Question

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

Short answer

When a supersonic gas flow enters an adiabatic diverging duct, its velocity decreases, while the temperature, pressure, and density all increase as the gas flows through the duct. This occurs due to the conservation of mass, isentropic relations, and the behavior of an adiabatic diverging duct.

Step-by-step solution

(1) Defining the problem

Consider a gas flow with an initial supersonic velocity that enters an adiabatic diverging duct. An adiabatic process is one in which there is no heat exchange between the system (i.e., the gas flow) and the surroundings (i.e., the duct walls). The main characteristics to investigate are the changes of velocity, temperature, pressure, and density as the gas flows through the duct.

(2) Applying the conservation of mass

For a steady, isentropic, and one-dimensional flow, the continuity equation states that the product of the area 'A' along the duct, the density '\rho', and the velocity 'V' is constant: \begin{align} A \rho V = \text{constant}. \end{align}

(3) Examining the effects on velocity

In a diverging duct, the area 'A' increases along the flow direction. Since the right-hand side of Eq. (1) is constant and '\rho V' must also increase for the equality to hold, thus the velocity 'V' must decrease. Therefore, the gas velocity decreases as it flows through the adiabatic diverging duct.

(4) Applying the isentropic gas relations

Since the flow is adiabatic and supersonic, the isentropic gas relations can be applied. The isentropic relations for an ideal gas with ratio of specific heats '\gamma' are: \begin{align} \frac{T}{T_0} = \left(\frac{P}{P_0}\right)^{\frac{\gamma - 1}{\gamma}} = \frac{1}{\rho^{\gamma - 1}}, \end{align} where subscripts '0' denote initial conditions before entering the duct.

(5) Examining the effects on temperature

From step (3), we know that the gas velocity decreases as it flows through the adiabatic diverging duct. This decrease in velocity implies an increase in the pressure ratio 'P/P_0'. Referring to the isentropic gas relations, when the pressure ratio increases, the temperature ratio 'T/T_0' also increases. Therefore, the temperature of the gas increases as it flows through the adiabatic diverging duct.

(6) Examining the effects on pressure

As mentioned earlier, when the gas velocity decreases in the diverging duct, the pressure ratio 'P/P_0' increases. This means that the pressure of the gas increases as it flows through the adiabatic diverging duct.

(7) Examining the effects on density

Referring back to the isentropic gas relations in Eq. (2), an increase in the pressure ratio 'P/P_0' also leads to an increase in the density ratio '1/\rho^{\gamma - 1}$. Therefore, the density of the gas increases as it flows through the adiabatic diverging duct. In summary, as a supersonic gas flow enters an adiabatic diverging duct, its velocity decreases, while its temperature, pressure, and density all increase throughout the duct.