# 3.4: Approximation of the Darcy-Weisbach Friction Factor

It is obvious that the Darcy-Weisbach friction factor λl depends on the pipe diameter Dp and the line speed vls. This may be confused with a direct influence of the pipe diameter Dp and the line speed vls. So it is interesting to see how the Darcy-Weisbach friction factor λl depends on the pipe diameter Dp and the line speed vls. Figure 3.7-1 shows the Darcy-Weisbach friction factor for smooth pipes as a function of the line speed vls at a number of pipe diameters, while Figure 3.7-2 shows the Darcy-Weisbach friction factor as a function of the pipe diameter Dp at a number of line speeds. In both figures, the Darcy-Weisbach friction factor can be well approximated by a power function

$\ \lambda_{1}=\alpha \cdot\left(\mathrm{v_{ls}}\right)^{\alpha_{1}} \cdot\left(D_{p}\right)^{\alpha_{2}}$

With:

$\ \alpha=0.01216 \quad\text{ and }\quad \alpha_{1}=-0.1537 \cdot\left(D_{p}\right)^{-0.089} \quad \text{ and }\quad \alpha_{2}=-0.2013 \cdot\left(\mathrm{v_{1 s}}\right)^{-0.088}$

For laboratory conditions both powers are close to -0.18, while for real life conditions with higher line speeds and much larger pipe diameters this results in a power for the line speed of about α1=-0.155 and for the pipe diameter of about α2=-0.168. This should be considered when analyzing the models for heterogeneous transport, where in real life these adjusted powers should be used.