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

Pierwsza zasada termodynamiki mówi, że szybkość zmian całkowitej energii zmagazynowanej w systemie jest równa szybkości netto ciepła i pracy przeniesionej do systemu.
W przypadku objętości regulacyjnej, która w danym momencie zbiega się z objętością regulacyjną, ciepło netto i praca wprowadzana do systemu są równe ciepłu netto i pracy wprowadzanej do objętości regulacyjnej.
Dla tego systemu i zbieżnej objętości kontrolnej stosuje się twierdzenie Reynoldsa o transporcie.
Całkowita szybkość zmiany energii w objętości kontrolnej jest określana zarówno przez szybkość zmian zmagazynowanej energii w czasie, jak i przez przepływ energii netto przez granicę.
Energia przepływająca przez granicę obejmuje wkład ciepła i pracy, takiej jak praca szybu i praca ciśnieniowa.
Praca wału wynika z obracających się elementów, takich jak turbiny, podczas gdy naprężenia płynów powodują pracę ciśnieniową.
Ogólne równanie energii łączy wszystkie elementy: zmagazynowaną energię, ciepło, pracę wału i pracę ciśnieniową spowodowaną naprężeniami normalnymi.
Równanie to reprezentuje zachowanie energii w układach płynów, obejmujące wymianę ciepła, pracę wału i siły nacisku na powierzchni sterującej.

zasada zachowania energii w objętości kontrolnej

Rozważmy turbinę pracującą w warunkach ustalonego przepływu. Objętość kontrolna otacza turbinę, z płynem wpływającym w jednym punkcie i wypływającym w innym. Turbina pobiera energię z płynu, który wykonuje pracę mechaniczną (pracę wału)

W przypadku układów o ustalonym przepływie pochodna czasowa energii zgromadzonej w objętości kontrolnej wynosi zero, ponieważ nie zachodzi akumulacja energii. Uproszcza to równanie energii do:

<img alt="Equation 1" src="/files/ftp_upload/18054/18054_Equation_1.svg" style="text-align: center; height: 27px;">

Ponieważ skupiamy się na pracy wykonanej przez wał turbiny i zakładamy pomijalny transfer ciepła, Q̇ ≈ 0. Dlatego równanie energii upraszcza się dalej do:

<img alt="Equation 2" src="/files/ftp_upload/18054/18054_Equation_2.svg" style="text-align: center; height: 27px;">

Przepływ jest podzielony na dwa składniki: płyn wchodzący do turbiny (wlot) i wychodzący z turbiny (wylot). Natężenie przepływu masy w obu punktach jest stałe, a strumień energii na wlocie i wylocie jest określony przez:

<img alt="Equation 3" src="/files/ftp_upload/18054/18054_Equation_3.svg" style="text-align: center; height: 27px;">

Gdzie:

-ṁ to przepływ masy,

-ũ1 , V_1 i z_1 odnoszą się do energii wewnętrznej, prędkości i wysokości na wlocie,

-ũ2 , V_2 i z_2 odnoszą się do energii wewnętrznej, prędkości i wysokości na wylocie,

W typowej turbinie energia wewnętrzna i kinetyczna cieczy maleje, gdy przechodzi przez turbinę, co skutkuje wykonaniem użytecznej pracy wału. Załóżmy, że turbina pracuje z prędkością wlotową V_1=50 m/s, prędkością wylotową V_2= 20 m/s, wysokością wlotu z_1=100 m i wysokością wylotu z_2=90m. Załóżmy, że zmiana energii wewnętrznej jest niewielka w porównaniu ze zmianami energii kinetycznej i potencjalnej, a natężenie przepływu masy wynosi 10 kg/s.

<img alt="Equation 4" src="/files/ftp_upload/18054/18054_Equation_4.svg" style="text-align: center; height: 27px;">

W rezultacie turbina wytwarza około 11,48 kW mocy mechanicznej, wydobywając energię z przepływającego płynu.

Zasada zachowania energii w objętości kontrolnej

Pierwsza zasada termodynamiki mówi, że szybkość zmiany całkowitej zmagazynowanej energii w układzie jest równa szybkości netto ciepła i pracy przeniesionej do ...

Education

JoVE Core

Civil Engineering

Inżynieria lądowa

Unit 1

Jednostka 1

Finite Control Volume Analysis

Analiza skończonej objętości kontrolnej

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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 and Velocity in Pipe Flow

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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.

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Turbine Work from Energy Changes

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