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

The continuity equation is derived from the principle that the mass within a system remains constant.
To analyze fluid movement, we observe the mass within a fixed control volume, accounting for the mass inside and crossing the control surface.
The total change in mass includes the rate at which mass accumulates inside the control volume and the rate at which mass flows through the control surface.
In steady flow, all field properties, including density, remain constant over time, and the mass within the control volume does not change.
The fluid velocity and density at the control surface determine the net mass flow, which must balance the mass entering and leaving the volume.
By integrating mass over the control volume and control surface, the general form of the continuity equation is derived, which applies to both steady and unsteady flows.
Engineers use the relationship between velocity and cross-sectional area in pipe systems to ensure the flow rate remains constant when the pipe&#39;s diameter changes.
This method allows for optimal drainage design, preventing urban flooding and effectively managing variable flow rates.

massabehoud in een eindig controlevolume

Conservation of Mass in Finite Cotrol Volume

Het principe van behoud van massa is een fundamentele wet in de vloeistofmechanica en wordt toegepast met behulp van de continuïteitsvergelijking. We passen het concept toe op een eindig controlevolume om de continuïteitsvergelijking af te leiden. Een systeem wordt gedefinieerd als een verzameling van onveranderlijke inhoud. De wet van behoud van massa stelt dat de massa van een systeem constant is. <img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;"> Waarbij Msys de massa van het systeem is. De massa van het systeem kan worden geschreven als: <img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;"> Beschouw een vast, niet-vervormend controlevolume dat samenvalt met het systeem op een bepaald moment. <img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;"> Met behulp van de Reynolds-transportstelling drukken we de tijdsafhankelijke verandering van de systeemmassa uit: <img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;"> Deze vergelijking verdeelt de massaverandering in twee delen: <ol> <li>Accumulatie van massa binnen het controlevolume.</li> <li>Netto massastroom door het controleoppervlak rondom het controlevolume.</li> </ol> Bij een stationaire stroming, waarbij de vloeistofeigenschappen (zoals dichtheid) in de loop van de tijd constant blijven, verdwijnt de tijdsafgeleide term: <img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;"> Dit stelt dat de netto massastroom door het controleoppervlak nul moet zijn; wat in het controlevolume stroomt, moet er ook uit stromen. De continuïteitsvergelijking behoudt beide termen voor onstabiele stromen of stromen waarbij eigenschappen zoals dichtheid in de loop van de tijd veranderen. Deze vorm wordt gebruikt voor complexere situaties, zoals tijdelijke stroming in pijpleidingen of fluctuerende stroming in rivieren als gevolg van damafvoeren

In toepassingen in de echte werreld, zoals het ontwerp van pijpsystemen, gebruiken ingenieurs de continuïteitsvergelijking om een ​​consistente stroomsnelheid te behouden. Wanneer de diameter van een pijp verandert, past de snelheid van de vloeistof zich dienovereenkomstig aan om ervoor te zorgen dat dezelfde massastroomsnelheid behouden blijft. Dit principe is cruciaal voor het ontwerpen van systemen die variabele stroomsnelheden beheren, zoals drainagesystemen.

Massabehoud in een eindig controlevolume

The continuity equation is derived from the principle that the mass within a system remains constant.
To analyze fluid movement, we observe the mass ...

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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