# Designing a Thermal Radiation Oven for Smart Phone Panels

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## Abstract

**:**

## 1. Introduction

## 2. Design Strategy

_{1}is the vertical distance between the center of a side lamp and the panel surface, while h

_{2}is the vertical distance between the center of the middle lamp and the panel surface.

_{1}/H, and h

_{2}/H) considered in modeling. Each control factor has three levels, and these levels form an arithmetic series. For l/L, the first level (Level 1) is zero. The center line of a side lamp is aligned along with a panel edge. As the level increases, side lamps get close to the middle one. The other two control factors are the vertical distance between lamps and the panel surface. h

_{1}/H and h

_{2}/H are the gaps associated with the side lamps and the middle one, respectively. The gap is enlarged as the level increases. Once the levels of all control factors were specified, the configuration of the oven was determined for testing. A test was then conducted numerically by heating a panel inside. Results were the temperature distribution at the top surface of the panel. Key quantitative data about the distribution was also recorded for analysis. One was ΔT

_{max}, the maximum temperature difference within the surface, to reflect temperature uniformity. Another was T

_{avg}, the average temperature over the surface. This value should be higher than panel’s initial temperature T

_{ini}after the panel is heated. Here, T

_{ini}= 25 °C was set to be the same as the temperature in the surroundings T

_{surr}and the commonly employed “room temperature”. The temperature difference (T

_{avg}− T

_{ini}) can be viewed as the heating capability of an oven. For an ideal oven, temperature uniformity of the heated panel should be high, and heating capability of the oven should be as large as possible.

_{max}. If it was larger than 20 °C, the temperature of the panel varied too much for the oven to be considered in further performance evaluation. As a result, Q was set to null. On the other hand, Q was calculated for the remaining tests with ΔT

_{max}≤ 20 °C. The calculation not only needs the denominator ΔT

_{max}, but the numerator T

_{avg}− T

_{ini}. Q is expected to be a large positive number for an ideal oven, which brings about uniform and high temperature within panels.

## 3. Numerical Model

#### 3.1. Thermophysical Properties

_{L}= 1). The emission was partly absorbed by the glass (SiO

_{2}) panel. Its thermal conductivity k and diffusivity a at room temperature were k = 1.51 W/m·K and a = 8.34 × 10

^{−7}m

^{2}/s, respectively [19]. Their variations with temperature were not considered in modeling because the amount was trivial. Clearly, the panel was not a good thermal conductor. Its temperature uniformity became critical for an oven using only thermal radiation.

_{2}[20]. The spectral range was from 0.2 μm to 20 μm, covering most of the emission spectra. These constants are employed later for calculating the spectral absorptivity α

_{λ}. When the wavelength was between 0.2 and 8 μm, the extinction coefficient κ was almost zero. The penetration depth δ = λ/4πκ was much greater than the thickness of the panel (0.7 mm), such that the panel was semi-transparent to emission. Conversely, as the wavelength λ was longer than 8 μm, the penetration depth δ was almost null. The panel became opaque to incident radiation, and the energy was either absorbed or reflected.

_{λ}was equal to the spectral emissivity ε

_{λ}according to Kirchhoff’s law [19]. The spectrum could therefore be employed to obtain the glass emissivity. Note that the total absorptivity α

_{total}is not the same as total emissivity ε

_{total}because temperature was different for the panel and lamps. The wavelength λ = 8 μm serves as a demarcation point in the figure. When λ < 8 μm, the glass absorptivity was low, but radiation power of the lamp was high, indicating that the absorption effect of the glass was poor in this range. When λ > 8 μm, the glass emissivity was high, indicating that the glass had high energy loss in this range. In particular, when 8 μm ≦ λ ≦ 10 μm or 20 μm ≦ λ ≦ 25 μm, the absorptivity attenuated significantly. The reason is the curves n and κ intersected in these two ranges. The radiative properties of the panel switched between those of a dielectric and a metal.

_{b}= 500 °C and 300 °C were treated as the ideal radiation intensity of lamp and glass, respectively. The peak values of the two radiation intensities were at λ = 3.75 μm and λ = 5.06 μm, respectively, indicating that the main distribution range of radiation energy was around the peak point. In addition, the combined effect of the demarcation point of absorptivity and the spectral distribution of the black body radiation intensity was considered, and the blackbody radiative energy ratio (Equation (2)) was used as the basis for selecting the average band for total absorptivity and total emissivity of the glass.

^{−8}W/m

^{2}·K

^{4}is the Stefan–Boltzmann constant, and C

_{1}= 3.742 × 10

^{8}W·μm

^{4}/m

^{2}and C

_{2}= 1.439 × 10

^{4}μm·K are the first and second radiation constants, respectively. According to calculation, when 0.2 μm ≦ λ ≦ 8 μm, the radiative energy ratio is ${F}_{0.2\text{}\mathsf{\mu}\mathrm{m},8\text{}\mathsf{\mu}\mathrm{m}}(500\text{}\xb0\mathrm{C})=0.76$. When 5 μm ≦ λ ≦ 25 μm, the radiative energy ratio is ${F}_{5\text{}\mathsf{\mu}\mathrm{m},25\text{}\mathsf{\mu}\mathrm{m}}(300\text{}\xb0\mathrm{C})=0.71$, indicating that the two bands are representative.

_{total}= 0.17 in the band of 0.2 μm ≦ λ ≦ 8 μm was selected as representative of the glass absorptivity, and the total emissivity ε

_{total}= 0.79 in the band of 5 μm ≦ λ ≦ 25 μm was selected as representative of the glass emissivity.

#### 3.2. Finite-Difference Method

_{total}. At the same time, the surfaces emitted thermal radiation to the environment with the total emissivity ε

_{total}. Boundary conditions for these surfaces therefore included these parts and conduction as listed in Equation (5):

## 4. Results and Discussion

#### 4.1. Program Convergence

_{i}

_{,j,k}obtained from two successive iterations is less than 0.1%. Criterion two is to prevent the program from infinite iterations. The maximum number of iterations was set to 400,000. If the loop number does not provide temperature convergence, an error message will pop out.

_{1}/H = 0.5, and h

_{2}/H = 0.5, and the grid sizes for the test were Δy = 1.75, 3.5, 7, and 14 mm, respectively. The upper and lower subgraphs show the convergence of the average temperature T

_{avg}and the maximum temperature difference ΔT

_{max}with the iteration number m, respectively. Results showed that the results of the previous four mesh sizes were all convergent when the number of iteration times m was about 300,000. In addition, the average temperature T

_{avg}was consistent with the maximum temperature difference ΔT

_{max}when the size of the grid was Δy ≤ 7 mm. To ensure convergence and consider the spatial temperature distribution resolution, the grid was set as Δy = 1.75 mm.

#### 4.2. Temperature Distribution

_{1}/H, and the three rows correspond to h

_{1}/H = 0.2, 0.5, and 0.8, respectively. Each column has the same number of h

_{2}/H, and the three columns from left to right correspond to h

_{2}/H = 0.2, 0.5, and 0.8, respectively. Each temperature distribution corresponding to the control factor is also marked blue in the upper-left corner, and the highest temperature T

_{max}, the lowest temperature T

_{min}, and the average temperature T

_{avg}are marked in green in the lower-left corner.

_{1}/H = h

_{2}/H = 0.2 (the temperature distribution in the lower-left corner of the figure), the glass had the highest average temperature (T

_{avg}= 170 °C), because all the lamps were close enough to the glass surface that the panel was effectively heated. In contrast, the temperature distribution in the upper-right corner (h

_{1}/H = h

_{2}/H = 0.8) showed the lowest average temperature (T

_{avg}= 99.6 °C) when all the lamps were far away from the glass. However, once the lamp was near the panel, the heat was concentrated below the lamps and the heat diffusion capacity of the glass is very low, forming a partial high temperature, such as the high temperature occurring under the central lamp in the third column (h

_{2}/H = 0.2), and the first line (h

_{1}/H = 0) showing the high temperature below the edge lamp. So, the compromise scheme is to improve the height of the lamp to h

_{1}/H = h

_{2}/H = 0.5, and the temperature distribution is shown in the second rows and second columns. In this figure, the temperature not only increased, but the distribution was also relatively uniform, indicating that the heat radiation energy emitted by all lamps was more evenly irradiated on the panel surface.

_{1}/H = h

_{2}/H = 0.5) has a large pattern of high temperature in the center. The second effect was that the average temperature was also increased, so when h

_{1}/H = h

_{2}/H = 0.2, the maximum average temperature could even reach 201.5 °C. The third effect was that the temperature at the edge of the glass was also increased. The main reason is that the side lamps shrank, and more heat could be absorbed by the edge of the glass.

_{1}/H = h

_{2}/H = 0.8) increased to 114.5 °C, but the three lamps were too close to the glass, which made the heat concentrated in the middle of the glass line. This concentration caused the highest and lowest temperatures to be at the center and edge, respectively, so the temperature difference became larger. For example, in the first row of the third line (h

_{1}/H = h

_{2}/H = 0.2), the temperature difference could reach 138.5 °C, which is unacceptable for an oven.

#### 4.3. Main Effects and Fitness Function

_{avg}and ΔT

_{max}, respectively. The numbers in the figure show the control factor’s effect, illustrating its calculation method: the T

_{avg}value of l/L at Level 1 was 108.7, meaning the average value of T

_{avg}in the l/L in line number 1 in Table 2. The results show that T

_{avg}increased by 20 °C when the level of l/L rose from Level 1 to Level 2, but the ΔT

_{max}only increased by 0.1 °C. When the level of l/L increased to Level 3, the variation of T

_{avg}was not significant (<2 °C), but the ΔT

_{max}obviously increased (>38 °C). This indicates that a moderate reduction in the distance between lamps could obviously increase T

_{avg}without ΔT

_{max}of the panel, but further decrease of the distance between lamps would not only yield a minor increase the average temperature, but lead to a sharp rise in the temperature difference. The best level of l/L was therefore l/L = 1/6 (Level 2). In terms of h

_{1}/H and h

_{2}/H performance, it was found that T

_{avg}decreased monotonically when their levels increased. ΔT

_{max}decreased obviously when the Level 1 was raised to Level 2, but ΔT

_{max}changed little when it continued to increase to Level 3. This indicates that increasing the distance between the lamps and the glass is helpful to reduce the temperature difference, but will also decrease the average temperature. The best level of h

_{1}/H and h

_{2}/H was therefore h

_{1}/H = h

_{2}/H = 0.5 (Level 2).

_{1}/H = h

_{2}/H = 0.5, the figure in the center of Figure 6b shows that ΔT

_{max}was larger than 20 °C, so the combination l/L = 0 and h

_{1}/H = h

_{2}/H = 0.5 should be taken into account. Its temperature distribution figure shows more uniformity and the T

_{avg}was not very low. So, the value of fitness function Q for each combination was calculated and listed for analysis below.

_{1}/H, and h

_{2}/. Furthermore, the T

_{avg}and ΔT

_{max}of panel and fitness function Q for each combination are shown in the table. It was found that the highest and lowest T

_{avg}were in Test 10 (201.5 °C) and Test 9 (99.6 °C), respectively. The highest and lowest ΔT

_{max}were in Test 19 (138.5 °C) and Test 9 (7.7 °C), respectively. The highest and lowest fitness function Q were in Test 5 (9.93) and Test 25 (1.15), respectively. Obviously, the number is quite discrete, so the effect of each oven was quite different. In addition, Test 5 which was treated as the choice of optimal combinations had the highest Q. So, the optimal design level was l/L = 0 and h

_{1}/H = h

_{2}/H = 0.5.

## 5. Conclusions

_{1}= 50 mm, and h

_{2}= 50 mm performed the best with fitness function Q = 9.93. This work has given a preliminary but systematic way of developing a thermal radiation oven. A comprehensive study taking into account more practical issues than this work will be followed up soon.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a | thermal diffusivity, m/s^{2} |

C_{1} | first radiation constant, 3.742 × 10^{8} W μm^{4}/m^{2} |

C_{2} | second radiation constant, 1.439 × 10^{4} μm·K |

D | diameter of infrared lamp, m |

E | emissive power, W/m^{2} |

${F}_{{\mathsf{\lambda}}_{1},{\mathsf{\lambda}}_{2}}$ | fraction of the total emission in a wavelength interval λ_{1} ≦ λ ≦ λ_{2} |

G | irradiation, W/m^{2} |

H | height, m |

h_{1} | vertical distance between the upper/lower side of oven wall and panel top/bottom surface, m |

h_{2} | vertical distance between center of the middle lamp and panel, m |

k | thermal conductivity, W/m·K |

L | length of glass panel |

l | lateral distance between center of a side lamp and the closest edge of glass panel, m |

Q | fitness function |

T | temperature, K |

W | width of panel, m |

x, y, z | Cartesian coordinate system |

Superscript | |

m | number of iterations |

Subscripts | |

avg | average |

b | blackbody |

down | bottom surface of panel |

ini | initial temperature |

i,j,k | incidence dummy index for x, y, and z |

L | heating lamp |

max | maximum |

min | minimum |

surr | surrounding |

total | total radiative property |

up | top surface of panel |

Greek symbols | |

α | absorptivity |

Δ | difference |

δ | penetration depth, m |

ε | emissivity |

θ | incident angle, degree |

κ | extinction coefficient |

λ | wavelength, m |

σ | Stefan–Boltzmann constant, 5.67×10^{-8} W/m^{2}·K^{4} |

Abbreviations | |

CFD | computational fluid dynamics |

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**Figure 1.**Configuration sketch of the proposed thermal radiation oven. Subgraphs

**I**,

**II**, and

**III**are the oven from stereoscopic perspective view, top view, and front view, respectively.

**Figure 2.**Optical constants (n and κ) of glass (SiO

_{2}) and the penetration depth (δ) of a 0.7-mm-thick glass panel at the spectral range 0.2 μm ≤ λ ≤ 25 μm.

**Figure 3.**Absorptivity spectrum of the glass panel when the incident angle of irradiation was θ = 0°, 15°, 30°, and 60°.

**Figure 4.**Spectra of spectral emissive power (E

_{λ}) from a blackbody at temperature T

_{b}= 300 °C and T

_{b}= 500 °C. The wavelength (λ

_{max}) corresponding to peak of each spectrum is also listed.

**Figure 6.**Temperature distribution of a glass panel within different ovens. Subgraphs (

**a**–

**c**) are associated with l/L = 0, 1/6, and 1/3, respectively. Each subgraph contains nine distributions. Three in the same row are results corresponding to h

_{1}/H = 0.2, 0.5, and 0.8 from left to right. Three in the same column are results corresponding to h

_{2}/H = 0.2, 0.5, and 0.8 from top to bottom.

**Figure 7.**Main effects of three control factors on T

_{avg}and ΔT

_{max}. Three subgraphs from top to bottom show the main effects of l/L, h

_{1}/H, and h

_{2}/H.

Factor | Level 1 | Level 2 | Level 3 |
---|---|---|---|

l/L | 0 | 1/6 | 1/3 |

h_{1}/H | 0.2 | 0.5 | 0.8 |

h_{2}/H | 0.2 | 0.5 | 0.8 |

**Table 2.**Average temperature T

_{avg}(°C), maximum temperature difference ΔT

_{max}(°C), and fitness function Q in all tests. Each test used an oven configuration specified with levels in the same row.

Test | l/L (Level) | h_{1}/H (Level) | h_{2}/H (Level) | T_{avg} | ΔT_{max} | Q |
---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 170.2 | 40.9 | 0 |

2 | 1 | 1 | 2 | 149.1 | 61.2 | 0 |

3 | 1 | 1 | 3 | 134.9 | 82.5 | 0 |

4 | 1 | 2 | 1 | 151.0 | 73.0 | 0 |

5 | 1 | 2 | 2 | 130.3 | 10.6 | 9.93 |

6 | 1 | 2 | 3 | 115.4 | 18.4 | 4.91 |

7 | 1 | 3 | 1 | 137.3 | 93.2 | 0 |

8 | 1 | 3 | 2 | 115.7 | 29.1 | 0 |

9 | 1 | 3 | 3 | 99.6 | 7.7 | 9.69 |

10 | 2 | 1 | 1 | 201.5 | 34.8 | 0 |

11 | 2 | 1 | 2 | 185.1 | 47.2 | 0 |

12 | 2 | 1 | 3 | 174.4 | 61.1 | 0 |

13 | 2 | 2 | 1 | 164.6 | 84.4 | 0 |

14 | 2 | 2 | 2 | 146.5 | 27.3 | 0 |

15 | 2 | 2 | 3 | 133.5 | 11.2 | 9.69 |

16 | 2 | 3 | 1 | 144.2 | 97.9 | 0 |

17 | 2 | 3 | 2 | 123.9 | 36.7 | 0 |

18 | 2 | 3 | 3 | 108.8 | 16.7 | 5.02 |

19 | 3 | 1 | 1 | 199.4 | 138.5 | 0 |

20 | 3 | 1 | 2 | 187.8 | 98.3 | 0 |

21 | 3 | 1 | 3 | 155.4 | 89.2 | 0 |

22 | 3 | 2 | 1 | 169.3 | 115.0 | 0 |

23 | 3 | 2 | 2 | 153.4 | 63.4 | 0 |

24 | 3 | 2 | 3 | 141.6 | 78.0 | 0 |

25 | 3 | 3 | 1 | 148.0 | 106.5 | 0 |

26 | 3 | 3 | 2 | 128.9 | 47.8 | 0 |

27 | 3 | 3 | 3 | 114.5 | 29.3 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Gu, M.-J.; Yang, S.; Wu, Y.-C.; Chiu, C.-J.; Chen, Y.-B.
Designing a Thermal Radiation Oven for Smart Phone Panels. *Inventions* **2018**, *3*, 36.
https://doi.org/10.3390/inventions3020036

**AMA Style**

Gu M-J, Yang S, Wu Y-C, Chiu C-J, Chen Y-B.
Designing a Thermal Radiation Oven for Smart Phone Panels. *Inventions*. 2018; 3(2):36.
https://doi.org/10.3390/inventions3020036

**Chicago/Turabian Style**

Gu, Min-Jhong, Shuai Yang, Yen-Cheng Wu, Chien-Jui Chiu, and Yu-Bin Chen.
2018. "Designing a Thermal Radiation Oven for Smart Phone Panels" *Inventions* 3, no. 2: 36.
https://doi.org/10.3390/inventions3020036