The table should be filled in until the distance to the required section is determined, or if the distance is known, it should be filled in until the depth of flow with the assumed rate of discharge has been checked.
If only the depth of flow at some section is known and it is required to know the maximum rate of flow with a free discharge, or a discharge with a submergence at the outlet less than the depth of flow with the maximum rate of discharge, it is necessary to make a preliminary estimate of the maximum rate of flow in order to fill in the quantity _Q_ at the head of the table. The procedure should be as follows:
1st. Assume a depth of flow at the outlet.
2nd. Compute the area (_A_) and the hydraulic radius (_R_) at the known section and at the outlet.
3rd. Determine the area and the hydraulic radius half way between these two sections as the mean of the areas and the hydraulic radii of the two sections.
4th. Determine the rate of flow through the sewer from the condition that the difference in head at the two sections is the head lost due to friction caused by the average velocity of flow between the sections (equals (_lV_^2)⁄(_C_^2_R_)) plus the gain in velocity head (equals _V__{ 2}^2 − (_V__{ 1}^2)⁄(2_g_)), which then combined and transposed result in the expression:
_Q_ = _AA__{ 1}_A__{ 2} √(2_Rgh_⁄(2_A__{ 1}^2_A__{ 2}^2_gl_ + (_A__{ 1} − _A__{ 2})(_A_^2_C_^2_R_)))
_A_ = the area determined in the 3rd step;
_A__{ 1} = the area at the upper cross-section;
_A__{ 2} = the area at the lower cross-section;
_C_ = the coefficient in the Chezy formula;
_h_ = the difference in elevation of the surface of the stream at the two cross-sections;
_l_ = the distance between the cross-sections;
_R_ = the hydraulic radius determined in the third step.
5th. Continue this process by assuming different depths at the outlet until the maximum rate of discharge has been found by trial.
With this rate of discharge and depth of flow at the outlet, the depth of flow at the known section can be checked. If appreciably in error a correction should be made by the assumption of a different depth of flow at the outlet. The approximate character of the method is scarcely worthy of the refinement in the results which will be obtained by checking back for the depth of flow at the known section. It will be sufficiently accurate to assume the rate of flow obtained by trial from the preceding expression, as the maximum rate of discharge from the sewer.