1、1. IntrodctionStainless steels play an important part in modern livingsince they have good corrosion resistance and a pleasingappearance. There are several types of stainless steel (typi-cal compositions are given in Table 1):3 series contain Cr and Ni and usually contain Mo; theyhave a metastable,
2、austenitic structure4 series contain Cr (but no Ni since this stabilises theaustenite) and have a ferritic structure but martensiticphases can be formed in some cases (e.g. 410) when oil-quenched or air-cooled 6-series are precipitation-hardened alloys and tend tocontain Cr and Ni in lower amountsMa
3、thematical modelling has become an established toolfor improving process control and product quality. How-ever, it has been shown that accurate thermophysical prop-erty data are needed for reliable predictions of defects1)orimproved product quality (e.g. Weld profiles in TIG/GTAwelding2,3). This stu
4、dy seeks to provide reliable thermo-physical property data for all types of stainless steels. The principal objective of this work was to establishequations which provide reliable values for thermo-physicalproperties for the full range of stainless steels. The generalapproach adopted here makes use
5、of fact that the thermo-physical properties of Fe, Cr and Ni are very similar; thusdifferences in Ni% and Cr% in different stainless steels donot have a significant effect on the property value.It should be noted that ferritic and martensitic (i.e. 4-se-ries alloys) undergo a magnetic transition and
6、 subsequentlytransform to austenite. The values of properties, such asdensity and enthalpy, are little affected by these transitionsbut their temperature coefficients, i.e. heat capacity (Cp)and thermal expansion coefficient (a), respectively, alongwith the thermal diffusivity do vary appreciably. C
7、on-sequently, in these cases, it is not always possible to giveone equation to cover all the properties for all types ofstainless steel.Thermal and electrical conductivity (or resistivity) valuesare affected by the microstructure of the solid sample whichis, in turn, dependent upon the heat treatmen
8、t and the coldwork given to the alloy. In these cases the thermal conduc-tivity and electrical resistivity refer to samples with themaximum conductivity or minimum resistance. Propertyvalues based on the various recommended equations aregiven at the end of the paper in Tables 3 and 4. The liquidus t
9、emperatures, Tliq, of the various types ofstainless steels were estimated using the coefficients recom-mended by Howe.4)The calculated values are shown inTable 1. It has been assumed in Table 3 that liquidus tem-perature of 3- and 6-series alloys occurs at 1 723 K and for4-series alloys at 1 773 K.2
10、. Density (r), Thermal Expansion Coefficient (a)2.1. DatabaseThermal expansion data have been reported by Bogaardet al.5)for austenitic 304 steel and ferritic 430 stainlessISIJ International, Vol. 44 (2004), No. 10, pp. 166116681661 2004 ISIJEquations for the Calculation of the Thermo-physical Prope
11、rtiesof Stainless SteelKenneth C. MILLS, Yuchu SU, Zushu LI and Robert F. BROOKS1)Department of Materials, Imperial College, London, SW7 2BP, UK. E-mail: , yuchu.suimperial.ac.uk,z.liimperial.ac.uk 1) National Physical Laboratory, Teddington, TW11 OLW, UK. E-mail: Rob.Brooksnpl.co.uk(Received on May
12、 6, 2004; accepted in final form on July 8, 2004)Equations have been derived to calculate values of the thermophysical properties of all stainless steels fortemperatures between 300 and 1 800 K (austenitic 3 series, ferritic-4 series and precipitation-hardened 6-se-ries alloys). Values of the follow
13、ing properties are given in both figures and tables: density (r), thermal ex-pansion coefficient (a), heat capacity (Cp), enthalpy (HTH11002H298), thermal conductivity (l) and thermal diffusivi-ty (a), electrical resistivity (R), viscosity (h) and surface tension (g).KEY WORDS: stainless steels; the
14、rmo-physical properties; density; surface tension; viscosity; heat capaci-ty; enthalpy; thermal and electrical conductivity.Table 1. Typical chemical compositions (in mass%) of exam-ples of 3-, 4-series and 6-series stainless steels andcalculated liquidus temperatures.steels. Density data for solid
15、304, 316, 410 and 446 stain-less steels are cited by Touloukian6)and by Mills7)for 304and 316. Density data for the liquid 316 alloy have been re-ported by McCormick and Brooks8)and for 430 by Li etal.9)Experimental uncertainties for measurements are proba-bly H110212% for density (r) and H1102110%
16、for the linear thermalexpansion coefficient (a).2.2. Analysis of Data Thermal Expansion Coefficient (a)Bogaard et al.5)report data which gives the followingequation for austenitic 304 steel:a (KH110021)H1100515.8H110010.6H1100310H110022(TH11002298) .(1)As can be seen from Fig. 1, at temperatures bel
17、ow1 000 K, the ferritic phase (in 430) has a lower a than thatfor austenite (in 304); above 1 000 K the austenite formedresults in a higher a value.5)Thus the thermal expansionfor ferritic alloys can be calculated using Eqs. (2) and (3).TH11005(298H110021 000 K)a (KH110021)H1100510.2H110010.6H110031
18、0H110022(TH11002298 K) (2)TH11005(1 000H110021 700 K)a (KH110021)H1100514.2H110012.4H1100310H110022(TH110021 000 K) .(3) Densities (r)Densities were first calculated by assuming that the fol-lowing equation applied:(4)where xH11005mole fraction and that 1, 2, 3 etc. refer to differentmetallic compon
19、ents. In actual fact this relation is morecorrect when using the molar volume, V, than for density,r, but the errors associated with the above assumption aresmall.Numerical analysis of the experimental density data gavethe following equations:Solid:rTH11005(79.6%Fe)H11001(78.3%Cr)H11001(85.4%Ni)H110
20、01(76.9%Mn)H11001(60.2%Mo)H11001(47.1%Si)H110020.5(TH11002298 K).(5)Liquid: rTH11005(69.4%Fe)H11001(66.3%Cr)H11001(71.4%Ni)H11001(57.2%Mn)H11001(51.5%Mo)H11001(49.3%Si)H110020.86(TH110021 823 K).(6)Equation (5) makes use of an average value for a; moreexact values can be obtained by removing theH110
21、020.5(TH11002298 K) term from Eq. (5) and dividing the remain-der of the equation by (1H110013a(TH11002298 K) where a can becalculated by Eq. (1) for austenitic alloys and by Eqs. (2)and (3) for ferritic alloys. However, values calculated usingEq. (5) with this procedure were found to lie within 0.5
22、%of those calculated by the more rigorous treatment.The calculated densities are in good agreement with theexperimental values (Fig. 2). It should be noted that the val-ues were calculated using Eq. (5). The liquid density valuesfor 4-series alloys shown in Fig. 3 and Table 3(b) may bevery slightly
23、low for temperatures between 1 723 and1 900 K. Vinet10)has reported a value of rH110056 980 kgmH110023for 316L at Tliqwhich is about 1% higher than the valuescalculated from Eq. (6), but lies within the band of experi-mental uncertainty.3. Heat Capacity (Cp), Enthalpy (HTH11546H298)3.1. DatabaseHeat
24、 capacity data for 304 and 316 stainless steels havebeen reported by Bogaard et al.5)and by Mills.7)The exper-imental uncertainty is probably about H110062% at temperaturesbelow 1 000 K and H110065% at temperatures in the range 1 000to 1 700 K. Ferritic alloys exhibit a magnetic transforma-tion6)lea
25、ding to a peak in the Cparound 1 000 K as can beH11005H11001H11001xx x12 3ISIJ International, Vol. 44 (2004), No. 10 2004 ISIJ 1662Fig. 1. Thermal expansion coefficient of stainless steels as afunction of temperature.Fig. 2. Density of solid stainless steels as a function of tempera-ture.Fig. 3. Den
26、sity of liquid stainless steels as a function of temper-ature.seen from Fig. 4; the peak Cpvalue and the temperature ofthe peak (Tpeak) show some deviation in the values collectedby Touloukian.6)The values shown in Fig. 5 were selectedbecause they had the same Tpeakas Tminin the thermal diffu-sivity
27、 data for ferritic steels.3.2. Analysis of Data The temperature dependencies of Cpand enthalpy(HTH11002H298) are usually expressed in the form of Eqs. (7) and(8), respectively. Kopps Rule is found to apply well for al-loys and the constants aH11032, bH11032, cH11032, can be estimated reason-ably acc
28、urately from relations such as Eq. (9)11); the sub-script ss refers to the stainless steel.CpH11005aH11001bTH11002c/T2(7)(HTH11002H298)H11005aH11032(T)H11001(bH11032/2)T2H11001cH11032/TH110021H11002dH11032 .(8)where dH11032H11005a(298)H11001(b/2)2982H11001c/298(9)where x is mole fraction and subscri
29、pts 1, 2, 3 etc. representthe different elements in the steel, respectively. Values of Cpestimated in this way tend to be ca. 23% lower thanexperimental values at temperatures below 800 K but agreewell at higher temperatures. Consequently, the Cpand(HTH11002H298) values for austenitic 3-series alloy
30、s, given inEqs. (7) to (9), were derived from a best fit of the experi-mental CpT relations. It is difficult to derive a relation forthe Cpof ferritic alloys between 500 and 1 100 K becauseboth the magnitude of Cpand the temperature of the peaktend to vary. However, the enthalpy, (HTH11002H298) valu
31、es arelittle affected and apply reasonably well to both ferritic andaustenitic alloys. Values for the liquid were derived fromthe experimental values reported by Chapman et al.12,7)The entropy of fusion, DSfusfor the steel can be estimat-ed from values of DSfusfor different elements using a simi-lar
32、 relation to that in Eq. (4).3.3. Results and DiscussionSolid:Cp(J KH110021kgH110021)H11005472H1100113.6H1100310H110022TH110022.82H11003106/T2.(10)(HTH11002H298)(JkgH110021)H11005472TH110016.8H1100310H110022T2H110012.82H11003106/TH11002156 000 .(11)Liquid:CpH11005800H1100650 J KH110021kgH110021(12)D
33、SfusH11005DHfus/TliqH11005160 J KH110021kgH110021(13)It can be seen from the results shown in Figs. 4 and 5 thatthere is good agreement between the values calculated withEqs. (10) to (13). The uncertainty associated with the calcu-lation of Cpand (HTH11002H298) is probably less than 5%.4. Thermal Co
34、nductivity (l)Thermal conductivity data for most solid alloys are sensi-tive to the microstructure. Consequently, these values are,in turn, dependent upon the thermal and mechanical histo-ries of the samples. These effects are relatively small in thecase of stainless steels because of their relative
35、ly low ther-mal conductivities (cf. Al alloys, where it can make a sig-nificant difference).4.1. DatabaseThermal conductivities have been reported by the follow-ing: Bogaard et al.5)for alloys, 304, 321 and 430; Chu andHo13)for 410 and 430; Bogaard14)for 3-series and 631 andMills7)for alloys 304 and
36、 316. The values cited below re-late to the maximum thermal conductivity. Values for theliquid phase are based on those reported by Mills.7)4.2. Analysis of Data 4.2.1. Method 1The thermal conductivity contains contributions fromboth lattice and electronic conductivities (Eq. (14) and theelectronic
37、conductivity at 300 K can be calculated from theelectrical resistivity, R by use of the WiedemannFranzLorenz relation (Eq. (15) where Lois constant with a valueof 2.445H1100310H110025.l300H11005lel300H11001llat300(14)(15)The resistivity, R300can be calculated from Eq. (16) usingthe values given in T
38、able 2 due to Mills et al.11)The unit ofR300is 10H110026W mH110021.(16)H11001H11001()wt% .R300 3RRR300 300 1 300 21 100H11005H11001(/ ) ( ) ( )wt% wt%300el oH11005H11003LTR300610axaxaxassH11005H11001H11001H1100111 2 2 3 3ISIJ International, Vol. 44 (2004), No. 101663 2004 ISIJFig. 4. Heat capacity o
39、f stainless steel as a function of tempera-ture.Fig. 5. Enthalpy (HTH11002H298) of stainless steel as a function oftemperature.where subscripts 1, 2, 3 refer to Fe, Cr, Ni etc.The lattice conductivities (llat300) can be calculated in asimilar manner using Eq. (17) and the values given in Table2 due
40、to Mills et al.11).(17)The thermal conductivity of the alloy at 300 K is then cal-culated using Eq. (14).The thermal conductivitytemperature relation was con-structed by joining l300to the following values: 27.6WmH110021KH110021at 1 273 K, 30.2 W mH110021KH110021at 1 473 K and 32.8WmH110021KH110021a
41、t 1 673 K.Experimental thermal conductivity values for both 3- and4-series stainless steels are similar at high temperatures(H110221 100 K), therefore an average value at each temperaturehas been used.4.2.2. Method 2Equation (18) is a “best fit” equation for the experimen-tal l data for austenitic 3
42、-series alloys as a function of tem-perature.l (W mH110021KH110021)H110059.2H110010.0175TH110022H1100310H110026T2(18)In the case of 4 series alloys there are appreciable differ-ences between the values reported for alloys 410 and 430.Consequently, mean values were used in deriving a best fitequation
43、Ferritic: Below 1 100 Kl (W mH110021KH110021)H1100523.5H110010.0016(TH11002300 K) (19)All alloys above 1 100 Kl (W mH110021KH110021)H1100525.4 H110011.3H1100310H110022(TH110021 100 K) .(20)4.3. Results and DiscussionThe results show that the differences between experimen-tal and calculated values ar
44、e within 5%, for the 3-series al-loys and for the alloy 410 but are more than 10% divergentfor 430. It can be seen from Figs. 6 and 7 that the thermalconductivities of ferritic, 4-series alloys are higher thanthose for austenitic 3-series alloys.5. Thermal Diffusivity (a)Thermal diffusivities for th
45、e solid state, like thermal con-ductivities, are dependent upon the microstructure and arethus influenced by the thermal and mechanical history ofthe sample. Furthermore, thermal diffusivity values for fer-ritic alloys are influenced by the magnetic transition(aH11005l/rCp). Values for the liquid we
46、re derived from valuesof thermal conductivity.5.1. DatabaseThermal diffusivity values have been reported for 304 byBogaard et al.,5)Monaghan15)and Szelagowski16)(as re-ported by Mills7) and by Seetharaman.17)Thermal diffusiv-ities can be calculated by aH11005l/r Cpbut are frequentlymeasured directly
47、 (with the laser pulse method). The mea-surements of thermal diffusivities of ferritic steels gothrough a minimum around 1 000 K (aH11005l/Cpr): this corre-sponds to the peak in the CpT relation. Touloukian6)hasreported thermal diffusivity values for 403, 410, 416, 420and 430 alloys, all of which ex
48、hibit a minimum valuearound 970 K.5.2. Results and DiscussionThe thermal diffusivity values shown in Figs. 8 and 9refer to austenitic, 3-series and to ferritic, 4-series alloys,respectively, these were calculated from the recommendedvalues of l, Cp, and r. In Fig. 8 calculated values for 3 se-ries a
49、lloys are compared with the experimental values re-ported by Mills7)and by Seetharaman.17)It can be seen thatthe calculated values are within 10% of the experimentalH11001()wt%300 3300latwt% wt%H11005H11001(/ ) ( ) ( )1 100300 1 300 2ISIJ International, Vol. 44 (2004), No. 10 2004 ISIJ 1664Table 2. Coefficients used in the calculation of thermal conductivity.11)Fig. 6. Thermal conductivity as a function of temperature for 3-series stainless steels (systems 304 and 316).Fig. 7. Thermal conductivity as a function of temperat
