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

L'equazione di continuità è derivata dal principio che la massa all'interno di un sistema rimane costante.
Per analizzare il movimento del fluido, osserviamo la massa all'interno di un volume di controllo fisso, tenendo conto della massa all'interno e attraversando la superficie di controllo.
La variazione totale della massa include la velocità con cui la massa si accumula all'interno del volume di controllo e la velocità con cui la massa scorre attraverso la superficie di controllo.
In flusso stazionario, tutte le proprietà del campo, compresa la densità, rimangono costanti nel tempo e la massa all'interno del volume di controllo non cambia.
La velocità e la densità del fluido sulla superficie di controllo determinano il flusso di massa netto, che deve bilanciare la massa in entrata e in uscita dal volume.
Integrando la massa sul volume di controllo e sulla superficie di controllo, si ricava la forma generale dell'equazione di continuità, che si applica sia ai flussi stazionari che a quelli instazionari.
Gli ingegneri utilizzano la relazione tra velocità e area della sezione trasversale nei sistemi di tubazioni per garantire che la portata rimanga costante al variare del diametro del tubo.
Questo metodo consente di progettare in modo ottimale il drenaggio, prevenendo gli allagamenti urbani e gestendo efficacemente le portate variabili.

conservazione della massa in un volume di controllo finito

Il principio di conservazione della massa, è una legge fondamentale della meccanica dei fluidi e viene applicato utilizzando l'equazione di continuità. Si applichi il concetto, a un volume di controllo finito per derivare l'equazione di continuità.

Un sistema, è definito come un insieme di contenuti immutabili e la conservazione della massa, afferma che la massa di un sistema è costante.

<img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;">

Dove Msys è la massa del sistema. La massa del sistema può essere scritta come:

<img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;">

Si consideri un volume di controllo fisso e non deformabile, che coincide con il sistema in un particolare istante.

<img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;">

Utilizzando il teorema del trasporto di Reynolds, si esprime la velocità di variazione temporale della massa del sistema:

<img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;">

Questa equazione scompone la variazione di massa in due parti:

<ol style="display: flex; flex-direction: column; gap: 10px;">
 <li>Accumulo di massa all'interno del volume di controllo.</li>
 <li>Flusso di massa netto attraverso la superficie di controllo che circonda il volume di controllo.</li>
</ol>

Per il flusso in stato stazionario, in cui le proprietà del fluido (come la densità), rimangono costanti nel tempo, il termine della derivata temporale svanisce:

<img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;">

Ne deriva che il flusso di massa netto attraverso la superficie di controllo deve essere zero; ciò che fluisce nel volume di controllo deve fluire fuori. L'equazione di continuità, mantiene entrambi i termini per flussi non stazionari o flussi in cui proprietà come la densità cambiano nel tempo. Questa forma è utilizzata per situazioni più complesse, come il flusso transitorio nelle condotte o il flusso fluttuante nei fiumi a causa di rilasci di dighe.

Nelle applicazioni del mondo reale, come la progettazione di sistemi di tubazioni, gli ingegneri utilizzano l'equazione di continuità per mantenere una portata costante. Quando il diametro di una tubazione cambia, la velocità del fluido si regola di conseguenza per garantire che venga preservata la stessa portata di massa. Questo principio è fondamentale per la progettazione di sistemi che gestiscono portate variabili, come i sistemi di drenaggio.

Conservazione della Massa in un Volume di Controllo Finito

L'equazione di continuità è derivata dal principio che la massa all'interno di un sistema rimane costante.
Per analizzare il movimento del fluido, osserviamo la massa ...

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

Education

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JoVE Core

Civil Engineering

Ingegneria Civile

Unit 1

Unità 1

Finite Control Volume Analysis

Analisi a volume di controllo finito

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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...

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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 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.

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Continuity Equation for a Fixed Control Volume

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