4.2 Friction Losses: Atkinson Expression
89
A new parameter is obtained, which is called the Atkinson friction factor (K 1.2 )
and is a function of Darcy’s friction factor
3 (f ) (Eq. 4.4), which, in turn, can be
obtained by Eqs. 1.25 to 1.29.
4
According to the paragraphs above, the equation for pressure losses can be written
as (Eq. 4.5):
P = K 1.2
O L
A
v
2
(4.5)
Nevertheless, expressing the flow as a function of the cross section and air speed,
the Atkinson equation (1862) (Eq. 4.6) is obtained:
P = K 1.2
O
L + L eq
Q
2
A 3
(4.6)
This expression states that the friction pressure losses in the airway are a function
of the fluid velocity, the characteristics of its inner surface (K 1.2 ) and its dimensions
(O, L, L eq and A).
where
• P: Pressure difference (Pa), (inch water);
• K 1.2 : Atkinson friction factor for 1.2 g cm
−3 air density (N s
2 m
−4 ó kg m
−3 ), (lb
min
2 ft
−4 );
• O: Perimeter of the airway (m), (ft);
• L and L eq : Length and equivalent length (m), (ft);
• Q: Volumetric air flow rate (m
3 s
−1 ), (ft
3 s
−1 );
• A: Cross-sectional area of the airway (m
2 ), (ft
2 ); and
• ρ: Air density (kg m
−3 ), (lb ft
−3 ).
The airflow is affected by the average air density, which for the purposes of the
above calculations has been considered to be 1.2 kg m
−3 . So, if you wish to correct
possible density variations due to altitude or temperature, you can use a corrected
coefficient (K c ), which is expressed as (Eq. 4.7):
K c = K 1.2
ρ
1.2
kg
m 3
(4.7)
3 The “f ” used in this text corresponds to the friction factor of Darcy’s law, sometimes denoted as
f D . Although the former is the most widespread, it is possible to find bibliographic sources using
Fanning’s friction factor (f F ), for example, Table 5.1 in McPherson (1993), p. 138. The relationship
between the two is: f D = 4f F . So in this case: K 1.2 = 0.6 f F .
4 Note that f also depends on roughness. The same values can be found for different types of mining
air conduits, for example, in Montecinos and Wallace (2010).
89
A new parameter is obtained, which is called the Atkinson friction factor (K 1.2 )
and is a function of Darcy’s friction factor
3 (f ) (Eq. 4.4), which, in turn, can be
obtained by Eqs. 1.25 to 1.29.
4
According to the paragraphs above, the equation for pressure losses can be written
as (Eq. 4.5):
P = K 1.2
O L
A
v
2
(4.5)
Nevertheless, expressing the flow as a function of the cross section and air speed,
the Atkinson equation (1862) (Eq. 4.6) is obtained:
P = K 1.2
O
L + L eq
Q
2
A 3
(4.6)
This expression states that the friction pressure losses in the airway are a function
of the fluid velocity, the characteristics of its inner surface (K 1.2 ) and its dimensions
(O, L, L eq and A).
where
• P: Pressure difference (Pa), (inch water);
• K 1.2 : Atkinson friction factor for 1.2 g cm
−3 air density (N s
2 m
−4 ó kg m
−3 ), (lb
min
2 ft
−4 );
• O: Perimeter of the airway (m), (ft);
• L and L eq : Length and equivalent length (m), (ft);
• Q: Volumetric air flow rate (m
3 s
−1 ), (ft
3 s
−1 );
• A: Cross-sectional area of the airway (m
2 ), (ft
2 ); and
• ρ: Air density (kg m
−3 ), (lb ft
−3 ).
The airflow is affected by the average air density, which for the purposes of the
above calculations has been considered to be 1.2 kg m
−3 . So, if you wish to correct
possible density variations due to altitude or temperature, you can use a corrected
coefficient (K c ), which is expressed as (Eq. 4.7):
K c = K 1.2
ρ
1.2
kg
m 3
(4.7)
3 The “f ” used in this text corresponds to the friction factor of Darcy’s law, sometimes denoted as
f D . Although the former is the most widespread, it is possible to find bibliographic sources using
Fanning’s friction factor (f F ), for example, Table 5.1 in McPherson (1993), p. 138. The relationship
between the two is: f D = 4f F . So in this case: K 1.2 = 0.6 f F .
4 Note that f also depends on roughness. The same values can be found for different types of mining
air conduits, for example, in Montecinos and Wallace (2010).