5.3 Natural Ventilation
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5.3.2 Calculation of the Logarithmic Formula Method
Despite the simplifications in the previous section, pressure does increase with depth,
and its increase takes place in a nonlinear manner. In order to consider this fact, the
following expression was developed. A height differential (dL), of a dry air column
of height h, increases the hydrostatic pressure at its base (A) by a value dP. Therefore,
establishing the balance of forces, we have:
A dP = γ A dL
where γ is the specific weight of air. This value can be obtained from the general
gas equation, therefore:
A dP =
P
R w T
A dL
where
R: Constant of gases (per unit mass)
J
K kg
.
Simplified:
dP =
P
R T
dL
Integrating between the pressure at the top of the downcast shaft (P 1 ) and the pressure at the bottom (P 2 ), for a depth of the shaft (h), and considering the temperature
as an integration constant (T md ), we have:
P 2
∫
P 1
dP
P
=
h d
∫
0
dL
R T md
Therefore, the pressure at the base of the downcast shaft is (Eq. 5.6):
log P 2 = log P 1 + 0.03415
h d
T md
(5.6)
In a similar way, the pressure at the base of the upcast shaft can be obtained
(Eq. 5.7):
log P 3 = log P 1 + 0.03415
h u
T mu
(5.7)
where
• h u : Depth of upcast shaft (m), and
• T mu : Average temperature of the upcast shaft air (K).