FLOW IN PIPES AND CHANNELS 27
Writing
√
128 k/d^3 Da: 8 u^2 o/kdD[a^2 d^3 / 128 k][8 Q^2 oð 16 /kd^2 d^4 ]DQ^2 o/k^2 d^2 ^2 a^2 and: P^2 oD[Q^2 o/k^2 d^2 ^2 a^2 sinh^2 al]/ 1 Csinh^2 al
or: PDPoDQo/kdatanhal
PROBLEM 3.10
A petroleum product of viscosity 0.5mNs/m^2 and density 700 kg/m^3 is pumped through
a pipe of 0.15 m diameter to storage tanks situated 100 m away. The pressure drop along
the pipe is 70 kN/m^2. The pipeline has to be repaired and it is necessary to pump the
liquid by an alternative route consisting of 70 m of 200 mm pipe followed by 50 m of
100 mm pipe. If the existing pump is capable of developing a pressure of 300 kN/m^2 ,
will it be suitable for use during the period required for the repairs? Take the roughness
of the pipe surface as 0.05 mm.
Solution
This problem may be solved by using equation 3.23 and Fig. 3.8 tofind the volumetric
flowrate and then calculating the pressure drop through the alternative pipe system. From
equation 3.23:R/ u^2 Re^2 DPfd^3 / 4 l ^2
D 70 , 000 ð 0. 153 ð 700 / 4 ð 100 ð 0. 52 ð 10 ^6 D 1. 65 ð 109From Fig. 3.8, ReD 8. 8 ð 105 D 700 ð 10. 15 u/ 0. 5 ð 10 ^3 and the velocity uD
4 .19 m/s.
Cross-sectional areaD/ 4 0. 152 D 0 .0177 m^2.
VolumetricflowrateD 4. 19 ð 0. 0177 D 0 .074 m^3 /s.
The velocity in the 0.2 m diameter pipeD 0. 074 // 4 0. 22 D 2 .36 m/s.
The velocity in the 0.1 m diameter pipeD 9 .44 m/s.
Reynolds number in the 0.2 m pipeD 700 ð 2. 36 ð 0. 2 / 0. 5 ð 10 ^3 D 6. 6 ð 105.
Reynolds number in the 0.1 m pipeD 700 ð 9. 44 ð 0. 1 / 0. 5 ð 10 ^3 D 1. 32 ð 106.
The values ofe/dfor the 0.2 m and the 0.1 m pipes are 0.00025 and 0.0005 respectively.
From Fig. 3.7,R/ u^2 D 0 .0018 and 0.002 respectively, and from equation 3.18:
PfD[4ð 0. 0018 70 / 0. 2 700 ð 2. 362 ]C[4ð 0. 002 50 / 0. 1 700 ð 9. 442 ]D 255 ,600 N/m^2 D 255 .6kN/m^2In addition, there will be a small pressure drop at the junction of the two pipes although
this has been neglected in this solution. Thus the existing pump is satisfactory for this
duty.