The analysis of heat transfer from finned surfaces involves solving second-order differential equations and is often a subject of researches including also the variable heat transfer coefficient as a function of temperature or the fin geometrical dimensions. To analyse the heat transfer problem, a set of assumptions is introduced so that the resulting theoretical models are simple enough for the analysis. Analytical investigations and search activities, which allow finding the optimal profile of the fin, are available under assumptions that simplify the problem of heat transfer.
These basic assumptions are proposed by Murray (1938) and Gardner (1945) and are called Murray-Gardner assumptions (Kraus et al., 2001): – the heat flow in the fin and its temperatures remain constant with time – the fin material is homogeneous, its thermal conductivity is the same in all directions, and it remains constant
– the convective heat transfer on the faces of the fin is constant and uniform over the
entire surface of the fin.
– the temperature of the medium surrounding the fin is uniform.
– the fin thickness is small, compared with its height and length, so that temperature
gradient across the fin thickness and heat transfer from the edges of the fin may be
neglected.
– the temperature at the base of the fin is uniform.
– there is no contact resistance where the base of the fin joins the prime surface.
– there are no heat sources within the fin itself.
– the heat transferred through the tip of the fin is negligible compared with the heat
leaving its lateral surface.
– heat transfer to or from the fin is proportional to the temperature excess between the fin and the surrounding medium.
– radiation heat transfer from and to the fin is neglected.
In general, the study of the extended surface heat transfer compromises the movement of the heat within the fin by conduction and the process of the heat exchange between the fin and the surroundings by convection.