Friction Correlations

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

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).

If Re_Dh < Re_Turb, 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 2^nd 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:

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:

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

Darcy type friction is calculated as

Fanning type friction is calculated as:

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.

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

Swamee-Jain Approximation of the Colebrook-White Equation (Moody Diagram)

The Darcy and Fanning type frictions are calculated as:

User-specified Friction Factor

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∗R_a,	R_a=Average Absolute Roughness
ε=3.100∗R_rms,	R_rms=Root Mean Square Roughness
ε=0.978∗R_zd,	R_zd=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.

D h = \frac{4 * A r e a}{P e r i m e t e r}

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	$\frac{{D h}_{e f f}}{D h} = 1$	AR=1
Rectangle	$\frac{{D h}_{e f f}}{D h} = \frac{2}{3} + \frac{11}{24} * A R * (2 - A R)$	AR=b/a
Ellipse	$\frac{{D h}_{e f f}}{D h} = 1 - . 2109 * {(1 - A R)}^{2}$	AR based on area and perimeter of the ellipse.
Isosceles Triangle	$\frac{{D h}_{e f f}}{D h} = \frac{64}{48 + 11.442 * A R - 6.0026 * {A R}^{2}}$	AR based on area and perimeter of the triangle.
Annulus	$\frac{{D h}_{e f f}}{D h} = \frac{1 + {A R}^{2} + \frac{(1 - {A R}^{2})}{l n (A R)}}{{(1 - A R)}^{2}}$	AR=b/a
Freeform (Arbitrary- Shape)	$\frac{{D h}_{e f f}}{D h} = \frac{\sqrt{A R} * (1 + A R) * (1 - (\frac{192}{π^{5}}) * A R * \tanh (\frac{π}{2 * A R}))}{. 75 * \frac{D h}{\sqrt{A r e a}}}$	AR based on area and perimeter of freeform shape using a rectangle equation. $A R = \frac{(P - \sqrt{(P^{2} - 16 * A)})}{(P + \sqrt{(P^{2} - 16 * A)})}$

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).

\emptyset = \frac{{(\frac{Δ P}{Δ L})}_{2 - p h a s e}}{{(\frac{Δ P}{Δ L})}_{l i q u i d o n l y}}

\emptyset = E + \frac{3.24 F H}{F r^{0.045} W e^{0.035}}

E = {(1 - x)}^{2} + x^{2} (\frac{ρ_{l} f_{g}}{ρ_{g} f_{l}})

F = x^{0.78} {(1 - x)}^{0.224}

H = {(\frac{ρ_{l}}{ρ_{g}})}^{0.91} {(\frac{μ_{g}}{μ_{l}})}^{0.19} {(1 - \frac{μ_{g}}{μ_{l}})}^{0.7}

F r = F r o u d e N u m b e r = \frac{U^{2}}{g L}

W e = W e b e r N u m b e r = \frac{ρ U^{2} D_{h}}{σ} (ρ i s d e n s i t y o f t h e m i x t u r e)

x = \frac{\dot{m_{g}}}{\dot{m_{g}} + \dot{m_{l}}}

(quality)

Recommended to use if $\frac{μ_{l}}{μ_{g}} < 1000$

Turbulator Friction

The friction factor for a turbulated surface can be calculated for the advanced or incompressible tube element. These friction factors are available when the tube's wall surface finish input is set to “Turbulated Surface”. The turbulated friction correlations are used for the tube wall sides that have turbulators. The walls without turbulators will use the smooth wall correlations from above.

The friction factor can come from four different references based on tube and turbulator geometry. The correlations and suggested use cases are described below.

Webb Circular Tube (ref. 1)
Use this correlation for circular tubes with square shaped ribs. This correlation and the Han correlations use the law-of-the-wall similarity for flow over rough surfaces. See the reference for more explanation.
$f_{F a n n i n g} = 2 * {[2.5 * l n (\frac{D_{h}}{2 e}) + 0.95 * P e_T E R M - 3.75]}^{- 2}$
$P e_T E R M = {(\frac{P}{e})}^{.53} if α = 90 (perpendicular to flow)$
$P e_{T E R M} = {(\frac{P}{e})}^{.53} {(\frac{α}{90})}^{- (\frac{.98 α}{90} + .5 l n (.5 e^{+}))} if α < 90$
TS Ravi Circular Tube (eq 11.2 in ref 2.)
Use this correlation for circular tubes and all rib profiles. This correlation uses a statistical approach to correlate many experimental results. It is good for a wide range of geometries but may not be as accurate as the other correlations for geometries specific to them. This correlation calculates a multiplier to a smooth tube friction. The smooth friction correlation also comes from reference 2.
$f_{F_s m o o t h} = \frac{1}{{(1.58 l n (R e) - 3.28)}^{2}}$
$e x p 1 = 0.67 - 0.06 \frac{P}{D_{h}} - .49 \frac{α}{90}$
$e x p 2 = 0.37 - 0.157 \frac{P}{D_{h}}$
$e x p 3 = - 1.66 x 10^{- 6} R e - 0.33 \frac{α}{90}$
$e x p 4 = 4.59 + 4.11 x 10^{- 6} R e - 0.15 \frac{P}{D_{h}}$
$f_{m u l t} = {\{1 + {[29.1 R e^{e x p 1} {(\frac{e}{D_{h}})}^{e x p 2} {(\frac{P}{D_{h}})}^{e x p 3} {(\frac{α}{90})}^{e x p 4} (1 + \frac{2.94}{n}) s i n β]}^{\frac{15}{16}}\}}^{\frac{16}{15}}$
$f_{F a n n i n g} = f_{m u l t} * f_{F_s m o o t h}$
Limits:
$0.01 < \frac{e}{D_{h}} < 0.2, .1 < \frac{P}{D_{h}} < 4, 6, 000 < R e < 160, 000, .66 < P r < 10, 25 < α < 90$
Han 90 deg, 2-sided rectangular tube (ref 3.)
Use this correlation for a rectangular shaped passage with ribs on two sides. This is for ribs that are perpendicular to the flow only.
$R = 3.2 * {(\frac{P}{10 e})}^{0.35}$
$Z = \frac{2 W}{W + H}$
$f_{F a n n i n g_4} = \frac{2}{{(R - 2.5 l n (2 Z \frac{e}{D_{h}}) - 2.5)}^{2}}$
Limits:
$e^{+} > 50, 0.021 < \frac{e}{D_{h}} < 0.078, 10 < \frac{P}{e} < 20, 1 < \frac{W}{H} < 4, 8000 < R e < 80, 000, P r ~ 0.7$
Han Angled, 2-sided rectangular tube (ref 4 and 5)
Use this correlation for a rectangular shaped passage with ribs on two sides. This is for ribs that are 30 to 90 degrees (perpendicular) to the flow.
$α_{t e r m} = 12.31 - 27.07 \frac{α}{90} + 17.86 {(\frac{α}{90})}^{2}$
$R = α_{t e r m} * {(\frac{P}{10 e})}^{0.35} {(\frac{w}{h})}^{m}$
$f_{F a n n i n g_4} = \frac{2}{{(R - 2.5 l n (2 Z \frac{e}{D_{h}}) - 2.5)}^{2}}$
Exponent, m, depends on the rectangle aspect ratio.
For 0.25<w/h<1 (ref 4):
$m = - .5 for α > 60, m = 0 for α < 30 and linear between 30 and 60$
For 1<w/h<4 (ref 5):
$m = .35 for α 〈80, m = 3.15 - .035 * α for α〉 80$
Limits:
$0.02 < \frac{e}{D_{h}} < 0.078, 10 < \frac{P}{e} < 20, .25 < \frac{W}{H} < 4, 3000 < R e < 60, 000$
Where:
$R e = \frac{\dot{m} D_{h}}{A r e a μ} = a x i a l R e y n o l d s N u m b e r$
$D_{h} = H y d r a u l i c D i a m t e r P = r i b p i t c h, e = R i b H e i g h t (a n d w i d t h), α = r i b a n g l e$
$w = r e c t a n g l e s i d e w a l l l e n g t h w i t h r i b s, h = r e c t a n g l e s i d e w a l l l e n g t h s m o o t h$
$e^{+} = \frac{e}{D_{h}} R e \sqrt{\frac{f}{2}} = r o u g h n e s s R e y n o l d s N u m b e r$
$N u = S t * P r * R e$
$H T C = \frac{N u * k}{D_{h}} w h e r e k = f l u i d c o n d u c t i v i t y a t f i l m t e m p e r a t u r e$

Friction Correlation References

Webb, R. L., Eckert, E. R. G., and Goldstein, R. J. "Heat Transfer and Friction in Tubes with Repeated-Rib Roughness", Int. Journal of Heat and Mass Transfer, 14 (1971).
Ravigururajan, T.S., "General correlations for pressure drop and heat transfer for single-phase turbulent flows in ribbed tubes", Iowa State Univ, Thesis, 1986.
Han J.C., "Heat Transfer and Friction Characteristics in Rectangular Channels with Rib Turbulators", Journal of Heat Transfer, ASME (1988).
Han, J. C., Ou, S., Park, J. S. and Lei, C. K. " Augmented Heat Transfer in Rectangular Channels of Narrow Aspect Ratios with Rib Turbulators" , International Journal of Heat Mass Transfer, 32, (1989).
Han, J. C. and Park, J. S. "Developing Heat Transfer in Rectangular Channels with Rib Turbulators", International Journal of Heat Mass Transfer, 31, (1988).