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Heating and Cooling Times by the Numbers

Biot and Fourier numbers, lumped and one-term solutions, surfaces, corners, finite differences and furnace cycles, worked through one hardened steel shaft

Estimate how long parts take to heat through and cool down: Biot and Fourier numbers, lumped and one-term solutions, semi-infinite surfaces, product solutions and finite differences.

10 chapters · About 102 minutes · Free to read · Intermediate heat transfer for manufacturing and plant engineers

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KEVOS® LibraryMANUFACTURINGCUEROUTINEREWARD21 DAYSHeating andCooling Times bythe NumbersKEVOS® ORIGINAL BOOK10 CHAPTERS

About this book

Every furnace, oven, quench tank, curing press and cooling conveyor runs on a time. Somebody decided how long a part stays in, and the decision is usually a rule handed down with the procedure: an hour for this, twenty minutes for that. A rule like that may be right for the part it was written for and badly wrong for the next one, because the time a part needs to heat through depends on its size, its material and how hard the surroundings push heat into it, and those three things combine in ways a single rule cannot follow. Too short a time leaves the centre unready; too long a time wastes energy and furnace hours, and in some processes damages the surface.

This book shows how to calculate those times. It starts with the two dimensionless numbers that organise the whole subject, the Biot number, which compares the resistance inside the part with the resistance at its surface, and the Fourier number, which measures time against the part’s size. It then works through the models in order of effort: the lumped part with one temperature, the one-term solutions for slabs, cylinders and spheres, the semi-infinite solid for surfaces and short times, product solutions for ends and corners, and an explicit finite-difference model that can be built in a spreadsheet and used to check everything else. The last three chapters turn the numbers into practice: what the surface condition really is in a furnace or a quench, how to measure and prove a cycle, and how to plan a cycle’s time, energy and throughput.

It suits manufacturing, process, quality and design engineers. It assumes school algebra, exponentials and logarithms, and the basic ideas of conduction, convection and radiation. Heat Transfer by the Numbers covers those from first principles and introduces the lumped model in its last chapter, Heating up and cooling down; this book starts there and goes further.

How to use the book

Read Chapters 1 and 2 first: they decide which model a problem needs. Chapters 3 to 6 can be read in order or picked up as the shape of a part requires. Chapter 7 builds the numerical model used to check the others. Chapters 8 to 10 bring it together for a real cycle. Each chapter ends with an exercise and a worked answer. Every number in the book was calculated, not read from a chart, and the methods are given so the calculations can be repeated in a spreadsheet.

The running case

Every chapter follows one fictional part. This is an illustration. DS-100 is a drive shaft in AISI 4140 (42CrMo4) chromium-molybdenum steel: a plain body 100 mm in diameter and 500 mm long, with an integral flange 220 mm in diameter and 40 mm thick at one end. The body weighs about 30.8 kg and the flange about 11.9 kg. A heat-treatment shop hardens the shafts by heating them to 850 °C in an electric box furnace, holding them, quenching them in oil at 60 °C, and then tempering them at 550 °C. The shop wants to know how long each stage really takes, because its present cycle was written for a different part.

For the calculations, the steel is given a density of 7,850 kg/m³, a mean thermal conductivity of 33 W/m·K and a mean specific heat of 650 J/kg·K over the hardening heat, which includes an allowance for the heat the steel absorbs as it transforms. Those are stated assumptions, chosen to fit the published trend for this steel, which conducts less heat and stores more as it gets hotter. Furnace, quench and price figures are illustrations too. The part, the shop and its numbers are invented; none describes KEVOS® work or a real installation. Prices are illustrations in Australian dollars including GST.

Source limits and professional responsibility

This book is educational. Its models assume constant properties, simple shapes and uniform surface conditions, and real parts depart from all three: properties change with temperature, steels absorb heat as they transform, surface coefficients change during a quench, and loads shade one another in a furnace. The models give good first estimates and show which variables matter; they do not replace measurement. Heat-treatment cycles for real parts are set by the material specification, the heat treater’s qualified procedures and the customer’s requirements, proved with load thermocouples and metallurgical testing, and approved by competent people. Furnaces, quench tanks and their atmospheres are serious hazards and are governed in Australia by work health and safety laws and the standards they call up.

Who this book is for

Manufacturing, process, quality and design engineers who set furnace, oven, quench, cure or cooling times, and anyone who has to decide how long a part must stay in before its centre is ready.

What you will learn

  • Calculate Biot and Fourier numbers and use them to decide which heating-time model a part needs.
  • Estimate centre and surface temperatures in slabs, cylinders and spheres with lumped and one-term solutions.
  • Use the semi-infinite solid for surface heating, short contacts and the transfer delay before a quench.
  • Combine one-dimensional solutions for ends and corners, and check them with an explicit finite-difference model.
  • Turn the calculations into a furnace cycle that is measured, proved and costed.

Inside this book

Chapters

  1. 01Biot and Fourier: the two numbers behind every heating timeTwo dimensionless numbers decide how long a part takes to heat through and which model to use. They are applied to the DS-100 shaft, with a rule-of-thumb scaling error they expose.
  2. 02The lumped part: when one temperature is enoughThe lumped model treats a part as one temperature with a time constant. It is applied to DS-100's coupon and flange, with radiation stepped by hand, and shown failing in the quench.
  3. 03The plane wall: heating the flange throughThe one-term plane-wall solution gives centre and surface temperatures in plates and flanges. It is applied to DS-100's flange in furnace and quench, with a one-sided-heating error.
  4. 04Long cylinders and spheres: heating the shaft bodyOne-term solutions for cylinders and spheres give the DS-100 body's heating and quench, show how shape changes time, and catch a radius-for-diameter slip that doubled the cycle.
  5. 05Surfaces first: the semi-infinite solidIn the first moments only a thin skin changes temperature. The semi-infinite solid gives that skin's behaviour, applied to DS-100's transfer to the quench and to contact with a cold rack.
  6. 06Corners, ends and junctions: product solutionsMultiply one-dimensional solutions to get ends, edges, corners, square bars and short cylinders. Applied to DS-100's free end, flange rim and coupon, with a thermocouple placed too near an end.
  7. 07A finite-difference model you can checkBuild an explicit finite-difference model of DS-100's body in a spreadsheet, check it against the one-term solution, add radiation and changing conductivity, and avoid an unstable time step.
  8. 08The surface condition: radiation, loads and quenchantsWhere surface coefficients come from in furnaces, loaded baskets, tempering ovens and oil quenches, and how a tempering cycle using the hardening coefficient ran half as long as needed.
  9. 09Thermocouples, surveys and proving the cycleHow to place load thermocouples at the slowest point, size them for the speed of the process, compare traces with the model, and avoid a thick sensor that made good oil look worn out.
  10. 10Planning the cycle: time, energy and throughputTurn heating times into a furnace plan: load energy, furnace power and recovery, the cost per part and shafts per hour, and a bigger load that would not have raised output at all.

Related Tools and Examples

References and source notes

The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 27 October 2026.

  • Donald R. Pitts and Leighton E. Sissom, McGraw-Hill. Schaum's Outline of Theory and Problems of Heat Transfer, second edition (1998). Supplied source, in copyright, and a scanned copy without a text layer. Used only as a subject map of time-varying conduction: Biot and Fourier numbers, lumped analysis, semi-infinite bodies, slabs and cylinders with convective surfaces, chart solutions, multidimensional products and numerical methods. No text, problems, figures, charts or tables are reproduced; this book's case, calculations, figures and exercises are original and were computed for it.
  • M. P. Heisler, Transactions of the ASME, vol. 69, pp. 227–236. Temperature Charts for Induction and Constant-Temperature Heating (1947). The origin of the well-known charts of centre temperature against Fourier number for slabs, cylinders and spheres. This book does not reproduce the charts; it computes the same one-term series directly, with eigenvalues and coefficients solved in its own calculations.
  • M&W Components (Elgin material sheets). Alloy Steel Grade 4140 Fact Sheet (AISI 4140, normalised) (2024). Source for the thermal conductivity of 4140 steel falling from 42.6 W/m·K at 100 °C to 42.2 at 200 °C, 37.7 at 400 °C and 33.0 at 600 °C, and for specific heats of 0.519 J/g·°C at 150–200 °C and 0.561 J/g·°C at 350–400 °C. The mean values used in the running case are stated assumptions informed by this trend.
  • The Engineering ToolBox. Emissivity Coefficients of Common Materials (current web page). Source for the emissivity of oxidised steel (0.79), rounded to 0.8 in the furnace calculations.
  • National Institute of Standards and Technology. CODATA value: Stefan-Boltzmann constant (current). Source for σ = 5.670 374 419 × 10⁻⁸ W/m²·K⁴, used in every radiation calculation.
  • Thermal Processing magazine. Determining Grossmann H-value from cooling curve data; The effect of agitation on oil quenchants (current web pages). Background for Chapter 8: the Grossmann H-value expresses quench severity through the surface heat transfer coefficient and the steel's conductivity, oils are characterised by cooling-curve tests to ASTM D6200, and modern oils differ from Grossmann's original straight oil. No H-values are quoted; the quench coefficient in the case is a stated assumption. The second article is at thermalprocessing.com/?p=26827.
  • SAE International; ASTM International. AMS2750, Pyrometry; ASTM D6200, Standard test method for determination of cooling characteristics of quench oils by cooling curve analysis (current editions). Named by number and scope only in Chapters 8 and 9, as the pyrometry specification many heat treaters work to and the standard cooling-curve test for quench oils. No requirements, classes, tolerances or intervals are quoted.

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