M. J. Peet , H. S. Hasan and H. K. D. H. Bhadeshia1
Published in the International Journal of Heat and Mass Transfer Vol. 54 (2011) page 2602-2608 doi:10.1016/j.ijheatmasstransfer.2011.01.025 Abstract A model of thermal conductivity as a function of temperature and steel composition has been produced using a neural network technique based upon a Bayesian statistics framework. The model allows the estimation of conductivity for heat transfer problems, along with the appropriate uncertainty. The performance of the model is demonstrated by making predictions of previous experimental results which were not included in the process which leads to the creation of the model. Keywords: Thermal Conductivity, Steel, Bayes, Neural Network, Heat treatment, Mathematical models, Physical properties, Temperature, Commercial alloys, Matthiessen’s rule
Prediction of thermal conductivity of 2steel 1∗
There are many situations in design or in process modelling where it would be useful to know the thermal conductivity of the steel being used, and how it would change as a function of temperature. With the lack of any quantitative model the usual recourse is to look for a similar composition contained in published tables of data [1, 2, 3]. However, in the absence of a quantitative model it is not possible to assess the validity of this procedure. Thermal conductivity controls the magnitude of the temperature gradients which occur in components during manufacture and use. In structural components subjected to thermal cycling, these gradients lead to thermal stresses. During heat treatment the conductivity limits the size of components that can be produced with the desired microstructure, since transformation depends on cooling rate and temperature. A suitable model of thermal conductivity should help to improve the design of steels and understanding of heat treatment, solidiﬁcation and welding processes, design of steel structures and components, and prediction of thermo–mechanical fatigue. The original motivation of the authors was to estimate thermal conductivity of a range of steels to assess the validity of lump–theory approximation in the design of a novel probe used to measure heat transfer coeﬃcient [4, 5]. The model presented here was developed using neural network software to model the thermal conductivity http://www.msm.cam.ac.uk/phase-trans/ Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge, CB2 3QZ, UK. 2 Department of Electromechanical Engineering, University of Technology, Baghdad, Iraq. 1
as function of composition and temperature. Subsequently the model was combined with experimentally determined heat transfer coeﬃcient in a ﬁnite–element scheme to predict the instantaneous temperature proﬁle in a cylinder of steel during quenching . Calculated cooling curves and transformation kinetics were used to calculate the resultant distribution of hardness using a quench factor .
In metals electrons provide an additional contribution to the thermal conductivity, which can therefore be much greater than in non–metals in which only phonons contribute. Interactions between phonons and electrons determine the thermal conductivity in a pure metal. In alloys additional lattice distortions by alloying elements cause similar disturbances. Both relying on electron transport, thermal and electrical conductivity behave analogously, and in the ideal case are related by the Wiedermann–Franz law . At temperatures above the Debeye temperature phonons begin to have wavelengths similar to the inter–atomic spacing and increasingly scatter electrons. For iron this is 398±9 or 418±4 K from X–ray measurements, or calculated to be 467 K from the elastic–constants . The maximum thermal conductivity occurs at cryogenic temperatures. Due to phonon...