# 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.

$$ A = A_w + A_c $$