A common problem today is aging infrastructure. For example, in many parts of the United States wooden water mains still exist $($dating back to the $1800\text{'}$s$).$ For this problem we will consider the benefits of replacing our aging water mains, solely from a frictional perspective. In this case we will consider replacing an old wooden water main with a more modern material $($PVC$).$ The old wooden main was $25$ meters long with an equivalent sand roughness $(k{s}_s)$ of $0\mathrm{.}005$ meters, and diameter of $450mm.$ The new pipe will be made of PVC with an equivalent sand roughness of $0\mathrm{.}00045$ meters. The new pipe will have the same length and diameter as the old pipe. For both pipes the change in head along the length of pipe will be maintained at $0\mathrm{.}5$ meters. Perform the following tasks assuming water temperature of $10C($kinematic viscosity, $v=1\mathrm{.}306{10}^{-6}($:$\frac{m^2}{s}\}.$

Hint: When in doubt about which power formula to use, always start by assuming flow in Manning's Range. Sometimes, based upon available information, you can use the ratio $\frac{R}{k}$ and the power formula sheet to provide an initial estimate for the appropriate rough turbulent flow range $($e$.$g$.$ Lower Manning, Manning, or Upper

Manning$)-$ this indicates which power formula is likely applicable. However, as a final check you should always confirm that the proper flow range and power formula were used.

a$)$ What is the slope of the energy grade line $($SEGL$)?$

b$)$ Using the appropriate power formula, determine the discharge capacity $($Q$)$ of the new pipe.

c$)$ Using R$/$ks$,$ Re$\text{'},$ and Rew along with the power formula sheet, confirm that the correct flow range was used to calculate $Q$ for the new pipe.

d$)$ What is the discharge capacity $(Q)$ of the old pipe?

e$)$ Calculate and compare friction factors for the old and new pipes. $($For the online assessment, you will be asked to consider the relationship between friction factor, pipe roughness, and flow$).$

f$)$ What are the characteristic $P_L$ values for both the old and new pipe?