1.5.1. Practice Problem on Continuity Equation

In one dimensional analysis, what we see is, in a steady flow mass rate of flow into the stream tube is equal to the mass rate of flow out of the tube. So, whatever goes comes in goes out of the tube for example, in a steady flow. So, in one dimensional the continuity equation says, this is the continuity equation. For incompressible fluid under a steady flow, steady flow we know that not a function of time. But what is incompressible fluid? Density remains constant, so, the density does not changes.

The continuity equation is fundamental in fluid mechanics, describing how mass is conserved in a fluid flow system. It states that, in a steady, one-dimensional flow, the mass flow rate into a section of a pipe must equal the mass flow rate out. This can be expressed mathematically as A1V1 = A2V2, where A is the cross-sectional area and V is the fluid velocity. For incompressible fluids, which have a constant density, this equation simplifies the analysis as density can be factored out. Essentially, this principle shows that if fluid enters a pipe at one rate, it must exit at another rate that maintains the balance of mass.

Think of a water slide at a theme park. When a person enters the slide at the top (inlet), the amount of water that enters must equal the amount that exits at the bottom (outlet) to keep the slide flowing smoothly. If the exit is narrower than the top, the water must flow faster at the exit to accommodate the same volume of water per second.

When there is a variation of velocity across the cross section of the conduit for an incompressible fluid discharge like this, you see. So, if these velocity, you see, this is the velocity component, this is V, and this is cross sectional area 1 this is cross sectional area 2 and if this is the profile, we can simply write, still the equation of continuity will hold but what we do instead because the velocity is varying across the cross section, so, instead of writing V A we do is we integrate the velocity with area for the entire surface.

In certain situations, the velocity of a fluid might vary across the cross-section of a pipe. To account for this, you cannot just use a simple V1A1 = V2A2 equation, because V is not constant across the entire area. Instead, the continuity equation can be expressed in an integrated form. This means summing or integrating the velocity over the entire cross-sectional area to find the total discharge. This method ensures that every part of the cross-section is considered, which is crucial when analyzing complex flow scenarios.

Imagine a garden hose that you squeeze near the end. When you do this, the water velocity increases at the narrowed section. To understand how much water is coming out accurately, you'd have to consider not just the speed of the water, but how the pressure is distributed across the wider area of the hose versus the narrower section.

Now, we will see a practice problem. Now, this figure we are going to show, shows a pipe network with junction nodes at A, this is junction node B, C, D, E and F. The numerals in the figure like this 100, 30, indicates the discharges at the node or in the pipes as the case is and the arrows indicate the direction of the flows... If we assume, this is we called Q. So, 100 is coming in and 70 is going out and Q also going out. If we consider this node...

To solve the practice problem, you need to apply the continuity equation at each junction in the pipe network. At each node (like A, B, C, etc.), you will sum the inflows and outflows. For node A, if water flows into the node at 100 units and out at 70 and another Q units, according to the continuity principle, the total inflow should equal the total outflow, which leads to the equation: 100 - 70 - Q = 0. From this, you can find the value of Q. This process is repeated at each node until all unknown flows are determined.

Consider a traffic intersection where cars are flowing in and out from various roads. If 100 cars enter from one road and 70 cars leave through another, there must be an unknown flow (like cars coming from another direction). By tracking in and out flows, you can determine how many cars are flowing in multiple directions, ensuring none gets 'lost' in the intersection.

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