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

تستمد معادلة الاستمرارية من مبدأ أن الكتلة داخل النظام تظل ثابتة.
لتحليل حركة السوائل ، نلاحظ الكتلة داخل حجم تحكم ثابت ، مع مراعاة الكتلة الموجودة داخل سطح التحكم وعبور سطح التحكم.
يشمل التغير الكلي في الكتلة معدل تراكم الكتلة داخل حجم التحكم ومعدل تدفق الكتلة عبر سطح التحكم.
في التدفق المستمر ، تظل جميع خصائص المجال ، بما في ذلك الكثافة ، ثابتة بمرور الوقت ، ولا تتغير الكتلة داخل حجم التحكم.
تحدد سرعة السائل وكثافته على سطح التحكم صافي تدفق الكتلة ، والذي يجب أن يوازن بين الكتلة التي تدخل وتخرج من الحجم.
من خلال دمج الكتلة على حجم التحكم وسطح التحكم ، يتم اشتقاق الشكل العام لمعادلة الاستمرارية ، والذي ينطبق على كل من التدفقات الثابتة وغير المستقرة.
يستخدم المهندسون العلاقة بين السرعة ومساحة المقطع العرضي في أنظمة الأنابيب لضمان بقاء معدل التدفق ثابتا عندما يتغير قطر الأنبوب.
تسمح هذه الطريقة بتصميم الصرف الأمثل ، ومنع الفيضانات الحضرية وإدارة معدلات التدفق المتغيرة بشكل فعال.

بقاء الكتلة في حيز دراسة محدود.

<div style="direction:rtl">يمثل مبدأ بقاء الكتلة قانون أساسي في ميكانيكا الموائع ويتم تطبيقه باستخدام معادلة الاستمرارية. يتم تطبيق هذا المبدأ على حيز دراسة محدود لاستنتاج معادلة الاستمرارية. يتم تعريف النظام على أنه مجموعة من المحتويات الغير متغيرة، وينص مبدأ بقاء الكتلة على ثبات كتلة النظام. <img alt="Equation 1" src="/files/ftp_upload/18048/18048_Equation_1.svg" style="text-align: center; height: 30px;"> حيث Msys هي كتلة النظام. يمكن كتابة كتلة النظام على النحو التالي: <img alt="Equation 2" src="/files/ftp_upload/18048/18048_Equation_2.svg" style="text-align: center; height: 30px;"> على سبيل المثال حيز دراسة ثابت غير مشوه يتزامن مع النظام في لحظة معينة. <img alt="Figure 1" src="/files/ftp_upload/18048/18048_Figure_1.jpg" style="text-align: center; height: 300px;"> باستخدام نظرية الانتقال لرينولدز، يتم التعبير عن معدل التغير الزمني لكتلة النظام: <img alt="Equation 3" src="/files/ftp_upload/18048/18048_Equation_3.svg" style="text-align: center; height: 30px;"> تقوم هذه المعادلة بتقسيم التغير في الكتلة إلى جزأين: <ol style="padding-right: 35px; margin: 0;"> <li style="text-align: right;">تراكم الكتلة داخل حيز الدراسة.</li> <li style="text-align: right;">تدفق الكتلة الصافية عبر السطح المرجعي المحيط بحيز الدراسة.</li> </ol> في حالة التدفق المستقر، حيث تظل خصائص السوائل (مثل الكثافة) ثابتة بمرور الوقت، يتلاشى المشتق الزمني: <img alt="Equation 4" src="/files/ftp_upload/18048/18048_Equation_4.svg" style="text-align: center; height: 30px;"> تنص هذه المعادلة على أن التدفق الصافي للكتلة عبر السطح المرجعي لحيز الدراسة يجب أن يكون صفرًا؛ بمعنى ان ما يتدفق إلى حيز الدراسة يجب أن يتدفق لخارج الحيز. تعبر معادلة الاستمرارية عن كلا التدفقين للتدفقات الغير مستقرة حيث تتغير خصائص مثل الكثافة بمرور الوقت. يستخدم هذا النموذج للحالات الأكثر تعقيدًا، مثل التدفقات متغيرة الخصائص مع الزمن في خطوط الأنابيب أو التدفق المتقلب في الأنهار الناتج عن فتح السدود. في التطبيقات العملية، مثل تصميم أنظمة الأنابيب، يستخدم المهندسون معادلة الاستمرارية للحفاظ على معدل تدفق ثابت. عندما يتغير قطر الأنبوب، تتغير سرعة السائل وفقًا لذلك لضمان الحفاظ على نفس معدل تدفق الكتلة. وهذا المبدأ بالغ الأهمية لتصميم الأنظمة التي تدير معدلات التدفق المتغيرة، مثل أنظمة الصرف.</div>

تستمد معادلة الاستمرارية من مبدأ أن الكتلة داخل النظام تظل ثابتة.
لتحليل حركة السوائل ، نلاحظ الكتلة ...

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

Education

JoVE Core

Civil Engineering

الهندسة المدنية

Unit 1

الوحدة الأولى

Finite Control Volume Analysis

تحليل حجم التحكم المحدود

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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...

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

conservation-of-mass-in-moving-nondeforming-control-volume

Stormwater Detention Basin Mass Balance

Stormwater detention basins use mass balance to manage runoff during heavy rainfall. These basins are especially important in urban areas, where impervious surfaces can increase flooding risk. Engineers use the conservation of mass to track how water enters, is stored in, and leaves the basin. In a detention basin, the control volume is the water inside the basin. Its boundaries are the basin walls, the inflow pipes, and the outflow structures. During a rain event, water comes in through the...

linear momentum in control volume

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...

application of the linear momentum equation

application-of-the-linear-momentum-equation

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

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...

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 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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