# An Analysis, Numerical Modeling and Experimental Verification of Low-Temperature Thermofoil Heaters

## Abstract

**:**

## 1. Introduction

## 2. Analysis of the Geometrical and Electrical Parameters of Thermofoil Heater

_{th}is the thermofoil resistance (Ω), ρ

_{m}is the specific resistance of the used material (Ωm), L

_{av}is the average length (m) and A

_{st}is the cross section of the heater (m

^{2}). It is assumed that all traces have equal cross-sections, which are a product of the width (B) and foil thickness (D) and can be calculated from A

_{st}= D × B (m

^{2}).

_{av}(m), comprised of N

_{st}number of traces, is calculated from

_{th}and ρ

_{m}, depends on the geometric parameters B and D, and the overall heater’s borders W (Width), H (Height) shown in Figure 1. The calculated power of the heater P

_{th}at given voltage V can be derived from $\mathrm{P}=\raisebox{1ex}{${\mathrm{V}}^{2}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{R}}_{\mathrm{th}}$}\right.\left(\mathrm{W}\right)$ and Equation (1) as

_{h}of the heater is experimentally verified in part 4.

- The number of traces can vary in a wide range, in this case between 12 and 155, depending mainly on the specific resistance of the material. Hence, it is necessary for the suggested preliminary design to be conducted in order for the heater geometry to be determined at the given input parameters as a first approximation, i.e., before heater FEM modeling and simulation.
- The match between the power of the designed heater and the calculated power is satisfactory in a wide range of materials specific resistance. This result provides analytical verification of the consistency of Equation (4), as Equation (3) is calculated by the cross-section ${\mathrm{A}}_{\mathrm{st}}$, and, respectively, the thermofoil width $\mathrm{B}$ and its specified value ${\mathrm{B}}_{0}$. With this, a satisfactory match between the heater modeling and thermal simulation can be expected.
- For the chrome-nickel based alloy material, the specific resistance defers from the first three materials in a power of two. The calculated relative error is marginally over the assumed 10% error of the calculated power. In order for the precision to be improved, it can be recommended that the thickness of the foil can be increased 2 to 5 times, which lowers the foil width B, increases the foil length ${\mathrm{L}}_{\mathrm{av}}$ and the number of traces ${\mathrm{N}}_{\mathrm{st}}$ and eventually lowers the error ${\mathrm{p}}_{\mathrm{err}}$.
- The designed thermofoil heaters are oversized for the given power. An acceptable precision is achievable in the range of the fill factor ${\mathrm{FF}}_{\mathrm{ps}}=0.7-0.85$. The fill factor can be altered by varying the isolation distance η according to Equation (8). Additionally, it is a good design practice for $\mathsf{\eta}$ to be accepted as little as possible at the beginning of the design procedure, as it will be resized after the final number of traces is calculated.

## 3. Analysis of the Power Density

## 4. Numerical Modeling of Thermofoil Heaters

^{2}C) is the convective heat transfer (depending the type of the convection), the surface and its orientation; T

_{amb}is the ambient temperature.

_{i}toward the ambient environment is described by

_{0}is the resistivity, α is the resistivity temperature coefficient and T and T

_{0}are the current temperature and the reference temperature.

_{e}is external current density $\text{}\left(\frac{\mathrm{A}}{{\mathrm{m}}^{2}}\right)$; ε

_{0}, ε

_{r}are the permittivity of the free space and relative permittivity, respectively; and ${\mathrm{Q}}_{\mathrm{J}}\text{}\left(\frac{\mathrm{A}}{{\mathrm{m}}^{2}}\right)$ is current source, described by equation

_{ini}and the final temperature T

_{f}of the heatsink, i.e., $\u2206\mathrm{T}={\mathrm{T}}_{\mathrm{ini}}\text{}-{\mathrm{T}}_{\mathrm{f}}$ (°C); $\mathrm{m}$ (grams) is the mass of the heatsink, obtained from the material density and volume; $\mathrm{t}$(sec) is the time; and $\mathrm{C}$ is the heat capacity as it is shown in Equation (18). Respectively, the necessary time in seconds that the temperature ${\mathrm{T}}_{\mathrm{f}}$ needs to reach is

## 5. Experimental Verification of the Thermal Field and Simulation-Experimental Data Comparison

## 6. Conclusions

## Funding

## Conflicts of Interest

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**Figure 2.**Experimental models of thermofoil flexible (

**A**,

**B**) and solid-state Printed Circuit Board (PCB) (

**C**,

**D**) heaters.

**Figure 3.**Mounting of a PCB heater to heatsinks (

**A**,

**B**) 1—thermofoil traces; 2—thermal conductive electrical insulation between the thermofoil traces and the heat sink; 3—heat sink.

**Figure 4.**Experimental results showing power density for two different heaters and isolation materials. Graphics 1, 2—flexible thermofoil heater with thermal isolation mica. Graphics 3–8—PCB heater with thermal isolations as follows: graphics 3, 4—mica; graphics 5, 6—silicone rubber; graphics 7, 8—thermal fabric. All solid lines (1, 3, 5 and 7) are obtained from experiment, all pointed lines (2, 4, 6 and 8) are obtained from trend analysis.

**Figure 5.**3D models of two thermofoil heaters and heatsinks with different parameters. (

**A**) Thermofoil heater assembled to aluminum ribbed heatsink. (

**B**) Thermofoil heater assembled to a monolithic aluminum heatsink. (

**C**) Transient process of heating: (1) temperature on the thermofoil traces, (2) temperature on the heatsink surface between the ribs and (3) temperature at the end of the rib.

**Figure 6.**Numerical analysis of the allocation of the thermal depending on the geometry model parameters’ traces width and insolation gap. (

**A**) B = 20 mm; (

**B**) B = 60 mm; (

**C**) B = 68 mm.

**Figure 7.**Experimental verification. Graphics 1 and 4 experimental measurements, respectively, over the thermofoil heater and the heatsink; Graphics 2 and 5 FEM simulations according to the presented geometry analysis; Graphics 3 and 6 Fem simulations without geometry model compensation; Graphic 7 temperature at the heatsink rib.

**Figure 8.**Experimental verification of the temperature distribution on the heatsink surface. Graphic 1—experimental measurements, graphic 2—FEM simulation.

Calculated Parameter | Equation | Thermofoil Heater Material | |||
---|---|---|---|---|---|

Cooper | Aluminum | Cooper-Nickel Based Alloy | Chrome-Nickel Based Alloy | ||

P = 100 W; W = 200 mm; H = 200 mm; V = 24 V; η = 1 mm; D = 0.1 mm | |||||

Specific resistance (Ω/m) | - | 1.72 × 10^{−8} | 2.65 × 10^{−8} | 36.3 × 10 ^{−8} | 2.35 × 10 ^{−6} |

Trace width (B, mm) | 4 | 0.74 | 1 | 4.79 | 16.28 |

Number of traces (N_{st}) | 5 | 114.97 | 100.18 | 34.57 | 11.57 |

Rounded number of traces (N_{st(rnd)}) | - | 115 | 100 | 35 | 12 |

Recalculated trace width (B_{o}, mm) | 6 | 0.74 | 1 | 4.71 | 15.67 |

Average length of the trace (L_{av}, m) | 2 | 23.12 | 20.10 | 0.83 | 2.41 |

Thermofoil resistance (R_{th}, Ω) | 1 | 5.38 | 5.33 | 5.43 | 5.16 |

Thermofoil cross-section (mm^{2}) | 7 | 0.07 | 0.1 | 0.47 | 1.57 |

Electric current through the heater (I, A) | I = V/R | 4.46 | 4.51 | 4.43 | 4.65 |

Power of the heater (P_{th}, W) | 3 | 107.08 | 108.14 | 106.33 | 111.68 |

Relative power error (|p_{err}|, %) | 10 | 7.08 | 8.14 | 6.33 | 11.68 |

Fill factor (FF_{ps}) | 8 | 0.43 | 0.5 | 0.83 | 0. 9 |

Active surface of the heater (S_{h}, cm^{2}) | 12 | 170.84 | 200.99 | 332.60 | 377.72 |

Power density of the heater (Q_{h}, W/cm^{2}) | 11 | 0.63 | 0.54 | 0.32 | 0.3 |

Heater width verification (W_{th}, mm) | 13 | 85 | 100 | 165 | 188.00 |

Calculated Parameter | Equation | Thermofoil Heater Material | |
---|---|---|---|

Cooper | Aluminum | ||

2 (W/cm^{2}), P = 800 W | |||

Specific resistance (Ω/m) | - | 1.72 × 10^{−8} | 2.65 × 10^{−8} |

Trace width (B, mm) | 4 | 2.77 | 3.55 |

Number of traces (N_{st}) | 5 | 53.04 | 43.97 |

Rounded number of traces (N_{st(rnd)}) | - | 53 | 44 |

Recalculated trace width (B_{o}, mm) | 6 | 2.77 | 3.55 |

Average length of the trace (L_{av}, m) | 2 | 10.65 | 8.84 |

Thermofoil resistance (R_{th}, Ω) | 1 | 0.66 | 871.36 |

Thermofoil cross-section (mm^{2}) | 7 | 0.28 | 0.35 |

Electric current through the heater (I, A) | I = V/R | 36.33 | 36.31 |

Power of the heater (P_{th}, W) | 3 | 871.89 | 871.36 |

Relative power error (|p_{err}|, %) | 10 | 8.99 | 8.92 |

Fill factor (FF_{ps}) | 8 | 0.74 | 0.78 |

Active surface of the heater (S_{h}, cm^{2}) | 12 | 295.44 | 313.52 |

Power density of the heater (Q_{h}, W/cm^{2}) | 11 | 2.95 | 2.78 |

Heater width verification (W_{th}, mm) | 13 | 147 | 156 |

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**MDPI and ACS Style**

Dimitrov, B.
An Analysis, Numerical Modeling and Experimental Verification of Low-Temperature Thermofoil Heaters. *Eng* **2020**, *1*, 249-264.
https://doi.org/10.3390/eng1020017

**AMA Style**

Dimitrov B.
An Analysis, Numerical Modeling and Experimental Verification of Low-Temperature Thermofoil Heaters. *Eng*. 2020; 1(2):249-264.
https://doi.org/10.3390/eng1020017

**Chicago/Turabian Style**

Dimitrov, Borislav.
2020. "An Analysis, Numerical Modeling and Experimental Verification of Low-Temperature Thermofoil Heaters" *Eng* 1, no. 2: 249-264.
https://doi.org/10.3390/eng1020017