## 4 Results and Discussion
### Hydrodynamics
#### 4.3.3 Packed-Bed Pressure Drop  
The pressure drop of the stacking arrangement in the particle bed was investigated in detail. The porosity and bed lamination drop were measured in a cylindrical device filled with air as the working fluid. Numerous variables can influence the pressure drop in a filled bed. The pressure drop was quantified as follows [23]:  
\[f_{p}=\frac{-\Delta Pd_{p}}{L\rho v^{2}}, \tag{10}\]  
where \(v\) is the apparent velocity. This velocity is usually related to the bed void \(\varepsilon\), tube-to-particle size ratio \(N\), and dimensionless Reynolds number:  
\[\text{Re}=\frac{\rho vd_{p}}{\mu}, \tag{11}\]  
where \(d_{p}\) is the hydraulic radius. The model gives the form of the dependence of pressure drop on porosity; if the hydraulic radius model is used, the coefficient of friction \(f_{k}\) is defined as  
\[f_{k}=\frac{-\Delta Pd_{p}}{L\rho v^{2}}\frac{\varepsilon^{3}}{(1-\varepsilon)}. \tag{12}\]  
The corrected Reynolds number (\(\text{Re}_{m}\)) is defined as  
\[\text{Re}_{m}=\frac{\rho vd_{p}}{\mu(1-\varepsilon)}=\frac{\text{Re}}{(1- \varepsilon)}. \tag{13}\]  
The coefficient of friction \(f_{k}\) is useful when \(\text{Re}_{m}\geq 20\). However, when \(\text{Re}_{m}\) decreases below \(20,f_{k}\) increases by orders of magnitude. For low \(\text{Re}_{m}\), [PERSON] [24] introduced a second friction factor defined as  
\[f_{v}=\frac{-\Delta Pd_{p}^{2}}{\mu vL}\frac{\varepsilon^{3}}{(1-\varepsilon) }=f_{k}\text{Re}_{m}. \tag{14}\]  
In contrast to \(f_{k},f_{v}\) exhibits only a modest increase with Re; its value only doubles when \(\text{Re}_{m}\) increases from \(0.01\) to \(100\).  
The method of calculating \(f_{k}\), \(\text{Re}_{m}\), and \(f_{v}\) shown in Equations (12)-(14) was used to determine the relationship between the coefficient of friction and Re, and the results were compared with those obtained by [PERSON] et al. [23] using the well-known KTA equation:  
\[f_{k}=\frac{160}{\text{Re}_{m}}+\frac{3.0}{\text{Re}_{m}^{0.1}}. \tag{15}\]  
The asymptotic correlation for the middle range depicted in Figure 7 exhibits good alignment with that derived from the KTA equation while also agreeing with the range of data from nine previous studies. Thus, the obtained correlation can be used to describe the pressure drop in porous media within this specific KTA flow rate range. However, the data accuracy is rapidly compromised at higher Reynolds numbers. It is worth mentioning that this compromised accuracy is not attributed to wall effects or shape effects; rather, it is due to the improper identification of high Reynolds numbers. Thus, this method is restricted to low Reynolds numbers [25].  
As shown in Figure 7, when the particle size ratio increases from 4.44 to 6.25, the coefficient of friction decreases within increasing Re and remains close to the KTA curve.  
Figure 8: Velocity streamlines in beds with different particle size ratios at the inlet velocity \(v=0.8\) m/s: (**a**) \(D/d_{p}=4.44\); (**b**) \(D/d_{p}=5.6\); and (**c**) \(D/d_{p}=6.25\).  
Figure 9: Fluid velocity and temperature distribution in a 180 mm packed beds with a particle size ratio of 4.44: (**a**) v\({}_{\rm inlet}\) = 0.5 m/s; (**b**) v\({}_{\rm inlet}\) = 0.8 m/s; and (**c**) v\({}_{\rm inlet}\) = 1.0 m/s.  
Figure 10: _Cont._