174
5 Main Ventilation
Bearing in mind the equations for an adiabatic system with self-compression of
air (Eq. 2.9), Voropaev (1950) stated that: “The work performed by 1 kg of air in a
natural ventilation system can be approximated with the area of a closed contour in
a coordinate system depth-temperature (h-T ), divided by the absolute temperature
of its centroid
6 ” (e.g. Novitzky 1962, p. 211) (Eq. 5.11):
P n =
S
T c
ρ
(5.11)
where
• P n : Natural ventilation pressure (NVP) (kp m
−2 ),
• S: Surface of the figure
kp m
kg
K
,
• T c : Centroid temperature (K), and
• ρ: Air density (usually taken as 1.25 kg m
−3 ).
The limitations of the above expression in the presence of a fan have been discussed
by Lepikhov (1975).
Surface decomposition can be systematized by the shoelace formula. This procedure consists of defining a polygonal surface from a series of lines determined by
pairs of points (x i–1 , y i–1 ), (x i , y i ) of a Cartesian coordinate system. In this nomenclature, i is the point number. It is important that the last point in the series coincides
with the first for the polygonal to be closed. Also, in order to not complicate the
method, the coordinates defining the polygonal should not produce line crossings.
The area of each element defined by coordinates is calculated as:
A n = x i y i−1 − x i−1 y i
Then the total area of the closed polygonal is:
A =
1
2
A i
The centroid of any triangle is given by the following expression:
X g =
1
3
(x 1 + x 2 + x 3 )
Y g =
1
3
(y 1 + y 2 + y 3 )
Exercise 5.4 The figure represents the scheme of a simplified mine with sidehill
shafts. The data corresponding to the coordinates of each point are included in the
table. You are asked to:
6 The centroid or barycenter is a geometric concept which defines the centre of symmetry of a
geometric figure. For a body with uniform density, it coincides with the centre of masses.
5 Main Ventilation
Bearing in mind the equations for an adiabatic system with self-compression of
air (Eq. 2.9), Voropaev (1950) stated that: “The work performed by 1 kg of air in a
natural ventilation system can be approximated with the area of a closed contour in
a coordinate system depth-temperature (h-T ), divided by the absolute temperature
of its centroid
6 ” (e.g. Novitzky 1962, p. 211) (Eq. 5.11):
P n =
S
T c
ρ
(5.11)
where
• P n : Natural ventilation pressure (NVP) (kp m
−2 ),
• S: Surface of the figure
kp m
kg
K
,
• T c : Centroid temperature (K), and
• ρ: Air density (usually taken as 1.25 kg m
−3 ).
The limitations of the above expression in the presence of a fan have been discussed
by Lepikhov (1975).
Surface decomposition can be systematized by the shoelace formula. This procedure consists of defining a polygonal surface from a series of lines determined by
pairs of points (x i–1 , y i–1 ), (x i , y i ) of a Cartesian coordinate system. In this nomenclature, i is the point number. It is important that the last point in the series coincides
with the first for the polygonal to be closed. Also, in order to not complicate the
method, the coordinates defining the polygonal should not produce line crossings.
The area of each element defined by coordinates is calculated as:
A n = x i y i−1 − x i−1 y i
Then the total area of the closed polygonal is:
A =
1
2
A i
The centroid of any triangle is given by the following expression:
X g =
1
3
(x 1 + x 2 + x 3 )
Y g =
1
3
(y 1 + y 2 + y 3 )
Exercise 5.4 The figure represents the scheme of a simplified mine with sidehill
shafts. The data corresponding to the coordinates of each point are included in the
table. You are asked to:
6 The centroid or barycenter is a geometric concept which defines the centre of symmetry of a
geometric figure. For a body with uniform density, it coincides with the centre of masses.