# Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Temperature-Dependent Properties of Air

#### 2.2. Analytical Formulas

#### 2.2.1. Surfaces of the Smooth and Finned Surface

_{0}, which corresponds to the area of the finned array, is calculated as

_{f}, the cross-sectional area A

_{f}

_{,cs}, the surface area A

_{f}

_{,s}, total fin area A

_{f,}and the fin base area A

_{fb}are calculated using the formulas below. The end surfaces of the fin or the finned array are neglected.

#### 2.2.2. Parameters for Plates

_{fluid}, and the characteristic length L

_{spe}according to the following formula [14].

_{p}, thermal expansion coefficient β, kinematic viscosity ν, as well as the acceleration of gravity g and the temperature difference between the surface T

_{S,i}and the environment using the following formula [13].

#### 2.2.3. Parameters for Vertical Rectangular Finned Arrays

_{Ro}can take values between 1 and 1.32, and C

_{Ro}has to be calculated according to the following equation [21].

_{real}in ratio to that one of an ideal fin Q

_{ideal}. It is defined as [13]:

#### 2.3. Hardware and Software

#### 2.4. CFD Simulation Modeling

^{+}at the solid-fluid interface was near 1 [12].

#### 2.5. Solid-State Simulation Modeling

## 3. Results

#### 3.1. Definition of the Fin Substitution Factor

- Identical temperatures exist for the smooth surface and the fin base
- Identical heat flows of the smooth and the ripped surfaces

_{0}of a smooth surface is calculated by

_{S}and the ambient temperature T

_{∞}, the size of the surface itself A

_{0}, and the heat transfer coefficient h

_{0}[13,14]. The heat conduction in the fins must also be taken into account with a finned surface. This can be done via the thermal fin efficiency η. The heat flow of a finned surface Q

_{f}can be determined using the heat transfer coefficient h

_{f}, the areas of the fins A

_{f}and the fin base A

_{fb}and the temperature difference between the fin base T

_{fb}and the ambient [13,14].

_{fb}, A

_{f}, A

_{0}), heat conduction (η), and heat transfer (h

_{f}, h

_{0}).

#### 3.2. Method for Determining Fin Substitution Factor

- Shape of the fins
- Orientation of the finned array
- Type of surrounding fluid
- Type of the thermal boundary conditions

- Geometric dimensions of the finned array
- Fluid and Material parameters
- Values of the thermal boundary conditions

_{fb,i}, which is given off by the finned array under the given boundary conditions.

_{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.

_{i}. This vector is determined according to the subprocess also shown in Figure 3 with the following steps:

- Determination of the case-specific, temperature-dependent air properties
- Calculation of the surfaces of the smooth base body and the finned array
- Calculation of the parameters of the smooth surface
- Calculation of the finned array parameters
- Determination of the FSF
_{i}vector

_{ref}(step 4) is compared with each entry in the vector of the fifth step. The evaluation can be done in different ways. In this work, the absolute deviation and the relative deviation according to the amount for each value FSF

_{i}are calculated. The deviation can be calculated overall as well as low-level parameter-specifically. The end of this process step is a listing of calculation path-specific deviations.

_{i}has the smallest deviations from the different FSF

_{ref}. It may be necessary to make low-level parameter-specific definitions of the calculation path.

#### 3.3. Case Study: Natural Convection on Vertical Rectangular Finned Arrays

#### 3.3.1. Specifications

- Shape of the fins: rectangular
- Orientation of the finned array: vertical
- Type of surrounding fluid: air
- Type of the thermal boundary conditions: constant temperature

#### 3.3.2. Objectives

#### 3.3.3. Determination of the Calculation Path for Vertical Rectangular Finned Arrays

_{ref,i}(output step 4), and a vector of analytically determined FSF

_{i}(output step 5). In the course of each process step 6, the absolute and relative deviation according to the amount concerning the respective FSF

_{ref,i}was determined for each FSF

_{i}. The deviations are determined using the heat flows. The heat flow, which resulted from the solid-state simulation in connection with the FSF of the respective calculation path, is always related to that heat flow, which was determined by the corresponding CFD simulation.

_{Ro}. The deviations are specified in both overall and low-level parameters. The five FSF calculation paths with the most accurate results are highlighted for each low-level parameter.

#### 3.3.4. Validation

_{A,h,Ai}was determined. Each FSF

_{A,h,Ai}was multiplied by the heat transfer coefficient of the corresponding smooth surface. The products were specified as the heat transfer coefficients for the convection boundary condition of the corresponding adapted solid-state simulation model.

## 4. Discussion and Conclusions

- FSFs were determined based solely on geometric ratios, and the heat conduction showed the largest deviations both overall and concerning low-level parameters. Thus, these calculation paths were unsuitable.
- The same applies to the calculation paths, which consider all three components. Looking at all, the results FSF
_{A}_{,η,h,EL}were the exception, as this calculation path was among the five most accurate ones. - The paths considering the geometric and heat transfer components gave the best results. Four of the five most accurate calculation paths were among these concerning the overall mean deviation and the overall standard deviation. The FSF
_{A,h,Ro}_{1}path had the best results for low fin base temperatures. There was no single calculation path for higher temperatures and different fin heights that turned out to be particularly accurate. The performance of the FSF_{A,h,El}, FSF_{A,h,Ai}, FSF_{A,h,Ba}and FSF_{A,h,Ro}_{1}paths were very similar for these parameters. For longer fin lengths and almost all examined fin spacing, the FSF_{A,h,Ai}path was the best one. - The FSF
_{A,h,Ai}path gave the best results with an 8% overall mean deviation and an overall standard deviation of 0.5. For this reason, this calculation path is recommended for the analytical determination of the FSF for vertical rectangular finned arrays considering natural convection with air as surrounding fluid and an isothermal boundary condition.

_{A,h,Ai}path was verified using three different fin arrays at three different temperatures. The deviation between the results of the CFD simulations and the solid-state simulations (FEA) using the FSF

_{A,h,Ai}were between −6.47% and 13.77%. The average relative deviation according to the amount of overall simulations was 6.2%. In light of these results, the approach presented in this work is suitable for accelerating thermal simulations and achieves sufficiently accurate results simultaneously.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A_{0} | area smooth surface |

A_{fa} | area finned array |

A_{fb} | area fin base |

A_{f} | area of all fins |

A_{f,cs} | cross-sectional area of one fine |

A_{f,s} | area of one fin |

B | fin width |

b | fin spacing |

Bi | Biot number |

c_{Ro} | Rohsenow parameter |

C_{Ro} | Rohsenow parameter |

CFD | Computational fluid dynamics |

c_{p} | specific heat capacity |

FEM | Finite-Element-Method |

FSF | fin substitution factor |

g | acceleration of gravity |

H | fin hight |

h | heat transfer coefficient |

h_{0} | heat transfer coefficient smooth surface |

h_{f} | heat transfer coefficient finned array |

k_{fluid} | conductivity fluid |

k_{f} | conductivity fins |

L | fin length |

L_{spe} | specific length |

n | number of fins |

Nu | Nusselt number |

P_{f} | perimeter of one fin |

Pr | Prandtl number |

Q_{0} | Heat flux smooth surface |

Q_{CFD} | Heat flux flow simulation |

Q_{f} | Heat flux finned array |

Q_{FEM+FSF} | Heat flux solid-state simulation |

Ra | Rayleigh number |

s | base plate thickness |

T_{∞} | ambient temperature |

T_{S} | surface temperature |

T_{fb} | fin base temperature |

W | base plate width |

y+ | dimensionless wall distance |

α | thermal diffusivity |

β | thermal expansion coefficient |

η_{f} | fin efficiency |

μ | fin parameter |

ν | kinematic viscosity |

ρ | density |

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**Figure 2.**CFD simulation model (

**a**), discretized CFD computational domain (

**b**), a detailed CFD grid view (

**c**), solid-state simulation model, and the meshed domain (

**d**).

**Table 1.**Temperature-dependent material properties of air [13].

T (K) | ρ (kg/m ^{3}) | ν (kg/m s) | k (W/m K) | c_{p}(J/kg K) | β (1/K) |
---|---|---|---|---|---|

273 | 1.276 × 10^{0} | 1.722 × 10^{−5} | 2.436 × 10^{−2} | 1.006 × 10^{3} | 3.674 × 10^{−3} |

293 | 1.189 × 10^{0} | 1.821 × 10^{−5} | 2.587 × 10^{−2} | 1.006 × 10^{3} | 3.421 × 10^{−3} |

373 | 9.333 × 10^{−1} | 2.190 × 10^{−5} | 3.162 × 10^{−2} | 1.011 × 10^{3} | 2.684 × 10^{−3} |

473 | 7.359 × 10^{−1} | 2.605 × 10^{−5} | 3.823 × 10^{−2} | 1.025 × 10^{3} | 2.115 × 10^{−3} |

573 | 6.075 × 10^{−1} | 2.981 × 10^{−5} | 4.442 × 10^{−2} | 1.045 × 10^{3} | 1.745 × 10^{−3} |

Parameter | Setting | Parameter | Setting |
---|---|---|---|

Solution Method | Residuals | ||

Pressure-Velocity-Coupling | Coupled | continuity | 10^{−6} |

Spatial Discretization | x-velocity | 10^{−6} | |

Gradient | Least Square Cell Based | y-velocity | 10^{−6} |

Pressure | PRESTO! | z-velocity | 10^{−6} |

Momentum | Second Order Upwind | energy | 10^{−6} |

Turbulent Kinetic Energy | Second Order Upwind | k | 10^{−6} |

Specific Dissipation Rate | Second Order Upwind | ω | 10^{−6} |

Energy | Second Order Upwind | ||

Pseudo Transient |

Low-Level Parameters | Unit | Value |
---|---|---|

W | mm | 180 |

s | mm | 5 |

B | mm | 3 |

k_{fin} | W/m K | 130 |

T_{amb} | K | 293 |

H | mm | 12.5/25/50/100 |

L | mm | 125/250/500 |

b | mm | 8.8/14.7/32.4/56 |

n | - | 4/6/11/16 |

T_{fb} | K | 303/323/373/423/473 |

**Table 4.**The relative deviation according to the amount of the heat flows results from the different FSFs concerning the respective reference heat flows FSF

_{ref,i}. The smallest deviations are highlighted.

FSF | Overall | T_{fb} (K) | H (mm) | L (mm) | b (mm) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

303 | 323 | 373 | 423 | 473 | 12.5 | 25 | 50 | 100 | 125 | 250 | 500 | 8.8 | 14.7 | 32.4 | 56.0 | ||

FSF_{A} | 14.12 | 21.62 | 17.74 | 12.97 | 9.83 | 8.45 | 17.28 | 13.89 | 11.97 | 13.35 | 8.17 | 12.52 | 21.68 | 23.43 | 14.06 | 9.63 | 9.37 |

FSF_{A,η,EL} | 12.86 | 17.93 | 12.57 | 9.41 | 10.55 | 13.84 | 15.78 | 11.97 | 11.77 | 11.91 | 10.65 | 11.47 | 16.45 | 19.80 | 11.98 | 9.53 | 10.12 |

FSF_{A,η,Ai} | 12.67 | 17.95 | 12.59 | 9.25 | 10.20 | 13.37 | 15.84 | 12.01 | 11.61 | 11.24 | 10.44 | 11.27 | 16.30 | 19.75 | 11.83 | 9.24 | 9.88 |

FSF_{A,η,Ba} | 12.82 | 17.92 | 12.55 | 9.36 | 10.49 | 13.78 | 15.78 | 11.96 | 11.73 | 11.81 | 10.61 | 11.43 | 16.42 | 19.85 | 11.94 | 9.44 | 10.04 |

FSF_{A,η,Ro1} | 12.35 | 18.00 | 12.64 | 9.00 | 9.59 | 12.55 | 15.94 | 12.08 | 11.32 | 10.07 | 10.06 | 10.96 | 16.04 | 19.75 | 11.55 | 8.68 | 9.42 |

FSF_{A,η,Ro1}._{32} | 13.42 | 17.89 | 12.63 | 9.94 | 11.51 | 15.12 | 15.61 | 11.87 | 12.30 | 13.90 | 11.24 | 12.02 | 17.00 | 20.02 | 12.54 | 10.32 | 10.79 |

FSF_{A,η,Ol1} | 13.42 | 17.99 | 12.72 | 9.98 | 11.44 | 14.96 | 15.64 | 11.91 | 12.32 | 13.82 | 11.22 | 12.00 | 17.05 | 20.12 | 12.69 | 10.26 | 10.61 |

FSF_{A,η,Ol1}._{32} | 14.94 | 18.13 | 13.14 | 11.51 | 13.83 | 18.08 | 15.29 | 11.87 | 14.02 | 18.56 | 12.70 | 13.58 | 18.53 | 20.48 | 14.37 | 12.42 | 12.48 |

FSF_{A,h,EL} | 9.64 | 15.62 | 13.99 | 8.41 | 4.62 | 5.57 | 12.85 | 9.20 | 7.49 | 9.01 | 9.92 | 9.66 | 9.33 | 8.75 | 11.51 | 10.16 | 8.14 |

FSF_{A,h,Ai} | 8.04 | 12.15 | 9.88 | 4.64 | 4.94 | 8.59 | 10.40 | 8.29 | 7.50 | 5.97 | 8.00 | 7.74 | 8.39 | 7.82 | 8.74 | 8.75 | 6.84 |

FSF_{A,h,Ba} | 9.51 | 15.51 | 13.44 | 7.98 | 4.57 | 6.06 | 12.80 | 9.19 | 7.47 | 8.59 | 9.74 | 9.34 | 9.46 | 9.16 | 11.62 | 9.70 | 7.56 |

FSF_{A,h,Ro1} | 9.65 | 6.53 | 4.09 | 6.95 | 12.18 | 18.50 | 9.80 | 10.49 | 12.12 | 6.19 | 9.34 | 8.58 | 11.02 | 9.25 | 8.42 | 10.41 | 10.52 |

FSF_{A,h,Ro1}._{32} | 17.92 | 23.84 | 23.33 | 18.99 | 14.01 | 9.44 | 21.87 | 17.02 | 13.97 | 18.83 | 18.87 | 18.53 | 16.37 | 14.68 | 21.94 | 18.22 | 16.86 |

FSF_{A,h,Ol1} | 16.73 | 22.89 | 21.52 | 17.45 | 12.76 | 9.00 | 20.39 | 15.71 | 13.20 | 17.60 | 17.68 | 16.91 | 15.59 | 11.92 | 22.67 | 17.75 | 14.56 |

FSF_{A,h,Ol1}._{32} | 33.61 | 35.83 | 37.27 | 35.34 | 31.71 | 27.87 | 36.95 | 32.92 | 30.29 | 34.27 | 35.91 | 34.13 | 30.77 | 22.28 | 39.48 | 37.53 | 35.14 |

FSF_{A,η,h,EL} | 11.58 | 11.91 | 9.42 | 8.49 | 11.46 | 16.62 | 11.52 | 9.20 | 11.62 | 13.97 | 8.70 | 10.26 | 15.79 | 12.79 | 12.08 | 11.50 | 9.95 |

FSF_{A,η,h,Ai} | 12.71 | 9.42 | 7.98 | 9.69 | 15.12 | 21.37 | 9.89 | 9.73 | 14.03 | 17.21 | 9.01 | 11.24 | 17.89 | 13.81 | 12.62 | 12.75 | 11.68 |

FSF_{A,η,h,Ba} | 11.83 | 12.06 | 9.29 | 8.57 | 11.97 | 17.25 | 11.54 | 9.31 | 12.04 | 14.43 | 8.84 | 10.33 | 16.32 | 12.89 | 12.30 | 11.77 | 10.36 |

FSF_{A, η,h,Ro1} | 18.25 | 9.21 | 9.44 | 16.56 | 24.17 | 31.87 | 10.41 | 14.04 | 22.63 | 25.92 | 13.95 | 16.47 | 24.32 | 18.02 | 16.76 | 19.12 | 19.11 |

FSF_{A,η,h,Ro1}._{32} | 13.53 | 19.45 | 16.72 | 11.95 | 9.78 | 9.75 | 19.73 | 13.18 | 10.08 | 11.13 | 13.75 | 12.39 | 14.45 | 13.23 | 15.56 | 13.44 | 11.90 |

FSF_{A,η,h,Ol1} | 13.47 | 19.14 | 15.88 | 11.60 | 10.00 | 10.74 | 18.34 | 12.81 | 10.72 | 12.01 | 12.88 | 12.05 | 15.48 | 13.78 | 15.85 | 13.27 | 10.98 |

FSF_{A,η,h,Ol1}._{32} | 25.78 | 31.66 | 30.62 | 26.30 | 21.80 | 18.51 | 34.80 | 28.26 | 21.44 | 18.62 | 30.41 | 25.55 | 21.38 | 18.26 | 29.64 | 28.37 | 26.83 |

**Table 5.**The standard deviation of the heat flows results from the different FSFs concerning the respective reference heat flows FSF

_{ref,i}. The smallest deviations are highlighted.

FSF | Overall | T_{fb} (K) | H (mm) | L (mm) | b (mm) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

303 | 323 | 373 | 423 | 473 | 13 | 25 | 50 | 100 | 125 | 250 | 500 | 8.8 | 14.7 | 32.4 | 56.0 | ||

FSF_{A} | 1.40 | 2.06 | 1.43 | 1.07 | 0.91 | 0.80 | 0.44 | 0.70 | 1.22 | 2.13 | 0.77 | 1.20 | 1.78 | 2.19 | 0.82 | 0.44 | 0.33 |

FSF_{A,η,EL} | 1.18 | 1.75 | 0.91 | 0.56 | 0.63 | 0.77 | 0.42 | 0.66 | 1.13 | 1.86 | 0.81 | 1.07 | 1.46 | 1.95 | 0.81 | 0.42 | 0.35 |

FSF_{A,η,Ai} | 1.17 | 1.75 | 0.91 | 0.55 | 0.60 | 0.74 | 0.42 | 0.66 | 1.13 | 1.85 | 0.80 | 1.06 | 1.45 | 1.93 | 0.78 | 0.40 | 0.35 |

FSF_{A,η,Ba} | 1.19 | 1.77 | 0.91 | 0.56 | 0.63 | 0.77 | 0.42 | 0.66 | 1.13 | 1.87 | 0.81 | 1.08 | 1.47 | 1.97 | 0.80 | 0.41 | 0.35 |

FSF_{A,η,Ro1} | 1.16 | 1.77 | 0.93 | 0.53 | 0.56 | 0.69 | 0.42 | 0.66 | 1.13 | 1.84 | 0.78 | 1.05 | 1.45 | 1.93 | 0.74 | 0.38 | 0.33 |

FSF_{A,η,Ro1}._{32} | 1.22 | 1.78 | 0.92 | 0.60 | 0.70 | 0.85 | 0.42 | 0.66 | 1.15 | 1.92 | 0.85 | 1.12 | 1.51 | 2.02 | 0.87 | 0.45 | 0.38 |

FSF_{A,η,Ol1} | 1.24 | 1.80 | 0.96 | 0.62 | 0.70 | 0.85 | 0.42 | 0.67 | 1.16 | 1.95 | 0.86 | 1.13 | 1.53 | 2.03 | 0.89 | 0.45 | 0.37 |

FSF_{A,η,Ol1}._{32} | 1.32 | 1.82 | 0.99 | 0.73 | 0.85 | 1.02 | 0.42 | 0.67 | 1.19 | 2.05 | 0.96 | 1.22 | 1.61 | 2.12 | 1.06 | 0.54 | 0.43 |

FSF_{A,h,EL} | 0.55 | 0.51 | 0.50 | 0.44 | 0.39 | 0.40 | 0.23 | 0.31 | 0.49 | 0.69 | 0.63 | 0.50 | 0.45 | 0.68 | 0.59 | 0.46 | 0.26 |

FSF_{A,h,Ai} | 0.47 | 0.38 | 0.34 | 0.30 | 0.31 | 0.41 | 0.22 | 0.30 | 0.49 | 0.62 | 0.51 | 0.44 | 0.44 | 0.63 | 0.49 | 0.39 | 0.23 |

FSF_{A,h,Ba} | 0.56 | 0.58 | 0.55 | 0.46 | 0.38 | 0.37 | 0.25 | 0.33 | 0.50 | 0.74 | 0.65 | 0.51 | 0.47 | 0.72 | 0.61 | 0.44 | 0.25 |

FSF_{A,h,Ro1} | 0.49 | 0.29 | 0.19 | 0.24 | 0.39 | 0.55 | 0.21 | 0.31 | 0.52 | 0.61 | 0.49 | 0.47 | 0.51 | 0.70 | 0.50 | 0.34 | 0.24 |

FSF_{A,h,Ro1}._{32} | 0.96 | 0.98 | 1.02 | 0.98 | 0.87 | 0.77 | 0.30 | 0.40 | 0.61 | 1.17 | 1.14 | 0.89 | 0.71 | 1.18 | 1.06 | 0.71 | 0.40 |

FSF_{A,h,Ol1} | 0.99 | 1.00 | 1.02 | 1.00 | 0.93 | 0.86 | 0.32 | 0.44 | 0.70 | 1.34 | 1.15 | 0.89 | 0.79 | 1.14 | 1.11 | 0.71 | 0.37 |

FSF_{A,h,Ol1}._{32} | 2.07 | 1.91 | 2.08 | 2.17 | 2.11 | 2.01 | 0.47 | 0.72 | 1.22 | 2.51 | 2.44 | 1.90 | 1.53 | 2.39 | 2.43 | 1.47 | 0.85 |

FSF_{A,η,h,EL} | 0.86 | 0.40 | 0.44 | 0.67 | 0.87 | 1.06 | 0.23 | 0.34 | 0.66 | 1.23 | 0.58 | 0.78 | 1.04 | 1.24 | 0.91 | 0.54 | 0.37 |

FSF_{A,η,h,Ai} | 0.91 | 0.38 | 0.47 | 0.73 | 0.95 | 1.15 | 0.22 | 0.34 | 0.67 | 1.23 | 0.65 | 0.86 | 1.09 | 1.32 | 0.96 | 0.55 | 0.39 |

FSF_{A,η,h,Ba} | 0.85 | 0.46 | 0.46 | 0.65 | 0.84 | 1.03 | 0.25 | 0.35 | 0.64 | 1.20 | 0.57 | 0.76 | 1.03 | 1.22 | 0.92 | 0.55 | 0.38 |

FSF_{A, η,h,Ro1} | 1.01 | 0.46 | 0.59 | 0.87 | 1.09 | 1.28 | 0.22 | 0.34 | 0.69 | 1.24 | 0.81 | 0.99 | 1.18 | 1.45 | 1.08 | 0.60 | 0.44 |

FSF_{A,η,h,Ro1}._{32} | 0.83 | 0.67 | 0.60 | 0.63 | 0.75 | 0.90 | 0.29 | 0.41 | 0.70 | 1.35 | 0.60 | 0.61 | 0.93 | 1.17 | 0.89 | 0.57 | 0.35 |

FSF_{A,η,h,Ol1} | 0.89 | 0.76 | 0.69 | 0.73 | 0.84 | 0.98 | 0.31 | 0.44 | 0.75 | 1.43 | 0.60 | 0.69 | 1.03 | 1.24 | 0.88 | 0.58 | 0.36 |

FSF_{A,η,h,Ol1}._{32} | 1.28 | 1.35 | 1.27 | 1.19 | 1.15 | 1.17 | 0.46 | 0.68 | 1.12 | 2.08 | 1.31 | 0.83 | 1.00 | 1.64 | 1.36 | 0.87 | 0.49 |

Parameter | Unit | Examples from Literature | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Source | - | [25] | [26] | [27] | ||||||

s | mm | 5.00 | 25.00 | 4.00 | ||||||

W | mm | 180.00 | 150.00 | 250.00 | ||||||

B | mm | 3.00 | 2.00 | 3.00 | ||||||

H | mm | 25.00 | 50.00 | 15.00 | ||||||

L | mm | 340.00 | 200.00 | 100.00 | ||||||

b | mm | 14.70 | 16.50 | 16.00 | ||||||

n | - | 11.00 | 9.00 | 14.00 | ||||||

k_{f} | W/m K | 130.00 | 200.00 | 200.00 | ||||||

T_{∞} | K | 293.00 | 301.30 | 293.00 | ||||||

T_{fb} | K | 314.5 | 339.0 | 360.0 | 314.3 | 333.8 | 351.8 | 323.0 | 338.0 | 353.0 |

h_{vp} | W/m^{2} K | 4.37 | 5.34 | 5.86 | 4.01 | 5.12 | 5.73 | 5.68 | 6.27 | 6.70 |

FSF_{A,h,Ai} | - | 3.66 | 3.64 | 3.62 | 7.01 | 6.94 | 6.89 | 2.85 | 2.83 | 2.82 |

Q_{CFD} | W | 18.48 | 50.20 | 82.26 | 10.99 | 35.89 | 63.54 | 11.06 | 18.76 | 27.50 |

Q_{FEA+FSF} | W | 21.03 | 54.74 | 87.01 | 10.93 | 34.50 | 59.43 | 12.11 | 19.96 | 27.38 |

Dev. | % | 13.77 | 9.04 | 5.77 | −0.57 | −3.88 | −6.47 | 9.48 | 6.39 | −0.45 |

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**MDPI and ACS Style**

Roppel, M.; Rieg, F.; Tremmel, S. Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations. *Appl. Sci.* **2022**, *12*, 4449.
https://doi.org/10.3390/app12094449

**AMA Style**

Roppel M, Rieg F, Tremmel S. Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations. *Applied Sciences*. 2022; 12(9):4449.
https://doi.org/10.3390/app12094449

**Chicago/Turabian Style**

Roppel, Matthias, Frank Rieg, and Stephan Tremmel. 2022. "Definition and Determination of Fin Substitution Factors Accelerating Thermal Simulations" *Applied Sciences* 12, no. 9: 4449.
https://doi.org/10.3390/app12094449