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

Die Kontinuitätsgleichung leitet sich aus dem Prinzip ab, dass die Masse innerhalb eines Systems konstant bleibt.
Um die Fluidbewegung zu analysieren, beobachten wir die Masse innerhalb eines festen Kontrollvolumens, wobei die Masse im Inneren und beim Durchqueren der Kontrollfläche berücksichtigt wird.
Die Gesamtmassenänderung umfasst die Rate, mit der sich die Masse innerhalb des Steuervolumens ansammelt, und die Rate, mit der die Masse durch die Steuerfläche fließt.
Bei stationärer Strömung bleiben alle Feldeigenschaften, einschließlich der Dichte, über die Zeit konstant, und die Masse innerhalb des Kontrollvolumens ändert sich nicht.
Die Geschwindigkeit und Dichte des Fluids an der Steuerfläche bestimmen den Nettomassenstrom, der die ein- und austretende Masse ausgleichen muss.
Durch die Integration der Masse über das Regelvolumen und die Steuerfläche wird die allgemeine Form der Kontinuitätsgleichung abgeleitet, die sowohl für stationäre als auch für instationäre Strömungen gilt.
Ingenieure nutzen die Beziehung zwischen Geschwindigkeit und Querschnittsfläche in Rohrsystemen, um sicherzustellen, dass die Durchflussrate konstant bleibt, wenn sich der Rohrdurchmesser ändert.
Diese Methode ermöglicht eine optimale Entwässerungsplanung, verhindert städtische Überschwemmungen und verwaltet effektiv variable Durchflussmengen.

erhaltung der masse in einem begrenzten kontrollvolumen

Das Prinzip der Massenerhaltung ist ein grundlegendes Gesetz der Strömungsmechanik und wird mithilfe der Kontinuitätsgleichung angewendet. Wir wenden das Konzept auf ein endliches Kontrollvolumen an, um die Kontinuitätsgleichung abzuleiten.

Ein System wird als Sammlung unveränderlicher Inhalte definiert und die Massenerhaltung besagt, dass die Masse eines Systems konstant ist.

<img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;">

Wobei Msys die Masse des Systems ist. Die Masse des Systems kann wie folgt geschrieben werden:

<img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;">

Betrachten Sie ein festes, sich nicht verformendes Kontrollvolumen, das zu einem bestimmten Zeitpunkt mit dem System übereinstimmt.

<img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;">

Mithilfe des Reynolds-Transporttheorems drücken wir die zeitliche Änderungsrate der Systemmasse aus:

<img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;">

Diese Gleichung zerlegt die Massenänderung in zwei Teile:

<ol style="display: flex; flex-direction: column; gap: 10px;">
 <li>Ansammlung von Masse innerhalb des Kontrollvolumens.</li>
 <li>Nettomassenstrom durch die das Kontrollvolumen umgebende Kontrollfläche.</li>
</ol>

Bei stationärer Strömung, bei der die Fluideigenschaften (wie die Dichte) über die Zeit konstant bleiben, verschwindet der Term der Zeitableitung:

<img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;">

Dies besagt, dass der Nettomassenstrom durch die Kontrollfläche Null sein muss; was in das Kontrollvolumen hineinfließt, muss auch wieder hinausfließen. Die Kontinuitätsgleichung behält beide Terme für instationäre Strömungen oder Strömungen bei, bei denen sich Eigenschaften wie die Dichte im Laufe der Zeit ändern. Diese Form wird für komplexere Situationen verwendet, wie z. B. vorübergehende Strömung in Rohrleitungen oder schwankende Strömung in Flüssen aufgrund von Staudammfreigaben.

In realen Anwendungen, wie etwa bei der Konstruktion von Rohrsystemen, verwenden Ingenieure die Kontinuitätsgleichung, um eine konstante Durchflussrate aufrechtzuerhalten. Wenn sich der Durchmesser eines Rohrs ändert, passt sich die Geschwindigkeit der Flüssigkeit entsprechend an, um sicherzustellen, dass die gleiche Massendurchflussrate erhalten bleibt. Dieses Prinzip ist entscheidend für die Konstruktion von Systemen, die variable Durchflussraten bewältigen müssen, wie etwa Entwässerungssysteme.

Erhaltung der Masse in einem begrenzten Kontrollvolumen

Die Kontinuitätsgleichung leitet sich aus dem Prinzip ab, dass die Masse innerhalb eines Systems konstant bleibt.
Um die Bewegung des Fluids zu analysieren, beobachten wir die Masse ...

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

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Civil Engineering

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Unit 1

Einheit 1

Finite Control Volume Analysis

Finite-Kontrollvolumen-Analyse

KEY TERMS AND CONCEPTS

WICHTIGE BEGRIFFE UND KONZEPTE

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...

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...

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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.

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Continuity Equation for a Fixed Control Volume

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