Solved Examples Based on Bernoullis Equation for Ideal Fluid

SOLVED EXAMPLES BASED ON BERNOULLIS EQUATION FOR IDEAL FLUID

Example - 54

A pipe of diameter 230mm conveys a discharge of 220 litres of water per minute and has a pressure of 20kN/m2 at certain section. Find the total energy head with respect to datum 4m below the pipe.

Given data:

Pipe diameter ( d) = 230mm = 0.230m

Discharge (Q) = 220 lit/min = 2.2 m3/min = 0.0366 m3/s (1 litre = 1/1000m3) Pressure (P) = 20kN/m2 = 20 × 103N/m2        (1KN = 1000N)

Datum head (Z) = 4m

To find:

Total energy head (H)

Solution :

Result :

Total energy head (H) = 6.0783 m

Example - 55

A pipe line is carrying water. At a point A in the inlet pipe line of the diameter is 500 mm, the pressure 70kN/m2 and the velocity is 2m/s. At another point B which is 2m higher than A, the diameter is 200mm and the pressure is 14KN/m2. Determine the direction flow.

Given data:

At a point A

Diameter (da) = 500 mm = 0.5m Velocity (Va) = 2 m/s

Pressure (Pa) = 70 kN/m2 = 70 × 103N/m2 Assume (Za) = 0

At a point B

Diameter (db) = 200 mm = 0.2m

Pressure (Pb) = 14 kN/m2 = 14 × 103 N/m2    

Datum level (Zb) = 2 m

To find:

Direction of flow

Solution:

5. Direction of flow

Total energy at point A(Ha) < total energy at point B(Hb) 

Hence flow is taking place from B to A

Result :

Flow is taking place from B to A.

Example - 56

A pipe 250m long has a slope of 1 in 50 and tapers from 1.2 diameter at the higher end and 0.50m diameter at the lower end. The discharge of water through the pipe is 200 lit/sec. If the pressure at higher end is 80 kN /m2, find pressure at the lower end neglecting friction.

Given data:

Length of the pipe (l) = 250 m 

Slope = 1in50

Diameter of the higher end (d1) = 1.2m

Diameter of the lower end (d2) = 0.50 m

Discharge (Q) = 200 lit/sec = 0.2m3/s

Pressure at higher end (P1) = 80 kN/m2 = 80 × 103 N/m2

Datum head at higher end (Z1) = 250 × 1/50 = 5 m

Datum head at lower end Z2 = 0

To find:

Pressure at the lower end (P2)

Solution :

Result :

Pressure at the lower end (P2) = 128.54 kN/m2.

Example - 57

Water is flowing through the pipe having diameter 400 mm and 250 mm at the bottom and upper end respectively. The intensity of pressure at bottom end is 23 N/cm2 and the pressure at upper end is 9 N/cm2. Determine the difference in the datum head if the rate of flow through the pipe is 30 lit/sec.

Given data:

Bottom end

Diameter (d1) = 400 mm = 0.4 m

Pressure (P1) = 23 N/cm2 = 23 × 104 N/m2

Upper end

Diameter (d2) = 250 mm = 0.250 m 

Pressure (P2) = 9 N/cm2 = 9 × 104 N/m2

Rate of flow (Q) = 30 lit/sec = 0.03m3/s

To find:

Difference in datum head

Solution:

Result :

Difference in datum head (Z2 – Z1) = 14.25 m.

Example - 58

A vertical pipe line which is 1.5 m long and tapering from 150 mm to 75 mm diameter is discharging water at the rate of 20 lit/sec. Find the difference of pressure between the top and bottom of pipe.

Given data:

Length of the pipe (L) = 1.5 m

Diameter of larger end (d1) = 150 mm 

Diameter of smaller end (d2) = 75 mm 

Discharge (Q) = 20 lit/sec = 0.02m3/s

Datum head at height end (Z1) = 1.5 m. 

Vertical position datum head at lower end (Z2) = 0

To find:

Pressure difference between top and bottom (P1 - P2)

Solution :

Result: 

Result :

Pressure difference between top and bottom (P1 - P2) = -5107.09 N/m2.

Example - 59

Water is flowing through the pipe having diameter 40 cm and 20 cm at section 1 and 2 respectively. The rate of flow through pipe is 30 lit/sec. The section 1 is 8 m above datum and section 2 is 5 m above datum. If the pressure at section 1 is 40 N/cm2, find the intensity of pressure at section 2.

Given data:

At section 1:

Pipe diameter (d1) = 40cm = 0.4 m 

Pressure (P1) = 40 N/cm2 = 40 × 104 N/m2

Datum head (Z1) = 8m

At section 2:

Pipe diameter (d2) = 20cm = 0.2m 

Datum head (Z2) = 5m

Discharge (Q) = 30lit/sec

To find: 

The pressure at section 2 (P2) 

Solution:

Result :

Result :

The pressure at section 2, P2 = 429.01 KN/m2