Question

Consider subsonic flow in a converging nozzle with fixed inlet conditions. What is the effect of dropping the back pressure to the critical pressure on ( \(a\) ) the exit velocity, ( \(b\) ) the exit pressure, and ( \(c\) ) the mass flow rate through the nozzle?

Short answer

Answer: By dropping the back pressure to the critical pressure in a converging nozzle with subsonic flow, there are three main effects: (a) the exit velocity increases to the speed of sound, (b) the exit pressure equals the critical pressure, and (c) the mass flow rate reaches its maximum possible value for this type of nozzle.

Step-by-step solution

Identify the knowns and unknowns

We are given a subsonic flow in a converging nozzle, and we need to determine the effect of dropping the back pressure to the critical pressure on the exit velocity, exit pressure, and mass flow rate. We will represent the exit velocity as \(V_e\), the exit pressure as \(P_e\), and the mass flow rate as \(\dot{m}\). The critical pressure is represented as \(P_{cr}\).

Calculate critical pressure

To find the critical pressure, we will use the isentropic flow equation: $$ P_{cr} = P_0 \left( \frac{2}{\gamma +1} \right)^\frac{\gamma}{\gamma -1} $$ Where \(P_0\) is the total pressure, and \(\gamma\) is the ratio of specific heats.

Determine the effect on exit velocity

As the back pressure drops to the critical pressure, it causes the flow to accelerate in the converging nozzle. Since the flow is subsonic, the exit velocity will increase until it reaches the speed of sound, which is the maximum possible velocity for this type of nozzle. Thus, by decreasing the back pressure to the critical pressure, we have increased the exit velocity to the speed of sound: $$ V_e = M_e a = a $$ Where \(M_e\) is the Mach number at the exit (which equals 1 for critical flow) and \(a\) is the speed of sound.

Determine the effect on exit pressure

When the back pressure drops to the critical pressure, it results in critical flow conditions at the exit of the nozzle. In critical flow, the exit pressure will be equal to the critical pressure: $$ P_e = P_{cr} $$

Determine the effect on mass flow rate

To find the mass flow rate, we will use the following equation: $$ \dot{m} = \rho A V $$ Where \(\rho\) is the density, \(A\) is the cross-sectional area of the nozzle, and \(V\) is the velocity. Since the exit velocity has increased to the speed of sound, the mass flow rate will also increase, reaching its maximum value for a converging nozzle. In this case, the density remains constant, so the mass flow rate can be calculated as: $$ \dot{m}_{max} = \rho A a $$ In summary, by dropping the back pressure to the critical pressure in a converging nozzle with subsonic flow, ( \(a\) ) the exit velocity increases to the speed of sound, ( \(b\) ) the exit pressure equals the critical pressure, and ( \(c\) ) the mass flow rate reaches its maximum possible value for this type of nozzle.