Name: continuity equation for a fixed control volume
Uploaded: 2025-02-13T15:43:21.000Z
Duration: 1 min 16 s
Channel: JoVE

A equação de continuidade é derivada do princípio de que a massa dentro de um sistema permanece constante.
Para analisar o movimento do fluido, observamos a massa dentro de um volume de controle fixo, contabilizando a massa interna e cruzando a superfície de controle.
A mudança total na massa inclui a taxa na qual a massa se acumula dentro do volume de controle e a taxa na qual a massa flui através da superfície de controle.
No fluxo constante, todas as propriedades do campo, incluindo a densidade, permanecem constantes ao longo do tempo e a massa dentro do volume de controle não muda.
A velocidade e a densidade do fluido na superfície de controle determinam o fluxo de massa líquido, que deve equilibrar a massa que entra e sai do volume.
Ao integrar a massa sobre o volume de controle e a superfície de controle, a forma geral da equação de continuidade é derivada, que se aplica a fluxos constantes e instáveis.
Os engenheiros usam a relação entre velocidade e área da seção transversal em sistemas de tubulação para garantir que a vazão permaneça constante quando o diâmetro do tubo muda.
Este método permite um projeto de drenagem ideal, evitando inundações urbanas e gerenciando efetivamente as taxas de fluxo variáveis.

conservação de massa em um volume de controle finito

O princípio da conservação de massa é uma lei fundamental na mecânica dos fluidos e é aplicado utilizando a equação da continuidade. Aplicamos o conceito a um volume de controle finito para derivar a equação da continuidade.

Um sistema é definido como uma coleção de conteúdos inalteráveis, e a conservação de massa afirma que a massa de um sistema é constante.

<img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;">


Onde Msys é a massa do sistema. A massa do sistema pode ser escrita como:

<img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;">


Considere um volume de controle fixo e não deformável que coincide com o sistema em um instante específico.

<img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;">


Usando o teorema de transporte de Reynolds, expressamos a taxa de mudança temporal da massa do sistema:

<img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;">


Esta equação divide a mudança na massa em duas partes:

<ol style="display: flex; flex-direction: column; gap: 10px;">
 <li>Acúmulo de massa dentro do volume de controle.</li>
 <li>Fluxo de massa líquida através da superfície de controle ao redor do volume de controle.</li>
</ol>

Para fluxo em estado estacionário, onde as propriedades do fluido (como densidade) permanecem constantes ao longo do tempo, o termo da derivada temporal desaparece:

<img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;">

Isso afirma que o fluxo de massa líquido através da superfície de controle deve ser zero; o que entra no volume de controle deve sair. A equação de continuidade mantém ambos os termos para fluxos não estacionários ou fluxos onde propriedades como a densidade mudam ao longo do tempo. Esta forma é usada para situações mais complexas, como fluxo transitório em tubulações ou fluxo flutuante em rios devido a liberações de barragens.

Em aplicações do mundo real, como um projeto de sistemas de tubulação, os engenheiros utilizam a equação da continuidade para manter uma taxa de vazão consistente. Quando o diâmetro de um tubo muda, a velocidade do fluido ajusta-se de acordo para garantir que a mesma taxa de fluxo de massa seja preservada. Este princípio é crucial para o design de sistemas que gerenciam taxas de vazão variáveis, como sistemas de drenagem.

Conservação de massa em um volume de controle finito

A equação de continuidade é derivada do princípio de que a massa dentro de um sistema permanece constante.
Para analisar o movimento do fluido, observamos a massa ...

conservation-of-mass-in-finite-cotrol-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 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 accumulation inside the control volume and net mass flow through the control surface around it.

For steady-state flow, fluid properties such as density do not change with time. In that case, the time derivative term becomes zero. This means the net mass flow through the control surface must also be zero, so what flows into the control volume must flow out.

For unsteady flow, the continuity equation keeps both terms. It is also used when density changes with time. This form is useful for situations such as transient flow in pipelines or changing river flow after dam releases.

Engineers use the continuity equation in real pipe systems to keep flow rates consistent. When a pipe diameter changes, the fluid velocity changes so the same mass flow rate is preserved. That principle is important in designs like drainage systems that must handle variable flow rates.

Continuity Equation for a Fixed Control Volume

continuity-equation-for-a-fixed-control-volume

conservation of mass in finite cotrol 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-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...

conservation of mass in moving, nondeforming control volume

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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...

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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

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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...

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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 conservation of mass leads to the continuity equation for a fixed control volume, including steady and unsteady flow in pipes and rivers.

Continuity Equation for a Fixed Control Volume | 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 Equation for a Fixed Control Volume

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