2.3. Designing Steel Pipe Diameter

Design the diameter of a steel pipe to carry water with mean velocity of 1 meters per second. The head loss is to be limited to 10 centimeter per 100 meter length. The effective roughness height can be taken as 0.45.

This chunk introduces the problem of designing a steel pipe for water transportation. We have specific criteria: a mean flow velocity of 1 m/s and a maximum allowable head loss of 10 centimeters over 100 meters. The effective roughness of the pipe is given as 0.45 mm. This means when calculating, we need to consider these parameters to ensure efficient and safe pipe design.

Imagine you are designing a water slide at a theme park. You want the slide to allow waters to flow smoothly down without splashing too much (limiting loss) and you need to ensure it is steep enough to keep the flow rate fast (like setting a mean velocity). The roughness of the slide's surface will impact how fast the water can slide down, similar to how pipe roughness affects water flow.

So, first we find the Reynolds number. So, VD by mu and V 1 meters per second, diameter D, let it be this and mu is 1 into 10 to the power -6, so it becomes 10 to the power 6D in terms of diameter.

In fluid mechanics, the Reynolds number (Re) helps us determine the flow regime in the pipe—either laminar or turbulent. The formula for Reynolds number is Re = VD/ν, where V is the flow velocity, D represents the pipe diameter, and μ (the dynamic viscosity of water) is given as 1 x 10^-6. By substituting the known values, we can express Re in terms of D to assess flow characteristics.

Think of it like a river. If water flows calmly and smoothly, it's like laminar flow (small Reynolds number), and if it rushes chaotically with waves and turbulence, it's like turbulent flow (high Reynolds number). Just like gauging how wild or calm the river is helps us understand the best path for a boat, knowing the Reynolds number helps engineers design better pipes.

Now, if we substitute, this in this value, the f which we have given, because most of the things we know already, into 0.45 into 10 to the power 3 by D. Because D is we do not know, that is what we have to find out, into D.

The friction factor (f) is essential for understanding the energy loss due to friction in the pipe. It is calculated with the help of roughness height (ε), which, in this case, is 0.45 mm. Since the diameter (D) is unknown, we will derive an equation for f that relates it to the diameter and then find D based on our maximum allowable head loss.

Imagine trying to push a toy car across different surfaces. If you push it on a smooth floor, it rolls easily (low friction). But on a carpet, it struggles more (high friction). In pipes, the roughness changes how easily water can flow, and the friction factor helps us quantify that struggle, just like we’d measure how hard you have to push on those different surfaces.

hf will be f * L * V^2 / (2 * g * D). So, head loss has already been given, it says 10 in 100, that is, what it said.

The head loss (hf) due to friction in the pipe can be represented by the Darcy-Weisbach equation: hf = f * L * V² / (2 * g * D). Here, L is the pipe length, V is the mean water velocity, g is the acceleration due to gravity, and D is the diameter of the pipe. The problem states that the head loss should be limited to 10 cm for every 100 m length of the pipe.

Think of this like trying to drink from a straw. The longer the straw, the harder it gets to suck the liquid up (more head loss). If the straw is wider (bigger D), it’s easier to drink. In our pipe, we need to balance the length, width, and the speed of water to ensure we don’t struggle against friction too much.

The value of D is obtained by trial and error, from above equations of f and f is found out to be 0.0133 and corresponding D comes out to be 0.678 meters.

After setting the relevant equations based on our initial conditions (mean velocity, head loss, and friction), we utilize a trial and error method to solve for diameter (D). By substituting likely diameter values into our derived equations for friction factor (f) and head loss, we converge on a feasible value for D, which comes out to be approximately 0.678 meters.

It’s like testing different sizes of water balloons to find just the right size that allows you to toss it without it breaking (finding D). Each time you try a different size (D value), you observe how it impacts your throw (f), adjusting until you find the optimal choice that achieves your goal with minimal frustration.

If there is a practical problem in practice, we would use next larger standard size, because the pipe diameters does not come, it is very difficult to design 0.678 meter.

In practice, if the calculated diameter of the pipe does not coincide with standard pipe sizes available in the market, engineers often round up to the next largest size. In this case, because 0.678 meters is not a standard dimension, they may select the nearest higher standard diameter, ensuring that their design remains practical and implementable.

Imagine trying to buy shoes. If your foot measures exactly 10.5 sizes but stores only carry size 11, you’d opt for the bigger size. Similarly, engineers must adjust their calculations to fit standard pipe sizes to ensure they can successfully implement their designs.

--