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

O princípio da conservação da massa afirma que, dentro de um volume de controle fixo, qualquer mudança na massa ao longo do tempo deve ser equilibrada pelo fluxo de massa através das superfícies de controle.
Considere um cano de esgoto como um exemplo de um volume de controle fixo e não deformante, onde a massa que entra é igual à massa que sai.
Nesse caso, o volume de controle representa a seção da tubulação de esgoto entre sua entrada e saída, com superfícies de controle em cada extremidade.
Como o fluxo é constante e incompressível, não há mudança na massa ao longo do tempo, o que significa que a taxa de fluxo de massa do fluido é constante nas superfícies de controle.
À medida que a água flui através do tubo, a velocidade pode mudar dependendo da área da seção transversal do tubo na entrada e na saída.
Quando a área da seção transversal na saída diminui, a velocidade da água aumenta para garantir a mesma taxa de fluxo de massa
Os engenheiros usam esse princípio para analisar taxas de fluxo e velocidades, garantindo o projeto adequado do sistema sem violar a conservação de massa.

conservação de massa em volume de controle fixo e não deformável

O princípio da conservação de massa é fundamental na dinâmica dos fluidos e é crucial para a análise do fluxo dentro de volumes de controle fixos, como tubos ou dutos. Este princípio afirma que a massa total dentro de um volume de controle permanece constante, a menos que seja alterada pela entrada ou saída de massa através das superfícies de controle. Isso resulta em uma relação vital para o fluxo estacionário e incompressível, onde a massa que entra em um sistema é igual à massa que sai dele.

No caso de um tubo de esgoto, que pode ser modelado como um volume de controle fixo e não deformável, a conservação de massa dita que a taxa de vazão de massa na entrada deve ser igual à taxa de vazão de massa na saída. As superfícies de controle, nesse cenário, estão na entrada e na saída do tubo, envolvendo a região de fluxo. Dado que o fluxo é incompressível, a densidade do fluido permanece constante. Isso significa que a relação entre a velocidade e a área da seção transversal torna-se crucial.

Taxa de Vazão de Massa e Área da Seção Transversal

A taxa de vazão de massa (ṁ) é expressa em termos da densidade do fluido (Ρ) e da vazão (Q). Aqui, a vazão é determinada multiplicando a velocidade (V) pela área da seção transversal (A).

<img alt="Equation 1" src="/files/ftp_upload/18049/18049_Equation_1.svg" style="text-align: center; height: 12px;">

Para o fluxo estacionário, onde a densidade permanece constante, esta equação se simplifica para indicar que qualquer mudança na velocidade deve corresponder a uma mudança na área da seção transversal do tubo. Quando a área da seção transversal do tubo diminui, como pode acontecer na saída de um tubo de esgoto, a velocidade do fluido deve aumentar para conservar a taxa de vazão de massa. A equação de continuidade descreve esse fenômeno:

<img alt="Equation 2" src="/files/ftp_upload/18049/18049_Equation_2.svg" style="text-align: center; height: 12px;">

Onde V1 e A1 denotam a velocidade e a área da seção transversal na entrada e V2 ​e A2 ​ na saída. Esta relação prevê o comportamento do fluxo e projeta sistemas que mantêm taxas de vazão apropriadas, ao mesmo tempo em que aderem ao princípio da conservação de massa.

Conservação de massa em volume de controle fixo e não deformável

O princípio da conservação da massa afirma que, dentro de um volume de controle fixo, qualquer mudança na massa ao longo do tempo deve ser equilibrada pelo fluxo de massa através de ...

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

Education

JoVE Core

Civil Engineering

Engenharia Civil

Unit 1

Unidade 1

Finite Control Volume Analysis

Análise de Volume de Controle Finito

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

Conservation of Mass in Fixed, Nondeforming Control Volume

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

conservation of mass in finite cotrol volume

conservation-of-mass-in-finite-cotrol-volume

Continuity Equation for a Fixed Control Volume

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 in fixed, nondeforming control volume

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...

conservation of mass in moving, nondeforming control volume

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

Stormwater Detention Basin Mass Balance

Stormwater detention basins use mass balance to manage runoff during heavy rainfall. These basins are especially important in urban areas, where impervious surfaces can increase flooding risk. Engineers use the conservation of mass to track how water enters, is stored in, and leaves the basin. In a detention basin, the control volume is the water inside the basin. Its boundaries are the basin walls, the inflow pipes, and the outflow structures. During a rain event, water comes in through the...

linear momentum in control volume

Momentum Balance for Fluid Control Volumes

Linear momentum balance for a fluid control volume links Newton’s second law to moving fluid. The law says that the rate of change of linear momentum equals the sum of the external forces acting on the system. In a fluid system, this idea helps explain how motion and force are related. A control volume can match the fluid system at a specific moment. When that happens, the forces on the control volume and the system are the same. Reynolds transport theorem explains this connection by splitting...

application of the linear momentum equation

application-of-the-linear-momentum-equation

Forces in a 180° Pipe Bend

Forces in a 180-degree pipe bend can be found by applying the linear momentum equation to flowing water. The pipe has a constant cross-sectional area of 0.01 square meters, and the water moves at 15 meters per second. The pressure at the inlet is 0.2 megapascals, and the pressure at the outlet is 0.16 megapascals. The goal is to find the force components in the x and y directions that hold the bend in place. In this case, the flow is only in the y direction at both the entrance and the exit.

moment-of-momentum equation

Wind Turbine Torque and Angular Momentum

The moment-of-momentum equation helps explain how wind turbine blades create torque. Torque is the turning effect of a force around an axis. In rotating systems, this equation links the motion of the wind with the force acting on it. The equation begins with Newton's second law for a fluid particle. That law says the rate of change of linear momentum equals the external force on the particle. To find torque, the position of the particle relative to the rotation axis is combined with that force.

conservation of energy in control volume

Turbine Work from Energy Changes

A turbine can turn fluid energy into shaft work when flow passes through a control volume. The fluid enters the turbine at one point and leaves at another. Under steady-flow conditions, there is no energy buildup inside the control volume, so the stored energy does not change with time. This steady-flow setup makes the energy equation simpler. The turbine is treated as doing mechanical work, and heat transfer is taken as negligible, so Q̇ is approximately zero. The flow at the inlet and outlet...

application of the energy equation

Centrifugal Pump Flow Rate Calculation

Centrifugal pump flow rate can be calculated with the energy equation. In this problem, water moves between two reservoirs at different elevations, and the pump provides the energy needed for the transfer. The pump input is 7500 joules per second, the elevation difference is 10 meters, and the head loss from friction and other resistances is 5 meters. Both reservoirs are open to the atmosphere, so the pressure at the inlet and outlet is the same. That means the pressure terms in the energy...

design example: forces in sluice gate

Sluice Gate Water Pressure and Reaction Force

Sluice gates control water flow in channels, reservoirs, and irrigation systems. They work as vertical barriers with an opening that can be raised or lowered. Changing the opening height changes the water velocity and pressure below the gate. The forces on a sluice gate depend on the water depths upstream and downstream, written as z1 and z2. When the gate is closed, the upstream water depth is greater. That creates a strong pressure difference across the gate and a net force toward the...

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.

Continuity and Velocity in Pipe Flow | JoVE K12

JoVE Core Civil Engineering provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

civil engineering

Continuity and Velocity in Pipe Flow

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