Consider uniform horizontal pipe as shown in and Equation derived in Darcy-Weisbach equation.

Where is,   h_f = f’/ γ. P/A X L V²

We know, hydraulic radius is the ratio of area of flow to wetted perimeter. It is denoted by ‘m’.

m = A/P = π/4d² / πd = d/4

P/A = 1/m

put value of P/A in Equation,

h_f = f’/  γ. 1/m. L V²

V²  = h_f.  γ.m / f’.L

V =  √ γ / f'. h_f / L. m

Consider √ γ / f' = C i.e. Chezy’s constant and

h_f / L = I i.e. loss of head per unit length of pipe.

Put the above value in Equation,

V = C√m i

This is known as Chezy’s formula.

Relation between the freiction factor f and the chezy’s constant C:

Head loss due to Darcy Equation,

hf = fLV²/ 2gd

From Chezy’s Equation, V = C√m i

Where, m = d/4,  I = hf/L put in Equation,

V = C √m i

Squaring both side,    V² = C² x d/4 x h_f / L

h_f = 4V²L / C²d

Equate Equation

f LV²/2gd  = 4V²L/C²d

f = 8g / C²

This is the required relation between Darcy’s coefficient and Chezy’s Constant.

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