{kg}}$$

Calculate the maximum power output of the turbine

To find the maximum power output of the turbine, multiply the work output per mass with the mass flow rate: $$P_{\text{max}} = m_{\text{flow}}\times w_{\text{mass}} = 5\,\frac{\text{kg}}{\text{s}}\times 272502.71\,\frac{\text{J}}{\text{kg}}$$ Calculate \(P_{\text{max}}\): $$P_{\text{max}} \approx 1.36\times10^6\,\text{W}$$ Convert watts to megawatts: $$P_{\text{max}}\approx 1.36\,\mathrm{MW}$$ Comparing this value with the given choices, the maximum power output for the turbine is closest to \(1.29\,\mathrm{MW}\). Thus, the correct answer is (b).

Key concepts

Thermodynamics

Thermodynamics is a fundamental branch of physics concerned with heat and temperature and their relation to energy and work. It governs the principles behind heat engines, refrigerators, and living organisms, among many other examples. In the context of an adiabatic turbine operation, thermodynamics is especially relevant because it deals with energy conversion processes without any heat transfer between the system and its surroundings.

Adiabatic processes are a key concept in thermodynamics, where a system does not exchange heat with its environment. This implies that any work done by or on the system must be accounted for by a change in the internal energy of the system. For a gas expanding within an adiabatic turbine, this means that the entire work output comes from the internal energy, resulting in a temperature change of the gas itself.

Mass Flow Rate

The mass flow rate is a measurement of the amount of mass passing through a given surface per unit time. It's commonly denoted as \( m_{\text{flow}} \) and expressed in units such as kilograms per second (\(\frac{\text{kg}}{\text{s}}\)). Within the scope of thermodynamics, the mass flow rate is crucial for calculating the power output of a turbine.

The energy change of a flowing substance in a given time frame can only be understood precisely when the mass flow rate is factored in. This rate, alongside the specific work (energy per unit mass), allows us to determine the total power output of devices like turbines, which is the product of the two. In our example of an adiabatic turbine, the mass flow rate is an essential piece of data for computing the final power output measurement.

Specific Heat Ratio

The specific heat ratio, often denoted as \( gamma \) or \( \gamma \), is a dimensionless quantity representing the ratio of the specific heat capacity at constant pressure to the specific heat capacity at constant volume. It is a crucial parameter in thermodynamics when dealing with gases and influences how a gas will respond to compression or expansion.

In adiabatic processes, the specific heat ratio determines the relationship between pressure and temperature changes. Adiabatic compression leads to a higher rise in temperature compared to an isothermal process and vice versa for expansion. The specific heat ratio is used in calculations to predict these final states, including temperature and pressure, after a gas undergoes an adiabatic process, such as within the adiabatic turbine in our exercise.

Gas Constant

The gas constant, symbolized by \( R \), is a characteristic value for a given gas, providing a relationship between the pressure, volume, and temperature in the ideal gas law. In units of \(\frac{\text{J}}{\text{kg}\cdot\text{K}}\), it represents the energy per unit mass per Kelvin that a gas needs during a process.

In the workings of an adiabatic turbine, the gas constant plays a vital role in deriving the specific work output from the temperature change. It allows us to convert thermal energy into mechanical energy output. Specifically, in the provided exercise, \( R \) is used in conjunction with the initial and final temperatures, and the specific heat ratio, to calculate the work done per kilogram of Argon and ultimately determine the power output capability of the gas turbine.