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

Der erste Hauptsatz der Thermodynamik besagt, dass die Änderungsrate der gesamten gespeicherten Energie in einem System gleich der Nettorate von Wärme und Arbeit ist, die in ein System übertragen wird.
Bei einem Regelvolumen, das zu einem bestimmten Zeitpunkt mit dem System zusammenfällt, entsprechen die Nettowärme und die in das System eintretende Arbeit der Nettowärme und der Arbeit, die in das Regelvolumen eintreten.
Für dieses System und das koinzidente Kontrollvolumen wird das Reynolds-Transporttheorem angewendet.
Die Gesamtrate der Energieänderung innerhalb des Kontrollvolumens wird sowohl durch die zeitliche Änderungsrate der gespeicherten Energie als auch durch den Nettoenergiefluss durch die Grenze bestimmt.
Die Energie, die über die Grenze fließt, umfasst Beiträge von Wärme und Arbeit, wie z. B. Schachtarbeit und Druckarbeit.
Wellenarbeit entsteht durch rotierende Komponenten wie Turbinen, während Fluidspannungen Druckarbeit erzeugen.
Die allgemeine Energiegleichung kombiniert alle Elemente: gespeicherte Energie, Wärme, Wellenarbeit und Druckarbeit aufgrund normaler Spannungen.
Diese Gleichung stellt die Energieeinsparung in Fluidsystemen dar, wobei Wärmeübertragung, Wellenarbeit und Druckkräfte an der Steuerfläche berücksichtigt werden.

energieerhaltung im kontrollvolumen

Stellen Sie sich eine Turbine vor, die unter stationären Strömungsbedingungen betrieben wird. Das Kontrollvolumen ist um die Turbine herum angelegt, wobei die Flüssigkeit an einem Punkt ein- und an einem anderen austritt. Die Turbine entzieht der Flüssigkeit Energie, die mechanische Arbeit (Wellenarbeit) verrichtet.

Bei stationären Strömungssystemen wird die zeitliche Ableitung der gespeicherten Energie Null, da es innerhalb des Kontrollvolumens keine Energieansammlung gibt. Dies vereinfacht die Energiegleichung zu:

<img alt="Equation 1" src="/files/ftp_upload/18054/18054_Equation_1.svg" style="text-align: center; height: 27px;">

Da wir uns auf die von der Turbinenwelle geleistete Arbeit konzentrieren und eine vernachlässigbare Wärmeübertragung annehmen, Q̇ ≈ 0. Daher vereinfacht sich die Energiegleichung weiter zu:

<img alt="Equation 2" src="/files/ftp_upload/18054/18054_Equation_2.svg" style="text-align: center; height: 27px;">

Die Strömung wird in zwei Komponenten aufgeteilt: die Flüssigkeit, die in die Turbine eintritt (Einlass) und die aus der Turbine austritt (Auslass). Die Massendurchflussrate ist an beiden Punkten konstant und der Energiefluss am Einlass und Auslass ergibt sich aus:

<img alt="Equation 3" src="/files/ftp_upload/18054/18054_Equation_3.svg" style="text-align: center; height: 27px;">

Wobei:

ṁ- die Massendurchflussrate ist,

-ũ1 , V_1 und z_1 sich auf die innere Energie, Geschwindigkeit und Höhe am Einlass beziehen,

-ũ2 , V_2 und z_2 sich auf die innere Energie, Geschwindigkeit und Höhe am Auslass beziehen,

In einer typischen Turbine sind die innere und kinetische Energie des Fluids Die Energie nimmt beim Durchgang durch die Turbine ab, was dazu führt, dass nützliche Wellenarbeit geleistet wird. Angenommen, eine Turbine arbeitet mit einer Einlassgeschwindigkeit von V_1 = 50 m/s, einer Auslassgeschwindigkeit von V_2 = 20 m/s, einer Einlasshöhe von z_1 = 100 m und einer Auslasshöhe von z_2 = 90 m. Nehmen wir an, dass die interne Energieänderung im Vergleich zu den kinetischen und potenziellen Energieänderungen gering ist und dass der Massenstrom 10 kg/s beträgt.

<img alt="Equation 4" src="/files/ftp_upload/18054/18054_Equation_4.svg" style="text-align: center; height: 27px;">

Dadurch erzeugt die Turbine ungefähr 11,48 kW mechanische Leistung, indem sie der fließenden Flüssigkeit Energie entzieht.

Energieerhaltung im Kontrollvolumen

Der erste Hauptsatz der Thermodynamik besagt, dass die Änderungsrate der gesamten gespeicherten Energie in einem System gleich der Nettorate von Wärme und Arbeit ist, die in ein System übertragen wird.

Education

Lehre

JoVE Core

Civil Engineering

Bauingenieurwesen

Unit 1

Einheit 1

Finite Control Volume Analysis

Finite-Kontrollvolumen-Analyse

KEY TERMS AND CONCEPTS

WICHTIGE BEGRIFFE UND KONZEPTE

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

conservation of mass in finite cotrol volume

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

Continuity Equation for a Fixed Control Volume

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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

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Stormwater Detention Basin Mass Balance

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linear momentum in control volume

Momentum Balance for Fluid Control Volumes

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application of the linear momentum equation

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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.

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

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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