Name: continuity and velocity in pipe flow
Uploaded: 2025-02-13T15:43:21.000Z
Duration: 1 min 7 s
Channel: JoVE

질량 보존 원리는 고정된 제어 부피 내에서 시간에 따른 질량 변화는 제어 표면을 가로지르는 질량 흐름에 의해 균형을 이루어야 한다고 말합니다.
하수관을 고정된 비변형 제어 체적의 예로 생각해 보십시오. 여기서 유입되는 질량은 나가는 질량과 같습니다.
이 경우 제어 볼륨은 입구와 출구 사이의 하수관 섹션을 나타내며 각 끝에 제어 표면이 있습니다.
흐름이 일정하고 비압축성이기 때문에 시간이 지남에 따라 질량에 변화가 없으며, 이는 유체의 질량 유량이 제어 표면 전체에서 일정하다는 것을 의미합니다.
물이 파이프를 통해 흐를 때 입구와 출구에서 파이프의 단면적에 따라 속도가 변할 수 있습니다.
출구의 단면적이 감소하면 물의 속도가 증가하여 동일한 질량 유량을 보장합니다
엔지니어는 이 원리를 사용하여 유속과 속도를 분석하여 질량 보존을 위반하지 않으면서 적절한 시스템 설계를 보장합니다.

고정된 비변형 제어 볼륨의 질량 보존

Conservation of Mass in Fixed, Nondeforming Control Volume

질량 보존의 원리는 유체 역학에서 기본이며 파이프나 덕트와 같은 고정된 제어 볼륨 내의 흐름을 분석하는 데 중요합니다. 이 원리는 제어 표면을 통한 질량의 유입 또는 유출에 의해 변경되지 않는 한 제어 볼륨 내의 총 질량이 일정하게 유지된다고 명시합니다. 이는 시스템에 유입되는 질량이 시스템에서 유출되는 질량과 같은 안정적이고 압축 불가능한 흐름에 대한 중요한 관계를 초래합니다.

고정된 비변형 제어 볼륨으로 모델링할 수 있는 하수관의 경우 질량 보존은 유입구의 질량 유량이 유출구의 질량 유량과 같아야 함을 지시합니다. 이 시나리오에서 제어 표면은 파이프의 유입구와 유출구에 있으며 흐름 영역을 둘러싸고 있습니다. 흐름이 ​​압축 불가능하므로 유체의 밀도는 일정하게 유지됩니다. 즉, 속도와 단면적 간의 관계가 중요해집니다.

질량 유량 및 단면적

질량 유량()은 유체 밀도()와 유량(Q)으로 표현됩니다. 여기서 유량은 속도(V)에 단면적(A)을 곱하여 결정됩니다.

<img alt="Equation 1" src="/files/ftp_upload/18049/18049_Equation_1.svg" style="text-align: center; height: 12px;">

밀도가 일정하게 유지되는 정상 흐름의 경우 이 방정식은 속도의 변화가 파이프의 단면적의 변화에 ​​해당해야 함을 나타내도록 간소화됩니다. 파이프의 단면적이 감소하면(하수관 출구에서 발생할 수 있음) 유체의 속도가 증가하여 질량 유량을 보존해야 합니다. 연속 방정식은 이 현상을 설명합니다.

<img alt="Equation 2" src="/files/ftp_upload/18049/18049_Equation_2.svg" style="text-align: center; height: 12px;">

여기서 와 는 입구에서의 속도와 단면적을 나타내고 와 는 출구에서의 속도와 단면적을 나타냅니다. 이 관계는 질량 보존의 원칙을 고수하면서 적절한 유량을 유지하는 유동 거동과 설계 시스템을 예측합니다.

질량 보존의 원리는 고정된 제어 부피 내에서 시간에 따른 질량의 모든 변화는 질량의 흐름에 의해 균형을 이루어야 한다고 말합니다.

conservation-of-mass-in-fixed-nondeforming-control-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

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 incompressible, its density stays constant as it moves through the pipe.

This makes cross-sectional area an important part of the analysis. Mass flow rate, written as m-dot, depends on fluid density and flow rate. The flow rate is found by multiplying velocity by cross-sectional area, so a change in area must be matched by a change in velocity when density stays the same.

The continuity equation describes this relationship. At the inlet and outlet, V1 and A1 at one end must balance V2 and A2 at the other end for steady flow. If the pipe becomes narrower, the fluid speed increases to keep the mass flow rate constant. This relationship helps predict flow behavior and supports the design of systems that maintain the right flow rate while following mass conservation.

Continuity and Velocity in Pipe Flow

conservation of mass in finite cotrol volume

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...

conservation of mass in fixed, nondeforming control volume

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 mass conservation links pipe area and fluid speed. See why steady, incompressible flow follows the continuity equation in a fixed control volume.

Continuity and Velocity in Pipe Flow | 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 and Velocity in Pipe Flow

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