Name: continuity equation for a fixed control volume
Uploaded: 2025-02-13T15:43:21.000Z
Duration: 1 min 16 s
Channel: JoVE

Süreklilik denklemi, bir sistem içindeki kütlenin sabit kalması ilkesinden türetilir.
Akışkan hareketini analiz etmek için, içindeki kütleyi hesaba katarak ve kontrol yüzeyini geçerek sabit bir kontrol hacmi içinde gözlemleriz.
Kütledeki toplam değişim, kütlenin kontrol hacmi içinde birikme hızını ve kütlenin kontrol yüzeyinden akma hızını içerir.
Sabit akışta, yoğunluk da dahil olmak üzere tüm alan özellikleri zamanla sabit kalır ve kontrol hacmi içindeki kütle değişmez.
Kontrol yüzeyindeki akışkan hızı ve yoğunluğu, hacme giren ve çıkan kütleyi dengelemesi gereken net kütle akışını belirler.
Kütleyi kontrol hacmi ve kontrol yüzeyi üzerine entegre ederek, hem sabit hem de kararsız akışlar için geçerli olan süreklilik denkleminin genel formu türetilir.
Mühendisler, borunun çapı değiştiğinde akış hızının sabit kalmasını sağlamak için boru sistemlerinde hız ve kesit alanı arasındaki ilişkiyi kullanır.
Bu yöntem, optimum drenaj tasarımına, kentsel taşkınların önlenmesine ve değişken akış hızlarının etkin bir şekilde yönetilmesine olanak tanır.

sonlu bir kontrol hacminde kütlenin korunumu

Kütlenin korunumu ilkesi akışkanlar mekaniğinde temel bir yasadır ve süreklilik denklemi kullanılarak uygulanır. Süreklilik denklemini türetmek için kavramı sonlu bir kontrol hacmine uygularız.

 Bir sistem, değişmeyen içeriklerin bir koleksiyonu olarak tanımlanır ve kütlenin korunumu, bir sistemin kütlesinin sabit olduğunu belirtir.

<img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;">

 Burada Msys sistemin kütlesidir. Sistemin kütlesi şu şekilde yazılabilir:

<img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;">

 Belirli bir anda sistemle çakışan sabit, deforme olmayan bir kontrol hacmi düşünün.

<img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;">

 Reynolds taşıma teoremini kullanarak, sistem kütlesinin zaman içindeki değişim oranını ifade ediyoruz:

<img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;">

 Bu denklem kütledeki değişimi iki parçaya ayırır:

<ol style="display: flex; flex-direction: column; gap: 10px;">
 <li>Kontrol hacmi içindeki kütle birikimi.</li>
 <li>Kontrol hacmini çevreleyen kontrol yüzeyinden geçen net kütle akışı.</li>
</ol>

 Akışkan özelliklerinin (yoğunluk gibi) zaman içinde sabit kaldığı sabit durum akışı için, zaman türevi terimi sıfıra iner:

<img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;">

 Bu, kontrol yüzeyinden geçen net kütle akışının sıfır olması gerektiğini belirtir; kontrol hacmine akan şey dışarı akmalıdır. Süreklilik denklemi, sabit olmayan akışlar veya yoğunluk gibi özelliklerin zaman içinde değiştiği akışlar için her iki terimi de korur. Bu form, boru hatlarındaki geçici akış veya barajların boşalması nedeniyle nehirlerdeki dalgalanan akış gibi daha karmaşık durumlar için kullanılır.

 Boru sistemlerinin tasarımı gibi gerçek dünya uygulamalarında, mühendisler tutarlı bir debiyi korumak için süreklilik denklemini kullanırlar. Bir borunun çapı değiştiğinde, akışkanın hızı aynı kütle akış hızının korunmasını sağlamak için buna göre ayarlanır. Bu ilke, drenaj sistemleri gibi değişken debileri yöneten sistemleri tasarlamak için çok önemlidir.

Sonlu Bir Kontrol Hacminde Kütlenin Korunumu

Süreklilik denklemi, bir sistem içindeki kütlenin sabit kalması ilkesinden türetilir.
Sıvı hareketini analiz etmek için kütleyi gözlemliyoruz ...

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

Education

JoVE Core

Civil Engineering

İnşaat Mühendisliği

Unit 1

Ünite 1

Finite Control Volume Analysis

Sonlu Kontrol Hacmi Analizi

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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 accumulation inside the control volume and net mass flow through the control surface around it.

For steady-state flow, fluid properties such as density do not change with time. In that case, the time derivative term becomes zero. This means the net mass flow through the control surface must also be zero, so what flows into the control volume must flow out.

For unsteady flow, the continuity equation keeps both terms. It is also used when density changes with time. This form is useful for situations such as transient flow in pipelines or changing river flow after dam releases.

Engineers use the continuity equation in real pipe systems to keep flow rates consistent. When a pipe diameter changes, the fluid velocity changes so the same mass flow rate is preserved. That principle is important in designs like drainage systems that must handle variable flow rates.

Continuity Equation for a Fixed Control Volume

continuity-equation-for-a-fixed-control-volume

conservation of mass in finite cotrol 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...

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key terms and concepts

Learn how conservation of mass leads to the continuity equation for a fixed control volume, including steady and unsteady flow in pipes and rivers.

Continuity Equation for a Fixed Control Volume | JoVE K12

JoVE Core Civil Engineering provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

civil engineering

Continuity Equation for a Fixed Control Volume

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