# Chézy Equation for Laminar Flow in Open Channels

$$v = C \sqrt{mi}$$

Where $$v$$ is the flow velocity, $$C$$ is the Chézy coefficient, $$m$$ is the hydraulic radius, $$i$$ is the hydraulic gradient.

$$m = \frac{A_w}{P_w}$$

Where $$A_w$$ is the wetted cross-sectional area, $$P_w$$ is the wetted perimeter.

$$i = \frac{\Delta Height}{\Delta Length}$$

Given a water level as height of a chord in the channel, chord parameters can be found by

$$P_t - P_w = \alpha r$$

$$h = r ( 1 - \cos \frac{\alpha}{r} )$$

$$A_c = \frac{1}{2} r^2 ( \alpha - \sin \alpha )$$

Where $$P_t$$ is the total perimeter, $$\alpha$$ is the chord angle, $$h$$ is the chord height, $$A_c$$ is the segmented area. The segmented area is summed with the wetted area for the total cross-sectional area of the channel.

$$A = A_w + A_c$$