Normal Depth Calculation for Pipes Flowing Less Than Half Full
The Excel template shown at the left can be used to calculate the normal depth, yo, for specified pipe diameter, D; full pipe Manning roughness, nfull; pipe bottom slope, S; and volumetric flow rate, Q, for a partially full pipe that is flowing less than half full.
The diagram and equations given in the third section (page 2 of this article) for partially full pipe flow with flow depth less than half of the pipe diameter apply here also. The determination of normal depth is an iterative (trial and error) calculation for this situation, just as it is for a rectangular channel or a trapezoidal channel.
In this case of partially full pipe flow, for each assumed value of flow depth, yo, it is necessary to calculate h, A, P, R, and n, so that the parameter, AR2/3 can be calculated. Assuming values of yo and calculating values for AR2/3, allows one to zero in on the yo value that comes closest to the target value for AR2/3, as shown in the Excel template example. The example calculation in this spreadsheet template resulted in yo = 0.35 m, accurate to 2 significant figures.
Note that if the answer comes out to be more than the pipe radius, then the spreadsheet template in the next section should be used, because it uses the slightly different equations that apply for depth greater than the pipe radius.