# Outlet Control

Together, inlet and outlet control equations form the basis for culvert design. To determine the headwater, both inlet and outlet control equations are used, and the higher result governs. Inlet control will govern in supercritical flow, and outlet control will govern in subcritical flow.  This article will give a brief overview of outlet control. For inlet control, see my previous post on the subject.

For outlet control, additional factors influence the headwater such as barrel roughness and slope.  The analysis method is a simple energy balance equation. The first, and hardest, part is the head loss through the barrel:

$H_L = H_e + H_f + H_o + H_b + H_j + H_g$

He = Entrance Loss
Hf = Friction Loss in barrel
Ho = Exit Loss
Hb, Hj, Hg = Losses in bends, junctions, and grates

Entrance Loss

$H_e = k_e\frac{V^2}{2g}$

ke = Entrance Loss Coefficent, looked up on tables for the inlet configuration type
V = barrel velocity (ft/s or m/s), calculated as Q/A.
g = gravitational constant (32.2 ft/s2 or 9.81 m/s2)

Friction Loss

$H_f = (\frac{K_Un^2L}{R^{1.33}})\frac{V^2}{2g}$

KU = Conversion factor, 29 in U.S. units, 19.63 in SI
n = Manning’s roughness coefficient of the culvert barrel
L = Length of the culvert (ft or m)
R = Hydraulic radius of the full culvert barrel (ft or m)
V = barrel velocity (ft/s or m/s), calculated as Q/A.
g = gravitational constant (32.2 ft/s2 or 9.81 m/s2)

Exit Loss

$H_o = 1.0[{\frac{V^2}{2g}-\frac{V_d^2}{2g}}]$

Once all the losses are calculated, simply add them to the tailwater, and also add the drop due to culvert gradient, like this:

$HW_o = TW + H_L - LS$

HWo = Headwater depth at culvert entrance (ft or m)
TW = Tailwater (ft or m)
HL = Head Losses (ft or m)
LS = Drop in culvert due to gradient (ft or m)