Tube furnace rework

Figure 1. The build as the model sees it, to scale. The dimensions marked “measure” are placeholders until a tape measure and a multimeter replace them.

Summary

Toronto Hacker Fab’s first tube furnace was built by three members of the chapter from a build video: 22 AWG nichrome wound directly on a 5/8 in quartz tube, refractory cement over the wire, ceramic fibre blanket in a carbon steel box, one type K thermocouple and a ramp-and-soak PID switching a solid-state relay. It reaches 1000 °C and glows. I took on the rework after it was built. The audit found mains wiring running unsheathed inside the earthed case, and nobody knew the furnace’s temperature profile, its ramp, its losses or its oxide rate. Before running it again I wrote a 1-D simulator of the build as it stands and made it predict every number the first logged runs will produce, with a band for each from the dimensions I have not yet measured. The model is verified against analytic limits and conservation by 230 tests and cross-checked by hand. It says the ±5 °C flat zone is about 26 mm, that the furnace runs away at 2 K/s and reaches the quartz limit 48 s after a relay fails closed, that the element sits near 1150 °C at full power, and that room humidity oxidises silicon in this furnace about as fast as the oxygen in the air does. A ranked measurement list and a designed set of five experiments follow from the model’s own sensitivities. The same simulator, run over 36 worst cases, chooses the heater architecture for the furnace that comes after this one. There are no measurements yet. That is the point of the page: the runs will be a test, not a survey.

Item	Detail
Organisation	Toronto Hacker Fab
Period	October 2026 to present
Role	Furnace rework lead: the hazard audit and fix, the specification, the simulator, the measurement plan
Deliverables	The furnace simulator (eight physics modules, parameter store with provenance, 230 verification tests, browser UI), the hand-calculation script that pins its numbers, the first-run prediction sheet, the measurement ranking, the experiment design, the v2 architecture study
Tools	Python, NumPy, SciPy, SALib, matplotlib, pytest, FastAPI, Plotly, Playwright

1. The furnace as inherited

Item	As built	Known or assumed
Tube	Fused quartz, 5/8 in outside diameter, 12 in long	known
Heater	22 AWG (0.65 mm) nichrome wound on the tube, under refractory cement	known; turns, pitch and wound length assumed (108, 1.4 mm, 150 mm)
Element resistance, supply	18 Ω cold at 120 V, about 750 W hot	assumed. The build post says a 120 V outlet and lists a NEMA 6-15P cord, which is a 240 V plug
Insulation	Ceramic fibre blanket and board in a carbon steel box bonded to protective earth	known; box 150 mm square assumed
Control	Ramp-and-soak PID, type K thermocouple, 40 A solid-state relay at a 2 s cycle, 20 °C/min to 1000 °C, hold 30 min	known; thermocouple position unknown
Ends, ambient	Both tube ends open to the room, no gas system	known

The furnace exists to grow oxide, and later to diffuse dopants. Both are Arrhenius rates, so what the wafer sees is the temperature at its position, held for a time, in an ambient. The questions a process tool has to answer are where that temperature is flat, how well it is held, and what the ambient is. The theory behind the oxide, from the flux balance to the Deal-Grove law and its activation energies, is worked through by hand in the notebook entry on tube furnace theory.

2. The hazard and the fix

Inside the case, the mains conductors to the element run unsheathed past the steel. The case is bonded to protective earth, which turns a short into a tripped breaker rather than a live box, but it does not stop the short. The fix specified before the next run:

• sleeving rated above the temperature it sees on every conductor near the hot zone, ceramic terminal blocks, mains routed away from the tube;
• strain relief at every case exit, and a residual-current device at the plug;
• a resistance check from each mains conductor to the case with the furnace unplugged (anything under 1 MΩ is a fault) and from the case to the plug’s earth pin (under 0.1 Ω);
• an over-temperature limiter with its own sensor driving a contactor, independent of the controller, because the solid-state relay fails closed. Section 5 gives the number it has to beat.

3. What the furnace has to do

These are the acceptance numbers I will hold the build to. Each is a furnace specification in the form a datasheet would state it, and each has a test.

Requirement	Target	Test
Flat zone at ±5 °C	long enough for a piece and its neighbour, stated as a length	traversing thermocouple scan at 700 and 1000 °C
Overshoot at the end of the ramp	reported; ≤ 2 °C for v2	logged thermocouple
Ramp	20 °C/min held by the controller, within the quartz tube’s limit	logged ramp
Box surface at 1000 °C	below the accessible-surface limit of the lab equipment standard	infrared survey, six faces
Runaway on a failed relay	limiter trips before the quartz limit	fault injection at a 300 °C setpoint
Oxide at 1000 °C for 60 min in room air	predicted thickness, two runs, repeatability stated	colour chart against a bare reference, humidity logged
Oxide as thermometer	temperature backed out of thickness agrees with the thermocouple	pieces at five positions against the scan

4. The model

The simulator is one-dimensional along the tube. Each axial cell carries the tube, the cement and the winding, with the insulation as a column of radial nodes out to the box, so the blanket’s slow charging during a run is resolved rather than lumped. The square box enters through the conduction shape factor for a cylinder in a square, which replaces a 2-D solve. Bore radiation is a conductance that scales with the cube of the bore radius, small here because the bore is 12.7 mm. The wafer side (Deal-Grove with the oxidant fluxes of O₂ and H₂O in parallel, dopant diffusion, junction depth, sheet resistance) takes the furnace temperature at each position and runs separately.

Every physical value lives in a parameter file with its unit, source and a status: literature, assumed, or measure. Every relation has a test against an analytic limit or a conservation law, 230 in all. A script computes the hand estimates from the same parameter file, so the model is checked against arithmetic I can follow, and the differences are explained rather than hidden.

The one argument that shapes everything is the decay length. The tube obeys a fin equation with axial conductance and radial conductance :

An end deficit decays as with . For this build is 6.6 cm from the tube and bore alone and 8.8 cm with the blanket’s own axial conduction, against a wound length of 150 mm. The two end regions overlap completely. There is no plateau. The profile is a single arch, and the only flat zone is the top of it.

Figure 2. The predicted profile at two setpoints. The grey band is the winding. At 1000 °C the winding ends sit 200 to 300 K below the centre, depending on where the open-bore radiation is taken to leave.

5. Predictions for the first run

Each row is what the logged run will show, from the model at the assumed dimensions. The band is the minimum and maximum over a screening sample of every unmeasured input, not a confidence interval. The hand value is the same quantity by the closed-form estimate.

Video 1. The run the table describes, as the model sees it. Top, the profile along the tube with the pieces coloured by their oxide. Bottom, the centre temperature against the setpoint, the relay power, and the oxide at the centre and ±2 and ±4 cm. The pieces go in at the start of the hold; the 25 nm they show at once is the thin-oxide offset the Deal-Grove model applies, which is the part of the model the oxide runs will test first.

Response at 1000 °C	Model	Band over the unmeasured inputs	Moved most by	Hand
Steady relay duty	8%	5 to 34%	element cold resistance	13%
Loss	61 W	51 to 163 W	blanket conductivity	95 W
Box surface temperature	80 °C	42 to 212 °C	shell film coefficient	64 to 88 °C
Time to 1000 °C from cold	49 min	49 to 63 min	the controller’s own ramp
Overshoot at the end of the ramp	7 °C with a plain PID, 0.2 °C with ramp feed-forward	0 to 2 °C	blanket participation
Duty at the start and end of the 30 min hold	12% falling to 9%		the blanket is still charging
Cool-down rate, heater off, at 900 and 700 °C	35 and 17 °C/min	6 to 59 at 1000 °C	box size	18 at 1000 °C
Flat zone at ±5 °C	26 mm	17 to 41 mm	wound length	about 30 mm
Centre minus winding end	190 K	81 to 538 K	wound length
Element temperature at full power	1153 °C	1040 to 1326 °C	element cold resistance	1153 °C
Element surface load at full power	6.9 W/cm²	1.5 to 17 W/cm²	element cold resistance	6.9 W/cm²
Runaway from 1000 °C to the quartz limit, relay failed closed	48 s	19 to 310 s	element cold resistance	102 s to 1200 °C
Oxide at the centre, 60 min, room air at 50% RH	44 nm	39 to 55 nm	room humidity
Oxide at the centre, 30 min	35 nm	33 to 41 nm	room humidity

Three of these are findings in their own right.

The relay fails closed and the heater is seven times the losses, so the furnace runs away at about 2 K/s. The limiter has to act within tens of seconds. The fault-injection test does this at a 300 °C setpoint, where a failure does no harm.

Room humidity matters as much as the oxygen. At 1000 °C the parabolic rate constant per atmosphere is about 30 times larger for water than for oxygen, and room air at 20 °C and 50% RH holds 0.012 atm of water against 0.21 atm of oxygen. The two contributions come out nearly equal, so a “dry” oxidation in this furnace depends on the weather, and a hygrometer reading per run is the cheapest valuable instrument on the list.

The colour chart cannot do what I wanted it to. Its 10 nm resolution is 7% at 150 nm, but the model predicts 35 to 44 nm oxides at 1000 °C, and the position-to-position differences that would make the pieces a thermometer are 1 to 9 nm. That check needs a reflectometer or holds of four hours. Both options are in the experiment design.

6. The measurements, ordered by what they buy

A Morris screening over the eighteen unmeasured inputs ranks the cold measurements by how much of each prediction’s spread they remove. The whole list takes an hour with the furnace unplugged.

Rank	Measurement	What it pins
1	Wound length, turn count, start and end of the winding	the flat zone
2	Room temperature and humidity, at every run	the oxide thickness
3	Box inner dimensions and blanket thickness	the box temperature
4	Element cold resistance, lead resistance	the duty, the power, the element temperature
5	Blanket grade and density	the losses
6	Supply voltage and the plug actually used	everything dynamic, by a factor of four. It ranks sixth only because the resistance range is wider. It is the first thing to settle

Each run is logged at 1 s: setpoint and reading over the controller’s RS485 port if it exposes them, otherwise from a second thermocouple on a logger, and the heater power from a clamp meter on the element lead. The run sheet records room temperature, humidity, mains voltage and the element’s cold resistance, so a drift between runs is a regressor and not noise.

Then five experiments, each designed to estimate named parameters with a stated resolution rather than to produce curves.

Experiment	Levels	Estimates	What the design showed
Logged ramp, hold and free cool-down	700, 850 and 1000 °C, one repeat	losses against temperature, effective heat capacity, the controller’s plant model	on its own it cannot separate the blanket conductivity from the shell film coefficient (correlation 0.98). Logging the heater power with a clamp meter to 0.5% instead of reading 2% off the controller is what separates them
Profile scan with a second thermocouple	1 cm steps at 700 and 1000 °C, both directions, the two thermocouples co-located at the centre first so the scan reads differences to ±1 K	axial and radial conductance and the end losses separately	the scan points outside the winding are what make it work: without them the parameter uncertainty is three times larger
Oxidation of silicon pieces in room air	1000 °C for 30 and 240 min, 900 °C for 240 min, plus a run at 5% RH behind a desiccant	rate constants and the thin-oxide offset at this ambient	the plan’s original 30 and 60 min cells leave the parabolic constant unidentified with the colour chart; the long holds and the dry run fix that
Element life	cold resistance before and after every run	resistance drift against the model’s element temperature	the element runs near its limit at full power
Safety verification	infrared survey of the box, earth continuity, insulation resistance, limiter fault injection at 300 °C	the box temperature and the limiter’s response against the runaway time

Each of these rehearses a v2 acceptance test. When the next furnace is specified, every line of its sheet will have been measured once at this scale, with a known resolution.

7. What it feeds: choosing the next furnace

The same simulator, with the v2 geometry from the system specification (a 60 cm heated length, a 25-wafer boat, 2.4 kW), was run over 36 worst-case scenarios for three heater architectures. The criterion is the spread of junction depth and oxide thickness across the boat, which is what the process cares about.

Design	Junction depth spread, goal 3%	Oxide spread, goal 2%	Flat zone at ±1 K with 2 slm of gas	Overshoot
Uniform winding	81%	50%	0.8 cm	2.7 K
Graded winding	79%	48%	0.8 cm	24.8 K
Three zones	0.9%	0.5%	48.5 cm	3.1 K

Figure 3. The v2 geometry at 1000 °C with 2 slm of gas. Left, nominal end losses. Right, end losses tripled. Only independent zones stay within ±1 K across the boat.

Three pieces of physics force the result. The insulation that keeps the shell touch-safe makes the decay length longer than the furnace, so the ends reach the centre. Process gas is a heat pump: 2 slm of nitrogen carries about 42 W from the inlet end to the outlet end and tilts the profile by 2.6 K/cm across the boat. And a controller can only null as many shapes as it has sensor-heater pairs: one sensor fixes the level, a graded winding cancels one symmetric droop, and only three zones correct the tilt as well. The conclusion holds whether the uncalibrated bore-radiation factor is halved or doubled. The same cheap experiments above, the scans at two flows and the pieces as a thermometer, are what calibrate it.

8. Status

As of 2026-10-11: the simulator and the plan exist, the hazard fix is specified, and no run has been logged. Next, in order: the cold measurements, the wiring fix and the earth and insulation checks, the limiter and its fault-injection test, then the logged ramp and the scans. The oxide runs wait on silicon pieces and on a reflectometer or the long holds.

Open items only the hardware can settle: the supply voltage, where the thermocouple sits (against the winding it reads the element, in the bore it reads the process, and the oxide runs will tell which), and whether the 2 W/cm² surface-load limit I am using for the nichrome is the right vendor number. The build cost, parts list and a comparison against a commercial tube furnace’s datasheet are to come with the measurements.

9. Further work

The model’s next jobs, after the first logged run. The list is ordered by what each buys against the acceptance numbers of Section 3, and everything on it waits on the plant being identified from a measured ramp, a scan and a cool-down. Until then a better controller model or a better winding is a prediction about a furnace whose heat loss is known to a factor of three.

Item	What exists	What it adds	Waits on
The controller as built	A generic PI tuned from the identified plant, with continuous power per 2 s tick	The box’s own algorithm from its manual: proportional band in °C, integer I and D, filtering, 1 °C resolution, its autotune; the relay as on/off time proportioning, or a number showing a 2 s cycle is invisible to a plant with minutes of time constant	the controller’s model number, then the logged run
Emulator over the unmeasured inputs	A Morris screening of 18 inputs, minutes per sheet	A fitted surrogate so every band on this page updates in seconds when a cold measurement lands	profiling one prediction sheet first
Graded pitch for this furnace	The v2 study, where a graded winding fixes only the symmetric droop	With no gas flow the droop here is symmetric, which is the case a graded pitch fixes; the optimum pitch against the end-loss factor, and the robust choice. Rewinding under cement is a rebuild, so this is a number before it is a decision	measured and
Model against measurement, in time	stored every 10 s	The scan points and the logged ramp over the predicted field, as one animation	the scan
Controller comparison	Overshoot, settling and duty metrics; feed-forward already cuts the overshoot from 7 to 0.2 °C	PI, PI with feed-forward, PID and a constrained predictive controller on the identified plant under the quartz ramp and element limits. On one actuator and one sensor they should converge, because the ramp is limited by power minus loss over heat capacity, not by the algorithm; the table is there to show it. What the box can actually be set to bounds the study	the identified plant
Shell temperature against insulation	Shell temperature predicted, 80 °C with a 42 to 212 °C band	One curve of shell temperature against blanket thickness for the v2 box, with the standard’s touch limit drawn on	the film coefficient from the infrared survey
Commercial datasheets	Nothing	Flat-zone length and tolerance, ramp and overshoot from two commercial tube furnaces beside this one’s measured numbers	the measurements

Not on the list: the room as a model (the screening says measure the film coefficient instead), and predictive control for its own sake, which earns its place with three coupled zones and hard constraints, that is, with v2.