How it cools? Studying the heat flow out of a semi-infinite slab in welding: An analytical approach

This paper presents a novel analytical framework using Laplace transforms and Fourier series to derive closed-form solutions for transient and steady-state heat flow in semi-infinite slabs with Newtonian cooling, offering a computationally efficient and accurate alternative to existing models for optimizing thermal management in welding and additive manufacturing.

The Gist

Imagine you are baking a very complex, multi-layered cake in a kitchen. You have a laser oven that moves around, melting the batter layer by layer to build a solid structure. This is similar to 3D printing (Additive Manufacturing) or welding, where a laser melts metal powder to build a part.

The problem? Heat.

When the laser melts the metal, it gets incredibly hot. As soon as the laser moves away, that spot starts to cool down. But here's the catch: the metal doesn't cool evenly. Some parts cool fast, some slow. This uneven shrinking creates invisible "tension" inside the metal, like a rubber band being stretched too tight. If the tension gets too high, the metal cracks, warps, or breaks.

For decades, engineers have used a famous, old recipe (called Rosenthal's Model) to predict how this heat moves. But that recipe has a major flaw: it assumes the metal is infinitely big. It forgets that real metal parts have edges, and heat escapes out of those edges into the air.

This paper introduces a new, smarter recipe that finally accounts for the edges and the cooling air.

The Core Idea: The "Leaky Bucket" Analogy

Think of the metal part as a bucket of water (the heat).

• The Old Model (Rosenthal): Imagine the bucket has no bottom and no sides. The water (heat) just spreads out forever into an infinite ocean. It never runs out, and it never hits a wall. This works okay for a tiny drop of water in a giant ocean, but it fails miserably for a real bucket sitting on a table.
• The New Model (This Paper): This model realizes the bucket has sides and a bottom. It knows that water leaks out the sides (cooling) and that the bucket has a specific width. It calculates exactly how much water is lost to the air at the edges.

How They Did It: Two Magic Tricks

The authors wanted to solve a very complicated math puzzle (the "Heat Equation") that describes how heat flows. They used two different "magic tricks" (mathematical tools) to solve it, and they proved both tricks lead to the exact same answer.

1. The Time-Travel Trick (Laplace Transform):
   Imagine you want to know the future of the heat. Instead of watching the heat move second-by-second, this trick "pauses" time and looks at the heat as a static picture in a different dimension. It's like taking a photo of a moving car and analyzing the blur to figure out its speed. This method is great for handling the "start" and "stop" of the laser.

2. The Lego Block Trick (Fourier Series):
   Imagine the heat distribution isn't a smooth wave, but a stack of Lego blocks of different sizes. The biggest block is the main heat, and the tiny blocks are the little ripples near the edges. This method breaks the problem down into these blocks, solves each one, and stacks them back together. It's perfect for dealing with the "walls" of the metal part.

The authors showed that whether you use the "Time-Travel" view or the "Lego Block" view, you get the same result. This gives them a super-reliable tool.

Why This Matters: Predicting the "Crack"

Why do we care about this new recipe?

• Stopping Cracks: By knowing exactly how heat escapes the edges, engineers can predict exactly when and where the metal will get stressed enough to crack. They can then adjust the laser speed or the cooling fans to prevent it.
• Saving Money and Time: Currently, to figure out how to weld a new part, companies often have to build it, test it, see if it breaks, and then try again. This new math model acts like a super-accurate simulator. You can run the numbers on a computer first, optimize the process, and only build it once.
• AI Training: The model can generate thousands of "fake" heat scenarios instantly. This is like feeding a robot chef a million practice recipes so it learns to cook the perfect cake without burning a single real one.

The "Aha!" Moment

The paper shows that for a short time, the old model and the new model look the same. It's like when you drop a pebble in a pool; the ripples spread out the same way whether the pool is small or huge.

But, as time goes on, the ripples hit the edge of the pool.

• In the Old Model, the ripples just keep going forever.
• In the New Model, the ripples hit the edge, bounce back, and the water level drops because it's leaking out.

This difference is crucial for long processes like welding a car frame. If you ignore the "leaking" (cooling at the edges), your predictions will be wrong, and your car part might fail.

Summary

In simple terms, this paper says: "Stop pretending metal parts are infinite. They have edges, and heat escapes from them. We have built a new, faster, and more accurate math tool that accounts for this, helping us build stronger, crack-free metal parts for planes, medical implants, and cars."

Technical Summary

1. Problem Statement

Additive Manufacturing (AM) and welding processes are highly sensitive to thermal management. Improper heat dissipation leads to residual stresses, distortion, and cracking. While traditional analytical models (e.g., Rosenthal's solutions) exist, they suffer from significant limitations:

• Geometry: They typically assume open-domain (infinite) geometries, failing to account for finite-width constraints.
• Cooling: They often neglect realistic cooling boundary conditions (BCs), such as Newton's Law of Cooling (NLOC) or radiation.
• Transient Behavior: Many models focus on steady-state solutions, missing the critical transient phase where cracks often initiate.
• Computational Cost: Numerical methods (e.g., Finite Element Method) can handle these complexities but are computationally expensive and offer less physical intuition.

The authors aim to develop a rigorous analytical framework for solving the 3D heat equation in a finite-width slab with cooling boundary conditions, capturing both transient and steady-state behaviors efficiently.

2. Methodology

The study models heat transfer using the linear diffusion-convection equation in a co-moving frame (moving with the heat source velocity $v$). The domain is semi-infinite in $x$ and $y$, but finite in $z$ (width $w$).

Key Mathematical Steps:

1. Boundary Conditions: The authors impose Newton's Law of Cooling (NLOC) at the top ($z=0$) and bottom ($z=w$) surfaces, representing heat flux loss to the environment. They also note the framework can accommodate Stefan-Boltzmann radiation.
2. Dual Analytical Approaches: To solve the Partial Differential Equation (PDE), the authors employ two mathematically equivalent methods:
   • Approach 1: Laplace Transform (LT) + Fourier Transform (FT):
     • Applies FT in spatial dimensions $x$ and $y$ (open domains).
     • Applies LT in time $t$ to handle initial conditions and transform the PDE into a second-order Ordinary Differential Equation (ODE) in $z$.
     • Solves the ODE using Green's function techniques.
     • Inverts the LT using the Residue Theorem and Contour Integration (Bromwich contour), carefully handling branch cuts and an infinite series of poles on the negative real axis.
   • Approach 2: Fourier Series (FS) + FT:
     • Expands the solution in the finite $z$-dimension using discrete eigenfunctions that satisfy the cooling BCs.
     • Reduces the problem to a first-order ODE in time, solved by direct integration.
     • Proves that as the width $w \to \infty$, the FS summation converges to the standard Fourier Transform integral, recovering the open-domain case.
3. Source Terms: The derived Green's function (Heat Kernel) is convolved with various heat source profiles:
   • Point sources (on/off switching).
   • Spatially distributed sources: Gaussian, Ellipsoidal, and Double-Ellipsoidal (mimicking the asymmetry of welding arcs).
   • Time-dependent power inputs.

3. Key Contributions

• Closed-Form Analytical Solutions: The paper derives exact, closed-form series solutions for transient and steady-state temperature profiles in a finite 3D geometry with cooling.
• Equivalence Proof: It rigorously proves the equivalence between the Laplace Transform approach (spectral space) and the Fourier Series approach (eigenfunction expansion), offering flexibility in implementation.
• Generalization of Rosenthal's Models: The framework naturally recovers Rosenthal's classic 2D (with cooling) and 3D (without cooling) solutions as limiting cases ($w \to 0$ and $w \to \infty$, respectively), validating the new model's robustness.
• Transient Dynamics: Unlike many steady-state models, this approach explicitly captures the time evolution from the initial state to equilibrium, which is critical for predicting crack formation.
• Computational Efficiency: The analytical solutions significantly reduce computational costs compared to numerical simulations (FEM) while providing deeper physical insight.

4. Results

• Validation: The analytical solutions were validated against:
  1. Fully numerical implementations (Finite Difference Method).
  2. Numerical inversion of the Laplace Transform.
  3. The Residue Theorem.
  • Outcome: Strong agreement was found across all methods, confirming the accuracy of the derived series solutions.
• Cooling Effects: Comparisons between the new finite-width model and traditional open-domain models (Rosenthal) show that:
  • At early time scales, both models coincide (heat has not reached boundaries).
  • As time progresses, the finite-width model with cooling diverges significantly, showing lower temperatures near boundaries due to heat loss.
  • This divergence is crucial for processes like welding where the system approaches a steady state.
• Source Profiles: The model successfully handles complex source geometries (Gaussian, double-ellipsoidal), providing temperature profiles that account for the asymmetry of the heat source relative to the motion direction.
• Pole Structure: The analysis reveals that the singularities (poles) of the system lie on the negative real axis in the spectral space, are infinitely countable, and are not evenly spaced, which dictates the decay rates of the thermal modes.

5. Significance and Future Outlook

• Process Optimization: The framework provides a scalable tool for optimizing thermal predictions in metal-based manufacturing, helping to mitigate residual stresses and cracking without expensive experimental trials.
• Machine Learning: The ability to generate synthetic datasets efficiently makes this framework ideal for training Physics-Informed Machine Learning (PIML) models to predict heat distribution.
• Future Work: The authors propose extending the model to curved geometries (cylindrical, spherical), incorporating finite boundaries in $x$ and $y$, and integrating the analytical solver with experimental data for benchmarking.

In conclusion, this work bridges the gap between simplified analytical models and computationally heavy numerical simulations, offering a precise, efficient, and physically intuitive tool for understanding heat dissipation in finite manufacturing geometries.