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

连续性方程是从系统内质量保持不变的原理推导出来的。
为了分析流体运动，我们观察固定控制体积内的质量，考虑内部并穿过控制表面的质量。
质量的总变化包括质量在控制体积内累积的速率和质量流经控制表面的速率。
在稳态流中，所有场属性（包括密度）都随时间保持恒定，并且控制体积内的质量不会改变。
控制表面的流体速度和密度决定了净质量流量，净质量流量必须平衡进出体积的质量。
通过在控制体积和控制面上对质量进行积分，可以推导出连续性方程的一般形式，该方程适用于稳态和非稳态流动。
工程师在管道系统中使用速度和横截面积之间的关系来确保当管道直径发生变化时，流速保持不变。
这种方法允许优化排水设计，防止城市洪水并有效管理可变流量。

有限控制体积中的质量守恒定律

质量守恒定律是流体力学的基本定律，使用连续性方程进行应用。我们将这一概念应用于有限控制体积，以推导出连续性方程。 系统定义为一组不变的物质，质量守恒定律表明系统的质量是恒定的。 <img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;"> 其中 Msys 是系统的质量。系统的质量可以写成： <img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;"> 考虑一个固定的、不变形的控制体积，它在特定时刻与系统重合。 <img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;"> 使用雷诺输运定理，可以将系统质量随时间的变化率表示为： <img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;"> 该方程将质量变化分为两部分： <ol> <li>控制体积内的质量累积。</li> <li>通过控制体积的控制表面的净质量流。</li> </ol> 对于流体特性（如密度）随时间保持不变的稳态流动，时间导数项消失： <img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;"> 这表明通过控制表面的净质量流必须为零；流入控制体积的必定流出。对非稳定流或密度等属性随时间变化的流，连续性方程保留两个项。此形式用于更复杂的情况，例如管道中的瞬态流动或由于大坝泄水而导致的河流流量波动。 在实际应用中，例如管道系统设计，工程师使用连续性方程来保持流速一致。当管道直径发生变化时，流体的速度会相应调整，以确保保持相同的质量流速。这一原理对于设计管理可变流速的系统（例如排水系统）至关重要。

连续性方程是从系统内质量保持不变的原理推导出来的。
为了分析流体运动，我们观察质量 ...

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

Education

教育

JoVE Core

Civil Engineering

土木工程

Unit 1

第一单元

Finite Control Volume Analysis

有限控制体分析

KEY TERMS AND CONCEPTS

关键术语与概念

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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