estimation of friction function
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E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
doi:10.3850/38WC092019-0188
4309
Estimation of Friction Function
BALKRISHNA SHANKAR CHAVAN(1), (1) Reservoir and Appurtenant Structures,
Central Water and Power Research Station,Pune-411046, Maharashtra, India
balkrishna0@gmail.com
ABSTRACT Estimating Friction function Bs is complex because it is sensitive to changes in size, shape, spacing
and spacing pattern that is It is sensitive to conveyance and shape factors. Many researchers such as
Colebrook- White, Afzal, Yalin and Garcia improved its estimation. Various parameters of fluid flow and
geometrical section are combined to describe the friction function. Estimation of friction loss in a flow is
important step in finalizing discharge passing through a water conductor system. surface roughness is
present, there is no universality of scaling of the friction factor with respect to the traditional Reynolds
number Re, and different expressions are evolved. Friction factor important role in the computation of
discharge, deciding geometry of the flow section. In the past attempts were made to express Friction
function Bs as a function of Reynolds number. Re can be estimated with empirical equation. Because
of its complexity so far published literature is not enough to arrive at generalized equation for the
computation of friction function accurately. There is need to analyze and develop empirical equations
for friction function. In this paper attempt has been made to analyze 878 published data collected from
various sources such as various project sites globally(123 observations). and from laboratory (755
observations). Data collected include geometrical parameters of flow i.e. width, depth and longitudinal
slope. Other parameters are grain size, roughness, shear velocity, discharge and fluid characteristics
such as density, viscosity and specific weight. Present works include; the analysis of data for the
computation of Reynolds number for the estimation of friction function with respect to depth of flow and
grain roughness. Mathematical simple equations are proposed obtained from plots were based on the
analysis of data for depth of flow and roughness.
Keywords: Empirical, Friction function, Friction factor Roughness Reynolds number,
1 INTRODUCTION
Friction function Bs term originates in a law of logarithmic distribution for velocity first given by Karman
Prandtl. Friction function Bs is complex because it is sensitive tohydraulic parameters and bed-material
particles, channel gradient in natural stable channels, depth of flow, suspended material, and the bed
level. Naturally occurring flow and its interaction within fluid and with flow boundaries result in two
unavoidable problems. One is that there are a great variety of semi-empirical and semi- theoretical or
full-empirical friction formulae based on the same, similar or different theories and data. Comparisons
of these formulae through theoretical analysis, and with laboratory and field data leads to interesting
conclusions and information. Methods for finding the friction factor f was using a diagram, such as
the Moody chart, or solving implicit equations such as the ColebrookโWhite equation upon which the
Moody chart is based, or explicit equations proposed by various investigators as given in Table 1 in
Annexure A.
The ASCE Task Force on Friction Factors in Open Channels (1963) highlighted usefulness of Darcy Weisbach formulation of friction factor in computing resistance to flow in open channels. Task Force noted formulation as fundamental and mentioned that the Manning equation could be used for fully rough conditions. Task Force presented figure for variation of resistance with Reynolds number, which showed with Mannings equation, there is continual decay of resistance with Reynolds number, even in the limit of large values. From this, one could deduce that Mannings equation is fundamentally flawed. The recommendations of the Task Force have almost entirely been ignored, and the Gauckler-Manning-Strickler formulation continued to dominate, even though, with the exception of the Strickler formula,
E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
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there are few general research results available and use of tables/graphs continued. Even though the Task Force presented a number of experimental and analytical results, there was no simple solution path to follow for friction factor problems.
2. REVIEW OF LITERATURE:
Law of logarithmic distribution for velocity first given by Karman - Prandtl - velocity distribution.๐ข
๐ขโ=
1
๐ ln (๐ด๐
๐ง
๐๐ ) + ๐ต๐ (1)
In a typical engineering applications, there will be a set of given or known quantities. The acceleration of gravity g and the dynamic/kinematic viscosity of the fluid ฮผ ,ฮฝ are known, as are the hydraulic radius of the R and its roughness height ฮต.
1
โ๐= ๐ด๐๐๐
๐
๐ฆโฒ + ๐ต๐
(2) The constant in resistance equation is due to the numerical integration, and is a function of shape of the cross-section. For circular section A = 2.0, B = -0.62 For rectangular section: A = 2, B = -0.79 (for large ratio of width/depth.
The Moody diagram proposed in 1940 is still relevant especially at remote sites where limited computing facility is available. A simplification and a rough approximation: the conditions it describes (fully developed, isothermal, incompressible, dissipative, pseudo-steady-state flow) are never quite attained in practice. Conditions within pipelines are inherently non-uniform over the flow cross section and over a given length. Fluids are more or less compressible, rather than incompressible, and have finite thermal conductivity. The flowing fluid within pipelines is not isothermal in the radial direction because of the very dissipation that is quantified by the Darcy-Weisbach equation. Frictional dissipation and finite thermal conductivity together give rise to differences in temperatureโand density and viscosityโbetween different parts of the flow. Inevitably too, any length of pipe for which the fully-developed flow requirement is approximated must be connected to entry and exit systems within which this is not the case. When the flow is turbulent, vortices form and collapse relentlessly over time. In the flow of liquids, vapour pressure may play a role. Nevertheless, the diagram is a very useful design tool. For computational purposes, it is easily represented as a data set of discrete points, or by the equations that define it along with appropriate solution algorithms. The same four zones that were labeled by Moody (laminar, critical, transition and complete turbulence) are shown.
Figure 1 Moody Diagram
E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
4311
Yalin (1992) suggested equation
1
โ๐= ๐ด๐๐๐
๐
๐ฆโฒ + ๐ต๐ (3)
It follows that As and Bs are related by
๐ด๐ = ๐๐ ๐ต๐ (4)
If as well the head loss per unit length S is a known quantity, then the friction factor f can be calculated directly from the chosen fitting function. Solving the DarcyโWeisbach equation:
Head loss due to channel friction, calculated with the help of Darcy Weisbach
Head loss = โ๐ = ๐๐ฟ๐2
2๐๐ท
(5)
3.Semi-Empirical Equations
3.1Colebrook and White
The smooth, rough and transition laws for Nikuradse's sand roughness in which the grains are of uniform size and closely packed together do not resemble with roughness of real pipes. Experimental results of Nikuradse considerably differ in the region of the transition curve for a pipe having a roughness
composed of isolated particles. Semi-Empirical Equation suggested by Colebrook and White overcome
variations in transition region is: 1
โ๐= 1.74 โ 2๐๐๐ (
๐
๐0+
18.7
๐ ๐โ๐) (6)
when Re is small, it is consistent with smooth pipe flow, when Re is large, it is consistent with rough pipe flow. However its performance in the transitional domain overestimates the friction factor by a substantial margin.[12] Colebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes. Indeed, such pipes are very different from those carefully prepared by Nikuradse: their surfaces are characterized by many different roughness heights and random spatial distribution of roughness points, while those of Nikuradse have surfaces with uniform roughness height, with the points extremely closely packed.
Estimation of friction factor Implicit equation for estimation of friction factor was given by Colbrook in 1939
1
โ ๐= ๐๐๐ [
๐
๐ท
3.7+
2.51
๐ ๐โ๐] (7)
๐ ๐ =๐๐ท๐
๐ (8)
Where ๐ = Friction factor
Re = Reynolds Number given by
ฮผ = Coefficient of Dynamic viscosity
Alternative equations to Equation 7 for the friction factor of smooth pipes have been proposed and there is some evidence to suggest that the smooth pipe friction factor curve should be a little higher at high Reynolds numbers. However, corroborating data are still scarce. Also, where friction factor has been measured at very high Reynolds numbers the fluid has been a gas. Equation (7) is implicit in nature and needs time consuming iterative steps for its solution.
3.2Two power explicit equation given by the author Chavan (2017) is simple and convenient:
E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
4312
๐ =0.0075(
(4000)
๐ ๐
0.05)
0.25+(4000
๐ ๐)
5 +20
๐ ๐ (9)
Many investigators (some of them are listed in Annexure A) proposed approximate solution of the Colbrook-White equation in a form of explicit equation. Plots given by various researchers is shown in Figure 2.
Figure 2 โFriction factor plots given by various researchers
3.3 Garcia Friction function Bs given by Garcia as a function of Roughnes (friction) Reynolds number Re* as
following empirical equation:
๐ต๐ = 8.5 + [2.5ln (๐ ๐โ) โ 3]๐โ0.121[ln(๐ ๐โ)]2.42
(10) Where Roughness Reynold Number Re* is
Reโ =uโ ks
ฮฝ (11)
In above expression friction function is expressed in only in terms of roughness Reynolds number, no
effect or consideration for ๐
๐ท for higher values of Reynolds number which is logical. Plot is shown in
Figure 3
E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
4312
Figure 3 Plot of Friction function Bs and Re* edited by Garcia
above graph exhibit following features
when ks = 0, then Re* is identically zero: flow is in the smooth regime. The data for thesepoints lie to the left extreme of the abscissa and are not within the frame of the graph.
When Re* < 5, the data lie on the line Bs = Re*; flow is in the smooth regime.
When Re* > 100, the data asymptotically approach a horizontal line; they are independentof Re, f and ks/D
While the ColebrookโWhite relation is, in the general case, an iterative method needs more computing time. In the filed it is very convenient to use two power explicit equation
Roughness Reynolds Number R* is defined by
Rโ =1
โ8. ๐ ๐ โ๐
โ
๐ท (12)
Friction(Roughness) Function Bs is defined by Afzal as
๐ต๐ =1
9.3โ๐+ ๐ฟ๐๐ (
1.9
โ8
โ
๐ท) (13)
E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
4313
Figure-4. Plot of Roughness Reynolds Number and Friction Function given by Afzal.
From the above graph exhibit following features:
When ฮต = 0, then Re* is identically zero: flow is always in the
smooth pipe regime. The data for these points lie to the left extreme of the abscissa and arenot within the frame of the graph.
When Re* < 5, the data lie on the line B(R) = Re*; flow is in the smooth pipe regime.
When Re* > 100, the data asymptotically approach a horizontal line; they are independentof Re, f and ฮต/D
The intermediate range of 5 < Re* < 100 constitutes a transition from one behavior to theother. The data depart from the line B(R) = R very slowly, reach a maximum near Re* = 10,then fall to a constant value.
A fit to these data in the transition from smooth to rough flow employs an exponential expression in Re* that ensures proper behavior for 1 < Re* < 63 (the transition from the smooth regime to the rough regime) function shares the same values for its term in common with the KรกrmรกnโPrandtl resistance equation, plus one parameter 0.34 to fit the asymptotic behavior for Re* โ โ along with one further parameter, 11, to govern the transition from smooth to rough flow. It is exhibited in Figure 3.
4. Data Collection:878 runs of basic flow parameters such as width, slope, discharge, depth of flow and sediment size offlumes, channel and rivers collected by different investigators has been compiled and analysed in thispaper. Experimental data from laboratory flumes, irrigation channels and Natural Rivers were collectedfrom publication of following sources.
a. Laboratory Flumes : USGS(1966), Bharat Singh(1971), J. R. Barton & P. N. Lin(1955), P. K.
Pande(1960), G. P. Williams(1970), E. M. Laursen(1957), E. O. J. Plate (1957) and H. P. (1996)
b. Irrigation Channels : U S channels (1970), Japanese channels (1978) and Pakistan channels
(1931)
c. Natural Rivers: Amezon, Rio Orinoco, Luznic, Ishikari, Skive-Karup and Tigus rivers.
Above data is analyzed for velocity, friction factor and roughness Reynolds, is plotted to obtain
proposed equation is shown in Figure 5
E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
4314
Figure-5. Plot of Roughness Reynolds Number and Friction Factor.
.5.0 Equatons obtained by analysis of data for flume data and field data are :
5.1 For Flume data 1
โ๐= โ0.57๐๐(๐ ๐) + 9.379 (14)
5.2 For Field data
1
โ๐= โ0.59๐๐(๐ ๐) + 8.011 (15)
uncertainty of collected data (Chien and Wan, 1999) (although laboratory-collected data has a relatively high accuracy, the scale problem has limited its extensive use). For example, in order to collect field data (in natural river course) to validate hydraulic parameter formulae, flow and sediment transport conditions are assumed to satisfy โuniform flow and equilibrium sediment transportโ, which is rarely the case in practice. Another inevitable problem for the collected data is โit is one dimensionalityโ. When these formulae are used in two-dimensionally, numerically modeling of sediment transport and bed deformation in natural rivers, the results are seldom expected.
Thus, it is important to examine these formulaeโs performance when Data mentioned above was plotted
for Roughness Reynolds Number Vs Friction factor Bs as shown in Figure 6.
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
1.00 10.00 100.00 1000.00 10000.00 100000.00
Roughness Re*
Variation of friction with Roughness Reynolds no
Re1/Sqrtffield
Re1/sqrtfflume
Log. (Re1/Sqrtffield)
Log. (Re1/sqrtfflume)
1
โ๐= โ0.57๐๐(๐ ๐) + 9.379
1
โ๐= โ0.59๐๐(๐ ๐) + 8.011
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September 1-6, 2019, Panama City, Panama
4315
Figure 6 Plot of Roughness Reynolds Number and Friction Function for data in Figure 5
Plot of Friction function Bs vs. roughness Reynolds number Re, the data fall on a single
trajectory when plotted.. The regime Re < 1 is effectively that of smooth pipe flow. For large Re,
the friction function Bs approaches a constant value. of 8.5
5. 3 Proposed equation for Bs
๐ต๐ = 8.5 + [2.6ln (๐ ๐โ) โ 3]๐โ0.12[ln(๐ ๐โ)]2.4 (16)
From the above plot exhibit following features:
When ฮต = 0, then Re* is identically zero: flow is always in the smooth pipe regime. The datafor these points lie to the left extreme of the abscissa and are not within the frame of thegraph.
When Re* < 7, the data lie on the line Bs = Re*; flow is in the smooth pipe regime.
When Re* > 30, the data asymptotically approach a horizontal line; they are independentof Re, f and ฮต/D
The intermediate range of 7 < Re* < 30 constitutes a transition from one behavior to the other.The data depart from the line Bs = Re*; very slowly, reach a maximum near Re* = 11, thenfall to a constant value.
Limitations of studies- As naturally flowing water has limited range of hydraulic parameters,
data collected has limited range of Reynolds number. There is need of additional data at
higher Roughness Reynolds number to refine the result of studies
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
1.00 10.00 100.00 1000.00 10000.00 100000.00
Bs
Roughness Reynolds number=Re* = u*ks/v
Bs Curve
ObservedValue
HPFLUME
E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama
4316
Figure-7. Plot of Roughness Reynolds Number Calculated and Observed.
6. CONCLUSIONS:
1 The proposed equation (16) is logarithmic exponential equation, explicit in nature. Friction function varies directly with log Reโ and inversely with exponent of ReโRoughness Reynolds Number.
2. The proposed equation (14) and (15) are logarithmic equations, simple explicit in nature.
Friction factor varies inversely with square of Roughness Reynolds Number. Equation (15) and
(16) would be of help in simulation of complicated water conductor networks in predicting flow
parameters.
3. Equation (14) give values of friction factor for laboratory flume lower than Field values
predicted by equation (15)
4. Figure 7 shows 25% and 50% envelop curve for Roughness Reynolds Number. It is seen that
99.66% of data is covered in 50% envelop, while 72 % data is covered in 25 % envelop.
5. Data collected has limited range. There is need of additional data at higher Roughness
Reynolds number to refine the result of studies.
6. Further scope to improve with additional parameters as pointed out by Hunter Rouse
e.g.Froude Number, porous bed & banks which affects friction factor after getting additional
data
7. REFERENCES
Afzal Noor 2007 Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow Journal of Fluids Transactions of the ASME Vol. 129, October 2007. Pp 1255-1267.
ASCE Task Force Report, 1963 Jour. Hydraulics Division, Amer. Soc. of Civ. Engs., vol. 89, no HY2, March,pp 97-143.
Bawdy, D.R., 1979, Flood frequency estimates on alluvial fans: American Society of Civil Engineers, Journal of Hydraulics Division, HY11, v. 105, p. 1407-1413.
Bobbie, C.H., and Wolf, P.O., 1953, The Lynmouth flood of August 1952:London, Proceedings of the Institution of Civil Engineers, v. 2, Part 3,Dec. 1953, pp. 522-588.
0
5000
10000
15000
20000
25000
30000
0 5000 10000 15000 20000 25000 30000
Calculated Vs Observed
50% Lower envelop
50% Upper envelop
25% envelop
25% Envelop
Series6
ReFrlaboratory
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September 1-6, 2019, Panama City, Panama
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Colebrook, C. F. (1939) Turbulent Flow in Pipes with particular reference to Transition Regions between Smooth and Rough Pipe Laws, Journal of Institution of Civil Engineers, Paper No. 5204 pp 133-156 + tables
Chavan Balkrishna Shankar (2017), Unified Equation for Estimation of Friction Factor, Proceedings of 37th IAHR World Congress, Aug 13-18, held at Kuala Lumpur, Malaysia Pages 5885-5893
Chow, V.T., 1959, Open Channel Hydraulics, New York, McGraw-Hill, 680 p.
Costa, J.E., and Jarrett, R.D., 1981, Debris flows in small mountain stream channels of Colorado and their hydrologicimplications: Bulletin of the Association of Engineering Geologists, v. 18, no. 3, p. 309-322.
Cowan, W.L.,1956, Estimating hydraulic roughness coefficients: Agricultural Engineering, v. 37, no. 7, p. 473-475.Davidian, Jacob, 1984, Computation of water-surface profiles in open channels: U.S. Geological Survey Techniques of Water-Resources Investigations, Book 3, Chapter A15, 48 p.
Gol'din, B.M., and Lyubashevskiy, L.S., 1966, Computation of velocity of mudflows for Crimean Rivers: Soviet Hydrology, v. 5, no. 2, p. 179-181.
Golubtsov, V.V., 1969, Hydraulic resistance and formula for computing the average flow velocity of mountain rivers: Soviet Hydrology, v. 8, no. 5, p. 500-510.
Hejl, H.R., Jr., 1977, A method for adjusting values of Manning's roughness coefficients for flooded urban areas: U.S. Geological Survey, Journal of Research, v. 5, no. 5, p. 541-545.
Herbich, J.B., and Shulits, Sam, 1964, Large-scale roughness in open-channel flow: Journal of the Hydraulics Division, American Society of Civil Engineers, v. 90, HY6, p. 203-230.
Jarrett, R. D. (1985) Determination of Roughness Coefficients for Streams in Colorado Lakewood, Colorado
Moody LF. 1944. Friction factors for pipe flow. Trans. A.S.M.E 66: p671-684.
Yalin M. S. (1992) River Mechanics, Pergamon Press, Newyork
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ANNEXURE-A
TABLE-1 Application of explicit Friction Factor Formulae
No Investigator Year Formula Validity Range R2
1 Chavan 2017 ff =20
Re+ [
0.0075 (4000
Re)
0.05
0.25 + (4000
Re)
5 ]
Valid for wide range of Reynolds number
0.9777
2 Morrison 2013 f =16
Re+ [
0.0076 (3170
Re)
0.165
1 + (3170
Re)
7 ]
Inaccuracy crept in for higher values of Reynolds numbers
3 Barenblatt 2005 f =8
ฯ2(1+ฮฑ) Re< 13ร106
4 Mckeon 2005
1
โf= 1.920 log (Reโf โ 0.475
โ7.04
(Reโf)0.55
5 Yen 1991 f =
1
4[โlog (
e
12R) +
5.2
(4Re)0.9]
โ2 30000<Re , e
D< 0.05
6 Barr 1981 f =1
4[โlog (
e
14.8R) +
5.2
(4Re)0.89]
โ2
7 Chen 1979 1
โf= โ2log [
1
3.7605[
e
D] โ
5.0452A
Re]
4000<Re<108
0.000001< e
D
0.9148
8 Churchil 1977 f = 8 [ (8/Re)12 + 1
(A + B)1.5]
112
0.9066
9 Haaland 1983 f =
0.308642
[log โจ(e
3.7D)
1.11
+ |6.9Re
โฉ]2 0.8509
10 Colebrook 1938-39
1
โf= โ2log [
eD
3.7+
2.51
Reโf]
0.90416
11 Serghides 1984 1
โf= โA โ
(B โ A)
C โ 2B + A
2
โ Re>2100 and any
e/D
12 Swami-Jain 1976 1
โf= 1.14 โ 2log [[
e
D] +
21.25
Re0.9]
5000<Re<108
0.000001หe
Dห0.05
13 Barr 1972 f =1
4[log (
e
14.8R) +
5.76
(4Re)0.9]
โ2
0.9051
14 Zigrang & Sylvester
1982 1
โf= โ2log [
1
3.7[
e
D] โ
5.02A
Re]
4000<Re<108 and
0หe
Dห0.04
0.9041
15 Wood 1966 f = 0.094 (
e
D)
0.223
+ 0.53 (e
D)
+ 88 (e
D)
0.44
(Re)โฯ
4000หReห5*108
0.00001หe
Dห0.04 0.7075
16 Moody 1947 f = 0.0053 [1 + (2 โ 104ฮต
D+
106
Re)
13
]
4000หReห5*108
0หฮต
Dห0.01
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