Does anyone have or know of a program to calculate the volume of a circular pipe on a slope in user definable increments. Our office is developing a spreadsheet to do this but it is turning into a lot of work. This would be used in calculating a stage-storage curve for routing an underground stormwater management system. We currently analyze pipe systems as if they are flat. Any help?
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Stage-Storage of Sloped Pipes
- Thread starter sapaeng
- Start date
Here's a handy little formula whipped up through a simple double integration which should work. Only the second time I've used calculus since college! You should verify the formula against the calculations you've already made.
For a partially full circular pipe, with a depth at the end of D, slope of S, and radius of R
Where: y = D - R (can be negative)
[((PI*y*(R^2))/2) - (SQRT(((R^2) - (y^2))^3)) +
((R^2)*((yARCSIN(y/R)) + R*SQRT(1-((y/R)^2))))]/S
Nothin' to it! You can email me at parkscad@parks.state.nv.us if necessary
For a partially full circular pipe, with a depth at the end of D, slope of S, and radius of R
Where: y = D - R (can be negative)
[((PI*y*(R^2))/2) - (SQRT(((R^2) - (y^2))^3)) +
((R^2)*((yARCSIN(y/R)) + R*SQRT(1-((y/R)^2))))]/S
Nothin' to it! You can email me at parkscad@parks.state.nv.us if necessary
Or, maybe you'd like a simple approach.
Assuming that the lower end of the pipe is full and the you know the slope and the elevation of the water surface: Calculate the length to the point where the top of the pipe is at the water/air interface. Then, calculate the length to the point where the bottom of the pipe is at the water/air interface. Add these lengths, divide by two, and multiply by the cross section area of the pipe.
Assuming that the lower end of the pipe is full and the you know the slope and the elevation of the water surface: Calculate the length to the point where the top of the pipe is at the water/air interface. Then, calculate the length to the point where the bottom of the pipe is at the water/air interface. Add these lengths, divide by two, and multiply by the cross section area of the pipe.
BillHolt's solution for the specific condition with the lower end the pipe full is correct. If you plug Depth = twice Radius (D = 2R) and then (y = 2R - R) or (y = R) into the formula, for the lower end full condition, you end up with the simplified formula
PI*(R^2)*(R/S) or (1/2)*PI*(R^2)*(D/S)
where depth divided by slope is the length, which is the same as BillHolt's solution
The full formula should give a solution for situations where the pipe is not full at he lower end; i.e. Depth at the lower end of the pipe is less than the diameter of the pipe. Note that depth should be measured normal to the flowline of the circular pipe for the formula to work properly, if depth is measured on a plumb line then a slight error is introduced due to the slope of the pipe, but the error is going to be negligible for most practical purposes at slopes less than 10%.
PI*(R^2)*(R/S) or (1/2)*PI*(R^2)*(D/S)
where depth divided by slope is the length, which is the same as BillHolt's solution
The full formula should give a solution for situations where the pipe is not full at he lower end; i.e. Depth at the lower end of the pipe is less than the diameter of the pipe. Note that depth should be measured normal to the flowline of the circular pipe for the formula to work properly, if depth is measured on a plumb line then a slight error is introduced due to the slope of the pipe, but the error is going to be negligible for most practical purposes at slopes less than 10%.
If you have access to Haestad's PondPack program for stormwater management, the program will perform the calculations for you for one or more underground pipes on a slope. The only drawback is all the pipes have to be identical size and slope.
You might also want to check out They have a helpful demo called Drainage Runoff and Flow Design, version 5.2 that I occasionally use. It has four modules: Runoff Calculator, Channel Capacity Calculator, Culvert Capacity Calculator, and Storm Drain Inlets and Grates Calculator. Also try They have a ready-made spreadsheet program called Civil Tools version 1.1 that looks promising.
Good luck!
Good luck!
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