$16-$Civ$-$A$5-$Dec $2019$

An upstream reservoir with starting water level of $80m$ drains into a downstream reservoir with starting water level of $40m$ via a pipe. The pipe has a $450mm$ diameter, a $1,000m$ length, and a $\text{'}C\text{'}$ factor of $120.$ The upstream and downstream reservoirs each have a cross sectional area of $5m^2.$

a$)$ Determine the steady state flow in the pipe at time $t=0.$

b$)$ Use the rigid water column model to determine the flow in the pipe and the water levels in the upstream and downstream reservoirs for the first two time steps.

c$)$ The plots in Figures Q$4-$ A and B show the system response when the pipe has a diameter of $450mm($Figure Q$4-$ A$)$ and when the pipe diameter is increased to a $750mm$ diameter $($Figure Q$4-$ B$).$ Explain the difference between these two responses. Base your explanation on the physical meaning of the mathematical terms in the rigid water column model and the initial amount of linear momentum of the water column at $t=0.$

Figure Q$4-$ A$.$ System response in a pipe with a $450mm$ diameter