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FIGURE 5.70 Temperature distribution in infinite length plate.
conduction of heat by contact of wires, the convection of heat in the air clearance
between wires, and heat radiation between wires.
Usually some effective heat conduction models on the packed bed are reported
in manuals of heat transfer engineering. These models can only be used for spherical
solids and therefore there exists no model that can be applied for wire. We can
calculate the real thermal conductivity of wires by means of a method of inverse
calculation of the λ through the differential equation of heat conduction (cylindrical
axis indication) by using measured data of heat distribution in coiled wires.
5.4.3.2 Furnace Scale-Up for Commercial Production
While the tests were conducted in an experimental furnace, the method of scale-up
for commercial application by applying the rule of similarity is important to obtain
high efficiency in a production furnace. In this section, the conditions to realize the
rule of similarity between the test furnace and the high efficiency production heat
treatment furnace are discussed. The conditions to realize the rule of similarity
between the test furnace and the highly efficient production heat treatment furnace
and the condition to realize the rule of similarity of heat transfer phenomena between
the two furnaces compared are as follows:
1. To realize the geometrical similarity
The relation, D 1 / W 1 = D 2 / W 2 , in Figure 5.71 must be realized. As the dimensions of
the above-mentioned experimental furnace are D 1 (average value) = 1.37 m, W 1 =
2 m, the relation, D 1 / W 1 = 1.37/2 = 0.685, can be obtained. If D 2 / W 2 = 0.685 in
another high performance furnace, then consistency of the geometrical similarity
between the two furnaces can be realized.
2. To derive the nondimensional control equations
The relation of geometric positions of heat-treating material and RT alone determines
the term of radiation, because the φ CH model that is applied to radiation of RT furnace
has no term for the gas surface area. Because the rule of geometric similarity between
the experimental furnace and the high performance heat treatment furnace is realized
by the relation of previously described items in this section, the two furnaces have
the same value of φ CH . The rule of similarity on heat transfer in the furnace can
therefore be decided only by the term of convection.
 
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