## 2 Wax Deposition Mechanism  
Among the six wax deposition mechanisms that have been suggested by researchers, molecular diffusion has been agreed by most researchers to be the dominant deposition mechanism [33]. This is because the pipe walls in deep water pipelines are influenced by cold subsea conditions. This causes \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Software** & **Description** & **References** \\ \hline [PERSON], [PERSON] and [PERSON] & _OLGA_TM & Multiphase flow wax deposition model which predicts wax deposition in wells as well as pipelines. & [31] \\ \hline Matzain & _OLGA_TM & Semi empirical model which takes into account shear stripping, molecular diffusion and shear dispersion to predict wax deposition. & [5] \\ \hline Heat Analogy & _OLGA_TM & Computes the mass transfer rate of wax utilizing the heat transfer analogy. & [30] \\ \hline University of Michigan Model & _LedaFlow_ & Models the wax crystallization and the wax deposition on pipe wall. & [30] \\ \hline FloWax & _FloWax_ & A complete compositional wax deposition model that considers the thermodynamics wax precipitation, wax diffusion based on heat and mass transfer analogy and the shearing effect. & [32] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Examples of models used in wax deposition studies. the crude oil flowing in the pipeline to experience a radial thermal gradient. This means that the temperature of crude oil situated closer to the pipe wall is lower compared to the temperature of the crude oil at the center of the pipeline [34]. When the temperature of the crude oil reaches the cloud point, the crystallization of wax takes place in the cold region closer to the pipe wall. The crystallization of wax thereby changes the equilibrium of the liquid and solid phases. As the solubility of wax in the crude oil decreases with respect to thermal energy, there will be a concentration gradient present. Cold regions situated closer to the pipe wall will have a lower wax concentration in the liquid phase. This is due to the solid wax depositing out of the bulk liquid crude oil on to the colder pipeline wall. Hence, wax molecular diffusion occurs from the bulk fluid to the pipeline wall [35; 36; 37]. The wax molecular diffusion or the rate of deposited wax in mass can be predicted by utilizing [PERSON]'s law of diffusion:  
\[\frac{dm_{m}}{dt}=\rho_{d}D_{m}A\frac{dC}{dr} \tag{1}\]  
where \(m_{m}\) is the deposited wax mass, \(\rho_{d}\) is the solid wax density, \(D_{m}\) is the liquid wax diffusion coefficient, A is the deposition surface area, C is the wax concentration in solution, and r is the radial coordinate. Equation (1) can also be written in terms of the wax solubility coefficient of the crude oil:  
\[\frac{dm_{m}}{dt}=\rho_{d}D_{m}A\left(\frac{dC}{dT}\right)\left(\frac{dT}{dr}\right) \tag{2}\]  
where \(dT/dr\) is the radial thermal gradient [38]. Expanding the molecular diffusion equation will give the molecular diffusion coefficient [38; 39] as follows:  
\[D_{m}=7.4\,\times\,10^{-9}\left[T(\varepsilon M)^{1/2}\right]/\mu V^{0.6} \tag{3}\]  
where \(T\) is defined as the absolute temperature, \(\varepsilon\) is a parameter for association, \(M\) is the molecular weight of crude oil, \(\mu\) is the dynamic viscosity of crude oil, and \(V\) is the molecular volume of wax. Subsequently, the second deposition mechanism based on thermal gradient is Soret diffusion [40]. Soret diffusion occurs when large molecules and small molecules are dispersed under thermal gradient regions. As yet, the employment of Soret diffusion in the wax deposition model is not fully understood and is recommended as the subject of future studies. In one case, Soret diffusion was negligible in the wax deposition model [41], while in another case, Soret diffusion was taken into account in the development of the wax deposition model [42]. This notwithstanding, implementing both molecular diffusion and Soret diffusion will develop an accurate wax deposition model.