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

החוק הראשון של התרמודינמיקה קובע כי קצב השינוי של סך האנרגיה המאוחסנת במערכת שווה לקצב נטו של חום ועבודה המועברים למערכת.
עבור נפח בקרה החופף למערכת ברגע נתון, החום נטו והעבודה הנכנסת למערכת שווים לחום נטו ולעבודה הנכנסת לנפח הבקרה.
עבור מערכת זו ונפח בקרה מקרית מוחל משפט התחבורה של ריינולדס.
הקצב הכולל של שינוי האנרגיה בנפח הבקרה נקבע הן על ידי קצב הזמן של השינוי באנרגיה המאוחסנת והן על ידי זרימת האנרגיה נטו דרך הגבול.
האנרגיה הזורמת מעבר לגבול כוללת תרומות מחום ועבודה, כגון עבודת פיר ועבודת לחץ.
עבודת פיר נובעת מרכיבים מסתובבים כמו טורבינות, בעוד שלחצים נוזליים מייצרים עבודת לחץ.
משוואת האנרגיה הכללית משלבת את כל האלמנטים: אנרגיה מאוחסנת, חום, עבודת פיר ועבודת לחץ עקב לחצים רגילים.
משוואה זו מייצגת שימור אנרגיה במערכות זורמים, ומשלבת העברת חום, עבודת פיר וכוחות לחץ על משטח הבקרה.

שימור אנרגיה בנפח שליטה

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 נבחן טורבינה הפועלת בתנאי זרימה יציבה. נפח השליטה מוגדר סביב הטורביה, כאשר הנוזל נכנס בנקודה אחת ויוצא בנקדוה אחרת. הטורבינה מפיקה אנרגיה מהנוזל, שמבצע עבודה מכנית (עבודת ציר). 

 עבור מערכות זרימה יציבה, הנגזרת בזמן של האנרגיה המאוחסנת שווה לאפס, מכיוון שאין הצטברות אנרגיה בתוך נפח השליטה. הדבר מפשט את משוואת האנרגיה ל: <img alt="Equation 1" src="/files/ftp_upload/18054/18054_Equation_1.svg" style="text-align: center; height: 27px;"> 

מכיוון שההתמקדות היא בעבודה המבוצעת על ידי ציר הטורבינה ונניח כי מעבר החום זניח ( Q=0), משוואת האנרגיה מתפשטת עוד יותר: <img alt="Equation 2" src="/files/ftp_upload/18054/18054_Equation_2.svg" style="text-align: center; height: 27px;"> 

הזרימה מחולקת לשני רכיבים: הנוזל הנכנס לטורבינה (כניסה) והנוזל היוצא מהטורבינה (יציאה). קצב זרימת המסה בנקודות אלו נשאר קבוע, ושטף האנרגיה בכניסה וביציאה מוגדר כ: <img alt="Equation 3" src="/files/ftp_upload/18054/18054_Equation_3.svg" style="text-align: center; height: 27px;"> 

כאשר: 

-ṁהוא קצב זרימת המסה,

- ũ1, V1, ו-z1 מתייחסים לאנרגיה הפנימית, מהירות, וגובה בכניסה,
 

- ũ2, V2, ו-z2 - מתייחסים לאנרגיה הפנימית, מהירות, וגובה ביציאה.
 

בטירבונה טיפוסית, האנרגיה הפנימית והקינטית של הנוזל פוחתות במהלך מעבר דרך הטורבינה, מה שמביא לביצוע עבודת ציר מועילה. נניח שטורבינה פועלת עם מהירות כניסה _v1=50 מ׳ לשנייה, מהירות יציאה _v2=20 מ׳ לשנייה, גובה הכניסה _z1=100 מ׳, וגובה יציאה `2=90 מ׳. נניח שהשינוי באנרגיה הפנימית זניח ביחס לשינויים באנרגיה הקינטית והפוטנציאלית, וקצב זרימת המסה הוא <img src="data:image/png;base64,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" style="height: 12px;"/> = 10 ק״ג לשנייה. <img alt="Equation 4" src="/files/ftp_upload/18054/18054_Equation_4.svg" style="text-align: center; height: 27px;"> 

כתוצאה מכך, הטורבינה מפיקה כ-11.48 קילוואט של אנרגיה מכנית על ידי הפקת אנרגיה מהנוזל הזורם.</div>

החוק הראשון של התרמודינמיקה קובע כי קצב השינוי של סך האנרגיה המאוחסנת במערכת שווה לשיעור נטו של חום ועבודה המועברת לתוך ...

Education

JoVE Core

Civil Engineering

הנדסה אזרחית

Unit 1

יחידה 1

Finite Control Volume Analysis

ניתוח נפח בקרה סופי

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

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

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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

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...

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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.

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...

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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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