Question

For the isentropic flow of an ideal gas with constant specific heat ratio \(k\), the ratio of the temperature \(T^{*}\) to the stagnation temperature \(T_{\mathrm{o}}\) is \(T^{*} / T_{o}=2 /(k+1)\). Develop this relationship.

Short answer

\( T^* / T_0 = \frac{2}{k+1} \).

Step-by-step solution

- Understanding the Isentropic Flow

In isentropic flow, the process is both adiabatic and reversible. This means there is no heat transfer and entropy remains constant.

- Ideal Gas Law

Recall the ideal gas law: \( PV = nRT \)

- Stagnation Temperature

The stagnation temperature is the temperature an ideal gas attains when brought to rest adiabatically. It includes both kinetic energy and internal energy.

- Applying Energy Conservation

Using the energy conservation principle for an ideal gas, the stagnation temperature \( T_0 \) can be expressed as:\[ T_0 = T + \frac{v^2}{2c_p} \]where: - \( T \) is the static temperature - \( v \) is the flow speed - \( c_p \) is the specific heat at constant pressure.

- Relating Temperature Ratio

For isentropic processes of an ideal gas with specific heat ratio \( k \), the temperature ratio can be simplified. The relationship between the static temperature \( T \) and the stagnation temperature \( T_0 \) is given by:\[ \frac{T}{T_0} = \left( \frac{2}{k+1} \right) \]

- Conclusion

Thus, the ratio of the temperature \( T^* \) of the flow to the stagnation temperature \( T_0 \) simplifies to \( T^* / T_{0} = \frac{2}{k+1} \).

Key concepts

Thermodynamics

Thermodynamics is the study of energy, heat, work, and how they interact with each other. The key concept in thermodynamics related to this problem is the isentropic process. In an isentropic process, entropy remains constant, meaning there is no loss or gain of energy due to friction or other forms of dissipation. In the context of isentropic flow, we are dealing with an ideal gas, where the process is both adiabatic (no heat transfer) and reversible. This means we can use thermodynamic relationships to determine various properties like temperature and pressure without loss of energy due to heat.

Stagnation Temperature

Stagnation temperature is a critical concept in fluid dynamics and thermodynamics. It is the temperature that a moving fluid would reach if it were brought to a complete stop adiabatically. This includes both the kinetic energy of the fluid and its internal energy. In mathematical terms, the stagnation temperature (\ref{T_0}) for an ideal gas can be expressed using the energy conservation principle as: \[ T_0 = T + \frac{v^2}{2c_p} \] where:

• \ref{T} is the static temperature.
• \ref{v} is the flow speed.
• \ref{c_p} is the specific heat at constant pressure.

Understanding this helps in determining how temperature and energy are distributed in the flow.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transferred or converted from one form to another. For the isentropic flow of an ideal gas, this principle allows us to relate different forms of energy within the system, like kinetic energy and thermal energy. In the context of the given problem, the stagnation temperature (\ref{T_0}) formula: \[ T_0 = T + \frac{v^2}{2c_p} \] illustrates how kinetic energy (from motion) and thermal energy (internal energy) contribute to the fluid's total energy when it is brought to rest adiabatically. This relationship helps in understanding the energy dynamics of the gas flow.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure (\ref{P}), volume (\ref{V}), and temperature (\ref{T}) of an ideal gas to the number of moles (\ref{n}) and the universal gas constant (\ref{R}). The equation is written as: \[ PV = nRT \] This law is crucial for understanding how gases behave under various conditions. In the context of the isentropic flow problem, the ideal gas law helps in determining how temperature changes due to changes in pressure and volume. This relationship is vital for further calculations, such as determining the temperature ratio (\ref{T/T_0}) for isentropic processes.

Specific Heat Ratio

The specific heat ratio (\ref{k}, also known as the adiabatic index) is a crucial parameter in thermodynamics, particularly for ideal gases. It is the ratio of specific heat at constant pressure (\ref{c_p}) to specific heat at constant volume (\ref{c_v}): \[ k = \frac{c_p}{c_v} \] This ratio is important for characterizing the thermodynamic behavior of gases under different processes. For the isentropic flow of an ideal gas, the specific heat ratio (\ref{k}) helps simplify the temperature ratio relationship. In the problem, the ratio of the temperature (\ref{T^*}) of the flow to the stagnation temperature (\ref{T_0}) is given by: \[ \frac{T^*}{T_0} = \frac{2}{k+1} \] This relationship helps understand how temperature and energy distribute in the gas flow during isentropic processes.