The heat-up time of thin metals can, therefore, be calculated by applying the principles of radiation only and neglecting the small contribution by convection as follows: the constant heat flow of a hot surface against a cool body of air is given by Equation 1.
Q = Total heat energy radiated (BTU)
A = Area of the radiating surface (ft2)
t = Time (hours)
T1 = Absolute temperature of the hot (furnace chamber) surface (°F)
T2 = Absolute metal temperature (°F)
e = Emissivity or absorption factor of the metal surface
Formula 1 is plotted on “Curve No. 6.” The ordinate is given as heat energy radiated (BTU/hr-ft2). The abscissa is given as the surface temperature (°F). It is assumed that the radiating surface or body is initially at room temperature (70°F).
When the cold metal is put into a hot furnace, the metal temperature T2 will increase and eventually reach the furnace (chamber) temperature, T1. This means that the rating of radiated energy will be very large at the beginning and near zero at the end of the heating cycle. In order to obtain the correct heat-up time, this radiated energy must be integrated from the beginning to the end of the heating cycle.
t2 = Metal temperature minus room temperature (°F)
tr = Room temperature or metal temperature when the metal is placed into the furnace (°F).
(4) dQ = G°-°c°-°dt2
G = Metal weight (pounds)
c = Specific heat of the metal (BTU/lb-°F)
dt2 = Average increase of metal temperature (°F)
The heat energy, which is radiated to the metal surface, must be equal to the heat energy that is put into the metal, and formula (3) equals to formula (4), or Equations 5 or 6a or 6b.
where the function r(t) represents the remaining part of formula (3). Thus formula (9) 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 (hrs) 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)
T1 = Absolute furnace (wall) temperature (°F)
t2 or t = Metal temperature (°F) minus room temperature (or metal temperature of the parts for t=0)
tr = Room temperature (or metal temperature (°F) when the parts are put into the furnace.
This subject continues in next week’s blog.