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Chapter 5: Problem 46

Figure \(\mathrm{PS} .46\) shows a lateral pipe fitting. This particular fitting has a mainline diameter of 4.0 in. The diameter of the lateral is 3.0 in. and the lateral angle is \(45^{\circ} ; 60^{\circ} \mathrm{F}\) water is flowing in the lateral. Measurements show that the pressure at point 1 is 34.0 psig. the pressure at point 2 is 35.0 psig. the pressure at point 3 is 33.5 psig. and the flow rate at point 2 is \(1.0 \mathrm{ft}^{3} / \mathrm{s}\). Determine the horizontal and vertical force components \(\left(F_{x}\) and \right. \(F_{y}\) ) required to hold the lateral fitting stationary. Neglect gravity. \(Q_{1}=1.63 \mathrm{ft}^{3} / \mathrm{s}\)

Short Answer

Expert verified
The step-by-step process involves calculating the inflow and outflow mass rates, applying Bernoulli's equation between key points to find the velocities, and then applying force balance to find the vertical and horizontal forces. The exact values would depend on the numerical calculations from the given values. All calculations must ignore the gravitational force as stated in the problem.

Step by step solution

01

Calculating Inflow and Outflow Mass Rates

The first step is to calculate the inflow and outflow mass rates using the given flow rates \(Q_1\) and \(Q_2\). This can be done using the formula: \(\dot{m} = Q \times \rho\), where \(\rho\) is the density of water, which is approximately \(998 \, \mathrm{Kg/m^3}\). Since \(1 \, \mathrm{ft^3/s}\) is approximately equal to \(0.0283 \, \mathrm{m^3/s}\), we can convert the given values of \(Q_1\) and \(Q_2\) into m\(^3\)/s before proceeding.
02

Applying Bernoulli's Equation

Bernoulli's equation, \(P_1 + \frac {1}{2}\rho v_1^2 = P_2 + \frac {1}{2}\rho v_2^2\), can be applied between points 1 and 2 and between points 1 and 3. Note that the velocity of fluid at a point can be obtained using the formula \(v = Q / A\), where A is the cross-sectional area of the pipe at that point, which can be found using the formula \(A = \pi d^2 / 4\). Here, \(d\) is the diameter of the pipe. This should give us the velocities at points 1, 2, and 3.
03

Applying Force Balance

Considering the forces acting vertically and horizontally due to pressure and flow rate, from both sides of the pipe, the balance is then applied to determine the vertical and horizontal components of the force required to hold the lateral fitting stationary. Forces due to pressure can be calculated by multiply the pressure by the area they act on vertically and horizontally. The term for the force due to momentum change is given by \(F = m \cdot \Delta v\), where \(\Delta v\) is the change in velocity of the fluid flowing in the x and y directions.

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Most popular questions from this chapter

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