## 2 Experimental
### Effective Density of Pellets  
Because the pellets are porous, it is crucial to determine their effective density in water, which could then be used in a CFD analysis. It can be determined in an experiment based on measuring the settling velocity. The settling velocity of particles in a fluid is derived from the balance of gravitational, drag and lift (buoyancy) forces. For spherical particles, the drag coefficient is determined by the following equation [15]  
\[C_{\rm d}=\frac{4}{3}\frac{d_{\rm p}(\varrho_{\rm p}-\varrho_{\rm f})g}{u^{2} \varrho_{\rm f}}, \tag{5}\]  
where \(d_{\rm p}\) is particle diameter, \(u\) the characteristic velocity, \(\varrho_{\rm p}\) and \(\varrho_{\rm f}\) are effective density of the particle and liquid phase, respectively. The drag coefficient \(C_{\rm d}\) is usually expressed in terms of Reynolds number, \(\mathrm{Re}=ud_{\rm p}/\
u\), where \(\
u=\mu/\varrho\) is kinematic viscosity. An example of a correlation describing this dependency could be  
\[C_{\rm d}=a_{1}+\frac{a_{2}}{\mathrm{Re}}+\frac{a_{3}}{\mathrm{Re}^{2}}\, \tag{6}\]  
where constants \(a_{1-3}\) are chosen according to corresponding range of Reynolds number [12]. [PERSON] and [PERSON] [13] presented the following equation, which takes into account the non-spherical shape of particles  
\[C_{\rm d}=\frac{24}{\mathrm{Re}}\left(1+b_{1}\mathrm{Re}^{b_{2}}\right)+\frac {b_{3}}{1+\frac{b_{3}}{\mathrm{Re}}}. \tag{7}\]  
The parameters \(b_{1-4}\) in (7) are defined as follows  
\[\begin{array}{l}b_{1}=\exp\left(2.3288-6.4581\varphi+2.4486\varphi^{2}\right) \\ b_{2}=0.0964+0.5565\varphi\\ b_{3}=\exp\left(4.905-13.8944\varphi+18.4222\varphi^{2}-10.2599\varphi^{3} \right)\\ b_{4}=\exp\left(1.4681+12.2584\varphi-20.7322\varphi^{2}+15.8855\varphi^{3} \right)\end{array} \tag{8}\]  
where shape factor \(\varphi\) (sphericity) represents the ratio of the surface of a sphere with the same volume as the real particle (s) and surface \(S\) of the real particle, i.e., \(\varphi=s/S\). The shape factor cannot exceed a value of 1, and it is \((2/3)^{1/3}=0.874\) for a cylinder with height equal to diameter. Equation (7) is used in ANSYS Fluent solver [16], which was used to perform CFD simulations in this work. This correlation is reported to be valid for \(\mathrm{Re}<2.5\times 10^{4}\).  
Figure 3: Experimental data for pellets TM0 and fitted Rosin–Rammler distribution curve (Equation (2)).  
Settling velocities of several particles in a water column (see Figure 4, left) were measured [17]. An HD camera with 29.879 fps recorded the particle trajectory within a water column of total height 300 mm. The measured settling trajectory was then within the bottom part of it (224 mm), where constant (terminal) velocity could be expected. The settling trajectory measurement accuracy was estimated as \(\pm\)0.14 mm (based on the camera resolution). The time necessary to evaluate the settling velocity was determined from the recorded movie with accuracy limited to one frame. The frame rate accuracy of the camera was estimated as 0.06%; therefore, the maximum time measurement inaccuracy is 0.004 s (for max. time 7.1 s, see Table 2), and the inaccuracy for the determined settling velocity should be below 1%. Assuming that the characteristic size (diameter) of the particle is known, the effective density of the particle could be expressed from Equation (5) as  
\[\varrho_{\rm P}=\varrho_{\rm f}+\frac{3}{4}\frac{\varrho_{\rm f}C_{\rm d}u^{2}} {gd_{\rm P}}. \tag{9}\]  
where \(u\) is the determined settling velocity. The characteristic size \(d_{\rm P}\) was based on a static camera picture with estimated accuracy 0.06 mm.