A 50 cm diameter pipe carries water (v = 0.294 × 10–6 m2/s) at an average velocity of 10 m/s . The roughness of pipe surface is 0.8 mm. For fully rough turbulent flow, What is the boundary shear stress.a)275b)277Correct answer is between '275,277'. Can you explain this answer?

Question

Answer

To calculate the boundary shear stress for fully rough turbulent flow in a pipe, we can use the Darcy-Weisbach equation:

𝑓 = (1/√𝑓) + 2 log10((𝜖/𝑅)√𝑓)

Where:
𝑓 = Darcy-Weisbach friction factor
𝜖 = roughness height on the pipe surface
𝑅 = hydraulic radius of the pipe

In this case, we are given the diameter of the pipe, so we can calculate the hydraulic radius:

𝑅 = 𝐷/4 = 50 cm / 4 = 12.5 cm = 0.125 m

Given that the roughness height is 0.8 mm, we need to convert it to meters:

𝜖 = 0.8 mm = 0.8 × 10^(-3) m

Now we can solve the Darcy-Weisbach equation for the friction factor. Since this is fully rough turbulent flow, we assume the friction factor is constant:

𝑓 = 0.02

Next, we can use the friction factor to calculate the boundary shear stress using the following equation:

𝜏 = 𝑓 × (𝜌 × 𝑢^2)/2

Where:
𝜏 = boundary shear stress
𝜌 = density of water
𝑢 = average velocity of water in the pipe

Given that the average velocity is 10 m/s and the density of water is approximately 1000 kg/m^3, we can substitute these values into the equation:

𝜏 = 0.02 × (1000 kg/m^3 × (10 m/s)^2)/2 = 0.02 × 5000 N/m^2 = 100 N/m^2

Converting to Pascal (Pa), we get:

𝜏 = 100 Pa

This is equivalent to 100 N/m^2, and since 1 N/m^2 is equal to 1 Pascal, the boundary shear stress is 100 Pa.

Therefore, the correct answer is not between 275 and 277. The correct answer is 100 Pa.

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