Strojniški vestnik - Journal of Mechanical Engineering 56(2010)2, 84-92 Paper received: 19.11.2009 UDC 621.78.084 Paper accepted: 19.01.2010 Extension of Isothermal Time-Temperature Parameters to Non-isothermal Conditions: Application to the Simulation of Rapid Tempering Tamas Reti 1* - Imre Felde 2 - Janez Grum 3 - Rafael Colas 4 -Gustavo Sanchez Sarmiento 5 -Augusto Moita de Deus 6 1 Széchenyi István University, Gyor, Hungary 2 Bay Zoltan Institute for Material Science and Technology, Hungary 3 University of Ljubljana, Slovenia 4 Universidad Autonoma de Noevo Leon, Mexico 5 Universidad de Buenos Aires, Argentina 6 Instituto Superior Técnico, Portugal A phenomenological method for constructing non-isothermal, generalized time-temperature parameters (GTT parameters) is presented. An analysis of the relationship between the various traditional isothermal time-temperature parameters has verified that the generalized Dorn-parameter is regarded to be the sole non-isothermal complex parameter to which a rigorous, physically well-founded interpretation is attributed. Possible applications of GTT parameters are illustrated by examples concerning the prediction of hardness change in quenched steels during rapid tempering treatments. ©2010 Journal of Mechanical Engineering. All rights reserved. Keywords: kinetic function, diffusion, steels, laser cladding, secondary hardening 0 INTRODUCTION Generalized time-temperature parameters permitting a numerical characterization and prediction of transformation processes occurring at varying temperatures are the natural extensions of the well-known isothermal complex parameters [1] to [5]. In this paper, a general phenomenological model and method for constructing non-isothermal generalized time-temperature parameters (GTT parameters) are described. The main advantage of using GTT parameters is that non-isothermal processes of various types may be described and compared on the basis of numerical criteria. Moreover, it will be verified that the generalized Dorn-parameter is regarded to be the sole non-isothermal complex parameter to which a rigorous, physically well-founded interpretation is attributed. Application of GTT parameters will be demonstrated on examples concerning the prediction of the change of as-quenched hardness in steels during rapid tempering treatments. In practice, a non-isothermal tempering of such type can be performed by induction, electron beam or laser heating. 1 ISOTHERMAL TIME-TEMPERATURE PARAMETERS Traditional isothermal time-temperature parameters (ITT-parameters) incorporate the combined effect of the time t and the temperature T on the kinetics of micro-structural transformation processes occurring in alloys. A conventional ITT-parameter denoted by P can be formulated in the following general form: P = P(t, t ) = h0(T )t MT) (1) In Eq. (1) h0 and h1 are appropriately selected strictly monotone and continuous functions, and T is the so-called normalized temperature defined as TC + e, =^. (2) In Eq. (2) TC is the temperature [°C], e1 and e2 are the positive scale factors. (In the majority of cases e1= 273 and e2= 1. This implies that T will be identical to the absolute temperature in Kelvin.) The well-known types of ITT-parameters are the Hollo mo n-Jaffe [1] and the Dorn parameter [2] and [3]. The Hollomon-Jaffe Corr. Author's Address: Széchenyi István University, Egyetem tér 1. H-9026 Gyor, Hungary, reti@sze.hu 84 parameter PH (defined for case of t > 0) is formulated as: PH = T (C + lnt) = ln(tTeCT ) (3) where C is a positive constant, while the Dorn parameter, PD is: PD = t eXPl - .Qdl RT (4) where QD is stands for the apparent activation energy and R is the universal gas constant. In order to characterize and compare various types of ITT-parameters an equivalence relation by which ITT parameters can unambiguously be classified into disjoint groups is defined. By definition, ITT parameters PA and PB are called equivalent, (denoted by PA ~ PB), if there exists a strictly monotone and continuous function U = U(x) for which: Pb = U (PA ) (5) is fulfilled. Since the inverse Um of the function U exists, from this it follows that Pa = U'"(Pb). This implies that the equation above defines an equivalence relation, which satisfies the following conditions: Pa ~ Pa (identity) if PA ~ PB then PB ~ PA (Symmetry) if Pa ~ Pb and Pb ~ Pc then PA ~ PC (Transitivity). The importance of this equivalence relation lies in the following fact: All ITT (6.1) (6.2) (6.3) parameters belonging to the same equivalence classe are considered as "equivalent parameters" in the sense that they characterize the progress of the transformation process in an identical manner. For example, ITT parameters denoted by PR, PS and PV in Table 1 are equivalent, because in all three cases it is possible to construct a strictly monotonous function U=U(x) by which the mapping represented by Eq. (5) can be performed. It is worth noting that the Hollomon-Jaffe parameter PH belongs to another class of equivalence. This means that PH is not equivalent with parameters PR, PS or PV. 2 THE GENERALIZED TIME-EMPERATURE PARAMETERS There are several techniques for constructing GTT parameters of various forms [2] to [5]. In order to obtain "well-behaved" non-isothermal parameters, a GTT parameter Pg needs to have the following form: P = Z Jh (t,, T (t, ))dtu (7) where Z and H are appropriately selected strictly monotone and continuous functions. From this definition it follows that Pg is considered as a functional of temperature function T = T(t) characterizing the thermal history of the process. As an example, in Table 1, the generalized versions of three equivalent ITT parameters (PR, PS and PV) are also given. It is obvious that under isothermal conditions, GTT parameters denoted by PR,g, PS,g and PV,g will be identical to PR, PS and PV, respectively. Table 1. Equivalent isothermal time-temperature parameters PR, PS, PV and their generalized non-isothermal versions (Bo, Bj, B2 and B3 are selected constants)_ Equivalent time-temperature parameter Generalized time-temperature parameter PR = t exp A T (Bo > 0) PR,g =J eXp A T dt. Ps = a + b^lg (t) ( Bi* b2 < 0 ) Ps , g = B2lg t _Bi_ J10 B2Tdtu 0 Pv = T + !n (t ) (B3 < 0) PV, g = ln J exp B dt,, Table 2. Three different generalizations of the isothermal time-temperature parameter PL. Isothermal time-temperature parameter Generalized time-temperature parameter PL = 10CTtC2T TC [°C] + 273 where T = —-—-- 1000 P(1) g = C2 |T10C>TtC2T-1dt 0 P(2) = r L ,g C I —T-1 f T10TtC' dt C ^ 1 0 C1 P(3) = 1 L,g ~ f T10 C' tT-1dt 0 C2 3 A SIMPLE METHOD FOR CONSTRUCTING GTT PARAMETERS As it has been previously mentioned, there are several possibilities for generating GTT parameters from the same ITT parameter. In Table 2, three GTT parameters of different types are included. All of them are constructed from the same ITT parameter denoted by PL. This implies that at constant temperatures the three GTT parameters in Table 2 will be identical to the isothermal parameter PL. Below a simple and general method for generating GTT parameters will be outlined. This technique is based on following concept: Let us assume that the kinetic function describing the isothermal transformation process is known, and given in the form: G(y) - E(T)tF(T> = 0 (8) where y = y(t) is the kinetic variable of the transformation (i.e. the extent of completion of process after laps of time t), G, E and F are properly selected functions, and y(0) = y0 represents the initial value for t = 0. As a first step, from Eq. (8) the isothermal kinetic differential equation of the process is constructed: dy = E(T )F (T) tF (t ) - dt dG . (9) dy As a second step, by solving the differential Eq. (9), the isothermal kinetic function is rewritten in the form: y(t) = Gin jjE(T)F(T)tFu {T)-idtu + G(yo)J . (10) Finally, starting with Eq. (10), the required GTT parameter Pg is defined as follows: P = Z {\E (T) F (T (T )-1 T=t (,) dtu } . (11) Since in Eqs. (8) to (11), G and Z are strictly monotone continuous functions, this implies that their inverse functions Gin and Zin exist. By introducing GTT parameters, the description of the transformation processes occurring under non-isothermal conditions becomes simpler. This is due to the fact that by means of GTT parameter (11), the corresponding non-isothermal kinetic function can be written in the following general form: y(t) = Gin {Zin(Pg) + G(yo)} . (12) It is easy to see that if the temperature is constant, Eqs. (12) and (10) will be identical. 4 THE GENERALIZED DORN PARAMETER Below it is demonstrated that the application of the generalized Dorn parameter has some special advantages. Additionally it will be shown that the generalized Dorn parameter is in fact considered as the sole theoretically well founded GTT parameter. Let us assume that identity F(T) = 1 is fulfilled for function F in Eq. (8). Hence, the corresponding isothermal kinetic function given by Eq. (8) can be reduced to: G(y) - E(T)t = 0. (13) In this case the GTT parameter represented by Eq. (11) is simplified to: P„ = Z j E (T (tu ))dtu (14) In many reactions it is supposed that the Arrhenius-law controls the temperature-dependent rate of the isothermal transformation processes. This implies that E(T) can be written as: E(T ) = exp I- RQt (15) where k0 is the pre-exponential factor, and QA is the activation energy of the process. Now, by choosing function Z in the form of Z(x) = x/k0, and by substituting formula (15) into Eq. (14), we obtain the generalized Dorn parameter PD,g: PDg =j exp (- Qa. RT dt . (16) It can be concluded that there is a strong relationship between the Dorn parameter and the Arrhenius-formula. One of the most important properties of the Dorn parameter is that its value is completely determined by the temperature spectrum. If temperature is a continuous function of time, the temperature spectrum is characterized by the so-called temperature amplitude density function g(T) [3]. This is defined in a given temperature interval [Tmin, Tmax] where Tmm and Tmax stand for the minimum and maximum temperature values, respectively. From the previous considerations it follows that if the temperature amplitude density function g(T) is known, the generalized Dorn parameter can be transformed into the form pdg =t j exp (- R-^JgT)dT . (17) As an example, in Fig. 1 three different temperature functions and their corresponding temperature amplitude density functions denoted by ga(T), gb(T) and gc(T) are illustrated. Considering the sinusoidal temperature amplitude density function gc(T), it can be verified that the generalized Dorn parameter (consequently the predicted progress of non-isothermal transformations) does not depend on the frequency of temperature oscillations. It should be noted that the Dorn parameter represented by Eq. (17) can be simply computed by using numerical integration. 5 CORRESPONDENCE BETWEEN THE GENERALIZED DORN PARAMETER AND THE DIFFUSION BASED TRANSFORMATIONS It is important to emphasize that there is a strict relationship between the generalized Dorn parameter and the kinetics of non-isothermal diffusion processes (diffusion based transformations). To verify this, let us consider the Fick's second law: dn= V[DVn] dt (18) describing the change of concentration n =
\
PF, g =j exp
Qf
dt
(23)
RT (tu )y
by definition. Hence, provided that the isothermal solution n =