This study generalizes the temperature distribution equation for finite metal solid, unifying previous separate models for thick and thin plates. Effects of surface scattering of free electrons on heat conduction is taken account. As plate thickness decreases, these scattering events increase, leading to elevated temperatures due to reduction in mean free paths. We propose a prediction model for temperature distribution incorporating these effects onto the existing Rosenthal solution. It exhibits excellent agreement with finite element analysis results across all thicknesses. Furthermore, from an engineering standpoint, two examples of how heat concentration occurs in material with geometry that promotes multiple surface scattering are presented.

Metals are known to have higher thermal conductivity than other materials because the effect of free electron movement is larger than the influence of phonons. According to the assumption of the classical Drude model

Drude Model (electron: small circle, metal ion: large circle)

When electrons in a conductor reach the grain boundary or external surface while traveling, boundary scattering occurs. It refers to a phenomenon that reflects the wave function of electrons due to a sudden change in electronic potential. Electrons change their direction as if a ball bounces off a wall. Due to this phenomenon, it was revealed that the size of the material affects heat conduction

Surface scattering and grain boundary scattering in a typical metallic nanowire

When the external boundary is under adiabatic condition, there is no heat exchange with the outside, which forms thermal symmetry with a heat flux of zero as shown in

Thermal symmetry boundary condition

Research efforts to predict the size of the heat affected zone (HAZ) and the fusion zone (FZ), the peak temperature, and cooling rates after the local heating of plates, i.e., welding and line heating, have advanced since the beginning of the welding process. A simple model that has been widely utilized is the Rosenthal solution

T : estimated temperature [°C]

T_{0} : initial temperature [°C]

t : time [sec]

Q : Total heat energy Q=νq [J/sec], q: effective heat flux [J/mm]

v : heat source speed [mm/sec]

k : thermal conductivity [J/mm∙sec∙°C]

r_{0} : radial distance from heat source center [mm], r_{0}=

α : thermal diffusivity [mm^{2}/sec]

x,y,z : coordinates of x,y and z from heat source center [mm]

It can predict the temperature over time at a point with a distance of _{0} from the heat source. A single optimal path is considered to represent various zigzag trajectories caused by the random motion of electrons. As illustrated in _{0}, lies along the optimal path. In the semi-infinite plate with one heat source, there exists only one optimal path to a specific point.

Prediction of temperature at a point in infinite plate,

Conventional heat conduction models for a semi-infinite solid assume an infinite thickness and were designed seperately for thick and thin plates. This study aims to propose a generalized heat conduction model for metals with a finite thickness by employing the surface scattering principle of free electrons. This mathematical model builds on conventional models and calculate FZ and HAZ distribution, the peak temperature distribution, and cooling rate during welding. In addition, precau- tions during the welding process are introduced through a case study on the heat concentration caused by surface scattering.

Since a finite plate has a thickness unlike an infinite plate, electrons and phonons scatter on the surface. Thus, the optimal paths from the heat source to a point include various paths by reflection. Theoretically, the number of the optimal paths can increase indefinitely. To quantify the heat conduction along these paths, this study assumed adiabatic condition in which the infinite plate was cut into the same thickness and there was no heat gain or loss on the cut surface. Since all electrons and phonons are reflected from the cut surface, the number of the optimal paths increases in proportion to the number of reflections as illustrated in

Prediction of temperature at a point in finite plate,

The temperature distribution at a point can be calculated by substituting the distances along all the optimal paths from the heat source to the point into _{0} of equation (1) and summing the temperatures calculated for each path. To apply this superposition principle, thermal properties were assumed to be independent of temperature. In the equation, the initial temperature _{0} is added to the final temperature for consistency. If the shortest distance on all the optimal paths that consider reflection is defined as _{n}, the temperature distribution can be expressed as equation (2). For simplicity, n=0 holds when there is no reflection. When the number of reflections is even, n increases by 1 to positive infinity. When it is anodd number, n decreases by 1 to negative infinity.

r_{n} : distance from heat source center along optimal paths [mm],

h : plate thickness [mm](z≤h)

n = −∞...,−3,−2,−1,0,+1,+2,+3,...,+∞

For n=0, equation (2) becomes equivalent to equation (1) as the case is the same as an infinite plate with no reflective surface. As the thickness decreases, the temperature at the same position becomes higher compared to the case of an infinite plate because the shortest length _{n} decreases despite the same number of reflections. This also means that MFP becomes shorter. Since manually calculating equation (2) is challenging, this study proposes an algorithm flow chart to enable fast computation as shown in

Flow chart of prediction algorithm for temperature distribution of a finite plate

In the heating-cooling cycle, the moment when the temperature rise halts momentarily before cooling process represents the condition for reaching the peak temperature. The peak temperature can be determined by setting the time derivative of equation (1) to zero(

Thick plate

Thin plate:

^{3}]

Unlike the above equations, the number of the

Regardless of the thickness, the back side temperature distribution of all plates can be calculated by substituting h for z in equation (2) as the z value of the bottom surface corresponds to the thickness h. Hand calculation, however, is impossible due to the need to compute the infinite series _{0} becomes very large, resulting in minimal influence on temperature. Consequently,

According to the above equations and equation (1), the bottom surface temperature of a finite plate with a sufficiently high thickness is always calculated to be twice the temperature at the same position of a semi-infinite plate. This is because thermal symmetry is formed at the bottom surface. It appears that the heat source located at a symmetrical position and the current heat source influence the bottom surface simultaneously as illustrated in

Thermal symmetry of a plate with finite thickness

However, the actual temperature is always more than twice as high because much more reflections inevitably occur in reality, which were not accounted for in the initial assumption. The temperature increment by the second reflection is calculated to be 1/9 of the first calculated temperature. The distance along the optimal path is three times the thickness in z-direction from the heat source and the peak temperature is inversely proportional to the square of the distance as in equation (3). Likewise, for the third reflection, the temperature increment is reduced by 1/25 due to five times the thickness. This is the value that cannot be neglected depending on the case. The bottom surface temperature of a thin finite plate is significantly affected by multiple reflections, it can be calculated through equation (2) and

To predict the peak temperature on the back side during welding, substituting equation (6) into

At the bottom surface (finite plate):

In the case of thin finite plate, however, the peak temperature on the bottom surface is significantly affected by multiple reflections. It can be achieved using equation (2) and

In the actual arc welding process, most plates are sufficiently thick finite plates. Therefore, the peak temperature on the back side can be calculated manually using equation (8). It is a simplified prediction formula for the peak temperature on the back side according to the material, welding conditions, and joint geometry. From equation (7) the constants along with the density and specific heat of the material were combined into the material constant C, and the heat input was separated into heat input parameters such as electric current, voltage, welding speed and heat input efficiency. Detailed input variables are listed in

Input variables for equation (8)

Variable | Variable name | Variable | Variable name | Variable | Variable name |
---|---|---|---|---|---|

T_{max} |
Back-side peak temperature | I | Welding current (A) | h | Thickness (mm) |

V | Welding voltage (V) | ν | Speed (cpm) | T_{0} |
steel plate initial temperature |

C | Material constant | W | Joint geometry | η | Heat input efficiency |

Steel: 777.3 Aluminum: 1131.4 Stainless Steel: 702.6 | Bead on plate: 1 T-joint: 0.5 | FCAW(CO_{2}):0.85 SAW(automatic): 1 GTAW(TIG): 0.7 |

An experiment was carreid out to verify equation (8) above.

Experiment for temperature measurement at the bottom of finite T-shaped plate

The temperature distribution in a finite plate over time by a point heat source can be calculated through equation (2) and

The common input conditions used for calculation are listed in

Input value for calculation

Input variables: unit | value |
---|---|

Density: |
7860×10^{-9} |

Specific heat: c(J/kg°C) | 460 |

Effective heat input: (J/mm) | 1000 |

Contour of peak temperature distribution with different thickness using the proposed model

To verify the assumptions in this study and subsequent calculation results, they were compared with the results under the same input conditions using Abaqus, a commercial finite element analysis (FEA) code. Adiabatic condition was imposed as boundary condition, and heat transfer analysis was conducted for thicknesses of 2mm, 6mm, and 12mm. DC2D4, a four-node heat transfer element, was applied and the total number of elements was 8,640. Since the geometry of the element is a 1mm × 1mm square from the center of the heat source, which has a high temperature gradient, to a distance of 40mm, the areas of FZ and HAZ were acquired by performing linear interpolation with the temperature values calculated at each node. For a comparison, the boundaries of FZ and HAZ were set at 1,450°C and 700°C, respectively.

Calculated length of fusion zone and heat affected zone

The proposed temperature distribution prediction formula for a finite plate and its principles were discussed from an application perspective. To assess potential painting damage on the back side of the plate during welding or line heating, it is essential to predict the peak temperature on the back side. Additionally, understainding the phenomenon of surface scattering in heat conduction is necessary to prevent heat concentration during local heating.

To verify the simplified prediction formula (8), an experiment was performed in accordance with the procedure outlined in section 2.3.2.

Temperature measured at the bottom of finite T-shaped plate

The heat conduction by electron behavior in consideration of surface scattering, which is used in this study, is easier to understand intuitively than the mechanical approach by the governing equations of heat conduction. It implies that heat is concentrated as the distance between the external surfaces decreases because the surface scattering effect increases and thus MFP decreases. During local heating, such as welding, attention must be paid to heat concentration in thin area or at corners(edges). Two representative cases are shown below.

Case 1) Temperature measurement experiment problem for specimens with hole drilling to insert a thermocouple

To measure the temperature inside a steel plate during welding or line heating, an experimental method that drills a hole and inserts a thermocouple on the back side (opposite side to the heat source) of the steel plate is often adopted. From a heat conduction perspective, it can be expected that heat will be concentrated due to surface scattering as the distance between the heat source and the hole surface decreases and the diameter of the hole increases.

FEA result of temperature distribution of a plate with a hole for inserting a thermocouple.: Estimated temperature on top of the hole with a length of 25mm and a diameter of 4mm is 965°C at 2.375sec, while estimated temperature at the same location without hole is 633°C

Case 2) Heat concentration around a corner

When the heat source is close to a corner or an edge, heat concentration is expected due to the surface scattering.

FEA result of temperature distribution: Estimated temperature at the corner with a distance of 10mm from the heat source is 1266°C at 2.436sec, while temperature at the same location in infinite plate is 633°C

A generalized model was developed to predict the temperature distribution regardless of the thickness using conventional semi-infinite solid models, which differentiate between thin and thick plates during local heating such as welding and line heating. The main conclusions are as follows.

1) When the thickness is finite, the transferred thermal energy overlaps because the heat conduction paths by electrons increase infinitely due to the surface scattering at the boundary. Based on this principle, a new prediction formula was proposed by infinitely adding the prediction formulas applied to the conventional models.

2) The validity of the model was confirmed through its consistency with the finite element analysis (FEA) results. In addition, an algorithm flow chart was presented for the efficiency of calculation. This model allows for the calculation of temperature distribution over time, from which the peak temperature distribution, the lengths of the fusion zone and heat affected zone, and the subsequent cooling rate can be achieved.

3) A simplified prediction formula for the peak temperature on the back side of heating was proposed for its practical utilization.

4) For sufficiently thick plates (HAZ does not exceed the center of the thickness), which are commonly used, it was shown that the peak temperature on the back side is always more than twice as high as the value predicted by the conventional model. This is attributed to the thermal symmetry effect.

5) Cases of heat concentration due to the surface scattering effect were presented.

i. Attention must be paid when measuring the temperature of a specimen with a hole for inserting a thermocouple, as heat concentration can lead to elevated temperature readings.

ii. Since heat concentration easily occurs at a corner (edge) near the heat source, this should be taken into account during design and processes.