Fluid Mechanics Lesson 05A: Conservation of Mass

Summary

In this lesson on the conservation of mass, the Reynolds Transport Theorem (RTT) is used to derive the mass conservation equation for a control volume. The video explores concepts such as mass flow rate, volume flow rate, and examples involving steady and unsteady flow. Through detailed explanations and examples, including a tank scenario with constant mass flow in, and a two-dimensional flow between plates, the lesson demonstrates how mass and volume flow rates are related in control volume analysis. The importance of understanding these principles for fluid dynamics is emphasized throughout the video.

Takeaways

• 😀 Conservation of mass is based on the principle that mass cannot enter or leave a system, ensuring mass remains constant within the system.
• 😀 The Reynolds Transport Theorem (RTT) is used to derive the conservation of mass equation for a control volume by transitioning from a system-based equation.
• 😀 In the Reynolds Transport Theorem, the rate of change of mass within a control volume is balanced by the net mass flow across the control surface.
• 😀 The mass flow rate across a control surface is defined as the integral of the product of density, velocity, and unit normal vector over the area of the surface.
• 😀 Mass flow rate (ṁ) can be either positive (outlet) or negative (inlet) depending on the direction of flow, with signs taken care of using the dot product.
• 😀 In steady flow, the time derivative term in the conservation of mass equation goes to zero, implying that the mass flow rate in equals the mass flow rate out.
• 😀 For steady, incompressible flow, the conservation of mass simplifies to the equation that the net volume flow rate in equals the net volume flow rate out.
• 😀 Volume flow rate (v̇) is similar to mass flow rate, except density is omitted, and it represents the amount of volume passing through a control surface per unit time.
• 😀 In incompressible flow, the density remains constant, so the conservation of mass equation simplifies further to relate the volume flow rates at inlets and outlets.
• 😀 In a steady flow example, the density of air in a tank increases linearly over time when mass is added at a constant rate, as the volume is fixed and the process is isothermal.

Q & A

What is the main concept of Lesson 5a?

The main concept of Lesson 5a is the conservation of mass, focusing on using the Reynolds Transport Theorem (RTT) to derive the conservation of mass equation for a control volume.

What is the Reynolds Transport Theorem (RTT) and how is it applied in this lesson?

The Reynolds Transport Theorem (RTT) is a mathematical framework used to relate the rate of change of a quantity within a control volume to the rate of flow of the quantity across the control surface. In this lesson, RTT is applied to the mass equation, converting it from a system equation to a control volume equation.

What does the conservation of mass equation for a system state?

The conservation of mass equation for a system states that the mass of a system does not change with time, i.e., the rate of change of mass within a system equals zero (dm/dt = 0).

What is the significance of the two terms in the conservation of mass equation for a control volume?

The two terms in the conservation of mass equation for a control volume are: (1) the rate of change of mass within the control volume, and (2) the net rate of mass flow out of the control volume through its surface. These terms ensure that mass is conserved within the control volume.

How does the sign convention work in the conservation of mass equation?

The sign convention dictates that if the rate of change of mass within the control volume is positive (i.e., mass is increasing), the net mass flow rate out of the control volume must be negative (i.e., mass is flowing in). Similarly, if the rate of change of mass is negative, the mass flow out is positive (i.e., mass is flowing out).

What is the definition of mass flow rate and how is it calculated?

Mass flow rate is the rate at which mass flows through a surface, and it is calculated as the integral of density (ρ) times velocity (v) over the area (A) of the control surface: m_dot = ∫(ρv·n) dA.

What is the difference between mass flow rate and volume flow rate?

Mass flow rate refers to the amount of mass passing through a surface per unit time, while volume flow rate refers to the volume of fluid passing through a surface per unit time. Mass flow rate is calculated using density, while volume flow rate is simply the velocity times the area.

What happens to the conservation of mass equation for steady flow?

For steady flow, the time derivative term in the conservation of mass equation becomes zero, simplifying the equation to the net rate of mass flow into the control volume being equal to the net rate of mass flow out of the control volume.

How does the conservation of mass apply to incompressible flow?

For incompressible flow, the density (ρ) is constant, and therefore the mass flow rate equation simplifies. In this case, the net volume flow rate into the control volume equals the net volume flow rate out, because the density cancels out in the mass flow rate equation.

Can you explain the example problem for unsteady flow in the lesson?

In the unsteady flow example, air is pumped into a rigid tank with a constant mass flow rate (m_dot in), and we are tasked with deriving the density in the tank as a function of time. The equation obtained shows that the density increases linearly with time as long as the mass flow rate remains constant.