Essentials of fluid dynamics
2.6 Friction in pipe flow
It is the ratio of fluid momentum (arising from ‘inertia forces’) to viscous friction which determines whether the flow becomes smooth (i.e. laminar) or turbulent. This ratio is usually characterised by the non-dimensional Reynolds number
=uX
(2.11)
Here u is the mean speed of the flow, X is a nominated characteristic length of the system (in this case the diameter of the pipe) and (= /; as in (2.10)) is the kinematic viscosity of the fluid. The value of is important for characterising types of fluid flow, e.g. turbulence. For instance in pipes, flow will be usually turbulent if is larger than about 2300 2300. Criteria for laminar or turbulent flow in heat transfer are discussed in Chapter 3.
In turbulent flow, random local fluctuations of velocity in three dimen-sions are imposed on the mean flow. Thus small elements of fluid moving along the pipe also move rapidly inwards and outwards across the pipe, as illustrated by Figure 2.5(b). Since fluid does not slip at the pipe surface (Section 2.4), the mean speed near the surface is smaller than the average and the mean speed near the centre of the pipe is correspondingly larger.
Therefore the effect of the sideways motions of the fluid elements is to carry fluid of larger velocity outwards, and fluid of smaller velocity inwards. This transfer of momentum by elements of fluid is much larger than the corre-sponding transfer by molecular motions described in Section 2.4 because an element of fluid may move significantly across the pipe in a single jump.
In this case with water as the fluid, the mean free path of a molecule in the liquid is of the order of nanometres.
This transfer of momentum from the fluid to the walls constitutes a sizeable friction force opposing the motion of the fluid. Thus the presence of turbulence increases friction as compared with laminar flow; it is important to appreciate this characteristic of turbulent flow as, for instance, in the aerodynamics of wind turbines.
If the walls of the pipe are hotter than the incoming fluid, then these rapid inward and outward motions transfer heat rapidly to the bulk of the fluid. An element of cold fluid can jump from the centre of the pipe, pick up heat by conduction from the hot wall, and then carry it much more rapidly back into the centre of the pipe than could molecular conduction. Thus turbulence increases heat transfer, as discussed in more detail in Section 3.4.
and as applicable in the design of active solar systems.
2.6 Friction in pipe flow
Due to friction, useful energy and pressure are ‘lost’ or ‘dissipated’ when a fluid flows through pipes. Such factors may cause significant inefficiency in hydropower (Chapter 8), in ocean thermal energy conversion (OTEC,
36 Essentials of fluid dynamics
Chapter 14) and in all applications where heat is transferred by mass flow (e.g. Section 3.7 and, for solar energy, Section 5.5).
Let p be the pressure overcoming friction, as fluid moves at average speed u, through the pipe of length L and diameter D. Since the flow is statistically uniform along the pipe, each meter of pipe is considered to contribute the same friction. Therefore p increases with L. Since much of the resistance to the flow originates from the no-slip condition at the walls (Section 2.4), moving the walls closer to the bulk movement of the fluid increases the friction. Therefore p increases as D decreases. Equa-tion (2.9) implies that fluid fricEqua-tion increases with flow speed, so that p increases with u. Bernoulli’s equation (2.3) shows that the quantity u2/2 has the same dimensions as p (i.e. kg m s2−1). All these characteristics can be expressed in the single equations
p= 2fL/Du2= 4fL/Du2/2 (2.12) or
p= fL/Du2/2 (2.13)
Here f and f=4f are dimensionless pipe friction coefficients that change value with experimental conditions. Two equations are given because there are (unfortunately) alternative conventions for the definition of friction coefficient. As with many non-dimensional factors in engineering, the mag-nitudes of f and f characterise the physical conditions independently of the scale, depending only on the pattern of flow, i.e. the shape of the streamlines.
This is because the factor u2/2 in (2.12) and (2.13) represents a natural unit of pressure drop in the pipe. The friction coefficient is the proportion of the kinetic energy u2/2 entering unit area of the pipe that has to be applied as external work p to overcome frictional forces. This will depend on the time that a typical fluid element spends in contact with the pipe wall, expressed as a proportion of the time the element takes to move a unit length along the pipe. This proportion is much larger for the turbulent paths (Figure 2.5(b)) than for the laminar paths (Figure 2.5(a)).
Fluid flow depends mainly on the dimensionless Reynolds number of (2.11). A plot of f or f against should give a single curve applying to pipes of any length and diameter, carrying any fluid at any speed. There is no particular reason why this curve should be a straight line, or even continuous. Indeed we might expect a discontinuity at ≥ 2000, where the flow pattern changes from laminar to turbulent. This curve, shown in Figure 2.6, does indeed have a discontinuity at ≈ 2000. If we consider real pipes with rough walls, it is reasonable to suppose that the flow pattern depends on the ratio of the height, , of the surface bumps to the diameter
2.6 Friction in pipe flow 37
Figure 2.6 Friction coefficient f for pipe flow (see (2.12)).
Table 2.1 Approximate pipe roughness
Material (mm)
PVC 0 (‘smooth’)
Asbestos cement 0.012
New steel 0.1
‘Smooth’ concrete 0.4
of the pipe, D. Plotting the experimental data on this basis, Figure 2.6, we obtain a series of curves, with one curve for each roughness ratio /D.
Provided the appropriate value of is used, these curves give a reasonable estimate of pipe friction. Typical values of are given in Table 2.1, but it should be realised that the roughness of a pipe tends to increase with age and, very noticeably, with accretion of sediments and encrustations. For example, this presents a significant maintenance factor for hydropower installations.
Example 2.1
What is the pressure head required to force 010 m3s−1 of water at 20C, through a concrete pipe of length 200 m and diameter 0.30 m?
Solution
The mean speed is u= Q
A = 01 m3s−1
015 m2 = 14 m s−1
38 Essentials of fluid dynamics
From (2.11), the Reynolds number
=uD
v =14 m s−103 m
10× 10−6m2s−1 = 04 × 106≥ 2000 Therefore flow is turbulent.
For concrete (from Table 2.1), = 04 mm. Thus the ratio
D= 04 mm
300 mm= 00013
For this and /D, Figure 2.6 gives f= 00050
and
f= 0020
Expressing (2.12) in terms of the head loss due to friction,
Hf= p
g =2fLu2
Dg (2.14)
Hence
Hf=250× 10−3200 m14 m s−12
03 m98 m s−2
= 13 m
Figure 2.6 shows only one curve for < 2000, indicating that the friction coefficient is independent of pipe roughness in this range. This is because the flow is laminar, and the bumps hardly disturb the smoothness of the flow. In this laminar case, the pressure drop p can be explicitly calcu-lated from (2.9) for viscous shear stress, as indicated in Problem 2.4. The corresponding expression for the friction coefficient is
f=16v
uD laminar (2.15)
or
f= 4f =64
uD laminar (2.15a)
2.7 Lift and drag forces: fluid and turbine machinery 39
2.7 Lift and drag forces: fluid and turbine