Institute of Metals Division - The Interface Temperature of Two Media in Poor Thermal Contact

The transient one-dimensional heat-conduction equation is solved for two semi-infinite media, at different initial temperatures, brought into (poor) thermal contact. It is shown that the two interface temperatures gradually approach the well-known, instantaneously assumed, common interface temperature of the case of zero contact resistance. The present analysis, with slight modifications, may be used in fair approximation also for the class of problems involving a change of phase. One particular application of the boundary value problem solved in this paper concerns the determination of the temerature distribution in metal castings and molds.P This problem has been treated extensively by analytical methods (see e.g., the literature listed in Refs. 1 to 4) by electrical analogs (see e.g., Refs. 11 to 14) and also by experimental methods (see the literature listed in Refs. 4, 5), but prior to the work of Pellini and Ruddle the experimental results were rather unreliable. Immediate motivation for the work described in this paper was to better understand the results of Pellini,' which are typified by the curves of Fig. 1. Pellini cast 20-in. high, 7-in. sq steel ingots into cast-iron molds of various thicknesses. He measured the temperatures at mid-height, at a number of stations in the casting and the mold, and obtained the cooling curves shown in Fig. 1. The most conspicuous thing about Fig. 1 is the large temperature discontinuity at the interface between casting and mold, due presumably to the shrinking away of the casting from the wall and the attendant creation of an airgap, as the casting cools. Such a contact resistance slows down the anticipated freezing rate; an analytical prediction of this is of much interest in foundry practice. The determination of the effect of contact resistance in the presence of a change of phase will be dealt with, by an approximate method, in a separate paper.' In the present paper the much simpler problem is considered in which there is contact resistance but no change of phase. This problem frequently arises in the analysis of aircraft structures.+ It can initial temperature, and may be regarded as semi-infinite. We suppose the thermal properties in the two media to be time, temperature, and space independent; there are no sources or sinks within the media. In Fig. 2(a) we illustrate, schematically, the case of no contact resistance. The semi-infinite media "0" and "2:' initially at temperatures 8 (- m) and 8(+ -), respectively, are brought into contact at time t =0. A common interface temperature 8,i = is instantaneously established, and thereafter remains constant. Temperature curves are drawn for three instants of time. When there is contact resistance between the two media, the temperature equalization occurs as sketched in Fig. 2(b). Fig. 2(c) illustrates a temperature curve, using the dimension-less notation Barber-Weiner-Boley6 have stated the equations [23a, b] given below, which govern the heat transfer in the two media, and solved them for a rather complicated special case. In their example both media "O" and "2" are finite, are insulated at their far ends, and there is steady heat input into medium 2. In Eq. [27] we give the general solution of the most important special case, where the two media are semi-infinite and there are no sources or sinks. In this case the results may be stated in simple closed form. In Fig. 3, to be discussed in the next section, we illustrate the variation of the interface temperature when steel at 2760" is brought into contact with iron at 100". This example, if we disregard the effect of the heat of fusion, is comparable to Pellini's experimental results, Fig. 1. (Unfortunately, however, experimental determination of the interface temperatures was difficult to carry out, and hence Fig. 1 is not very accurate near the interface.) In Figs. 4 and 5 of the following section, we similarly compare Pellini's sandcasting tests with analytical results for the case where steel at 2760" is brought into contact with sand at 100". It will be pointed out that, insofar as the interface temperatures are concerned, heat of fusion has a minor effect, and so the present approximation, with small modifications, may be