When heating thin metal parts that are placed into a hot furnace chamber, it can be assumed that the heat energy that enters the metal surfaces will be by radiation and that convective heat transfer to the center of the part will happen (almost) instantaneously. In practice this assumption is only correct because the heat conductivity of metal is very high and, in thin metals, the distance between the surface and the center is very short. In other words, a small temperature gradient exists between the center and the surface of the metal during the heat-up time.

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.

where:
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.

For a very small increment of time, dt, there will be radiated a very small heat energy, dQ, and formula (1) is transformed into Equations 2 or 3:

where:
t2 = Metal temperature minus room temperature (°F)
tr = Room temperature or metal temperature when the metal is placed into the furnace (°F).

The heat energy, dQ, that is put into the metal will increase the metal temperature (assuming uniform heating), and for increments of time this heat energy will be given as Equation 4:

(4) dQ = G°-°c°-°dt2

where:

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.

After integration of formula (6b), the solution becomes Equation 7 or 8, an finally Equation 9.

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).

where
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.