Mathematical Model for Temperature Field of Strip Coil in Cooling and Heating Process

Vol. 12 No. 2J. Iron & Steel Res. , Int,Mar. 2005Mathematical Model for Temperature Field of Strip Coilin Cooling and Heating ProcessSUN Jji-quan'，SUN Jing hong'，WU Bin2 ，LIAN Jia- chuang(1. University of Science and Technology Beijing, Beijing 100083, China; 2. Baoshan Iron and Steel Group Co,Shanghai 201900, China; 3. Yanshan University, Qinbuangdao 066004, China)Abstract : The convection between the strip coil boundary and the surrounding medium was studied, and the math-ematical model and boundary conditions for the temperature field of anisotropic strip coil was proposed, and thetemperature field of strip coil were calculated by the analytic method,Key words: strip coil; anisotropy; temperature field; mathematical modelWith the development of the automobile indus-try and electric appliances, the requirement on stripz▲tgquality, especially on surface quality, becomes strin-T.gent. The variations of the temperature field duringcooling cannot be neglected, because asymmetricalatemperature field will result in the thermal stressand change the strip shape. Especially during annea-ling, the cold rolled strip coils will bring about cohe-sion between coil layers. Therefore, the tempera-ture field should be calculated for both hot and coldO=0rolled strip coils. It is also the prerequisite for thecomputation of the thermal stress. The heat transferbetween strip surface and surrounding medium wasFig. 1 Heat transfer model of strip coilstudied in this paper.With different coil compacting pressure, con-erential directions of the cylindrical reference frametact between layers differs and there will be somerespectively, thermal conductivities in these direc-surface thermal contact resistance, so the heattions are the principal thermal conductivities. So thetransfer is anisotropic. That is, the thermal conduc-heat conduction equation can be described as[1]:tivity in radial direction is different from that in axialand cutting directions, and depends on the coil com-t_?(r \+20+n. rt(1)pc-北rr(pacting pressure. The heat transfer model of stripwhere p is density of the strip; c is specific. heat ofcoil is shown in Fig. 1.the material; and τ is time.1 Basic FormulaSince the circumferential thermal conductivityof the coil is similar to the axial one, i.e. ，λo=λ:=1. 1 Heat conduction differential equation of coilλo，while the radial thermal conductivity differs fromAs to heat conduction, the coil should be homo-others, the transforms can be obtained as2;geneous and anisotropic. The thermal conductivitydistribution of anisotropic matter takes an ellipsoid.n=r/空,0二0号,z=z√(2)The principal axes of this ellipsoidal surface ξ, η，ζare called the principal heat conductive axis, and the Let m,=A(3)thermal conductivity λe，入，, λ5 are the principal ther-Eqn. (2) can be changed to following form:mal conductivity. Because the principal heat conduc-tion axes accords with the axial, radial and circumf-中国煤化工(4)Biography; SUN Ji quan (1959-), Male, Doctor, Professor;E- mail: sjq19590MHCNMHGer2s,2003●34●J. lron& Steel Res.，Int.Vol. 12λo is optional and can be the thermal conductivi-coiler, it is cooled spontaneously in the air, and thety. Through the transformation, Eqn. (1) can beconvective heat transfer can be described by Eqn. (7)written as:to Eqn. (10). The convective heat transfer condi-at_0(ψ+1些+1+)tions between the coil and the surrounding mediuma pc(ri" ndn' r 30f' zijcan be considered as convective heat transfer be-The isotropic matter is the same as Eqn. (5)，sotween the large spaces, while that of the inside sur-the heat conduction of the anisotropic matter can beface can be viewed as spontaneous convective heatchanged to that of the isotropic matter and only thetransfer condition.reference frame has corresponding conversion.If the coil is put horizontally, az is given by Eqn.When the coil is cooled in the air or annealed in cap(11) and Eqn. (13)[3], and as is given by Eqn. (12) andcover furnace, the boundary condition is axisymmet-Eqn. (14)[3]. .rical, so the thermal conduction equation of coil is: .When the liquid flow is laminar flow, i.e. 10*a=a(+1 +(6)< 10* , ar and as can be written as;two interfaces of the coil are all the third kind ofaz=1. 24(tw - - t)1/3(13)boundary conditions. As the temperature distribu-a3=1.24(2- - t)1/3(14)tion is almost symmetrical in width, only half coil is .where Gr，Pr: are Grashof number and Prandtlstudied. The middle surface is the symmetrical onenumber respectively, tw is temperature of inside sur-and it is the second boundary condition, i.e. q= 0face of the coil, and D is outside diameter.(Fig. 1). The boundary conditions are as follows:The heat transfer condition is very complicatedr= r。(Inside surface of the coil)and can be considered as the spontaneous convectivea(t-t,-.)=-2o(斯(7)heat transfer in the limited space. After the modifi-cation based on Eqn. (11) and Eqn. (13), the flow ofr=r,(Outside surface of the coil)the air upon the surface of the coil is usually laminara2(4-L=b)=-20(n ).(8)flow, therefore:z=0 (Cross section of the coil), q=0(15)at_a-1.2(")“(号)“二=0(9)where d is equivalent diameter of the air, which isthe inside diameter of coil as to the inside surfacez= 5 (Side of the coil)and equivalent diameter of the gap between the heat-as(4-t.-2)=-20() )(10)ing cover and the outside surface of coil as to the :)-4outside surface and the equal quantum diameter ofwherea1, az, as are convection coefficients betweenthe convection boards flowing groove as to thecoil surface and surrounding medium; t is tempera-sides.ture of surrounding medium; and r。, r are radii ofThe coil is put upright no matter it is annealedinside and outside surface of the coil respectively.in the cap cover furnace or cooled in the air.If the preliminary condition t(r, x) anda1, a2,When the flow of the air is laminar flow, i. e.a3 are determined, Eqn. (5) can be solved according10* 2. 1 Strip in spontaneous cooling condition109[], a2 and a3 can be written as:After the strip is coiled and discharged from thear=0.95(tw - t)1/3(18)No. 2Mathematical Model for Temperature Field of Strip Coil in Cooling and Heating Processas=1.43(tw - ty)}/3(19) tion:S,U.- V。W.=0(24)2. 2 Forced heat transfer condition of coilThe norm N(η) is given by:The annealing process of the coil in the cap cov-12(% +a3)(25)er furnace is forced convective heat transfer in bothN(n)~ H(π +a4)+asthe heating and cooling processes. The convection where n is the eigenvalue of Eqn. (26):coefficient a between the coil surface and media cannitanyH=a3be described as5.6]:This formula is the expression of temperaturea=0. 023 ARei'Pr^'(台)(20)field for the coil. Since the Bessel function, expo-nential function and trigonometric function convergeRe=_wdfast in computation process. In this paper, ten fron-tal items are taken.where λ is thermal conductivity of the media; Re isReynolds number of the air; V is viscosity; and n is4 Resultexponent, n= 0.55 at heating time, n=0 at coolingThe eigenequations including two latent rootstime.are determined by boundary conditions. But theEqn. (12)一Eqn. (20) are computation expres-boundary conditions of coil, i.e. the heat transfersions of the convective heat transfer between coilcondition, are different at different time, so the la-surfaces and surrounding medium in all the formingtent roots depend on heat transfer conditions beprocesses of the coil, and they were all obtainedtween the coil boundaries and environment. Afterfrom experiments and experiences. In the actual apthe heat transfer condition and the heat transfer co-plication，the temperature field can be calculatedefficient are resolved, the latent roots at differentfirstly and the results can be compared with metricaltime can be known, and finally, substituting thevalues. If there are any differences, the equationsvalues of these latent roots into Eqn. (24) and Eqn.will be modified until the results are satisfactory.(26)，the variation condition of the temperaturefield at different time can be obtained.3 Solution of Basic FormulaThe temperature field at different time for an-The variable separation method is used to solve nealing treatment according to annealing curve canthe formula and boundary conditions, and the finalbe solved in this way. Fig.2 to Fig.4 show the in-form of the temperature field can be obtained:ternal temperature field of coil. Fig. 5 is the annea-- lsingh S(S6J(.)-ling curve, describing the variation of controlling,m7nN(m )N(n )bottom and central temperature with time.SoaJ[(Bma)]-[V。bY,(βb)-VoaY(Bma)]}●The temperature field of the coil in annealing is[S。Jo(Bmr)- VYo(βmr)]e 慌+"cosηp.z (21)calculated according to measured value of the con-The eigenfunction R。(Bm, r) is as follows'l :trolling and bottom temperatures.. The measuredRo(m,r)=SsJo(Bmr)-VYo(Bmr) (22)curve is used to describe the variation of the control-whereling and bottom temperatures with time which canSo=pnY'。(Bmb) +anY,(Bb)be used as boundary condition. The ostensible tem-Vo=βnJ '。(Bmb)+a]J(B.b)perature can be determined and the convective heatYo，J。, Y'。and J' are zeroth order, Bessel func-transfer coefficients between the boundary and pro-tions and their first derivations respectively.tective gas can be corrected using the boundary con-The norm N(pm) is given by:dition. So the controlling temperature and bottom(23)temperature are given by measured value, while theN()= 2 ‘B,U:-B,Vcoil's central temperature is obtained by calculation.5U。=R_J '。(Ra) - a1J。(B.a)中国煤化工Wo=βmY"o(Pma)- anYo(Bma)_pic property, the coilB:=ai+fisdfYHCN M H Gcal and aisoropic 3DB2=ai +β流.heat conduction model. By using this model, the ax-where Bm is the eigenvalues of the following equa- isymmetrical anisotropy heat conduction equation●36●. J. Iron & Steel Res.，Int.Vol. 12田)()70040-。660号200.620601 000" 4001 00400800880咖600200mm 600~400~ δ0(a) After annealed for 2 h;(b) After heat preservated for2 hFig. 2 Internal temperature field of coil17(a)691)16968968768516161683400600 800 1000 1 200400600 800 1000 1200H/mm(b) After heat preservated for 2 hFig. 3 Internal temperature distribution of coil on width700「b)2060 t弋200640 t80 t60L500900 1 100300001 100r/mm(a) After annealed for2 h;Fig. 4 Radial temperature distribution of coil on center sectionand boundary conditions are deduced, and the tem-ther calculated by the temperature field results. Mo-perature field is finally determined by the measuredreover, the time and position of the cohesion be-annealing curve.tween the coil layers can be determined correctly,The inner thermal stresses of the coil are fur-which has been proved by cold rolling plant 0Baosteel Co. Therefore the measure basis of avoi-ding coil cohesion is provided.600CentralReferences: .Bottom1] Necati Ozisik. 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