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Chapter 4: Problem 98

\(\mathrm{A}\) nozzle for a spray system is designed to produce a fiat radial sheet of water. The sheet leaves the nozzle at \(V_{2}=\) \(10 \mathrm{m} / \mathrm{s},\) covers \(180^{\circ}\) of arc, and has thickness \(t=1.5 \mathrm{mm}\). The nozzle discharge radius is \(R=50 \mathrm{mm}\). The water supply pipe is \(35 \mathrm{mm}\) in diameter and the inlet pressure is \(p_{1}=150 \mathrm{kPa}\) (abs). Evaluate the axial force exerted by the spray nozzle on the coupling.

Short Answer

Expert verified
The axial force exerted by the spray nozzle on the coupling can be computed by using Newton's second law, which involves finding the mass flow rate of water, the outlet and inlet momentum and the difference between them. The exact numerical value of the force depends on the given parameters.

Step by step solution

01

Calculation of mass flow rate

The mass flow rate (\(m\)) of water from the spray nozzle can be calculated using the formula \(m = \rho \cdot Q\), where \(\rho\) is the density of water (typically 1000 kg/m鲁) and Q is the volumetric flow rate. The volumetric flow rate can be calculated as \(Q=V_{2}\cdot A\), where \(V_{2}\) (10 m/s) is the water velocity and A is the cross-sectional area of the water sheet. For a flat radial sheet of water covering 180掳 of arc, the shape of the cross-section is a semi-circle. The area \(A\) of the semi-circle is given by \(\frac{1}{2} * \pi * R^{2}\) where \(R\) is the radius of the spray nozzle (50 mm or 0.05 m). We will convert all the units to SI units (m, kg, s).
02

Calculation of outlet momentum

The outlet momentum (\(P_{out}\)) is the product of the mass flow rate (\(m\)) and the outlet velocity (\(V_{2}\)). Using the value of \(m\) obtained in Step 1, \(P_{out}\) can be calculated as \(P_{out} = m * V_{2}\).
03

Calculation of inlet momentum

The inlet momentum (\(P_{in}\)) is the product of the mass flow rate (\(m\)) and the inlet velocity (\(V_{1}\)). Since the inlet pressure is given (150 kPa), the inlet velocity can be derived from Bernoulli's equation which describes the conservation of mechanical energy in a flowing fluid. Substituting the known values into Bernoulli's equation gives \(V_{1} = \sqrt{\frac{2(p_{2}-p_{1})}{\rho}}\), where \(p_{1}\) is the inlet pressure (150 kPa or 150000 Pa), \(p_{2}\) is the outlet pressure (assumed to be atmospheric pressure, i.e., 0 Pa), and \(\rho\) is the density of water. After calculating \(V_{1}\), \(P_{in}\) can be calculated as \(P_{in} = m * V_{1}\).
04

Calculation of axial force

The axial force (\(F\)) exerted by the spray nozzle on the coupling is the time rate of change of momentum, or equivalently, the difference between the outlet momentum and the inlet momentum. It can be calculated as \(F = P_{out} - P_{in}\). Using the values of \(P_{out}\) and \(P_{in}\) obtained from Steps 2 and 3, the axial force can be found.

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