Estimation of Temperature Dependences of Specific Heat Capacity of Low- Alloy Steels

Transcription

1 Estimation of Temperature Dependenes of Speifi Heat apaity of Low- Alloy Steels Е.А. Protopopov, S.S. Dobrykh, A.А. Protopopov Tula State University, 9 Lenin pr. Tula Russia. Abstrat The existing empirial polynomial temperature dependenes of the speifi heat of steels do not apply to the temperature range from 973 K to A 3. The paper desribes the solution of the problem of obtaining an equation that desribes the temperature dependene of the speifi heat of low-alloy steels in the temperature range from 973 K to A 3. The equation obtained is a saled equation for the speifi heat in the viinity of the urie temperature. The solution of the equation is ross-linked with known empirial polynomial dependenies. A omparison of the results of alulations on the saled equation for the speifi heat in the viinity of the urie temperature with known experimental data is presented. It is shown that the average error in estimating the speifi heat in the viinity of the urie temperature is about 1.5%. To evaluate the urie temperature of low-alloy steels, an equation is proposed for the first time that allows this estimate to be performed depending on the hemial omposition of the steel. The equation for estimating the urie temperature of low-alloy steels is based on the Vonsovskii equation for alulating the urie temperature of binary ferrimagneti alloys. Approximation of the exhange integrals in the Vonsovskii equation was arried out approximately using the onentration dependenes of the magneti onversion temperature aording to the diagrams of the state of the irondoping element systems. Examples of a number steels show satisfatory agreement of the alulated and experimental values of the urie temperature. Using a saled equation for the speifi heat in the viinity of the urie temperature, together with empirial polynomial temperature dependenes for the speifi heat of steels, it is possible to alulate the polytherms of the speifi heat of low-alloy steels in the temperature range from 300 K to the solidus temperature. Keywords: low-alloy steel, speifi heat apaity, estimated polyterm of heat apaity, estimation of urie temperature of low-alloy steel. INTRODUTION Speifi heat apaity in onjuntion with other thermophysial properties of the material affets the auray of the results of omputer simulations in alulations of temperature fields in produts sold in pakages of applied programs AE (omputer-aided engineering), providing automation of engineering alulations, implementation of imitating modeling of tehnologial proesses and physial phenomena, the optimization of produts, et. The temperature dependene of the speifi heat of low-alloy steel and tehnially pure iron has a pronouned maximum at the point of magneti transformation. For arbon steels urie point temperature is independent of the arbon ontent and is 770 [1-4]. This allows to approximate experimentally obtained polytherms of heat apaity of arbon steels by polynomial dependenes. This approah is used when arrying out the appliation of engineering alulations [5, 6]. For alloyed steels in AE uses a table setting of known experimental data or methods for prediting thermal properties, inluding speifi heat [7]. However, to date, the equation for alulating the urie temperature of low-alloy steels has not been developed. Therefore, the use of polynomial models gives an error of up to 50 K in the position of the maximum speifi heat of low-alloy steels [5]. The purpose of this work is to solve the problem of refining the existing method of alulating the temperature dependene of the speifi heat apaity of steels using polynomial models by developing and taking into aount when prediting the heat apaity of the equation that estimates the urie temperature in low-alloy steels, and the saled equation for the temperature dependene of the speifi heat apaity of steel in the viinity of the urie temperature. DESIGN PROEDURE (METHOD OF ALULATION) Speifi heat apaity of all low-alloy steels is determined by the empirial polynomial dependenies [8], in kj/(kg K): 4 1 0,443 5, (1) 6, , in the temperature range K with an error of less than.1% and 4 8, T 7, ( T ) 1,136 () 7,5410 with an auray of up to 1.7% in the temperature interval Т Ас3... T S, where Т, Т Ас3, T S is temperature of, respetively, the urrent, point А С3 and solidus (K). There are a number of equations for prediting А С3 and T S in steels. Based on the omparative analysis given in [9], it an be assumed that for low-alloy steels the best predition auray of А С3 is provided by the equation: T A Mn 39Si 31Ni 55Mo 41V (3) 3 where the symbols of the hemial elements represent the mass onentration in the steel

2 The solidus temperature of low alloy steel an be determined by the equation [10] (K): 1811 T S ,5Si 6,5Mn 500P 700S 5.5Al r 11,5 Niif С 0,09 0,5Si 6,5Mn 500P 700S 5.5Al r 11,5 Niif 0,09 С 0,17 187,5 0,5Si 6,5Mn 500P 700S 5.5Al r 11,5 Niif С 0, 17 (4) The least error in determining the liquidus temperature T L low alloy steel gives the equation [11] (K): T L Si 5Mn 1,5r 4Ni 5u Mo V 1,5W Al o 10Ti 150B 6r 30P 5S 80O 90N 1300H For binary ferromagneti alloys, the urie temperature an be determined by the equation Vonsovskogo [1]: A A n A A A T A1 n (6) 1 where is the oordination number, k is the Boltzmann onstant; А 1, А and А 1 are the exhange integrals, respetively, for the neighborhoods of type A A, B B, A B (A and B - are atoms of different omponents of the alloy); n is the onentration of atoms of grade B. Equation (6) an be written in the form: A F T 1 where F na1 A1 n A1 A A1 is the funtionality. For iron-based alloys of (7), if n 0 it is follows that the urie temperature of the iron T A. Given that for iron 1 k T 1043 K, for binary iron alloys equation (7) takes the form: T 1043 F. (8) The numerial values of the funtional F at different onentrations of alloying elements in the iron based binary alloy matrix an be estimated approximately from the onentration dependenes of the magneti transformation temperature aording to the diagrams of the state of the ironalloying element systems. When using the expression (8) for a multiomponent system, i.e. for low-alloy steel, it is neessary to take into aount the ontributions to the formation of Т с of various funtional F of alloying elements and impurities. In the first approximation, molar frations of these hemial elements an be used for this purpose. Taking into aount the above, the data given in [1-4, 13], based on the expression (8) the equation for prediting the urie temperature of low-alloy steel, depending on the ontent of the hemial elements determining this temperature, (K):. (5) (7) 7,5Mn Mn XMn 0,1Si 1,5Si XSi 36Ti XTi 4Al XAl 5 Mo 0,Mo XMo 0,5r Xr 10,5V 0,4V XV 14o 0,17o Xo 0,8r 0,4r Xr 4,8 Ni 0,6Ni 1 T XB i i 11u Xu 100N XN P XP 4As XAs 1043 XNi where XBi is the sum of the molar frations of the i hemial elements present in the steel, whih are onsidered in equation (9); B i is the symbol of the i th hemial element in (9). Equation (9) reflets the different onentration dependene of the temperature of the magneti transformation on the respetive diagrams of binary systems of iron-alloying element with orrelation oeffiients R > 0.95 and the ase when Mn,5 %; Si 3 %; Ti,5 %; Al 5 %; Mo 5 %; r 5 %; V,5 %; o 5 %; r 9 %; Ni 4 %; u 1 %; N 0,05 %; P 0,5 % and As 0,5 %. Experimental values of urie temperature for a number of steels (hemial omposition of steels is given in table. 1) and the results of alulations on the equation (9) are presented in table.. Table 1. The hemial omposition of the steels, mass % Element Low-alloy steel [14] Steel grade Modified steel [15] 9r- 1Mo [16] (9) 100r6 [17] r 1,50 8,55 8,44 1,40 Mo 0,50 0,88 0,94 V 0,10 0,1 0,001 Nb 0,10 0,08 0,5 0,10 0,10 1,0 Mn 0,0 0,51 0,46 0,31 u 0,18 0,11 0,10 Si 0,0 0,3 0,49 0, N 0,035 Ni 0,15 0,17 0,086 P 0,01 0,008 0,019 S 0,005 0,049 0,00 Ti 0,00 Al 0,007 0,011 r 0,001 o 3,00 B 0,

3 Table. Experimental and alulated values of the urie temperature, о С urie temperature Experimental data Low-alloy steel Steel grade Modified steel 100r6 790, 1 [14] 741 [15] 7405 [16] 769 [17] alulation data (8) 790,0 741, 741,5 768,4 Notes: 1 maximum temperature on the differential thermal analysis urve; the value is determined by the shedule given in [17]. As an be seen from on the table, the alulated urie temperature values are lose enough to the experimental data, whih indiates the possibility of using the equation (9) in determining the urie temperature of low-alloy steels. The heat apaity at urie temperature has a jump [18]: a T (10) b where a, b are the oeffiients of the deomposition of the thermodynami potential in Landau's theory by degrees of the order parameter. Taking into aount that in the onsidered method of alulation of speifi heat apaity at heating of various low-alloyed steels is ounted from one level defined by the equation (1), and the ontribution to the total heat apaity at T is predominant, then, on the basis of (10) it an be onsidered that the speifi heat apaity of low-alloy steel at T is approximately proportional to T, i.e. is given by the expression: k T (11) where k is the oeffiient. From (11) it follows that for iron at T 1043 К and 3 1,5 kj/(kg K) [19] k 1,48310 kj/(kg K ). This k value is assumed, in the first approximation, for lowalloy steels. Saling of the heat apaity in the viinity of the urie temperature an be arried out by the equations: T T 1 T T T 1 T 1 at T T at T T (1) where 1 and are oeffiients. Using the least squares method, it is obtained 1 0, 36 and 1 0, 36 that the best agreement of equation (1) with the experimental data is provided. The rosslinking of the solution of equation (1) and dependenies (1) and () is arried out by equating (1) and (1), and () and (1) to find the temperatures at whih these equations are performed. The heat apaity of various liquid steels is pratially independent of the temperature and is in a narrow range: from 0.75 to 0.85 kj/(kg K), and the spread of values does not go beyond the experimental errors, and the average value oinides with the heat apaity of pure iron, i.e kj/(kg K) [0]. Therefore, in the developed tehnique, the heat apaity of the liquid low-alloy steel is assumed to be equal to, kj/(kg K): 0,85. (13) L Assuming a linear hange in the speifi heat of steel in the interval (T S... T L) you an write the equation 4 L S S 3 3 TL TS T. (14) Solutions of equations (1) - (14) determine the temperature dependene of the speifi heat of low-alloy steels. RESULTS AND DISUSSION Solutions of the system of equations (1) - (14) are shown in Fig Sine low-alloy steel an be onsidered a dilute solution based on iron, the solution of equations (1) - (14) should be valid for pure iron. For Fig. 1 experimental data [19] and the alulated polytherm of speifi heat of iron are presented. Figure 1. Experimental data ( [19]) and the results of alulating the polytherm of the speifi heat of pure iron aording to equations (1) (14) Presented in Fig. 1 experimental data are reommended and obtained in [19] by statistial proessing of 41 iron speifi heat polytherms experimentally obtained by different researhers. As seen in Fig. 1 ompliane of experimental and alulated results an be onsidered satisfatory. For fig. -4 results of 13538

4 alulations on equations (1) (14) and experimental data for a number of steels are given. The average relative standard deviation of the alulated speifi heat polytherms from the experimental values for all steels and iron onsidered in the viinity of the urie temperature (Fig. 1-4) is about of 1.5%. Figure. alulated polytherm of speifi heat of steel 15 and experimental data: [0] ONLUSION In addition to the known empirial polynomial equation of temperature dependene of speifi heat apaity of low-alloy steels (1), (), valid in temperature ranges, respetively, K and А с3... T S, for the temperature range from 973 K to А с3, a saled equation of temperature dependene of the speifi heat of steel in the viinity of the urie temperature (1) is proposed. Using the obtained equation for alulating the urie temperature depending on the hemial omposition of steel (9), a refined method for alulating the speifi heat polytherms of low alloy steels aording to the equations (1) - (14) is developed, whih provides an overall average error in prediting the speifi heat apaity not exeeding the error of the empirial equations (1) and (). Figure 3. alulated polytherm of the speifi heat of steel 30Nir14 and experimental data: [0] Figure 4. alulated polytherm of speifi heat of 14MoV6-3 and experimental data: [1] REFERENES [1] Lyakishev N.P., 1996, Diagrams of the state of double metal systems: referene book, Vol. 1, Mashinostroenie, Mosow, Russia. [] Lyakishev N.P., 1997, Diagrams of the state of double metal systems: referene book, Vol., Mashinostroenie, Mosow, Russia. [3] Massalski T. B., Okamoto H., 1986, Subramanian P. R. Binary alloy phase diagrams. Vol., Ohio: Amerian Soiety for Metals. [4] Massalski T. B., Okamoto H., 1986, Subramanian P. R. Binary alloy phase diagrams. Vol. 1, Ohio: Amerian Soiety for Metals. [6] EN , 005, Euroode 3: Design of steel strutures. Part 1-. European ommittee for standardization. [7] Sidorov A., 015, «JMatPro software pakage for modeling properties of steels and alloys», SAPR i grafika, No 4(), pp [8] Aronov M.A., Krzhizhanovsky R. E. and Nemzer G.G., 1988, «Thermophysial properties of steel in the modes of heating and ooling», Kuznehno-shtampovoye proizvodstvo, No 8, pp [9] Trzaska J., 016, «alulation of ritial temperatures by empirial formulae», Arh. Metall. Mater., 61(B), [10] Xie. and Yang J., 015, «alulation of Solidifiation Related Thermophysial Properties of Steels Based on Fe Pseudobinary Phase Diagram», Steel researh international, 86(7),

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