Finding fluid flow with Numerical Method

[chronicals]

Homework Statement

For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor (f). The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which all can be represented by another dimensionless quantity, the Reynolds number (Re). A formula that predicts f given Re for turbulent flow is the von Karman equation,

1/sqrt(f) = 1.74* ln ( Re* sqrt(f)) - 0.4

Typical values for the Reynolds number for turbulent flow are 10 000 to 500 000 and for the Fanning friction factor are 0.001 to 0.01. For Reynolds numbers less than 2500 fluid flow type is called laminar flow and the Fanning friction factor is given by f = 16/Re. Use a numerical method to determine the friction factor for Re = 10 000 within 5% relative error.

Homework Equations

The Attempt at a Solution

our function is g(f) = 1.74* ln ( Re* sqrt(f)) - 0.4 - 1/sqrt(f)

how can i solve this problem?

 

[hotvette]

You want to find f such that g(f) = 0. Looks like a good candidate for the Newton-Raphson method.

1. Start with an initial guess f₀
2. Calculate g and g' = dg/df at f₀
3. Update your estimate f₁ = f₀ - g/g'
4. Repeat steps 2-3 until the solution is obtained within the desired accuracy