# Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations
## 1 Introduction  
The numerical simulation of natural convective heat transfer corresponds to the state of the art and is used in various cases. Especially when considering cooling fins, the focus is usually on analyzing the thermal processes or optimizing fin geometries and arrangements. [PERSON] et al. [1, 2] examined the heat transfer to annular finned cylinders. His work focused on developing a Nusselt number correlation for the horizontal and vertical orientation and the investigation of the optimal fin spacing and fin-to-cylinder diameter ratio for these two cases. [PERSON] et al. [3] carried out similar studies on horizontal annular finned cylinders, in which they also considered the influence of different fin materials. [PERSON] and [PERSON] [4, 5] investigated numerically flat plates with rectangular fins with different inclinations, whereby the fins were always oriented in the flow direction. Moreover, they defined correlations for estimating convective heat transfer rates. In contrast, [PERSON] et al. [6] considered cross-flow rectangular finned arrays at different inclinations and presented a Nusselt number correlation for such cases.  
In all the works mentioned above, numerical flow simulations were used. Depending on the fluid, its behavior, the geometry, and the level of detail, the computational effort can be very intensive. This is also because processes in the thermal and velocity boundary layer are decisive for the natural convective heat transfer [7]. To map the convective heat transfer with sufficient accuracy in numerical simulations, a sufficiently fine discretization of the flow domain and a suitable mapping of the near-wall turbulence are necessary. The larger and more complex the domain, the greater the calculation effort. Depending on the problem and the focus of the investigation, the computational effort can be reduced through various options, such as exploiting symmetry properties or using less computationally intensive material models and turbulence models [8].  
When considering heat transfer processes on finned surfaces, there are more options to reduce the modeling and calculation effort. The simplest option is to assume a smoothsurface and neglect the fins, whereby conservative results are achieved [9]. Another option is to simulate smooth surfaces using experimentally determined, fin-specific factors. The purpose of such factors is that a smooth surface behaves thermally sufficiently similar to a specific finned array with its use. With such factors, it is possible to model, discretize and simulate only the geometrically simpler base body of the finned array [10; 11]. However, the experimental determination of such factors is laborious, especially in the case of complex structures.  
Such factors to simplify simulation models correspond to the state of the art and are available in various simulation programs. With Ansys Fluent, for example, it is possible to define a convective augmentation factor (CAF) or a specific roughness for a surface. With both, the heat transfer can be manipulated [10; 12].  
Regardless of the designation of the respective factors, the challenge is to calculate them correctly as long as there is no reliable experimental data. Following the authors' research, there is neither a clear definition nor a corresponding analytical calculation method for such factors yet. The present work contributes to closing this research gap by defining both the fin substitution factor (_FSF_) and a process for determining the best _FSF_ analytical calculation method for certain boundary conditions. The approach is demonstrated in the example of vertical rectangular finned arrays considering natural convective heat transfer but can be applied to other fin profiles, fin orientations, and boundary conditions. With the approach presented in this work, thermal simulations of finned surfaces can be accelerated, both during modeling and during calculation.