Question

Consider fully developed flow in a circular pipe with negligible entrance effects. If the length of the pipe is doubled, the pressure drop will \((a\) ) double, \((b)\) more than double, \((c)\) less than double, \((d)\) reduce by half, or \((e)\) remain constant.

Short answer

Answer: (a) Double

Step-by-step solution

Recall the Hagen-Poiseuille equation

The Hagen-Poiseuille equation for fully developed laminar flow in a pipe is given by: \[\Delta P = \frac{8\mu LQ}{\pi R^4}\] where \(\Delta P\) is the pressure drop, \(\mu\) is the dynamic viscosity of the fluid, \(L\) is the length of the pipe, \(Q\) is the volumetric flow rate, and \(R\) is the radius of the pipe.

Double the length of the pipe

Let's denote the initial length of the pipe as \(L_1\) and the new length as \(L_2\). We are given that the length of the pipe is doubled, so we have: \[L_2 = 2L_1\]

Determine the pressure drop in the original pipe

Using the Hagen-Poiseuille equation, the pressure drop in the original pipe (with length \(L_1\)) is given by: \[\Delta P_1 = \frac{8\mu L_1Q}{\pi R^4}\]

Determine the pressure drop in the new pipe

Using the Hagen-Poiseuille equation, the pressure drop in the new pipe (with length \(L_2\)) is given by: \[\Delta P_2 = \frac{8\mu L_2Q}{\pi R^4}\]

Compare the pressure drops

To compare the pressure drops, we take the ratio of \(\Delta P_2\) and \(\Delta P_1\) \[\frac{\Delta P_2}{\Delta P_1} = \frac{8\mu L_2Q/\pi R^4}{8\mu L_1Q/\pi R^4} = \frac{L_2}{L_1}\] Since \(L_2 = 2L_1\), the ratio of the pressure drops is \[\frac{\Delta P_2}{\Delta P_1} = \frac{2L_1}{L_1} = 2\]

Answer the question

The pressure drop in the new pipe is double the pressure drop in the original pipe, so the correct answer is (a) double.

Key concepts

Fully Developed Laminar Flow

In the study of fluid mechanics, fully developed laminar flow refers to a flow regime where fluid moves in parallel layers with no disruption between them. This type of flow occurs in situations where the fluid is moving at a slow enough pace that kinetic disturbances are damped out, resulting in a smooth and orderly fluid motion. For this to happen in a pipe, the flow must be sufficiently far from the pipe entrance, allowing it to develop a parabolic velocity profile across the pipe's cross-section.

Understanding fully developed laminar flow is crucial in applications such as predicting the behavior of a fluid traveling through pipes, especially when it comes to calculating the pressure drop which is directly related to energy consumption in pumping systems, and optimizing flow systems for various industrial processes that require precise control of the fluid movement.

Pressure Drop in Pipes

The concept of pressure drop is a fundamental aspect of pipe flow analysis. It refers to the reduction in pressure as fluid moves through a pipe, caused by frictional forces and resistance encountered by the flowing medium. Calculating the pressure drop is essential for the design and efficient operation of piping systems. It influences the choice of pumping equipment and impacts the overall energy requirement to maintain flow.

To estimate the pressure drop, various factors are considered, including the fluid's viscosity, the length and diameter of the pipe, the flow rate, and whether the flow is laminar or turbulent. The Hagen-Poiseuille equation, which relates these factors, is specifically applied for laminar flow conditions in circular pipes. Through this equation, it's possible to predict how changes in the system, such as doubling the pipe's length, will affect the pressure drop.

Fluid Dynamics

The field of fluid dynamics encompasses the study of fluids in motion. It deals with the physical laws and equations that describe how liquids and gases (fluids) behave when they flow. This includes analyzing factors such as velocity, pressure, density, and temperature across different points in a flow field.

In engineering and physics, fluid dynamics is instrumental in designing systems that involve fluid transport, like pipelines, air conditioning systems, or blood circulation in the human body. Fluid dynamics also encompasses understanding different flow regimes, such as laminar and turbulent flow, and their impact on the fluid's behavior and energy distribution within the system. By mastering fluid dynamics concepts, engineers can create more efficient systems and predict how those systems will perform under various operating conditions.

Pipe Flow Analysis

The goal of pipe flow analysis is to understand and predict how fluids behave while traveling through a pipe or conduit. This involves assessing numerous factors, such as flow rate, pressure, temperature, and fluid properties. Engineers use this analysis to design pipes that can deliver fluids from one point to another with minimal energy loss and optimal efficiency.

During the analysis, they might utilize various principles from fluid dynamics, including the continuity equation, Bernoulli's equation, and the Hagen-Poiseuille equation for laminar flow scenarios. Additionally, the analysis can be further categorized based on the type of flow – whether it is single-phase (carrying only one fluid) or multiphase (carrying two or more fluids). With a firm grasp on pipe flow analysis, industries are better able to manage the transport of liquids and gases safely and efficiently, which is paramount in several applications ranging from municipal water supply to chemical processing plants.