Compute the cooling rate through the heat-affected zone (HAZ) of an arc weld using the Rosenthal-Adams analytical solution. Change the heat input, preheat and plate thickness to see the 540 °C cooling rate and the thick/thin regime in real time, and find welding parameters that avoid martensite hardening and cold cracking.
Parameters
Heat input H_net
kJ/mm
η·V·I/v (arc efficiency included)
Preheat temperature T_p
°C
Base-metal temperature before welding
Plate thickness t
mm
Thermal conductivity k
W/(m·K)
~40 for carbon steel, ~16 for stainless
Volumetric heat capacity ρc_p
kJ/(m³·K)
Density × specific heat. ~3.8 MJ/m³K for steel
Results
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Relative thickness τ
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Cooling regime
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Thick-plate R (K/s)
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Thin-plate R (K/s)
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Applied R (K/s)
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HAZ hardening risk
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Weld HAZ isotherms — torch traversal animation
The torch travels across the plate, forming HAZ isotherms (red → orange → yellow → green) around the bead. The right side sketches the thermal history at a fixed point and the cooling rate R at 540 °C.
Adams thick-plate (3D) and thin-plate (2D) cooling rates. T is the evaluation temperature (540 °C for steel HAZ), T_p the preheat temperature, H_net the heat input per unit length including arc efficiency, t the plate thickness, k the thermal conductivity and ρc_p the volumetric heat capacity. Raising either the heat input or the preheat lowers (T−T_p) and/or H_net in the denominator, so both reduce the cooling rate and the risk of martensite formation.
Relative thickness τ classifies the regime (τ>0.9 thick, τ<0.6 thin, in between transitional). This tool switches to the thick-plate formula when τ>0.75.
What is the weld cooling rate (Rosenthal-Adams)?
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I always thought welding was just "melt the steel and stick it together". Is the HAZ cooling rate really that big a deal?
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It is a huge deal. In an arc weld, the base metal right next to the bead is heated to over 1200 °C in a flash and then cooled rapidly as heat drains into the surrounding cold parent material. How fast it cools after the peak decides what microstructure the HAZ — the heat-affected zone — ends up with. Cool slowly and you get pearlite or bainite, soft and tough. Cool fast and you get martensite, hard and brittle. A martensitic HAZ is the classic source of hydrogen-induced cold cracking, where a weld can crack on its own a day or two after it has been laid down. That is the failure mode bridge and pressure-vessel engineers lose the most sleep over.
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So we just "cool it slowly", right? Why do we need names like Rosenthal and Adams to figure that out?
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Because "cool slowly" has to be quantified. In the 1930s Daniel Rosenthal solved the temperature field for a moving point heat source in an infinite plate analytically, and C. M. Adams simplified it into the cooling rate at any temperature T. The thick-plate result is R = 2πk(T−T_p)²/H_net. In practice we read R near 540 °C: under about 6 K/s is safe for low-alloy steel, above 20 K/s the HAZ is essentially martensitic and risky. Try moving the heat input slider down to 0.5 kJ/mm on the left — the cooling rate jumps up and the verdict flips to "HAZ hardening risk".
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Why do we need two different formulas for "thick" and "thin" plates?
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Because the geometry changes the heat flow. In a thick plate, heat drains three-dimensionally into a semi-infinite body — that is the thick-plate (3D) limit. In a thin plate it can only flow in-plane through a small cross-section, which is the thin-plate (2D) limit. The dependencies differ: thick scales as 1/H_net, while thin scales as 1/H_net², so on a thin plate adding heat input cuts the cooling rate much faster. The dimensionless relative thickness τ = t·√{ρc_p(T−T_p)/(2k·H_net)} picks the regime: τ>0.9 thick, <0.6 thin, transitional in between. Watch the "Relative thickness τ" card while you move the plate-thickness slider — it crosses into 2D somewhere around 6 mm.
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Got it. If I want to lower the cooling rate, what should I change first?
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The biggest lever is preheat. The (T−T_p)² term means that going from 25 °C ambient to 150 °C preheat cuts the thick-plate cooling rate by 43%. For high-CE steels like HT780, 150-200 °C preheat is basically mandatory. The next lever is raising heat input — pump up the current or slow the travel speed. Be careful though: too much heat input causes grain coarsening and HAZ softening (a real issue on quenched-and-tempered steels). Joint design — going from a thick to a thinner plate, or a shallower groove — is the last lever and belongs at the design stage. Play with H_net and T_p in this tool to find a procedure that lands R below 6 K/s.
Frequently Asked Questions
During welding the HAZ is heated above 1200 °C and then transforms from austenite back to a room-temperature microstructure on cooling. That transformation happens roughly between 800 and 500 °C, so the cooling rate right in the middle — around 540 °C — controls whether you end up with pearlite, bainite or martensite. Many specifications use the 800-500 °C transit time t8/5 instead, but the Adams analytical solution gives the cooling rate at any temperature T in one step, which makes 540 °C the convenient single-number basis for comparison.
For the same heat input and temperature difference, the thick-plate (3D heat flow) case cools faster. In a thick plate heat drains three-dimensionally into a semi-infinite body around the bead, while in a thin plate it can only escape laterally through a small cross-section. Adams classifies the regime by the relative thickness τ = t·√{ρc_p·(T−T_p)/(2k·H_net)}: τ > 0.9 is thick, τ < 0.6 is thin, in between is transitional. This tool uses τ > 0.75 as the switch to the thick-plate formula.
Rapid cooling does not give austenite enough time for diffusional transformation (pearlite, bainite); it transforms diffusionlessly into martensite — hard and brittle. Martensite often exceeds HV400 and becomes the starting point for hydrogen-induced cold cracking. For low-alloy steels a typical WPS keeps the 540 °C cooling rate below 6 K/s to hold the HAZ peak hardness under HV350. Above 20 K/s the microstructure is largely martensitic, and without preheat the joint is at high risk of delayed cracking within 24-48 hours after welding.
The Adams formulas show that the cooling rate is proportional to (T−T_p)² (thick plate) or (T−T_p)³ (thin plate), where T is the evaluation temperature 540 °C and T_p is the preheat temperature. Moving from 25 °C ambient to 150 °C preheat drops (T−T_p) from 515 K to 390 K, cutting the thick-plate R to (390/515)² = 0.573 — a 43% reduction. Preheat also gives hydrogen extra time to diffuse out, making it the single most reliable countermeasure against cold cracking. High-CE steels (carbon equivalent) need more preheat; JIS Z 3158 and AWS D1.1 set minimum preheat temperatures by thickness and restraint.
Real-World Applications
Field welding of bridges and large structures: On bridge girders, box columns and offshore structures fabricated from high-strength steels such as SM490 or SM570, HAZ cooling-rate control directly governs the safe service life of the joint. In cold-climate winter field welding without preheat the 540 °C cooling rate can exceed 30 K/s, and cold-cracking incidents are well documented. WPSs always specify a minimum preheat temperature as a function of thickness, ambient temperature and CE (for example, 100 °C preheat for a 40 mm plate at CE 0.45%).
Pressure vessel and boiler fabrication: JIS B 8265 and ASME BPVC Section VIII require strict HAZ hardness control for Cr-Mo low-alloy steels (such as 2.25Cr-1Mo), with limits commonly between HV248 and HV375. To meet those limits, heat input, preheat and interpass temperatures are all locked down in the WPS, and confirmed by a procedure qualification record (PQR) measuring HAZ hardness on production-representative coupons. A Rosenthal-Adams calculation like this tool is used to narrow down WPS parameters before PQR.
Thin-sheet welding in automotive bodies-in-white: MAG and laser welding of 1.0-2.0 mm sheet steel sits firmly in the thin-plate (2D) regime, where the cooling rate scales as 1/H_net². For 1500 MPa hot-stamped boron steel or TWIP steels, too low a heat input martensite-hardens the joint and the HAZ tears in a crash; too much heat input causes HAZ softening. Finding the sweet spot using the Adams thin-plate equation together with CCT diagrams is everyday production engineering.
Pre-study for welding CAE: Detailed weld FE simulations (SYSWELD, Simufact Welding, Abaqus Welding Interface) take days just for meshing and material setup. Before that investment, an analytical estimate like this tool answers questions such as "how much does adding 100 °C of preheat lower the cooling rate?" and serves as a sanity check on FEM output. If the FE result differs from Adams by an order of magnitude, suspect a boundary-condition or convection-coefficient mistake.
Common Misconceptions and Pitfalls
The biggest pitfall is forgetting that Rosenthal-Adams is an infinite-plate, quasi-steady, point-source analytical solution. Real joints have starts and stops, plate edges, T-joint intersections and multi-pass build-ups, where the boundary conditions differ from an infinite plate and the Adams estimate carries roughly 30-50% error. In multi-pass groove welds, preheating from previous passes lowers the actual cooling rate well below the Adams calculation. Treat the equation as an order-of-magnitude first-pass and lock in the final WPS with PQR hardness and microstructure data on production-like coupons.
Next, assuming that "as long as the cooling rate is low enough, no martensite will form". The martensite start temperature Ms depends strongly on the carbon equivalent CE, and in high-strength steels above CE 0.5% some martensite forms even at cooling rates below 1 K/s. Cooling-rate control is a necessary but not sufficient condition. For high-CE steels, combine (1) preheat, (2) low-hydrogen consumables and (3) post-weld heat treatment (PWHT). Concluding "I am under 6 K/s, so I am safe" purely from the Adams output is dangerous.
Finally, do not assume the tool works as-is for stainless steel. The default values here, k = 40 W/mK and ρc_p = 3.8 MJ/m³K, are for carbon steel. Austenitic stainless (304) sits around k ≈ 16 and ρc_p ≈ 4.0; ferritic Cr-Mo around k ≈ 25. More importantly, the HAZ failure mode for stainless is not martensite hardening but sensitisation (Cr-carbide precipitation and intergranular corrosion) or residual delta-ferrite, where the cooling-rate requirements can be the opposite. Re-evaluate both the thermal properties and the acceptance criteria for each material family before using the tool.
How to Use
Enter heat input (kJ/mm) from your welding procedure specification—typically 1.2–2.8 kJ/mm for GMAW on structural steel.
Set preheat temperature (°C) and plate thickness (mm); the simulator calculates relative thickness τ to determine if thick-plate or thin-plate Rosenthal solution applies.
Input thermal conductivity (W/m·K) for your material—200 W/m·K for mild steel, 16 W/m·K for austenitic stainless—and press Calculate to obtain cooling rates (K/s) between 800–500°C.
Worked Example
API 579 assessment of a 25 mm ASTM A516 Grade 70 plate with 1.8 kJ/mm heat input, 150°C preheat, and κ = 42 W/m·K. The simulator computes τ = 0.31, triggering thick-plate regime (R = 18.5 K/s). Since this exceeds the 15 K/s threshold for martensitic cracking risk, hardness screening of the HAZ is mandatory before in-service operation.
Thick plates (>20 mm) transition to 3D heat flow; ensure your thermal conductivity accounts for temperature-dependent variation, especially near austenite grain boundaries.
Cooling rate spikes near the fusion line (1500°C) then stabilizes; use the 800–500°C window for CCT diagram intersection checks and hardness prediction.