Name: continuity and velocity in pipe flow
Uploaded: 2025-02-13T15:43:21.000Z
Duration: 1 min 7 s
Channel: JoVE

The principle of conservation of mass states that, within a fixed control volume, any change in mass over time must be balanced by the flow of mass across the control surfaces.
Consider a sewer pipe as an example of a fixed, non-deforming control volume, where the mass entering equals the mass exiting.
In this case, the control volume represents the section of the sewer pipe between its inlet and outlet, with control surfaces at each end.
Since the flow is steady and incompressible, there is no change in mass over time, meaning the fluid&#39;s mass flow rate is constant across the control surfaces.
As water flows through the pipe, the velocity may change depending on the pipe&#39;s cross-sectional area at the inlet and outlet.
When the cross-sectional area at the outlet decreases, the water&#39;s velocity increases to ensure the same mass flow rate
Engineers use this principle to analyze flow rates and velocities, ensuring proper system design without violating mass conservation.

conservation of mass in fixed, nondeforming control volume

The principle of conservation of mass is fundamental in fluid dynamics and is crucial for analyzing flow within fixed control volumes, such as pipes or ducts. This principle states that the total mass within a control volume remains constant unless altered by the inflow or outflow of mass through the control surfaces. This results in a vital relationship for steady, incompressible flow where the mass entering a system equals the mass leaving it.
In the case of a sewer pipe, which can be modeled as a fixed, nondeforming control volume, mass conservation dictates that the inlet's mass flow rate must equal the outlet's mass flow rate. The control surfaces in this scenario are at the inlet and outlet of the pipe, enclosing the flow region. Given that the flow is incompressible, the density of the fluid remains constant. This means that the relationship between velocity and cross-sectional area becomes critical.
Mass Flow Rate and Cross-Sectional Area
The mass flow rate (&#7745;) is expressed in terms of fluid density (&#929;) and flow rate (Q). Here, the flow rate is determined by multiplying the velocity (V) by the cross-sectional area (A).
<img alt="Mass flow rate formula; m&#775;=&#961;Q=&#961;AV; fluid dynamics; equation; educational reference." src="/files/ftp_upload/18049/18049_Equation_1.svg" style="text-align: center; height: 12px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
For steady flow, where the density remains constant, this equation simplifies to indicate that any change in velocity must correspond to a change in the pipe's cross-sectional area. When the pipe's cross-sectional area decreases, as might happen at the outlet of a sewer pipe, the velocity of the fluid must increase to conserve mass flow rate. The continuity equation describes this phenomenon:
<img alt="Continuity equation, Q=A1V1=A2V2, fluid dynamics, mathematical concept, diagram." src="/files/ftp_upload/18049/18049_Equation_2.svg" style="text-align: center; height: 12px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
Where V1 and A1 denotes the velocity and cross-sectional area at the inlet and V2 and A2 at the outlet. This relationship predicts flow behavior and design systems that maintain appropriate flow rates while adhering to the principle of mass conservation.

Conservation of Mass in Fixed, Nondeforming Control Volume

The principle of conservation of mass states that, within a fixed control volume, any change in mass over time must be balanced by the flow of mass across ...

conservation-of-mass-in-fixed-nondeforming-control-volume

Education

JoVE Core

Civil Engineering

Unit 1

Finite Control Volume Analysis

KEY TERMS AND CONCEPTS

continuity and velocity in pipe flow

Conservation of mass explains how fluid moves through a fixed control volume such as a pipe or duct. In a fixed, nondeforming control volume, the total mass stays constant unless mass enters or leaves through the control surfaces. For steady, incompressible flow, the mass entering the system equals the mass leaving it.

A sewer pipe is a useful example of this idea. It can be treated as a fixed control volume with inlet and outlet control surfaces around the flow region. Because the fluid is incompressible, its density stays constant as it moves through the pipe.

This makes cross-sectional area an important part of the analysis. Mass flow rate, written as m-dot, depends on fluid density and flow rate. The flow rate is found by multiplying velocity by cross-sectional area, so a change in area must be matched by a change in velocity when density stays the same.

The continuity equation describes this relationship. At the inlet and outlet, V1 and A1 at one end must balance V2 and A2 at the other end for steady flow. If the pipe becomes narrower, the fluid speed increases to keep the mass flow rate constant. This relationship helps predict flow behavior and supports the design of systems that maintain the right flow rate while following mass conservation.

Continuity and Velocity in Pipe Flow

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The continuity equation describes conservation of mass for a fixed control volume. In fluid mechanics, a system is a collection of the same contents over time, and its mass stays constant. That idea is the starting point for deriving the equation. To analyze flow, a fixed, non-deforming control volume is chosen so it matches the system at one instant. The Reynolds transport theorem then links the system to the control volume. It shows that the rate of change of system mass has two parts: mass...

Conservation of mass explains how fluid moves through a fixed control volume such as a pipe or duct. In a fixed, nondeforming control volume, the total mass stays constant unless mass enters or leaves through the control surfaces. For steady, incompressible flow, the mass entering the system equals the mass leaving it. A sewer pipe is a useful example of this idea. It can be treated as a fixed control volume with inlet and outlet control surfaces around the flow region. Because the fluid is...

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key terms and concepts

Learn how mass conservation links pipe area and fluid speed. See why steady, incompressible flow follows the continuity equation in a fixed control volume.

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Continuity and Velocity in Pipe Flow

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