Draining a lake

[scubatwin]

How do I calculate the time it would take to lower a lake 4 feet through a pipe 18 inches high. The lake has an average depth of 10 feet and a circumference of 4 miles. Thank you.

[Administrator]

As the level of lake is going to change that flow is not stationary so the flow rate will drop from some point to zero. Some other calculation methods are required.
What is the difference in height between lake surface and pipe outlet - is it 10 feet as the height of lake?

[scubatwin]

The height is 10 feet from the bottom of the pipe and 8 1/2 feet from the top of the pipe. Thanks for your help.

[Administrator]

I have attached full derivation for lake draining problem BUT it has to be checked and some values are needed to be presumed like friction factor is constant which is not because Re number will change from some value to another during drain, but some average value can be used.

With last equation:
√(2(f L/D+K)H/g) D_L/D_p =t
total time t for lake drain can be calculated where is:

f – friction factor
L – pipe length
D – pipe diameter
K – resistance coefficient
g – gravity acceleration 9,81 m/s
H – height of lake to pipe axis
D_L – lake diameter – changing with height
D_p – pipe diameter

K – resistance coefficient for pipe entrance and exit K = 1,78
Friction factor f will mainly depend on the pipe roughness and for 0.1 mm roughness and pipe diameter of 18" it would be for fully turbulent flow f=0.014

Comments are very welcomed!