1. Friction Losses Flow through Conduits Incompressible Flow

2. Goals Calculate frictional losses for laminar and turbulent flow through circular and non-circular pipes Define the friction factor in terms of flow properties Calculate the friction factor for laminar and turbulent flow Define and calculate the Reynolds number for different flow situations Derive the Hagen-Poiseuille equation

3. Shear Stress When the lower plate is in motion, a force F is required to maintain the velocity V. This is because viscous forces in the fluid resists the deformation arising from the change in velocity with respect to y.

4. Shear Stress This give us Newton’s Law of viscosity: γ is the strain rate τ is a force per unit area

5. Flow Through Circular Conduits Consider the steady flow of a fluid of constant density in fully developed flow through a horizontal pipe and visualize a disk of fluid of radius r and length dL moving as a free body. Since the fluid posses a viscosity, a shear force opposing the flow will exist at the edge of the disk

6. Balances Mass Balance → Momentum Balance

7. Momentum Balance (contd) If we imagine that the fluid disk extends to the wall, F w is just due to the shear stress  w acting over the length of the disk. Equating and solving for  p over a length of pipe L.

8. Mechanical Energy Balance

9. Viscous Dissipation (Frictional Loss) Equation Combining the Momentum and MEB results: Applies to laminar or turbulent flow Good for Newtonian or Non-Newtonian fluids Only good for friction losses as result of wall shear. Not proper for fittings, expansions, etc.

10. The Friction Factor  w is not conveniently determined so the dimensionless friction factor is introduced into the equations.

11. Friction Factor Increases with length Decreases with diameter Only need L, D, V and f to get friction loss Valid for both laminar and turbulent flow Valid for Newtonian and Non-Newtonian fluids –

12. Calculation of f for Laminar Flow First we need the velocity profile for laminar flow in a pipe. We’ll rely on BSL for that result. Recall our earlier result:

13. Laminar Flow Find Bulk Velocity (measurable quantity).

14. Laminar Flow ←Laminar Flow ←Newtonian Fluid

15. Hagen-Poiseuille Recall again: Use: Measurement of viscosity by measuring  p and q through a tube of known D and L.

16. Turbulent Flow When flow is turbulent, the viscous dissipation effects cannot be derived explicitly as in laminar flow, but the following relation is still valid. The problem is that we can’t write a closed form solution for the friction factor f. Must use correlations based on experimental data.

17. Friction Factor Turbulent Flow For turbulent flow f = f( Re, k/D ) where k is the roughness of the pipe wall. Note, roughness is not dimensionless. Here, the roughness is reported in inches. MSH gives values in feet. Material Roughness, k inches Cast Iron0.01 Galvanized Steel0.006 Commercial Steel Wrought Iron 0.0018 Drawn Tubing0.00006

18. How Does k/D Affect f?

19. Friction Factor Turbulent Flow As and alternative to Moody Chart use Churchill’s correlation:

20. Friction Factor Turbulent Flow A less accurate but sometimes useful correlation for estimates is the Colebrook equation. It is independent of velocity or flow rate, instead depending on a combined dimensionless quantity

21. Flow Through Non-Circular Conduits Rather than resort to deriving new correlations for the friction factor, an approximation is developed for an ‘equivalent’ diameter D eq with which to calculate the Reynolds number and the friction factor. where: R H = hydraulic radius S= cross-sectional area L p = wetted perimeter Note: Do not use Deq to calculate cross-sectional area or for laminar flow situations.

22. Examples Circular Pipe How About Concentric Pipes? Rectangular Ducts

23. Example Water flows at a rate of 600 gal/min through 400 feet of 5 in. diameter cast-iron pipe. Find the average (bulk) velocity and the pressure drop.

24. I. Unknown Driving Force 1.Calculate Reynolds Number 2.Calculate k /D 3.Determine f from Moody chart or correlation 4.Find h f 5.Calculate necessary driving force W,  p,  z

25. II. Unknown Flow Rate Known: DF,  D, L, k Unknown: Q(V) First cancel unknown Q with: 1.Calculate fRe 2. 2.Use the Colebrook equation to get f. This assumes flow is turbulent. 3.Get Re from results of step 1 and 2 4.Use Moody diagram with Re and k/D to get f 5.Repeat until f doesn’t change 6.Calculate Q=  D  Re / 4 

26. III. Unknown Diameter Known: DF, Q(V), , L, k Unknown: D 1.Calculate f Re 5 from known parameters 2.Guess f =0.005 3.Calculate Re from value obtained in step 1 4.Get D from Re 5.Calculate k/D 6.Use Moody chart to get f and compare to that assumed 7.Repeat steps 3 through 7 until f agrees.