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

連続方程式は、システム内の質量が一定に保たれるという原理から導き出されます。
流体の動きを解析するために、固定された制御ボリューム内の質量を観察し、制御面内と制御面を横切る質量を考慮します。
質量の総変化には、制御体積内に質量が蓄積する速度と、制御表面を質量が流れる速度が含まれます。
定常流では、密度を含むすべてのフィールドプロパティが時間とともに一定に保たれ、制御ボリューム内の質量は変化しません。
制御面での流体の速度と密度によって正味の質量流量が決定され、ボリュームに出入りする質量のバランスをとる必要があります。
制御ボリュームと制御サーフェスの質量を積分することにより、連続方程式の一般的な形式が導出され、これは定常流れと非定常流れの両方に適用されます。
エンジニアは、パイプシステムの速度と断面積の関係を利用して、パイプの直径が変化したときに流量が一定に保たれるようにします。
この方法により、最適な排水設計が可能になり、都市の洪水を防ぎ、変動する流量を効果的に管理できます。

有限制御体積における質量保存

質量保存の原理は流体力学における基本法則であり、連続方程式を使用して適用されます。この概念を有限の制御体積に適用して、連続方程式を導出します。

 系は変化しない内容の集合として定義され、質量保存則は系の質量が一定であることを示します。

<img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;">

 ここで、MSYS は系の質量です。系の質量は次のように表すことができます。

<img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;">

 特定の瞬間に系と一致する、固定された変形しない制御体積を検討します。

<img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;">

 レイノルズの輸送定理を用いて、系質量の時間変化率を次のように表します。

<img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;">

 この式は質量の変化を 2 つの部分に分割します。

<ol style="display: flex; flex-direction: column; gap: 10px;">
 <li>制御体積内の質量の蓄積。</li>
 <li>制御容積を囲む制御面を通る正味質量流量。</li>
</ol>

 定常流では、流体の特性（密度など）は時間とともに一定であり、時間微分項は消えます。

<img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;">

 これは、制御面を通る正味の質量流量がゼロでなければならないこと、つまり制御体積に流入したものは流出しなければならないことを示しています。連続方程式は、非定常流、または密度などの特性が時間とともに変化する流れの両方の項を保持します。この形式は、パイプラインの過渡的な流れやダムの放流による河川の変動する流れなど、より複雑な状況で使用されます。

 配管系の設計などの実際のアプリケーションでは、エンジニアは連続方程式を使用して一定の流量を維持します。パイプの直径が変化すると、流体の速度がそれに応じて調整され、同じ質量流量が維持されます。この原理は、排水系などの可変流量を管理する系の設計に不可欠です。

連続方程式は、システム内の質量が一定に保たれるという原理から導き出されます。
流体の動きを分析するために、質量を観察します...

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

Education

JoVE Core

Civil Engineering

土木工学

Unit 1

ユニット1

Finite Control Volume Analysis

有限コントロールボリューム法による解析

KEY TERMS AND CONCEPTS

主要用語と概念

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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