Determine the friction factor f of the system using the

Question

Problem- The resistance to flow through pipes and tubes is parameterized by a dimensionless number called the friction factor (denoted as f, ranging from about 0.008 to 0.08), which can estimated using

1/ √f = -20.log₁₀(ε/3.7D + 2.51/Re √f)

where ε = the roughness (m), D = diameter (m), and Re = the Reynolds number defined as

Re = ΡVD/ μ

With Ρ = the fluid's density (kg/m³), V = the fluid's velocity, and μ = dynamic viscosity (N·s/m²).

Assume Ρ = 1.23 kg/m³, μ = 1.79×10^-5 N·s/m², D = 0.005 m, V = 40 m/s, and ε = 1.5×10^-6 m.

Part I: By-hand

(a) Determine the friction factor f of the system using the Secant method. Start with initial guesses f = 0.01 and 0.02, and perform two iterations.

Part II: Matlab: Find the friction factor f by performing iterations until the approximate relative error

ε_a = |f_i+1-f_i/f_i| · 100 % falls below 2 %

(b) Use the Newton's method with initial guess f = 0.01;

(c) Use the Secant method with initial guesses f = 0.01 and 0.02.