Optimizing the Size of a Moving Annular Hollow Laser Heat Source

Abstract

The physical phenomenon of the annular hollow laser surface treatment process is complex, and the internal mechanism involves multiple disciplines and fields. In addition to the general parameters of laser beams, such as laser power and scanning speed, an annular hollow laser beam exhibits unique physical characteristics, including hollow ratio and hollow area. The selection of the inner and outer annular radii of the laser plays a critical role in the study of metal surface heat treatment. From the point of view of heat transfer, the entransy dissipation theory is introduced in the metal surface treatment process with an annular hollow heat source. Firstly, using the principle of the extreme value of the entransy dissipation rate, under a constant heat flux boundary condition, the entransy dissipation rate is obtained through the temperature field distribution in the calculation area by numerical simulation. Secondly, the selection of the inner and outer ring radii of the annular laser is explored, and the average temperature difference of the heating surface is minimized to reduce the thermal stresses of the material. This paper seeks new insights into the geometric parameters of the inner and outer radii of the annular heat source.

1. Introduction

Over the past decade, considerable research has been devoted to the characteristics of laser surface treatment processes. However, due to the complexity of the physical processes involved in laser surface treatment, the aforementioned studies have primarily focused on the molten pool and its adjacent areas for both non-contact and contact measurements. Experimentally capturing the temperature and thermal stress distribution outside the molten pool presents significant challenges. Therefore, theoretical analysis and computational simulation play an indispensable role in the analysis of this process.

In numerous studies, when the heat source is small relative to the substrate to the extent that the shape and size of the heat source can be disregarded, the heat source can be considered a point or a line heat source. Devesse et al. [1] derived the control equations for isothermal line growth when heating a workpiece with a steady-state or moving-point heat source. Mendez [2] conducted a systematic scaling analysis of a steady-state point heat source on a semi-infinite solid, finding that all characteristic values related to the isotherms could be reduced to dimensionless expressions solely dependent on the Rykalin number. Heller et al. [3] established a simplified model for the point heat source to describe the temperature field outside the melt pool during the welding process, where energy is concentrated on the top surface of a plate with finite thickness. Based on the analytical solution for the moving-point heat source by Rosenthal, Gockel et al. [4] derived closed solutions for the temperature distribution near the free boundary and precise solutions for the free-edge effect using two closely spaced point heat sources. However, their results were applicable only to small-scale processes involving low laser power and low scanning speeds. Zhao [5] utilized the thermal lattice Boltzmann method to conduct a two-dimensional numerical simulation of natural convection in a heated enclosed cavity, analyzing the physical principles of mass and heat transfer with different heating surface areas.

Over the last decade, the interaction between laser beams and materials, especially those beams with Gaussian profiles (TEM00 mode), has been extensively studied. Due to the complexity of the physical processes involved, several models [6,7] are devoted to the impact of Gaussian beam modes in thermal conduction scenarios. Cline and Anthony [8] formulated a Gaussian heat source model based on the material’s thermal conductivity to ascertain its thermal distribution and cooling rates; the results showed the temperature distribution of a moving heat source was asymmetric relative to its direction of motion. Reséndiz-Flores et al. [9] presented the proposition of employing the Kuhnert Finite Pointset Method (FPM) [10] for conducting numerical simulations of heat transfer associated with a moving Gaussian thermal source during the processing of welding materials. Kar and Rath [11] established a transient model to predict the evolution of melt pools generated by moving lasers with Gaussian and transient trapezoidal distributions in space.

Beyond the conventional models of point and line heat sources, as well as the widely utilized Gaussian heat source model for simulation purposes, the diversity in laser processing conditions across various laser machining applications has led scholars to study the thermal physical mechanisms under specific conditions by either combining existing heat source models or developing new ones. Zhou et al. [12] employed a combined model of dual ellipsoids and a conical body as an equivalent heat source to describe the flow and heat transfer within the plasma arc welding melt pool. Flint et al. [13] derived a semi-analytical solution for the transient heat conduction equation of a moving dual ellipsoid-conical (DEC) heat source distribution with a surface Gaussian component. Yadaiah et al. [14] proposed a volumetric heat source model structured like an egg, which served as a heat source model for laser welding processes, and conducted a transient heat transfer analysis based on 3D finite element modeling. Velaga et al. [15] proposed a three-dimensional finite element analysis on thin-walled cylindrical specimens of AISI 304 stainless steel, investigating the impact of the geometric parameters of a dual-ellipsoidal moving heat source during the welding process. They found that changes in the heat source’s geometrical parameters exerted minimal impact on the dimensions and shape of the melt pool and the temperature distribution without affecting the distribution of residual stresses. However, in laser surface treatment, other beam modes, like TEM01*, might offer better advantages [16]. While point circular laser beam spots were suitable for most laser applications, there were cases where rectangular beam modes were preferred [17]. Differing from the Gaussian laser heat source or combined Gaussian-based models predominantly used in research, the research team led by Shi [18] developed a novel technique of annular hollow laser light powder feeding. Liu et al. [19] investigated the effects of the hollow ratio of the hollow laser beam on the temperature distribution and cooling rate of the melt pool, and the accuracy of their numerical calculations was validated by experiments. Compared to a Gaussian heat source, this annular hollow laser heat source model can generate higher temperatures at the edges, which helps to inhibit the formation of powder adhesion and reduce the meniscus height, thereby reducing the need for post-processing steps and saving time and energy.

On the other hand, starting from the analogy between heat conduction and electrical conduction processes, Guo et al. [20,21] introduced physical quantity “entransy” characteristics to describe an object’s heat transfer ability and proposed the principle of minimum entransy dissipation for optimizing heat conduction processes. Xia et al. [22] applied this principle to study a simple one-dimensional plate liquid–solid phase change process; they aimed to minimize entransy dissipation as the optimization object under the condition of a fixed total process time and derived the optimal heat transfer strategy corresponding to the phase change process. Chen et al. [23] derived the optimal heat transfer process path with the minimum entropy generation under the general heat transfer law.

In the past few decades, extensive research has been conducted on the laser and material interaction processes, with the majority of these studies focusing on laser beams with a Gaussian profile (TEM00 mode). However, due to their high central heat flux density, Gaussian laser beams produce non-uniform temperature gradient fields on the surface of the substrate during laser surface treatment. Exceeding the material’s ultimate strength, thermal stresses can lead to substrate fracture, making Gaussian laser beams unsuitable for substrates with uniform thermal stress requirements. In contrast to traditional Gaussian beams, annular hollow laser beams significantly enhance laser energy intensity at the periphery of the spot while slightly reducing it at the center. This beam mode is an emerging technology. Several studies have explored the heat transfer mechanisms of media subjected to irradiation by moving annular hollow laser heat sources. The parameters of annular lasers impact product performance, such as surface quality and mechanical properties. To the authors’ knowledge, the effect of selecting the inner and outer radii of the annular laser on the temperature distribution of the substrate has not been studied from a heat transfer perspective.

In this study, the temperature field distribution within the computational domain was determined through numerical simulation under a constant heat flux boundary condition, the entransy dissipation extremum theory was used, and the selection of the inner and outer ring radii of the annular laser was explored. By ensuring melting, the average heat transfer temperature difference on the heating surface was minimized, thereby the thermal stress was reduced. The minimum of this indicates the minimum loss of heat transfer capacity in the process.

2. Physical and Mathematical Model and Numerical Simulation

2.1. Principle of Entransy Dissipation Extremum

For a steady-state heat conduction process without internal heat sources, when the boundary heat transfer rate is given, minimization of the entransy dissipation leads to minimization of the temperature difference [21,24]. This principle is formally expressed mathematically as follows [21,24]:

$${\stackrel{·}{Q}}_{net}\Delta T_{eq}={{\displaystyle \underset{A}{\iint }k(\nabla T)}}^{2}dA$$

(1)

In Equation (1), $\nabla T$ denotes the temperature gradient. When the heat flux within the system is specified, the minimum entransy dissipation rate corresponds to the minimization of the heat flux-weighted equivalent temperature difference, and it is called the principle of minimum entransy dissipation. According to this principle, under a given heat flux condition, the minimization of entransy dissipation rate leads to the minimized average temperature difference during heat conduction. This implies that a reduction in material surface thermal stress can be achieved, so even material crack formation can be avoided.

2.2. Physical Models

As illustrated in Figure 1, the model depicts an annular hollow laser heat source impinging upon the substrate, with the orange-shaded region representing the computational area A designated for parameter optimization. To simplify the model, the top surface of the substrate is simplified to a rectangular shape, with the substrate’s height direction perpendicular to the plane of the paper. The annular hollow laser irradiates in a direction orthogonal to the top surface of the substrate.

To take the movement of the source into consideration, a standard coordinate transformation is applied. The fixed coordinates of the plate are transformed into a moving coordinate system centered on the annular heat source. Based on the quasi-steady-state assumption, the temperature distribution on the plate in this state is determined.

The annular hollow heat source moves along the x-axis to heat the top surface of the substrate. Through the analysis of the Gaussian laser beam power density distribution formula, the heat flux input by the annular hollow laser at the substrate interface is described as follows:

$$q(x,y)=\{\begin{array}{cc}0& 0\le r\le R_1\\ {\displaystyle \frac{2A_b·q_0}{\pi ({R_2}^2-{R_1}^2)}}\exp (-2{\displaystyle \frac{r^2}{{R_2}^2}})& R_1\le r\le R_2\\ 0& r>R_2\end{array}$$

(2)

$$r=\sqrt{x^2+y^2}$$

(3)

Given that the size of the annular hollow heat source is significantly smaller than that of the substrate, it is assumed for simplification that the other surfaces of the substrate are isothermal walls, with an initial temperature set to the ambient temperature T₀. At any given moment, the boundary conditions for the interaction between the top surface of the substrate and the other surfaces are as follows:

$$\{\begin{array}{cc}0\le r<R_1& {\displaystyle \frac{\partial T}{\partial r}}=-{\displaystyle \frac{h(T-T_0)}{k}}\\ R_1\le r\le R_2& {\displaystyle \frac{\partial T}{\partial r}}=-{\displaystyle \frac{q_0}{k}}\\ r>R_2& {\displaystyle \frac{\partial T}{\partial r}}=-{\displaystyle \frac{h(T-T_0)}{k}}\end{array}$$

(4)

$$t=0,T=T_0$$

2.3. Optimization of Annular Heat Source

Heat input, which is the ratio of laser power q₀ to scanning speed u, profoundly affects the structure and properties of the components in both meso- and micro-scales [25]. Given the heat input and a material’s absorptivity, it is possible to ascertain the heat flux on the material’s surface. That means under specified heat flux conditions, the objective is to minimize the entransy dissipation rate ${\stackrel{·}{\mathsf{\Phi }}}_g$ (it is the entransy dissipation rate per unit volume caused by heat conduction) within the calculation domain A. This optimization yields the optimal inner and outer radii of an annular heat source.

The temperature distribution within computational domain A is determined by numerical simulation. Based on the nodal positions and temperatures, the entransy dissipation rate within domain A can be calculated, as delineated in Equation (1). Due to the discrete nature of temperature, the acquisition of the entransy dissipation rate for the aforementioned domain A needs summation over temperatures in various directions at corresponding nodes:

$$\stackrel{·}{\mathsf{\Phi }}=k{\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n\left[{\left(\frac{\Delta T_i}{\Delta x_i}\right)}^2+{\left(\frac{\Delta T_j}{\Delta y_j}\right)}^2\right]}}$$

(5)

In the optimization process, the entransy dissipation principle is utilized to examine the impact of varying inner and outer radii of the laser on the uniformity of the temperature field within region A under the condition of constant heat input. The aim is to achieve the objective of reducing the average temperature difference within region A.

2.4. Mesh Independence and Computational Method Verification

To precisely calculate the substrate temperature distribution, it is imperative that the mesh size in the laser irradiated area is sufficiently small. To address the issue, this study implements a dense mesh arrangement in the region traversed by the thermal source. The fine meshes are configured where the heat source passes through, gradually transitioning to coarser meshes outward from this area, as shown in Figure 2.

To assess the grid independence for the selection of an appropriate mesh size, a case study was conducted employing an annular laser power of 500 W, heat source movement speed of 5 mm/s, and an annular heat source with inner and outer radii of R₁ = 3 mm and R₂ = 4 mm, respectively. Mesh sizes of 15,000; 30,000; 45,000; 60,000; and 90,000 were evaluated to determine the impact of the mesh size on the numerical solution. The grid independence analysis, as shown in

Table 1. Grid independent analysis results.

Parameters	Number of Grids	Maximum Temperature/k
Laser power: 500 W	15,000	607.88
Moving speed: 5 mm/s	30,000	618.05
	45,000	622.15
R1 = 3 mm	60,000	622.72
R2 = 4 mm	90,000	623.26

, indicates that the impact of changing the number of grids on the maximum temperature is less than 20 K, indicating that the difference in maximum temperature values between high grid numbers is very small, and the error within this range can be ignored. From Table 1 and Figure 3, it can be seen that under this parameter condition, the relative error of the highest temperature obtained by grid numbers 15,000 and 30,000 is relatively large; in contrast, when the grids are encrypted to 60,000 and 90,000, there is no significant change in the calculation results compared to the number of grids of 45,000, but the computational workload increases and the computation time increases. Therefore, 45,000 grids were selected for numerical simulation in order to maintain the best balance between computational efficiency and accuracy.

In the end, in order to fully assure the numerical solution method, the present work is compared with the experiments in the work [26], which is not repeated here.

3. Results and Discussion

3.1. The Impact of Inner Radius of Annular Hollow Laser on Entransy Dissipation Rate

In addition to parameters such as laser power and scanning speed, the unique inner ring radius R₁ and outer ring radius R₂ of the annular hollow laser beam also have a crucial impact on the temperature field. The ratio between the inner and the outer diameter of the annular laser spot is called the hollow ratio (η = R₁/R₂).

Regarding the radius of the inner ring of the annular laser, we discuss the Variation in the entransy dissipation rate with changes in R₂ when R₁ = 2.5 mm, 3.5 mm, and 4.5 mm under a certain heat input level, as shown in Figure 4.

As illustrated in Figure 4a, under the conditions of a power input q₀ of 500 W, a scanning speed u of 0.005 m/s, and an annular inner radius R₁ of 2.5 mm, it can be observed that the entransy dissipation rate decreases as the hollow ratio decreases. When R₂ is between 2.5 mm and 3 mm, corresponding to the hollow ratio approximately between 0.83 and 1, there is a sharp decline in the entransy dissipation rate. This trend of change also exists in Figure 4b,c. When R₁ is 3.5 mm, and R₂ is between 3.5 mm and 4 mm, which means the hollow ratio η is between 0.875 and 1, the entransy dissipation rate sharply decreases. When R₁ is 4.5 mm, and R₂ is between 4.5 mm and 5 mm, that is, the hollow ratio η is between 0.9 and 1, there is a marked decline in the entransy dissipation rate. Furthermore, within the range of R₁ values, as R₁ increases, the hollow ratio, which causes a sharp decrease in the entransy dissipation rate, also increases.

The smaller the entransy dissipation rate of the material, the more uniform the temperature gradient field [24], and the more effective the reduction in the average temperature difference of the material. In the context of surface heating, while ensuring the material reaches its melting point, it is advantageous to minimize the entransy value. When R₁ is 2.5 mm, and the outer radius R₂, with a lower entransy dissipation rate, ranges from 4.5 mm to 5 mm, this corresponds to a hollow ratio η approximately between 0.5 and 0.56. When R₁ is 3.5 mm, and the outer radius R₂, with a lower entransy dissipation rate, ranges from 5.8 mm to 7 mm, the hollow ratio η is roughly between 0.5 and 0.6. When R₁ is equal to 4.5 mm, and the outer radius R₂, with the lowest entransy dissipation rate, ranges from 7.3 mm to 9 mm, the hollow ratio η is between 0.5 and 0.62.

It can be inferred that within the range of R₁ values, when the hollow ratio is between 0.5 and 0.56, the entransy dissipation rate is relatively small, which means that laser heating under this parameter can obtain a more uniform temperature gradient field. In addition, when the heat input and hollow ratio are given, the maximum entransy dissipation rate for 3.5 mm is obtained when comparing R₁ with 2.5 mm, 3.5 mm, and 4.5 mm.

Given a constant inner annular radius R₁, a reduction in the outer annular radius R₂, resulting in an increased hollow ratio, leads to a decreased area directly exposed to laser irradiation; consequently, this induces an increase in the average heat flux density, and thus, the temperature gradient field in region A is more uneven, resulting in an increase in the entransy dissipation rate.

As depicted in Figure 5a–c, it can be observed that under constant heat input and fixed inner annular radius R₁, the peak temperature within the computational domain decreases with an increase in R₂, which means that the maximum temperature reduces as the hollow ratio diminishes. Furthermore, with constant heat input and a hollow ratio η, for R₁ = 2.5 mm, 3.5 mm, and 4.5 mm, the highest peak temperature is attained at R₁ = 3.5 mm. Among these three annular laser parameters, when R₁ is 3.5 mm, it not only has the highest entransy dissipation rate but also the highest temperature.

Meanwhile, under the condition of ensuring a certain heat input, when R₁ = 2.5 mm, the variation curve of its entransy dissipation rate with R₂ satisfies the minimum entransy dissipation rate and the minimum maximum temperature within the calculation area.

3.2. The Impact of Outer Radius of Annular Hollow Laser on Entransy Dissipation Rates

Regarding the annular hollow laser’s outer radius R₂, we explored the variations in entransy dissipation rates with the alterations in the inner annular radius R₁ under a scenario where the heat input (q₀/u) remained constant and the outer annular radius R₂ was set at 4 mm, 5 mm, and 6 mm, respectively. The results are depicted in Figure 6.

As illustrated in Figure 6, with constant heat input and a fixed outer annular radius R₂ of the annular heat source, the entransy dissipation rate generally exhibits an upward trend as the inner annular radius R₁ increases.

The calculation area is the inner ring area of the annular laser, and different inner ring radii result in different calculation areas. When the outer radius of the laser is constant, as the inner radius R₁ increases, the area directly irradiated by the annular heat source decreases, the average heat flux density increases, and the temperature is higher. Therefore, the temperature gradient field between the laser direct irradiation area and the calculation area A is more uneven, resulting in a higher dissipation rate. R₁ directly affects the calculation area, so the entransy dissipation rate varies more dramatically with the radius of the annular laser inner ring.

Figure 7 presents the temperature field of the substrate surface under the action of the annular heat source.

It can be observed from Figure 7 that the highest temperature always occurs in the direct heating zone of the annular laser. Figure 7 corresponds to the temperature field in Figure 6b when R₁ is 2.3 mm. It can be seen that the yellow high-temperature zone almost covers the entire circle formed by the outer radius of the annular laser, leaving only a small portion of the green low-temperature zone in the center. It can be seen that the temperature field distribution in the ring and the inner area is relatively uniform, resulting in a lower entransy dissipation rate.

As depicted in Figure 8, under the specified laser power of q₀ = 500 W, scanning speed of u = 0.005 m/s, outer annular radius R₂ = 5 mm, and inner annular radius R₁ = 4 mm, the temperature field exhibits a specific distribution pattern. There is no red high-temperature area; instead, it primarily consists of a small yellow temperature zone and a yellow-green temperature zone, with most of the computation area A presented as a green low-temperature zone. This distribution, compared to Figure 7 and Figure 9, indicates that the temperature field in Figure 7 is slightly more uneven than in Figure 8, while in Figure 8, it is relatively more uniform than in Figure 9. It can be inferred that under these conditions, the entransy dissipation rate is higher than in Figure 7 but lower than in Figure 9.

Furthermore, upon adjusting the inner radius to R₁ = 4.8 mm while keeping the other parameters constant, a significant change in the temperature field is observed, as shown in Figure 9. A small portion of the red high-temperature area and a section of the yellow temperature zone have developed, but the majority of the computation area still remains a green low-temperature zone, with even a blue low-temperature area present. This indicates that under this set of parameters, the temperature field distribution becomes even more uneven, and the entransy dissipation rate significantly increases.

4. Conclusions

From the perspective of heat transfer, with a fixed heat input and the aim of minimizing the dissipation rate, a model was established to select the parameters of an annular laser’s inner and outer radii. The optimal ranges for the inner and outer ring radii, corresponding to the minimization of the material temperature difference, are provided.

The range of hollow ratios that minimize dissipation rates are nearly identical, and within the parameter ranges under consideration, the hollow ratio is between 0.5 and 0.56, the entransy dissipation rate is the smallest, and the temperature field distribution is more uniform. The average heat transfer temperature difference of the parameter combination with the lowest entransy dissipation rate is effectively reduced, and the overall temperature difference is significantly improved. The essence is that the temperature gradient field of the parameter combination with the lowest entransy dissipation rate is more uniform. The minimum of the entransy dissipation rate indicates the minimum loss of heat transfer capacity in the process. This optimized energy utilization not only enhances processing efficiency but also reduces energy consumption, highlighting the energy-saving advantages of the annular laser.

When the heat input and outer annular radius are constant, the entransy dissipation rate increases with an increase in the inner annular radius, implying that the entransy dissipation rate rises with a higher hollow ratio. Conversely, when the heat input and inner radius are constant, the entransy dissipation rate decreases with an increase in the outer radius, indicating that the entransy dissipation rate declines with a lower hollow ratio. Moreover, the radius of the outer radius of the annular laser is constant, and changing the radius of the inner radius has a greater impact on the entransy dissipation rate.

The maximum temperature decreases with a reduction in the hollow ratio. Within the parameters studied, the maximum temperature for the annular laser hollow ratio between 0.5 and 0.56 is relatively small compared with other hollow ratios, aligning with the hollow ratio ranges that minimize the entransy dissipation rate. A higher entransy dissipation rate corresponds to a more uneven distribution of the temperature field.