$2.$ It is decided to place a gate valve at the beginning of the system in Problem $01$ to control the flow. Please answer the following assuming all the conditions of Problem $01$ with flow into the valve well developed $(35$ Points$)$:

a$.$ What is the expected loss of energy through the valve when it is fully opened?

b$.$ What is the expected loss of energy through the valve when the valve is $\backslash (1/2\backslash )$ opened?

c$.$ What is the change in inlet pressure between problem $1($no valve$)$ and the fully opened valve? Assume the system discharges to atmosphere.

d$.$ What is the change in inlet pressure between the valve fully opened and the valve fully closed? For both parts c and d$,$ you may report your answer in height units.