Air flows into the atmosphere from a nozzle and strikes a vertical plate as shown in the figure below. A horizontal force of 8 N is required to hold the plate in place. Determine the reading on the pressure gage. Assume the flow to be incompressible and frictionless.
Control volume analysis concepts:
Momentum flux is a result of fluid motion across the control surface.
External forces, including surface forces and body forces act on a control volume.
The resultant force on a control volume results from the interaction of the momentum fluxes, surfaces, and body forces.
Multiple control volumes may be used within a problem.
The conservation equations for a fluid are a system of equations.
Begin by examining what is known and considering the control volume(s) needed.
Carefully consider the control volume that would allow the determination of the velocity of the air exiting the nozzle.
(a) What is the cross-sectional area at the position of the pressure gage?
A = m^2
(b) What is the density of the fluid jet striking the plate?
? = kg/m^3
(c) What are the magnitude and direction of the horizontal force holding the plate?
F = N
(d) What is the cross-sectional area of the jet as it exits the nozzle, before it strikes the plate?
A = m^2
(e) What is the gage pressure of the jet at the nozzle exit and approaching the plate?
p = kPa
(f) What is the gage pressure of the jet traveling radially outward beyond the edges of the plate?
p = kPa
(g) What is the gage pressure acting on the right-hand face of the plate?
p = kPa
The correct answer and explanation is:
To determine the pressure gage reading, let’s analyze the problem using control volume analysis and the principles of momentum conservation.
Given Information:
- The fluid (air) exits a nozzle and impacts a vertical plate.
- A horizontal force of 8 N is required to hold the plate in place.
- The flow is incompressible and frictionless.
Control Volume Analysis:
- Velocity of the Jet:
- Applying Bernoulli’s equation between the nozzle entrance and exit: P1+12ρV12=P2+12ρV22P_1 + \frac{1}{2} \rho V_1^2 = P_2 + \frac{1}{2} \rho V_2^2 Since air exits the nozzle into the atmosphere, we assume P2=0P_2 = 0 (gage pressure).
If the upstream pressure is known, velocity can be determined.
- Applying Bernoulli’s equation between the nozzle entrance and exit: P1+12ρV12=P2+12ρV22P_1 + \frac{1}{2} \rho V_1^2 = P_2 + \frac{1}{2} \rho V_2^2 Since air exits the nozzle into the atmosphere, we assume P2=0P_2 = 0 (gage pressure).
- Momentum Conservation in the Horizontal Direction:
- Since the plate is held in place by an 8 N force, Newton’s second law in the horizontal direction gives: F=m˙VF = \dot{m} V where m˙=ρAV\dot{m} = \rho A V is the mass flow rate of air.
- Pressure Conditions:
- At the nozzle exit, pressure is near atmospheric (0 gage pressure).
- The jet striking the plate leads to a stagnation region, where the pressure increases.
- Air then flows radially outward beyond the plate with nearly atmospheric pressure.
Answers:
(a) Cross-sectional area at the pressure gage: A=m2A = \text{m}^2
(Requires problem-specific data)
(b) Density of the fluid jet (ρ\rho): ρ=kg/m3\rho = \text{kg/m}^3
(Assume air density at standard conditions: 1.225 kg/m³)
(c) Horizontal force holding the plate: F=8 NF = 8 \text{ N}
(Given in problem)
(d) Cross-sectional area of the jet at nozzle exit: A=m2A = \text{m}^2
(Requires data)
(e) Gage pressure of the jet at nozzle exit: p=0 kPap = 0 \text{ kPa}
(Since it exits into the atmosphere)
(f) Gage pressure of the jet beyond the plate edges: p=0 kPap = 0 \text{ kPa}
(Since it disperses into the atmosphere)
(g) Gage pressure on the right-hand face of the plate:
- The stagnation pressure can be estimated using Bernoulli’s equation: Ps=12ρV2P_s = \frac{1}{2} \rho V^2
- The force balance gives: PsA=8 NP_s A = 8 \text{ N}
- Solving for PsP_s, the gage pressure on the plate is: p=kPap = \text{kPa} (Requires numerical substitution)
Explanation
In fluid mechanics, momentum conservation is a crucial tool for analyzing flow interactions. Here, air exits a nozzle, forming a free jet that strikes a vertical plate. The force required to hold the plate is due to the change in momentum of the air.
Using Bernoulli’s equation, we determine the velocity of the air exiting the nozzle. Since the jet is exposed to the atmosphere before striking the plate, the gage pressure at the nozzle exit is zero. The plate acts as a stagnation surface, causing the air velocity to drop to zero, thereby increasing pressure. This stagnation pressure is determined from the kinetic energy of the air, using the relation Ps=12ρV2P_s = \frac{1}{2} \rho V^2.
The air then spreads radially beyond the plate, where it re-equilibrates with the surrounding atmospheric conditions, resulting in a gage pressure of zero. The horizontal force required to hold the plate is calculated using momentum conservation, where the force is equal to the rate of change of momentum of the incoming fluid.
By solving the force balance equation, we determine the stagnation pressure acting on the plate. The pressure gauge should read this stagnation pressure value, which depends on the velocity of the jet and air density.
Thus, control volume analysis enables precise determination of forces and pressures, essential for designing systems involving fluid jets and obstacles.
Now, I will generate an image illustrating the flow and forces involved.
Here’s the schematic diagram illustrating the air jet flow, forces, and control volume analysis. Let me know if you need any modifications or additional explanations!
