Making Industrial Processes More Cost Efficient
Residual resistivity ratio is a standard measure of the purity and low temperature conductivity of copper. Oxygen free copper for cryogenic applications can have a conductivity two to three orders of magnitude higher at 4.2 K than at room temperature. There are several papers in literature indicating the effects of various impurities on the conductivity or resistivity of copper. Quantitative effects are usually described as linear variations. The general wisdom regarding the low temperature conductivity has favoured higher and higher levels of purity. However, at even lower concentrations of impurities, their effects are not always negative. Secondly, the effects are not very linear in terms of either conductivity, resistivity or residual resistivity ratio.
New techniques of nonlinear modeling have come up in the last ten-twelve years which have made the development of nonlinear empirical models of a higher quality possible, without the necessity of knowing the type and severity of the nonlinearities present in the relations. These techniques have been used successfully for a variety of applications in materials science, particularly for the prediction of material properties from the composition of the material. This article describes nonlinear modelling for residual resistivity ratio of oxygen free copper for cryogenic applications from Outokumpu Poricopper’s production and experimental data.
While copper is used for a large variety of purposes including wiring, plumbing and architectural applications, it has alternatives in the common applications. The level of know-how required to produce copper for these purposes is also relatively low. Among the highest value-added copper materials are oxygen free cryogenic grades of copper, which require a much higher level of know-how to produce and process without losing its low temperature conductivity, and do not have any good alternatives. Such grades of copper are used as stabilizers for superconductors], needed for large magnet windings. The amount of copper used is several times that of the actual superconducting material, which makes the demand for these grades of copper very significant.
The key quality measure of this copper is the residual resistivity ratio (RRR), which is defined as the ratio of the resistance at room temperature, 293 K, to the resistance at helium’s boiling point, 4.2 K. The resistivity of copper at room temperature is not very sensitive to impurities. Typical values of RRR of oxygen free, cryogenic grade copper range between 200 and 700. RRR values of several thousand have also been reported. Most of the earlier literature shows linear effects of impurities and temperature on resistivity. Every impurity in solution was said to reduce the electrical conductivity. However, some of the impurities separate out as oxides forming another phase, which then do not influence the conductivity of the solution. Oxygen, thus, even reduces the effects of the impurities and has a positive effect on RRR upto a certain limit. Impurities like zinc which form such oxide inclusions do not reduce the conductivity. An interesting attempt to empirically model the RRR is based on atomic sizes of the impurities, in which the extra resistivity contributed by each impurity is a sum of linear and exponential terms of the atomic volume size factor. The main merit of this model is that the coefficients used on the linear and exponential terms are constants, and do not depend on the impurity.
New techniques of nonlinear modelling which have come up within the last ten-twelve years, have permitted the development of highly sophisticated nonlinear empirical models, without knowing the type and severity of nonlinearities present in the relations. It is also possible to combine process knowledge with this kind of empirical models, which often leads to better models. These new techniques have opened up new possibilities. It is now possible to develop accurate and reliable nonlinear models relating composition with material properties like RRR or tensile strength from production data, when the production data has sufficient information content.
There are hardly any processes or materials in this world which have absolutely linear characteristics. It is therefore wise to treat the nonlinearities rather than ignore them. To treat the nonlinearities, one can use new techniques of nonlinear modelling, like artificial neural networks. The proponents of linear techniques draw on their simplicity and the possibility of adding nonlinear terms in linear regression. Often this is not done, and is not efficient even if it is done. Nature does not follow the simplicities that we try to fit it in, using linear techniques.
Neural networks, on the other hand, have the so-called universal approximation capability which make them suitable for most function approximation tasks we come across in process industries. The user does not need to know the type and severity of nonlinearities while developing the models.