Question

Consider fully developed laminar flow in a circular pipe. If the diameter of the pipe is reduced by half while the flow rate and the pipe length are held constant, the pressure drop will \((a)\) double, \((b)\) triple, \((c)\) quadruple, \((d)\) increase by a factor of 8 , or \((e)\) increase by a factor of \(16 .\)

Short answer

Short Answer: The pressure drop will increase by a factor of 16.

Step-by-step solution

Write down the Hagen-Poiseuille equation

The Hagen-Poiseuille equation is expressed as: \[\Delta P = \frac{8 \mu L Q}{\pi R^4}\] Where \(\Delta P\) is the pressure drop, \(\mu\) is the dynamic viscosity, \(L\) is the length of the pipe, \(Q\) is the flow rate, and \(R\) is the radius of the pipe.

Adjust equation for initial conditions

Initially, let the radius of the pipe be \(R\). The initial pressure drop can be expressed as: \[\Delta P_1 = \frac{8 \mu L Q}{\pi R^4}\]

Adjust equation for final conditions

When the diameter of the pipe is halved, the radius becomes \(R/2\). The final pressure drop can be expressed as: \[\Delta P_2 = \frac{8 \mu L Q}{\pi (R/2)^4}\]

Find the relation between initial and final pressure drop

Divide \(\Delta P_2\) by \(\Delta P_1\) to find the factor by which the pressure drop has changed: \[\frac{\Delta P_2}{\Delta P_1} = \frac{\frac{8 \mu L Q}{\pi (R/2)^4}}{\frac{8 \mu L Q}{\pi R^4}}\] Simplify the expression to find the factor: \[\frac{\Delta P_2}{\Delta P_1} = \frac{8 \mu L Q \cdot \pi R^4}{8 \mu L Q \cdot \pi (R/2)^4}\] Cancel out terms that appear in both the numerator and denominator: \[\frac{\Delta P_2}{\Delta P_1} = \frac{R^4}{(R/2)^4}\] Further simplify the expression: \[\frac{\Delta P_2}{\Delta P_1} = \frac{R^4}{(R^4/16)} = 16\]

Conclusion

So, the pressure drop will increase by a factor of 16 (option \((e)\)) when the diameter of the pipe is reduced by half while keeping the flow rate and the pipe length constant.

Key concepts

Laminar Flow

Laminar flow is a fundamental concept in fluid dynamics where fluid particles move in parallel layers or streams. In this type of flow, there is no cross-currents or mixing between the layers, meaning the flow is smooth and orderly.
Laminar flow is commonly observed when a fluid moves at low velocities and in small diameter pipes. One can easily visualize it by thinking of stacking layers of paper, where each layer moves steadily along the same path.
Larger pipes or increased flow speeds usually result in turbulent flow, where the motion is chaotic. Hence, understanding laminar flow helps in predicting and controlling fluid behavior in engineering applications.

• Characterized by parallel motion of fluid substances.
• Does not involve mixing between different layers of the fluid.
• Occurs in pipes, particularly when the flow is slow and pipe diameter is small.

Pressure Drop

Pressure drop refers to the decrease in pressure as a fluid flows through a pipe. When fluid moves through a piping system, it encounters resistance because of friction with the pipe walls and internal viscosity, causing a reduction in pressure.
This phenomenon is critical in engineering because it affects the efficiency and functionality of fluid systems. If the pressure drop is too high, a more powerful pump may be required to maintain flow rate, increasing energy consumption.
The Hagen-Poiseuille equation provides a way to calculate this drop, showing how factors like pipe radius, viscosity, and flow rate contribute. In the case of laminar flow, the pressure drop is directly influenced by these variables:

• Lower pipe radius increases pressure drop significantly.
• Increased dynamic viscosity leads to higher resistance and pressure loss.
• Longer pipes exhibit more pressure drop because of the more extended contact with the pipe surface.

Circular Pipe

A circular pipe is the most common geometry for carrying fluid in engineering systems. Its round shape provides a uniform path, minimizing the risk of trapping air or other gases as fluid moves through.
Circular pipes are often chosen because their construction makes them more structurally stable under pressure compared to other shapes. This shape also facilitates analysis through mathematical equations like the Hagen-Poiseuille equation, which predicts fluid behavior and pressure changes.
In fluid dynamics, the pipe radius is a crucial factor, as demonstrated in problems involving pressure drop, such as when the diameter is altered:

• Reducing the diameter leads to increased flow resistance and pressure drop.
• Consistency in diameter ensures predictable laminar flow patterns.
• Pipes with larger diameters tend to have lower laminar flow resistance.

Dynamic Viscosity

Dynamic viscosity is a measure of a fluid's internal resistance to flow. It describes how "thick" or "sticky" a fluid is and is a crucial factor in determining flow characteristics in a pipe.
In practical terms, high dynamic viscosity means the fluid flows less easily (like honey), while low viscosity means the fluid flows more freely (like water). This property directly affects the energy required to transport fluids and contributes significantly to the pressure drop described by the Hagen-Poiseuille equation.
Dynamic viscosity is influenced by temperature; fluids typically become less viscous at higher temperatures.

• Higher viscosity means greater energy is needed to move the fluid.
• Affects the rate of flow and resistance encountered in a pipe system.
• Plays a key role in determining the efficiency and performance of fluid systems.