4. Discharge-Area-Velocity Relationships

Flow rate or discharge, Q, is the volume of water in cubic feet passing a flow section per unit time, usually measured in cubic feet per second (ft3/s). The distance, dv, in feet that water will travel at a given velocity in a pipe of constant diameter is velocity, V, in feet per second (ft/s) multiplied by time, t, in seconds, or:

Vi (2-7)

The volume, Vo, in cubic feet passing from the upstream to the downstream ends of this distance is the distance, dv, in feet times area, A, in square feet of the flow section. Thus:

eqn (2-8)

To get the time rate of flow or discharge, Q, in cubic feet per second, divide the right and left sides of equation 2-8 by time, t, in seconds, resulting in:

eqn (2-9)

Flow in open channels of rectangular cross section is often expressed in terms of unit discharge, q, in cubic feet per second per foot of width which is discharge, Q, in cubic feet per second divided by cross-sectional width, Lb, in feet or:

eqn (2-10)

The area, A, is LbD, where D is the depth of flow. The continuity concept is an important extension of equation 2-9. On the basis that water is incompressible and none is lost from a flowing system, then as the cross-sectional area changes, the velocity must adjust itself such that the values of Q or VA are constant:


where the subscript denotes any number of arbitrarily selected positions along the flowing system. This principle, known as continuity, is especially useful in the analysis of tube flow measurement devices such as the venturi meter.