5.1 Fourier Conduction, Rosenthal Moving Heat Source & 2D vs. 3D Heat Flow

5.1 Conduction Heat Transfer, Fourier's Law & Rosenthal's Heat Flow

Quick Answer: Conductive heat flow in fusion welding is modeled using Rosenthal's quasi-steady-state equations, which transform transient coordinates into a moving reference frame ($\xi = x - vt$). The transition between two-dimensional (thin-plate) and three-dimensional (thick-plate) heat flow is governed by the relative plate thickness $\tau = d \sqrt{\frac{\rho C_p (T_c - T_0)}{H_{\text{net}}}}}$, where $\tau < 0.6$ indicates 2D behavior and $\tau > 0.9$ indicates 3D behavior. In 3D conduction, the centerline cooling rate is inversely proportional to net heat input ($R \propto 1/H_{\text{net}}$) and independent of plate thickness, whereas in 2D conduction, it is inversely proportional to the square of both heat input and thickness ($R \propto 1/(H_{\text{net}} d)^2$). The critical cooling time $\Delta t_{8/5}$ dictates austenite decomposition kinetics and cold cracking susceptibility.

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Fourier's Law and Thermal Properties of Engineering Alloys

Conduction is the primary mode of thermal energy transport within the solid and semi-solid regions of a welded joint. The fundamental equation governing heat conduction is Fourier's Law, which states that the local heat flux vector $\mathbf{q}$ is directly proportional to the negative gradient of temperature:

$$\mathbf{q} = -k \nabla T$$

In one dimension (Cartesian $x$-direction):

$$q_x = -k \frac{dT}{dx}$$

where:

• $q_x$ = Heat flux per unit area ($\text{W/m}^2$ or $\text{J/(s}\cdot\text{m}^2\text{)}$)
• $k$ = Thermal conductivity of the material ($\text{W/(m}\cdot\text{K)}$ or $\text{J/(s}\cdot\text{mm}\cdot^\circ\text{C)}$)
• $\frac{dT}{dx}$ = Temperature gradient in the direction of heat flow ($^\circ\text{C/m}$ or $\text{K/m}$)

The negative sign represents the second law of thermodynamics: heat conducts spontaneously down the temperature gradient from higher-temperature regions to lower-temperature regions.

   Arc Heat Source (High T)
             │
             ▼
       ┌───────────┐      Heat Flux Vector (q = -k dT/dx)
       │ Weld Pool │ ──────────────────────────────────────►  Base Metal (Low T)
       └───────────┘      Steep Temperature Gradient (dT/dx < 0)

Volumetric Heat Capacity and Thermal Diffusivity

While thermal conductivity ($k$) dictates the instantaneous rate at which heat moves through a material under a static thermal gradient, welding is inherently transient. The rate at which the temperature field changes depends simultaneously on the material's capacity to store thermal energy. This is governed by the volumetric heat capacity ($C_v = \rho C_p$) and the thermal diffusivity ($\alpha$):

$$\alpha = \frac{k}{\rho C_p}$$

where:

• $\alpha$ = Thermal diffusivity ($\text{m}^2/\text{s}$ or $\text{mm}^2/\text{s}$)
• $\rho$ = Material mass density ($\text{kg/m}^3$ or $\text{g/cm}^3$)
• $C_p$ = Specific heat capacity at constant pressure ($\text{J/(kg}\cdot\text{K)}$ or $\text{J/(g}\cdot^\circ\text{C)}$)
• $\rho C_p$ = Volumetric heat capacity ($\text{J/(m}^3\cdot\text{K)}$ or $\text{J/(mm}^3\cdot^\circ\text{C)}$)

Thermal diffusivity measures the rate at which thermal disturbances propagate through a solid. A material with high thermal diffusivity (such as aluminum or copper) conducts thermal energy rapidly relative to its volumetric energy storage capacity, equalizing internal temperature gradients quickly. Conversely, a material with low thermal diffusivity (such as austenitic stainless steel or titanium) conducts heat slowly, resulting in localized heat accumulation, steep temperature gradients, high thermal stresses, and severe angular distortion.

Thermal Properties of Common Structural Alloys (at $20^\circ\text{C}$)

Welding Engineering Insight: The table above explains why welding austenitic stainless steel requires substantially lower heat input to achieve penetration compared to mild steel, yet results in higher residual distortion. Because $\alpha_{\text{SS304}}$ is less than one-third that of carbon steel, heat cannot conduct away rapidly; it concentrates near the fusion boundary. In contrast, welding copper or aluminum requires massive electrical power or elevated preheat ($150^\circ\text{C} - 250^\circ\text{C}$) to establish and sustain a stable weld pool because thermal energy rapidly dissipates away from the arc via conduction.

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Steady-State vs. Transient Heat Transfer & Moving Coordinate Transformation

In a stationary coordinate system $(x, y, z)$ fixed to the workpiece, the temperature at any location fluctuates continuously as the welding heat source approaches, passes, and recedes. The general three-dimensional transient heat conduction equation with no internal heat generation is:

$$\rho C_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k \frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k \frac{\partial T}{\partial y}\right) + \frac{\partial}{\partial z}\left(k \frac{\partial T}{\partial z}\right)$$

Assuming temperature-independent thermal properties (constant $k, \rho, C_p$):

$$\frac{\partial T}{\partial t} = \alpha \nabla^2 T = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right)$$

Solving this transient equation directly for a moving heat source is analytically intractable. In 1935–1946, Daniel Rosenthal introduced a seminal mathematical transformation that simplified welding thermal analysis: transforming the stationary coordinate system $(x, y, z)$ into a moving coordinate system $(\xi, y, z)$ fixed to the heat source traveling along the $x$-axis at constant velocity $v$.

       Stationary System (x, y, z)                Moving System (ξ, y, z)
       Workpiece fixed                            Heat source fixed at ξ = 0

             y                                          y
             ▲                                          ▲
             │   Weld Travel (v)                        │
             │      ───────►                            │      Leading Edge (ξ > 0)
             ├──────────────────► x                     ├───────────┼──────► ξ
            /   Arc Source                             /          Arc (ξ = 0)
           /                                          /   Trailing Wake (ξ < 0)
          z                                          z

                                  ξ = x - v·t

The Quasi-Steady-State Assumption

Let the moving coordinate in the direction of travel be:

$$\xi = x - vt$$

Applying the chain rule of multivariable calculus to relate time derivatives in the stationary frame to spatial derivatives in the moving frame:

$$\frac{\partial T}{\partial t} = \frac{\partial T}{\partial \xi} \frac{\partial \xi}{\partial t} = -v \frac{\partial T}{\partial \xi}$$

After an initial transient period following arc ignition (typically several centimeters of travel), the temperature distribution around the moving arc becomes invariant with respect to time for an observer moving with the torch. This condition is known as quasi-steady state:

$$\left( \frac{\partial T}{\partial t} \right)_{(\xi, y, z)} = 0$$

Substituting the moving coordinate relationship into the governing differential equation yields the Rosenthal Quasi-Steady-State Conduction Equation:

$$\frac{\partial^2 T}{\partial \xi^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = -\frac{v}{\alpha} \frac{\partial T}{\partial \xi}$$

This elliptic partial differential equation converts a transient problem into a steady-state spatial boundary value problem, enabling exact closed-form analytical solutions.

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Rosenthal's Classical Moving Heat Source Solutions

Rosenthal formulated two primary idealized solutions based on workpiece geometry:

1. Point Source Solution (3D) for thick plates (semi-infinite solid).
2. Line Source Solution (2D) for thin sheets/plates where temperature is uniform through the thickness.

Rosenthal Idealized Assumptions

1. The heat source is concentrated at an infinitesimal point or line.
2. Workpiece physical and thermal properties ($k, \rho, C_p, \alpha$) are homogeneous, isotropic, and temperature-independent.
3. Latent heat of fusion (melting) and latent heat of solid-state phase transformation are neglected.
4. Heat losses across the plate surfaces via convection and radiation are zero (adiabatic boundaries except at the source).
5. The welding velocity $v$ is constant, and the process has reached quasi-steady state.

3D Thick-Plate Point Source Solution

For a semi-infinite solid ($z \ge 0$) with a point heat source operating at the origin $(\xi = 0, y = 0, z = 0)$ on the surface, heat conducts hemispherically in three dimensions. The temperature field $T(\xi, y, z)$ is:

$$T - T_0 = \frac{q_{\text{net}}}{2 \pi k R} \exp\left[ -\frac{v (\xi + R)}{2\alpha} \right]$$

where:

• $T_0$ = Initial uniform preheat temperature of the workpiece ($^\circ\text{C}$ or $\text{K}$)
• $q_{\text{net}} = \eta V I$ = Net heat input delivered to the workpiece ($\text{W}$ or $\text{J/s}$)
• $\eta$ = Arc thermal efficiency factor (dimensionless)
• $V$ = Arc voltage ($\text{V}$), $I$ = Welding current ($\text{A}$)
• $R = \sqrt{\xi^2 + y^2 + z^2}$ = Radial distance from the point source to the calculation point ($\text{m}$ or $\text{mm}$)
• $\xi = x - vt$ = Longitudinal distance relative to the source ($+ \xi$ ahead of arc, $-\xi$ behind arc)

2D Thin-Plate Line Source Solution

When welding thin sheets of thickness $d$, the temperature gradient through the thickness ($z$-direction) is negligible ($\partial T/\partial z \approx 0$). The arc is modeled as a line source penetrating completely through the thickness $d$. The analytical solution is:

$$T - T_0 = \frac{q_{\text{net}}}{2 \pi k d} \exp\left( -\frac{v \xi}{2\alpha} \right) K_0\left( \frac{v r}{2\alpha} \right)$$

where:

• $d$ = Plate thickness ($\text{m}$ or $\text{mm}$)
• $r = \sqrt{\xi^2 + y^2}$ = Radial distance in the horizontal plane from the line source ($\text{m}$ or $\text{mm}$)
• $K_0$ = Modified Bessel function of the second kind of order zero

                      ISOTHERM CONTOURS (QUASI-STEADY STATE)

                     Leading Edge                     Trailing Thermal Wake
                     (Steep Gradient)                 (Elongated Tail)
                           │                                │
                           ▼                                ▼
                      ╭─────────╮                ╭────────────────────────╮
                   ───┼─────────┼───► v          │     Arc     1000°C 800°C 500°C
                      │  Ahead  │                │      ●        │      │     │
                   ───┼─────────┼───             ╰────────────────────────╯
                      ╰─────────╯                   ξ = 0       ─── -ξ ───►

Characteristics of Rosenthal Isotherms

• Ahead of the Heat Source ($\xi > 0$): Isotherms are tightly compressed. The temperature rises extremely rapidly as the arc approaches because conduction opposes the incoming convective flow of cooler base metal.
• Behind the Heat Source ($\xi < 0$): Isotherms are drawn out into elongated, teardrop-shaped or elliptical tails. The rate of cooling decreases as distance behind the arc increases.
• At High Travel Speeds ($v \to \infty$): Isotherms become increasingly elongated and slender, resembling narrow parabolas.

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2D vs. 3D Heat Flow: The Relative Plate Thickness Criterion ($\tau$)

Selecting whether to model heat flow using 2D or 3D equations is one of the most critical decisions in welding thermal analysis. The physical transition depends on whether thermal conduction reaches the back (root) surface of the plate and reflects back into the fusion zone during the critical cooling cycle.

The governing dimensionless parameter is the relative plate thickness ($\tau$):

$$\tau = d \sqrt{\frac{\rho C_p (T_c - T_0)}{H_{\text{net}}}}$$

where:

• $d$ = Plate thickness ($\text{m}$ or $\text{mm}$)
• $T_c$ = Critical temperature of interest (e.g., $540^\circ\text{C}$ or $500^\circ\text{C}$ for steel transformations)
• $T_0$ = Initial plate preheat temperature ($^\circ\text{C}$)
• $H_{\text{net}} = \frac{\eta V I}{v} = \frac{q_{\text{net}}}{v}$ = Net heat input per unit weld length ($\text{J/m}$ or $\text{J/mm}$)

Dimensional Regime Thresholds

       3D HEAT CONDUCTION (τ > 0.9)                 2D HEAT CONDUCTION (τ < 0.6)

              Arc Source (v)                               Arc Source (v)
                    │                                            │
            ────────▼────────                            ────────▼────────
           /  · · · · · · ·  \                          │   │   │   │   │ │
          / · · · · · · · · · \                         │   │   │   │   │ │  Uniform
         │ · · · · · · · · · · │ d (Thick)              │   │   │   │   │ │  through
         │ · · · · · · · · · · │                        │   │   │   │   │ │  thickness
         └─────────────────────┘                        └─────────────────┘  d (Thin)
          Hemispherical Heat Flux                        Planar Cylindrical Flux

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