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Hydraulic diameter is a useful concept in fluid mechanics, especially when dealing with complex duct shapes like rectangles. Unlike circular pipes, rectangular ducts don't have a true diameter. That's where hydraulic diameter comes in. It's a way to equate the size of any non-circular duct to a circular one.

The formula for calculating the hydraulic diameter,\[D = \frac{4A}{P}\]is used to transform the irregular shape into an equivalent circle. Here,

• \(A\) represents the cross-sectional area of the duct,
• \(P\) stands for the wetted perimeter, the boundary of the duct that comes in contact with the fluid.

For instance, in rectangular ducts, while the area \(A\) remains constant, the wetted perimeter \(P\) changes with aspect ratio. This means that the hydraulic diameter gives us a consistent way to measure how frictional effects change with different shapes.

Aspect ratio is the proportion comparing the width to the height of a duct. It is a significant factor when looking at fluid flow characteristics. For the given exercise, we have two ducts with aspect ratios of \(2\) (where width is double the height) and \(4\) (where width is four times the height).

The height can be expressed as width divided by the aspect ratio. This detail plays a crucial role when calculating hydraulic diameter since it affects the wetted perimeter. The formula for perimeter, \(P = 2(w + h)\), adjusts with changing aspect ratios.

Thus, the aspect ratio directly impacts the hydraulic diameter and ultimately the frictional losses of the duct. A higher aspect ratio tends to reduce the hydraulic diameter, increasing the frictional losses.

Frictional losses in fluid mechanics describe the loss of pressure or "head" due to the movement of fluid through a duct. This is influenced by the hydraulic diameter. For fluid flow, smaller hydraulic diameters increase the frictional losses.

Given that the frictional losses are inversely proportional to the hydraulic diameter, as calculated in earlier sections, the duct with a smaller hydraulic diameter will experience higher frictional losses.

• The duct with aspect ratio \(2\) will have a higher hydraulic diameter than the one with an aspect ratio \(4\) because its wetted perimeter decreases less relative to its increased width.
• Thus, the duct with an aspect ratio of \(4\) will have higher frictional losses owing to its smaller hydraulic diameter.

This fundamental understanding of how hydraulic diameter and aspect ratio interplay significantly aids in predicting and reducing energy losses due to friction.