3. Flow Through Pipes

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 Understanding Flow Through Pipes: Unit 3 of CE 4001

In the study of fluid mechanics, the flow of fluids through pipes is one of the most crucial aspects, especially in hydraulic systems. In this blog, we will break down Unit 3 of the Hydraulics course (CE 4001) for 4th-semester mechanical engineering students at Rajasthan Polytechnic. We will cover major and minor head losses in pipes, types of flow through pipes, and essential devices for measuring discharge in pipe systems.

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3. Flow Through Pipes

Fluid flow through pipes can be quite complex, depending on factors like the type of flow, friction, pipe characteristics, and fittings. Understanding the factors that affect flow through pipes helps engineers design efficient piping systems.

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3.1 Major Head Loss in Pipe

Head loss refers to the loss of energy (usually in the form of pressure) as fluid moves through a pipe. This loss can be attributed to friction and other resistances encountered by the fluid.

3.1.1 Frictional Loss and its Computation by Darcy’s Weisbach Equation

Frictional loss occurs due to the resistance created by the pipe walls as the fluid moves through the pipe. The frictional loss is significant in most pipe flow situations and is calculated using Darcy's Weisbach Equation:

$$h_f = f \left(\frac{L}{D}\right) \left(\frac{v^2}{2g}\right)$$

Where:

• $h_{\mathrm{f\; is\; the\; frictional\; head\; loss\; (m),}}$
• $f$ is the Darcy friction factor (dimensionless),
• $L$ is the length of the pipe (m),
• $D$ is the diameter of the pipe (m),
• $v$ is the velocity of the fluid (m/s),
• $g$ is the acceleration due to gravity (9.81 m/s²).

Example: To calculate the frictional loss in a pipe, if the pipe length is 100 meters, the diameter is 0.3 meters, and the fluid velocity is 2 m/s, you would first need to determine the value of $f$ (based on the Reynolds number and the roughness of the pipe) and then plug the values into the Darcy equation.

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3.2 Minor Losses in Pipe

While major losses primarily occur due to friction, minor losses are caused by fittings, valves, bends, and other disturbances in the flow path. These losses are typically less significant but still important in design calculations.

3.2.1 Loss at Entrance and Exit

• Entrance Loss: When a fluid enters a pipe, it decelerates and loses energy due to sudden expansion. This is called entrance loss and is given by the formula:

  $$h_{\text{entrance}} = K_{\text{entrance}} \left(\frac{v^2}{2g}\right)$$
  

  Where $K_{\text{entrance}}$ is a coefficient that depends on the geometry of the pipe entrance.

• Exit Loss: Similar to the entrance, the fluid experiences a loss of energy when exiting the pipe, especially if there is a sudden change in cross-section. The formula is similar to that of entrance loss:

  $$h_{\text{exit}} = K_{\text{exit}} \left(\frac{v^2}{2g}\right)$$
  

3.2.2 Sudden Contraction and Sudden Enlargement

• Sudden Contraction: When a pipe suddenly narrows, the velocity of the fluid increases, leading to a loss in pressure. The loss due to sudden contraction can be calculated using a loss coefficient $K_{\text{contraction}}$.

• Sudden Enlargement: When the pipe suddenly widens, the velocity of the fluid decreases, which also results in energy loss. The loss due to sudden enlargement is also quantified by a loss coefficient $K_{\text{enlargement}}$.

3.2.3 Fittings

Other minor losses in pipes come from fittings such as elbows, tees, and valves. Each fitting introduces a specific loss, which can be calculated using empirical formulas or reference tables that provide values for the loss coefficient $K$ for various types of fittings.

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3.3 Flow Through Pipes

In a piping system, multiple pipes may be connected to each other. The flow characteristics differ depending on whether the pipes are arranged in series or parallel.

3.3.1 Pipes in Series

When pipes are connected in series, the fluid flows through them one after the other. The total head loss in a series arrangement is the sum of the head losses in each pipe:

$$h_{\text{total}} = h_1 + h_2 + h_3 + \ldots$$

Where:

• $h_1, h_2, h_3, \dots$ are the head losses in each pipe.

3.3.2 Pipes in Parallel

In parallel arrangements, fluid can flow through multiple parallel pipes. The total head loss is the same in each pipe, but the discharge is divided among the pipes. The total discharge can be calculated by summing the discharges in each pipe.

3.3.3 Dupuit’s Equation for Equivalent Pipe

When pipes of different lengths and diameters are arranged in parallel, it is often useful to calculate an equivalent pipe that represents the combined system. Dupuit's equation helps to find this equivalent pipe based on the total head loss and discharge.

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3.4 Hydraulic Gradient Line and Total Energy Line

The Hydraulic Gradient Line (HGL) represents the height to which water would rise in piezometers along the pipe. It gives an idea of the pressure at different points along the pipe.

The Total Energy Line (TEL) includes the elevation head, velocity head, and pressure head. It is a graphical representation of the total energy at different points along the flow path.

• HGL is always below the TEL, because it only represents the potential and pressure energy.
• The difference between the TEL and HGL at any point represents the velocity head.

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3.5 Water Hammer in Pipes: Causes and Remedial Measures

Water hammer is a pressure surge or wave caused when the flow of fluid in a pipe is suddenly stopped or altered (such as when a valve is closed too quickly). The surge in pressure can cause pipes to burst or fittings to fail.

Causes:

• Sudden closing of a valve.
• Rapid changes in fluid velocity.

Remedial Measures:

• Use of surge tanks to absorb pressure fluctuations.
• Installation of slow-closing valves to gradually stop the fluid.
• Air chambers to cushion the pressure changes.

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3.6 Discharge Measuring Device for Pipe Flow: Venturimeter

A Venturimeter is a device used to measure the flow rate of a fluid through a pipe. It works on the principle of pressure difference: as the fluid flows through a constriction in the pipe (narrow section), its velocity increases, and the pressure decreases.

Construction:

• The Venturimeter consists of a converging section, throat (narrow part), and a diverging section. Pressure gauges are installed at the inlet and throat to measure pressure differences.

Working:

• The flow rate is determined using the Bernoulli equation and the pressure difference between the inlet and throat.

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3.7 Discharge Measurement Using Orifice, Hydraulic Coefficients of Orifice

An orifice is a hole or opening in the side of a container or pipe through which fluid flows. The flow rate through the orifice is influenced by the pressure difference across it and the size of the opening.

Discharge Equation:

$$Q = C_d A \sqrt{2gh}$$

Where:

• $Q$ is the discharge,
• $C_d$ is the discharge coefficient (depends on the shape of the orifice),
• $A$ is the area of the orifice,
• $g$ is the acceleration due to gravity,
• $h$ is the head or pressure difference driving the flow.

Hydraulic Coefficients of Orifice:

• Discharge coefficient $C_d$: This is a dimensionless number that accounts for losses due to the shape and roughness of the orifice.

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Sample Questions for Practice

1. Calculate the frictional head loss in a 200-meter-long pipe with a diameter of 0.5 m and a fluid velocity of 2 m/s. Assume the Darcy friction factor is 0.02.
2. What are minor losses in a pipe system? Calculate the exit loss for a pipe with a velocity of 3 m/s and an exit loss coefficient of 0.5.
3. If two pipes of lengths 100 meters and 150 meters are connected in series, calculate the total head loss in the system. Assume the head losses in each pipe are 5 m and 8 m, respectively.
4. Explain the working of a Venturimeter and how it is used to measure flow.
5. What is water hammer, and what measures can be taken to prevent it in a pipeline system?

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Conclusion

This unit introduces you to the critical aspects of flow through pipes, including head losses, flow arrangements, and key measurement techniques. Understanding these concepts will help you design and analyze piping systems effectively, whether in residential plumbing or industrial fluid transport systems.

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