Question

A gas initially at a supersonic velocity enters an adiabatic converging duct. Discuss how this affects ( \(a\) ) the velocity, \((b)\) the temperature, \((c)\) the pressure, and \((d)\) the density of the fluid.

Short answer

In an adiabatic converging duct with a supersonic gas, the following changes occur: (a) the velocity increases, (b) the temperature decreases, (c) the pressure decreases, and (d) the density decreases. These changes are primarily due to the conservation of total enthalpy resulting from the adiabatic process and the relationships between velocity, pressure, and density described by Bernoulli's equation and the ideal gas law.

Step-by-step solution

Understand the Adiabatic Process

An adiabatic process refers to a thermodynamic process in which no heat is exchanged between the system (in our case, the gas) and its surroundings. The primary consequence of an adiabatic process is the conservation of total enthalpy, which is a combination of the gas's internal and kinetic energy.

Use Bernoulli's Equation to Relate Velocity and Pressure

To analyze the change in velocity, we need to use Bernoulli's equation for compressible flow: \(P_g + \cfrac{\rho_{g}V^{2}}{2} = Constant\) where \(P_g\) is the pressure, \(\rho_{g}\) is the density, and \(V\) is the fluid velocity. As the duct converges, the gas must go faster to maintain the mass flow rate. Thus, a converging duct accelerates the gas. Since the gas is already supersonic, the velocity will increase even further.

Analyze Temperature Changes in the Converging Duct

For an adiabatic process, we can express the temperature changes in terms of pressure and density using the ideal gas law combined with the adiabatic relation: \(T_g P_g^{\gamma -1} = Constant\) In the converging duct, the gas accelerates (and hence, the velocity increases). Using Bernoulli's Equation, we see that with a constant enthalpy given by the adiabatic condition, the pressure must decrease. With decreasing pressure, temperature will also decrease.

Analyze Pressure Changes in the Converging Duct

To assess the change in pressure, we use the relation between pressure, velocity, and density given by Bernoulli's equation. Since the gas accelerates in the converging duct (as seen in Step 2), it is implied through Bernoulli's equation that the pressure of the gas must decrease in this process.

Analyze Density Changes in the Converging Duct

To analyze the change in density, we can rearrange the ideal gas law in terms of density: \(\rho_g = \cfrac{P_g}{R_gT_g}\) where \(P_g\) is the pressure, \(R_g\) is the specific gas constant, and \(T_g\) is the temperature. Since we have shown that the pressure and temperature decrease in a converging duct, the decrease in pressure is more significant than the decrease in temperature. Due to this, the overall density of the gas will decrease as well. To summarize, in an adiabatic converging duct with a supersonic gas: \((a)\) The velocity increases. \((b)\) The temperature decreases. \((c)\) The pressure decreases. \((d)\) The density decreases.