Find the loss of head when a pipe of diameter 200 mm is suddenly enlarged to a diameter of 400 mm. The rate of flow of water through the pipe is 200 lit/s.

Find the loss of head when a pipe of diameter 200 mm is suddenly enlarged to a diameter of 400 mm. The rate of flow of water through the pipe is 200 lit/s.

Solution: 

Given:

Diameter of smaller pipe

D1=200mm=2001000=0.20m$D_1=200mm={\displaystyle \frac{200}{1000}}=0.20m$

Area, A1=π4×D21$A_1={\displaystyle \frac{\pi }{4}}\times D_1^2$

A1=π4×(0.20)2=0.03141m2$A_1={\displaystyle \frac{\pi }{4}}\times (0.20{)}^2=0.03141m^2$

Diameter of larger pipe

D2=400mm=4001000=0.40m$D_2=400mm={\displaystyle \frac{400}{1000}}=0.40m$

Area, A2=π4×D22$A_2={\displaystyle \frac{\pi }{4}}\times D_2^2$

A2=π4×(0.40)2=0.12564m2$A_2={\displaystyle \frac{\pi }{4}}\times (0.40{)}^2=0.12564m^2$

Discharge, Q=250lit/sec=2501000=0.25m3/s$Q=250lit/sec={\displaystyle \frac{250}{1000}}=0.25m^3/s$

Velocity, V1=QA1=0.250.03141=7.96m/s$V_1={\displaystyle \frac{Q}{A_1}}={\displaystyle \frac{0.25}{0.03141}}=7.96m/s$

Velocity, V2=QA2=0.250.12564=1.996m/s$V_2={\displaystyle \frac{Q}{A_2}}={\displaystyle \frac{0.25}{0.12564}}=1.996m/s$

Now to find the loss due to enlargement

he=(V1−V2)22×g$h_e={\displaystyle \frac{(V_1-V_2{)}^2}{2\times g}}$

he=(7.96−1.99)22×9.81$h_e={\displaystyle \frac{(7.96-1.99{)}^2}{2\times 9.81}}$

he=1.816m

$h_e=1.816m$ of water