Question

The velocity of flow of water in a pipe of 150 mm dia is 0.3m/sec, a diaphragm with a central hole 80mm in diameter is placed in the pipe obstructing the flow. With coefficient od contraction Cc=0.60, the loss of head from Vena Contracta to a point downstream will be ?

Answer 1

Solution:

Diameter of pipe = 150 mm = 150 / 1000 = 0.15 m

Diameter of Diaphragm with a central hole = 80 mm = 80 / 1000 = 0.08 m

Area of pipe =  $\frac{\pi }{4} * (0.15)^2$ =  $\frac{3.14 }{4} * 0.0225$ = 0.0176 m²

Area of central hole present in diaphragm = $\frac{\pi }{4} * (0.08)^2$ = $\frac{0.02}{4}$ = 0.005 m²

$h_{obstruction} = \frac{v^{2} }{2g} [ \frac{A}{C_{c}(A-a) } ]^2$

where ,

v = Velocity = 0.3 m/s

g = Acceleration due to gravity = 9.8 m/s²

A = Area of pipe

a = Area of central hole present in diaphragm

$C_{c}$ = Coefficient of contraction = 0.60

$h_{obstruction} = \frac{0.3^{2} }{2 * 9.8} [ \frac{0.0176}{0.60(0.0176 - 0.005) } ]^2$

$h_{obstruction} = \frac{0.09 }{19.6} [ \frac{0.0176}{0.60 * 0.0126 } ]^2$

$h_{obstruction} = \frac{0.09 }{19.6} [ \frac{176 * 100 * 1000}{60 * 126 * 1000} ]^2$

$h_{obstruction} = \frac{0.09 }{19.6} [ \frac{176 * 10}{6 * 126} ]^2$

$h_{obstruction} = \frac{9 * 10 }{196 * 100} [ \frac{1760}{756} ]^2$

$h_{obstruction} = \frac{9 * 10 * 1760 * 1760 }{196 * 100 *756 *756}$

$h_{obstruction} = \frac{9 * 176 * 1760 }{196 *756 *756}$

$h_{obstruction} = \frac{2787840 }{112021056}$

$h_{obstruction} = 0.0248 m$

$h_{obstruction} = 24.8 mm$

The loss of head from Vena Contracta to a point downstream will be 0.0248 m or 24.8 mm

Answer 2

The value of loss of head is h = 24.8 mm

Explanation:

• The formula for velocity is given;
• h[obstruction] = V^2 / 2g [A / Cc (A-a)]^2
• Here  
• V = velocity
• A = area of pipe
• a = area of central pipe
• cc = Coefficient of contraction

Now put the values in the formula.

h[obs] = 0.3^2 / 2 x 9.8 [ 0.0176 / 0.06 x (0.0176 - 0.005)]^2

h[obs] = 0.09 / 19.6 [176 x 10 / 6 x 126]^2

h [obs] = 9 x 10 x 1760 x 1760 / 196 x 100 x 756 x 756

h[obs] = 0.0248 m

convert into millimeters b multiplying with 1000

h[obs] = 24.8 mm