Presets
Playback
Furnace, gas and floor
Furnace power is multiplied by 0.56 to give heat in the gas, the supply temperature follows from m cp, and the slab takes heat U A (T_gas - T_floor) with U = 3.2 W/m2K, capped at 92 % of the delivered heat. The slab loses heat to the room at 10.8 W/m2K, so its steady temperature balances the two.
Model checks
The science
What is modelled
Hot gas from the furnace enters the void under the floor, where it passes between the pillars and gives heat to the slab through a lumped heat-transfer coefficient. The gas leaves by the flue. The floor is a single lumped heat capacity that is heated from below and cools to the room air through its surface.
Equations
- Supply temperature:
Tin = Tamb + 0.56 P / (ρ V̇ cp). - Heat to the floor:
Q = min(0.92 · 0.56 P, U A (Tin - Tf)), exit gasTout = Tin - Q / (ṁ cp). - Slab:
C dTf/dt = Q - h A (Tf - Troom), time constantC / (A (U + h)).
Checks run in the page
- The gas-side heat balance used on the 3D page.
- Steady floor temperature against a hand solution.
- The warm-up time constant.
- Energy delivered minus lost equals energy stored.
Honest simplifications
A single lumped slab temperature, a constant heat-transfer coefficient, a constant gas specific heat and no heat loss through walls, no pillar thermal mass and no real combustion or draught model. Roman practice varied widely; this is a teaching model.
What to try
- Press Typical bath floor and watch the slab warm up over several hours.
- Press Strong fire, slow air: the floor overshoots the comfortable 29 C limit.
- Press Thick slab, big hall: more thermal mass means a much longer warm-up.