Table A1. Constants for the friction factor equation.
Water
Flow
Pipe
Reynolds
temp.
velocity
diameter
number
Max.
Avg.
(C)
106
(m/s)
(m)
error
error
min/max
min/max
min/max
min/max
a
b
c
(%)
(%)
50/130
0.5/3.3
0.05/0.77
0.04/11
0.123
0.146
0.0626
6.2
1.0
50/130
0.5/4.5
0.05/0.77
0.04/15
0.119
0.152
0.0568
6.9
1.1
70/150
0.3/6.3
0.03/0.93
0.02/29
0.129
0.156
0.0589
10.8
2.0
50/90
0.5/2.9
0.05/0.41
0.04/3.7
0.140
0.141
0.0762
4.1
0.8
90/130
0.5/2.9
0.10/0.46
0.16/5.9
0.116
0.150
0.0563
2.5
0.6
50/90
0.5/2.9
0.10/0.46
0.09/4.1
0.128
0.132
0.0751
3.4
0.7
50/90
0.5/3.7
0.10/0.46
0.09/5.3
0.125
0.137
0.0698
3.9
0.8
90/130
0.5/3.7
0.10/0.46
0.16/7.5
0.113
0.154
0.0520
2.8
0.6
n
A23 = A32 = ∑ X2,i X1,i
i =1
n
(X2,i )2
A33 = ∑
i =1
n
C1 = ∑ Yi
i =1
n
C2 = ∑ Yi X1,i
i =1
n
C3 = ∑ Yi X2,i .
i =1
This system of linear equations can be solved by forward elimination and
subsequent back solution. The resulting expressions for the parameters are
(
)
A32 - (A12 A31 / A11)
} [
]
{
C3 - (C1A31 / A11) - C2 - (C1A21 / A11)
(
))
(
2
A22 - A12 / A11
β2 =
(A14)
(
)
A32 - (A12 A31 / A11)
{
)} [
(
]
A33 - A13 / A11 - A23 - (A13 A21 / A11)
2
(
))
(
2
A22 - A12 / A11
{C2 - (C1A21 / A11)} - {[A23 - (A13 A21 / A11)][β2 ]}
β1 =
(A15)
{A - (
)}
2
A12 / A11
22
{
}
β0 = C1 - [A12β1] - [A13β2 ] / A11.
(A16)
A FORTRAN program FFCONST was written to evaluate the A's and C's in the
^
factor f can then be found. For this program, the "observed" friction factor f is found
using the Colebrook-White equation (Jeppson 1976)
f )]-2 .
f = [1.14 0.869 ln(RR + 9.35/Re
(A-17)
The Colebrook-White equation is implicit in the friction factor f and thus it cannot
be solved directly. A number of methods can be used to solve implicit equations such
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