21.4.1. Mass Conservation Equation

To do these integrations, we can consider $y = 1 \text{ at } r = 0$ $y = 0 \text{ at } r = R$ So we will change this upper limit and lower limit of the equations when we are converting from dr to the y integrations.

This chunk introduces how to set limits for integration when transforming radial coordinates (r) into a different variable (y). Here, the variable y is defined such that when r is at its minimum (0), y is 1, and when r is at its maximum (R), y is 0. This transformation is necessary for correctly calculating integrations across a circular cross-section.

Imagine trying to calculate the area of a circular pizza slice. Rather than measuring the slice directly, you can break it into smaller vertical segments. By redefining your measuring convention based on where you start taking your measurements (like changing from the outer edge to the middle), you can simplify the process.

In laminar flow case whatever this velocity distributions, the beta factor is called, comes to what one by third. That means, if you are computing the momentum flux using these average velocities, the actual momentum flux going through that surface if following these velocity distributions, will be the one third of that.

This chunk discusses the momentum flux correction factor in laminar flow, which is represented as beta (β). It states that for laminar flow, the average flow is typically one-third of the actual momentum flux through a surface calculated using the velocity distributions. This means that when using average values in calculations, one might be overestimating momentum flux without considering actual distributions.

Think of it like measuring the speed of cars in a narrow, crowded street. If you just use the average speed of a few cars without considering how slow they go when traffic is heavy, you might think the street can handle more traffic than it really can—this is an oversimplification much like using average velocity for calculating momentum.

If you look at that, if you have beta = 1/3. What it indicates that the momentum flux using velocity distributions divided by the momentum flux using average velocity.

This chunk emphasizes the importance of the momentum flux correction factor, specifically noting that when beta equals one-third, the adjustment indicates that the momentum calculated using average velocity greatly overestimates the actual motion through velocity distributions. This distinction is crucial especially in conditions with non-uniform flow profiles.

Consider trying to fill a bathtub with water from a faucet. If you only check how quickly the water is flowing once it's out of the tap (the average speed), you might think it fills up faster than it actually does because it splashes around and some water flows over the side. The lumped measure fails to capture reality, similar to using average velocity in fluid dynamics.

Now, if you look it, I will apply mass conservation equation, because it is single inlet and outlet conditions, the mass influx is equal to mass outflux.

This section discusses the application of the mass conservation equation in fluid systems. It states that in a system with a single inlet and outlet, the mass flowing into the control volume must equal the mass flowing out. This fundamental principle ensures the conservation of mass within the specified conditions and forms the basis for analyzing fluid flow.

Imagine a water park slide—a fixed amount of water flows down the slide at any moment. If more water were added to the top (inlet), you’d see more water pouring out the bottom (outlet) than just what was there initially. This helps depict how mass conservation works in fluid dynamics.

--