Question

Consider a converging nozzle with sonic speed at the exit plane. Now the nozzle exit area is reduced while the nozzle inlet conditions are maintained constant. What will happen to \((a)\) the exit velocity and \((b)\) the mass flow rate through the nozzle?

Short answer

Answer: (a) The exit velocity will increase when the exit area is reduced and the inlet conditions are constant. (b) The mass flow rate through the nozzle will remain constant when the nozzle exit area is reduced, and the nozzle inlet conditions are maintained constant.

Step-by-step solution

Understand the given conditions

The exercise tells us that we are dealing with a converging nozzle with sonic speed at the exit plane. The inlet conditions of the nozzle are maintained constant while the exit area is reduced.

Formulate necessary equations

To analyze the behavior of the nozzle, we will need two important equations: 1. The equation for mass flow rate (m_dot): \(m_dot = \rho A_1 V_1 = \rho A_2 V_2\) Here, ρ = density of the fluid \(A_1\) = area of the inlet plane \(V_1\) = velocity at the inlet plane \(A_2\) = area of the exit plane \(V_2\) = velocity at the exit plane 2. The equation relating the Mach number (Ma) to the exit velocity (assuming an isentropic flow): \(Ma = \frac{V_2}{a_2}\) Here, \(a_2\) = speed of sound at the exit plane \(Ma = 1\) at sonic speed

Analyze the exit velocity

Set the Mach number equal to 1, so we have \(V_2 = a_2\). Now we are told that the exit area (\(A_2\)) is reduced while the inlet conditions (\(\rho\), \(A_1\), and \(V_1\)) are maintained constant. Since we know that mass flow rate should remain constant, this implies that \(V_2\) must increase to compensate for the decreased \(A_2\) (according to the mass flow rate equation). Hence, the exit velocity will increase when the exit area is reduced and the inlet conditions are constant.

Analyze the mass flow rate

From step 2, we have the mass flow rate equation: \(m_dot = \rho A_1 V_1 = \rho A_2 V_2\) We are told that the nozzle inlet conditions (\(\rho\), \(A_1\), and \(V_1\)) are maintained constant. Therefore, the term \(\rho A_1 V_1\) on the left side of the equation remains constant. As mentioned previously, when \(A_2\) decreases, \(V_2\) must increase in order to maintain a constant mass flow rate. Thus, the mass flow rate through the nozzle is constant, as the product \(\rho A_2 V_2\) remains constant even when \(A_2\) is reduced. In conclusion: \((a)\) The exit velocity will increase when the exit area is reduced and the inlet conditions are constant. \((b)\) The mass flow rate through the nozzle will remain constant when the nozzle exit area is reduced, and the nozzle inlet conditions are maintained constant.