Thermal Diffusivity Simulator

What Is the Thermal Diffusivity Simulator?

How to use the thermal diffusivity simulator

1. Start from the state it opens in. The lab boots at rest on 1.000 W/(m K), 1995.3 kg/m3, 900 J/(kg K), 100 mm and a clock reading of 0.100, so every card is already filled before you touch anything. Reset returns all five sliders to exactly that and stops the clock if it is running.
2. Set the conductivity. Thermal conductivity is logarithmic, because the quantity spans four orders of magnitude between a still gas and a metal, and a linear slider from 0.010 to 398.107 would squeeze every insulator into the first hundredth of its travel. The card beside it always prints the conductivity itself, never the exponent: 0.010 W/(m K) at one end, 398.107 W/(m K) at the other. Fourier's law of thermal conduction is where that quantity comes from and what it does once the wall is steady.
3. Set the density and the specific heat capacity. Density is logarithmic for the same reason; Specific heat capacity is linear, from 100 to 5000 J/(kg K) in steps of 5. These two reach the answer only as their product, which the panel prints for you as Volumetric heat capacity in MJ/(m3 K).
4. Set the thickness. Slab thickness runs from 5 to 300 mm in steps of 5 and is the one slider that leaves every material card exactly where it was. What it moves is the clock: the time to 99 per cent goes from 22.0 s at 5 mm to 22.046 h at 300 mm for the same brick-like triple.
5. Move the clock. Time · Fourier number runs from 0 to 0.80 in steps of 0.002. It is dimensionless on purpose, so the same reading is the same picture for every material; Time now gives you that instant in seconds, minutes, hours or days beside it.
6. Read the four cards down the right. Thermal diffusivity carries its own formula line, α = k/(ρc), to four decimal places at every size. Time to 99 per cent and Mid-depth now sit under it, and Where the slab has got to prints one of exactly three phrases: Still filling, Past half way, Practically steady.
7. Read the ten cells under the graph. The three material figures you set, the effusivity built from them, the clock in both forms, the two times, and then the two percentages that matter most: Flux out of the far face and Energy stored, each against its own settled value. Steady-state heat flux closes the row with the figure a conduction calculation would have given you all along.
8. Read the sentence strip headed The band. It describes the shading between the two curves, and it changes its mind when there is nothing left to shade. Below a Fourier number of 0.558 it names the gap at mid-depth; at and above it, it says that no band is drawn.
9. Press Play the clock to let it run. The clock advances at 0.08 in the Fourier number every second and the button becomes Pause the clock while it is going. Only the clock moves; the three material sliders and the thickness stay exactly where you put them.

If the wall has already settled and all you want is the rate through it, this is more lab than you need. The Fourier's law lab draws the straight line on its own, with the area and the temperature difference that go with it, and that straight line is the dashed one here. Come back the moment the question turns into how long.

Worked example: change one thing at a time

Every row below is one setting of the five sliders, and every cell is a string the running lab printed there. The first six are the presets; the next four hold the brick-like triple still and walk the clock from its first step to its last; the final two change nothing but the thickness. Where a cell and the lab ever part company, believe the lab.

Rows 3 and 4 are the pair the lab exists for. The stainless-like plate conducts about 575 times better than the still-air-like gap, and the gap still diffuses a temperature change 5.71 times faster, at 21.7688 mm2/s against 3.8109. Both ratios come from the unrounded triples the sliders hold rather than from the rounded cards. A steady-state tool has nowhere to put a density or a specific heat, so it can never show you that inversion.

Rows 4 and 5 print the same temperatures on purpose. The still-air-like gap and the water-like layer both sit at a clock reading of 0.300, so both give Mid-depth now 66.7 C, Flux out of the far face 89.6 % and Energy stored 95.8 %, to the last digit. That is the whole point of a dimensionless clock, and it is not a copied row: the two differ by a factor of about 151 in real time, 3.76 min against 9.474 h.

Rows 7 to 10 hold one material still and move only the clock. The diffusivity and the time to 99 per cent print the same strings in all four, because neither depends on where the clock stands. What moves is the picture: 20.0 C at the first step, then 45.3, then 69.5, then 70.0 C, which is exactly the steady-state midpoint of the two fixed faces. The Energy stored column climbs 10.1, 68.6, 99.4, 100.0 per cent behind it.

Rows 1, 11 and 12 are the geometry lesson. Same material, same clock reading, the same temperature picture to the last digit — only the time moves, and it moves as the thickness squared: 9.19 min at 25 mm, 2.450 h at 100 mm and 22.046 h at 300 mm, which are ratios of exactly 16 and 9. If you would rather type a published triple than dial one in on a logarithmic slider, the thermal diffusivity calculator takes the same relation to more figures and runs it backwards for the conductivity, the density or the specific heat.

Formula and symbol reference

The lab works from one definition and one dimensionless group. The diffusivity is α = k/(ρc), and the clock is the Fourier number Fo = αt/L2, which is the only combination of material, time and thickness the temperature profile cares about. Everything else on the panel is that profile read at a particular depth, or the same clock read at a particular threshold.

The density and the specific heat capacity appear only as their product, so halving one and doubling the other leaves the diffusivity exactly where it was. That is algebra, not a discovery, and nothing here tests it: specific heat capacity is defined per kilogram while what conduction cares about is per cubic metre, and the density is simply the bridge between them.

Two of those rows are worth a second look. The Thermal diffusivity card keeps four decimal places from 0.0056 right up to 1111.1111, with no switch to fewer digits anywhere in between, which is deliberate: a magnitude branch is exactly where a rounding bug hides. And Thermal effusivity is printed as a whole number, which is honest at 37388 and lossy at 6 — a point the section below returns to.

The physics: why the profile bends before it straightens

Heat arriving at the hot face has two jobs, and every joule does one or the other. Some of it presses on into the material ahead; the rest is spent lifting the temperature of the material already behind it. The first is what the conductivity measures and the second is what the volumetric heat capacity measures, so the front advances at the rate the one outruns the other.

That is what the bend in the gold curve is. The dashed line is the state Fourier's law assumes, a straight fall from the hot face to the cold one; the solid curve is where the slab has actually got to; and the band between them is the temperature rise the slab has not made yet, because the heat that would have made it is still being absorbed on the way. As the absorption finishes, the curve straightens onto the line.

Mid-depth is 99 per cent of the way there at a Fourier number of 0.4911, which for a given slab is t = 0.4911 L2/α. The thickness enters squared, and the three brick-like rows of the table above show it doing so exactly: a quarter of the thickness gives a sixteenth of the wait, three times the thickness gives nine times. Nothing else in the relation can produce a factor like that.

Ninety-nine per cent in temperature is not ninety-nine per cent in everything. At the clock reading where the status card turns over, mid-depth has made 99 per cent of its rise, but Flux out of the far face reads 98.4 % and Energy stored reads 99.4 %. Temperature settles before the flux does, and the lab prints all three so you can see them disagree. The full account of thermal diffusivity works the constants through with diagrams and problems.

The constant 0.4911 is exact algebra for this threshold, because the mid-depth series contains only odd terms and the next one after the first is smaller than any arithmetic here can see. The companion figure for the far face is a solved root rather than a closed form, and it is the larger of the two. Both belong to one arrangement only: a slab that starts uniform, one face stepped, the far face held.

The status card is thresholded on the clock, not on the temperature beside it. The two thresholds are solved from the same series the curve is drawn from, and both land between slider stops rather than on one, so the phrase cannot flicker. The table below reads the stops either side of each, and adds the stop where the shading gives up.

Rows 3 and 4 are the interesting pair. The card changes from Past half way to Practically steady while Mid-depth now prints 69.5 C at both and the flux column does not move either. Nothing physical happened at that stop; a threshold on the clock was crossed, and the reading beside it had already rounded to its final value.

The band stops being drawn at a clock reading of 0.558. That is a pixel test on the plot the lab is actually drawing, not a statement about the physics: this canvas gives a plot 381 px tall at every column width, and at 0.558 the two curves are closer together than one pixel at mid-depth. The figure belongs to that plot. An independent check on a nominal 300 px plot puts the same test at 0.534, and the two are not interchangeable — a taller plot keeps the curves apart for longer.

Where the slab model breaks down

The lab solves its own model exactly, so nothing on the screen ever fails. Everything below is a limit of that model, of the situation it stands for, or of the way the figures are printed, and each item says what the lab does about it.

The hot face steps instantly, and nothing real does

At the first instant the model puts a 100 K jump across zero thickness, so the flux into that face is formally infinite and the profile there is vertical. A real surface is warmed through an air film, a contact resistance or a joint, all of which delay the whole picture by an amount this lab knows nothing about. Thin, fast layers suffer most from it, because their own clock is short enough for the delay to matter.

The three material figures are held constant

Conductivity, density and specific heat capacity all move with temperature, and the diffusivity inherits every one of those movements. Over a small range the drawing is fine; across a quench or a furnace it is not. Post 713 lists k is a fixed constant for each material among its misconceptions, and this is the same point seen from the transient side: run the lab at both ends of your range and treat the two answers as a band.

One dimension, two faces, no edges

Heat here flows straight through the slab and nowhere else. A corner, a stud bridging an insulated cavity or a pipe buried in a floor is a two- or three-dimensional problem, and the front arrives sooner than this drawing says because it has more than one route. There is no control for it, because the model has no term for it.

Nothing convects and nothing radiates

Both faces are held at fixed temperatures rather than exchanging heat with air or with a surrounding surface. A real far face loses heat to a room and so never quite reaches a fixed value, and the still-air-like preset is the worst case of the lot, because a real air gap circulates rather than merely conducting and this model has no term for that. The comparison of conduction, convection and radiation sets out what the other two do.

The three Fourier constants are not universal

The two thresholds behind the status card, 0.0946870 and 0.4910769, and the far-face figure that goes with them, belong to this boundary condition alone: uniform to start with, one face stepped, the other held for ever. Heat the slab from both sides, insulate its back face, or make it a rod or a sphere, and the same thickness-squared law carries a different number in front of it. The diffusivity does not change; only the schedule does.

The presets are idealised triples, not materials

Each is chosen to land near a published diffusivity and each lands a little off it: brick-like is 7.1 per cent high against 0.52 mm2/s, copper-like 2.1 per cent high against 111, stainless-like 9.3 per cent low against 4.2, water-like 0.7 per cent high against 0.143, pine-like 1.7 per cent high against 0.082, and the still-air-like gap 14.6 per cent high against 19. The air gap is the big one, and it is printed here rather than quietly closed by tuning the triple.

Dividing one printed figure by another will not always come out

Every card is rounded on its own and the arithmetic behind it is not. Thermal effusivity is the clearest case: it reads 7753 for the stainless-like plate and 6 for the still-air-like gap, and dividing those gives about 1292, while the unrounded triples behind them give 1375. The one-significant-figure card is carrying the whole of that error, which is why this page quotes 1375 and names where it came from.

Two readings that agree are sometimes only one reading

The Volumetric heat capacity cell and the Thermal diffusivity card are the same arithmetic seen twice, since the card is simply the conductivity divided by that cell. Their agreeing is a definition restated and proves nothing about the material. The same goes for the temperature difference: it cancels out of the clock, the times and every percentage, which is why there is no slider for it.

Nothing here has been measured

The five sliders are things you chose, and no reading on this page describes a particular wall, bar or board. The lab will happily draw a 300 mm slab of something ten times denser than steel with the specific heat of water. Whether such a thing exists is a separate question the arithmetic has no view on, so check any figure you intend to rely on against your own data first.

Where the diffusivity clock is actually used

Heavy walls that lag the afternoon

Load the brick-like wall and drag the thickness to 300 mm: Time to 99 per cent reads 22.046 h, which is most of a day, while the material cards do not move at all. That lag is why a thick stone building stays cool through a hot afternoon, and it is set by the diffusivity and the thickness rather than by the conductivity on its own. The steady-state account of the same wall names the effect and hands the quantity over; this is the quantity.

Deciding when a test rig has actually settled

A measurement that assumes steady conduction is wrong until the transient has finished, and the two are easy to confuse because the temperature stops moving first. Run the clock up and watch Flux out of the far face reach 98.4 % at the very stop where the card first says Practically steady. On the brick-like wall that is 2.454 h after the step, against a Time now of 29.93 min at the state the lab opens in.

Why a cavity fills quickly and still insulates

The still-air-like gap reaches 99 per cent of its steady state in 3.76 min, behind only the copper-like bar among these six, and its Steady-state heat flux is 26.3 W/m2 against 1000.0 for the brick-like wall. Filling fast and passing little are not in conflict: there is almost nothing in a gas to warm up, and almost nothing to carry heat either. It is the clearest case on these sliders of why the two quantities answer different questions.

Comparing two materials on the same clock

Set any two triples to the same Fourier number and the profiles are identical, so the comparison you are left with is the one that matters: how long each took. That is the discipline the dimensionless clock enforces, and it is why Time now sits beside Fourier number rather than replacing it. Use the specific heat calculator if the figure you are missing is the one under the line.

Sanity-checking a published diffusivity

A handbook may give you a conductivity, a density and a specific heat but no diffusivity, or a diffusivity and no triple. Dial in whichever three you have and compare the card with the published figure: the gap is real information about the sample, the temperature or the table. Every preset here is 0.7 to 14.6 per cent away from its published value, and none of them was tuned to close that gap.

Teaching how much against how soon

Load the still-air-like gap and then the stainless-like plate and the two readings go opposite ways, which is the inversion rows 3 and 4 of the table above set out in figures: the far better conductor is the slower diffuser. That is hard to believe on paper and immediate here, because one press of a preset moves both readings at once. The heat transfer lab puts conduction beside convection and radiation once the idea has landed.

Where to go next

For the quantity in full, with worked problems, the three questions one material answers differently and the diagrams that go with them, read Thermal Diffusivity: How Fast Heat Moves. If you would rather type your own conductivity, density and specific heat than dial them in, the calculator is the first card under Related tools below, and it will run the relation backwards for whichever of the three you are missing.

The steady state this transient ends in belongs to Fourier's Law of Thermal Conduction, which carries the conductivity table, R-value, U-value and layers in series, with the Fourier's law lab and the thermal conduction calculator beside it. For the quantity under the line there is Specific Heat Capacity, the specific heat lab and the specific heat calculator.

Further afield, Conduction, Convection and Radiation covers the two mechanisms this model leaves out, Heat vs Temperature separates the energy from the reading, and Thermal Expansion Explained and its lab take up what the temperature change does to size. The rest of the collection is in the library of physics simulations and on the blog, and the site search will find a topic by name.