piping - speed up pipe flow calculations - chemical engineering - dec 1998
TRANSCRIPT
Design
Speed Up Pipe Flowe 1 1 t· Bharat B. Gulyania cu a IonS Bikash MohantyUniversity of Roorkee
Account for pipe roughness, identify a flow regime,
and then select the best equation for friction factor
NOMENCLATURE
D Diameter of pipe, mf Darcy friction factor,
dimensionless
g Gravitationalacceleration, m/s2
Gu l?Ynamic roughness,dlmensionless
hf Frictional head loss, m
In logarithm to base e
log logarithm to base 10
L Length of pipe, mQ Volumetric
f10w rate, m3/s
Re Reynolds number,dimensionless
V Fluid velocity, mis
E Pipe roughness, m
v Kinematic viscosity, m2/s
(1)
... : ... -
At very low Re, flow is entirely laminar. The flow develops a velocity distribution across the pipe, which is always slowest near the wall. As Reincreases, a thick laminar sub-Iayercovers all the protrusions. but it isgradually getting thinner as it isdragged forward by the flow near thecenterline. At medium-high Re, manyprotrusions are disturbing the sublayer. Finally, a Re is reached whereall protrusions disturb the sub-Iayer.The pipe flow in this domain can beconsidered as totally rough.
The discussion thus far has beenall qualitative. To be of use, numbersand equations must be attached.But, figuring out what regime is inside a pipe is tricky. The commoncalculation methods require one toas sume a flow regime, calculate theflow, and then check to see if there gime assumed was correcto Carefuldesigners have learned to solve threecases and eliminate two. This is inef-
directly with velocity.We will be usingan alternative formula for Re, onebased on the kinematic viscosity oftheflowing fluids. This is the measuredviscosity divided by the density.
Re=DVv
Boundary layersAt low velocity, the entire fluid in apipe is in laminar flow. Roughnessdoes not break it up. At high velocity,the central portion of the fluid is experiencing turbulent flow. Many observers, however, have found a thinlayer of fluid at the outer boundarywhere the flow is laminar (acceptedterminology: laminar sub-Iayer).
Pipe-surface roughness interactswith this layer. The height of the protrusions is the variable. If the protrusions extend beyond the sub-Iayer,flow becomes broken up, and the surface is considered as hydrodynamically rough. If the surface protrusionsare completely covered by the laminarsub-Iayer, the surface is hydrodynamically smooth.
At moderate flow rates, the flow is amixture ofturbulent and laminar. Thisis the transition regime, and it occurswhen some,but not all, protrusions disturb the sub-Iayer. From a designstandpoint, it is best to avoid thisregime, as it is unstable.
Follow now the situation as a pipesees increasing fluid velocity in thetransition regime. Flow correlationsare usually based on the Reynoldsnumber (Re), a dimensionless parameter, defined in Equation 1. Re varies
Acetic acidv = 1.167 X 10-7Benzene
0.0015 < V < 0.1050.0026 < V < 0.1790.0032 < V < 0.2230.0085 < V < 0.5960.255 < V < 17.9v = 1.276 X 10-7
Ethanol0.00091 < V < 0.06390.00155 < V < 0.1090.00194 < V < 0.1360.00517 < V < 0.3620.155<V< 10.9
v = 0.776 X 10-7Hexane
0.0008 < V < 0.05740.0014 < V < 0.09760.0017 < V < 0.1220.00465 < V < 0.3250.139 < V < 9.76v = 0.697 X 10-7
Kerosene0.0032 < V < 0.2210.0054 < V < 0.3750.0067 < V < 0.4690.179 < V < 1.250.536 < V < 37.5
v = 2.679 X 10-7Water
0.0012 < V < 0.08470.0021 < V < 0.1440.0026 < V< 0.1800.0069 < V < 0.4800.206 < V < 14.4v = 1.028 X 10-7
Piping pressure-drop calcula
tions are among the most tedious design assignments toexecute. Conventional calcu-
--1 ations require the engineer to assumewhich flow regime applies -laminar,transitional or turbulent - thenmake the calculation and check thatthe assumption was correctoIn this article we present a quicker methodbased on surface roughness, that requires no assumptions or recalculation.
Real pipes in actual service haverough internal walls. Unfortunately,there is no surface that can be regardedas perfectlysmooth.A pipe's resistanceto fluid flowincreases as its roughnessincreases. Countless tests have shownthat this is from the formation of eddies behind protrusions.
The extent of roughness varies,depending on the manufacturingprocess, the finishing steps used, andthe material itself. There is a complexinteraction between the roughnessand the bulk flow regime. Observershave generalized the flows into threedifferent hydrodynamic regimes: laminar, transitional and turbulentoSometimes the roughness will changethe regime. Sometimes, there is no effect. We will show you a quick way tosort out the regimes.
CHEMICAL ENGINEERING / DECEMBER 1998 145
Design
(7)
The most commonly accepted approximation for f is the Colebrook-Whiteequation [6]. Rowever, this equation isimplicit in f and calls for a trial-anderror solution. Many explicit approximations ofthis equation are available inliterature [7-12]. Even these are quitecumbersome and computation-intensive. Equation 8 is the Colebrook-Whiteequation, and we used it in this study.
J..,=_1.7371n[1.256 +~] (8)fi Re.J7 3.707
3. Completely Rough Regime: Athigh velocity, flow is fully turbulent andthe laminar sub-layer is very small. Almost all the surface protrusions extendthrough the boundary layer. Thisregime is encountered at very high values of Re and El D. Flow is so disturbedby the rough surface that friction factoris a function of El D only. The criterion todetermine this regime is Equation 9 [3].
eRe# > 100 (9)D
For the completely rough regime,von Karman's equation [13] approximates very well the Colebrook-Whiteequation.
J.., = -1.7371n[ % ] (lO).J7 3.707
These equations force a tedious calculation process. Even the most helpfulcomputer program is loaded with decision loops. Determining the appropriateequation for friction factor requiresknowingthe flowregime. Unfortunately,this is not possible from the above-mentioned criteria (Equations. 4, 7 and 9),since these require the value off, landingthe designer in a Catch-22 situation!
J¡ = 1.810g(~e) Colebrook (6)
2. Transition Regime: Once theflow is fast enough to be affected byboth Re and El D, it is considered to bein a transitional regime. Rere the friction factor is a function ofbothRe and
El D. Some protrusions are disturbingthe boundary layer. The test for thisregime is by this criterion [3]:
8< eRe{1 (100D
J..,=2.01og(Re{l)-0.S Prandtl- (5)# vonKarman
(4)
(3)
Pipes also can have turbulent flowin the central part of the pipe with thesurface protrusions completely submerged in the laminar boundary layer.The test criterion is Equation 4 El].
eRe{l (SD
Essentially, the pipe is smooth sincethe protrusions do not disturb theflow. This is an uncomplicated case,and is a favorite research topic. Oneresult is that the literature is full ofreliable equations [2, 3, 4, 5] for estimating friction factor. The Prandtlvon Karman equation [4] and its explicit approximation given byColebrook [5] are the most widelyused. We used Equation 5 in thisstudy and ignored Equation 6.
full pipe in laminar flow (Re < 2,100).The best correlation is' the RagenPoiseuille Equation (Equation 3).
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The equations offluid flowFor the analysis of steady-state Newtonian fluid flow in pipes, the DarcyWeisbach Equation is most commonlyused (Equation 2). This relates headloss to surface resistance, fluid properties and pipe geometry.
SfLQ2
hf = ¡r2gD5 (2)
The Darcy friction factor, f, is commonly presented as a function of theReynolds number and the relativeroughness, E'ID.
1. Hydrodynamically SmoothRegime: Flow is smooth when protrusions do not disturb the flow. The friction factor does not depend on El D,but only on Re. This will happen withlow flowrates and small El D.
The easiest regime to visualize is a
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ficient, We have developed a shortcut method that homes in the bestequation the first time.
146 CHEMICAL ENGINEERING I DECEMBER 1998
Predict fluid velocities that will sim
ulate smooth-pipe behavior, by substituting Gu < 10 into Equation 15:
The designer has a circular process.First, assume a flow regime, then estimate f and then back-check whetherthe assumed flow re gime is correcto Thetest criteria become part ofthe problemrather than part ofthe solution. Thereis a way out ofthe morass, with a criterion based on dynamic roughness.
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(11)
(15)
We calculated friction factors usingEquations 5, 8 and 10 for the ranges 4X 103 < Re < 1 X 108 and 10-6 < ElD <10-2. We accounted for overlappingregimes by assuming that Equation 8was the most accurate, and calculatedthe errorwhen using Equations 5 or 10.We are out ofthe range ofEquation 1.
We found that for Gu < 10, at appropriate values of Re and El D, Equation5 was less than 2% different from
Equation 8. Equation 5 implies thehydraulically smooth regime. Wefound that for Gu > 700, at appropriate values of Re and El D, Equation '10was less than 2% different from Equation 8. Equation 10 implies the roughregime. This can be summarized in arule for characterizing flow regimes:
Gu < 10
Smooth regime, use Equation 5. (12)
10 < Gu < 700
Transition regime, use Equation 8. (13)
Gu > 700
Rough regime, use Equation 10. (14)
Thus, these three equations replacethe implicit criteria of Equations 4, 7and 9. We will take one more simplifYing step, and eliminate calculating Gu.Rearrange Equation 11, solve for V.
Gu = Re(~) = e,%
Introducing dynamic roughnessCalculations of the flows in commer
cial pipes are not easy when using static roughness (El D), which dependsonly on pipe geometry and not on flowconditions. It becomes easy with anew parameter, dynamic roughness.Dynamic roughness, Gu, is defined as.;he product of Reynolds number andsta tic roughness:
V=Gu(%)
CHEMICAL ENGINEERING / DECEMBER 1998 147
Design
ApplicationWe used the criteria of Equations16-18 for six fluids and five pipes. Thevalues of E are from Coker [14]. The
Predict fluid velocities that will simulate rough-pipe behavior, by substituting Gu > 700 into Equation 15:
V> 700(%) (17)
Predict fluid velocities that will sim
ulate transitional behavior, by substituting 10 <Gu < 700 into Equation 15:
10(%) < V < 700(%) (18)
These equations are a simple and reliable tool for the design engineer. For agiven fluid there are two inputs. One iskinematic viscosity and the other issome knowledge of pipe materials.1t becomes easy to assess the effect of probable fluid velocities on the flow regime,even before selecting the pipe size.
v <10(%) (16) values ofv at 20°C are from Yaws [15].The table shows those velocity rangesthat are in the transition regime,thereby directly defining the upperlimit of smooth flow and the lowerlimit ofrough flow.
Another check is to take a problemfrom Coulson and Richardson [16]."Water at 20°C flows in a 50-mm-dia.
pipe, 100 m long, whose roughness isequal to 0:013 mm. Calculate the maximum velocity for which the pipe willbehave as smooth."
.':'= 1.028X 10-7 - .0079e 1.3x 10-5
From Equation 16, smooth flow isfound for V < 0.079 mis. and fromEquation 17, completely rough flow isfound for V > 5.53 mis. Usual engineering practice is to avoid the transitio n regime. The length and diameter of the pipe will be used when thepressure drop is calculated. •
Edited by Peter M. Silverberg
References1. Schlichting, H., "Boundary Layer Theory,"
4th ed., p. 585, McGraw-Hill, NewYork. 1960.2. von Karman, T.J., Turbulence and Skin fric
tion, Aeronautical Science, 7, pp.1-20, 1934.3. Rouse, H., "Engineering Hydraulics," Chap.
VI: Steady flow in pipes and conduits, Chapman and Hall, NewYork, 1950.
4. B1asius, H., Das Ahnlichkeitsgesetz beiReibungsvorgangen in Flüssigkeiten, Forschungsarbeit Arb. Ing.- Wes, No. 131, Berlin,1913.
5. Colebrook, C.F., Turbulent Flow in Pipeswith Particular Reference to the TransitionRegion Between the Smooth and Rough PipeLaws, Jl. Inst. Civil Engineers (London), 11,1, pp. 133-156, 1939.
6. Jain, A., Gulyani, B.B., Kumar, S., and Mohanty, B., Mathematieal Analysis ofCorrelation for Pipeflow Friction Factors, "Conference on Mathematics and its Applications inEngineering and Industry," Roorkee, India,Dec. 16-18, 1996.
7. Swamee, P.K, and Jain, A.K, Explicit Equa·tions for Pipeflow Problems, J. Hydraulic.Eng., 102, 5, pp. 657-664. 1976.
8. Round, G.F., An Explicit Approximation forthe Friction Factor - Reynolds Number Relation for Rough and-8mooth Pipes, Canadian.Journal ofChem. Eng., 58, pp. 122-123, Feb.1980.
9. Moody, M.L., An Approximate Formula forPipe Friction Factors, Trans. ASME, 69, p.1005,1947.
10. Chen, N.H., An Explicit Equation for Frietion Factor in Pipe, Ind. Eng. Chem. Fund.,18,3,p.296,1979.
11. Zigrang, J., and Sylvester, N.D., Explicit Approximation to the Solution of Colebrook'sFriction Factor Equation, AIChE Journal,28, 3, p. 514, May 1982.
12. Churchill, S.W., Friction-factor EquationSpans AlI Fluid Flow Regimes, Chem. Eng.,91,24, pp. 91-92, Nov. 7, 1977.
13. Moody, L.F., Friction Factors for Pipe Flow ,Trans. ASME, pp. 671-684, Princeton, N. J.,Nov. 1944.
14. Coker, A.K, Program Evaluates PressureDrop for Single-phase Fluids, Hydrocarbon
.Processing, 70, Feb. 1991.i5. Yaws, C. L., "Physical Properties," pp. 58,
112,182,222, McGraw-Hill, NewYork, 1977.16. Coulson, J.M, and Richardson, J.F., "Chemical
Engineering,", Vol. One, 3rd edition, Example3.2, Pergamon Press, Oxford, U. K, 1977.
AuthorsBharat Bhushan GuIyani isa research associate inthe Chemical EngineeringDepartment of the Univ. ofRoorkee (Roorkee, 247667India; Phone: 91-1332-75649;Fax: 91-1332-73560, E-mail:[email protected]). Hehas a B.E. and an M.E. inchemieal engineering, bothfrom Roorkee. He has written
over 20 artides in various joumals. His researchinterests indude process integration, modelingand simulation, and expert system development.He is a member ofthe AIChE.
Bikash Mohanty is associate professor of chemical engineering at the Universityof Roorkee. He has B.E.,M.E. and Ph.D. in chemicalengineering, all from Roorkee. He has published over60 papers. His research interests are process control,artificial intelligence, dynamic simulation and computer-aided designo
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