Modelling for Temperature Distribution Calculation using Surface Scattering of Free Electrons in Finite Metal Solid
Article information
Abstract
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.
1. Introduction
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 model1,2) used to explain the heat conduction of metals, free electrons behave like gas diffusion in classical mechanics as illustrated in Fig. 1. When electrons with high thermal energy near a heat source travel in a straight line and collide with metal cations, they transfer energy, change their direction by scattering, and travel again. They diffuse in disorder while repeating this collisionchange of direction. The average distance traveled by electrons before a collision is referred to as mean free path (MFP), and the temperature increases as MFP decreases. Since electrons perform random motion, some electrons in high temperature zones transfer energy to low temperature zones while some electrons in low temperature zones also travel to high temperature zones to transfer energy. Such movement, however, does not mean that energy flows from low temperature zones to high temperature zones. A net energy always flows from high temperature zones to low temperature zones because the average active energy of electrons in high temperature zones is relatively higher than that in low temperature zones. After the Drude model, various quantum mechanical models were presented considering interactions between electrons, between electrons and defects, and between electrons and boundaries1,2).
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 conduction3). In particular, when electrons traveling inside a metal hit the external surface, surface scattering occurs and the velocity of the electrons changes with no change in their quantity as shown in Fig. 2. Phonons also cause surface scattering despite insignificant influ ence. Such scattering occurs in the form of total reflection (probability p) and diffuse reflection depending on the surface condition and interference effect. After reflection, the electrons again diffuse in disorder.
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 Fig. 3. This means that the temperature gradient does not occur in a direction perpendicular to the surface near the surface. From a microscopic perspective, unlike the reflection of light from a mirror, the reflection of electrons has different incidence and reflection angles depending on the surface condition. Overall thermal energy becomes symmetrical with respect to the surface because electrons enter in random directions and also proceed in random directions after reflection. If the external boundary is not under adiabatic condition, heat exchange with the outside occurs, resulting in a temperature gradient (directionality) for which the heat flux is not zero. In particular, when the thickness is low compared to the heat input, the heat exchange by convection on the external surface has significant influence. On the other hand, when heat energy is locally given onto a plate in the case of welding or line heating and the thickness is moderately high, the cooling effect by convective heat transfer on the external surface is low at the beginning. This is attributed to the fact that the rate of conduction is much faster than that of convection. In practical applications, assuming adiabatic conditions during the initial cooling phase does not yield substantial discrepancies with experimental results.
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 solution4,5), which is expressed as equation (1). It represents the temperature distribution inside a semiinfinite solid with a point heat source on the surface. Derived from the governing equations of heat conduction, it has been widely adopted by researchers.
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 r_{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 Fig. 4, the shortest distance to a point in the plate, r_{0}, lies along the optimal path. In the semiinfinite plate with one heat source, there exists only one optimal path to a specific point.
Conventional heat conduction models for a semiinfinite 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.
2. Method
2.1 Temperature distribution in a finite solid
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 Fig. 5. This study incorporated this effect into the Rosenthal solution, and proposed a method applicable to all general thicknesses rather than semiinfinite solids.
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 r_{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 T_{0} is added to the final temperature for consistency. If the shortest distance on all the optimal paths that consider reflection is defined as r_{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 r_{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 Fig. 6. Although the value of n theoretically ranges from negative infinity to positive infinity , it was set to range from 200 to 200 for both calculation efficiency and accuracy. A wider range is also possible, but it was judged to be meaningless from an engineering perspective because the temperature result changed by less than 1°C. By incrementing n from 200 to 200 and calculating each f term, then summing these to find the temperature T, and adding the initial temperature, the final temperature distribution can be predicted. The initial temperature is the temperature of the metal before heating, so the preheating temperature or interpass temperature can be applied if in use.
2.2 Peak temperature of a finite plate
In the heatingcooling 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(∂T/∂t=0). It is summarized as equation (3) for an infinite plate and equation (4) for a thin plate. Since the equations are the results of previous researches6), the solving process is omitted here.
Thick plate
Thin plate:
ρ:density[kg/mm^{3}]
c:specific heat[J/kg°C]
Unlike the above equations, the number of the f terms in equation (2) is theoretically infinite. Thus, differentiation to obtain the peak temperature is not feasible. The algorithm in Fig. 6, however, enables to derive the temperature matrix at each point. Given that the heat source moves, the peak temperature distribution matrix is determined by comparing each component of the yz matrix over the xaxis, which is the movement path of the heat source, and selecting the maximum value. The results are shown in chapter 3.
2.3 Temperature distribution on the back side of a thick finite plate
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 f term. If the thickness is sufficiently large, a simpler expression of the formula can be derived because most terms can be neglected. To this end, it was assumed that the first surface scattering effect on the bottom surface was the most dominant, while the effect of the subsequent reflections was insi gnificant. Here, the sufficient thickness can be seen as a degree at which the depth of HAZ does not exceed the center of the thickness. Therefore,
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 semiinfinite 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 Fig. 7. From a quantum perspective, it can be understood that electrons and phonons affect the temperature both when they are incident upon and reflected from the surface.
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 zdirection 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 Fig. 6.
2.3.1 Peak temperature on the back side of a thick finite plate
To predict the peak temperature on the back side during welding, substituting equation (6) into ∂T/∂t=0 yields equation (7). As described above, the peak temperature on the back side of a sufficiently thick finite plate is calculated to be more than twice as high compared to an infinite plate. When the variable h is substituted into z in equation (3), the solution to equation (7) is twice as high.
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 Fig. 6.
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 Table 1. For the Tshaped joint geometry with no significant thickness difference, W was set to 0.5 by assuming that heat is distributed to the horizontal and vertical plates by 50%.
2.3.2 Backside peak temperature measurement experiment
An experiment was carreid out to verify equation (8) above. Fig. 8(a) shows the experimental specimen. For the Tshaped geometry, multipass partial penetration welding was underwent with flux cored arc welding (FCAW). The heat input of 220A, 22V, and 20cm/min was applied for the first pass while the heat input of 290A, 26V, and 35cm/min was used for the second to fifth passes. The environment was controlled to minimize the heat loss by airinduced convection or contact with other materials. As shown in Fig. 8(b), a thermocouple was attached to the center of the bottom surface to measure the temperature over time. To compare the bottom surface temperature under the constant thickness condition, the temperature increments during the first, second, and fourth passes, which correspond to the first layer, were considered.
3. Results and Discussion
The temperature distribution in a finite plate over time by a point heat source can be calculated through equation (2) and Fig. 6 proposed in this study. In this chapter, the calculation results for the peak temperature distribution and the areas of FZ and HAZ were presented, and the model was compared with other models for its verification from an application perspective
The common input conditions used for calculation are listed in Table 2 below. They were set to the material properties of mild steel. Effective heat input is a value that considers both the heat input efficiency and the heat input by the welding process.
3.1 Calculation results by models
3.1.1 Peak temperature distribution results for a finite plate: model results
Fig. 9 shows the calculation results by the proposed model. The peak temperature distribution with respect to the thickness is displayed with contour lines. For a thickness of 5mm, perfect thin plate behavior is observed, as the temperature distribution is uniform through thickness. As the thickness increases, the bottom surface maintaines constant temperature in the thickness direction, indicated by contour lines with zero heat flux. For a thickness of 35mm, it can be seen that the temperature distribution of approximately 240°C or higher shows a circular heat source distribution as with infinite plates. The bottom surface with lower temperatures shows contour lines with zero heat flux. The peak temperature at the bottom surface was calculated to be 105°C. For all thicknesses, the bottom surface exhibited thermal symmetry. The results of the proposed model well represented thermal symmetry regardless of the thickness.
3.1.2 Sizes of FZ and HAZ in a finite plate: comparison between models
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 fournode 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. Fig. 10 shows the results of the FZ and HAZ lengths with respect to the thickness. Fig. 10(a) shows the calculation results of the proposed model and the FEA results with good agreement regardless of the thickness. This indicates that the proposed temperature prediction formula serves as a generalized model that does not differentiate between thin and thick plates. Fig. 10(b) shows the results of previous models that are categorized into equation (3) for a thick plate and equation (4) for a thin plate, with the values calculated by Ueda et al.6). When the thickness is between approximately 5mm and 12mm, the results overlap. Thus, it is necessary to select between the thin plate and thick plate models. This highlights the challenge in establishing a clear criterion for distinguishing between the two models. The uncertainty increases further when considering various combinations of heat input and thickness.
3.2 Model utilization
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.
3.2.1 Validation of simplified prediction model for backside peak temperature
To verify the simplified prediction formula (8), an experiment was performed in accordance with the procedure outlined in section 2.3.2. Fig. 11 shows the temperature results measured over time. The measured temperature increments were 46.1°C, 54.0°C, and 60.4°C, respectively. The initial temperature of the steel plate was recorded at 12.2°C. Meanwhile, the temperature increments by equation (8) were calculated to be 40.0°C, 58.1°C, and 58.1°C. The constant values used for calculation are summarized in Table 1. Mild steel was applied as material, fillet welding as geometry, and FCAW as heat input efficiency. A comparison between the experiment and calculation results shows that the temperature is predicted in a reasonable manner despite some errors.
3.2.2 Understanding of heat concentration
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.
Fig. 12 shows the results of conducting twodimensional thermal analysis through FEA. The input conditions are shown in Table 2. The hole has a diameter of 4mm and a depth of 25mm from the back side of the heat source on the central axis. The analysis results predict that the temperature at the node located 10mm below from the center of the heat source, which is the top surface of the hole, was predicted to be 965°C. Under the same conditions without the hole, however, the temperature at the same position was 633°C, showing a significant temperature difference between the two cases. The temperature difference may decrease as the diameter or depth of the hole decreases, but it is important to note that the method of measuring the temperature inside a steel plate by drilling a hole has low accuracy.
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. Fig. 13 shows the temperature distribution when a point 10mm away from the corner is heated. The temperature at the corner node was predicted to be 1,266°C, but it was 633°C when the heat source and corner were sufficiently distant from each other under the same conditions. Heat concentration commonly occurs in lap joint welding among others, often resulting in corner(or edge) melting.
4. Conclusion
A generalized model was developed to predict the temperature distribution regardless of the thickness using conventional semiinfinite 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.