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

La ecuación de continuidad se deriva del principio de que la masa dentro de un sistema permanece constante.
Para analizar el movimiento del fluido, observamos la masa dentro de un volumen de control fijo, teniendo en cuenta la masa en el interior y cruzando la superficie de control.
El cambio total en la masa incluye la velocidad a la que la masa se acumula dentro del volumen de control y la velocidad a la que la masa fluye a través de la superficie de control.
En el flujo constante, todas las propiedades del campo, incluida la densidad, permanecen constantes a lo largo del tiempo y la masa dentro del volumen de control no cambia.
La velocidad y la densidad del fluido en la superficie de control determinan el flujo másico neto, que debe equilibrar la masa que entra y sale del volumen.
Al integrar la masa sobre el volumen de control y la superficie de control, se deriva la forma general de la ecuación de continuidad, que se aplica tanto a flujos estacionarios como no estacionarios.
Los ingenieros utilizan la relación entre la velocidad y el área de la sección transversal en los sistemas de tuberías para garantizar que el caudal permanezca constante cuando cambia el diámetro de la tubería.
Este método permite un diseño óptimo del drenaje, evitando inundaciones urbanas y gestionando eficazmente los caudales variables.

conservación de la masa en un volumen de control finito

El principio de conservación de la masa es una ley fundamental en mecánica de fluidos y se aplica mediante la ecuación de continuidad. Aplicamos el concepto a un volumen de control finito para derivar la ecuación de continuidad. Un sistema se define como una colección de contenidos inmutables, y la conservación de la masa establece que la masa de un sistema es constante. <img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;"> Donde Msys es la masa del sistema. La masa del sistema se puede escribir como: <img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;"> Consideremos un volumen de control fijo, no deformable, que coincide con el sistema en un instante particular. <img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;"> Usando el teorema de transporte de Reynolds, expresamos la tasa temporal de cambio de la masa del sistema: <img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;"> Esta ecuación divide el cambio de masa en dos partes: <ol> <li>Acumulación de masa dentro del volumen de control.</li> <li>Flujo neto de masa a través de la superficie de control que rodea el volumen de control.</li> </ol> Para el flujo en estado estacionario, donde las propiedades del fluido (como la densidad) permanecen constantes a lo largo del tiempo, el término de la derivada temporal se anula: <img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;"> Esto establece que el flujo neto de masa a través de la superficie de control debe ser cero; lo que fluye hacia el volumen de control debe fluir hacia afuera. La ecuación de continuidad conserva ambos términos para flujos inestables o flujos donde las propiedades como la densidad cambian a lo largo del tiempo. Esta forma se utiliza para situaciones más complejas, como flujo transitorio en tuberías o flujo fluctuante en ríos debido a liberaciones de presas. En aplicaciones del mundo real, como el diseño de sistemas de tuberías, los ingenieros utilizan la ecuación de continuidad para mantener un caudal constante. Cuando cambia el diámetro de una tubería, la velocidad del fluido se ajusta en consecuencia para garantizar que se mantenga el mismo caudal másico. Este principio es fundamental para diseñar sistemas que gestionen caudales variables, como los sistemas de drenaje.

Conservación de la masa en un volumen de control finito

La ecuación de continuidad se deriva del principio de que la masa dentro de un sistema permanece constante.
Para analizar el movimiento del fluido, observamos la masa ...

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

Education

Educación

JoVE Core

Civil Engineering

Ingeniería Civil

Unit 1

Unidad 1

Finite Control Volume Analysis

Análisis de Volumen de Control Finito

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

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 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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