In my Crane TP410 (25th printing in 1991) the derivation of their "Resistance of Bends" equation starts on page 2-12 and is presented in final form in Equation 2-20:
KB = (n-1)(0.25.fT.Pi.(r/d) + 0.5.K1) + K1
n = number of 90o bends
fT = fully turbulent Darcy friction factor
r = radius of bend
d = diameter of pipe (I use 4 x RH for 1/2 pipe)
K1 = K of 1 bend from Table on page A-29
r/d K1
1 20.fT
1.5 14.fT
2 12.fT
3 12.fT
4 14.fT
6 17.fT
8 8.fT
10 10.fT
12 12.fT
14 14.fT
16 16.fT
20 20.fT
When I put KB into a pressure drop equation I use d = equivalent flow area diameter, square root of (4.A/Pi), so the velocity is correct.
It has been field tested many, many times by me over the years.
Good luck,
Latexman