t=G°-°c / A°-°e °-° 0.01°-°ρ(t)
This formula can be used to calculate the heat-up time of a thin metal or the time to heat the outside of a thicker section of metal. (Note: This formula does NOT take into account the soak time required for the center of a thicker section to reach temperature).
t = Heating time (hours) for the metal to a mean temperature, T.
G = Weight of the metal (pounds)
c = Specific heat of the metal (BTU/lb-°F)
A = Radiation area (ft2) of the metal part exposed to the furnace chamber.
e = Emissivity (e = 1 for a black body)
Consider heating a 4.5-inch-diameter x 10-inch-long round steel bar under the following assumptions:
- 2100°F chamber temperature
- 2050°F final part temperature
- 45.07 pounds (weight of part)
- 0.166 BTU/ft2-°F specific heat (steel)
- radiation area = 9.60-10.80 square feet (80-90% of the area capable of "seeing" radiation)
- emissivity value of 0.787 (oxidized steel)
- ρ(t) = 4.58 (Curve 35)
Assuming that the bar will conduct heat to the center at a rate of (approximately) 1/8 inch per 5 minutes, and adding a factor (25%) for the additional time required to heat (by conduction) the 10-20% of the bar not directly exposed to radiant heat (but more than likely exposed to some re-radiated heat), we come up with a total time of 116 minutes.
Adding 20-60 minutes to allow the structure to stabilize is typical, so we reach a total time of:
3.5 minutes + 116 minutes + 30 minutes = 150 minutes (2.5 hours)
All in all, the "rule of thumb" that the heat-up rate is one hour per inch of cross-sectional thickness, or 2 1/2 inches x 1 hour/inch = 2.5 hours, is accurate (you use half the thickness since you are theoretically heating from two sides).