4.4. Problem-solving

Now, it is a good idea to solve one problem here, on this particular concept, and the solving the problem will give more understanding now. You know, the question is, determine the average height of roughness for a rough pipe of diameter 10 centimeter when the velocity at point 4 centimeter away from the wall is 40 percent more than the velocity at a point 1 centimeter from the wall.

This chunk introduces a specific problem related to turbulent flow in rough pipes. The problem requires determining the average height of roughness given certain parameters (the diameter of the pipe and the relationship between velocities at different distances from the wall). Understanding the initial requirements of the problem is crucial for structuring the solution approach.

Think of a hose pipe that is rough on the inside, which would affect how water flows through it. If you were to measure the speed of water at different points along the inner wall, you'd want to know how the roughness affects that speed, similar to what we are doing in this problem.

Given is, D is equal to 10 centimeters or 10 into 10 to the power minus 2 meters or 0.1 meter. It is given, u at y is equal to 4 centimeters or 4 is equal to 1.4 times u at y is equal to 1 and the surface is rough.

In this chunk, the problem provides specific values that will be used in calculations. The diameter of the pipe (D) is converted into meters for consistency with other measurements. Additionally, the relationship between the velocities at different distances from the wall is established, highlighting the importance of understanding how these distances affect flow speed.

Imagine you’re measuring the speed of a car at different points on a race track. Just like how you need to know the distance from the starting line to determine speed, knowing the distances from the wall helps us understand how the pipe's roughness influences water flow.

Therefore, we can write, u at y is equal to 4 by u star can be written as, 1.4 u at y is equal to 1 divided by u star.

This chunk outlines the formulation of the relationship between the velocities at two different points in the rough pipe. Essentially, it shows how to express one velocity in terms of another, aiding in the application of the relevant equations later in the solution process. This step is important for applying the principles of turbulent flow.

Think of this like comparing the speed of two runners in a race. If you know how fast one runner is going, you can express the speed of another runner relative to them, which helps you understand the dynamics of their competition.

So, we write on the left-hand side, 5.75 log base 10, 4 by k, because it is rough, plus 8.5 is equal to 1.4 times 5.75 log base 10 kappa plus 8.5.

In this step, logarithmic functions are used to express the relationship between the velocities and surface roughness in a more manageable form. By utilizing logarithmic properties, the equation can be solved systematically. This shows the validity and application of empirical relationships in hydraulic engineering.

Using logarithms can be likened to using a recipe in cooking. Just as recipes provide specific proportions for ingredients to achieve the desired taste, logarithmic equations help relate different variables in a flow problem to achieve a specific solution.

So, this was what was asked. So, k is coming out to be 0.9399 centimeter. This is the answer and how to solve?

The chunk concludes the problem by providing the calculated average height of roughness (k) for the pipe based on the earlier equations. It confirms the value obtained through proper substitutions and careful mathematical manipulations, emphasizing the importance of accuracy in hydraulic calculations.

Imagine finally getting the measurements right for a construction project. Just as precision is crucial in building to ensure stability and safety, accurate calculations in fluid mechanics lead to better designs and more efficient systems.

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