5.3 Convection, Stefan-Boltzmann Radiation & Emissivity of Weld Surfaces

5.2 Convection, Radiation & Boundary Layer Heat Dissipation

Quick Answer: External boundary heat dissipation from weldments occurs through combined convection and radiation. Convection is modeled by Newton's law of cooling ($q''{\text{conv}} = h(T_s - T\infty)$), with heat transfer coefficient $h$ increasing from $10 - 25\text{ W/(m}^2\cdot\text{K)}$ in natural convection up to $50 - 300\text{ W/(m}^2\cdot\text{K)}$ under forced cross-drafts. Radiation is governed by the Stefan-Boltzmann law ($q''{\text{rad}} = \varepsilon \sigma (T_s^4 - T{\text{sur}}^4)$), which dominates at molten pool and high-HAZ temperatures ($>600^\circ\text{C}$) due to its fourth-power absolute temperature dependence ($T^4$). Surface emissivity $\varepsilon$ ranges from $0.15$ for polished stainless steel to $0.90$ for heavily oxidized carbon steel. In preheat maintenance, thermal insulation blankets lower the effective heat transfer coefficient ($h_{\text{eff}}$), reducing electrical heating power requirements by over $90%$.

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Convective Heat Transfer & Newton's Law of Cooling

While internal conduction redistributes heat throughout the solid workpiece, thermal energy is ultimately rejected from the external surfaces of the plate and weldment to the ambient environment via convection and thermal radiation. Convective heat transfer occurs at the fluid-solid interface between the exposed weldment surface and the surrounding air.

                    Thermal Boundary Layer δ_t(x)
             Air (T_∞)  ▲ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 
                        │   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 
                        │     ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 
       ─────────────────┴───────────────────────────────────────────── Workpiece Surface
         Hot Plate (T_s)      q"_conv = h · (T_s - T_∞)

Newton's Law of Cooling

The fundamental relationship governing convective heat loss is Newton's Law of Cooling:

$$q''_{\text{conv}} = h (T_s - T_\infty)$$ $$q_{\text{conv}} = h A_s (T_s - T_\infty)$$

where:

• $q''_{\text{conv}}$ = Convective heat flux ($\text{W/m}^2$)
• $q_{\text{conv}}$ = Total convective heat transfer rate ($\text{W}$ or $\text{J/s}$)
• $h$ = Convective heat transfer coefficient ($\text{W/(m}^2\cdot\text{K)}$ or $\text{W/(m}^2\cdot^\circ\text{C)}$)
• $A_s$ = Exposed surface area of the weldment ($\text{m}^2$)
• $T_s$ = Surface temperature of the weldment ($^\circ\text{C}$ or $\text{K}$)
• $T_\infty$ = Ambient bulk air temperature ($^\circ\text{C}$ or $\text{K}$)

The Convective Heat Transfer Coefficient ($h$)

The parameter $h$ is not a thermodynamic material constant; it is a complex hydrodynamic parameter governed by fluid velocity, fluid density, viscosity, thermal conductivity, boundary layer thickness, and surface orientation. Convection around weldments falls into two distinct hydrodynamic regimes:

1. Natural (Free) Convection: Fluid motion is driven solely by density differences (buoyancy forces) caused by temperature gradients in the air adjacent to the hot plate. Hot air expands, decreases in density, and rises, drawing cooler ambient air across the plate surface.

   • Governed by the dimensionless Grashof number ($Gr$) and Rayleigh number ($Ra = Gr \cdot Pr$): $$Gr_L = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}$$ where $g$ is gravitational acceleration ($9.81\text{ m/s}^2$), $\beta$ is the volumetric thermal expansion coefficient of air ($\beta = 1/T_{\text{film}}$ for ideal gases, in $\text{K}^{-1}$), $L$ is characteristic plate length, and $\nu$ is kinematic viscosity.
   • Typical natural convection heat transfer coefficients for welded plates in quiescent shop air range from $5\text{ to }25\text{ W/(m}^2\cdot\text{K)}$.

2. Forced Convection: Fluid motion is driven by external mechanical forces, such as shop ventilation, draft fans, weld fume extraction hoods, or ambient wind during outdoor field welding.

   • Governed by the dimensionless Reynolds number ($Re_L = u_\infty L / \nu$) and Nusselt number ($Nu_L = h L / k_f$).
   • Under forced air velocities of $2\text{ to }10\text{ m/s}$ ($4.5\text{ to }22\text{ mph}$), $h$ increases dramatically to $50\text{ to }300\text{ W/(m}^2\cdot\text{K)}$.

Structural Welding Code Limits on Ambient Airflow

Because forced convection multiplies boundary cooling rates by up to an order of magnitude, major structural welding codes strictly limit ambient air velocity:

• AWS D1.1 (Structural Welding Code—Steel), Clause 7.11.1: Prohibits welding when ambient wind or drafts exceed $8\text{ km/h}$ ($5\text{ mph}$ / $2.2\text{ m/s}$) unless the work is protected by an approved wind screen or enclosure.
• AWS D1.5 (Bridge Welding Code), Clause 5.1.2: Mandates identical restrictions.

Engineering Consequence: When ambient wind exceeds $8\text{ km/h}$, forced convective heat extraction rapidly quenches the weldment exterior. In addition to disrupting shielding gas, this accelerated quenching dramatically shortens $\Delta t_{8/5}$, inducing hard, crack-susceptible martensitic HAZ microstructures in low-alloy and structural steels.

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Stefan-Boltzmann Thermal Radiation in Welding

Every body at a temperature above absolute zero emits electromagnetic radiation. In fusion welding, where temperatures span from ambient room temperature up to the liquidus melting point of steel ($>1530^\circ\text{C}$ / $1803\text{ K}$) and plasma arc temperatures exceed $15,000\text{ K}$, thermal radiation plays a massive role in boundary energy dissipation.

                               Thermal Radiation (q"_rad = ε·σ·(Ts^4 - Tsur^4))
                             ▲  ▲  ▲  ▲  ▲  ▲  ▲  ▲
                             │  │  │  │  │  │  │  │   Infrared & Visible Photons
       ──────────────────────┴──┴──┴──┴──┴──┴──┴──┴───────────────────
        Solidifying Pool / HAZ (Ts > 800°C)       Surroundings (T_sur = 20°C)

The Stefan-Boltzmann Law

The net rate of radiative heat exchange between a grey surface of area $A_s$ at absolute temperature $T_s$ and large surrounding walls at absolute temperature $T_{\text{sur}}$ is governed by the Stefan-Boltzmann Law:

$$q''_{\text{rad}} = \varepsilon \sigma \left( T_s^4 - T_{\text{sur}}^4 \right)$$ $$q_{\text{rad}} = \varepsilon \sigma A_s \left( T_s^4 - T_{\text{sur}}^4 \right)$$

where:

• $q''_{\text{rad}}$ = Radiative heat flux ($\text{W/m}^2$)
• $\varepsilon$ = Total hemispherical surface emissivity (dimensionless, $0 \le \varepsilon \le 1.0$)
• $\sigma$ = Stefan-Boltzmann constant = $5.670 \times 10^{-8}\text{ W/(m}^2\cdot\text{K}^4)$
• $T_s$ = Absolute surface temperature of the workpiece ($\text{Kelvin, K} = ^\circ\text{C} + 273.15$)
• $T_{\text{sur}}$ = Absolute temperature of the surrounding enclosure / atmosphere ($\text{Kelvin, K}$)

Critical Mathematical Rule: In all radiative heat transfer calculations, temperatures MUST ALWAYS be converted to absolute units (Kelvin). Evaluating $(T_s^4 - T_{\text{sur}}^4)$ in degrees Celsius results in catastrophic numerical errors.

The Radiation vs. Convection Temperature Regime

Because radiative heat transfer depends on the fourth power of absolute temperature ($T^4$), while convective heat transfer scales linearly with temperature difference $(T_s - T_\infty)$, their relative contributions shift radically across the welding temperature spectrum:

$$\frac{q''_{\text{rad}}}{q''_{\text{conv}}} = \frac{\varepsilon \sigma (T_s^4 - T_{\text{sur}}^4)}{h(T_s - T_\infty)}$$

• At Low Temperatures ($T_s < 200^\circ\text{C}$ / $473\text{ K}$): Convection dominates boundary dissipation ($60% - 80%$ of total heat loss). Radiative heat flux remains modest.
• At Intermediate Temperatures ($T_s \approx 500^\circ\text{C} - 600^\circ\text{C}$): Radiative and convective heat dissipation are approximately equal ($q''{\text{rad}} \approx q''{\text{conv}}$).
• At High Temperatures ($T_s > 1000^\circ\text{C}$ to Molten Pool $1550^\circ\text{C}$): Radiation overwhelmingly dominates boundary heat loss, accounting for over $85% - 92%$ of total surface heat dissipation. Consider a molten steel pool at $1550^\circ\text{C}$ ($1823\text{ K}$) radiating to surroundings at $20^\circ\text{C}$ ($293\text{ K}$): $$T_s^4 = (1823)^4 = 1.104 \times 10^{13}\text{ K}^4 \quad \gg \quad T_{\text{sur}}^4 = (293)^4 = 7.385 \times 10^9\text{ K}^4$$ The radiative emission rate reaches hundreds of kilowatts per square meter!

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Emissivity ($\varepsilon$) Dynamics Across Welding Alloys and Surface Oxides

Emissivity ($\varepsilon$) is the ratio of radiant energy emitted by a real material surface to that emitted by an ideal blackbody ($\varepsilon = 1.0$) at the identical temperature and wavelength. In welding engineering, emissivity is highly dynamic and depends on alloy composition, surface roughness, temperature, and—most critically—surface oxidation.

Surface Emissivity Values for Common Welding Engineering Alloys

Optical Infrared Pyrometry Calibration Pitfalls

In field and shop welding inspection, non-contact infrared (IR) thermometers and thermal imaging cameras are widely used to verify preheat and interpass temperatures. Most consumer and industrial pyrometers have a default factory emissivity preset of $\varepsilon = 0.95$ (calibrated for matte organic materials or heavy carbon deposits).

The Pyrometry Trap: If an inspector uses a default $\varepsilon = 0.95$ setting to measure the preheat temperature of clean, shiny stainless steel ($\varepsilon \approx 0.18$), the sensor detects only a fraction of the infrared flux it expects at a given temperature. The pyrometer displays an indicated temperature hundreds of degrees below the true surface temperature! Believing the part is under-heated, the operator applies excessive preheat, causing grain growth, loss of corrosion resistance (sensitization), burn-through, or massive distortion. Always calibrate pyrometers against contact thermocouples or apply high-emissivity calibration paint.