[Angelo]

good day ,
I took your advice and did a little research.

i wrote the program that works out the friction factor using approximation fd= 0.25/(log10(k/(3.7d) + 5.74/(re^0.9))^2 for re >= 4000.

with this answer as my initial guess i used derivatives to calculate the friction factor. reducing processing time by more 2/3 rd

thanx for your help