3. (25 pts) Water flows from left to right through a horizontal section of pipe of inner diameter D. A piezometer tube is tapped into the pipe and a Pitot probe is also inserted into the center of the pipe, facing upstream as sketched. The water rises in the piezometer tube to height HGL from the reference datum plane, which we take as the centerline of the pipe. The water rises in the Pitot tube to height EGL from the datum plane, where the HGL and EGL denote hydraulic grade line and energy grade line, respectively.
HGL
EGL
Pipe diameter D
In
Reference datum plane (z = 0)
Out
(a) Generate an expression for the centerline velocity in the pipe in terms of the given variables, as needed.
(b) For water density $\rho = 1000 \text{ kg/m}^3$, $D = 2.75 \text{ cm}$, HGL $= 22.5 \text{ cm}$, and EGL $= 37.5 \text{ cm}$, calculate the centerline velocity.
(c) Repeat if the liquid were oil with a density of $\rho = 854 \text{ kg/m}^3$, and all other dimensions and heights remain the same.
(d) As sketched, the velocity profile in the pipe is not uniform, but has the profile shown. The flow is fully developed, and the average velocity at any cross section is $75.5\%$ of the velocity at the centerline. For the water case (Part b), estimate the mass flow rate of water through this pipe in units of kg/s.
