Numerical study of the antibiotic transport and distribution in the Laizhou Bay, China

Abstract

A large number of antibiotic residues are discharged into aquatic environments due to the widespread use of antibiotics in daily life. After a series of processes in the water, the residues will potentially have adverse impacts on water ecosystem and human health. Therefore, it is critical for mediating the pollution to know how the antibiotics transport and distribute in the water. This study utilizes a two-dimensional lattice Boltzmann model to find out the transport and distribution of antibiotics in a highly polluted area, the Laizhou Bay in China. Furthermore, the model was used to simulate two scenarios in the Laizhou Bay, the antibiotics from sewage treatment plants and the contamination of mariculture. The simulated results show that the model as an effective tool can provide a useful basis for the management of antibiotics-related environmental issues in the Laizhou Bay.

Appendix: The Dry-wet boundary

When it comes to the dry-wet boundary, two cases with n < 4 and n ≥ 4 are included, n is the number of the adjacent wet nodes of a dry node (Fig. 12). The undetermined f_α at dry nodes moving to wet nodes can be calculated by Eq. (21) (Liu et al. 2016). Take Fig. 12a for instance, f₁,f₂, and f₈ at t + Δt can be calculated from

$$ \begin{array}{r} f_{\alpha}=f_{\alpha}^{(0)}-\tau_{a}{\varDelta} t[(\frac{\partial}{\partial t}+e_{\alpha j}\frac{\partial}{\partial x_{j}})f_{\alpha}^{(0)}+3\omega_{\alpha}\frac{g\overline{h}e_{\alpha j}\partial Z_{b}}{e^{2}\partial x_{j}}-3\omega_{\alpha}\frac{e_{\alpha i}F_{i}}{e^{2}}]+\\\tau_{a}^{2}({\varDelta} t)^{2}(1-\frac{1}{2\tau_{a}})[(\frac{\partial}{\partial t}+e_{\alpha j}\frac{\partial}{\partial x_{j}})^{2}f_{\alpha}^{(0)}+3\omega_{\alpha}\frac{ge_{\alpha j}}{{e^{2}}}(\frac{\partial h}{\partial t}+e_{\alpha j}\frac{\partial h}{\partial x_{j}})\frac{\partial Z_{b}}{\partial x_{j}}+\\3\omega_{\alpha}\frac{g\overline{h}e_{\alpha i}e_{\alpha j}\partial^{2} Z_{b}}{e^{2}\partial x_{i}\partial x_{j}}-3\omega_{\alpha}\frac{e_{\alpha j}}{{e^{2}}}(\frac{\partial F_{i}}{\partial t}+e_{\alpha j}\frac{\partial F_{i}}{\partial x_{j}})]. \end{array} $$

(21)

However, there are still undetermined f₃ and f₇, which can be calculated by the average value of its adjacent nodes for n < 4 as Eq. 22. If n ≥ 4 (Fig. 12b), only undetermined term f₀ can be obtained by Liu et al. (2016)

$$ f_{\alpha}=\frac{1}{8}\sum\limits_{\alpha=1}^{8}f_{\alpha}(x+e_{\alpha}{\varDelta} t). $$

(22)

Fig. 12

Sketch of the dry-wet interface (d and w represent dry and wet grids, respectively): an < 4; b n ≥ 4.

Therefore, the computing procedure can be summarized as:

• i. Initialize the h, u, and v in the wet area;

• ii. Calculate $f_{\alpha }^{eq}$ from Eq. (11);

• iii. In case of f_α > 0 at the dry grid next to the dry-wet interface (the flow at a wet node has enough momentum to arrive at the adjacent dry node), f_α at dry grid can be obtained by Eqs. (21) and (22). Aside from this case, the bounce-back boundary is implemented to determining f_α;

• iv. Calculate f_α using Eq. (4);

• v. Utilize step iii to obtain the undetermined f_α at the dry grid at t + Δt;

• vi.Update h, u, and v in the wet area with Eqs. (12)–(13);

• vii. Go back to step ii and repeat steps iii–vi until getting the results in the required time.

Similarly, for the ADR model, g₁,g₂, and g₈ at t + Δt can be calculated by

$$ \begin{array}{c} g_{\alpha}=g_{\alpha}^{(0)}-\tau_{a}{\varDelta} t[(\frac{\partial}{\partial t}+s_{\alpha j}\frac{\partial}{\partial x_{j}})g_{\alpha}^{(0)}-\frac{S_{c}}{b}]+\\\tau_{a}^{2}({\varDelta} t)^{2}(1-\frac{1}{2\tau_{a}})[(\frac{\partial}{\partial t}+s_{\alpha j}\frac{\partial}{\partial x_{j}})^{2}g_{\alpha}^{(0)}-\frac{S_{c}}{b}], \end{array} $$

(23)

and

$$ g_{\alpha}^{(0)}=0, \frac{\partial g_{\alpha}^{(0)}}{\partial t}=\frac{g_{\alpha}^{(0)}(t)-g_{\alpha}^{(0)}(t-1)}{{\varDelta} t}=0, $$

(24)

g₀ can be calculated from

$$ g_{0}=\frac{1}{8}\sum\limits_{\alpha=1}^{8}g_{\alpha}(x+s_{\alpha}{\varDelta} t). $$

(25)