1. Viscous Flow in Pipes

2. PART I Introduction Laminar Pipe Flow

3. Introduction
Transport of fluid in a pipe system is extremely important in engineering practice.
Water pipes and distribution systems
Oil pipelines
Gas pipelines

4. Basic Components of a Pipe System1 2 3 5 4
Individual straight pipes
Pipe connectors
Flow rate control devices (valves)
Inlet and outlet
Pump or turbines that add energy or remove energy from the fluid
Items 2,3 and 4 are often called pipe components.

5. Bernoulli’s Equation - Revisit
For INVISCID, INCOMRESSIBLE, STEADY and IRROTATIONAL flows, we have
i.e.
(E6-1)

6. Real Pipe Flows For real pipe flows, we need to consider
the pressure drop in a pipe system due to the viscous nature of real fluid, , including
(1) pressure drop in straight pipes
i.e. major loss,
(2) pressure drop in pipe components
i.e. minor loss,
Energy input and output by pumps or turbines,

7. Real Pipe Flows – Energy Equation
The energy equation for a real pipe system:
where
These equations have significant implications to engineering design of a pipe system
(E6-2)
(E6-3)

9. We Need to Sufficient Knowledge On…
How to determine major loss,
How to determine minor loss,
Energy gain by devices such as pumps,
here
later
Last section
To determine this factors, we need to know the properties of real
flow in pipes, and this can be explain by Reynolds experiment

10. Pressure Drop Test
Figure 6-2 illustrates is a simple experimental setup for measuring pressure drop across a pipe. Liquid flew from the tank (by elevation energy) to the pipe. There would be a long section where the flow was not uniform, before the fluid entering the test section to produce a uniform flow. The test section has a length of Δx and the pressures at both ends were measured as P1 and P2.
A flow-regulating valve was introduced to control the flow rate so that tests could be conducted to find the correlation between the pressure gradient across the test section and the fluid flow rate in pipe. Figure 6-3 presents the original experimental results of Osborne Reynolds in 1883 of one specific fluid and one specific pipe [3]. Extensive experiments showed that these observations are generic for pipe flow, regardless of the type of liquid and kind of pipe used in such experiments.
Figure 3 measured pressure gradient (ΔP/ Δx) of a specific pipe as a function of volumetric flow rate Q of specific fluid (this data are extracted from Reynolds original experiment.

11. Pressure Drop Test
Fig 2
The simple but significant pressure-drop test done by Osborne Reynolds (1842 – 1912) in 1883.

12. III I II
Figure 3

13. Laminar, transitional and turbulent flow in pipe [1]
Depending on the flow rate Q, Figure 3 shows that the measured pressure gradient of the
test section may fall into one of the three regions:
Region I: At a low flow rate, the pressure gradient is proportional to flow rate;
Region II: At an intermediate flow rate, the flow behaviour seemed to be unpredictable so that the experimental data are scattered within the regions of the two curves which are extrapolated from those in Region I and III.
Region III: At a high flow rate, the pressure gradient is proportional to the flow rate to the 1.8 (for smooth pipe) or 2.0 power (for rough pipe); The questions are: why are there three regions in Figure 3? How should we interpret Figure 3 and obtain enough knowledge on pipe flows to guide the design of pipe systems?
Laminar, transitional and turbulent flow in pipe [1]
In 1883, Osborne Reynolds [3] did the pioneer work to understand Figure 6-3. Flow in the
three regions were visualised using a transparent pipe with a dye streak as a tracer. The flow
patterns of the three regions in Figure 3 are shown in Figure 4 while the time dependence
of fluid velocity at point A is shown in Figure Laminar flow: the dye streak remains a steady line as it flows through the pipe. All the
flow motion is in axial direction, there is no mixing perpendicular to the axis of the
pipe. This is observed at low flow rates, i.e. Region I in Figure 3.
2. Transitional flow: the dye streak fluctuates in time and space, and intermittent bursts
of irregular behaviour appear along the streak. The flow motion is not solely in axial
direction anymore. Some mixing perpendicular to the axis of the pipe may occur. This
is observed at higher flow rates, i.e. Region II in Figure 3.
3. Turbulent flow: the dye streak spreads across the entire pipe in a random fashion. The
flow motion is chaotic in all directions, causing rapid, crosswise mixing. This is
observed at high enough flow rates, i.e., Region III in Figure 3.

14. Real Pipe Flow
Figure 4

15. Real Pipe Flow
Figure 5

16. Real Pipe Flow Reynolds Number, Re For pipe flow, we have
Re < 2100, laminar flow, Region I in Figure 6-3
Re > 4000, turbulent flow, Region II in Figure 6-3
2100 <Re< 4000, transitional flow, Region III in Figure 6-3

17. Entrance Region Flow and Fully Developed Pipe FlowLe
We focus on fully developed pipe flow.

18. Example
Define the regime of the fluid?

22. 22

23. Laminar Pipe Flow - Force Balance

24. Laminar Pipe Flow - Force Balance
Pressure force:
Shear force at r:
No fluid acceleration, we have
i.e.
Newton’s 2nd Motion Law!!

25. Laminar Pipe Flow For a given pipe :
dP/dx is independent of r, shear stress is linearly distributed with r;
Shear stress is zero at pipe centre-line while reaches maximum at the wall;
The wall shear stress is given by

26. Laminar Pipe Flow – Velocity Profile
For laminar flow of a Newtonian fluid, we have
Recall that
Therefore
Integrate and apply boundary conditions: V(R) = 0

27. Laminar Pipe Flow – Velocity Profile
V(r)
D=2R
The velocity at wall is zero;
The maximum velocity is at the centre-line;
The velocity distribution is parabolic;
The pressure gradient is independent of the fluid density, but is proportional to fluid velocity and viscosity.

28. Laminar Pipe – Flow Rate

29. Laminar Pipe – Flow Rate
We have
Average velocity, or superficial velocity, can be calculated as

30. Summary laminar flow in straight pipe

31. Summary laminar flow in inclined pipeӨ Ө

32. pressure of oil in a pipe which discharges into the atmosphere is measured at a certain location. Assume laminar flow conditions. (a) The flow rates are to be determined? Proof the flow is laminar.
The pressure at the distance 15 m found 88 kPa
Properties The density and dynamic viscosity of oil are given to be  = 876 kg/m3 and  = 0.24 kg/ms.
(b) If the pipe is designed to produce uphill flow with an inclination of 8, find flow rates
(c) If the pipe is designed to produce down hill flow with an inclination of 8, find flow rates

33. (a)

37. Example
(kinematic viscosity)

39. Example 3

40. Exam question: A Newtonian fluid having a specific gravity of 0
Exam question: A Newtonian fluid having a specific gravity of 0.92 and a kinematic viscosity of ( ) flows past a fixed surface. The velocity profile near the surface is shown in Fig. 2. Determine the magnitude of the shear stress developed on the surface of the plate. (Hint the flow is Laminar)
Fig.2
(Last Exam)

42. (Last Semester Exam)