How to select friction factor

There are many friction factors that have been developed. Listed below is a summary of the most popular friction factors, along with a commentary on their application and relevance.

Hagen and Poiseuille equation (1840)

This is the most common formula to estimate the friction factor in laminar flow. It is used to construct a Moody chart for the laminar flow regime.

f=\frac{64}{Re}

Where Re = \frac{U\rho _{m}D}{\eta_{p} }

Application:

Blasius correlation (1913)

The Balusis formula is used to determine the friction factor for Newtonian fully turbulent flow in smooth pipes (Munson 1990). It is valid for only smooth pipes and up to the Reynolds number 105. The friction factor is:

f=0.316\left ( Re \right )^{^{\frac{-1}{4}}}                        for    2,100 < Re \leqslant 2\times 10^{4}

f=0.184\left ( Re \right )^{\frac{-1}{5}}                        for  Re \geqslant 2\times 10^{4}

 

Application:

Buckingham-Reiner equation (1921)

Buckingham was the first person to develop a precise description of friction loss for Bingham plastics in fully developed laminar pipe flow. A solution to the Buckingham-Reiner equation can be obtained, but due to the complexity it is rarely used. Explicit approximations are generally used instead.

f_{L}=\frac{64}{Re}\left [ \left ( 1+\frac{He}{6Re}-\frac{64}{3}\frac{He^4 }{f_L^3 Re^7} \right ) \right ]

Where

He=\frac{\tau _y \rho_mD^2 }{\eta _p}

Colebrook-White equation (1933)

Where relative roughness is negligible the following implicit formula applies for smooth pipes:

\frac{1}{\sqrt{f}}=2\textup{log}_{10} \left ( \frac{2.51}{Re\sqrt{f}} \right )

This is frequently presented in its explicit form (1939):

\frac{1}{\sqrt{f}}=1.8\textup{log}_{10} \left (\frac{Re}{6.9} \right )

Application:

Prandtl correlation (1935)

The Prandtl correlation is not explicit, and it must be solved iteratively.

\frac{1}{\sqrt{f}}=2\textup{log}_{10}\left ( Re\sqrt{f} \right )-0.8

Churchill friction factor (1977)

Churchill proposed a friction factor correlation for both laminar and turbulent flow regimes.

Darby and Melson (1981)

Darby and Melson modified the Buckingham-Reiner equation for Fanning friction factor (C_f) in terms of a modified Reynolds number (Re_{mod}) :

C_f=\frac{16}{Re_{mod}}

Where:

Re_{mod}=\frac{6Re_B ^2}{6Re_B + He}

Darby and Melson also developed an empirical expression for turbulent flow as:

f_T=10^aRe^{-0.193}

Where:

a=-1.47\left [ 1+0.146e^{-2.9\times 10^{-5}He} \right ]

To get a single explicit friction factor equation that is valid for all flow regimes, they developed the following:

f=\left [ f_L^{m}+f_T^m \right ]^{1/m}

Where:

m=1.7+\frac{40,000}{Re}

Wilson and Thomas (1985)

Wilson and Thomas developed an analytical equation for turbulent flow of non-Newtonian fluids based on the drag reduction associated with the thickening of viscous sub-layer. The required flow parameters can be determined directly from a rheogram without employing correlations based on pipe-flow data.

For Bingham plastic fluids in turbulent flow:

\frac{V}{U_*}=2.5\textup{ln}Re'_*+2.5\textup{ln}\left [ \frac{\left ( 1-\xi \right )^2}{\left ( 1+\xi \right )} \right ]+\xi\left ( 14.1+1.25\xi \right )

Where:

Re'_*=\frac{\rho D U_*}{\eta_p}

\xi =\frac{\tau_{yB}}{\tau_w}

U_* =\sqrt{\frac{\tau_w}{\rho }}

Note that f is the Stanton-Moody friction factor.

When Re'_* >830 and \frac{\rho D V}{\eta_t}>1.7\times 10^4 the equation can be simplified so that:

\frac{V}{U_*}=2.5\textup{ln}\left ( Re'_* \right )+32

Slatter correlation (1995)

Slatter modified the Reynolds number for determining the transition from laminar to turbulent flow for Herschel-Bulkley (and Bingham plastic) slurries (where laminar ceases at Re_{mod}\leq 2100):

Re_{mod}=\frac{8\rho V_{ann}^{2}}{\tau_{YH}+K\left ( \frac{8V_{ann}}{D_{shear}} \right )^n}

 

Where:

V_{ann}=\frac{Q-Q_{plug}}{\pi\left ( R^{2}-R_{plug}^{2} \right )}

D_{shear}=D-D_{plug}=2\left ( 2-R_{plug} \right )

V_{plug}=\frac{nR}{\left ( n+1 \right )}\left ( \frac{\tau_w}{K} \right )^{1/n}\left ( 1-\varnothing \right )^{\frac{\left ( n+1 \right )}{n}}

\varnothing =\frac{D_{plug}}{D}=\frac{\tau_{YH}}{\tau_w}

Q=\pi R^3 n \left ( \frac{\tau_w}{K} \right )^{1/n}\left ( 1-\varnothing \right )^{\frac{(n+1)}{n}}\left \{ \frac{\left ( 1-\varnothing \right )^2}{3n+1} + \frac{2\varnothing \left ( 1-\varnothing \right )}{2n+1}+\frac{\O ^2}{n+1}\right \}

Q_{plug}=\pi D_{plug}^2V_{{plug}}

Re_{mod} = Slatter modified Reynolds number

V_{ann} = velocity in the sheared annulus (m/s)

D_{shear} = equivalent diameter of sheared region (m)

Q = transitional flow rate (m^3/h)

R{_{HB}}=\frac{\rho V^{n-1}D^n}{8^{n-1}K}\left ( \frac{4n}{1+3n} \right )^n

f_f=\frac{16}{\Psi R_{HB}}

\Psi =\left ( 1+3n \right )^n \left ( 1+\varnothing \right )^{1+n}\left [ \frac{(1-\varnothing)^n}{1+3n}+\frac{2\varnothing (1-\varnothing )}{1+2n}+\frac{\O ^2}{1+n} \right ]

For Bingham plastics n=1, and \tau_{HB}=\tau_{YB}, and K=\eta _p

Application:

Danish-Kumar solution (2011)

Danish provided an explicit procedure to calculate the friction factor by using the Adomian decomposition method. Danish provided an equation for laminar and turbulent flows.

Swamee-Aggarwal equation (2011)

The Swamee-Aggarwal equation is used to solve directly for the Darcy-Weisbach friction factor f_L for laminar flow of Bingham plastic fluids. It is an approximation of the Buckingham-Reiner equation. The accuracy is very good.

Application:

Morrison correlation (2013)

Morrson developed a correlation that spans the entire range of Reynolds numbers, from laminar flow through transitional flow and reaching the highest values of Reynolds number. It is for smooth pipes.

C_f=\left [ \frac{0.0076\left ( \frac{3170}{Re} \right )^{0.165}}{1+\left (\frac{3170}{Re} \right )^{7.0}} \right ]+\frac{16}{Re}

 

Application:

Fanning friction factor 

 

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