Mathematically the discharge coefficient may be related to the mass flow rate of a fluid through a straight tube of constant cross-sectional area through the following: Where: This parameter is useful for determining the irrecoverable losses associated with a certain piece of equipment (constriction) in a fluid system, or the "resistance" that piece of equipment imposes upon the flow.
This flow resistance, often expressed as a dimensionless parameter,
, is related to the discharge coefficient through the equation: which may be obtained by substituting
Due to complex behavior of fluids around some of the structures such as orifices, gates, and weirs etc., some assumptions are made for the theoretical analysis of the stage-discharge relationship.
Therefore, engineers assume that the pressure is zero at the gate opening and following equation is obtained for discharge: where: However, the pressure is not actually zero at the gate; therefore, discharge coefficient, Cd is used as follows: