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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(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:chemi@rurkiu.emet.in). 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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