# Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations
## 3 Results
### Method for Determining Fin Substitution Factor  
The challenge in determining the _FSF_ is to define a suitable calculation path for a specific case. Therefore, a process is presented, which is shown schematically in Figure 3.  
The task is to define an analytical _FSF_ calculation path for a specific fin geometry and for specific thermal boundary conditions. The measure of the quality of the calculated _FSF_ is its ability to produce the greatest possible thermal similarity between a smooth surface and the corresponding fin array. In the first process step, many parameters must be specified. Thereby a distinction is necessary between high-level parameters and low-level parameters. The analytical formulas or empirical correlations used later in the process are decisive for this classification. Because some of these formulas only have a limited scope of application and validity. All parameters that are decisive for these restrictions are high-level parameters. These are:  
* Shape of the fins
* Orientation of the finned array
* Type of surrounding fluid
* Type of the thermal boundary conditionsAll other parameters of the analytical or analytical-empirical equations are low-level parameters. These parameters have to be varied to find the best \(FSF\) calculation path. These include:  
* Geometric dimensions of the finned array
* Fluid and Material parameters
* Values of the thermal boundary conditions  
After defining the high-level parameters, any number \(j\) of configurations can be generated by varying the low-level parameters. The further process has to be run through once for each of these configurations. Basically, the larger the interval and the number of values of each low-level parameter, the more configurations can be considered, and the more general is the resulting analytical \(FSF\) calculation path of this process. Starting with the first configuration, \(i=1\), the following process steps have to be completed:  
The second process step calculates the respective configuration in a detailed CFD simulation. It is important to ensure that the fluid domain around the finned array is large enough not to affect the heat transfer process. The discretization must be fine enough not to affect the result. The output of this step is a particular heat flow \(Q_{\beta,i}\), which is given off by the finned array under the given boundary conditions. Figure 3: Process for determining the specific analytical \(FSF\) calculation method. The third process step aims to determine a vector of manipulated heat flows from the smooth surface to the ambient air. Therefore, a large number of solid-state simulations are carried out. Only the smooth surface and the corresponding thermal boundary conditions at the fin base are considered within those simulations. The side where the fins are located obtains a convection boundary condition. The heat transfer coefficient of this boundary condition has to be varied in a certain interval. Investigations of the interval limits for this parameter for different cases have shown that it is sufficient to vary the heat transfer coefficient between 1 and 500 times that value, which can be determined analytically for the corresponding smooth surface. The output of this step is a vector of manipulated heat flows, depending on the heat transfer coefficient. In the fourth process step, the heat flow of the second step is compared with each entry in the vector of the third step. This step aims to find out at which heat transfer coefficient the heat flow of the smooth surface (step 3) corresponds to that of one of the finned arrays (step 2). Interpolation is possible within certain limits. If a corresponding transfer coefficient is found, this ratio to the analytically determined one of the smooth surfaces has to be formed. This ratio is denoted as \(FSF_{ref,i}\) and corresponds to the reference value to which the analytically determined \(FSF\) of the finned arrays (step 5) should deviate as little as possible.