Problem of Heating in a Zone Refining Process

From the mathematical viewpoint, it is important to assume that a very rapid heat transfer occurs at the extremities of the stick, and that a rapid cooling system is activated when the heating source is stopped. In addition, as far as we only take 

From Fig. 3.36 we observe that the stick is characterized by its density p, thermal conductivity X and sensible heating capacity Cp. The geometric dimensions of the stick are radius R and length 1. The temperature distribution inside the stick results from relation (3.125) as a particularization of Eq. (3.6)  

The univocity conditions that complete this general mathematical model can be written as follows  

For a correct perception of relation (3.128), we must notice that this is a heat sink that keeps its constant temperature due to a rapid heat exchange between the surface with a cooling medium maintained at constant (t j,) temperature. The assembly of relations (3.124)-(3.126) represents in fact an abstract mathematical model for the above described heating case because the numerical value is given neither for the system geometry nor for the material properties. Apart from the temperature, all the other variables of the model can be transformed into a dimensionless form introducing the following dimensionless coordinates  

With these transformations, the abstract model can now be described by assembling the following relations (3.129)-(3.133)  

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