Question

A horizontal pipe has an abrupt expansion from D1 = 5 cm to D2 = 10 cm. The water velocity in the smaller section is 8 m/s and the flow is turbulent. The pressure in the smaller section is F1 400 kPa. Taking the kinetic energy correction factor to be 1.06 at both the inlet and the outlet, determine the downstream pressure F2, and estimate the error that would have occurred if Bernoulli's equation had of water to be p 1000 kg/m3 Di = 5 cm Water 10 cm The downstream pressure iskPa The error in Bernoulli's equation is kt ta.

Answer 1

Consider the water inlet point is 1 and the end of larger section is 2.

From the continuity equation

A1 x V1 = A2 x V2

V2 =( D1/D2)² x V1

=( 0.05/0.10)² x 8 = 2 m/s

For sudden expansion

Loss coefficient KL = (1- A1/A2)² = [1- (D1/D2)²]²

= (1 - 0.5²)² = 0.5626

Head loss hL = KL x V1² / 2g

= 0.5625 x 64/2*9.81

= 1.835 m

From the Bernoulli's theorem

Energy at point 1 = Energy at point 2

head loss occur due to sudden expansion and the pressure increases from 400 to 413.8 kPa because of dynamic pressure converted to static pressure due to decrease in velocity.

V. P V 2g Pg Solving for Pi and substituting, 2 (1.06(8m/s)'-1.06 2 ms-oSlm/s'y«sm) 1000kg-m/s\IN/m 1kN 1kPa

Without head loss, from Bernoulli's theorem

P2 = 400 +( 64 - 4) /2

= 430 kPa

Error = 430 - 413.8 = 16.2 kPa