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Chapter 4: Problem 82

A flat plate orifice of 2 in. diameter is located at the end of a 4 -in.-diameter pipe. Water flows through the pipe and orifice at $20 \mathrm{ft}^{3} / \mathrm{s}$. The diameter of the water jet downstream from the orifice is 1.5 in. Calculate the external force required to hold the orifice in place. Neglect friction on the pipe wall.

Short Answer

Expert verified
The external force required to hold the orifice in place can be found by using the Bernoulli's equation and equation of motion. After computing the velocities and pressures by using the given diameters and flow rate, the internal force exerted by water can be found. This will be the force required externally to hold the orifice in place, but in the opposite direction.

Step by step solution

01

Calculate velocity and pressure before and after passing through the orifice

Firstly we need to calculate the velocity \(v_1\) and \(v_2\) and pressure \(P_1\) and \(P_2\). The formula to calculate the velocity of water in a pipe is \(v_0 = Q/A\), where \(Q\) is the flow rate and \(A\) is the cross-sectional area of the flow. The cross-sectional area \(A = (\pi d^2)/4\). So the cross-sectional areas of the pipe and the orifice can be obtained as \(A_1 = (\pi (4^2))/4 = \pi\) in^2 and \(A_2 = (\pi (2^2))/4 = \pi/2\) in^2, respectively. The velocities \(v_1 = Q/A1\) and \(v_2 = Q/A2\) can now be calculated.
02

Calculate the internal force exerted by the water on the orifice

The force that the water exerts on the orifice internally is equal to the change in momentum per unit time. Considering that the friction on the pipe wall is negligible and that we have steady flow, we can write, by Newton's second law, that Force exerted = mass flow rate * (velocity downstream - velocity upstream). To obtain the mass flow rate, 蟻 * Q can be utilized, where 蟻 is the density of the fluid. Thus, the force exerted can be computed as \(F_i = \rho Q (v_2 - v_1)\). Considering a density of water \(\rho\) = 62.4 lb/ft^3.
03

Calculate the external force required to hold the orifice in place

The force required to hold the orifice in place will be equal to the internal force exerted by water on the orifice. The force will act in the direction opposing the force exerted by water. So external force \(F_{ext} = -F_i\).

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