OPEN-CHANNEL HYDRAULICS:  LECTURE 014 - FRICTION RELATION

1. DARCY-WEISBACH FORMULA
1.01
The Darcy-Weisbach friction equation, applicable to closed-conduit flow, is:
1.02

         L   v2
hf = fD 
 
         do  2g

1.03
in which hf = head loss, fD = Darcy-Weisbach friction factor, L = length of conduit, do = pipe diameter, v = flow velocity, g = gravitational acceleration.
1.04
The hydraulic radius can be expressed in terms of pipe diameter as follows:
1.05

     A     (1/4) π do2      do
R = 
  = 
 = 
     P         π do         4

1.06
Therefore, the head loss in terms of hydraulic radius is:
1.07

         L  v2
hf = fD 
 
        4R  2g

1.08
The energy slope is:
1.09

     hf      fD v2
S = 
  = 
     L      8 g R

1.10
Multiplying by hydraulic depth D:
1.11

      A
D = 
 
      T

1.12
in which T = top width.
1.13

     fD  D   v2
S = 
 
 
     8   R  gD

1.14
The most general definition of the Froude number is in terms of hydraulic depth:
1.15

       v
F = 
 
     (gD)1/2

1.16
Thus, the energy slope is:
1.17

     fD  D
S = 
 
 F2
     8   R


2. FRICTION RELATION
2.01
A hydraulically wide channel is that in which the ratio of top width to depth is greater than 10.
1.19
Most natural channels are hydraulically wide.
2.02
For a hydraulically wide channel, the hydraulic radius can be approximated by the hydraulic depth.
2.03
Thus, the friction relation reduces to:
2.04

     fD  
S = 
 F2
     8   

2.05
For simplicity, we define a friction factor f equal to 1/8 of the Darcy-Weisbach friction factor.
2.06
Then, the energy slope reduces to its simplest form:
2.07

S = f F2

2.08
Thus, the Froude number defines the relation between gravity and friction.
2.09
The energy slope is proportional to the Froude number, with the friction factor as the proportionality factor.
2.10
For constant friction, an increase in energy slope causes an increase in Froude number; conversely, a decrease in energy slope causes a decrease in Froude number.
2.11
For constant Froude number, an increase in friction causes an increase in energy slope; conversely, a decrease in friction causes a decrease in energy slope.
2.12
For constant energy slope, an increase in friction causes a decrease in Froude number, and viceversa.

3. FRICTION FACTOR
3.01
In laminar flow, the friction factor is inversely proportional to the Reynolds number.
3.02

      K
fD = 
 
      Re

3.03
For smooth channels, K varies in the range 14-24.
3.04
For rough channels, K varies in the range 33-60.
3.05
In turbulent flow, K is less dependent on the Reynolds number.
3.06
For turbulent flow in smooth channels, the Darcy-Weisbach friction factor fD is typically in the range 0.016-0.040.
3.07
Thus, the friction factor f is in the range 0.002-0.005.
3.08
For turbulent flow, the relation between Darcy-Weisbach friction factor and Reynolds number can be approximated with the implicit Prandtl-von Karman equation:
3.09

  1
 = 2 log (Re √fD) + 0.4
 √fD

3.10
which can be expressed explicitly as follows:
3.11

       1
Re = 
 10 (1 - 0.4 √fD)/(2 √fD)
      √fD

3.12
4.00
4.01
4.02
Narrator: Victor M. Ponce

Music: Fernando Oñate

Editor: Flor Pérez

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