15.5.1. Conservation of Mass

Welcome all of you to this fluid mechanics course. Today, we are going to plan about conversion of mass. As you could remember it, then, in the last class we discussed about Reynolds transport theorem. So, the same Reynolds transport theorem will be used to derive mass conservation equation which is an important equation for any fluid flow problems.

The conservation of mass is a fundamental principle in fluid mechanics that states that mass cannot be created or destroyed in a closed system. This principle is essential for analyzing fluid flow problems. In this course, the Reynolds transport theorem is a key tool used to derive the mass conservation equation. By applying this theorem, we can relate mass transfer between a control volume and its surroundings, ensuring that the mass within the system remains constant unless acted upon by external forces or changes.

Imagine a water tank that is being filled with water from a hose while simultaneously leaking water from a small hole at the bottom. The principle of conservation of mass tells us that the amount of water in the tank will change based on the rate of water flowing in from the hose compared to the rate of leaking out. If more water flows in than leaks out, the water level in the tank will rise, and vice versa.

...you can have three type of control volumes, fixed control volume, that means with reference to the space it remains at the same location. As the time goes, it does not move it. That is what fixed control volume is. It has definite control surfaces. The other one is moving control volume, like the ship movement in a sea or the river. The ship and the adjacent fluid can be considered as moving control volume. And we solved many of the problems with moving control volume, considering this control volume is moving with a particular velocity.

Control volumes are essential in fluid mechanics to analyze how mass, momentum, and energy move in a fluid flow. There are three main types of control volumes: fixed, moving, and deformable. A fixed control volume remains stationary in space, providing a consistent boundary for analysis over time. A moving control volume shifts with the fluid, like a ship moving through water, allowing us to study the interaction between the fluid and the object. Deformable control volumes can change shape, adapting to the flow characteristics, which is crucial in analyzing complex fluid behaviors.

Think of a fixed control volume as a large clear tank sitting on a table that captures water poured into it from above but does not move. A moving control volume is like a boat traveling on a river while the water flows past it. As the boat moves, it experiences the water’s movement relative to its own motion, which impacts how the boat floats and interacts with the water.

In case of moving control volume, the control volume moves with a particular velocity. That is what is moving control volume. And fixed control volume, as you can see it remains stationary at that space regions. Today, we will talk about how we can derive the conservation of mass.

When dealing with moving control volumes, it's essential to consider the velocity of the fluid as well as the control volume's velocity. This motion affects how mass flows into and out of the control surface. In deriving the conservation of mass, we account for the mass entering and leaving the control volume over time, ensuring a balance between inflow and outflow.

Consider a fire engine pumping water to extinguish a fire while moving. The rate of water being pumped (inflow) must equal the rate at which the water leaves the hose (outflow) for it to maintain a consistent spray pattern. If the fire truck accelerates or decelerates, the dynamics of how the water is delivered changes, requiring calculations that account for both the water flow and the movement of the truck.

Now, let us go to very interesting applications. As you know it Indian Space Research Organisation launched the satellite which is called MOM programmes, that means Mars Orbiter Mission programme.

The concept of conservation of mass is not only theoretical; it has practical applications in various fields, including aerospace engineering. The Mars Orbiter Mission (MOM) exemplifies how understanding fluid mechanics and mass conservation can successfully design trajectories for spacecraft. By accurately calculating the forces and fluid interactions that the satellite would face, scientists can predict its path from Earth to Mars.

Imagine launching a paper airplane. To get it to fly far, you need to consider how much air it can push down as it goes up - this is similar to how spacecraft like MOM must manage airflow and fuel to navigate through space. Efficiently managing the mass of the spacecraft, whether it’s fuel or the satellite structure, determines how successfully it can reach its intended orbit.

If I put B = m and b = 1, and no doubt that or for any systems, that is 0, unless otherwise you have nuclear energy or additional energies coming to the system. What in general you do not consider in this case.

In mathematical modeling of fluid movement, the conservation of mass is often expressed using integrals to analyze how mass enters or leaves a control volume. When applying the Reynolds transport theorem, we can simplify problems by assuming certain conditions, such as steady flow, which allows us to focus mainly on net outflow and storage instead of complex calculations. This makes it easier to analyze fluid systems without the overwhelming need for complex calculations every time.

Think of measuring how much water flows into and out of a garden hose. If you know the water flow rate into the hose (like measuring pressure) and the hose’s length and diameter, you can quickly determine how much water is available at your garden faucet without having to constantly measure every drop. This is similar to how we apply integrals to simplify mass conservation calculations in fluid mechanics.

If you consider one dimensional flow. I know the average velocity of this things. If it is that, then very easy problem now. I can, instead of doing this surface integrals, okay, if I take this is my control volume, so I can just get the mass flux per unit time.

In a steady flow situation, understanding one-dimensional flow allows for simpler calculations of mass flux, where the flow is constant in one direction. By focusing on the average velocity and the cross-sectional area, we can efficiently calculate the mass entering or leaving the control volume without needing complex integrals. This approach helps streamline many practical engineering problems.

Consider a long driveway with a single lane. If you know that cars can only enter and exit at a certain rate (the number of cars per minute), and the width of the driveway (the area), then you can easily compute how many cars are going to pass through during rush hour without measuring each car. This is akin to simplifying mass flow calculations for through a one-dimensional flow in a pipe.

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