Lining Math

How to Calculate Refractory Lining Volume, Brick Count, and Dryout Time

Step-by-step math for the five calculations that govern furnace lining installs: castable volume, brick count, dryout time, thermal loss, and wear rate.

Start with castable volume, because material shortfall stops an install cold. For an annular lining, volume equals pi times (R_outer squared minus R_inner squared) times height. A furnace with a 1.2 m inside radius, 0.15 m lining thickness, and 2.0 m height gives pi times (1.35^2 minus 1.20^2) times 2.0, or 3.1416 times (1.8225 minus 1.44) times 2.0, which is 2.40 cubic meters. Multiply by the material bulk density, typically 2,400 to 2,900 kg per cubic meter for dense castable, so 2.40 times 2,650 equals about 6,360 kg. The Castable Volume calculator handles tapered and irregular shapes where hand math gets error prone.

Brick count follows from wall area divided by the exposed face area of one brick, then adjusted for waste. A standard straight brick measures 230 by 114 by 65 mm, so its 230 by 65 mm face covers 0.01495 square meters. A cylindrical wall 1.2 m inside radius and 2.0 m tall has an inner surface of 2 times pi times 1.2 times 2.0, or 15.08 square meters. Dividing gives 1,009 bricks per course arrangement before waste. Add 5 to 8 percent for cutting and breakage, so budget roughly 1,070 bricks. The Brick Count calculator lets you set courses, joint thickness, and arch shapes directly.

Dryout and cure time are separate calculations, and confusing them ruins linings. Cure is the hydraulic set of the cement bond, usually 24 to 48 hours at ambient before any heat. Dryout drives off free and chemically bound water on a controlled ramp, commonly 25 to 40 degrees C per hour with holds at 110 degrees C and 315 degrees C. For a 150 mm castable section, a typical schedule runs 55 to 70 hours total. Use the Cure Time calculator for the ambient set window and the Dryout Time calculator to build the heat ramp against section thickness and water content.

Thermal loss through the lining tells you steady-state heat flow and hot-face behavior. For a plane wall, Q equals area times temperature difference divided by the sum of thickness over conductivity for each layer. Take a 0.23 m dense firebrick layer at k equals 1.5 W per m K backed by 0.10 m insulating board at k equals 0.25. The resistance is 0.23 over 1.5 plus 0.10 over 0.25, or 0.153 plus 0.400, equals 0.553 m^2 K per W. With a 1,300 degrees C hot face and 60 degrees C cold face over 15 square meters, Q equals 15 times 1,240 divided by 0.553, about 33.6 kW.

The Thermal Loss Estimate calculator adds convective and radiative shell losses that the plane-wall formula ignores, which matters because shell radiation at 80 degrees C can add 400 to 700 W per square meter. Interface temperatures come from the same resistance chain: the temperature drop across each layer is proportional to its share of total resistance. In the example above, the firebrick carries 0.153 over 0.553, or 27.7 percent of the 1,240 degree drop, so the brick-to-board interface sits near 957 degrees C. Check that value against the backup material's rated service limit before finalizing the layer stack.

Lining wear rate converts thickness loss into remaining life, the number that schedules relines. Measure residual thickness at fixed points with a laser or probe across campaigns. If a 150 mm working lining reads 96 mm after 210 heats, wear is 54 mm over 210, or 0.257 mm per heat. Set a minimum safe thickness, say 40 mm, so remaining wear is 56 mm, giving 56 divided by 0.257, about 218 more heats. The Lining Wear Rate calculator tracks this per zone, since slag lines and tap holes erode 2 to 4 times faster than sidewalls.

Batch mix yield closes the loop between dry material and placed volume. A castable at 2,650 kg per cubic meter placed density, mixed with 6.5 percent water by weight, yields slightly more wet volume than the dry powder implies. For a 25 kg bag, water adds 1.625 kg, and placed yield is roughly 25 plus 1.625 divided by 2,650, about 0.010 cubic meters per bag after allowing for entrained air and rebound. For the 2.40 cubic meter job above, that is 240 bags. The Batch Mix Yield calculator adjusts for water ratio, waste, and gunning rebound losses of 10 to 25 percent.

Chain these calculations in order and each feeds the next: volume sets material tonnage, tonnage sets bag count and batch yield, geometry sets brick count, section thickness sets the dryout schedule, and the layer stack sets thermal loss and interface temperatures. Wear rate then governs when you repeat the cycle. Carry units at every step, since a millimeter left as a meter turns a 6-ton order into a 6,000-ton absurdity that a spreadsheet will happily accept. Recheck any interface temperature that lands within 100 degrees C of a material limit before you commit the design.

Published 2026-07-01.