22.6.3. Flow in Rectangular Conduits

Now coming to the other case, many of the times also we do not go for only the circular conduits or the circular pipes from one point to other points okay. But when you go for a noncircular case you need to define as a equivalent flow okay. So that means we introduce a hydraulic diameters okay, it is called as hydraulic diameters, 4$D_h = \frac{4 \times Area}{Wetted \, Perimeter}$

When we talk about flow in noncircular conduits such as rectangular or triangular shapes, we have to introduce the concept of 'hydraulic diameter.' The hydraulic diameter is a way to relate the noncircular shape to a circular one by considering the area of flow and the wetted perimeter (the part of the perimeter that is in contact with the fluid). This means that for a rectangular section, for example, you would calculate the flow area produced by the width and height, then calculate the wetted perimeter by summing up the lengths of the sides that are in contact with the fluid. The hydraulic diameter is used to help calculate flow characteristics like velocity and shear stress.

Imagine a garden hose and a rectangular fish tank. If you want to compare the flow of water from both, you can't just compare their diameters. Instead, you consider the entire area of water flowing through the hose and the tank's base, which is more complex. Just like you would use the hydraulic diameter to understand water flow from a rectangular conduit, you would consider the total area and perimeter of water flow in the fish tank to evaluate its efficiency.

Now come back to the noncircular conduit like you may have a conditions where you have a only flow through this ones okay. That is called circular annulus diameter. 4$D_{h} =\frac{(A_{1} - A_{2})}{(P_{1} - P_{2})}$ where A is area and P is wet perimeter.

When you are dealing with shapes other than circular pipes, like annulus shapes or rectangles, the calculation of the hydraulic diameter becomes vital. For example, in an annulus (the space between two circles), you calculate the flow areas A1 and A2, and the perimeters P1 and P2. The hydraulic diameter gives you a measure of the effective diameter to use in equations for flow rate and velocity to help ensure accuracy in your calculations.

Think of a double-walled coffee mug where you fill water between the two walls. The space where the water flows resembles an annulus. To understand how fast the water can flow through that gap, you would need to calculate the area available between the walls and its perimeter, similar to calculating the hydraulic diameter for accurate flow predictions.

Now let us come it with that when you have a noncircular pipe like if you have a pipe like a triangular shape okay. So in this case what will happen the if you have a laminar flow you will have a wall stress will be maximum near the mid points of the sides.

The shape of the conduit significantly affects flow behavior. In a triangular conduit with laminar flow, you can expect the highest wall shear stress to be near the mid-point of each side. This means the fluid experiences more friction and drag against the wall at these points, which can affect overall flow characteristics and efficiency.

Consider a slide at a playground. When sliding down a slide that has a steep slope, you experience a lot of speed (analogous to flow), but you also feel pressure (analogous to shear stress) more at the edges where the slide touches you. Similarly, fluid flowing in a triangular shape will experience the greatest shear stress at certain points against the wall, impacting how smoothly the fluid flows.

Now if you look it that if you have the turbulent flow same case you have a turbulent flow the velocity distributions as well as the wall shear stress distributions exchanges it.

In turbulent flow within noncircular conduits, the behavior changes significantly compared to laminar flow. The velocity distribution is more uniform, and wall shear stress is less concentrated at specific points. This means fluids can flow through the conduit much faster and with more efficiency than laminar flow, but it may also produce more energy loss due to increased turbulence and mixing.

Imagine a busy highway during rush hour as opposed to a quiet side street. On the busy highway (analogous to turbulent flow), cars are moving quickly and closely together, resulting in mixed traffic patterns. However, on the side street (analogous to laminar flow), cars are spaced out and moving one after the other more smoothly. The busy highway allows for higher overall speeds, but there's also greater potential for delays (energy loss) due to traffic congestion.

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