Every resistance heating element starts from the same three inputs: the supply voltage, the power required, and the alloy already chosen for the atmosphere and temperature. Everything else — wire diameter, length, coil geometry — follows from arithmetic rather than from guesswork. This article sets out the three formulas and works two complete examples, one in NiCr and one in FeCrAl.
1. The Three Quantities That Decide the Design
- Resistance (Ω) — fixed by the supply voltage and the power you need
- Length (m) — fixed by that resistance and the wire cross-section
- Watt density (W/cm²) — the power released per unit of wire surface, which decides how hot the wire actually runs and how long it survives
Buyers frequently order on length alone. Two coils of the same alloy and the same length behave completely differently on the same supply if their diameters differ, which is why a purchase order should state the target resistance with a tolerance — not only a dimension.
2. Step One: From Power and Voltage to Resistance
For a purely resistive circuit, Ohm’s law and the power equation combine into one useful form:
- R = V² / P
- P = V² / R
A 2,000 W element on a 220 V supply therefore needs 220 × 220 ÷ 2,000 = 24.2 Ω. Two consequences of the square deserve attention: a +5% supply voltage raises output by about +10%, and a coil delivered 5% below its nominal resistance also draws about 5% more power. Both push the wire hotter than the design intended.
On three-phase banks the element voltage depends on the connection: in delta each element sees the full line-to-line voltage, while in star it sees that voltage divided by √3 — roughly 220 V per element on a 380 V supply. Calculate each element separately.
3. Step Two: From Resistance to Length
Resistance grows with length and shrinks with cross-section:
- R = ρ × L / A, therefore L = R × A / ρ
- A = π × d² / 4
Here ρ is the resistivity at 20°C, listed on every Ipace product page and expressed in Ω·mm²/m (numerically identical to μΩ·m). Since ρ/A is a constant for a given diameter, it is usually faster to work from a resistance-per-metre figure.
For Cr20Ni80, with ρ = 1.09 Ω·mm²/m and a density of 8.40 g/cm³:
| Diameter (mm) | Cross-Section (mm²) | Resistance (Ω/m) | Mass (g/m) |
|---|---|---|---|
| 0.50 | 0.196 | 5.55 | 1.65 |
| 0.60 | 0.283 | 3.86 | 2.38 |
| 0.80 | 0.503 | 2.17 | 4.22 |
| 1.00 | 0.785 | 1.39 | 6.60 |
| 1.20 | 1.131 | 0.96 | 9.50 |
| 1.50 | 1.767 | 0.62 | 14.8 |
| 2.00 | 3.142 | 0.35 | 26.4 |
| 2.50 | 4.909 | 0.22 | 41.2 |
| 3.00 | 7.069 | 0.15 | 59.4 |
Calculated from the nominal resistivity at 20°C used by our calculator; the Cr20Ni80 datasheet quotes 1.09–1.14 Ω·mm²/m depending on wire diameter, so a few per cent spread is normal. Certified values from a production lot differ within the usual ±2–5% resistance tolerance, and the Ω/m figures above are rounded — use the full value (1.09 ÷ A) for long coils.
4. Step Three: Check the Watt Density
The wire surface is what disposes of the heat, so the surface load is:
- S = π × d × L (with d and L in cm, S is in cm²)
- Watt density = P / S
This is the number that governs element life, and it is far more sensitive to diameter than most buyers expect. At a fixed power and voltage, a thicker wire is longer and has more surface per metre, so the surface load falls with the cube of the diameter: doubling d reduces the watt density roughly eightfold.
Practical maxima for continuous duty in air:
| Application | Max Surface Load (W/cm²) |
|---|---|
| Glass melting furnace | 2.5–3.5 |
| Ceramic kiln (oxidising) | 2.0–3.0 |
| Industrial oven (air circulation) | 3.0–5.0 |
| Heat treatment furnace | 2.5–4.0 |
| Vacuum furnace | 1.5–2.5 |
These are workmanship and life limits, not temperature limits: the alloy’s own maximum operating temperature still applies, and the two are compared in Nichrome vs FeCrAl.
5. Worked Example One: a 2 kW Kiln Element in Cr20Ni80
- Resistance: R = 220² ÷ 2,000 = 24.2 Ω
- Diameter: 1.2 mm is the natural starting point for a kiln element of this size, giving A = π × 1.2² ÷ 4 = 1.131 mm²
- Length: L = 24.2 × 1.131 ÷ 1.09 = 25.1 m (cross-check with the table: 25.1 m × 0.96 Ω/m ≈ 24.2 Ω)
- Surface: S = π × 0.12 cm × 2,510 cm = 947 cm²
- Watt density: 2,000 ÷ 947 = 2.1 W/cm² — comfortably inside the 2.0–3.0 W/cm² kiln band
- Material: 25.1 m × 9.5 g/m ≈ 240 g of wire per element
The same element, calculated at three diameters:
| Diameter | Required Length | Watt Density | Verdict |
|---|---|---|---|
| 1.0 mm | 17.4 m | 3.7 W/cm² | Above the kiln band — short element life |
| 1.2 mm | 25.1 m | 2.1 W/cm² | Selected |
| 1.5 mm | 39.2 m | 1.1 W/cm² | Safe but wasteful — more wire, larger coil |
6. Worked Example Two: the Same Method at Higher Voltage
A 5,000 W heat-treatment element on a 380 V supply, in 0Cr25Al5 (ρ = 1.42 Ω·mm²/m, density 7.1 g/cm³, maximum service temperature 1,300°C):
- Resistance: R = 380² ÷ 5,000 = 28.9 Ω
- 1.5 mm — A = 1.767 mm², L = 28.9 × 1.767 ÷ 1.42 = 35.9 m, surface 1,694 cm² → 2.9 W/cm², inside the 2.5–4.0 W/cm² heat-treatment band
- 1.2 mm — only 23.0 m of wire, but 867 cm² of surface → 5.8 W/cm², which will overheat and fail early
- 2.0 mm — 63.9 m and 1.25 W/cm²: reliable and long-lived, but about 1.4 kg of alloy per element and a much bigger coil to support
Raising the supply voltage is the standard way to run thick, long-lived wire at a given power — which is why industrial furnaces rarely use 220 V element circuits when 380 V or higher is available.
7. Four Effects That Shift the Numbers
- Temperature. All of the above uses resistivity at 20°C, which is what a supplier measures on the coil as delivered and what your ohmmeter reads cold. The change across the working range is small for NiCr (typically a few per cent) and somewhat larger for FeCrAl, whose resistance dips slightly at intermediate temperatures before rising. Ask for the alloy’s temperature–resistance curve before freezing a design.
- Coiling. A helix has the same surface area as the straight length it was formed from, so coiling by itself does not change the watt density. Spacing does: tightly packed turns radiate into one another, and the wire then runs hotter than the average surface load suggests. A pitch of about 2–3 times the wire diameter is the usual starting point.
- Aging. As an element scales, its effective metallic diameter decreases, so resistance rises and power falls on a constant-voltage supply. A furnace circuit drawing noticeably less current than when new normally points to worn elements rather than to a faulty controller.
- Terminations. The last turns conduct heat into the terminals, so designs allow a short non-radiating length at each end and keep the connections below the temperature at which the clamp material creeps.
8. Doing the Arithmetic in Seconds
The wire resistance calculator on this site holds the resistivity of every alloy we supply — Cr20Ni80, Cr30Ni70, Cr15Ni60, Cr20Ni35, the eight FeCrAl grades and pure nickel — and returns resistance from diameter and length directly, which is the same equation rearranged. The SWG and AWG converters cover drawings that specify a gauge number instead of millimetres; Understanding Wire Gauge explains why those two systems disagree with each other. For the furnace-side decisions that come before this calculation — alloy family, atmosphere compatibility, coil geometry — see the Heating Wire Selection Guide.
9. What to Put on the Purchase Order
- Alloy grade (for example Cr20Ni80) and the standard or datasheet it must meet
- Wire diameter in mm, with its tolerance
- Total length and total resistance at 20°C with a ± tolerance
- Form of supply: coil, straight cut, spool, or pre-formed to a core diameter
- Surface condition: bright, acid-pickled or oxidised
- Application: furnace type, target watt density, expected element life
The last line is not bureaucracy. It is what allows the mill to match surface finish and temper to how the wire will actually be used, and it turns any later conversation about element life into something concrete.
10. Conclusion
Three formulas carry most of the work: R = V²/P sets the resistance, L = R×A/ρ sets the length, and P / (π × d × L) checks whether the wire can actually shed the heat. The diameter that satisfies the resistance is rarely free — it is chosen against the watt density band your application can tolerate, and then verified against the alloy’s temperature limit.
Our engineering team can run this calculation for your element, confirm the surface load against the furnace design, and supply the wire to a certified resistance tolerance. Send us your voltage, power and temperature for a recommendation.