научная статья по теме THERMAL DIFFUSIVITY, VISCOSITY AND PRANDTL NUMBER FOR MOLTEN IRON AND LOW CARBON STEEL Физика

Текст научной статьи на тему «THERMAL DIFFUSIVITY, VISCOSITY AND PRANDTL NUMBER FOR MOLTEN IRON AND LOW CARBON STEEL»

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ТРОЦЕНКО

THERMAL DIFFUSIVITY, VISCOSITY AND PRANDTL NUMBER FOR MOLTEN IRON AND LOW CARBON STEEL

© 2013 D. Ceotto

Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica (DIEGM) Université degli Studi di Udine, Udine (ITALY) E-mail: diego.ceotto@uniud.it Received February 21, 2012

Abstract—This paper investigates fundamental properties of liquid iron as well as molten low carbon steel. Scope of this work is to formulate simple, even if approximated, equations useful to simulate heating and solidification processes. Attention is paid first to thermal diffusivity which is determined using known empirical equations (density, free energy and electrical resistivity) and well-accepted relationships like the Wiedemann-Franz-Lorentz law. Then, viscosity is estimated from a multiple linear regression analysis of available data. In such a way an estimation of Prandtl number is calculated. The proposed equations are tested by comparing with existing data. Such modelling may be useful for simulations in the field of process metallurgy, in steel-making industry and also in some more recent manufacturing technologies such as selective laser and electron beam melting, when experimental values of the physical parameters are missing.

THERMAL DIFFUSIVITY "VISCOSITY AND PRANDTL NUMBER

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INTRODUCTION

Metal properties are deeply influenced by primary solidification processes, even if further plastic processing is used. It is then important to evaluate material properties evolution by means of experiments and, or, simulation. Solidification processes related to metal industry have been extensively studied in a scientific manner since the works of Chalmers and Flemings and there is currently very extensive literature available in several textbooks (e.g. Flemings [1], Campbell [2], Stefanescu [3]). Solidification is a very complex subject which only recently has been investigated on a physical ground by means of Sinchrotron radiation which involves different scales from the microscopic to the macroscopic and not only physical transformations, but may also be interrelated with chemical reactions which occur during solidification and dynamical interaction with heat transfer. The whole process has been studied in the past mainly by physical simulation of transparent organic materials and according to the specific solidification pattern (in pure metals, eutec-tic, peritectic and monotectic) different physical variables have shown to play some role [3]. In the classical theory of solidification of metals — which is related mostly in evaluating and controlling the undercooling in a way that has been opened by Turnbull [4] — viscosity has been one of the main parameters involved for a kinetic theory of solidification. In industrial processes which involve melting and solidification, thermal properties, and — related to the Fourier heat conduction equation — thermal diffusivity are key variables. As far as viscosity is concerned, it is generally well accepted that an exponential Arrhenius trend works for both pure metals and alloys. This behaviour heavily influences mass flow and heat transfer by inducing or limiting diffusion. An elementary approach, which is useful for rules of thumb and industrial simulation evaluation of casting processes, a fluent flow permits to obtain the shape desired to obtain the designed dimensions. While it is well known that viscosity of melt influences the solidification structure, elementary theories of micro- and macrosegregation, do not involve viscosity directly; this is likely due to a not thorough understanding of this physical parameter, which in turn has been thoroughly modelled in rapid solidification processes since the eighties of the previous century with special attention to the undercooled liquid metal regime. In industrial modelling, solidification is simply modelled as heat extraction process, for instance in continuous casting. In this case, the enthalpy exchange drives the phase change and determines the solid structure. Cooling process control (both in the mould and in the secondary cooling zone) is conducted by taking into account time and space variables, which are the key variables to control the cooling rate and temperature gradient. The parameters (i.e. in continuous casting the quantity of water circulating in the mould or the amount ofwater and water/air mist in the secondary zone) determinate solidification patterns which are influencing the solidification zones (in con-

tinuous casting primary equiaxed zone, dendritic zone and central zone) and also changing the quantity of phases and the microstructures (e.g. grain size). In the thermal properties related to the materials subject to solidification, along with latent heat, one of the most significant parameters in industrial process modelling is the thermal diffusivity. Any kind of simulation of solidification must take into account heating history, cooling processes and related heat transfer, mass flow, specific shapes and time, which may be faced with a thermal fluid dynamic approach in which mass flow and heat transfer are combined. In the macro-mass transport several adimensional parameters are proposed in this field [3]: Thermal Grashof Number, So-lutal Grashof Number, Prandtl Number, Schmidt Number; of these, we will draw our attention to the Prandtl number. This number is the simple ratio of kinematic viscosity to thermal diffusivity. Such are key variables for evaluating the interaction of mass movement with heat exchange in a conduction transfer

THERMAL DIFFUSIVITY: SEMI-EMPIRICAL EQUATION

Thermal diffusivity is an intrinsic property of a material under thermal rates stress. Fourier law models the temperature behaviour in function of time and space and it is applied by engineer when they investigate the temperature behaviour in a molten metal thermal flux during all kinds of solidification, cooling and heat transfer processes. The thermal diffusivity parameter is essential for describing solidification in many industrial processes as continuous casting and foundry factory, but also during heating in blast furnace, in EAF (Electric Arc Furnace) and in metal powder metallurgy. This important parameter is not well known even for solid iron dilute alloys i.e. steel [5]. However, molten metals diffusivity values and its behaviour are very relevant to understand and consequently simulate physical processes at high temperature. The purpose of this work is to give in primis a simple, even if approximated, model of thermal diffusivity at high temperatures for molten iron and low carbon steel. Such a simple, semi-empirical approach may be then improved by comparison to published data and data that will be available in the future. Basically, not too many measures are available for molten iron (e.g. Monaghan [6], Taluts [7], Zagrebin [8] and Nishi [9]) and, typically in a limited melting temperature range. In particular this is due to the technological difficulties for conducting experiments at so high temperatures. It is well established that by combining a theoretical approach with empirical functions one can simulate high temperatures regime and obtain thermal diffusivity values in a wide range. Due to these severe experimental limitations, it is a common practice in simulation modelling on an industrial ground to use extrapolation of data obtained in solid state. Such a procedure could be questioned as thermal properties — undoubtedly entropy — do change in solid to liquid transformations.

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2 -1 a, mm2s 1

5.76 5.72 5.68 5.64 5.60

5.56

1800 1840 1880 1920 1960 2000

T, K

Fig. 1 Thermal diffusivity for pure liquid iron.

Thus, a specific thermal diffusivity model, related to a simple but more physical ground could be more useful and specific in this field.

When pressure is constant, thermal diffusivity a is defined as in equation (1).

k Pcp

, (m 2s)■

(1)

cp = -T

д 2Gl

dT

2 '

(3)

Finally, free energy GL equation for molten Fe-C alloy is taken by Turkevich [11] study and it is based on the regular solution model. Equations used in the present work are as follows:

Gl = x

GFe +

Fe "Fe

+ XC°GC + RT (XFelnXFe + ^C^C) + XFe-^FeC oGFe = 13265 + 117.576T - 23.5143T ln T -- 4.39752 x 10-3T2 - 5.89269 x 10-8T3 -- 3.6751551 x 10-21T7 + 77358.5 x T_1,

°OcL = 105 + 146.1T - 24.3T lnT -

- 0.4723 x 10-3T2 + 2562600/T -

- 2.643 x 108T~2 + 1.2 x 1010T~3,

LFeC =-124320 + 28.5T +

+ 19300(xC - xFe) + (49260 - 19T)(xC - xFe)2. The thermal conductivity may be calculated with the Wiedemann-Franz-Lorentz law (temperature in K). For example, Nishi [9] experimentally verified and proposed the following law

2.445 x 10-8T P,

k =

In order to calculate it, we will consider different physical variables separately. Then, the behaviour of the following independent variables: k — thermal conductivity (W m-1 K-1), p — density (kg m-3) and cp — heat capacity (J kg-1 K—1) will be taken into consideration.

Density of liquid steel can be calculated by an equation presented by Miettinen [10] after a Jablonka et al. data analysis. A simplified version will be used for our purpose. In this version density is approximated to a function only of temperature, T (°C), and carbon content, xC (wt.%).

p = 8319.49 - 0.835T + (-83.19 + 0.00835T)xC. (2)

Then heat capacity is calculated deriving the free Gibbs energy for the liquid steel phase with the according to the well-known equation (3).

where pe is the electrical resistivity. While thermal conductivity is difficult to measure, electrical resistivity is not. For pure iron, it has been calculated by Kita et al. [12] (pe is expressed in ^flcm and T is considered in °C)

p e = 0.0154T + 112.3.

Miettinen [10] already reported that "at high temperature the solute effect upon the thermal conductivity is known to be small". For this reason, in the present paper, the thermal conductivity calculated in this way will be considered valid also for carbon steel. Combining all th

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