Viscous Flow Questions & Answers  
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Yes, you can assume $dP/dx$ to be the same when solving the rusted pipe and the smooth pipe. For this problem the pump can be assumed to yield constant pressure increase, and this pressure increase leads to a certain $dP/dx$ in the pipe which won't change as long as the pressure at the pipe exit and the pressure of the fluid entering the pump doesn't change. In this case, the pressure of the fluid entering the pump as well as the pressure at the exit of the pipe are atmospheric, hence why $dP/dx$ can be assumed not to vary. I'll give you 2 points bonus boost.




You can proceed iteratively by trying out a correlation and checking if the correlation is valid once the answer is obtained. Or you can use the Moody diagram instead of using the correlations. I'll give you 1 point bonus boost.




This is a good question. What you have to do is to make a table of $u_{\rm b}$ as a function of the number of nodes as follows:





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