Friction Correlations

The friction in the Flow Simulator element can be calculated by the given equation, where:

(1)

stands for the calculated friction based on the user-selected friction mode (Abauf, Swamee), friction type (Fanning or Darcy), and Re (to determine whether it is in a turbulent region, or in a laminar region).(2)

If ReDh < ReTurb, laminar friction calculations take place, otherwise, the turbulent friction calculation routine is used.

Nomenclature Subscripts
f: friction F: fanning
Re: Reynolds number D: darcy
FMULT: friction multiplier turb: turbulent flow
ε: sand grain roughness lam: laminar flow
A: Cross sectional area Abuaf: Abuaf friction relation
L+: Inlet station + 1/9 of 2nd station Smooth: smooth surface
XMU: dynamic viscosity Rough: rough surface
W: mass flow rate SJ: Swamee-Jain approximation
X: station length Dh: hydraulic diameter
L: equivalent diameter

Laminar Friction

Calculates the friction coefficient for laminar flow in shaped ducts based on the references Yunus A. Cengel, 2006 and Bruce Munson, 2005.

Friction coefficient for hydrodynamically fully developed flow can be calculated as:(3)

For a Tube Element, Laminar Friction Inlets effects can be accounted. Friction coefficient for hydrodynamically developing flow with “Muzychka Yovanovich Laminar Inlet Effects” can be calculated as:

(4)

The friction coefficient for combining developing flow and fully developed flow can be calculated as:

(5)

Darcy type friction is calculated as

(6)

Fanning type friction is calculated as:

(7)

Turbulent Friction

Calculates the turbulent friction for smooth or rough walls.

Abuaf Friction Relation
The Abuaf friction relation should generally be used for smooth walled tubes.
(8)

In Flow Simulator, you have the option to use the Abuaf friction relation together with wall roughness. The following adjustment equation is used:

(9)
Swamee-Jain Approximation of the Colebrook-White Equation (Moody Diagram)
(10)
The Darcy and Fanning type frictions are calculated as:(11)
User-specified Friction Factor
(12)

Roughness

Surface roughness values can be entered in four different measurement types. The roughness values are converted to sand grain roughness equivalents using the following equations from table 1 of reference 63.
ε=5.863∗Ra, Ra=Average Absolute Roughness
ε=3.100∗Rrms, Rrms=Root Mean Square Roughness
ε=0.978∗Rzd, Rzd=Peak to Valley Roughness

Non-Circular Shapes in Flow Simulator Tubes

The friction factor and heat transfer coefficient (HTC) correlations were developed for circular pipes. The traditional method to use these correlations on non-circular shapes is to calculate a hydraulic diameter based on the shape area and perimeter.

(13)
Dh=4 *AreaPerimeter

The errors associated with this method can be +/-40% for laminar flow but less for turbulent flow, +/-15% (see White, ref 3).

A more accurate option is to adjust the hydraulic diameter with a friction factor ratio (see White, ref 3). The effective hydraulic diameter can then be used in the friction factor and HTC correlations.

The following table summarizes the relationship between the Dh based on 4*A/P and the effective hydraulic diameter.

Shape Effective Dh Equation Aspect Ratio (AR)
Circle DheffDh=1 AR=1
Rectangle

DheffDh=23+1124*AR*(2-AR) AR=b/a
Ellipse

D h e f f D h = 1 - . 2109 * ( 1 - A R ) 2 AR based on area and perimeter of the ellipse.

Isosceles Triangle

DheffDh=6448+11.442*AR-6.0026*AR2 AR based on area and perimeter of the triangle.

Annulus

DheffDh=1+AR2+1-AR2ln(AR)1-AR2 AR=b/a
Freeform

(Arbitrary- Shape)

DheffDh=AR*1+AR*1-192π5*AR*tanhπ2*AR.75*DhArea AR based on area and perimeter of freeform shape using a rectangle equation.

A R = ( P - P 2 - 16 * A ) ( P + P 2 - 16 * A )

See Blevins (ref 15) and Muzychka et al. (ref 50) for additional information.

If the compressible tube, advanced orifice, and incompressible tube have a cross sectional shape that is not circular, the equations in this table are used for the effective hydraulic diameter equation.

Two-phase Flow Friction

The friction factor for two-phase flow (liquid and gas) in an incompressible tube can be calculated using two options. The homogenous approach uses the laminar and turbulent equations shown above with fluid properties based on the liquid/gas mixture. The second approach uses friction equations developed by Friedel (ref 64).

(14)
= Δ P Δ L 2 p h a s e   Δ P Δ L l i q u i d   o n l y MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHfiIXcqGH9aqpdaWcaaWdaeaapeWaaeWaa8aabaWdbmaalaaa paqaa8qacqqHuoarcaWGqbaapaqaa8qacqqHuoarcaWGmbaaaaGaay jkaiaawMcaa8aadaWgaaWcbaWdbiaaikdacqGHsislcaWGWbGaamiA aiaadggacaWGZbGaamyzaiaacckaa8aabeaaaOqaa8qadaqadaWdae aapeWaaSaaa8aabaWdbiabfs5aejaadcfaa8aabaWdbiabfs5aejaa dYeaaaaacaGLOaGaayzkaaWdamaaBaaaleaapeGaamiBaiaadMgaca WGXbGaamyDaiaadMgacaWGKbGaaiiOaiaad+gacaWGUbGaamiBaiaa dMhaa8aabeaaaaaaaa@5876@
(15)
=E+ 3.24 F H F r 0.045  W e 0.035 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHfiIXcqGH9aqpcaWGfbGaey4kaSYaaSaaa8aabaWdbiaaioda caGGUaGaaGOmaiaaisdacaGGGcGaamOraiaacckacaWGibaapaqaa8 qacaWGgbGaamOCa8aadaahaaWcbeqaa8qacaaIWaGaaiOlaiaaicda caaI0aGaaGynaaaakiaacckacaWGxbGaamyza8aadaahaaWcbeqaa8 qacaaIWaGaaiOlaiaaicdacaaIZaGaaGynaaaaaaaaaa@4DE9@
(16)
E= 1x 2 + x 2 ρ l f g ρ g f l MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGfbGaeyypa0ZaaeWaa8aabaWdbiaaigdacqGHsislcaWG4baa caGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaikdaaaGccqGHRaWkca WG4bWdamaaCaaaleqabaWdbiaaikdaaaGcdaqadaWdaeaapeWaaSaa a8aabaWdbiabeg8aY9aadaWgaaWcbaWdbiaadYgaa8aabeaak8qaca WGMbWdamaaBaaaleaapeGaam4zaaWdaeqaaaGcbaWdbiabeg8aY9aa daWgaaWcbaWdbiaadEgaa8aabeaak8qacaWGMbWdamaaBaaaleaape GaamiBaaWdaeqaaaaaaOWdbiaawIcacaGLPaaaaaa@4CEB@
(17)
F= x 0.78 1x 0.224 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGgbGaeyypa0JaamiEa8aadaahaaWcbeqaa8qacaaIWaGaaiOl aiaaiEdacaaI4aaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcaWG4b aacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaicdacaGGUaGaaGOm aiaaikdacaaI0aaaaaaa@4462@
(18)
H= ρ l ρ g 0.91 μ g μ l 0.19 1 μ g μ l 0.7 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGibGaeyypa0ZaaeWaa8aabaWdbmaalaaapaqaa8qacqaHbpGC paWaaSbaaSqaa8qacaWGSbaapaqabaaakeaapeGaeqyWdi3damaaBa aaleaapeGaam4zaaWdaeqaaaaaaOWdbiaawIcacaGLPaaapaWaaWba aSqabeaapeGaaGimaiaac6cacaaI5aGaaGymaaaakmaabmaapaqaa8 qadaWcaaWdaeaapeGaeqiVd02damaaBaaaleaapeGaam4zaaWdaeqa aaGcbaWdbiabeY7aT9aadaWgaaWcbaWdbiaadYgaa8aabeaaaaaak8 qacaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaaicdacaGGUaGaaGym aiaaiMdaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqaa8 qacqaH8oqBpaWaaSbaaSqaa8qacaWGNbaapaqabaaakeaapeGaeqiV d02damaaBaaaleaapeGaamiBaaWdaeqaaaaaaOWdbiaawIcacaGLPa aapaWaaWbaaSqabeaapeGaaGimaiaac6cacaaI3aaaaaaa@5AB4@
(19)
Fr=Froude Number= U 2 gL   MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGgbGaamOCaiabg2da9iaadAeacaWGYbGaam4BaiaadwhacaWG KbGaamyzaiaacckacaWGobGaamyDaiaad2gacaWGIbGaamyzaiaadk hacqGH9aqpdaWcaaWdaeaapeGaamyva8aadaahaaWcbeqaa8qacaaI YaaaaaGcpaqaa8qacaWGNbGaamitaaaacaGGGcaaaa@4B2B@
(20)
We=Weber Number= ρ U 2 D h σ   ρ is density of the mixture MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGxbGaamyzaiabg2da9iaadEfacaWGLbGaamOyaiaadwgacaWG YbGaaiiOaiaad6eacaWG1bGaamyBaiaadkgacaWGLbGaamOCaiabg2 da9maalaaapaqaa8qacqaHbpGCcaWGvbWdamaaCaaaleqabaWdbiaa ikdaaaGccaWGebWdamaaBaaaleaapeGaamiAaaWdaeqaaaGcbaWdbi abeo8aZbaacaGGGcWaaeWaa8aabaWdbiabeg8aYjaacckacaWGPbGa am4CaiaacckacaWGKbGaamyzaiaad6gacaWGZbGaamyAaiaadshaca WG5bGaaiiOaiaad+gacaWGMbGaaiiOaiaadshacaWGObGaamyzaiaa cckacaWGTbGaamyAaiaadIhacaWG0bGaamyDaiaadkhacaWGLbaaca GLOaGaayzkaaaaaa@6B10@
(21)
x = m g ˙ m g ˙ + m l ˙ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG4bGaeyypa0ZaaSaaa8aabaWaaCbiaeaapeGaamyBa8aadaWg aaWcbaWdbiaadEgaa8aabeaaaeqabaWdbiaacMTaaaaak8aabaWaaC biaeaapeGaamyBa8aadaWgaaWcbaWdbiaadEgaa8aabeaaaeqabaWd biaacMTaaaGccqGHRaWkpaWaaCbiaeaapeGaamyBa8aadaWgaaWcba WdbiaadYgaa8aabeaaaeqabaWdbiaacMTaaaaaaaaa@452E@
(quality)

Recommended to use if μ l μ g <1000 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeqiVd02damaaBaaaleaapeGaamiBaaWdaeqa aaGcbaWdbiabeY7aT9aadaWgaaWcbaWdbiaadEgaa8aabeaaaaGcpe GaeyipaWJaaGymaiaaicdacaaIWaGaaGimaaaa@4061@