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

연속성 방정식은 시스템 내의 질량이 일정하게 유지된다는 원리에서 파생됩니다.
유체의 움직임을 분석하기 위해 고정된 제어 체적 내에서 질량을 관찰하여 내부의 질량을 고려하고 제어 표면을 가로지릅니다.
질량의 총 변화에는 제어 체적 내부에 질량이 축적되는 속도와 제어 표면을 통해 질량이 흐르는 속도가 포함됩니다.
정상 흐름에서는 밀도를 포함한 모든 자기장 특성이 시간이 지남에 따라 일정하게 유지되며 제어 체적 내의 질량은 변하지 않습니다.
제어 표면의 유체 속도와 밀도는 순 질량 흐름을 결정하며, 이는 체적에 들어오고 나가는 질량의 균형을 맞춰야 합니다.
제어 체적과 제어 표면에 질량을 적분함으로써 연속성 방정식의 일반적인 형식이 도출되며, 이는 정상 흐름과 불안정 흐름 모두에 적용됩니다.
엔지니어는 파이프 시스템의 속도와 단면적 간의 관계를 사용하여 파이프의 직경이 변경될 때 유속이 일정하게 유지되도록 합니다.
이 방법을 사용하면 최적의 배수 설계가 가능하여 도시 홍수를 방지하고 가변 유속을 효과적으로 관리할 수 있습니다.

유한 제어 볼륨에서의 질량 보존

질량 보존의 원리는 유체 역학의 기본 법칙이며 연속성 방정식을 사용하여 적용됩니다. 우리는 유한 제어 볼륨에 이 개념을 적용하여 연속성 방정식을 도출합니다.

계란 변하지 않는 내용의 모임으로 정의되며, 질량 보존 법칙은 계의 질량이 일정하다는 것을 나타냅니다.

<img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;">

여기서 Msys는 시스템의 질량입니다. 시스템의 질량은 다음과 같이 쓸 수 있습니다.

<img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;">

특정 순간에 시스템과 일치하는 고정되고 변형되지 않는 제어 볼륨을 생각해 보세요.

<img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;">

레이놀즈 수송 정리를 사용하여 시스템 질량의 시간적 변화율을 다음과 같이 표현합니다.

<img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;">

이 방정식은 질량의 변화를 두 부분으로 나눕니다.

<ol style="display: flex; flex-direction: column; gap: 10px;">
 <li>제어 볼륨 내부의 질량 축적.</li>
 <li>제어 볼륨을 둘러싼 제어 표면을 통과하는 순 질량 흐름.</li>
</ol>

정상 상태 흐름의 경우, 유체 속성(예: 밀도)이 시간에 따라 일정하게 유지되는 경우 시간 미분 항은 사라집니다.

<img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;">

이는 제어 표면을 통과하는 순 질량 흐름이 0이어야 하며 제어 볼륨으로 유입되는 것은 유출되어야 함을 나타냅니다. 연속 방정식은 비정상 흐름 또는 밀도와 같은 속성이 시간에 따라 변하는 흐름에 대해 두 항을 모두 유지합니다. 이 형태는 파이프라인의 과도 흐름이나 댐 방류로 인한 강의 변동 흐름과 같은 더 복잡한 상황에 사용됩니다.

파이프 시스템 설계와 같은 실제 응용 분야에서 엔지니어는 연속 방정식을 사용하여 일관된 유량을 유지합니다. 파이프의 직경이 변하면 유체의 속도가 그에 따라 조정되어 동일한 질량 유량이 유지됩니다. 이 원리는 배수 시스템과 같이 가변 유량을 관리하는 시스템을 설계하는 데 중요합니다.

연속성 방정식은 시스템 내의 질량이 일정하게 유지된다는 원리에서 파생됩니다.
유체 움직임을 분석하기 위해 우리는 질량을 관찰합니다 ...

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

Education

교육

JoVE Core

Civil Engineering

토목공학

Unit 1

제1단원

Finite Control Volume Analysis

유한 제어 체적 해석

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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 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.

civil engineering

Continuity Equation for a Fixed Control Volume

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