Relation between Hydraulic Coefficients – Coefficient of Contraction, Coefficient of Velocity, and Coefficient of Discharge

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Hydraulic coefficients are defined in the study of fluid flow through orifices, nozzle, etc.. There are three main hydraulic coefficients

• Coefficient of contraction (Cc)
• Coefficient of velocity (Cv)
• Coefficient of discharge (Cd)

Coefficient of contraction

Coefficient of contraction is defined as the ratio of the area of jet at Vena contracta to the area of orifice (theoretical area). It is denoted by Cc.

Mathematically, Cc = Area of jet at Vena contracta/ area of orifice
Cc = Ac / A
Ac = area of jet at vena-contracta
A = area of orifice

The value of Cc varies with the available head of liquid and shape, size of orifice. In general for an orifice Cc = 0.61 to .69

Coefficient of Velocity

It is defined as the ratio of actual velocity of jet of fluid at vena-contracta to the theoretical velocity of jet. The coefficient of velocity is denoted by Cv
The mathematical expression for Cv is given below

Cv = Actual velocity of jet at Vena contracta/ theoretical velocity of jet
Cv = Va / √(2gH)

Here Va is actual velocity and √(2gH) is theoretical velocity
g = Acceleration due to gravity
H = Head of liquid

The variation in fluid velocity is due to friction of orifice. For orifice Cv = 0.95 to 0.99

Coefficient of discharge

Coefficient of discharge is defined as the ratio of actual discharge of fluid to the theoretical discharge. It is denoted by Cd.

Let Qa = Actual discharge
Qth = Theoretical discharge
Then
Cd = Qa/ Qth

For an orifice Cd = 061 to 065

Relationship between Cc, Cv, Cd

Actual discharge, Qa = Actual velocity of jet, V x Actual area of jet, ac
Theoretical discharge, Qth = Theoretical velocity, √(2gH)   x Theoretical area, a
Coefficient of discharge Cd = Qa/Qth
= (Actual velocity of jet x Actual area of jet) / (Theoretical velocity x Theoretical area)
= (Actual velocity of jet/ Theoretical velocity) x (Actual area of jet/ Theoretical area)
Cd = Cv x Cc
Coefficient of discharge = Coefficient of velocity x Coefficient of contraction