**2. Physical Models**

## *2.1. Turbulent Flow Model*

The "standard" *k* − *ε* low Reynolds number (LRN) eddy-viscosity turbulence model in STAR-CD was adopted to model the flow and mass transfer [25]. The mass transfer is modelled by solving the mass transport equation together with the hydrodynamic equations, where the turbulent Schmidt number, *Sct* = *μt*/*ρDt*, defines the ratio of turbulent momentum transport to turbulent diffusive transport. In the present study, *Sct* was assigned a value of 0.9. The model is founded on the *k* − *ε* model proposed by Launder and Spalding [26], in which modelling of fully turbulent bulk flow is achieved through the transportation equations of turbulent kinetic energy (TKE), *k*, and its dissipation rate, *ε*. Launder and Spalding model the near-wall region, where the viscous effect is dominant, by way of algebraic wall functions. However, this approach uses a velocity profile that is not applicable to recirculating flow, where there is flow separation and reversal, and does not take the detailed mass transfer characteristics in the diffusion boundary layer into account [27]. It is therefore necessary to use a turbulence model that directly solves the flow field and mass transfer deep inside the viscous sublayer to account for the changes due to the wall. The LRN model does so by solving the transport equations of *k* and *ε* everywhere, including the near-wall regions.

To extend the *k* − *ε* high Reynolds number model to model low Reynolds number conditions, Lien et al. [28] have proposed damping functions that modify the transport equations of *k* and *ε* in the near-wall region. One such is the semi-viscous near-wall effect *fu*, which modifies the eddy-viscosity term *μt*. An additional term for turbulence generation *Pk* , which vanishes as *Rey* approaches the order of 100, is added to ensure that the correct level of TKE dissipation is returned, and the destruction of TKE dissipation is modified with another damping factor *f*2. These are formulated as follows in STAR-CD:

$$P\_k^{\prime} = 1.33 \left[ 1 - 0.3 \varepsilon^{-R\_t^{\prime} 2} \right] \left[ P\_k + 2 \frac{\mu}{\mu\_t} \frac{k}{\mathcal{Y}^2} \right] \varepsilon^{-0.00375 Re\_{\mathcal{Y}}^2},\tag{2}$$

$$f\_{\mu} = \left[1 - e^{-0.0198 R c\_{y}}\right] \left(1 + \frac{5.29}{R c\_{y}}\right) \tag{3}$$

$$f\_2 = 1 - 0.3e^{-Rc\_l^2},\tag{4}$$

where the turbulent Reynolds numbers are *Rey* = *ρy* <sup>√</sup>*k*/*<sup>μ</sup>* and *Ret* <sup>=</sup> *<sup>ρ</sup>k*2/*με*.
