Reynolds Transport Theorem and Control Volume Analysis

Control volume momentum balance on jets and bends

The linear momentum equation for a control volume states that the sum of external forces acting on the fluid inside the control volume equals the net rate of momentum leaving through the control surface minus the net rate entering. For steady flow with uniform properties at each one-dimensional section, this reads \sum \vec F = \sum_{out} \dot m \vec V - \sum_{in} \dot m \vec V, where \dot m = \rho A V. The force that the fluid exerts on a solid body, such as a bend, vane, nozzle or plate, equals the force the body exerts on the fluid in magnitude, but is opposite in direction. The support reaction is therefore found after including pressure forces and the momentum change.

Key points:

• Pressure forces act on each section as pA. Gage pressure is used when the exit is open to atmosphere, which makes the exit pressure force zero. For a pipe bend, the component of the support force along a direction is the sum of the inlet and exit pressure forces and the momentum change \dot m(V_{2x}-V_{1x}).
• Gravity is usually negligible for horizontal jets and bends, so the balance is taken only in the plane of the flow.
• For a jet striking a plate and spreading parallel to it, the momentum normal to the plate falls to zero, so F = \rho A V^2. Equating this to a weight mg gives the minimum jet velocity needed to hold the mass, V = \sqrt{mg/(\rho A)}.
• For a jet engine in which the inlet and exhaust flows are in the same direction, the thrust is \dot m(V_{out}-V_{in}) when the pressure difference is neglected.
• For a rotating sprinkler arm, the torque equals the outflow moment of momentum, T = \sum \rho Q r V_\theta, where V_\theta is the absolute tangential velocity. Fluid entering on the axis carries no angular momentum.
• When a vane turns a jet, the force along a chosen direction equals the momentum change along that direction, so the incoming component and the outgoing speed must both be counted with the correct sign.
• For a plate with a central orifice, if the fluid through the hole has the same speed as the jet, the force is F = \rho V^2 (A_1 - A_2), where A_2 is the hole area. Radially outflowing fluid carries no axial momentum and does not change the force. Enlarging the hole reduces the force, while increasing the jet area or the velocity increases it.