Thermal Diffusivity Calculator

What Is the Thermal Diffusivity Calculator?

How to use the thermal diffusivity calculator

1. Choose the unknown. The Solve for menu opens on Thermal diffusivity. The other three choices are Thermal conductivity, Density and Specific heat capacity, and whichever you pick vanishes from the boxes below.
2. Type the conductivity. Thermal conductivity is the figure a handbook lists in watts per metre per kelvin, and the box also takes mW/(m K) for insulation and still air. It is the only one of the three material figures that carries heat forward; the other two decide how much of it is spent warming the material on the way.
3. Type the density and the specific heat capacity. These two enter only as their product, so the boxes are asking for one number in two parts. Density takes kg/m3 or g/cm3 and Specific heat capacity takes J/(kg K) or kJ/(kg K).
4. Set the layer thickness. Layer thickness is never the unknown, because it is not a property of the material at all. It sets the clock: every time printed on the chips scales with the thickness squared.
5. Read the answer and the chips. The headline is the quantity you asked for, carrying up to six significant figures with trailing zeros trimmed. The chips give the thermal effusivity, the volumetric heat capacity, the time to 99 per cent of steady state, the time to half at mid-depth, the Fourier number after one hour and the steady flux per kelvin, per square metre.
6. Read the two times for what they are. Both belong to one boundary condition: a slab that starts at a single temperature, with one face stepped to a new one and the far face held there. A slab heated from both sides, or backed by insulation, keeps a different schedule and neither figure carries over to it.
7. Compare two materials rather than reading one. Load the Still-air-like gap, note the diffusivity, then load the Stainless-like plate. The conductivity climbs by a factor of 575 and the diffusivity falls by 5.71, which is the comparison a steady-state tool has no way of showing.
8. Open Show working. The steps restate the relation, list your figures in SI base units whatever the menus say, and end on the two times. Seeing 100 mm come back as 0.1 m in that second line is the quickest way to see what the conversion did.

This page starts where a steady-state tool stops. Once nothing is changing any more, the heat flow through a slab is Fourier's law and the thermal conduction calculator is the right tool, with its area, its temperature difference and its thickness. What it cannot tell you is how long the wall took to get there, because it has nowhere to put a density or a specific heat.

For the underlying law, the conductivity table and the way resistances add in series, the guide to Fourier's law of thermal conduction covers all of it, and this page assumes it. The one quantity it names and then hands on is this one. If the specific heat capacity in the denominator is the part you are unsure of, the guide to specific heat capacity defines it properly and the specific heat calculator works it out from an energy and a temperature rise.

Two mistakes account for most wrong answers here, and both are about what was typed. The first is a specific heat capacity entered per gram rather than per kilogram, which makes the material a thousand times slower than it is; the kJ/(kg K) option is there because that is how many tables print it. The second is a unit menu left on the wrong option after a preset, which the second line of Show working will always catch.

Worked example: change one thing at a time

The table starts at the defaults and moves one thing at a time: which quantity is the unknown, then the material, then the thickness, then the unit the figures are typed in. Every Result cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool. The input column is simply what you type, in the order the boxes appear.

Rows 1 to 4 are one state read four different ways. Each answer returns the figure the row above it started from, so the diffusivity gives back the conductivity, then the density, then the specific heat capacity. Four unknowns, one relation, and no arrangement more fundamental than another.

That round trip is the genuine check on this page, and it is worth saying why the volumetric heat capacity chip is not. In row 1 the answer is the conductivity divided by that chip, so the two agreeing is the definition restated. Rows 2 to 4 do different arithmetic in the other direction and still land on 1 W/(m K), 1995.26 kg/m3 and 900 J/(kg K), which is a claim that can fail.

Row 5 halves the density and doubles the specific heat capacity, and the diffusivity does not move. That is algebra rather than a discovery: the two appear only as their product, and the tool is restating its own definition rather than testing it. It is still worth seeing, because it is the reason a light foam and a dense solid can share a diffusivity.

Rows 6 and 7 are the demonstration this page exists for. The still-air-like gap conducts 575 times worse than the stainless-like plate and diffuses a temperature change 5.71 times faster, so the two quantities put the same pair of materials in opposite orders. A steady-state conduction tool has no density and no specific heat capacity, so it can never show that.

Rows 8 to 10 run from the quickest of the six presets to the slowest, with the water-like layer sitting between them: a copper-like bar at 113.381 mm2/s against a pine-like board at 0.0834155, a factor of about 1359 across that pair. The times beside them say the same thing more usefully: 43.3 s against 16.353 h for the same 100 mm.

Rows 11 and 12 change only the thickness, and the headline does not move, because thickness is not a material property. The time does: 25 mm takes 9.19 min and 300 mm takes 22.046 h, which are the same 2.450 h multiplied by a sixteenth and by nine. Rows 13 and 14 are the edges — the same brick typed as 1.99526 g/cm3 returns the identical answer, and a density of zero is refused rather than answered with infinity.

Formula and symbol reference

The calculator uses one relation in four arrangements. The diffusivity is alpha = k / (rho × c), so the conductivity is k = alpha × rho × c, the density is rho = k / (alpha × c) and the specific heat capacity is c = k / (alpha × rho). The effusivity chip beside them is e = sqrt(k × rho × c), the same three properties arranged the other way round.

Those two carry everything the three do, for conduction: k = e × sqrt(alpha) and rho × c = e / sqrt(alpha). It is worth knowing because the two say opposite things about the same material — of the six presets only the copper-like bar diffuses faster than the still-air-like gap, and not one of them has an effusivity anywhere near as low as that gap's.

The physics: why conductivity alone cannot tell you how long

Heat arriving at one face of a slab does two things at once: some of it moves on through the material, and some of it stays behind to warm the material it has passed. Conductivity governs the first and the volumetric heat capacity governs the second, and the speed of the front is the ratio of the two. That ratio is the diffusivity, and it is why a good conductor with a great deal of heat to absorb can be slow.

The model behind the two times on the chips is the simplest one that ends in Fourier's law. A slab starts at a single temperature; one face is stepped to a new one and the far face is held at the old one for ever. The temperature profile then climbs from flat to the straight line that the steady-state law assumes from the outset, and the whole of the transient is carried by one dimensionless group.

That group is the Fourier number, Fo = alpha × t / L2. Two different materials at the same Fourier number have identical temperature profiles, differing only in how long they took to get there, which is why the chips print one after an hour: it says where a given layer is in its own story. The thickness enters squared, so it is the most powerful thing on the page.

Mid-depth reaches 99 per cent of its final rise at Fo = 0.4911, and that number is exact algebra rather than a fit. The mid-depth series contains only odd terms, so the one-term form is wrong by the third term, a matter of 4e-20. The companion figure for the far face is a solved root rather than a closed form, and it is larger: the far face is still delivering only 98.43 per cent of its final flux when the mid-depth is 99 per cent of the way there, so temperature settles before flux does.

Both constants belong to that boundary condition and to no other. Heat the slab from both faces, insulate the back of it, or make it a cylinder, and the transient has different numbers in front of the same L2/alpha. What survives every version is the L2 itself, which is why the 25 mm and 300 mm rows of the table differ by exactly a factor of 144.

The effusivity chip answers a different question again: not how fast the front moves, but what happens in the instant two surfaces touch. Two bodies brought into contact settle their interface immediately at a temperature weighted by their effusivities, so equal effusivities meet exactly in the middle and a much more effusive body keeps the interface close to its own temperature. The copper-like bar's chip reads 37387.8 against the pine-like board's 346.24, a factor of 108, which is the honest version of the familiar observation that metal feels colder than wood at the same temperature.

None of this brings in a property the material did not already have. The same three figures make the diffusivity and the effusivity, the temperature difference cancels out of every percentage and every time on this page, and the Fourier's law simulator shows what the same slab is doing once all of it has finished happening.

Where the thermal diffusivity calculator breaks down

The arithmetic is short and hard to get wrong. What fails is the relation being pressed onto a situation it does not describe, or a constant being carried somewhere it does not belong.

The two times belong to one boundary condition

A slab that starts uniform, with one face stepped and the far face held for ever. Heat it from both sides, insulate the back face, or make it a rod or a cylinder and the same L2/alpha carries a different number in front of it. The diffusivity itself is unaffected; only the schedule changes.

Symptom: a measured time that is a fixed multiple of the figure here, rather than randomly wrong.

The properties all move with temperature

Conductivity, density and specific heat capacity are each functions of temperature, and the diffusivity inherits all three. Over a small range the tool is fine; over the range of a quench or a furnace it is not, and the honest approach is to run it at both ends and treat the two answers as a band.

A layered wall has no single diffusivity

Each layer has its own, and the transient depends on which one is on the outside as well as on the thicknesses. Running each layer alone gives the individual timescales and the slowest usually dominates, but the combination is not the average of the parts.

Moisture changes everything at once

Water in a porous material raises the conductivity, the density and the specific heat capacity together, and moving water carries heat in a way this relation does not model at all. A figure for dry brick says little about the same brick after a week of rain.

Anything that melts, boils or sets breaks the specific heat capacity

A phase change absorbs energy at constant temperature, which is a latent heat rather than a specific heat, and no single value of c describes a material passing through one. The same goes for a curing resin or a hydrating cement, where the material is generating heat of its own.

Contact resistance and surface films are not modelled

The model steps the face itself to the new temperature. A real surface is warmed through an air film or a joint, which delays everything by an amount this page knows nothing about, and thin fast layers are affected most.

Wood, composites and rolled metals are not the same in every direction

Conductivity along the grain and across it can differ by a factor of two or more, so a single diffusivity describes one direction only. Note which figure a table is quoting before you use it.

The presets are idealised triples, not materials

Each is chosen to land near a published diffusivity, and each lands a little off it: the still-air-like gap is 14.6 per cent above the published 19 mm2/s, and the stainless-like plate about 9 per cent below its 4.2. That is stated rather than hidden, and it is why every name carries a "-like".

Rounded figures in, rounded figures out

Every number printed is rounded and the arithmetic behind it is not, so multiplying two figures off the screen need not reproduce a third in its last digit. The working lines carry the intermediate products to nine figures for exactly that reason.

Where this calculator is actually used

Typing a thickness and reading the wait off the chips

A heavy wall delays heat rather than stopping it, and that delay is the whole reason the Layer thickness box is on the page at all. Load the Brick-like wall and the chips read 2.450 h to 99 per cent of steady state with 28.34 min to half way at mid-depth; retype the thickness as 300 mm and the first of those becomes 22.046 h while the headline does not stir.

Putting a section thickness against a treatment time

Whether a quenched or case-hardened section changes right through or only near its face is a question about how quickly the front crosses it, so the figure to read here is one of the times rather than the headline. Load the Stainless-like plate for 3.81092 mm2/s and 21.48 min at 100 mm, then put the section you actually have into the thickness box and read the pair again.

Putting a food-like layer on the clock

Most foods sit near the water-like end of the range, and the Water-like layer preset returns 0.143981 mm2/s with 9.474 h to 99 per cent at 100 mm, so the thickness of a joint counts for a great deal more than its weight. Turning the oven up does get the middle to any chosen temperature sooner; what it cannot do is shorten the time to a given fraction of the final rise, because the temperature difference cancels out of that clock — which is why no box on this page asks you for one.

Turning a flash-method diffusivity back into a conductivity

The standard laboratory technique pulses one face of a thin disc and watches the other, so what it measures directly is the diffusivity; the conductivity has to be rebuilt from it with a separately measured density and specific heat capacity. That is this page with Solve for moved to Thermal conductivity, which is exactly what the Copper-like bar, back to k preset sets up: 113.3807 mm2/s in, 398.107 W/(m K) out. The diffusivity box takes cm2/s and m2/s as well, because instruments do not all report in the same one.

Reading the Fourier number to tell a transient from a settled layer

A heat sink is sized on conductivity, but whether a short burst of power lifts a junction temperature depends on how fast the heat spreads out of it. The Fourier number after one hour chip is the quick test: the still-air-like gap reads 7.83679 and finished long ago, while the water-like layer reads 0.0518331 and has barely begun.

Reading the effusivity chip when the question is contact

If what you want to know is how a surface behaves in the instant something touches it, the headline is the wrong number and the Thermal effusivity chip is the right one. The physics section above sets the copper-like bar's chip against the pine-like board's for precisely that reason, and the chip is printed beside the answer in every one of the four solve modes, so you never have to switch to reach it.

Where to go next

For the method in full, with worked problems and the diagrams that go with them, read Thermal Diffusivity: How Fast Heat Moves. For the steady state this transient ends in — the conductivity table, the R-value and how layers add up — Fourier's Law of Thermal Conduction is the place to go, with the thermal conduction calculator beside it. The specific heat calculator handles the denominator on its own, and the whole physics lab library is open if you would rather watch a temperature front than type one.