Every thermal process in heat treatment is solving the same partial differential equation under different boundary conditions. Quenching is where that equation becomes most consequential.
The Equation
The heat equation (Equation 1) governs transient temperature distribution in solids:
The local rate of temperature change equals thermal diffusivity multiplied by the spatial curvature of the temperature field [1]. Where the temperature profile curves steeply, heat moves fast. Where it is flat, heat barely moves.
The underlying constitutive relation is Fourier’s law of heat conduction (Equation 2):
Heat flux q is proportional to the temperature gradient and flows in the direction of decreasing temperature [1]. The negative sign enforces that direction. The heat equation is derived directly from Fourier’s law combined with conservation of energy.
Thermal diffusivity determines how fast temperature changes propagate (Equation 3):
Thermal conductivity k describes how readily heat moves through a material. Density ρ and specific heat capacity cp describe how much energy must be extracted before the material cools. Thermal diffusivity combines all three into a single quantity [1]. Steel has low thermal diffusivity relative to aluminum and copper. Significant internal temperature gradients develop during rapid cooling and persist [1, 6].
What Quenching Is
Steel is hardened by driving a phase transformation. Heating above the austenitizing temperature, typically between 800°C and 950°C depending on alloy composition, transforms the crystal structure to austenite and dissolves carbon into a solid solution [2]. Slow cooling allows carbon to diffuse, and the steel ends up soft. Rapid cooling suppresses diffusion and forces formation of martensite, a hard supersaturated phase that gives quenched steel its mechanical properties [2].
Martensitic transformation does not happen at a single temperature. It initiates at the martensite start temperature (Ms) and progresses to the martensite finish temperature (Mf). Different regions of the component reach those temperatures at different times depending on local cooling history. Thermal gradients and transformation stresses are coupled throughout the quench [5].
Ms temperature is not fixed. It is a function of alloy composition. Andrews (1965) proposed the following empirical relation for predicting Ms in degrees Celsius [7] in Equation 4:
Carbon has the largest influence by a significant margin. Increasing carbon content depresses Ms steeply, meaning a high carbon steel begins its martensitic transformation at a lower temperature than a low carbon steel. Manganese, nickel, chromium, silicon, and molybdenum all depress Ms as well, though less aggressively than carbon [7]. Alloying elements are added to improve hardenability, corrosion resistance, and toughness, but every addition moves Ms downward and narrows the window between Ms and Mf.
The practical consequence is direct. A steel with a low Ms temperature may still be undergoing martensitic transformation at or near room temperature at the end of the quench. If Mf falls below room temperature, transformation is incomplete and retained austenite remains in the microstructure. Retained austenite is dimensionally unstable and can transform during service, causing distortion or cracking in finished components [2, 5]. Alloy selection determines where the transformation window sits relative to the quench medium temperature. The thermal field inside the part does the rest.
The part is pulled from the furnace and immersed in water, oil, or a polymer solution. The surface loses heat immediately. The core cannot cool until heat conducts outward first. The heat equation governs that conduction.
The Boundary Condition Is Everything
The governing equation is identical for every quench. The boundary condition at the surface determines the outcome.
The convective boundary condition is [1] seen in Equation 5:
Heat flux leaving the surface equals the convective heat transfer coefficient h multiplied by the temperature difference between the surface and the quench medium. Every engineering decision made at the quench tank is a decision about h.
The Biot number determines how much that boundary condition matters (Equation 6):
Where L is a characteristic length of the component [1]. A low Biot number means surface resistance dominates, and the internal temperature gradient is small enough to ignore. A high Biot number means internal conduction resistance dominates, and the gradient across the part is significant. Quenching steel in water or oil produces high Biot numbers. The internal temperature field cannot be treated as uniform, and the full heat equation must be solved.
Water produces higher heat transfer coefficients than oil and removes heat more aggressively [3]. A hot steel component entering a quench passes through three distinct regimes: vapor film boiling, nucleate boiling, and single-phase convection. Each regime transfers heat at a different rate. The effective boundary condition changes throughout the quench [3, 5].
Water cools the surface fast enough that it contracts while the core is still hot and expanded. Thermal and transformation stresses combine. The part cracks. Oil cools more gradually; the interior follows the surface more closely, and cracking risk drops [3, 4].
What the Solution Looks Like
For a solid cylinder, the transient solution to the heat equation takes the form of a Bessel function series with eigenvalues set by the surface boundary condition [1].
The Fourier number tracks how far the transient has progressed (Equation 7):
It is dimensionless time scaled by the thermal diffusivity and the square of the characteristic length [1]. At low Fourier numbers, the temperature gradient across the part is steep, and the surface and core are thermally disconnected.
At high Fourier numbers, the gradient has largely flattened. The Fourier number at which a steel component passes through the martensitic transformation range determines whether surface and core transform together or sequentially, which directly governs residual stress and cracking risk.
At immersion, the temperature is nearly uniform throughout. The surface drops immediately. A steep gradient develops between surface and core. As heat conducts outward, the gradient flattens until the component reaches quench medium temperature. Thermal diffusivity controls how long the gradient persists. Low diffusivity means the gradient survives longer. Regions passing through the martensitic transformation range at different times generate transformation stresses that layer onto thermal stresses. That combination drives quench cracking [2, 4].
Whether the cooling rate suppresses diffusional products depends on alloy composition. Continuous cooling transformation (CCT) diagrams map cooling history against resulting microstructure, showing where pearlite, bainite, and martensite form as a function of cooling rate [2, 5]. Cracking is the worst outcome of uneven cooling. Differential thermal contraction also causes distortion. Shafts bend. Plates warp. Precision components require straightening or finish machining after heat treatment. Quench media selection, bath temperature, and agitation intensity all modify h and, therefore, the outcome [4].
Why This Matters
The boundary condition is the engineering decision. Quench media, bath temperature, and agitation set h; h determines the Biot number. The Biot number determines whether internal gradients are significant. The Fourier number tracks how those gradients evolve over time. The temperature field controls phase transformations. Phase transformations establish hardness, toughness, distortion, and residual stress.
References
- T.L. Bergman, A.S. Lavine, F.P. Incropera, D.P. DeWitt. Fundamentals of Heat and Mass Transfer, 7th ed. John Wiley & Sons: Hoboken, NJ, 2011.
- G. Krauss. Steels: Processing, Structure, and Performance. ASM International: Materials Park, OH, 2005.
- G.E. Totten, C.E. Bates, N.A. Clinton. Handbook of Quenchants and Quenching Technology. ASM International: Materials Park, OH, 1993.
- ASM International. ASM Handbook, Volume 4: Heat Treating. ASM International: Materials Park, OH, 1991.
- G.E. Totten (Ed.). Steel Heat Treatment: Metallurgy and Technologies. CRC Press.
- ASM International. ASM Handbook, Volume 1: Properties and Selection: Irons, Steels, and High-Performance Alloys. ASM International: Materials Park, OH, 1990.
- K.W. Andrews. Empirical Formulae for the Calculation of Some Transformation Temperatures. Journal of the Iron and Steel Institute, 203 (1965), pp. 721-727.
- L.S. Thomas, D.K. Matlock, J.G. Speer. The Effects of Heating Rate and Cold Work on the Development of Dual-Phase Steel Microstructures. Advanced Materials Research, 922 (2014), pp. 755-760, 10.4028/www.scientific.net/AMR.922.755.
