A valve controls the flow of benzene from a supply tank through a pipeline, as illustrated in FigureQ1. The pipeline has diameter 0.75 m, wall thickness 10 mm and wall

roughness O.245 mm. A cylindrical surge tower protects the pipeline between the supply tank and the tower. The tower has diameter 3.0 m and its base is 20 m below the fluid level in the supply tank. The valve is 500 m from the tower and is at the same elevation as the tower base. The pipe length between the supply tank and the tower is 1000 m. The density, dynamic viscosity and bulk modulus of the benzene for the conditions are 875 kg/m', 0.6 x 10' Ns/m'and 1.05 x 10" N/m' respectively; the Young's Modulus for steel is 210 x 10' N/m. (a) Under steady flow conditions with the valve open, the fluid level in the tower is 4.518 m above the tower base. ignoring all losses other than head loss due to wail friction, show that the steady flow discharge in the pipeline is 1.7 m/s.(5 marks! The minimum allowable time to fully close the valve during operation is specified to be 1 s. Check that this is reasonable given that the maximum allowable hoop stress for the pipeline is 150 MPa.(Hoop stress is ah pd/2r,where p.d,t are pipe pressure, diameter and wall thickness respectively). \frac{d u}{d r}=-\frac{g}{L}\left(y-H+\frac{K}{2 g} u|u|\right) \text { and } A u \equiv A_{r} \frac{d y}{d t} where k"u/2g represents the hydraulic losses due to wail friction and other terms are as shown in Figure Q1. Use a finite difference scheme with At = 25 s to calculate and plot the time variation of fluid level in the tower for the first 200 s following the 1-s valve closure.