## 3 Methods
### Models of Radial and Flat Diffusion Based on Fick's Laws
#### 3.2.1 Model of Radial Flow Diffusion and Determination of the Emission Rate G
2.2 Model of Flat Upward Flow Diffusion and Determination of Mass Transfer Coefficient K\({}^{\prime}\)  
The diffusion velocity v\({}_{A}\) is constant at any plane perpendicular to the direction of flow and its value is:  
\[\mathrm{v_{A}}=\frac{\mathrm{N_{A}}}{\mathrm{C_{A}}} \tag{9}\]  
v\({}_{A}\): diffusion velocity (m/s)  
N\({}_{A}\): molar flux of gas A per unit area perpendicular to flow (mol/s m\({}^{2}\))  
C\({}_{A}\) molar concentration of gas A (mol/m\({}^{3}\)), since the velocity in a particular plane at a given height is constant, in that plane there will be a proportionality between the gas flow through it and the gas concentration above it:  
\[\mathrm{N_{A}}=\mathrm{v_{A}}\mathrm{C_{A}}=\mathrm{K}\mathrm{C_{A}} \tag{10}\]  
K: coefficient of proportionality (m/s)  
It will be assumed that this proportionality also exists in the upward plane flow produced by the emission of gas from a solid surface, area A (m\({}^{2}\)), as is the case in this study, with an emission rate G (ng/s). The concentration C\({}_{i}\) (ng/m\({}^{3}\)) at a given height above the surface will be proportional to the flux F (ng/s m\({}^{2}\)) at any time:  
\[\mathrm{F}=\frac{\mathrm{G}}{\mathrm{A}}=\mathrm{K^{\prime}}\mathrm{C_{i}} \tag{11}\]  
K\({}^{\prime}\): coefficient of proportionality (m/s)  
If the movement of the gas in the plane of the surface was due to the pure diffusion of the gas (e.g., if it were the surface of the liquid), the constant K would be equal to the diffusion velocity, which, in the case of the plane flow, would be v\({}_{A}\)= D/z, where D is the diffusivity coefficient of the gas and z is the distance from the plane to the focus. However, the movement of the gas from the debris surface is due to a phenomenon of emission from a solid and is therefore not exclusively due to diffusion. The coefficient of proportionality between concentration and flux, K\({}^{\prime}\), is not equal to the diffusion rate, so it must be estimated. The following relationship is fulfilled at the focus:  
\[\mathrm{F}=\frac{\mathrm{G}}{\mathrm{A}}=\mathrm{K^{\prime}}\mathrm{C_{10}} \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**0** (\({}^{\circ}\)**C)** & **T (K)** & **A (m\({}^{2}\))** & **C\({}_{9}\) (ng/m\({}^{3}\))** & **D (m\({}^{2}\)/s)** & **G (ng/s)** & **F (ng/s m\({}^{2}\))** & **F (ng/m\({}^{2}\) h)** \\ \hline
29 & 302 & 314 & 20,867 & \(1.18\times 10^{-5}\) & 15.51 & 0.0494 & 177.8 \\
30 & 303 & 314 & 15,000 & \(1.19\times 10^{-5}\) & 11.22 & 0.0357 & 128.6 \\
6.5 & 279.5 & 314 & 5524 & \(1.03\times 10^{-5}\) & 3.57 & 0.0114 & 40.9 \\
7.5 & 280.5 & 314 & 1330 & \(1.04\times 10^{-5}\) & 4.53 & 0.0028 & 9.9 \\
12.5 & 285.5 & 314 & 6488 & \(1.07\times 10^{-5}\) & 4.36 & 0.0139 & 49.9 \\
10.5 & 283.5 & 314 & 4689 & \(1.06\times 10^{-5}\) & 3.11 & 0.0099 & 35.6 \\
4 & 277 & 314 & 7205 & \(1.01\times 10^{-5}\) & 4.58 & 0.0146 & 52.5 \\
11 & 284 & 314 & 7153 & \(1.06\times 10^{-5}\) & 6.06 & 0.0152 & 54.5 \\
14 & 287 & 314 & 6426 & \(1.08\times 10^{-5}\) & 5.62 & 0.0139 & 49.9 \\
14.5 & 287.5 & 314 & 11,493 & \(1.08\times 10^{-5}\) & 7.82 & 00249 & 89.6 \\
24 & 297 & 314 & 19,681 & \(1.15\times 10^{-5}\) & 14.19 & 0.0452 & 162.7 \\
21 & 294 & 314 & 8785 & \(1.13\times 10^{-5}\) & 6.22 & 0.0198 & 71.3 \\
13.5 & 286.5 & 314 & 4387 & \(1.08\times 10^{-5}\) & 2.96 & 0.0094 & 34.0 \\
3 & 276 & 314 & 3402 & \(1.01\times 10^{-5}\) & 2.15 & 0.0068 & 24.6 \\
6 & 279 & 314 & 4538 & \(1.03\times 10^{-5}\) & 2.92 & 0.0093 & 33.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation of emission rate G from experimental data T and C\({}_{9}\).