Consider two section 1-1 and 2-2 before and after the contraction for a horizontal pipe.

Let, V₁ = velocity of flow at 1-1
        A₁ = cross-sectional area at 1-1
        V₂ = velocity of flow at 2-2
        A₂ = cross-sectional area at 2-2
        V_c = velocity of flow at vena-contract
        A_c = cross-sectional area at 3-3
        h_c = loss of head due sudden contraction.

The loss of head occurring between section 1 to section 3 is appreciable. The major loss occurs when the jet of liquid enlarges from section 3 to section 2 i.e. mainly due to enlargement between section 3 to section 2.

From Equation, head loss is given by,

h_c = (V_c – V₂)² / 2g

= V₂²/2g [ V_c/V₂ – 1]²

According to the continuity equation,

A_c V_c = A₂ V₂

V_c / V₂ = A2/Ac

V_c / v₂ = 1/C_c

Putting the value Vc/V2 in Equation

h_c = V₂² / 2g [1/C_c – 1]

If the value of Cc = 0.62 (Assume) then,

h_c = V₂² / 2g [1/0.62 -1]

h_c = 0.375 V₂² / 2g

If the value of Cc is not given, then

h_c = 0.5 V₂² / 2g

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