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Volume 1, Issue 1, Year 2012  Pages 6671
A Simple Algorithm to Relate Measured Surface Roughness to Equivalent Sandgrain Roughness
Thomas Adams, Christopher Grant RoseHulman Institute of Technology, Department of Mechanical Engineering 5500 Wabash Ave., Terre Haute, Indiana, USA adams1@rosehulman.edu; grantcl@rosehulman.edu
Heather Watson James Madison University, School of Engineering 801 Carrier Drive, Harrisonburg, Virginia, USA watsonhl@jmu.edu
Abstract One of the most important resources available in the field of fluid mechanics, the Moody Chart gives Darcy friction factor as a function of Reynolds number and relative roughness. The experimentalists who generated the data correlated in the Moody Chart, however, roughened pipe surfaces by coating their internal surfaces with a monolayer of sand, the pipe wall roughness being defined as the average diameter of the sand grains. Thus, the sandgrain roughness values required for use with the Moody Chart are not derived from any direct measure of roughness using modern surface characterization equipment, such as an optical profilometer. Using direct measurements of surface roughness in fluid flow calculations may therefore result in significant error. In this paper we present a simple algorithm with which various measured surface roughness parameters can be converted to equivalent sandgrain roughness. For nearly every surface roughness value converted to equivalent sandgrain roughness using the algorithm, better agreement with fluid flow experiments is seen over using the raw roughness value.
Keywords: Surface Roughness, SandGrain Roughness, Moody Chart, Darcy Friction Factor
© Copyright 2012 Authors  This is an Open Access article published under the Creative Commons Attribution License terms (http://creativecommons.org/licenses/by/2.0). Unrestricted use, distribution, and reproduction in any medium are permitted, provided the original work is properly cited.
1. Introduction The Moody
Chart (Moody, 1944) represents one of the most widely used resources in fluid
mechanics. Relating Darcy friction factor to Reynolds number and relative
roughness, the Moody Chart correlates extensive experimental data obtained
earlier by Nikuradse (1933), in which pipe surfaces
were roughened by coating their internal surfaces with a layer of sand. Pipe
wall roughness, ε, was thus
defined as the average diameter of a sand grain. Actual pipe surfaces, however,
do not have such a regular surface geometry as that created by a monolayer of
sand grains. Rather, they are replete with hills and valleys of various sizes
and shapes. Thus, direct measurements of surface roughness may not be
appropriate for fluid flow calculations and could lead to significant error.
And though profilometry techniques for measuring
surface roughness represent a mature technology, the algorithms employed for
calculating an average roughness do not coincide with the idea of sandgrain
roughness. Values of
roughness reported in tables in fluid mechanics texts and handbooks typically
reflect an "equivalent sandgrain" idea of roughness, having been
backcalculated by comparing pressure drop data from fluid flow experiments to the
Moody Chart. These equivalent sandgrain roughness values do not result from
any direct measurement of surface roughness using standard surface
characterization equipment and the various definitions of surface roughness
thereof. When encountered with a pipe surface for which no such data exist,
especially when utilizing new materials and/or fabrication techniques, one
usually measures surface roughness directly using any number of available profilometers. A link between measured surface roughness
and the sandgrain roughness required for friction factor purposes would
therefore be highly useful. A number of
researchers have recognized the shortcomings of using measured surface
roughness parameters in conjunction with the Moody Chart. The work of Kandlikar et al. (2005) was partly motivated by the very
large relative roughnesses (up to 14%) encountered in
microchannels. They replotted the Moody Chart using
the idea of a constricted flow diameter. Bahrami et
al. (2005) assumed pipe wall roughness to have a Gaussian distribution, and
found frictional resistance using the standard deviation in the roughness
profile. Pesacreta and Farshad
(2003) showed that measured peaktovalley roughness, R_{zd}, better represent
sandgrain roughness than the more common arithmetic average roughness, R_{a}. Taylor et al. (2005)
gives an excellent review of much of this work. 2. Proposed Roughness Conversion Algorithm A simple
solution to the problem or relating surface measurements to sand grain
roughness is to calculate the roughness of a hypothetical surface assuming it
to be made up of a uniform monolayer of samediameter spheres while employing
the same integration techniques as in standard profilometer
software. The resulting algebraic expressions can then be inverted and solved
for sphere diameter in terms of "measured" roughness. When applied to actual
surface roughness measurements, these expressions give an approximate value for
equivalent sandgrain roughness. 2. 1. Measured Surface Roughness Parameters A number of different parameters have
been defined to characterize the roughness of surface such as that illustrated
in Fig. 1. By far the most common is the arithmetic average of absolute values,
where y_{i} is
the distance from the average height of a profile (the mean line) for
measurement i,
and n is the number of measurements.
Two other parameters considered in the present work are the root mean squared
and the peaktovalley values:
In the
peaktovalley parameter R_{pi}
and R_{vi}
refer to the largest distances above and below the mean line for one of five
measurements, all of equal scan length in the xdirection. Fig. 1. A rough surface of arbitrary profile. 2. 2. Illustration of Conversion Algorithm Figure 2 gives
a schematic diagram of a single row of spheres of diameter ε on a flat surface as viewed from the side. For a scan in the
xdirection across the tops of the
spheres, the surface as seen by a profilometer would
appear as a uniform row of halfcircles (Fig. 3). In the limit as the number of
measurements goes to infinity, (1) becomes the integral
For the
profile in Fig. 3
and
Substituting
(5) and (6) into (4) and performing the integration gives
Solving (7)
for ε and simplifying gives
Equation (8) shows that if a profilometer were used to measure R_{a} for a surface consisting of a layer of spheres of
diameter ε, the resulting value
of R_{a} could be as much as
an order of magnitude smaller than sandgrain roughness appropriate for
friction factor calculations. Fig. 2. A row of uniform spheres on a flat surface Fig. 3. The surface as seen by a profilometer 3. Generalized Algorithm It is highly
unlikely that a profilometer measuring the roughness
of a surface comprised of a monolayer of uniform spheres would scan atop the
peak of each sphere. Therefore, a more sophisticated model in which integrals
of the type given in (4) can be performed for different scan directions was
developed using the software package MATLAB. Fig. 4 shows a model of
hexagonally packed spheres created in MATLAB for this purpose. Fig. 4. Model of hexagonally packed spheres created in MATLAB Using the
algorithm outlined in 2.2, the MATLAB model was used to relate R_{a}, R_{RMS}, and R_{zd} as given in (1)(3) to the diameter of the
spheres, ε. For each parameter
the MATLAB model averaged scans made in three directions: atop the peaks of
each sphere, over the points of contact between the spheres (the lowest point),
and midway between those two directions. The results are given in Table 1. Table 1. Estimated sandgrain roughness based on measured surface roughness parameters.
4. Experimental Validation In order to
validate the estimates of ε given
in Table 1, a A Zygo NewView 6300 interferometer
was used to measure R_{a}, R_{RMS}, and R_{zd} for pipes of
several different materials, including copper, aluminium, steel, and galvanized
steel. The values were then converted to their respective sandgrain roughness
estimates given in Table 1 and then compared to equivalent sandgrain roughness
values obtained from turbulent fluid flow experiments performed on the same
length of pipe. 4. 1. Fluid Flow Experiments Experimental
values for equivalent sandgrain roughness were obtained for the various pipes
via fluid flow experiments. In effect, measured values of head loss and flow
rate were used to calculate experimental values of friction factor and Reynolds
number, which in turn were used in the equation developed by Haaland (1983) to estimate equivalent sandgrain roughness:.
The Haaland equation was used rather than other curve fits for
the Moody Chart since the equation can be explicitly solved for ε. Figure 5
gives a schematic diagram of the flow apparatus itself along with the
physically measured parameters. The various pipe materials and dimensions are
given in Table 2, and typical values of the other measurands
are given in Table 3. Water at room temperature was used in all experiments. Fig. 5. Fluid flow experiment to measure equivalent sandgrain roughness. A measured volume of water discharged to the atmosphere along with time measurements yielded flow rates. Table 2. Pipe materials and dimensions.
Table 3. Typical measure and values for fluid flow experiments.
In order to reduce
the data to find equivalent sandgrain roughness, experimental values of
friction factor and Reynolds number were determined first. Equations (10) and
(11) give the familiar relations for head loss in terms of friction factor and
Reynolds number, respectively:
where f is friction factor, ϱ, is density, and μ is viscosity. In terms of the measurands given in Tables 2 and 3, (10) and (11) can be rearranged
to give the data reduction equations for f
and Re as
These values
were then used in the friction factor equation given in (9) and backsolved for
ε to give the experimental value
of sandgrain roughness:
4. 2. Results and Discussion Figure 6
gives a comparison of the various roughness values for the 2.05cm diameter
copper pipe obtained from the optical profilometer, the
estimated sandgrain roughness based on those values using the algorithm, and
the value of equivalent sandgrain roughness found from the fluid flow
experiments. Circles represent roughness values obtained from the profilometer, with filledin circles giving the
corresponding estimated sandgrain roughness using the algorithm. The shaded
grey region on either side of the measured sandgrain roughness indicates its
experimental uncertainty. The measured value of equivalent sandgrain roughness
from the fluid flow experiments is considered the true value. Table 4 gives the
corresponding numerical values. Standard uncertainty propagation techniques
were used to estimate all experimental uncertainties. Fig. 6. Comparison of measured surface roughness, estimated sandgrain roughness obtained via the algorithm, and equivalent sandgrain roughness obtained from fluid flow experiments for 2.05cm diameter copper pipe. Table 4. Comparison of various roughness values.
Figure 6 and Table 4 show that of
the measured roughness parameters, R_{a}
does the worst job of estimating equivalent sandgrain roughness and R_{zd} does
the best. Once the algorithm is applied to a surface measurement, R_{RMS} does the worst job of
estimating ε, whereas R_{zd}
again performs the best. The superior estimate of ε resulting from the use of R_{zd} is consistent with previous research.
By comparison, the expected value for equivalent sandgrain roughness for
copper as given in Binder (1973) is 1.5 μm. In all cases, however, it should
be noted that the converted roughness values always come closer to the true
value of sandgrain roughness than the raw measured values, with the converted
value of R_{zd}
falling within the range of experimental uncertainty of the true value. Figures 711 show the same
comparisons for the remaining pipe materials and dimensions. The figures appear
in order of increasing sandgrain roughness as obtained via experiment. Fig. 7. Comparison of roughness values for 1.44cm copper pipe. Fig. 8. Comparison of roughness values for 4.22cm aluminium pipe. Fig. 9. Comparison of roughness values for 1.55cm steel pipe. Fig. 10. Comparison of roughness values for 2.10cm galvanized steel pipe. Fig. 11. Comparison of roughness values for 1.57cm galvanized steel pipe. Figures 711 all show the same
trends as does Fig. 6 in terms of raw measured surface parameters; that is, R_{a} does the worst job of
estimating sandgrain roughness whereas R_{zd} does the best. Furthermore, the estimated
values of sandgrain roughness found by applying the algorithm to R_{a} and R_{RMS} always come closer to ε than do the raw values, with the converted R_{a} consistently outperforming
the converted R_{RMS}. Less consistent, however, are
trends in R_{zd}.
At lower values of sandgrain roughness R_{zd} under predicts ε, whereas at higher roughnesses R_{zd}
over predicts ε. As the
conversion factor for estimating ε
based on R_{zd}
is less than one, the converted R_{zd} value therefore does a better job of
estimating ε at low roughnesses than does the raw measured value, but a worse
job at high roughnesses. And at the two highest
sandgrain roughnesses tested, the converted R_{a} value actually slightly
outperforms both the measured and converted R_{zd} values. Also seen in Figs. 611 is that
the conversion algorithm generally does the best job of estimating ε at the lowest roughness values.
That is, the rougher the surface, the less benefit there is in applying the
algorithm. These trends can also be inferred from Fig. 12, in which a
comparison of all measured and converted roughness parameters is given for all
pipe materials and dimensions. Fig. 12. Comparison of normalized roughness values for all geometries and materials. The normalized roughness on the vertical axis is a given roughness parameter divided by equivalent sandgrain roughness, so that a perfect estimate of ε has a normalized roughness of one. It is seen, then, that when the
only available information regarding surface roughness consists of profilometerobtained values of R_{a} and/or R_{RMS},
the two most common surface roughness measurements, a correction should always be applied. This ensures a better
estimate sandgrain roughness over the raw value, the converted R_{a} value being the preferred
parameter. When R_{zd}
information is also available, however, the algorithm may be of less benefit.
This stems from both the correction factor for R_{zd}_{ }being
close to unity as well as the uncertain trends of R_{zd} over large ranges
of roughness. 5. Conclusion A simple
algorithm to convert various measured surface roughness parameters to
sandgrain roughness has been developed. The algorithm assumes that the relationship
between measured surface roughness parameters and sandgrain roughness can be
approximated by applying roughness parameter definitions to a hypothetical
surface consisting of a monolayer of spheres of uniform diameter. Based on
fluid flow experiments, arithmetic average, root mean square, and
peaktovalley surface roughness parameters almost always better approximate
equivalent sandgrain roughness after the algorithm has been applied. Acknowledgements The authors
wish to thank Drs. Scott Kirkpatrick and Azad Siahmakoun of the Department of Physics and Optical
Engineering at RoseHulman Institute of Technology
for their support in this work. Bahrami, M., Yovanovich, M. M., Cullham, J. R. (2005). Pressure drop of fullydeveloped laminar flow in rough microtubes, Third International Conference on Microchannels and Minichannels ICCMM 200575108. View Article Binder, R. C. (1973). "Fluid Mechanics" PrenticeHall. Farshad, F. F., Rieke, H. H., (2005). Technology innovation for determining surface roughness in pipes, Journal of Petroleum Technology, 57 (10), 8286. View Article Haaland, S. E. (1983). Simple and explicit formulas for the friction factor in turbulent flow, Journal of Fluids Engineering No. 103 (5), 8990. View Article Kandlikar, S. G., Schmitt, D., Carrano, A. L., Taylor, J. B. (2005). Characterization of surface roughness effects on pressure drop in singlephase flow in minichannels, Physics of Fluids, 17 (10). View Article Moody, L. F. (1944). Friction factors for pipe flow. Transactions ASME., 66, 671–683. View Article Nikuradse, J. (1937). Laws of flow in rough pipes, NACA Technical Memorandum 1292. View Article Pesacreta, C., Farshad, F. (2003). Coated Pipe Interior Surface Roughness as Measured by Three Scanning Probe Instruments, AntiCorros. Methods and Materials 50, 6. View Article Taylor, J. B., Carrano, A. L., Kandlikar, S. G., (2006). Characterization of the effect of surface roughness and texture on fluid flow: past, present, and future. International Journal of Thermal Sciences, 45, 962–968. View Article 
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