1 Fluid dynamic principles
1.5 Pressure losses
1.5.2 Influence of roughness on friction losses
1.5 Pressure losses 21
grain diameter, since gaps were present between the individual grains. The rough- ness coefficients thus determined apply to this specific surface structure only, which is called “sand roughness” ε (frequently called ks). The sand roughness is described by the grain diameter ds (ε≡ ks = ds).
In the transition domain between hydraulically smooth and hydraulically rough, this type of regular roughness leads to a minimum in the curve λR = (Re) brought about by the fact that the varnish-coated grains do not generate an appreciable form drag when their tops protrude only slightly from the laminar sub-layer.
While the sand roughness possesses a uniform surface structure, technical (i.e.
ground, cast or machined) surfaces have an irregular roughness which can be characterized, for example, by the maximum roughness depth εmax (“technical roughness”). As the Reynolds number increases, the boundary layer thickness is reduced and more roughness peaks gradually rise through the laminar sub-layer.
With such surfaces, the friction coefficients continuously fall from hydraulically smooth to the fully rough domain. This behavior is found in Fig. 1.13 and in the graph in Table 1.6. Technically rough surfaces do not exhibit a minimum in the function λR = (Re) in the transition region, as it is observed with regular surface structures.
The roughness depth εmax according to Fig. 1.14 is obtained from the envelope of all roughness peaks and troughs. Since the measurement of the effective rough- ness of technical surfaces is quite involved, surface standards (e.g. “Rugotest”) are frequently used in practice. The roughness is determined by touching the surface and comparing with roughness specimens (plates with different roughnesses). The roughness is then defined by the classes N1, N2, N3 etc. The roughness increases by a factor of two from one class to the next. The measured roughness is charac- terized as arithmetic mean roughness εa (corresponding to CLA = center line aver- age or AA = arithmetic average); εa is defined by (Fig. 1.14):
dx L y 1L a= 0³
ε (1.36a)
The relationship between the maximum roughness depth εmax and the arithmetic mean roughness εa is typically εmax = (5 to 7)×εa. On average we can expect ap- proximately: εmax = 6×εa, [1.12].
x
y εmax
Fig. 1.14. Definition of the maximum roughness depth εmax
If the friction coefficient λR is determined from a pressure loss measurement in a pipe with a given roughness depth εmax, the equivalent sand roughness ε can be
1.5 Pressure losses 23
calculated by means of Eq. (1.36). From this analysis an “equivalence factor” ceq
is obtained as the ratio εmax/ε :
ε
≡εmax
ceq (1.36b)
Dividing the maximum roughness depth εmax of any given surface by the equivalence factor, we get that equivalent sand roughness which leads to the same friction coefficient as the measurements which were used as a basis for Eq. (1.36).
The equivalence factor depends on the structure of the roughness – i.e. the ma- chining process – and the orientation of the finishing marks relative to the flow di- rection; ceq can therefore vary within wide boundaries. Of particular influence in this regard is the number of roughness peaks per unit area as shown in Fig. 1.15.
0 2 4 6 8 10
0 5 10 Pitch/heigt: 15t/εmax
ceq
[35] Spheres [27] Ground surface [27] Sand
[35] Calottes [35] Cones [35] Short angles
Fig. 1.15. Equivalence factor ceq as function of the roughness density
To determine the friction coefficient cf or λR of a component with the rough- ness εa or εmax, the equivalent sand roughness ε must be calculated from Eq. (1.36c):
eq max c
=ε
ε or with εmax = 6 εa from
eq a c 6ε
=
ε (1.36c)
With that calculated equivalent sand roughness, use Eq. (1.33) and Fig. 1.13 or Eq. (1.36) or Table 1.6 to determine the friction coefficient.
Scanty information for ceq is available in the literature. Table D1.2 provides some data, see also [1.12].
Table D1.2 Roughness equivalence factors ceq ceq
Manufacturing marks perpendicular to the flow direction 2.6 Manufacturing marks parallel to the flow direction 5
Drawn metal tubes 2 to 2.6
Smooth coating (e.g. paint) 0.63
Table D1.3 shows the roughness values εa of the surface standards, the maxi- mum roughness values εmax obtained with Eq. (1.36c) and the equivalent sand roughness ε (εa is the upper limit of the respective roughness class; the values in Table D1.3 are rounded).
Table D1.3 Roughness classes
Arithmetic average roughness
Maximum roughness (Fig. 1.14)
Equivalent sand roughness Roughness
class
εa (μm) εmax (μm) ε (μm)
N5 0.4 2.4 1
N6 0.8 4.8 2
N7 1.6 9.6 4
N8 3.2 19 8
N9 6.3 38 16
N10 12.5 75 32
N11 25 150 64
N12 50 300 128
N13 100 600 256
The diagrams for resistance coefficients with sand roughness and technical roughness differ only in the domain of transition from smooth to rough; therefore, they yield identical resistance coefficients in the fully rough region for any given value of ε/d. In order to determine the friction coefficient from Eqs. (1.33) or (1.36), we must use that “technical roughness” ε which leads to the same resis- tance coefficient as the equivalent sand roughness. Some examples are listed in Table D1.4. These values originate from test results which can be found in the lit- erature, for instance in [1.5, 1.6, 1.11].
One of the main uncertainties when calculating the friction losses of turbulent flows lies in determining the physically relevant roughness: if the roughness is as- sessed incorrectly by the factor of 2, the uncertainty of the loss calculation is ap- proximately 15 to 35%. The impact of the roughness on pump efficiencies is cov- ered in Chap. 3.10.
Table D1.4 Equivalent sand roughness ε ε (mm)
Glass, coatings, plastic, drawn metal tubes, polished surfaces 0.001 to 0.002
Drawn steel pipes, new 0.02 to 0.1
Drawn steel pipes, lightly rusted 0.15 to 1
Steel pipes, severely rusted or with deposits 1 to 3
Cast iron pipes and pump components 0.3 to 1
Concrete pipes 1 to 3
1.5 Pressure losses 25
The flow resistance is greatly increased through vortex shedding caused by regular roughness patterns which are perpendicular to the direction of flow. This type of roughness can be generated, for example, by magnetite deposits in boilers where it creates a marked increase of pressure loss (“ripple roughness”). The phe- nomenon has also been observed in pipelines with ripple-type deposits.