An ideal gas flows adiabatically through a duct. At section...

An ideal gas flows adiabatically through a duct. At section $1, p_{1}=140 \mathrm{kPa}, T_{1}=260^{\circ} \mathrm{C},$ and $V_{1}=75 \mathrm{m} / \mathrm{s},$ Farther downstream, $p_{2}=30 \mathrm{kPa}$ and $T_{2}=207^{\circ} \mathrm{C}$. Calculate $V_{2}$ in $\mathrm{m} / \mathrm{s}$ and $s_{2}-s_{1}$ in $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ if the gas is $(a)$ air, $k=1.4$ and $(b)$ argon, $k=1.67$

Key Concepts

Steady-Flow Energy Equation

The steady-flow energy equation is fundamental in analyzing the energy balance of a fluid flowing through a control volume, such as a duct. In adiabatic flow, where no heat is transferred into or out of the system, this equation relates the change in kinetic energy and enthalpy between two sections of the flow. Specifically, it is used to determine the change in velocity by equating the sum of the static enthalpy and dynamic kinetic energy at the inlet and outlet of the duct.

Ideal Gas Law

The ideal gas law is crucial for relating the pressure, temperature, and density of the flowing gas. This relationship allows one to compute thermodynamic properties like specific enthalpy and is used in conjunction with the energy equation. Understanding the ideal gas behavior is essential when dealing with flow problems involving gases under various thermodynamic states.

Specific Enthalpy

Specific enthalpy is a measure of the total energy content per unit mass due to the thermal state of a gas. For an ideal gas, enthalpy is directly proportional to temperature, and this proportionality is represented by the specific heat capacity at constant pressure. It plays a key role in the energy balance for adiabatic flows, as the conversion between thermal and kinetic energy is mediated by the enthalpy differences between sections.

Specific Entropy Change for Ideal Gases

The change in specific entropy for an ideal gas can be calculated using the relationship that involves the logarithmic changes of temperature and pressure, along with the specific heats and the gas constant. This concept is important for assessing the thermodynamic irreversibility of the process, especially when the process is adiabatic but not necessarily isentropic.

Specific Heat Ratio and Gas Properties

The specific heat ratio, often denoted as k (or ?), influences the relationship between pressure and temperature changes in a gas. Different gases have different specific heat ratios, which affect how energy is partitioned between kinetic and thermal forms during flow. This concept is essential when comparing flows of different gases, such as air and argon, under similar thermodynamic conditions.

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