Rosenthal Solution Simulator — Quasi-Stationary Welding Heat Transfer
Compute heat input Q, line energy q, cooling time t8/5 and HAZ width from a moving arc using Rosenthal's thick-plate solution. Sliders for voltage, current, travel speed and arc efficiency update everything in real time.
Welding Parameters
Voltage V
V
Current I
A
Travel speed v
mm/min
Arc efficiency η
SAW≈0.95 / SMAW≈0.80 / GMAW≈0.78 / GTAW≈0.60
Fixed Constants
Ambient T₀ = 25 °C
Conductivity k = 45 W/(m·K) (steel)
Diffusivity α = 1.25×10⁻⁵ m²/s
Results
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Heat input Q
—
Line energy q
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Cooling time t8/5
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HAZ width
Welding side view — torch, weld pool, HAZ
Line energy q vs. cooling time t8/5
Theory & Key Formulas
Heat input: $$Q = \eta \cdot V \cdot I$$
Line energy: $$q = Q / v$$
Cooling time (thick plate): $$t_{8/5}=\frac{q}{2\pi k}\left(\frac{1}{T_5-T_0}-\frac{1}{T_8-T_0}\right)$$
What exactly is the "Rosenthal solution"? Is it a quick way to estimate weld temperatures without running a full FEM?
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Exactly. Rosenthal assumed a moving point heat source in an infinite plate, with a quasi-stationary temperature field travelling along with the arc. That gives a closed-form solution that lets you guess t8/5 or HAZ width in seconds. In this tool, the four sliders — voltage, current, travel speed, arc efficiency — feed Q = ηVI and q = Q/v, and the right-hand stats update live.
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OK, but what is "t8/5"? Why exactly 800 and 500 °C, not some other range?
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For carbon and low-alloy steels, the time spent between 800 °C and 500 °C controls the HAZ microstructure — martensite, bainite or ferrite — and therefore hardness and toughness. So 800/500 is the "industry standard" cooling window. The tool uses $t_{8/5} = q/(2\pi k)\,[1/(T_5-T_0)-1/(T_8-T_0)]$ from the thick-plate Rosenthal solution.
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So if I speed up the travel, q drops and t8/5 should fall. Can I check that with the "Speed sweep" button?
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Yes, that's the right intuition. Sweeping v from 50 to 1000 mm/min, q = Q/v falls inversely, so t8/5 and the HAZ width shrink in proportion. In practice, when you need a longer t8/5 (low-hydrogen steels) you slow down the arc or add preheat to raise T₀. The tool gives you a feel for "below ~700 J/mm it gets risky" before you commit to any procedure.
Physical Model and Key Equations
Heat input is $Q = \eta V I$ and line energy is $q = Q/v$. Rosenthal's 3D (thick-plate) solution gives a quasi-stationary temperature field around a moving point source; the time to cross a specific temperature interval scales linearly with q. This tool evaluates $t_{8/5} = q/(2\pi k)\,[1/(T_5-T_0)-1/(T_8-T_0)]$ and estimates the HAZ width with $w_{HAZ}\approx 0.5\sqrt{\alpha\,t_{8/5}}$ (α = k/(ρc) ≈ 1.25×10⁻⁵ m²/s, k = 45 W/(m·K) for steel).
Real-World Applications
Low-hydrogen steel control: Grades such as S355, X70 or HT780 have specified t8/5 windows. Sweep current, speed and efficiency to land Q inside the allowed window before committing to a WPS.
WPS / pWPS drafting: Combine this with preheat and interpass-temperature rules to bracket the line-energy limits for your procedure qualification record.
HAZ-width-driven fatigue: Weld-toe fatigue and brittle fracture both scale with HAZ width and peak hardness. The side-view canvas visualises how HAZ width grows with q.
Laser and electron-beam welding: These high-energy-density processes approach the thin-plate (line-source) regime, but the trend of t8/5 with Q/v shown here is still useful for first sanity checks.
Common Misconceptions and Pitfalls
First, treat Rosenthal as a trend tool, not an absolute one. Convection and latent heat are ignored, so near-pool temperatures are over-predicted; integrated quantities such as t8/5 and the spatial gradient are far more reliable.
Second, arc efficiency η varies strongly by process: SAW ≈ 0.95, SMAW 0.75–0.85, GMAW 0.70–0.85, GTAW 0.5–0.7. Using the default η = 0.80 for a TIG-fine job overstates line energy by 20–30%. Always sweep the η slider to bound the realistic range.
Third, this tool assumes the thick-plate (3D) regime. The thin-plate (2D) solution gives t8/5 ∝ q² rather than q, so for very thin sheets the prediction here is on the non-conservative side. Check the dimensionless plate thickness before quoting absolute numbers.
FAQ
Yes. Because q = Q/v, raising v lowers q and the thick-plate Rosenthal solution gives t8/5 proportional to q, so it shrinks together. Use the Speed sweep button to see t8/5 and HAZ width drop side by side.
In the formula the bracket (1/(T5−T0)−1/(T8−T0)) grows as T₀ rises, so t8/5 lengthens (cooling becomes slower). This is exactly why low-hydrogen steels require a minimum preheat to keep hardness inside the acceptable range.
It uses the diffusion length scale w_HAZ ≈ 0.5·√(α·t8/5). For steel we use α = 1.25×10⁻⁵ m²/s. Real HAZ width depends on the local transformation temperature, but this rule-of-thumb captures the trend well.
Aluminium and copper have k and α several times larger than steel, so t8/5 would be shorter. This tool is fixed to steel (k = 45 W/(m·K), α = 1.25×10⁻⁵ m²/s); for other metals scale the result by the property ratios.