Name: turbine work from energy changes
Uploaded: 2025-02-13T15:43:22.000Z
Duration: 1 min 14 s
Channel: JoVE

K12 Standards

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

High School

Chemistry

Energy

C-PS3-4

Conservation of Energy in Control Volume

열역학 제1법칙에 따르면 시스템에 저장된 총 에너지의 변화율은 시스템으로 전달되는 열과 작업의 순 속도와 같습니다.
주어진 순간에 시스템과 일치하는 제어 볼륨의 경우, 시스템에 유입되는 순 열 및 작업은 순 열 및 제어 볼륨에 들어가는 작업과 같습니다.
이 시스템과 동시 제어 볼륨에 대해 Reynolds Transport Theorem이 적용됩니다.
제어 체적 내의 총 에너지 변화율은 저장된 에너지의 시간 변화율과 경계를 통한 순 에너지 흐름에 의해 결정됩니다.
경계를 가로질러 흐르는 에너지에는 샤프트 작업 및 압력 작업과 같은 열과 작업의 기여가 포함됩니다.
샤프트 작업은 터빈과 같은 구성 요소를 회전시켜 발생하는 반면 유체 응력은 압력 작업을 생성합니다.
일반 에너지 방정식은 저장된 에너지, 열, 샤프트 작업 및 수직 응력으로 인한 압력 작업과 같은 모든 요소를 결합합니다.
이 방정식은 유체 시스템의 에너지 보존을 나타내며, 열 전달, 샤프트 작업 및 제어 표면의 압력력을 통합합니다.

제어 볼륨에서의 에너지 보존

정상 흐름 조건에서 작동하는 터빈을 생각해 보세요. 제어 볼륨은 터빈 주위로 그려지고, 유체는 한 지점에서 유입되고 다른 지점에서 유출됩니다. 터빈은 유체에서 에너지를 추출하여 기계적 작업(샤프트 작업)을 수행합니다.

정상 흐름 시스템의 경우 제어 볼륨 내에 에너지가 축적되지 않으므로 저장된 에너지의 시간 미분은 0이 됩니다. 이렇게 하면 에너지 방정식이 다음과 같이 간소화됩니다.

<img alt="Equation 1" src="/files/ftp_upload/18054/18054_Equation_1.svg" style="text-align: center; height: 27px;">

터빈 샤프트에서 수행되는 작업에 초점을 맞추고 무시할 수 있는 열 전달을 가정하기 때문에 에너지 방정식은 다음과 같이 더욱 간소화됩니다 Q̇ ≈ 0 

<img alt="Equation 2" src="/files/ftp_upload/18054/18054_Equation_2.svg" style="text-align: center; height: 27px;">

흐름은 두 가지 구성 요소로 나뉩니다. 터빈에 유입되는 유체(입구)와 터빈에서 유출되는 유체(출구)입니다. 두 지점의 질량 유량은 일정하고 입구와 출구의 에너지 플럭스는 다음과 같습니다.

<img alt="Equation 3" src="/files/ftp_upload/18054/18054_Equation_3.svg" style="text-align: center; height: 27px;">

여기서:

-ṁ 은 질량 유량입니다.

-ũ1 , V_1, z_1은 입구의 내부 에너지, 속도, 높이를 나타냅니다.

-ũ2 , V_2, z_2는 출구의 내부 에너지, 속도, 높이를 나타냅니다.

일반적인 터빈에서 유체의 내부 에너지와 운동 에너지는 다음과 같이 감소합니다. 터빈을 통과하여 유용한 샤프트 작업이 수행됩니다. 터빈이 V_1=50m/s의 입구 속도, V_2=20m/s의 출구 속도, 입구 높이 z_1=100m, 출구 높이 z_2=90m로 작동한다고 가정합니다. 내부 에너지 변화가 운동 에너지와 위치 에너지 변화에 비해 작고 질량 유량 이 10kg/s라고 가정합니다.

<img alt="Equation 4" src="/files/ftp_upload/18054/18054_Equation_4.svg" style="text-align: center; height: 27px;">

결과적으로 터빈은 흐르는 유체에서 에너지를 추출하여 약 11.48kW의 기계적 전력을 생산합니다.

열역학 제1법칙에 따르면 시스템에 저장된 총 에너지의 변화율은 열과 작업으로 전달되는 순 속도와 같습니다.

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

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 is then described by the mass flow rate and by the fluid’s internal energy, velocity, and height at each location.

For the inlet and outlet, the mass flow rate is constant. The energy flux depends on ṁ, u1, V1, and z1 at the inlet, and on ṁ, u2, V2, and z2 at the outlet. Here, u is internal energy, V is velocity, and z is height.

In a typical turbine, the fluid loses internal energy and kinetic energy as it moves through the device. That energy loss becomes useful shaft work. For example, if V1 = 50 m/s, V2 = 20 m/s, z1 = 100 m, z2 = 90 m, and the mass flow rate is 10 kg/s, with a small internal energy change, the turbine produces about 11.48 kW of mechanical power.

Turbine Work from Energy Changes

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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

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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

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Stormwater Detention Basin Mass Balance

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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...

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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.

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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

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 a steady-flow turbine uses changes in velocity, height, and internal energy to produce shaft work, with a worked 11.48 kW example.

Turbine Work from Energy Changes | JoVE K12

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Turbine Work from Energy Changes

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