1. 18. Pipe Flow 1: Shear Stress, Pressure Drop
CH EN 374: Fluid Mechanics

2. What makes fluid flow in a pipe?
𝑷 𝟏
𝑷 𝟐
𝑷 𝟏
𝑷 𝟐
A pressure drop
An elevation drop
(These are sort of the same thing.)

3. 𝑷 𝟐 𝑷 𝟏 𝑃 2 − 𝑃 1 𝜌 + 1 2 𝑣 2 2 − 𝑣 1 2 +𝑔 𝑧 2 − 𝑧 1 = w s − w f
𝑃 2 − 𝑃 1 𝜌 =− 𝑤 𝑓
Δ𝑃 =𝜌 𝑤 𝑓

4. 𝑷 𝑳 𝑷 𝟎 𝝉 𝒘 𝑃 0 𝜋 𝐷 2 2 − 𝑃 𝐿 𝜋 𝐷 2 2 − 𝜏 𝑤 𝜋𝐷𝐿 =0
𝑃 0 𝜋 𝐷 2 2
− 𝑃 𝐿 𝜋 𝐷 2 2
− 𝜏 𝑤 𝜋𝐷𝐿 =0
𝑃 0 − 𝑃 𝐿 𝐷 − 𝜏 𝑤 𝜋𝐷𝐿=0
𝜏 𝑤 = 𝑃 0 − 𝑃 𝐿 𝐿 𝐷 4
𝜏 𝑤 = |∆𝑃| 𝐿 𝐷 4

5. Dynamic Pressure

6. Dimensional Analysis, the Return
𝜏 𝑤 in a pipe depends on 𝜌, 𝜇, 𝑣, and 𝐷. Use these parameters to find a group of dimensionless numbers for pipe flow.

7. Fanning Friction Factor
𝑓= 2 𝜏 𝑤 𝜌 𝑣 2

8. Darcy Friction Factor
𝑓= 8 𝜏 𝑤 𝜌 𝑣 2

9. WE WILL USE THE FANNING FRICTION FACTOR IN THIS CLASS.
Fanning friction factor more commonly used by chemical engineers
WE WILL USE THE FANNING FRICTION FACTOR IN THIS CLASS.
𝑓= 2 𝜏 𝑤 𝜌 𝑣 2

10. 𝑓=𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛(𝑅𝑒)

11. 𝐶 𝐷 = 2 𝐹 𝐷 𝜌 𝑣 2 𝐴 =𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛(𝑅𝑒)
Drag:
𝑓= 2 𝜏 𝑤 𝜌 𝑣 2 =𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛(𝑅𝑒)
Pipe Flow:
𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑏𝑎𝑙𝑎𝑛𝑐𝑒
𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑎𝑛𝑑
𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒𝑠
𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑓𝑙𝑜𝑤 𝑖𝑛𝑒𝑟𝑡𝑖𝑎

12. Pressure Drop
𝑓= 2 𝜏 𝑤 𝜌 𝑣 2
𝜏 𝑤 = |∆𝑃| 𝐿 𝐷 4
Δ 𝑃 𝐿 =2𝜌𝑓 𝐿 𝐷 𝑣 2

13. Head Loss
ℎ 𝐿 =2𝑓 𝐿 𝐷 𝑣 2 𝑔

14. Finding Friction Factor
Moody Chart:

15. Finding Friction Factor
Moody Chart, smooth cylindrical tubes:

16. Finding Friction Factor
Re ≤ 2100 (laminar flow)
𝑓= 16 𝑅𝑒

17. Problem a) determine the head loss across this section of pipe.
Consider the fully developed flow of glycerin at 40°𝐶 (𝜌=1252 𝑘𝑔 𝑚 3 , 𝜇=0.3073𝑃𝑎∙𝑠) through a 70-m-long, 4-cm-diameter, horizontal, circular pipe. If the flowrate through the pipe is 3.77 L/s:
a) determine the head loss across this section of pipe.
b) determine the pressure difference across this section of pipe

18. Non-Horizontal Pipes 𝑷 𝟏 𝑷 𝟐
𝑃 2 − 𝑃 1 𝜌𝑔 + 1 2𝑔 𝑣 2 2 − 𝑣 𝑧 2 − 𝑧 1 = h s − h L
𝑷 𝟏
𝑷 𝟐

19. Problem
If the pipe in the last problem is tilted at 20°𝐶, and the flowrate through remains the same:
a) determine the head loss across this section of pipe.
b) determine the pressure difference across this section of pipe

20. Finding Friction Factor
Re ≥ 3x103 (turbulent flow)
Smooth pipe
Prandtl-Karman:
1 𝑓 =4.0 log 𝑅𝑒 𝑓 −0.4

21. Finding Friction Factor
Re ≥ 5x103 , ≤ 5x107 (turbulent flow)
Explicit simplification of PK, pretty accurate
Colebrook-Holland:
𝑓= 3.6𝑙𝑜𝑔 𝑅𝑒 −2

22. Why Laminar v. Turbulent Difference?
Velocity profiles!

23. Problem a) determine the head loss across this section of pipe.
Consider the fully developed flow of glycerin at 40°𝐶 (𝜌=1252 𝑘𝑔 𝑚 3 , 𝜇=0.3073𝑃𝑎∙𝑠) through a 70-m-long, 4-cm-diameter, horizontal, circular pipe. If the flowrate through the pipe is 377 L/s:
a) determine the head loss across this section of pipe.
b) determine the pressure difference across this section of pipe