Equation 1

When cooling thin, hot metal parts that are placed into a cold (black-body) furnace chamber or still-air cooled in a room-temperature environment, it can be assumed that the heat energy that is radiated from the metal surfaces will be by radiation and convection. In this assumption, heat will be transferred from the center of the part to the surface (almost) instantaneously due to the high conductivity and short distance between the surface and center of the thin metal.

The cooling time can be calculated applying the principles of radiation only as in Equation 1. The energy radiated to the black body can be expressed as:

Q = Total heat energy radiated (BTU)
A = Area of the radiating surface (ft2)
t = Time (hours)
T1 = Absolute temperature of the hot surface (°F)
T2 = Absolute temperature of the cool body of air (°F)
e = Emissivity or absorption factor of the metal surface

Equation 2a & 2b

When the metal is placed into the cold chamber or into the room for cooling, the metal temperature T1 will decrease and after a certain time will reach room temperature T2. At the beginning of the cooling time, the rate of radiant energy is quite large and almost zero towards the end of the cooling cycle. In order to obtain the correct cooling time this radiated energy must be integrated from the beginning to the end of the cooling cycle.

For a very small increment of time, dt there will be radiated a very small heat energy, dQ, and Equation 1 is transformed into Equation 2a or 2b.

where:

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

Equation 3

The heat energy that is removed from the metal will decrease the metal temperature (assuming uniform cooling), and for increments of time this heat energy will be given as Equation 3.

where:

G = Metal weight (pounds)
c = Specific heat of the metal (BTU/pound-°F)
dt1 = Average decrease of metal temperature (°F)

Due to its complex nature, this discussion will conclude next week.