## The energy balance equation

Equation (14.25) has two unknowns of velocity and depth at each cross-section. Thus another equation is required to effect a solution. The second fundamental equation can be derived from considering either the energy or momentum equation for the short length of channel, Ax. In Fig. 14.16, u is a mean velocity averaged over the cross-section at distance x in the reach, and g is the gravity acceleration. The loss in head over the length of the reach, Ax, has two main components:

the head loss due to friction, and: 1 dv ha = —— Ax = SaAx (14.27)

a g dt a the head loss due to acceleration.

Fig. 14.16 Definition diagram for the energy equation expressed in terms of energy per unit weight or head (which has units of length); (z + y) represents the potential head, v2/2g represents the velocity head at a point; while So is the bed slope, hf represents the friction head loss, and ha represents the acceleration head loss over a short reach of length Ax.

Fig. 14.16 Definition diagram for the energy equation expressed in terms of energy per unit weight or head (which has units of length); (z + y) represents the potential head, v2/2g represents the velocity head at a point; while So is the bed slope, hf represents the friction head loss, and ha represents the acceleration head loss over a short reach of length Ax.

If it is assumed that the channel bed slope is small and the vertical component of the acceleration force is negligible, then the combined loss of head is (hf + ha). Using the Bernoulli expression for total head, H:

2g then the change in H over Ax is - AH = hf + ha. Thus: AH d ( v2 \

from which:

Equations (14.25) and (14.30) provide two equations in the unknowns v and y. The friction slope, Sf, however also depends on depth and velocity. To complete the system of equations it is common to assume that the rate of head loss due to friction under dynamic flow conditions is the same as if the flow was steady and uniform with a water surface slope equal to the friction slope. Under this assumption, one of the uniform flow equations can be used to derive the friction slope. For example, the Manning equation is widely used in hydraulic routing models in the form

where Rh is the hydraulic mean radius given by cross-sectional area divided by wetted perimeter (R^ = A/P) and n is the Manning roughness coefficient that depends on the nature of the channel. For more rigorous derivations of the equations describing unsteady open channel flow, the reader is referred to the text on open channel flow and the solution of the St Venant equations (e.g. Chow, 1959; Abbott and Minns, 1998). The specification of channel and flood plain roughness values is considered in Section 14.4.2 below.