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

L’équation de continuité est dérivée du principe selon lequel la masse à l’intérieur d’un système reste constante.
Pour analyser le mouvement des fluides, nous observons la masse à l’intérieur d’un volume de contrôle fixe, en tenant compte de la masse à l’intérieur et traversant la surface de contrôle.
La variation totale de masse comprend la vitesse à laquelle la masse s’accumule à l’intérieur du volume de contrôle et la vitesse à laquelle la masse s’écoule à travers la surface de contrôle.
En écoulement régulier, toutes les propriétés du champ, y compris la densité, restent constantes dans le temps et la masse dans le volume de contrôle ne change pas.
La vitesse et la densité du fluide sur la surface de contrôle déterminent le débit massique net, qui doit équilibrer la masse entrant et sortant du volume.
En intégrant la masse sur le volume de contrôle et la surface de contrôle, la forme générale de l’équation de continuité est dérivée, qui s’applique à la fois aux écoulements stationnaires et instationnaires.
Les ingénieurs utilisent la relation entre la vitesse et la section transversale dans les systèmes de tuyauterie pour s'assurer que le débit reste constant lorsque le diamètre du tuyau change.
Cette méthode permet une conception optimale du drainage, en prévenant les inondations urbaines et en gérant efficacement les débits variables.

conservation de la masse dans un volume de contrôle fini

Conservation of Mass in Finite Cotrol Volume

Le principe de conservation de la masse est une loi fondamentale de la mécanique des fluides et s'applique à l'aide de l'équation de continuité. Nous appliquons le concept à un volume de contrôle fini pour dériver l'équation de continuité.

Un système est défini comme un ensemble de contenus immuables, et la conservation de la masse stipule que la masse d'un système est constante.

<img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;">

Où Msys est la masse du système. La masse du système peut s'écrire comme suit:

<img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;">


Considérons un volume de contrôle fixe et non déformable qui coïncide avec le système à un instant donné.

<img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;">


En utilisant le théorème de transport de Reynolds, nous exprimons le taux de variation temporelle de la masse du système:

<img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;">


Cette équation divise le changement de masse en deux parties:

<ol style="display: flex; flex-direction: column; gap: 10px;">
 <li>Accumulation de masse à l’intérieur du volume de contrôle.</li>
 <li>Débit massique net à travers la surface de contrôle entourant le volume de contrôle.</li>
</ol>

Pour un écoulement à l'état stationnaire, où les propriétés du fluide (comme la densité) restent constantes dans le temps, le terme de dérivée temporelle s'annule:

<img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;">

Cette équation stipule que le débit massique net à travers la surface de contrôle doit être nul ; ce qui s'écoule dans le volume de contrôle doit en sortir. L'équation de continuité conserve les deux termes pour les écoulements instables ou les écoulements où les propriétés comme la densité changent au fil du temps. Cette forme est utilisée pour des situations plus complexes, telles que les écoulements transitoires dans les pipelines ou les écoulements fluctuants dans les rivières en raison des lâchers de barrages.

Dans les applications concrètes, comme la conception de systèmes de canalisations, les ingénieurs utilisent l'équation de continuité pour maintenir un débit constant. Lorsque le diamètre d'un tuyau change, la vitesse du fluide s'ajuste en conséquence pour garantir le maintien du même débit massique. Ce principe est essentiel pour la conception de systèmes qui gèrent des débits variables, comme les systèmes de drainage.

Conservation de la masse dans un volume de contrôle fini

L’équation de continuité est dérivée du principe selon lequel la masse à l’intérieur d’un système reste constante.
Pour analyser le mouvement des fluides, on observe la masse ...

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

Education

JoVE Core

Civil Engineering

Unit 1

Finite Control Volume Analysis

KEY TERMS AND CONCEPTS

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

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 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 offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

Continuity Equation for a Fixed Control Volume

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