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Chapter 6: Problem 67

\(\mathrm{A}\) fire nozzle is coupled to the end of a hose with inside diameter \(D=75 \mathrm{mm} .\) The nozzle is smoothly contoured and its outlet diameter is \(d=25 \mathrm{mm}\). The nozzle is designed to operate at an inlet water pressure of 700 kPa (gage). Determine the design flow rate of the nozzle. (Express your answer in L/s.) Evaluate the axial force required to hold the nozzle in place. Indicate whether the hose coupling is in tension or compression.

Short Answer

Expert verified
The design flow rate of the nozzle is calculated using Bernoulli's equation, the continuity equation, and the known parameters, to be expressed in L/s. The axial force required to hold the nozzle in place is determined using the momentum equation and indicates whether the hose coupling is in tension or compression.

Step by step solution

01

Determine the velocity at nozzle inlet

Using Bernoulli's equation, let point 1 be the nozzle inlet and point 2 be the nozzle outlet. Generally, the equation is \(P_1 + 0.5蟻v_1^2 + 蟻gh_1 = P_2 + 0.5蟻v_2^2 + 蟻gh_2\), where \(P\) is the pressure, \(蟻\) is the fluid density, \(v\) is the velocity, \(h\) is the height, and the subscripts denote their respective points. For this specific problem \(P_1 = 700 \, kPa\), \(P_2 = 0\) (since it's open to the atmosphere), \(蟻 = 1000 \, kg/m^3\), all heights are at the same level so \(h_1 = h_2 -> 蟻gh_1 = 蟻gh_2\). Thus this cancels out and can be removed from the equation. Solving the equation, velocity at point 1 \(v_1\) is \(v1 = sqrt{(P1 - P2) * 2/蟻}\), where \(sqrt\) is the square root function.
02

Determine the velocity at nozzle outlet

For steady, incompressible flow, the equation of continuity (\(Q = A_1v_1 = A_2v_2\), where \(Q\) is the flow rate, \(A\) is the cross-sectional area) can be used to find the velocity at the nozzle outlet \(v_2 = A_1v_1/A_2\). The cross-sectional areas \(A_1\) and \(A_2\) can be determined using \(A = 蟺r^2\), where \(r\) is the radius.
03

Calculate the design flow rate of the nozzle

The flow rate \(Q\) can be calculated using \(Q = A_2v_2\). However, remember to convert the answer to \(L/s\). This can be done by noting that 1 cubic meter is equivalent to 1000 liters.
04

Calculate the axial force

The axial force \(\Delta F\) required to hold the nozzle in place can be calculated by using the momentum equation \(\Delta F = \dot{m}(v_2 - v_1)\), where \(\dot{m}\) is the mass flow rate. The mass flow rate can be determined by \(\dot{m} = 蟻Q\). Here, the force is expected to be positive, indicating that the force is in the same direction as the fluid flow and hence the hose is in tension.

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