# Characterisation of the Pump-Suction Flow Field of Antarctic Krill and Key Influencing Factors
## 2 Materials and Methods
### DPM Model  
The body length of adult krill typically ranges from 42 mm to 65 mm, with the largest being 90 mm. The body length of juvenile krill is about 25 mm to 40 mm [44; 45]. The body length of the krill studied in this paper is 60 mm, and the maximum thickness of the krill body is 10 mm, which is considerably smaller than the dimensions of a centrifugal pump. Krill are not fast swimmers and are usually found suspended in clusters in seawater. The body of the krill bends when stimulated, which can be calculated for a circle with a diameter between 12 and 20 mm based on the above lengths. For the purpose of a simulation analysis, krill is simplified as a spherical particle of equal volume with a particle diameter of 20 mm (Figure 1). Currently, the main mathematical descriptions of CFD in the study of solid-liquid two-phase flow are categorised into three main computational techniques, namely the two-fluid model (Euler-Euler method), the discrete-phase model (Euler-Lagrange method), and the fluid-simulated particle method (Lagrange-Lagrange method). In the numerical simulation of solid-liquid two-phase flow in pumps, the Euler-Euler two-fluid model has significant limitations in resolving the interaction mechanism between particles, especially in accurately capturing the characteristics of particle collision dynamics. The continuous medium assumption of the model leads to significant numerical deviations when there is a localised increase in particle concentration in the flow field. In contrast, the Eulerian and Lagrangian coordinate systems allow the particles to be treated as independent entities and their dynamic behaviour to be resolved by the application of [PERSON]'s second law of motion, while the fluid is treated as a continuous medium; the Continuous-Discrete-Phase Model (CDPM), described by the Reynolds-averaged Navier-Stokes (RANS) equations, is used. This approach allows for the exchange of momentum between the solid particles and the fluid. The representative models of this approach include the discrete-phase model (DPM) and the discrete element method (DEM). [PERSON], [PERSON], and [PERSON] et al. (2016); [PERSON] et al. (2017); [PERSON] et al. (2017) conducted a more in-depth study of centrifugal pumps through the CFD-DPM coupling. The present study focuses on the movement of a mixture of krill and water when subjected to the action of a centrifugal machine. If the density of krill is close to that of water, the relative motion of krill and water is not significant. Given the modest mixing density of krill-water, both the DPM and DEM models are applicable ([PERSON], 2016; [PERSON] et al., 2017). The DEM model is known to require more time than the DPM model to calculate interaction forces and is moreover only applicable to small-scale systems. Therefore, the DPM model was used to ensure the reliability of results and enhance computational efficiency. Fluent 2023 R1 predicts the trajectory of a discrete-phase particle (droplet or bubble) by integrating the force balance on it, which is described in the Lagrangian coordinate system. Its force balance equation can be written as the following equation ([PERSON], 2013):  
\[m_{p}\frac{\overset{\rightarrow}{u}_{p}}{dt}=m_{p}\frac{\overset{\rightarrow} {u}-\overset{\rightarrow}{u}_{p}}{\tau_{r}}+m_{p}\frac{\overset{\rightarrow} {S}(\rho_{p}-\rho)}{\rho_{p}}+\overset{\rightarrow}{F} \tag{4}\]  
where \(m_{p}\) is the mass of the particle, \(\overset{\rightarrow}{u}\) is the velocity of the continuous phase, \(\overset{\rightarrow}{u}_{p}\) is the particle velocity, \(\rho\) is the density of the continuous phase, \(\rho_{p}\) is the density of the particle, \(\overset{\rightarrow}{F}\) is the added mass force, \(m_{p}\frac{\overset{\rightarrow}{u}-\overset{\rightarrow}{u}_{p}}{\tau_{r}}\) is the particle flow resistance (drag force), and \(\tau_{r}\) is the particle relaxation time.