PROCESSING TECHNOLOGY 1

1. PROCESSING TECHNOLOGY 1 Fluid Dynamics 3 – Pipe/Duct Flow

2. Introduction In this unit we will study the flow of fluids within pipes and ducts by introducing the concept of laminar and turbulent flows together with their different velocity profiles.

3. Types Of Flow There are three fundamentally different types of flow that will occur in pipes.: Laminar flow (also called streamline) Turbulent flow Transitional flow (between laminar and turbulent)

4. Types Of Flow Laminar flow is a smooth well-ordered flow. Any flow may be laminar provided it occurs at low velocities or on a small scale.

5. Types Of Flow Turbulent flows are highly disturbed and occur with significant eddy formation. Many industrial scale flows are turbulent. Shear sensitive fluids such as yeast suspensions or wort are best transported in laminar flow to minimise shear damage.

6. Types Of Flow Transitional flow is flow, which has characteristics of both laminar and turbulent flows

7. Reynolds’ Experiment In the 1880’s Osborne Reynolds conducted a series of experiments where he used a dye injection system to introduce a fine filament of dye into a flowing water stream in a tube.

8. Reynolds’ Experiment In Moderate flow vels small random fluctuations. Higher flow vels - dye filament rapidly disperses into well mixed region Low flow velocities – Continuous, smooth dye filament.

9. Reynolds Number (Re), Reynolds number (Re), r = fluid density (kg/m3) u = mean fluid velocity (m/s) d = pipe internal diameter (m) m = fluid dynamic viscosity (kg/ms)

10. Reynolds Number (Re), • For flow within circular cross-section pipes • the following limiting Reynolds numbers • characterise the type of flow, • Re < 2100 laminar (streamline) flow2100 < Re < 4000 transitional flow • Re > 4000 turbulent flow.

11. Reynolds Number (Re), • Re < 2100 laminar (streamline) flow2100 < Re < 4000 transitional flow • Re > 4000 turbulent flow.

12. Non-circular Pipes/ducts – Turbulent Flow For non-circular pipes and ducts, the diameter used can be estimated using the *equivalent hydraulic diameter dh calculated as follows,

13. Non-circular Pipes/ducts – Turbulent Flow For a rectangular cross section duct of breadth b and height a, the equivalent hydraulic diameter is given by b Cross sectional area = ab Wetted perimeter = 2a + 2b a

14. Non-circular Pipes/ducts – Turbulent Flow For a rectangular cross section duct of breadth b and height a, the equivalent hydraulic diameter is given by

15. Equivalent hydraulic diameters dh for non-circular geometries.

16. Velocity u Pipe radius R u0 Radius r Fluid Velocity Profiles In Pipes/ducts

17. Velocity u Pipe radius R u0 Radius r Fluid Velocity Profiles In Pipes/ducts The velocity profile is symmetrical across the pipe centreline. The centreline velocity u0 is the maximum velocity, hence u0=umax. The velocity decreases as we move away from the centreline until it falls to zero at the pipe wall. i.e uR=0.

18. Velocity ur Diameter d Radius r u0 Laminar Pipe Flow The velocity profile of fully developed laminar flow takes the form of a parabola

19. Laminar Pipe Flow Mathematically this profile takes the form of, ur = velocity at radius r from the centreline u0 = velocity at r = 0 (the centreline) d = pipe diameter

20. Laminar Pipe Flow It can be shown that the mean velocity u is half of the maximum velocity umax i.e, u = 0.5u0 = 0.5umax Or rearranging for umax gives, umax = 2u

21. Velocity ur Diameter d Radius r u0 Turbulent Pipe Flow The velocity profile of fully developed turbulent flow takes the form of a 1/7th power law

22. Turbulent Pipe Flow Mathematically this profile takes the form of, ur = velocity at radius r from the centreline u0 = velocity at r = 0 (the centreline) d = pipe diameter. This expression is frequently termed Prandtl’s one-seventh power law.

23. Turbulent Pipe Flow Mean velocity u is related to maximum velocity umax by, u = 0.82u0 = 0.82umax Or rearranging for umax gives, umax = 1.22u