# Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations
## 2 Materials and Methods
### Analytical Formulas
#### 2.2.3 Parameters for Vertical Rectangular Finned Arrays  
In contrast to flat plates, many Nusselt number correlations exist for vertical rectangular finned arrays. Regardless of the correlation, the characteristic length applies to vertical finned arrays [14]:  
\[L_{spe,yfa}=L \tag{15}\]  
Vertical channel flows can approximate flow and heat dissipation within vertical rectangular finned arrays. With such channels, the Rayleigh number has to be determined using the special formulation below, where thermal diffusivity \(\alpha\) is used [17].  
\[Ra_{vfa,T}=\frac{g~{}\beta~{}(T_{s}-T_{\infty})~{}b^{3}}{v~{}\alpha}\frac{b}{L} \tag{16}\]  
There are different Nusselt number correlations for vertical rectangular finned arrays based on experiments. In the course of this work, it will be worked out which of these correlations can be used to determine the _FSF_ so that there is the highest possible thermal similarity between a smooth and a specific finned surface. Below, the correlations for isothermal boundary conditions are listed. The respective index refers to the author.  
[PERSON] [18]:  
\[Nu_{El}=\frac{Ra_{vfa,T}}{24}\bigg{(}1-e^{-\frac{35}{Ra_{vfa,T}}}\bigg{)}^{0.75} \tag{17}\]  
[PERSON] [19]:  
\[Nu_{Ai}=\frac{Ra_{vfa,T}}{24}\bigg{(}1-e^{-\frac{32.7}{Ra_{vfa,T}}}\bigg{)}^{0.75} \tag{18}\]  
[PERSON] et al. [20]:  
\[Nu_{Ba}=\left(\frac{576}{Ra_{vfa,T}}^{2}+\frac{2.873}{\sqrt{Ra_{vfa,T}}}\right) ^{-0.5} \tag{19}\]  
[PERSON] et al. [21]:  
\[Nu_{Ro}=\left(\bigg{(}\frac{Ra_{vfa,T}}{24}\bigg{)}^{-1.9}+\left(c_{Ro}C_{Ro} Ra_{vfa,T}^{0.25}\right)^{-1.9}\right)^{-\frac{1}{13}} \tag{20}\]  
[PERSON] [22]:  
\[Nu_{Ol}=\left(\bigg{(}\frac{Ra_{vfa,T}}{24}\bigg{)}^{-1.3}+\left(c_{Ro}C_{Ro} Ra_{vfa,T}^{0.25}\right)^{-1.3}\right)^{-\frac{1}{13}} \tag{21}\]  
Concerning Formulas (20) and (21) \(c_{Ro}\) can take values between 1 and 1.32, and \(C_{Ro}\) has to be calculated according to the following equation [21].  
\[C_{Ro}=\frac{0.671}{\bigg{[}1+\left(\frac{0.492}{Pr}\right)^{\frac{9}{16}} \bigg{]}^{\frac{3}{9}}} \tag{22}\]An ideal fin would have an infinitely high thermal conductivity, and therefore, the temperature at every point of the fin would be the same as at the fin base. As a result, a maximum heat flow could be given off to the environment. The fin efficiency \(\eta\) sets the transferable heat of a real fin \(Q_{real}\) in ratio to that one of an ideal fin \(Q_{tidal}\). It is defined as [13]:  
\[\eta_{f}=\frac{Q_{real}}{Q_{ideal}}=\frac{Q_{real}}{h_{f}\ A_{f,s}\left(T_{fb}- T_{\infty}\right)}\leq 1 \tag{23}\]  
In addition to the heat transfer from solid to fluid, the heat transport in the fins must also be considered. This is possible via fin efficiency. It depends on the temperature distribution in the fin and the fin geometry. For rectangular fins with heat transfer via all fin surfaces, fin efficiency can be determined using the Biot Number Bi and in parameter \(\mu\)[16].  
\[\eta_{f}=\frac{1}{\mu\left(L+\frac{A_{f,cs}}{p_{f}}\right)}\ \frac{sinh(\mu L)+\frac{Bi}{\mu L}\cosh(\mu L)}{cosh(\mu L)+ \frac{Bi}{\mu L}\ sinh(\mu L)} \tag{24}\]  
\[\mu=\sqrt{\frac{h_{f}\ P_{f}}{k_{f}\ A_{f,cs}}} \tag{25}\]  
\[Bi=\frac{h\ L_{spe}}{k_{fluid}} \tag{26}\]