Friction Factor Calculation
Content
The Darcy-Weisbach equation
Weisbach first proposed the equation we now know as the Darcy-Weisbach formula or
Darcy-Weisbach equation:
hf = f (L/D) x (V^2/2g)
where:
hf = head loss (m)
f = friction factor
L = length of pipe work (m)
D = inner diameter of pipe work (m)
v = velocity of fluid (m/s)
g = acceleration due to gravity (m/s²)
Determination of the friction factor used in the Darcy -Weishbach equation
A) Smooth pipe / laminar flow
f =64/Re
B)
Re <2100
Friction factor determination ( for turbulent flow )
Numerical resolutions:
B-1
1)
using
Colebrook's equation
Newton Raphson's method
1/f^1/2= -2 *log10(eps/3.7/D+2.51 /Re/f^1/2)
fn+1=fn-F(fn)/F'(fn)
equations :
F(f)= 1/f^1/2+2* log10(eps/3.7/D+2.51 /Re/f^1/2) =0
a)
F(f)=f^-1/2+2 *ln(eps/3.7/D+2.51 /Re*f^-1/2)/ ln(10)=0
b)
F'(f)=-1/2*f^-3/2 *( 1+2 *2.51/ Re/ln(10)/(eps/3.7/D+2.51 /Re*f^-1/2))
initialization : f=
Input Data
data
pipe diameter D
rugosity
flow rate
density
eps
mm
4.00E-05
mm
8
m3/h
1000
kg/m3
1
cpo
Velocity
4.39
m/s
111395
> 4000
turbulent
Output Data
f
B-2
1.7591E-02
Swamee& Jain's equation
f= 0.25/(log10(eps/3.7/D+5.74 /Re^0.9))^2
Input Data
data
pipe diameter D
rugosity
eps
calculation
25.4
mm
4.00E-05
mm
flow rate
8
m3/h
density
1000
kg/m3
Viscosity
1
cpo
Velocity
4.39
m/s
Re
111395
> 4000
turbulent
Output Data
f
C)
1.7467E-02
Churchill's equation
(full range from laminar to turbulent)
f=8*((8/Re)^12+((37530/Re)^16+B)^-1.5)^1/12
with
B=(-2.457*ln((7/Re)^0.9+0.27*(eps/D)))^16
Input Data
data
pipe diameter D
rugosity
eps
calculation
25.4
mm
4.00E-05
mm
flow rate
8
m3/h
density
1000
kg/m3
Viscosity
1
cpo
Velocity
4.39
m/s
Re
B
111395
1.91723E+21
Output Data
f
fn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.01
1.50E-02
1.73E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
0.01 f=
F(fn)
1.7591170583E-02
F'(fn)
fn+1
initialization X=
Δ
calculation
25.4
Viscosity
Re
Iteration
Number
1.7489E-02
2100 >
laminar
Re
>4000
turbulent
2.71E+00
7.01E-01
7.00E-02
8.28E-04
1.18E-07
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
Colebrook's equation
2)
using
Alternative Method
iterative calculation
X=F(X)
with
X = 1/(f^1/2)
F(X)= -2 *log10(eps/3.7/D+2.51 /Re*X)
-543.43
-301.74
-244.81
-239.06
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
1.50E-02
1.73E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
0.004979
0.002323
0.000286
3.46E-06
4.94E-10
0.00E+00
0
0
0
0
0
0
0
0
0
iteration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
X
3
8.33E+00
7.45E+00
7.55E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
3 f = 1/X^2
F(X)
8.33E+00
7.45E+00
7.55E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
X-F(X)
-5.33E+00
8.84E-01
-9.72E-02
1.12E-02
-1.29E-03
1.48E-04
-1.70E-05
1.96E-06
-2.25E-07
2.59E-08
-2.97E-09
3.42E-10
-3.93E-11
4.52E-12
-5.20E-13
6.04E-14
-7.11E-15
1.760027025475E-02
PVC, CPVC Pipe Same I.D. as Steel Pipe of Comparable Schedule.
Pipe Type
Lined Pipe
FRP
Steel
Liner
PTFE
N/A
N/A
D, Nom.
1
1.5
2
3
4
6
8
10
12
1
1.5
2
3
4
6
8
10
12
14
D, Nom.
1
1.5
2
3
4
6
8
10
12
14
DOW
ID
0.78
1.31
1.82
2.8
3.76
5.61
7.46
9.73
11.69
Crane
ID
0.78
1.27
1.78
2.76
3.76
5.51
7.48
9.49
11.29
Matl. Spec. for lined Pipe: ASTM F 1545-95
19.81
19.81
33.27
32.26
46.23
45.21
71.12
70.10
95.50
95.50
142.49
139.95
189.48
189.99
247.14
241.05
296.93
286.77
Fibercast Fibercast
RB-2530 CC-II-EP
ID
ID
CC-II-EP limit to 225 F
0.935
X
23.75
1.48
1.62
37.59
41.15
1.875
2.135
47.63
54.23
2.98
3.26
75.69
82.80
3.92
4.26
99.57
108.20
6.025
6.345
153.04
161.16
8.025
8.285
203.84
210.44
10.15
10.37
257.81
263.40
12.45
12.37
316.23
314.20
13.4
13.56
340.36
344.42
Sch-40
ID
1.049
1.61
2.067
3.068
4.026
6.065
7.981
10.02
11.938
13.124
Sch-80
ID
0.957
1.5
1.939
2.9
3.826
5.761
7.625
9.562
11.374
12.5
Sch-40
ID mm
26.64
40.89
52.50
77.93
102.26
154.05
202.72
254.51
303.23
333.35
Sch-80
ID mm
24.31
38.10
49.25
73.66
97.18
146.33
193.68
242.87
288.90
317.50
RB-2530 limit to 250 F.
absolute rugosity
PTFE limit to -20 to 500 F
glass
lead
copper
laiton
mm
0.0015
0.0015
0.0015
0.0015
steel
new
very old
rusty
v.much rusty
galvanized
0.04
0.2
0.4
3.35
0.15
Concrete
0.025-0.250
cast iron
new
dirty
rusty
v. much rusty
0.5
0.2
1.5
3
PTFE lined
0.0015
Fiber glass
0.005
Weisbach first proposed the equation we now know as the Darcy-Weisbach formula or
Darcy-Weisbach equation:
hf = f (L/D) x (V^2/2g)
where:
hf = head loss (m)
f = friction factor
L = length of pipe work (m)
D = inner diameter of pipe work (m)
v = velocity of fluid (m/s)
g = acceleration due to gravity (m/s²)
Determination of the friction factor used in the Darcy -Weishbach equation
A) Smooth pipe / laminar flow
f =64/Re
B)
Re <2100
Friction factor determination ( for turbulent flow )
Numerical resolutions:
B-1
1)
using
Colebrook's equation
Newton Raphson's method
1/f^1/2= -2 *log10(eps/3.7/D+2.51 /Re/f^1/2)
fn+1=fn-F(fn)/F'(fn)
equations :
F(f)= 1/f^1/2+2* log10(eps/3.7/D+2.51 /Re/f^1/2) =0
a)
F(f)=f^-1/2+2 *ln(eps/3.7/D+2.51 /Re*f^-1/2)/ ln(10)=0
b)
F'(f)=-1/2*f^-3/2 *( 1+2 *2.51/ Re/ln(10)/(eps/3.7/D+2.51 /Re*f^-1/2))
initialization : f=
Input Data
data
pipe diameter D
rugosity
flow rate
density
eps
mm
4.00E-05
mm
8
m3/h
1000
kg/m3
1
cpo
Velocity
4.39
m/s
111395
> 4000
turbulent
Output Data
f
B-2
1.7591E-02
Swamee& Jain's equation
f= 0.25/(log10(eps/3.7/D+5.74 /Re^0.9))^2
Input Data
data
pipe diameter D
rugosity
eps
calculation
25.4
mm
4.00E-05
mm
flow rate
8
m3/h
density
1000
kg/m3
Viscosity
1
cpo
Velocity
4.39
m/s
Re
111395
> 4000
turbulent
Output Data
f
C)
1.7467E-02
Churchill's equation
(full range from laminar to turbulent)
f=8*((8/Re)^12+((37530/Re)^16+B)^-1.5)^1/12
with
B=(-2.457*ln((7/Re)^0.9+0.27*(eps/D)))^16
Input Data
data
pipe diameter D
rugosity
eps
calculation
25.4
mm
4.00E-05
mm
flow rate
8
m3/h
density
1000
kg/m3
Viscosity
1
cpo
Velocity
4.39
m/s
Re
B
111395
1.91723E+21
Output Data
f
fn
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.01
1.50E-02
1.73E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
0.01 f=
F(fn)
1.7591170583E-02
F'(fn)
fn+1
initialization X=
Δ
calculation
25.4
Viscosity
Re
Iteration
Number
1.7489E-02
2100 >
laminar
Re
>4000
turbulent
2.71E+00
7.01E-01
7.00E-02
8.28E-04
1.18E-07
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
Colebrook's equation
2)
using
Alternative Method
iterative calculation
X=F(X)
with
X = 1/(f^1/2)
F(X)= -2 *log10(eps/3.7/D+2.51 /Re*X)
-543.43
-301.74
-244.81
-239.06
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
-238.99
1.50E-02
1.73E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
1.76E-02
0.004979
0.002323
0.000286
3.46E-06
4.94E-10
0.00E+00
0
0
0
0
0
0
0
0
0
iteration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
X
3
8.33E+00
7.45E+00
7.55E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
3 f = 1/X^2
F(X)
8.33E+00
7.45E+00
7.55E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
7.54E+00
X-F(X)
-5.33E+00
8.84E-01
-9.72E-02
1.12E-02
-1.29E-03
1.48E-04
-1.70E-05
1.96E-06
-2.25E-07
2.59E-08
-2.97E-09
3.42E-10
-3.93E-11
4.52E-12
-5.20E-13
6.04E-14
-7.11E-15
1.760027025475E-02
PVC, CPVC Pipe Same I.D. as Steel Pipe of Comparable Schedule.
Pipe Type
Lined Pipe
FRP
Steel
Liner
PTFE
N/A
N/A
D, Nom.
1
1.5
2
3
4
6
8
10
12
1
1.5
2
3
4
6
8
10
12
14
D, Nom.
1
1.5
2
3
4
6
8
10
12
14
DOW
ID
0.78
1.31
1.82
2.8
3.76
5.61
7.46
9.73
11.69
Crane
ID
0.78
1.27
1.78
2.76
3.76
5.51
7.48
9.49
11.29
Matl. Spec. for lined Pipe: ASTM F 1545-95
19.81
19.81
33.27
32.26
46.23
45.21
71.12
70.10
95.50
95.50
142.49
139.95
189.48
189.99
247.14
241.05
296.93
286.77
Fibercast Fibercast
RB-2530 CC-II-EP
ID
ID
CC-II-EP limit to 225 F
0.935
X
23.75
1.48
1.62
37.59
41.15
1.875
2.135
47.63
54.23
2.98
3.26
75.69
82.80
3.92
4.26
99.57
108.20
6.025
6.345
153.04
161.16
8.025
8.285
203.84
210.44
10.15
10.37
257.81
263.40
12.45
12.37
316.23
314.20
13.4
13.56
340.36
344.42
Sch-40
ID
1.049
1.61
2.067
3.068
4.026
6.065
7.981
10.02
11.938
13.124
Sch-80
ID
0.957
1.5
1.939
2.9
3.826
5.761
7.625
9.562
11.374
12.5
Sch-40
ID mm
26.64
40.89
52.50
77.93
102.26
154.05
202.72
254.51
303.23
333.35
Sch-80
ID mm
24.31
38.10
49.25
73.66
97.18
146.33
193.68
242.87
288.90
317.50
RB-2530 limit to 250 F.
absolute rugosity
PTFE limit to -20 to 500 F
glass
lead
copper
laiton
mm
0.0015
0.0015
0.0015
0.0015
steel
new
very old
rusty
v.much rusty
galvanized
0.04
0.2
0.4
3.35
0.15
Concrete
0.025-0.250
cast iron
new
dirty
rusty
v. much rusty
0.5
0.2
1.5
3
PTFE lined
0.0015
Fiber glass
0.005
