2.2. Control Volume

So, now we are moving from a system to a control volume. So, we will see, how when we apply B system to mass, what is going to happen.

The shift from a system to a control volume signifies the transition from studying a fixed mass of fluid (system) to observing the flow of fluid through a defined space (control volume). This allows us to analyze the behavior of fluids entering and exiting a space over time, which is fundamental in fluid mechanics.

Think of a control volume like a fish tank. When you place a fish in the tank, the tank becomes the control volume. You can observe how much water the fish displaces, how it moves through the water, and how the fish interacts with the water, which is similar to how fluid flows in a defined space.

So, now conservation of mass, how can we apply, what we have learned in Reynolds transport theorem to the conservation of mass.

The conservation of mass states that mass cannot be created or destroyed in a closed system. In the context of control volumes, this means that the amount of mass entering the control volume must equal the amount of mass leaving, minus any accumulation within the volume. The Reynolds transport theorem provides the mathematical framework to express this principle quantitatively.

Imagine filling a bathtub. If you turn on the faucet, water pours in, but if you also drain the bathtub at the same rate, the water level remains constant. This is analogous to the conservation of mass where the amount of water going in equals the amount going out.

This gives us ρV·n̂ dA + ∫ρdV ∂/∂t = 0. This equation tells us that mass leaving minus mass entering is minus rate of increase of mass in the control volume.

In this equation, ρ is the fluid density, V is the velocity vector, n̂ is the unit normal vector (indicating direction), and dA is the differential area element on the surface of the control volume. This formula represents the continuity equation, illustrating that the rate of mass change within the control volume is equal to the net mass flow out of the volume.

Think of this like a water bottle with a straw. As you drink, water is leaving the bottle (mass exiting the control volume) while the amount inside decreases. If you don’t stop drinking, soon enough, the water level drops, which aligns with the mass conservation principle.

For constant density and uniform velocity, we can simply write ∫ρV·n̂ dA𝟏 + ∫ρV·n̂ dA𝟐 = 0.

This implies that the mass flow rates must equal at different cross-sections in steady-state flow. Given constant density and uniform velocity, the integrals simplify, allowing us to express continuity simply as the product of density, velocity, and area across two cross-sections. It means the mass flow rate at one cross-section equals that at another.

Imagine two parallel garden hoses. If one hose has a wide opening and the other a narrow one but both are used to fill the same bucket, the water speed increases in the narrower hose to maintain the same amount of water flowing through. This maintains the mass flow rate according to the continuity principle.

The flow out of a reservoir is 2 liters per second. The reservoir surface is 5 meters into 5 meters, the question is how fast is the reservoir surface dropping.

In this scenario, the conservation of mass is applied to predict how the water level in the reservoir changes as it drains. By calculating the outflow and the area of the reservoir, we can use the equation of continuity to find the rate at which the water level drops.

This is akin to a water jug with a constant hole. As water flows out, you can measure how fast the water level drops based on the size of the hole and the amount of water already present in the jug.

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