The formula to calculate the friction factor (f) using the Colebrook equation is:
\[ \frac{1}{\sqrt{f}} = -2 \log_{10}\left(\frac{\epsilon/D}{3.7} + \frac{2.51}{Re \sqrt{f}}\right) \]
Where:
Let's say the relative roughness (\( \epsilon/D \)) is 0.005 and the Reynolds number (\( Re \)) is 100,000. Using the Colebrook formula and an iterative method:
\[ \frac{1}{\sqrt{f}} = -2 \log_{10}\left(\frac{0.005}{3.7} + \frac{2.51}{100000 \sqrt{f}}\right) \]
After solving iteratively, we find the friction factor (\( f \)) to be approximately 0.031.
The Colebrook formula is widely used in fluid mechanics to estimate the friction factor for turbulent flow in smooth and rough pipes. It is an empirical formula derived from experimental data and is considered to be accurate for a wide range of Reynolds numbers and relative roughness values.
Definition: The Colebrook equation is used to calculate the Darcy-Weisbach friction factor for turbulent flow in pipes.
Formula: \( \frac{1}{\sqrt{f}} = -2 \log \left( \frac{\epsilon/D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right) \)
Example: \( \frac{1}{\sqrt{0.02}} = -2 \log \left( \frac{0.0001/0.05}{3.7} + \frac{2.51}{5000 \sqrt{0.02}} \right) \)
Definition: Solving the Colebrook equation involves iterative methods due to its implicit nature.
Formula: \( \frac{1}{\sqrt{f}} = -2 \log \left( \frac{\epsilon/D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right) \)
Example: \( \frac{1}{\sqrt{0.03}} = -2 \log \left( \frac{0.0002/0.1}{3.7} + \frac{2.51}{10000 \sqrt{0.03}} \right) \)
Definition: Solving the Colebrook equation for roughness involves isolating the roughness term.
Formula: \( \epsilon = 3.7 D \left( 10^{\frac{1}{-2 \log \left( \frac{1}{\sqrt{f}} - \frac{2.51}{Re \sqrt{f}} \right)}} - \frac{2.51}{Re \sqrt{f}} \right) \)
Example: \( \epsilon = 3.7 \times 0.05 \left( 10^{\frac{1}{-2 \log \left( \frac{1}{\sqrt{0.02}} - \frac{2.51}{5000 \sqrt{0.02}} \right)}} - \frac{2.51}{5000 \sqrt{0.02}} \right) \)
