# 3.4: Approximation of the Darcy-Weisbach Friction Factor

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It is obvious that the Darcy-Weisbach friction factor **λ _{l} **depends on the pipe diameter

**D**

**and the line speed**

_{p}**v**

_{ls}. This may be confused with a direct influence of the pipe diameter

**D**

**and the line speed**

_{p}**v**

_{ls}. So it is interesting to see how the Darcy-Weisbach friction factor

**λ**

**depends on the pipe diameter**

_{l}**D**

**and the line speed**

_{p}**v**

_{ls}. 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

**D**

**at a number of line speeds. In both figures, the Darcy-Weisbach friction factor can be well approximated by a power function**

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

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 **α**_{1}**=-0.155 **and for the pipe diameter of about **α**_{2}**=-0.168**. This should be considered when analyzing the models for heterogeneous transport, where in real life these adjusted powers should be used.