In Dungeons & Dragons 5e, the spell Heat Metal is a low-level staple for cooking enemies in their own armor. The spell description states: “you cause a metal object to glow red hot.” If we assume the magic maintains this “red hot” state regardless of external cooling (a necessity if it’s to remain damaging while underwater or in high winds), we have discovered probably the highest energy output per level of any spell in DnD.
By wrapping this enchanted metal into a heat exchanger, we can design a steam turbine capable of generating significant wattage. Let’s estimate exactly how much power a single caster can contribute to the fantasy industrial revolution.
The Physical Constraints
To find the upper bound of our heating element, we look to the heaviest common metal object a caster can target: Plate Armor.
- Mass of the Heating Element: According to the Player’s Handbook, a suit of plate armor weighs 65 lbs (29.48 kg).
- Molar Equivalence: Plate is typically steel (mostly iron). Using the atomic mass of iron (\(M_{Fe} \approx 55.85\) g/mol): \[n_{Fe} = \frac{29,480\text{ g}}{55.85\text{ g/mol}} \approx 527.9\text{ moles}\]
- Copper Conversion: For maximum efficiency, our fantasy engineers would use copper for its superior thermal conductivity. We’ll assume the spell targets an equivalent molar amount of copper (\(M_{Cu} \approx 63.55\) g/mol): \[m_{Cu} = 527.9\text{ mol} \times 63.55\text{ g/mol} \approx 33,550\text{ g} \approx 33.55\text{ kg}\]
Engineering the Boiler: Metal Selection
While copper is the “standard” for conductivity, different metals offer different limits. In a boiler, the “red hot” temperature is capped by the metal’s melting point.
| Metal | Thermal Conductivity (\(k\)) | Melting Point | Max Safe Temp | Notes |
|---|---|---|---|---|
| Silver | 429 W/m·K | 961°C | ~900°C | Best conductivity, but “soft” at high heat. |
| Copper | 401 W/m·K | 1085°C | ~1000°C | The industry standard for high-flux boilers. |
| Gold | 315 W/m·K | 1064°C | ~1000°C | Chemically inert; great for corrosive alchemical steam. |
| Steel | 15–50 W/m·K | ~1500°C | ~1300°C | Poor conductivity but allows for “White Hot” temperatures. |
| Tungsten | 173 W/m·K | 3422°C | ~3000°C | For 9th-level casts; can sustain incredible temperatures. |
Selection: For our baseline, we stick with Copper for 2nd-level slots, but transition to Tungsten for anything higher.
Engineering the Boiler: Maximizing Surface Area
Using a density of \(8,960\text{ kg/m}^3\), our 33.55 kg copper mass occupies a volume of approximately 3.74 Liters. To maximize heat transfer, we must draw this metal into the largest possible surface area \(A_s\) while maintaining structural integrity for a high-pressure boiler.
If we draw the metal into a high-density bundle of 1/4-inch nominal Type M copper tubing (OD 9.5 mm, wall 0.6 mm):
Cross-sectional area of metal per tube: \(\approx 17.6\text{ mm}^2\)
Total length of tubing possible: \(\approx 212\text{ meters}\)
Internal Surface Area (\(A_s\)): \(\approx 5.5\text{ m}^2\)
By utilizing a multi-tube heat exchanger (similar to a fire-tube or water-tube boiler), we can achieve a surface area of 5.5 m\(^2\) per caster. This optimization significantly increases the raw wattage captured compared to a simple pipe.
Scaling Up: Upcasting and Temperature
Heat Metal deals an extra 1d8 damage for every slot above 2nd. We can model this increased “potency” as an exponential increase in maintained temperature. While 2d8 corresponds to a “Red Hot” 800°C, a 9th-level cast represents a truly sun-like energy density:
- 2nd Level (2d8): 800°C (Dull Red)
- 4th Level (4d8): 1150°C (Bright Red-Orange)
- 6th Level (6d8): 1500°C (Yellow-White)
- 9th Level (9d8): 2000°C (White Hot)
The Spellcaster’s Endurance: Total Energy
A Level 20 caster is a walking nuclear reactor. By cycling through their spell slots, they can provide sustained power. We assume a 30% turbine efficiency for our baseline generation.
| Slot Level | Duration | Temp (°C) | Metal | Raw Thermal | Generated (@30%) |
|---|---|---|---|---|---|
| 2nd (x3) | 3 min | 800 | Copper | 825 kW | 247.5 kW |
| 3rd (x3) | 3 min | 1000 | Tungsten | 1,320 kW | 396.0 kW |
| 4th (x3) | 3 min | 1150 | Tungsten | 1,930 kW | 579.0 kW |
| 5th (x3) | 3 min | 1300 | Tungsten | 2,650 kW | 795.0 kW |
| 6th (x2) | 2 min | 1500 | Tungsten | 3,840 kW | 1,152.0 kW |
| 7th (x2) | 2 min | 1650 | Tungsten | 4,940 kW | 1,482.0 kW |
| 8th (x1) | 1 min | 1850 | Tungsten | 6,710 kW | 2,013.0 kW |
| 9th (x1) | 1 min | 2000 | Tungsten | 8,280 kW | 2,484.0 kW |
| TOTAL | 18 min | AVG 1350 | - | ~2,930 kW | ~880 kW (1,180 HP) |
A Level 20 caster can provide an average of 1,180 Horsepower for 18 minutes, dumping nearly one Gigajoule of usable electrical energy into the grid.
Mid-Level Industrialist: The 10th-Level Cleric
A 10th-level Forge Domain cleric is still a formidable power source for local industry.
| Slot Level | Duration | Temp (°C) | Metal | Raw Thermal | Generated (@30%) |
|---|---|---|---|---|---|
| 2nd (x3) | 3 min | 800 | Copper | 825 kW | 247.5 kW |
| 3rd (x3) | 3 min | 1000 | Tungsten | 1,320 kW | 396.0 kW |
| 4th (x3) | 3 min | 1150 | Tungsten | 1,930 kW | 579.0 kW |
| 5th (x2) | 2 min | 1300 | Tungsten | 2,650 kW | 795.0 kW |
| TOTAL | 11 min | AVG 1060 | - | ~1,590 kW | ~475 kW (640 HP) |
Even at level 10, a single cleric can generate over 300 MJ of usable energy—enough to run a fleet of heavy locomotives for a short-haul route.
Estimating Heat Flux (\(q''\)) and Bromley’s Correlation
When our boiler pipe is maintained at extreme temperatures (\(T_s\)), water in contact with it enters film boiling. In this regime, the heat transfer is governed by Bromley’s Correlation for a horizontal cylinder:
\[h_{conv} = 0.62 \left[ \frac{k_g^3 \rho_g (\rho_l - \rho_g) g (h_{fg} + 0.4 c_{pg} \Delta T)}{D \mu_g \Delta T} \right]^{1/4}\]
Total heat flux \(q''\) also includes radiation, which becomes dominant at these temperatures: \[h_{total} = h_{conv} + \frac{3}{4} h_{rad}\] \[h_{rad} = \frac{\epsilon \sigma (T_s^4 - T_{sat}^4)}{T_s - T_{sat}}\]
The \(T^4\) Shift: Convection vs. Radiation
It is critical to note that \(h_{conv}\) and \(h_{rad}\) scale very differently. While convective film boiling (\(h_{conv}\)) stays relatively stable as temperature rises, radiative heat transfer (\(h_{rad}\)) is governed by the Stefan-Boltzmann Law, scaling with the fourth power of absolute temperature (\(T^4\)).
- Copper Core (2nd Level): Oxidized \(\epsilon \approx 0.8\). At 800°C, \(h_{rad}\) provides ~35% of the heat.
- Tungsten Core (9th Level): Oxidized \(\epsilon \approx 0.7\). At 2000°C, \(h_{rad}\) explodes to over \(600\text{ W/m}^2\text{K}\). Radiation accounts for over 75% of the heat transfer, resulting in a staggering 1.5 MW/m\(^2\) of flux.
The Universal Glow: Black Body Radiation
A common misconception in fantasy engineering is that different metals might “glow” differently at the same temperature. However, the “red hot” or “white hot” state described in the spell is governed by the principles of Black Body Radiation.
According to Planck’s Law, the spectral radiance of an object is a function of its temperature \(T\), not its chemical composition. While the intensity of the light can be modified by the material’s emissivity (\(\epsilon\)), the peak wavelength (color) is determined solely by the temperature.
Wien’s Displacement Law
The mathematical justification for mapping specific colors to temperatures comes from Wien’s Displacement Law:
\[\lambda_{max} = \frac{b}{T}\]
Where: - \(\lambda_{max}\) is the peak wavelength of light. - \(b \approx 2.897 \times 10^{-3}\text{ m}\cdot\text{K}\) is Wien’s displacement constant. - \(T\) is the absolute temperature in Kelvin.
As \(T\) increases, \(\lambda_{max}\) shifts left toward the visible spectrum:
Red Hot (\(800^\circ\text{C} \approx 1073\text{ K}\)): \(\lambda_{max} \approx 2.7\text{ }\mu\text{m}\). The peak is in the infrared, but the “tail” of the Planck distribution bleeds into the visible red spectrum (~700 nm).
Orange/Yellow (\(1300^\circ\text{C} \approx 1573\text{ K}\)): The distribution shifts further, covering more of the visible spectrum.
White Hot (\(2000^\circ\text{C} \approx 2273\text{ K}\)): The peak wavelength \(\lambda_{max} \approx 1.27\text{ }\mu\text{m}\). The distribution covers the entire visible spectrum, emitting all colors nearly equally. To the human (or elven) eye, the object appears as a blinding, brilliant white.
This is why our choice of metal changes the melting point and conductivity, but a Copper pipe and a Tungsten pipe at \(900^\circ\text{C}\) will both emit the exact same “Bright Red” hue.
Power Generation: The Rankine Cycle
Increasing the boiler pressure (\(P_{boiler}\)) improves thermal efficiency by allowing a higher temperature drop across the turbine.
- Efficiency Gains: A pressurized boiler (e.g., 20 bar) can push efficiency toward 20-40%, depending on the sophistication of the condenser and turbine design.
- The Pump Tax: For our ~8 MW boiler, we’d need to pump roughly 3 L/s. At 20 bar, this consumes about 6 kW—still negligible.
Interactive Power Grid
Adjust the parameters below to see what your adventuring party can power.
Assumptions & Limitations
- Magic as an Infinite Sink: This assumes the spell provides the energy required to maintain the temperature regardless of the power being siphoned off.
- Material Stress: Copper and Tungsten at these temperatures are significantly weakened. Engineers would likely need magical reinforcement (Mending or Permanency) to prevent bursting under the high pressures of a Rankine cycle.
- Concentration: Collectivized labor is a must. A caster must concentrate for the full duration (1 minute), but a group of only 200 low to mid level casters working in shifts could keep the lights on 24/7 at more than a constant megawatt of power.