Here, we reconsider the branched flow example, Where the flow is

produced because of the water level in the very large reservoir, as shown in the diagram

$($below$).$ Branch $\text{'}$a$\text{'}$ includes a $2$ inch Schedule $40$ steel pipe, $2$ standard $($flanged$)$ elbows $($K$=0\mathrm{.}3),$and a gate valve, while branch $\text{'}$b$\text{'}$ includes a $11/4$ inch schedule $40$ steel pipe, $2$ standard $($flanged$)$ elbows $($K$=0\mathrm{.}3).$ The piping in each branch is $6$m long and both pipes can be assumed to have a friction factor of f $=0\mathrm{.}024.$To solve this question, please make the following assumptions:

No losses, other than those associated with the branched flow, need to be determined, i$.$e$.,$ entrance losses and losses associated with the piping outside of the branch;

Plane $(2)$ is at atmospheric pressure;

The flow is steady and fully developed;

The reservoir level is $H=1\mathrm{.}5m$ above discharge point at Plane $(2).$

Using these assumptions, please answer the following questions:

a$)$ When the gate valve in branch $\text{'}a\text{'}$ is fully open $($use $K_v=0\mathrm{.}15),$ what are the branch

flow rates $Q_a$ and $Q_b$ and the total system flow rate $Q^2?$

b$)$ Determine $(P_1-P_2)$ when the gate valve in branch $\text{'}a\text{'}$ is fully open.

c$)$ In order to balance the flow rates through the branches, i$.$e$.,Q_a=Q_b,$ what value of

$K_v$ is required and what position $(\%$ open$)$ will the valve will be in$?$ Prove that this

value is independent of $Q.$

d$)$ Determine the flow rates and average velocities in each branch, as well as the total

flow rate, when the flow through the branches is balanced.

e$)$ Determine $(P_1-P_2)$ when the flow through the branches is balanced.