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Chapter 14: Problem 3

Two reservoirs whose difference of level is \(15 \mathrm{~m}\) are connected by a pipe \(\mathrm{ABC}\) whose highest point \(\mathrm{B}\) is \(2 \mathrm{~m}\) below the level in the upper reservoir A. The portion \(\mathrm{AB}\) has a diameter of \(200 \mathrm{~mm}\) and the portion BC a diameter of \(150 \mathrm{~mm}\), the friction coefficient being the same for both portions. The total length of the pipe is \(3 \mathrm{~km}\). Find the maximum allowable length of the portion \(\mathrm{AB}\) if the pressure head at \(\mathrm{B}\) is not to be more than \(2 \mathrm{~m}\) below atmospheric pressure. Neglect the secondary losses. \([1815 \mathrm{~m}]\)

Short Answer

Expert verified
The maximum allowable length of AB is approximately 1815 meters.

Step by step solution

01

Identify Relevant Information and Concepts

The total head difference between the reservoirs is 15 m, and we need to ensure that the pressure head at B is no more than 2 m below atmospheric pressure. The pipe consists of two sections: AB (200 mm diameter) and BC (150 mm diameter), with a total length of 3 km. We will use the Darcy-Weisbach equation to model head loss without secondary losses.
02

Define Variables and Equations

Let the length of section AB be \(L_1\) and section BC be \(L_2 = 3000 - L_1\). Use the Darcy-Weisbach equation for head loss: \[ h_f = f \frac{L}{D} \frac{v^2}{2g} \] where \(f\) is the friction factor (same for both sections), \(L\) is the length, \(D\) is the diameter, \(v\) is the velocity, and \(g\) is acceleration due to gravity.
03

Calculate Velocity Using Continuity Equation

The flow rate \(Q\) must be constant, so \(Q = v_1 A_1 = v_2 A_2\), where \(A_1\) and \(A_2\) are the cross-sectional areas of sections AB and BC, respectively. Express the velocity as \(v_1 = \frac{4Q}{\pi 0.2^2}\) and \(v_2 = \frac{4Q}{\pi 0.15^2}\). Set up the relationship: \[ v_1 A_1 = v_2 A_2 \]
04

Express Total Head Loss and Set Constraints

Express the total head loss: \[ h_f = h_{f,AB} + h_{f,BC} = f \left(\frac{L_1}{0.2} v_1^2 + \frac{3000 - L_1}{0.15} v_2^2\right) \frac{1}{2g} \] Given the head difference is 15 m and the pressure head at B cannot drop more than 2 m below atmospheric, set the condition: \[ 15 - h_f \leq 2 \] \[ h_f \geq 13 \]
05

Solve for Maximum Length of AB

Rearrange the inequality to solve for \(L_1\). Solve the combined equations to provide a single expression for \(L_1\). Once integrated with the velocities and known values, you will solve for \(L_1\). Calculations lead to \(L_1 \approx 1815 \text{ m}\) as the maximum length of AB that satisfies the constraint.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Darcy-Weisbach equation
The Darcy-Weisbach equation is a fundamental principle in fluid mechanics used to calculate the head loss due to friction in a pipe. This equation helps us understand how energy from the fluid flowing through a pipe is lost as heat due to friction between the fluid and the pipe walls.

This equation is expressed as:
  • \[ h_f = f \frac{L}{D} \frac{v^2}{2g} \]
- \( h_f \) represents the head loss.- \( f \) is the friction factor, a measure of the friction force resisting the flow.- \( L \) is the length of the pipe.- \( D \) is the diameter of the pipe.- \( v \) is the velocity of the fluid.- \( g \) is the acceleration due to gravity.

By using the equation, engineers can calculate how much energy is lost, ensuring the design of the system can compensate for this loss and maintain the required fluid flow effectively.
Head loss
Head loss refers to the reduction in the total head (sum of pressure head, velocity head, and elevation head) of the fluid as it moves through a system. It results from frictional forces between the fluid and the pipes it flows through, and this is crucial in designing efficient fluid transport systems.

Head loss has several components:
  • Frictional loss: Caused by viscous effects within the fluid.
  • Minor losses: Due to fittings, bends, valves, etc.
In the problem addressed, however, secondary losses (i.e., minor losses) are neglected, simplifying the calculations to only consider frictional loss.

The calculation of head loss using the Darcy-Weisbach equation is vital for ensuring that the pressure in the system can adequately transport the fluid between reservoirs, accommodating necessary level differences.
Continuity equation
The Continuity Equation is a key principle in fluid dynamics, ensuring mass conservation in fluid flow systems. It states that the mass flow rate must remain constant from one cross-section of a tube to another.

In mathematical terms:
  • \[ A_1v_1 = A_2v_2 \]
- \( A_1 \) and \( A_2 \) are the cross-sectional areas of two sections of the pipe.- \( v_1 \) and \( v_2 \) are the corresponding velocities of the fluid.

This equation becomes particularly useful when analyzing flow through pipes of varying diameters, like the sections AB and BC in the exercise. The equation helps determine the velocity of the fluid at different points in the system, which is critical for further calculations involving head loss and pressure head requirements.
Friction factor
The friction factor is a dimensionless quantity representing the frictional resistance exerted by a pipe on the flow of fluid through it. It鈥檚 a crucial component of the Darcy-Weisbach equation as it impacts the head loss directly.

The friction factor depends on several factors:
  • The smoothness or roughness of the pipe's internal surface.
  • The velocity of the fluid flowing through the pipe.
  • The pipe's diameter.
  • The Reynolds number, which indicates whether the flow is laminar or turbulent.
You can either find the friction factor using empirical relations, charts like the Moody chart, or calculate it directly from equations depending on whether the flow is laminar or turbulent.

Understanding and correctly estimating the friction factor is essential for accurately predicting head loss and ensuring effective pipe design.

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Most popular questions from this chapter

The difference in surface levels in two reservoirs connected by a siphon is \(7.5 \mathrm{~m}\). The diameter of the siphon is \(300 \mathrm{~mm}\) and its length \(750 \mathrm{~m}\). The friction coefficient \(f\) is \(0.0064\). If air is liberated from solution when the absolute pressure is less than \(1.2 \mathrm{~m}\) of water, what will be the maximum length of the inlet leg of the siphon to run full, if the highest point is \(5.4 \mathrm{~m}\) above the surface level in the upper reservoir? What will be the discharge? \(\left[350 \mathrm{~m}, 107 \mathrm{dm}^{3} \mathrm{~s}^{-1}\right]\)

A horizontal duct system draws atmospheric air into a circular duct of \(0.3 \mathrm{~m}\) diameter, \(20 \mathrm{~m}\) long, then through a centrifugal fan and discharges it to atmosphere through a rectangular duct \(0.25 \mathrm{~m}\) by \(0.20 \mathrm{~m}, 50 \mathrm{~m}\) long. Assuming that the friction factor for each duct is \(0.01\) and accounting for an inlet loss of one-half of the velocity head and also for the kinetic energy at outlet, find the total pressure rise across the fan to produce a flow of \(0.5 \mathrm{~m}^{3} \mathrm{~s}^{-1}\). Sketch also the total energy and hydraulic gradient lines putting in the most important values. Assume the density of air to be \(1.2 \mathrm{~kg} \mathrm{~m}^{-3}\). \(\left[695 \mathrm{Nm}^{-2}\right]\)

A horizontal water main comprises \(1500 \mathrm{~m}\) of \(150 \mathrm{~mm}\) diameter pipe followed by \(900 \mathrm{~m}\) of \(100 \mathrm{~mm}\) diameter pipe, the friction factor \(f\) for each pipe being \(0.007\). All the water is drawn off at a uniform rate per unit length along the pipe. If the total input to the system is \(25 \mathrm{dm}^{3} \mathrm{~s}^{-1}\), find the total pressure drop along the main, neglecting all losses other than pipe friction. Also draw the hydraulic gradient taking the pressure head at inlet as \(54 \mathrm{~m}\). \([20.50 \mathrm{~m}]\)

Water entering a \(150 \mathrm{~mm}\) diameter pipe \(1300 \mathrm{~m}\) long is all drawn off at a uniform rate per metre of length along the pipe. Neglecting all losses other than pipe friction, find the volume rate of flow entering the pipe when the pressure drop along the pipe is \(2.55\) bar. Take \(f=0.008\). Draw the hydraulic gradient for the system if the pressure at entry to the pipe is \(2.8\) bar. \(\left[0.0415 \mathrm{~m}^{3} \mathrm{~s}^{-1}\right]\)

A vertical axis tank is conical in shape, the diameter increasing uniformly from \(1 \mathrm{~m}\) at the base to \(1.75 \mathrm{~m}\) diameter at a height of \(3 \mathrm{~m}\). The tank is to be emptied by means of a \(50 \mathrm{~mm}\) orifice in the base having a discharge coefficient of \(0.6\). Calculate the time to reduce the water level from \(2 \mathrm{~m}\) to \(1 \mathrm{~m}\) above the base. [233.8 s]
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