3 Source modelling
3.2 Pool evaporation calculations
3.2.1 Yellow Book (as input to DEGADIS)
The time varying evaporation rate from the pool, q(t), is calculated by:
A
28 FFI-rapport 2010/00874
where Hc(t) and Ha are the heat fluxes from the subsoil and the air, Lv(Tb) is the latent heat of vaporisation at the boiling temperature Tb, and A is the area of the pool. The latent heat is a temperature dependent parameter of the evaporating substance, while the heat flux is a property of the subsoil and the surrounding air. The heat flux from the subsoil is calculated by:
( ) (
,0) /
c R s s b s
H t C T T a t
,where CR is a correction term to reflect freezing of the water in the subsoil, λs is the thermal conductivity of the subsoil, as the thermal diffusivity of the subsoil, and Ts,0 the initial subsoil temperature. The thermal diffusivity is related to the thermal conduction and the specific heat of the subsoil, Cs, by:
where Ta is the ambient temperature and kH,a the heat transfer coefficient to the atmosphere. The heat transfer coefficient can be estimated by:
r Nu a
H a
k
,
2 ,where λa is the thermal conductivity of air, 2r a characteristic length of the pool (in these calculations it is set equal to the pool diameter), and Nu the Nusselt number, which can be expressed by the Reynold’s number (Re) and Schmidt’s number (Sc) by:
Da
U is the air velocity, ν the kinematic viscosity of air, and Da the thermal diffusivity. For air, the Schmidt number is: Sc ≈ 0.8.
Figure 3.1 shows the calculated evaporation rate from the chlorine pool described above, when the ambient conditions are as described in Chapter 4.5. The temperature of the soil is taken to be that of air (14.2 °C). The evaporation rates calculated with heat flux from only the ground and atmosphere are also shown. Initially the heat flux from the subsoil is dominating, but after some time, heat flux from the air becomes comparable and even dominating.
4 The TNO Yellow Book [22] also lists other formulas for calculating the heat flux from the air, which give quite different results. A comparison between different methods, and an assessment of which method would give the most accurate result, are not given in this report. This formula is given as an example.
FFI-rapport 2010/00874 29
Figure 3.1 The evaporation rate from a chlorine pool with area 900 m2 as function of time. The calculated evaporation rate with only heat flux from the surrounding air, Ha (red), only from the ground, Hc (blue), and the total rate (black) are shown.
The heat flux from the air gives a constant evaporation rate. This will not be completely correct.
Firstly, it is assumed that the temperature of the air above the pool does not decrease. This is a simplification as the temperature of the air will decrease because heat is taken from the air by the evaporation process. Secondly, the evaporation rate will decrease toward the end as the area of the pool decreases. (The figure shows the evaporation rate for the first 40 minutes only.) Figure 3.2 shows the evaporated mass from the pool as function of time as calculated with the evaporation rates shown in Figure 3.1 with combined heat from the ground and air and heat only from the ground. It is clearly seen that the evaporation rates are equal in the first minutes.
However, after some time heat from the air stream is dominant. With only heat from the ground, the pool will evaporate in about four hours, but when heat from the passing air is included, the evaporation time decrease to about one hour5.
As mentioned above, however, the evaporation rate will decrease toward the end. Thus the time for the evaporation would be somewhat larger than shown in the figure, and in reality the curves for the evaporated mass will flatten out when approaching 14000 kg (the original mass of the pool).
5 As mentioned in footnote 4, there are other methods for calculating the heat flux from air, which would alter the results somewhat.
30 FFI-rapport 2010/00874
Figure 3.2 Evaporated mass from a chlorine pool as a function of time, for combined heat
transfer from the air and the ground, and from the ground only.
3.2.2 ARGOS
The pool evaporation in ARGOS assumes heat transfer mainly from the surface beneath the pool, but also from the air and from short-wave and long-wave radiation. The portion of the total heat flux of the radiation depends on solar angle, cloud coverage and other external conditions.
However, for cryogenics, thermal conduction from the surface beneath the pool will be dominating [23].
ARGOS computes the heat flux from the pool basin as heat diffusion from a semi-infinite solid with uniform material properties and no porosity. The thermal conduction from the surface, λs, is described by the ordinary heat conduction equation [23]:
2 2
z s T Tt s
s
C
,where ρs is the density of the soil, Cs is the heat capacity of the soil, and z is the depth into the subsoil. From this, an expression for the heat flux is found:
z s T c
t
H ( )
,which, for constant surface temperature, is simplified to:
t C
c t T s s s
H ( ) .
This is the same formula for the heat flux as in the TNO Yellow Book, except that the Yellow Book equation includes a factor to account for freezing of water in the subsoil.
The evaporation mass flux is then calculated by dividing the heat, H(t), by the latent heat of evaporation.
FFI-rapport 2010/00874 31 3.2.3 HPAC
The mass transfer rate, q, in the HPAC calculation of pool evaporation is given by the Sherwood number:
DWC0
Sh
q ,where D is the diffusivity of vapour in air, W is a width of an equivalent square pool (W=π1/2R, where R is the radius of the circular pool), and C0 the saturation concentration. HPAC uses expressions for the Sherwood number from the HGSYSTEM [24] for calculating the evaporation rate:
where the Reynold’s number based on the wind speed, U, is:
a
where ρa, νa and μa are the density and the kinematic and dynamic viscosity of air respectively.
This mass flux (evaporation rate) is coupled to the heat fluxes by the latent heat of evaporation:
)
The heat to the evaporation process are taken from heat conduction to the ground and convected heat flux from the air by similar expressions as for the Yellow Book calculations:
t
where Tp is the pool temperature (not necessarily equal to the boiling temperature) and the ambient temperature, Ta, is taken 10 meters from the pool. These are also equal to the Yellow Book formulas, except the correction factor for freezing of water in the subsoil. From the last three equations, the pool temperature is calculated.
The physical properties of the underlying surface are taken to be that of sand.