Modeling-of-heat-transfer-during-controlled-cooling-in-hot-rod-rolling-of-carbon-steel

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Mathematical model for heat transfer mechanism

Mathematical model for heat transfer mechanism

Mathematical model for heat transfermechanism for particulate systemA.R.Khan *,A.ElkamelDepartment of Chemical Engineering,Faculty of Engineering and Petroleum,Kuwait University,P.O.Box5969,13060Safat,KuwaitAbstractVarious theoretical models for fluidized bed to surface heat transfer have been considered to explain the mechanism of heat transport.The particulate fluidized bed which is the common case for liquid–solid fluidized bed is much simpler and homoge-neous and transport operation can be easily modeled.The heat transfer coefficient in-creases to a maximum and then steadily decreases as the bed void fraction increases from that of a packed bed to unity.The void fraction e max at which the maximum value of heat transfer coefficient occurs is a function of the solid–liquid system properties.An unsteady state thermal conduction model is suggested to describe the heat transfer process.The model consists of strings of the particles with entrained liquid moving parallel to surface,during the time interval heat conduction takes place.These strings are separated by liquid into which the principal mode of transfer is by convec-tion.The model shows a dependence of heat transfer coefficient on void fraction and on physical properties,which is consistent with the results of experimental work.Ó2002Elsevier Science Inc.All rights reserved.1.IntroductionThe high heat transfer coefficient from a hot surface to a fluidized bed fa-cilitates the addition and removal of heat to and from a process efficiently.The role of various parameters and the mechanism of heat transfer in this field have been the subject of extensive investigations during the last three decades.Many Applied Mathematics and Computation 129(2002)295–316*Corresponding author.Tel.:+965-481-7662;fax:+965-483-9498.E-mail address:rehman@.kw (A.R.Khan).0096-3003/02/$-see front matter Ó2002Elsevier Science Inc.All rights reserved.PII:S 0096-3003(01)00039-X296 A.R.Khan,A.Elkamel/put.129(2002)295–316A.R.Khan,A.Elkamel/put.129(2002)295–316297empirical correlations relating bed to surface heat transfer coefficients for a range of operating variables have been proposed.They are of restrictive va-lidity because they cannot make adequate allowance for different geometries of equipment used and varying degree of accuracy of the experimental techniques used.Furthermore,it is difficult to extrapolate outside the experimental range of variables studied.Different models have been proposed to explain the dif-ferent aspects of this complexproblem.There are particularly diverse concepts suggested by different workers regarding the mechanism of heat transfer be-tween afluidized bed and a heat transfer surface.In this paper an attempt has been made to explain the mechanism of heat transfer from bed to surface in liquidfluidized systems.The model results are compared with experimental data[1]and computed values of the existing models.2.Heat transfer models forfluidized bedBoth gas,and to a lesser extent liquidfluidized beds have been employed in chemical engineering practice particularly where the addition or removal of heat from the bed is required.In the case of gasfluidized beds the more important aspects have been collected and presented in detail by Zabrodsky [2].Most of the workers have examined a limited range of experimental variables and presented their results in the form of correlations;the power of any group in the correlations gave some indication of its importance within the experimental range investigated.It is clear that the scale of the equipment in which measurements were made has influenced the results.There is no sufficiently general theory of heat transfer influidized beds,although several different models have been proposed to explain various aspects of this problem.A brief description of some of the models is given in the following section.2.1.The limiting laminar layer modelLeva and Grummer[3]noted that the core of the bed was isothermal and offered negligible thermal resistance while the main resistance limiting the rate of heat transfer between the bed and the heat source lay in afluidfilm near the hot surface.They suggested that particles acted as turbulence promoters,which eroded thefilm reducing its resistive effects.Levenspiel and Walton[4]derived an expression in terms of modified Nusselt and Reynolds numbers for the ef-fectivefluidfilm thickness assuming that thefilm breaks whenever a particle touches the transfer surface.They have to modify the coefficients in the model to account for their own experimental data.Wen and Leva[5]correlated the published heat transfer results on the basis of a scouring action model in which particle movement was assumed to be vertical and parallel to the heat source.Nu¼consC S qSd1:5Pffiffiffig pk g0:4GdPElgR"#0:36:ð1ÞIn this correlation thefluidization parameters are defined as follows:1.E is thefluidization efficiencyðGÀG mfÞ=G mf.2.R is the expansion ration of the bed H=H mf.Richardson and Mitson[6]and Richardson and Smith[7]reported that for liquidfluidized beds the resistance to heat transport lay near the tube wall within the laminar sub-layer where the effective thickness is reduced by the presence of particles for two reasons:1.The particles cause turbulence in thefluid thereby reducing the thickness ofthe laminar boundary layer.2.The particles themselves transport heat as a result of the radial componentof their rapid oscillating motion.Wasan and Ahluwalia[8]proposed that heat transfer through afluidfilm was promoted byfluid eddies beyond thefilm boundary.They assumed that the solid particles were stationary and equally spaced and that heat was transferred through thefilm and then spread byfluid convection into the bulk of the bed.They compared the experimental results of various workers and found deviations of up toÆ44%.These models basically involve a steady-state concept of the heat transfer but Wasan and Ahluwala[8]included some dynamics transfer features for transfer through thefluid into the bed.298 A.R.Khan,A.Elkamel/put.129(2002)295–3162.2.Two resistancefilm modelWasmund and Smith[9]suggested a modified laminar layer model,in which they considered particle convective transfer due to radial motion of particles into the laminar layer.This mechanism contributed50–60%of the total heat transferred and the remainder was fromfluid convective transfer.Tripathi et al.[10]used the series model proposed by Ranz[11]for effective transport properties in packed beds.They compared results obtained by Wasmund and Smith[12]using radial velocities of the particles and observed deviation of Æ20%.Brea and Hamilton[13]and Patel and Simpson[14]used a two resis-tancefilm model and emphasized that thefluid eddy convection is the main contributing factor to the heat transfer.Zahavi[15]measured the effective diffusivity of thefluidized beds and also developed a semi-empirical correlation, which represented his results with a maximum deviation ofÆ34%.2.3.Unsteady state heat transferMickley and Trilling[16]suggested that the heat transfer process in a gas-fluidized bed was of an unsteady state ter Mickley and Fairbanks[17] developed a model of heat transfer on the assumption that at any time there is unsteady state heat transfer within thefluidized bed close to heat source;this can be broken down into components due to solid/solid,solid/surface,gas/solid, and gas/surface transfer.Packets of loosely locked particles which are assumed to have uniform thermal properties constitute thefluidized bed.The mean heat transfer coefficient can be calculated for packets of particles moving with a constant speed u passing rapidly along the length of the heat source,thenh¼2ffiffiffip pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik0q0C0Pu0L r:Thermal conductivity k0,density q0and heat capacity C0P for packets can beestimated by use of the Schumann and Voss[18]correlation.The assumption that the thermal properties of the bed are uniform in the neighborhood of the heat source is unrealistic when the source and the bulk of the bed are at considerably different temperatures.Mickley and Fairbanks[17]calculated the residence times of packets from resistancefluctuations recorded for a thin electrically heated platinum strip.The frequency of packets was of the order of two per seconds and the residence time of0.4s.Henwood[19],Catipovic et al.[20],Suarez et al.[21]and George and Smalley[22]used a small heat transfer surface to measure the variations in local heat transfer coefficient within and adjacent to a rising bubble.They concluded that heat transfer took place mainly through thefluid to the particle with the maximum rate in the vicinity of the contact part.A.R.Khan,A.Elkamel/put.129(2002)295–316299300 A.R.Khan,A.Elkamel/put.129(2002)295–3162.4.Simplified modelsThe heterogeneity offluidized beds is an important factor enhancing the value of the heat transfer coefficient up to50–100folds for a gas and5–8folds for liquidfluidized bed.Botterill and Williams[23]have proposed a model for heat transfer in gas-fluidized systems which are based upon the unsteady state conduction of heat to spherical particles adjacent to the transfer surface.The convective transfer through the gas is ignored because the effective diffusivity of thefluidized bed is much higher than the eddy diffusivity of the gas.The particle and gas,whose temperatures were initially the same,approached the surface,the temperature of which remained approximately constant because of its high heat capacity.The Fourier equations for thermal conduction were solved byfinite difference technique.Apart from the axes of symmetry through the particles,there were three space limits to the problem.(A)Close to the heater it was assumed that there was a continuous thin layer ofpurefluid with which a particle was in contact.(B)The temperature of the other end the particle was set at the sink temper-ature,taken as the bulk temperature.(C)Transfer of heat between particles in a direction parallel to the surfacecould be neglected because the temperature difference between particles in adjacent position was very small.The experimental results for heat transfer coefficients for the shortest resi-dence time of metallic particles were far less than the predicted values.This discrepancy was accounted for by assuming that a gasfilm of thickness equals to about10%of the particles around the heat source.Botterill[24]tried both models in which there was triangular and square packing on the particles and a fluidfilm between the particles and the heat transfer surface.The thickness of thefilm was related to the resistance limiting the heat transfer.Different workers made observations but no satisfactory conclusion was reached about the local variation of void fraction in the vicinity of the heat transfer surface.Davies[25]considered the unsteady state heat transfer by conduction be-tween the element at one temperature and spherical particles immersed in a liquid at another temperature.The particles were assumed to enter the thermal boundary layer with a mean radial velocity v r,approached the surface and left it again with the same velocity.The Fourier equations were solved by using the explicitfinite difference method and the same boundary condition as reported by Botterill and Williams[23].The very low values of particle convective heat transfer component predicted by the model indicated that only a very small proportion of the heat transferred from the heated surface to the bed was carried byfluidized particles.The experimental high value was attributable toA.R.Khan,A.Elkamel/put.129(2002)295–316301 the fact that the effective thickness of the thermal boundary layer had been substantially reduced;this was mainly due to the following causes:1.The scouring action of particles.2.The high interstitial velocity of the liquid.2.5.Particle replacement modelGabor[26]has proposed that heat has been absorbed by the particulate bed based on string of spheres of infinite length normal to heat transfer surface. Another simplified approach based on series of alternating gas and solid slabs also provided similar results as the spherical model.Gelperin and Einstein[27]have developed a more refined model taking into account other details of the process involved.They considered that heat is transferred from the heat transfer surface by packets of solid particles by gas bubbles and by gas passing between the packet and the surface.The total heat transfer coefficient is expressed ash¼h PðÀh convÞ1ðÀf0Þþh b f0þh r;ð2Þwhere h P;h conv;h b and h r are the heat transfer coefficients corresponding to packet,convection,bubble and radiation,respectively,and f0is the fraction of time for which the surface is covered by bubbles.They solved the basic equations for their models of bed to surface heat transfer in terms of two heat resistances:R WS the resistance offered by gas entrained by the particles close to the transfer surface and R a the resistance offered by the gas–solid packets.They have tabulated theirfinal equations for instantaneous and mean heat transfer coefficient for different boundaries in their publication[27].For isothermal conditions of heat source,which have already been proposed a simplified solution can be used with little error.Martin[28]has presented a particle convective energy transfer model for wall to bed heat transfer from solid surface immersed in gas-fluidized bed.In his model the following assumptions were applied.1.The contact time is regarded to be proportional to the time taken to coverthe path with the length of one particle diameter in freeflight.2.The wall to particle heat transfer coefficient is calculated by integrating thelocal conduction heatfluxes across the gaseous gap between sphere and plain surface over the whole projected area of the sphere[29].3.The average kinetic energy for the random motion of particles comes from acorresponding potential energy.302 A.R.Khan,A.Elkamel/put.129(2002)295–3163.Development of mathematical modelFor the purpose of establishing a simplified model of a liquidfluidized bed, the system is assumed to consist of strings of particles with liquidfilling the intervening spaces.It is proposed that unsteady state thermal conduction takes place into both the liquid and the solid particles in the string as reported by Gabor[26].Liquid layers into which the principal mode of heat transfer is forced convection as shown in Fig.1to separate the strings.The overall heat transfer coefficient for liquidfluidized bed from immersed surface constitutes solid conductive,liquid conductive and liquidconvective Array Fig.1.Mathematical representation offluidized particle entrained in liquid with initial and boundary conditions.components based on the void fraction determined by bed expansion charac-teristics and particle axial velocity V P;Axial.The convective component is cal-culated for liquid moving with interstitial velocity parallel to the heating surface.The conductive components for solid and liquid are evaluated based on contact time using unsteady state conduction equations for string of par-ticles with entrained liquid.3.1.Heat transfer across incompressible boundary layersThe simulated element is considered as aflat plate located on the axis of the column with the large faces parallel to liquidflow.The liquid with an average velocity u and uniform temperature T B passes over the hotflat plate at a constant temperature T E.At high Prandtl numbers the thermal boundary layer is always confined entirely within the laminar sublayer.This limiting case of forced convection across a turbulent boundary layer can be solved analytically. Kestin and Persen[30]based their analysis on the laminar form of the energy equation and confined their attention to the laminar sublayer only.The other assumption made is that the velocity varies linearly with distance perpendicular to theflat plate.The detailed solution of the energy equation for laminar form assuming linear velocity profile within laminar sublayer is expressed asy s WðxÞlo To xÀy2d s Wd xo To y¼ao2To y2ð3aÞby substitutingg¼y3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs W=lÞ3q9aR xx0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs W=lÞpdx;Eq.(3a)can be transformed to ordinary differential equation.d2T d g2þgþ23d Td g¼0;ð3bÞwhereT¼T EÀTT EÀT B¼1;for x¼0and all values of y>0,T¼1,for y¼/and all values of x>0and T¼0,for y¼0and all values of x>0.The solution is given in the form of incomplete c-function asTðgÞ¼c1=3;g ðÞCð1=3Þ:The calculated values of the heat transfer coefficient are presented together with experimental values[1]in Table2.The experimental data are obtainedA.R.Khan,A.Elkamel/put.129(2002)295–316303304 A.R.Khan,A.Elkamel/put.129(2002)295–316from immersed electrically wound heating surface in6-mm glass particleflui-dized bed dimethyl phthalate.The high experimental values are due mainly to the following causes:1.The plane element was not a trueflat plate.2.The edge effects of the element caused a high value.3.The temperature of the element might not be uniform.In the case of afluidized bed the presence of solid particles decreased the free area available forflow and caused an increase in the liquid velocity near the heat source.The contribution of heat transfer due to liquid alone was assessed on the basis of the interstitial velocity which was the factor determining the thickness of the thermal boundary layer.3.2.The contribution offluidized particles3.2.1.Particle velocities influidized bedsBy means of high speed photography several workers have measured the paths taken by a tracer particle in a transparentfluidized bed.Toomey and Johnstone[31]obtained particle velocities near the wall of dense phase gas-fluidized bed.For the particular case of0.376-mm diameter glass spheresflu-idized in air,they reported particle velocities from60to600mm/s.Kondukov et al.[32]used radioactive tracer particles and radiation detectors to measure the particle trajectories in an airfluidized bed.Their results were similar to those of Toomey and Johnstone[31]and gave additional information on particle behavior in the interior of the bed.Handley[33]fluidized1.1-and1.53-mm glass spheres with methyl benzoate in31-and76-mm diameter columns.He reported the radial and axial velocities of the particles for bed voidage ranging from0.67to0.905.He concluded that the velocity of the particles was completely random in a uniformlyfluidized bed.A more extensive study of particle velocities influidized bed was made by Carlos[34]and by Latif[35].Carlosfluidized9mm glass beads with dimethyl phthalate at30°C in a100mm diameter column.He reported the particle velocities(radial,angular,axial,horizontal and total)over the voidage range 0.53–0.7.A set of differential equations governing the mixing process was numerically solved by computer taking into account the effect of radial and axial tif[35]extended the work of Carlos to6-mm glass spheres and developed a simple relationship between particle velocity and axial and radial positions.The constants of these equations are listed in Table1.These equations are used to evaluate particle velocities as a function of bed expansion characteristics.3.2.2.Residence time of particle in the vicinity of hot surfaceDavies [25]assumed that the high heat transfer coefficient for the fluidized bed was attributable to effects arising from the radial velocity of the particles as reported by Figliola and Beasley [36].His model predicted very small values of heat transfer coefficient because the thermal boundary layer thickness was small compared with the diameter of the particle and the residence time in the thin boundary layer was short.However from the average calculated values of radial and axial particle velocity components at the center of the fluidized bed,it was obvious that the axial component of velocity was dominant.On this basis it appeared reason-able to assume,as a first approximation,that particles and fluid both at the bulk temperature of the bed approached the heat transfer surface at a velocity approximately equal to the average axial component of a particle velocity.The string of particles separated by intervening liquid thus traveled parallel to the hot surface and unsteady state heat conduction took place through the solid and liquid in parallel.The residence time for a particle could be given ast ¼L V P ;Axial:3.2.3.Unsteady state thermal conduction for liquid and solidIn the present model in which it is assumed that heat transfer is because of unsteady state thermal conduction in both liquid and solid,the following assumptions are made:1.The temperatures of the bulk of the fluidized bed ðT B Þand the temperature of the element ðT E Þare uniform.2.A string of particles with entrained liquid at a uniform temperature equal to T B arrives quickly in an axial direction at the element.Table 1Coefficients for calculation of axial velocity component [35]V P ;Axial ¼A Ár þB where r is normalized radial coordinate of particle A ¼a 0þa 1z þa 2z 2þa 3e a 4z2B ¼b 0þb 1z þb 2z 2þb 3e b 4z2e a 0a 1a 2a 3a 4b 0b 1b 2b 3b 40.55)0.767.25)8.74)87.4)20.20.42)3.08 3.8331.8)21.90.65)2.749.66)6.92)74.22)10.09 1.17)1.550.3846.5)17.030.75)1.42 4.78)2.84)128.2)15.110.96)3.17 1.9474.56)18.340.850.55)3.3 2.58)162.3)20.0)0.18 3.12)2.9286.7)22.70.95)4.27)17.9822.27)222.0)27.52.2212.04)15.09119.7)29.0A.R.Khan,A.Elkamel /put.129(2002)295–3163053.The particles and entrained liquid absorb heat by unsteady state conductionas they travel along the surface.Immediately the particles leave the vicinity of the element they exchange heat with the surrounding liquid.The model explains the way in which thefluidized particles contribute to-wards the transfer of heat between the hot surface and the bed.At any time the heat contents of both the liquid and the particles may be obtained from a knowledge of the temperature distribution within the particle and liquid.The temperature distribution in the solid and liquid in contact with the surface may be obtained by the heat conduction equation for both the liquid and solid over the residence time for which they are present at the hot surface.The heat conduction equation in spherical coordinates necessitates the use of three space dimensions;this may cause complications in solving the equation with its ap-propriate boundary conditions.The situation may be simplified by defining the system in terms of Cartesian coordinates and assuming that the particles may be replaced by cubes,the length of the side of each of which is equal to the diameter of the particle.Each cube moves with one face in contact with the surface and liquid occupies the intervening spaces.Symmetry is assumed along the plane perpendicular to the surface.The length of the liquid slug between the particles and the thickness of the liquid layer separating the strings will be calculated as follows: In Fig.1a cube of dimensionðX SþX LÞis considered and the void fraction in the vicinity of the surface assumed to be the same as in the bulk where X S and X L are dimensions of particle and liquid slug,respectively.The volume ofthe particle is X3S and of the liquid slug X2SX L and void fraction is expressed ase¼ðX SþX LÞ3ÀX3S ðX SþX LÞ:On rearranginge¼1ÀX SX SþX L 3orX L¼X S11Àe1=3"À1#:The Fourier equations for unsteady state thermal conduction within the two homogeneous phases of the system areFor solid phaseo2T P o x2þo2T Po y2¼1a Po T Po t:ð4aÞ306 A.R.Khan,A.Elkamel/put.129(2002)295–316For liquid phaseo2T f o x2þo2T fo y2¼1a fo T fo t:ð4bÞThese equations are put in dimensionless form by definingT P¼T EÀT PT EÀT B;T f¼T EÀT fT EÀT Band s¼tÁa P;s¼tÁa fo2T P o x2þo2T Po y2¼o T Po s;ð4cÞo2T f o x2þo2T fo y2¼o T fo s:ð4dÞInitial and boundary conditions,at t¼0,the particle and liquid slug both are divided to give a mesh nÂn and all points within the particle and liquid slug are at the bulk temperatureT P¼T f¼1;T P¼T f¼T B:ð5aÞThe temperature at the face of the liquid slug at x¼0and at all distances in the y-direction perpendicular to the surface is considered to be at the bulk temperature.The temperature of the similar face of the solid particle is taken as the computed values of the liquid temperature at the end of thefirst time step at x¼X Lat y¼0and x¼0;T f¼1;T f¼T B;T P¼T f and T P¼T fðAfter first time step at x¼X LÞ:ð5bÞAt t P0,the faces of both particle and liquid which are in contact with the surface are at all times at the surface temperature.At y¼0;T P¼T f¼0;T P¼T f¼T E:ð5cÞEqs.(4c)and(4d)are solved simultaneously usingfinite difference techniques for afixed and for a variable boundary.For t>0thefixed boundary is ex-pressed by Eq.(5c)and the variable boundary is aty¼0at x¼0for liquidT f¼T P and T f¼T PðAfter previous time step at x¼X SÞand for particleA.R.Khan,A.Elkamel/put.129(2002)295–316307T P¼T f and T P¼T fðAfter current time step at x¼X LÞ:ð5dÞThe explicit method used by Botterill and Williams[23]and by Davies[25] restricts the time and space increments to ensure stability according to the equationD s61ðD XÞÀ2þðD YÞÀ2 h i:Their difference equation isTði;jÞkþ1ÀTði;jÞkD s ¼TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞD X2kþTði;jÀ1ÞÀ2Tði;jÞþTði;jþ1ÞD Y2k:ð6aÞThe above-mentioned method is very sensitive to the value of the operator k h which is given ask hX¼a hD tD X2and k hY¼a hD tD Y2:For the sake of simplicity equal increments are taken in the X-and Y-directions,i.e.D X¼D Y.An implicit method can make the equations inde-pendent of the operator value as well as of space and time increments.The difference equation is thenTði;jÞkþ1ÀTði;jÞkD s ¼TðiÀ1;jÞÀ2Tði;jÞþTðiþ1;jÞD X2þTði;jÀ1ÞÀ2Tði;jÞþTði;jþ1ÞD Y2kþ1:ð6bÞOn rearranging one gets a pentadiagonal matrixÀTðiÀ1;jÞÀTðiþ1;jÞþ1k hþ4Tði;jÞÀTði;jÀ1ÞÀTði;jþ1Þkþ1¼1k hTði;jÞk;ð6cÞwherek h¼a h D t D X2;which can be solved by either the Gaussian elimination method or the Gauss-Seidel iterative method to givefive unknowns.The implicit alternating direction method discussed by Carnhan et al.[37], which avoids all the disadvantages discussed above,is thought to be most suited for this type of problem.The two difference equations are used in turn over successive time steps,each of duration D s=2.Thefirst Eq.(7a)is implicit 308 A.R.Khan,A.Elkamel/put.129(2002)295–316。

专业英语

专业英语

Definition of polymers A simple understanding of polymers can be gained by imaging them to be like a chain or, perhaps, a string of pearls, where the individual pearl represent small molecules that are chemically bonded together. Therefore, a polymer is a molecule made up of smaller molecules that are joined together by chemical bonds. The word polymer means „many parts or units.‟ The parts or units are the small molecules that combine. The result of the combination is, of course, a chainlike molecule (polymer). Usually the polymer chains are long, often consisting of hundreds of units, but polymers consisting of only a few units linked together are also known and can be commercially valuable.
Figure 1.1 Diagram illustrating the definition of plastics.
As Figure 1.1 shows, all materials can be classified as gases, simple liquids, or solids, with the understanding that most materials can be converted from one state to another through heating or cooling. If only materials that are structural solids at normal temperatures are examined, three major types of materials are encountered: metals, polymers, and ceramics. The polymer materials can be further divided into synthetic polymers and natural polymers. Most synthetic polymers are those that do not occur naturally and are represented by materials such as nylon, polyethylene, and polyester. Some synthetic polymers could be manufactured copies of naturally occurring materials (such as

Mathematical Modeling of Heat Transfer

Mathematical Modeling of Heat Transfer

A rc hi v eo fSI DMathematical Modeling of Heat Transfer for Steel Continuous Casting ProcessM. Alizadeh 1*, H. Edris 2 and A. Shafyei 31, 2, 3- Department of Materials Engineering, Isfahan University of Technology, Isfahan,8415683111, IranAbstractHeat transfer mechanisms and the solidification process are simulated for a continuous casting machine and the geometric shape of the liquid pool is predicted considering different conditions. A heat transfer and solidification model is described for the continuous casting of steel slabs. The model has been established on the basis of the technical conditions of the slab caster in the continuous casting unit of Mobarakeh Steel Company. This model involves a two-dimensional (2-D) transient energy equation. The governing equation was solved using the finite-volume procedure. The boundary conditions of the mold, water spray cooling, and air cooling regions have been defined. The mathematical model is able to predict the shell thickness, temperature distribution in the mold and shell, and the interfacial gap between shell and mold. The modeling results were verified by the measured slab surface temperatures and a reasonable agreement was achieved.Keywords: Heat transfer, Solidification, Continuous casting, Mathematical modeling.IntroductionA continuous casting process setup is shown inFigure 1. Different regions of cooling in the continuous casting machine are shown in this figure.Since heat transfer plays an important role incontinuous casting, it is desirable to have aquantitative understanding of the heat flow processeswhich will permit the prediction of the shell profileand temperature distribution as a function of thecasting variables.Fig. 1. Schematic of continuous casting process.∗Corresponding author:Tel: +98- 311- 3912750-1 Fax: +98- 311- 3912752 E-mail: alizadeh@ma.iut.ac.irAddress: Dept. of Materials Engineering, Isfahan University of Technology, Isfahan, 8415683111, IranIn the copper mold heat is transported to the cooling water. Below the mold, in the so-called secondary cooling zone, cooling is performed by water sprays, by contact with water cooled supportrolls, and in the lower part of the machine, by radiation. The mold must be able to extract sufficient heat from the incoming molten steel to solidify a shell which can support the liquid pool at the mold’s exit, and thereby minimize breakouts. It is also important for the mold to be able to remove the heat uniformly to avoid the formation of thin regions in the shell which could rupture and lead to breakouts or surface cracks. Therefore, the mold provides support for the newly solidified steel shell at a time when the shell is weak and sensitive to bulging due to ferrostatic pressure 1).Heat transfer in the mold region is controlled by: the convection of liquid superheat to the shell surface, latent heat evolution in the mushy zone, heat conduction through the solid shell, the thickness and other properties of the interface between the shell and the mold, heat conduction through the copper mold, and heat convection to the mold-cooling water 2).The mechanism of heat transfer from the surface of the strand to the cooling water is complex. It involves resistances in the layer of flux or resolidified mold powder, in the air gap, the mold wall, and in the mold/cooling water interface.1. Ph.D. Sstudent, Department of Materials Engineering, Isfahan University of Technology2. Associate Professor, Department of Materials Engineering, Isfahan University of Technology3. Assistant Professor, Department of Materials Engineering, Isfahan University of TechnologyA rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2Table 1: Comparison of thermal conductivities of materials present in the continuous casting mold [3].MaterialsTemperature(o C) Thermal conductivity(W m -1 K -1)Steel St 37 1200 29 Copper 30-130 385 Casting flux 1000-1300 0.5 to 1.2 Water 25 0.62Radiation conductivity of gas gap 1000 0.043Fig. 2. Different regions and simulation domain in continuous casting process.The thermal conductivities of different layers are compared in Table 1. As shown in this table, the air gap has the largest resistance to heat flow, while the other parts have a comparatively small resistance. Therefore, the pattern of heat removal in the mold is dependent largely upon the dynamics of gap formation. The air gap or contact resistances can be generated by the shrinkage of the steel shell away from the mold walls, especially after the flux is completely solid and unable to flow into the gaps. More researches were performed to show that the air gap was usually created in the lowest one-third of the mold length 3).In the upper part of the secondary cooling zone, the strand is usually sprayed by water emerging from nozzles arranged in the spaces between the rolls. The rate at which heat is extracted from the strand surface by water sprays has been measured by many researchers. These researchers have shown that under normal continuous casting conditions, in which shell surface temperatures range between 700 and 1200 o C, surface temperature has a little effect on the heat transfer coefficient. All studies agree that, in the stated temperature range, the spray water flux has the most effect on the heat transfer coefficient. Moreover, the temperature of sprayed water does not have a large influence on the heat transfer coefficient 4).In the lower part of the secondary cooling zone, heat transfer is preferred mainly by radiation and by roll contact. Therefore, it should be considered that the oxide scales generated on the surface of the strand could cause a thermal resistance in this zone 5).In this study, the mathematical model has been established on the basis of the technical conditions of the slab caster in the continuous casting unit of Mobarakeh Steel Company. In this model, steel heat capacity and steel thermal conductivity were considered as functions of steel temperature and chemical composition. Considering these functions, the governing equation is a non-linear equation. In this study the equation is solved in non-linear state. This model is also capable of predicting the temperature distribution, including the solidus and liquidus isotherm which defines the solid shell and mushy zone, respectively, as a function of section size, pouring temperature, steel composition, casting speed, mold length and spray conditions.Mathematical modelingFigure 2 shows the different regions of the continuous casting machine and the model considered for physical simulation of the caster. A typical method of modeling the strand thermal condition is shown in this figure. The mathematical model is applied to slices of strand that start at the meniscus and travel through the machine at the casting speed. New slices are generated periodically. A sufficient number of slices exist in each cooling zone to give an accurate representation of the thermal condition in each zone. In this model, only a quarter of the strand is considered due to the symmetry of the heat flow conditions (Figure 3).A rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2Fig. 3. A symmetric quarter of strand width section with physical coordinate.AssumptionsThe following assumptions are made to simplify the mathematical model 6):-Conduction can take place only in the transverse directions.-Forced convective heat flow in the liquid pool is considered by defining an effective liquid thermal conductivity as:l eff K K ⋅=7(1) -The density of steel is constant, but specific heatcapacity and heat conductivity of steel are functions of temperature and chemical composition and therefore not constant.Model formulationThe energy conservation equation can be written as 7):(2)()()()()⎟⎠⎞⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=∂∂+∂∂+∂∂+∂∂z T k z y T k y x T k x wH vH y uH x H t effeffeffρρρρ To simplify the equation, a transformation as wt z −=ζ is used. Therefore this is: (3)()ls ls s sH H H t y x H H q tHy H y x H x tH +==+∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=∂∂;,,,;0ζρααIn order to solve the governing equation, it is necessary to transform the physical domain into a computational domain. In general, this sort of transformation is used, and leads to a uniformly spaced grid in the computational domain but the points in physical domain may be unequally spaced. The original partial differential equation is transformed from physical coordinates (x, y ) to computational coordinates (ξ,η) by applying the chain rule of partial derivatives.;JS H H J H t x y s y x s s +⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=⎟⎠⎞⎜⎝⎛∂∂ξηηαηηξξαξ (4);tH S l ∂∂−= y x J ηξ= In the above equations "S " is a term for heat source due to the metal phase transformation (liquid to solid). In order to establish the region of phase change, the latent heat contribution is specified as a function of temperature i.e. 8):f l l L f H ⋅=(5) Where L f is the latent heat of the phase changeand the liquid fraction (f l ) is computed by:⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧≤≥≥−−≥=.......01sol sol liq sol liq sol liq l T T when T T T when T T T T T T when f (6)A typical 2D cartesian control volume is shownin Figure 4. This C.V contains a central node (P) with four neighborhood points (E, W, N, and S). The integral form of equation (4) is obtained on the control volume by the finite volume method 9,10): a P H sP = a E H sE + a W H sW + a N H sN + a S H sS + b (7) a P = a W + a E + a S + a N + a P oa E =ey x e ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔηξξηα a W =wyx w ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔηξξηα a N =nx y n ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔξηηξα a S =sx ys ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔξηηξαa P o =PJ t ⋅ΔΔ⋅Δηξ b = a P o (H sP o + H lP o - H lP ) + o Pq J ρηξΔ⋅Δ (8)Fig. 4. A typical control volume and the notation used for a Cartesian 2D grid [9].To approximate the variable values on the surface of the control volume, the Quadratic Upwind Interpolation (QUICK) algorithm is used 9). In the QUICK scheme, the variable profile between two points approximated by a parabola instead of aArc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2straight line (Figure 5), on a uniform Cartesian grid leads to 9):W E P e φφφφ818386−+= (9)Fig. 5. Approximation of gradients at cell faces [9].Boundary conditionsTo solve the above equation, the boundary conditions are needed for different regions include the mold, water spray cooling, and air cooling. Figure 6 shows some machine cooling layouts while the technical information belonging to each zone is shown in Table 2. A general form of the boundary condition can be expressed through an equation (10), in which the heat transfer coefficient, h , is estimated for different cooling zones.()water b b T T h n T k −=∂∂−| (10)To determine the temperature of the boundaries, the discretion of the bnT |∂∂ on boundary points isrequired. According to the QUICK scheme b nT |∂∂could be calculated by the following correlation [10]: nT T T nT W P b bΔ+−=∂∂398(11)Therefore, as seen in equation (10), finding the heat transfer coefficient for different regions such as: mold, water spray, and air cooling is necessary.Fig. 6. Secondary cooling of the slab caster.In the mold, several thermal resistance layers exist between the steel shell surface and the recirculation water. All the thermal resistances in the mold are shown in Figure 7. The effective thermal resistance of the water channel is estimated from the water channel heat transfer coefficient, thermal conductivity and the thickness of scale deposits on the surface of the cooling-water channel 11):⎟⎟⎠⎞⎜⎜⎝⎛+=w scale scale water h k d r 1 (12)Fig. 7. Thermal resistances existing between the shell surface and water channel in the mold.Table 2. Secondary cooling zones variables.No. Zone Length zone(m)SegmentNumber of spray nozzles Water flow rate(m 3/sec)Number rollin zoneRoll radius (m)1 0.439 - -2 0.220 - -3 0.303 - 30 0.003975 - -4 0.925 0 38 0.004967 5 0.140 5 1.470 0 38 0.0048426 0.200 6 1.475 1 10 0.004858 5 0.2507 1.725 2 10 0.003975 5 0.3008 1.725 3 10 0.003733 5 0.3009 3.950 4,5 20 0.005667 10 0.350 10 5.200 Roll 37-47 22 0.006483 11 0.380,0.440 11 9.400 Roll 48-63 - Air cooling 16 0.440A rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2 The heat transfer coefficient between the water and the side walls of the water channel (h w ) is calculated assuming a turbulent flow through an equivalent-diameter pipe (D ) using the empirical correlation of Sleicher and Reusse 11):()21Pr Re 015.05c c water w Dkh += (13)()Pr 6.0215.0333.0;Pr 424.088.0−+=+−=e c cOther thermal resistances shown in Fig 7 can be calculated by the following correlations [11]:;;slag slag slag moldmold mold k d r k dr == ;air air air k d r = (14)Which d slag can be found from the powder consumption per mass of product (M slag (kg/ton)): ()N W N W M d slagsteel slag slag +×××=2ρρ (15) Also, d air includes a gap due to shrinkage of the steel shell, which can be calculated by the thermal linear expansion relation of steel using following correlation for each control volume: T l l Δ⋅⋅=Δλ (16) Where, λ is linear thermal expansion coefficient of steel and l is length of each control volume. Moreover, the heat transfer coefficient due to radiation is calculated by: ()()212221T T T T h rad ++=εσ (17) Heat transfer mechanisms in the spray cooling zones below the mold are defined in Figure 8. The heat extraction due to the water sprays is a function of the water flux. The relationship between the rate of heat extraction by the water sprays and the spray variables has been established in a number ofexperimental studies. One of the most widely usedrelations has been presented by Nozaki's 12): ();0075.013925.055.0water water spray T Q h ×−××= (18)Fig. 8. Heat transfer mechanisms in the secondarycooling zones [12].Due to the high temperature of the strand surfaceand the exposure of water to the surface, an oxidescale is produced on the surface of the strand. Despite the low thickness, the scale can have animportant role in the heat transfer control. Therefore,the effective heat transfer coefficient should be considered as 12):sprayscsceff h k h 11+=δ (19)It has already been mentioned that the cooling ofthe strand in the lower part of the secondary cooling zone is mainly done by radiation 12). Therefore, the equation for the heat transfer coefficient is given as follows:()().2.2am s am s rad T T T T h ++=εσ (20) Besides the radiation, heat transfer is alsoachieved by natural convections, but this part is rather small and can be neglected in comparison to radiation cooling.The symmetrical boundary condition has been considered for midplanes as follows:0.,00.,0=∂∂−==∂∂−=yTk y xTk x (21)Computation and verificationThe algebraic equation of the boundary conditions has been solved with a Tridiagonal Matrix Algorithm (TDMA) solver 13). As seen in equations(7) and (8), to solve the algebraic equation, it is necessary to know the latent enthalpy in a new time step (H l ). To update the amount of the latent enthalpy an iterative solution is used for each time step:11++−+=k sP k sP k lP k lP H H H H (22)As seen in the above equation, by using the sensitive enthalpy that has been obtained by solving the energy equation (H sP k+1), latent enthalpy could be updated in the k+1th inner iteration in each time stepto achieve a certain convergence. The calculation mentioned has been programmed in the FORTRAN language. The mathematic simulation starts by setting the initial steel temperature at the pouring temperature. Input parameters, in Table 3, in the standard cases are verified by the measured temperatures on the shell surface of the strand.Equilibrium lever-rule calculations are performed on a Fe-C phase diagram in order to calculate steel phase fractions. By this means, phase field lines are specified as simple linear functions of carbon equivalent content. The carbon of the steel is applied as the carbon equivalent content that is calculated by the following correlation 14): wt%CE=wt%C+0.04(wt%Mn)-0.14(wt%Si)-0.04(wt%Cr)-0.1(wt%Mo)-0.24(wt%Ti)+0.1(wt%Cu) For a 0.16%C, 1.3%Mn, 0.5%Si, 0.05%Cr, 0.03%Mo and 0.01%Ti plain carbon steel, the carbon equivalent percentage calculated as 0.135 and also the equilibrium phase diagram model calculates T liq.=1528 o C, T sol = 1494 o C. The solid fraction-A rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2temperature curve in the mushy zone obtained from the model is shown in Figure 9. As seen in this figure the relation between temperature and solid fraction of steel in the mushy zone is non-linear.Solidified shell thickness is one of the most important calculated parameters in the model, and the influence of the grid spacing on this parameter should be considered. The influence of the gridspacing on the solidified shell thickness, in exit point of the mold, is shown in Table 4. The thickness amounts in width and narrow sides are presented in this table. It is clear from Table 4 that when grid spacing is reduced, solidified thicknesses are changed and they are stable in a narrow limit with reduction of grid spacing lower than a certain limit. It can be concluded that the solidified shell thickness is independent of the mesh size with the reduction of grid spacing lower than a certain limit.The mold zone is a complex and important area in continuous casting machine. The solid shell growth in this zone is complicated and the results of this study are compared to those of some other researchers. The comparison between the results of the model in this study and experimentalmeasurement by some other researchers is shown in Figure 10. The figure shows the variation of growth of the solid shell thickness on the ingot for low carbon steel (0.06% C). It is clear that the results from numerical solution in this study have a very good compatibility with those of three 3-D model of Thomas and experimental measurement of Alberny and co-worker.Table 3. Input data for standard conditions.Carbon equivalent content, CE pct 0.132 pctSteel density, ρ7500 kg/m 3 Steel emissivity, ε0.8 Mold copper plates thickness 0.043×0.030 m ×m Total mold length 0.704 m Mold copper plates width 2.220×0.215 m ×m Scale thickness on the surface of mold cold face 0.00001 m Mold conductivity, k mold 315 W/mK Mold powder conductivity, k slag 1.27 W/mK Air conductivity, k air 0.083 W/mKMold powder density, ρslag0.650 kg/m 3 Mold powder consumption rate, M slag 0.8 kg/ton steel Casting speed, V c 0.0167 m/secPour temperature, T in 1546 oCLiquidus temperature, T liq. 1528.6 oCSolidus temperature, T sol. 1494 oC Working mold length 0.659 m Slab geometry, W ×N 1.250×0.203 m ×m Scale conductivity on the surface of slab, k sc 0.5 W/mK Scale conductivity on the surface of mold, k scale 1.0 W/mK Water channel geometry, large & small plates 25×5×29 & 22×5×26mm 3 Average cooling water temperature in the mold 28 oC Water flow rate entering the mold small plate, 0.0061 m 3/sec Water flow rate entering the mold large plate, 0.0553 m 3/sec Latent heat of the steel phase change, L f 272140 J/kgWater conductivity , k water0.615 W/mKSteel specific heat capacity [11], C p ()()()()⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪⎪⎬⎫⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎨⎧≥=≤=≤+=≤=−==+=+=liqp liq sol p sol op p o p p op o p T T C T T T C T T C T C T C T C T C T C T C T C T C T C 7877721100*334.02681100850648850750*766.338497507001431700500*836.0268500*376.0456p p p p p p p p p p J/kgK Steel conductivity, k steelf l *k liq +(1-f l )*k sol W/mK Solid steel conductivity, k sol 33.0 W/mK Effective molten steel conductivity, k liq 7*43.0 W/mKScale thickness on the surface of slab , δsc0.001 mA rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2Table 4. Effect of mesh size on the shell thickness at mold exit.Grid system #nx ×n y Shell thickness at middle of wide face (m)Shell thickness at middle of narrow face (m)25×10 0.0145 0.0111 25×15 0.0135 0.0110 50×15 0.0136 0.0118 50×20 0.0133 0.0119 100×200.0136 0.0117 100×300.0131 0.0117Fig. 9. Phase fraction variation with temperature in mushy zone.Fig. 10. Comparison between model results and other references.Results and discussionThe calculated surface temperatures of a slab, for the Table 3 conditions as a function of the distance below the meniscus, are presented in Figure 11. This figure shows the calculated surface temperatures at the centers of the wide and narrow faces and at the corners of the slab caster. The central areas of the wide faces are cooled one-dimensionally, whereas the slab corners are subject to 2-D cooling. The slab corners can therefore become significantly colder than other parts of the inner wide face. At the beginning of straightening, the slab corner temperature is 230 o C less than the temperature at the center of the wide face. Control of corner cooling is critical for much of continuously cast products. The slab corners tend to have meniscus marks, which act as stress risers. A combination of temperature, stress risers and a low ductility region in the 700-900 o C temperature range during the straightening process often leads to cracks in the corners. One way to increase the corner temperature is to widen the strip of unsprayed strand at the corner. However, as the non-sprayed strip is widened, a hot spot will develop between the colder corner and the sprayed area. Thus, the design must be in a way to ensure that it does not cause other quality problems. Also, as seen in Figure 11, the intensity of heat transfer of mist spray cooling is less than cooling in the lower part of the mold for all of the curves, because of this, the model predicts a 200 o C reheating of the slab surfaceon leaving the mold. A similar situation also exists in water spray cooling and air cooling regions on the surface of the wide face slab.Fig. 11. Predicted surface temperature of strand. Figure 12 shows the solidified shell thickness profiles of both the narrow and wide faces of the slab. This figure shows a sudden change of slope at the beginning of the solidified shell growth curve. It is indicated that the rate of solidified shell growth is clearly high in the mold region. Figure 13 shows a set of data of the local heat flux density along the length of the mold [15]. There is a maximum of heat flux density somewhat below the meniscus. Downward, the heat flux density usually decreases. If the air gapA rc hi v eIDInternational Journal of ISSI, Vol.3 (2006), No. 2between the mold and strand approximately has uniform thickness, or it increases uniformly in the downward direction, the heat flux density along the length of the mold continuously decreases.Fig. 12. Predicted solidified shell thickness.Fig. 13. Heat flux density distribution along the length of the mold.Fig. 14. Effect of casting speed on the pool depth.The casting speed is the most effective parameter in changing the position of the solidified shell thickness profiles. The relation of the "metallurgical length" (maximum length of the liquid pool) with the casting speed is shown in Figure 14. Increase in casting speed decreases the holding time of the slab in the secondary cooling zones and increases the length of the liquid core. Therefore, casting speed is the most important factor in controlling mold heat extraction. The solidified shell thickness as a function of the casting speed of bothwide face and narrow face of the strand are shown in Figure 15. Since, a lower casting speed provides more time for the heat to be extracted from the shell, the shell thickness increases. Moreover, the shell thickness in the initial solidification stage decreases at the high casting speed, as shown in Figure 15, which often easily causes the breakout of strand. Therefore, to prevent this unfavorable defect, the casting speed is limited. It should be mentioned that steel composition and slab width in comparison to casting speed has no significant effect on the shell thickness. As seen in this figure, solidified shell thickness for wide face is higher than it is for narrow face. This is because of the different air gap sizes between mold powder and mold wall in both wide and narrow faces. The air gap size along the length of the mold is predicted by the model for both faces in Figure 16. As seen in this figure, total shrinkage value in narrow face is more than wide face as expected because width size of slab is much larger than thickness size in cross section of strand. Furthermore, this phenomenon could change Fig. 15. Effect of casting speed on shell thickness.Fig. 16. air gap size along the mold.Heat transfer in the mold is governed by these three resistances: the casting-mold interface, the mold wall, and the mold-cooling water interface. Although, thermal resistance due to the air gapA rc hi v e o fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2 should also be considered while the amount of air gap thermal resistance is usually quite large compared to the other resistances especially for the lower portion of the mold. Temperatures of three points: slag layer/mold wall interface (T 1), mold wall (T 2), and water channel wall (T 3) as shown in Figure 7 are predicted by the model. Since, the heat flux for steady state conditions will be constant and independent of distance: (23)mold rad airradair slag s rad airrad air slags slags total water s r rr r r r T T rr r r r T T r T T r T T q +⎟⎟⎠⎞⎜⎜⎝⎛+⋅+−=⎟⎟⎠⎞⎜⎜⎝⎛+⋅+−=−=−=′321The above relation represents three equations for the three unknowns, T 1, T 2, and T 3. The results of the model predictions are shown in Figure 17(a) and Figure 17(b). As seen in these figures, the existence of the air gap between the shell surface and mold wall causes the T 1 to increase both in wide and narrow faces of strand. Since the air gap thickness in the narrow face is larger than the wide face, T 1 in the narrow face is much higher than in the wide face. Therefore, the pattern of heat removal in the mold is dependent largely upon the dynamics of gap formation. It causes temperature difference to exist in solidified shell and has a strong influence on transverse cracks generation in the mold region.Fig. 17. Predicted temperatures for T 1, T 2 and T 3 points in Fig 7 -(a) wide face and (b) narrow face (T 1=slag layer/mold wall interface temperature, T 2=mold wall temperature, and T 3=water channel wall temperature).The effect of the amount of superheat temperature of the molten steel, entering the mold, on the solidified shell thickness is shown in Figure 18. At first, with increasing the superheat temperature, the superheat flux will increase and therefore, the solidification rate decreases. This, in turn, reduces the thickness of the steel shell. On the other hand, by increasing the superheat temperature, the driving force for the heat transfer will increase. Therefore, as seen in Figure 18, the influence of the superheat temperature is insignificant to shell growth, especially in the wide face of the strand. Low superheat, near liquidus temperature, is beneficial to continuous casting. Internal quality is improved as the equated zone is made significantly larger. Therefore, a more desirable structure with greater resistance to halfway cracks is produced. Centerline segregation and porosity are also reduced or eliminated.Fig. 18. Effect of steel superheat temperature on the shell thickness at mold exit.Conclusions1- A finite volume heat transfer and solidification model has been formulated to predict the temperature field and liquid pool position in the continuous casting process under different conditions. This has been verified with the temperature measurement of slab surface.2- Casting speed is the most effective parameter on mold heat removal. Therefore, it is the most important factor in controlling solidified shell thickness and slab temperature.3- Since, the air gap size in narrow face of mold is higher than the wide face, the breakout of strand often occurs in the narrow face.4- Air gap existing in the casting-mold interface causes a large thermal resistance for heat transfer from the solidified shell to the mold. Therefore, it has a strong influence on product quality and casting problems, especially for the narrow face of strand. 5- High superheat temperature may cause breakouts at the mold exit, especially for the narrow face, so it should be exactly kept at a low level.。

摩擦纳米发电激发光动力学

摩擦纳米发电激发光动力学

摩擦纳米发电激发光动力学英文回答:Frictional nanogenerators (FNGs) are devices that can convert mechanical energy into electrical energy throughthe process of triboelectric effect. This technology has gained significant attention in recent years due to its potential applications in self-powered systems and wearable electronics. In this article, I will discuss the exciting field of optokinetics, which involves the use of FNGs to generate light.Optokinetics is the study of light-induced motion andits interaction with matter. It explores the principles of photophysics and photochemistry to understand how light can be used to control the movement of objects at the nanoscale. FNGs play a crucial role in optokinetics by providing the necessary electrical energy to drive light-emitting devices.The process of optokinetics begins with the generationof electrical energy through the triboelectric effect in FNGs. When two different materials come into contact and then separate, a charge imbalance is created on their surfaces. This charge imbalance can be harnessed togenerate an electric current. By integrating FNGs withlight-emitting devices, such as light-emitting diodes (LEDs), the electrical energy can be converted into light energy.One example of optokinetics in action is the development of self-powered nanoscale light sources.Imagine a scenario where you are in a dark room and need to find your way to the bathroom without turning on the lights. With optokinetics, you can wear a bracelet equipped with FNGs that generate electricity as you move your arm. This electrical energy can then be used to power tiny LEDs embedded in the bracelet, creating a small but sufficient amount of light to guide you in the dark.Another application of optokinetics is in the field of wearable electronics. For instance, researchers have developed smart clothing that can generate light patternsin response to specific gestures or movements. By incorporating FNGs into the fabric of the clothing, the mechanical energy produced during body movements can be converted into electrical energy, which in turn powers the light-emitting components. This technology has the potential to revolutionize fashion and entertainment industries by enabling interactive and visually appealing garments.In addition to self-powered light sources and wearable electronics, optokinetics can also be applied in other areas such as biomedical devices and environmental monitoring. For example, FNGs integrated with biosensors can generate light signals in response to specific biological markers, enabling real-time monitoring of disease progression or drug efficacy. Similarly, optokinetic sensors can be used to detect environmental pollutants by converting mechanical energy from air or water flow into light signals.中文回答:摩擦纳米发电器(FNG)是一种通过摩擦电效应将机械能转化为电能的设备。

机械专业毕业论文中英文翻译--在全接触条件下,盘式制动器摩擦激发瞬态热弹性不稳定的研究

机械专业毕业论文中英文翻译--在全接触条件下,盘式制动器摩擦激发瞬态热弹性不稳定的研究

Frictionally excited thermoelastic instability in disc brakes—Transientproblem in the full contact regimeAbstractExceeding the critical sliding velocity in disc brakes can cause unwanted forming of hot spots, non-uniform distribution of contact pressure, vibration, and also, in many cases, permanent damage of the disc. Consequently, in the last decade, a great deal of consideration has been given to modeling methods of thermo elastic instability (TEI), which leads to these effects. Models based on the finite element method are also being developed in addition to the analytical approach. The analytical model of TEI development described in the paper by Lee and Barber [Frictionally excited thermo elastic instability in automotive disk brakes. ASME Journal of Tribology 1993;115:607–14] has been expanded in the presented work. Specific attention was given to the modification of their model, to catch the fact that the arc length of pads is less than the circumference of the disc, and to the development of temperature perturbation amplitude in the early stage of breaking, when pads are in the full contact with the disc. A way is proposed how to take into account both of the initial non-flatness of the disc friction surface and change of the perturbation shape inside the disc in the course of braking.Keywords: Thermo elastic instability; TEI; Disc brake; Hot spots1. IntroductionFormation of hot spots as well as non-uniform distribution of the contact pressure is an unwanted effect emerging in disc brakes in the course of braking or during engagement of a transmission clutch. If the sliding velocity is high enough, this effect can become unstable and can result in disc material damage, frictional vibration, wear, etc. Therefore, a lot of experimental effort is being spent to understand better this effect (cf. Refs.) or to model it in the most feasible fashion. Barber described the thermo elastic instability (TEI)as the cause of the phenomenon. Later Dow and Burton and Burton et al.introduced a mathematical model to establish critical sliding velocity for instability, where two thermo elastic half-planes are considered in contact along their common interface. It is in a work by Lee and Barber that the effect of the thickness was considered and that a model applicable for disc brakes was proposed. Lee and Barber’s model is made up with a metallic layer sliding between twohalf-planes of frictional material. Only recently a parametric analysis of TEI in disc brakes was made or TEI in multi-disc clutches and brakes was modeled. The evolution of hot spots amplitudes has been addressed in Refs. Using analytical approach or the effect of intermittent contact was considered. Finally, the finite element method was also applied to render the onset of TEI (see Ref.).The analysis of nonlinear transient behavior in the mode, when separated contact regions occur, is even accomplished in Ref. As in the case of other engineering problems of instability, it turns out that a more accurate prediction by mathematical modeling is often questionable. This is mainly imparted by neglecting various imperfections and random fluctuations or by the impossibility to describe all possible influences appropriately. Therefore, some effort aroused to interpret results of certain experiments in addition to classical TEI (see, e.g.Ref).This paper is related to the work by Lee and Barber [7].Using an analytical approach, it treats the inception of TEI and the development of hot spots during the full contact regime in the disc brakes. The model proposed in Section 2 enables to cover finite thickness of both friction pads and the ribbed portion of the disc. Section 3 is devoted to the problems of modeling of partial disc surface contact with the pads. Section 4 introduces the term of ‘‘thermal capacity of perturbation’’ emphasizing its association with the value of growth rate, or the sliding velocity magnitude. An analysis of the disc friction surfaces non-flatness and its influence on initial amplitude of perturbations is put forward in the Section 5. Finally, the Section 6 offers a model of temperature perturbation development initiated by the mentioned initial discnon-flatness in the course of braking. The model being in use here comes from a differential equation that covers the variation of the‘‘thermal capacity’’ during the full contact regime of the braking.2. Elaboration of Lee and Barber modelThe brake disc is represented by three layers. The middle one of thickness 2a3 stands for the ribbed portion of the disc with full sidewalls of thickness a2 connected to it. The pads are represented by layers of thickness a1, which are immovable and pressed to each other by a uniform pressure p. The brake disc slips in between these pads at a constant velocity V.We will investigate the conditions under which a spatially sinusoidal perturbation in the temperature and stress fields can grow exponentially with respect to the time in a similar manner to that adopted by Lee and Barber. It is evidenced in their work [7] that it is sufficient to handle only the antisymmetric problem. The perturbations that are symmetric with respect to the midplane of the disc can grow at a velocity well above the sliding velocity V thus being made uninteresting.Let us introduce a coordinate system (x1; y1)fixed to one of the pads (see Fig. 1) thepoints of contact surface between the pad and disc having y1 = 0. Furthermore, let acoordinate system (x2; y2)be fixed to the disc with y2=0 for the points of the midplane. We suppose the perturbation to have a relative velocity ci with respect to the layer i, and the coordinate system (x; y)to move together with the perturbated field. Then we can writeV = c1 -c2; c2 = c3; x = x1 -c1t = x2 -c2t,x2 = x3; y = y2 =y3 =y1 + a2 + a3.We will search the perturbation of the uniform temperature field in the formand the perturbation of the contact pressure in the formwhere t is the time, b denotes a growth rate, subscript I refers to a layer in the model, and j =-1½is the imaginary unit. The parameter m=m(n)=2pin/cir =2pi/L, where n is the number of hot spots on the circumference of the disc cir and L is wavelength of perturbations. The symbols T0m and p0m in the above formulae denote the amplitudes of initial non-uniformities (e.g. fluctuations). Both perturbations (2) and (3) will be searched as complex functions their real part describing the actual perturbation of temperature or pressure field.Obviously, if the growth rate b<0, the initial fluctuations are damped. On the other hand, instability develops ifB〉0.2.1. Temperature field perturbationHeat flux in the direction of the x-axis is zero when the ribbed portion of the disc is considered. Next, let us denote ki = Ki/Qicpi coefficient of the layer i temperature diffusion. Parameters Ki, Qi, cpi are, respectively, the thermal conductivity, density and specific heat of the material for i =1,2. They have been re-calculated to the entire volume of the layer (i = 3) when the ribbed portion of the disc is considered. The perturbation of the temperature field is the solution of the equationsWith and it will meet the following conditions:1,The layers 1 and 2 will have the same temperature at the contact surface2,The layers 2 and 3 will reach the same temperature and the same heat flux in the direction y,3,Antisymmetric condition at the midplaneThe perturbations will be zero at the external surface of a friction pad(If, instead, zero heat flux through external surface has been specified, we obtain practically identical numerical solution for current pads).If we write the temperature development in individual layers in a suitable formwe obtainwhereand2.2. Thermo elastic stresses and displacementsFor the sake of simplicity, let us consider the ribbed portion of the disc to be isotropic environment with corrected modulus of elasticity though, actually, the stiffness of this layer in the direction x differs from that in the direction y. Such simplification is, however, admissible as the yielding central layer 3 practically does not take effect on the disc flexural rigidity unlike full sidewalls (layer 2). Given a thermal field perturbation, we can express the stress state and displacements caused by this perturbation for any layer. The thermo elastic problem can be solved by superimposing a particular solution on the general isothermal solution. We look for the particular solution of a layer in form of a strain potential. The general isothermal solution is given by means of the harmonic potentials after Green and Zerna (see Ref.[18]) and contains four coefficients A, B, C, D for every layer. The relateddisplacement and stress field components are written out in the Appendix A.在全接触条件下,盘式制动器摩擦激发瞬态热弹性不稳定的研究摘要超过临界滑动盘式制动器速度可能会导致形成局部过热,不统一的接触压力,振动分布,而且,在多数情况下,会造成盘式制动闸永久性损坏。

Heat Transfer in the Mold

Heat Transfer in the Mold

Heat Transfer in the MoldHeat transfer from Steel, Mold to Cooling WaterHeat transfer in the mold is critical and complex. The predominant transverse heat transfer can be considered as a flow of heat energy through a series of thermal resistances, from the high-temperature source of liquid steel core in the mold to the sink of cooling water of the mold-cooling system. It includes:∙Heat transfer in the solidifying casting.∙Heat transfer from steel shell surface (skin) to inner copper-lining surface.∙Heat transfer through copper lining.∙Heat transfer from outer copper-lining surface to mold-cooling water.In the solidifying castingHeat transfer in the solidifying casting occurs in a complex way since the heat to be extracted originates from enthalpy changes in the steel strand both from temperature decreases and phase changes. The former is referred to as sensible heat change and the latter as latent heat. Moreover, phase changes involve not only the changes between solid phases, but also the conditions produced by the solidification of an alloy. For example, a "mushy zone" exists between theliquidus and solidus temperatures which depend on the carbon content of the steel. In addition, the thermal resistance increases as the shell thickness increases from the meniscus to the bottom of the mold. Heat transfer in this region is by conduction.From steel shell surface to inner copper-lining surfaceHeat transfer in this step is most complex and is the controlling step in the mold. It involves mainly two mechanisms of heat transfer: conduction and radiation. The salient feature of this heat-transfer step is the shrinkage of the solidifying steel (which is a function of steel grade and caster operating conditions), and the resulting tendency for an air gap to form between the steel shell and the mold surface.The formation of the air gap is complex and may vary both in the transverse and longitudinal direction. Thus, it has a variable effect on the heat-transfer mechanism and the magnitude of heat flux. For example, as the air gap is formed, the heat transfer proceeds mainly from conduction to radiation with a resulting decrease in heat flux. In general, this heat-transfer step represents the largest thermal resistance of all of the four steps, especially with respect to heat transfer through the copper lining and from the latter to the mold cooling water.The entire pattern of heat removal in the mold is dependent on the dynamics of gap formation. In general, gap width tends to increase with increasing distance from the meniscus as the steel shell solidifies and shrinks away from the mold surface. In addition, as the shell thickness increases with distance from the meniscus, it tends to withstand the opposing bulging effect of the ferrostaticpressure to reduce the gap.Heat transfer at the copper inner surface is further complicated by the effects of mold lubrication. Another factor influencing heat transfer at this mold surface is the mold taper, which tends to increase heat transfer because it opposes the effect of gap formation.In general, the local heat flux down the mold length reaches a maximum value at or just below the liquid steel meniscus, and decreases down the mold length. The average heat flux for the whole mold increases with increasing casting speed.Through copper liningHeat transfer in this step is by conduction. It is dependent on the thermal conductivity of the copper and its thickness; the greater the thickness, the higher the hot-face temperature of the copper lining.From outer copper-lining surface to mold-cooling waterHeat transfer in this step is accomplished by forced convection. Although the bulk temperature of the cooling water, typically about 40°C (90*F, is usually below its saturation temperature at a given water pressure, boiling is still possible at local regions at the mold outer surface if the local temperature of this surface is sufficiently high for water vapor bubbles to nucleate at the surface, pass to the colder bulk cooling water, and condense. This effect increases heat transfer. Nucleate boiling can result in cycling of the temperature field through the copper mold (both at the cold face and the hot face) and can result in deleterious product quality. Boiling can be suppressed by increasing the water velocity in the cooling system or by raising the water pressure. Incipient boiling is more likely in billet molds, which have higher cold-face temperatures than slab molds because of their thinner wall thicknesses. Typical values for cold-face temperature are in the range of 150°C (302°F) for billet molds and 100°C (212°F) for slab molds.Cooling Water SystemControl of heat transfer in the mold is accomplished by a forced-convection cooling-water system, which must be designed to accommodate the high heat-transfer rates that result from the solidification process. In general, the cooling water enters at the mold bottom, passes vertically through a series of parallel water channels located between the outer mold wall and a steel containment jacket, and exits at the top of the mold.The primary control parameters are:∙The volume of water at the required water temperature, pressure and quality.∙The flow velocity of water uniformly through the passages around the perimeter of the mold liner.Water Volume, temperature, pressure and qualityTypically, a pressurized recirculating closed loop system is employed. The rate of water flow should be sufficient to absorb the heat from the strand without an excessive increase in bulk water temperature. A large increase in temperature could result in a decrease in heat-transfer effectiveness and higher mold temperatures. For this same reason, the inlet water temperature to the mold should also not be excessive; a proper mold water pressure is also required. For example, as discussed previously, higher water pressures tend to suppress boiling but excessively high pressures may cause mechanical mold deformation.Water quality is an important factor with regard to scale deposition on the mold liner. Scale deposition can be a serious problem because it causes an additional thermal resistance at the mold-cooling water interface that increases the mold-wall temperature leading to adverse effects such as vapor generation and a reduction in strength of the copper liner. The type and amount of scale formed is mainly dependent on the temperature and velocity of the cooling water, the cold-face temperature of the mold, and the type of water treatment.Water flow velocityTo achieve the proper flow velocity, the cooling system is designed such that the velocity is high enough to produce an effective heat-transfer coefficient at the mold-cooling water interface. Too low a flow velocity will produce a higher thermal resistance at this interface, which may lead to boiling and its adverse effects. In general, the higher the cooling-water velocity, the lower is the mold temperature. The cooling system should also be designed to maintain the required flow velocity distribution uniformly around the mold and to maximize the area of the faces that are directly water-cooled. Uniform flow distribution can be achieved by the proper geometrical design of the water passages with the use of headers and bale plates. Monitoring the operating parameters of the mold cooling system provides an assessment of the casting process. For example, with a constant cooling-water flow rate, the heat removed from a mold face will be directly related to the difference between the inlet and outlet watertemperature, AT. Thus an excessively large DT may indicate an abnormally low flow rate for one or more mold faces, whereas an excessively small DT may indicate an abnormally large scale buildup for one or more mold faces. An unequal DT for opposite faces may result from an unsymmetrical pouring stream mold distortion, or from strand misalignment.Mold OscillationDuring casting as the strand moves down the mold, tensile forces are developed in the solidifying skin due to high friction and sticking of the casting skin to the working face of the mold. The friction and sticking can be further enhanced by increasing ferrostatic pressure. If these tensile forces exceed the cohesive forces of the solidifying steel, the skin will tear and a breakout may occur. Sticking can be exacerbated by local rough areas in the mold such as gouges.To reduce the mold-strand adhesion and the risk of breakouts, in which liquid steel breaks through the thin solidified shell either in or below the mold, the mold is oscillated and lubricated. Oscillation may be accomplished by:∙motor-driven cams∙levers and cranks∙hydraulic actuation∙etc.Motor-driven cams, which support and reciprocate the mold, are used primarily.Mold oscillating cycles are many and varied with respect to frequency, amplitude and form. Many oscillation systems are designed so that the cycle can be changed when different section sizes on steel grades are cast on the same machine. However, there is one feature that has been adopted, almost without exception, which applies a negative strip to the solidifying shell. Negative strip is obtained by designing the "down stroke" of the cycle such that the mold moves faster than the withdrawal speed of the section being cast. Under these conditions, compressive stresses are developed in the solidifyingshell which tend to seal surface fissures and porosity and thus enhance the strength of the shell. During the "up stroke" portion of the cycle, the mold is very rapidly returned to the starting position and the cycle then repeated. Thus the shape of the oscillating cycle is non-symmetrical with respect to time.Mold LubricantsMold oscillation alone is insufficient to prevent skin ruptures and the use of mold lubricants is essential. Mold lubricants can be divided into two categories:∙liquids (oil lubricants)∙ solids (mold fluxes or mold powders)Oil lubricants (used with open pouring) tend to wet the copper mold and permit greater heat transfer at the upper part of the mold. Liquid oil lubricants include those of mineral, vegetable, animal and synthetic origin. Rapeseed oil was commonly used but is being replaced bysemi-refined vegetable oils. Because of the casting environment, the oil lubricants require high- temperature properties, such as a high flash point, so that they can effectively lubricate the mold surface in contact with the steel. The oil is continually injected through a series of small holes or slots in the upper portion of the mold above the steel meniscus to form a thin continuous film over the surface of the mold walls. Oils are principally used in billet or bloom machines casting silicon-killed steels.Solid lubricants (mold fluxes or mold powders) are widely used with submerged refractory tube shrouds in casting aluminum-killed steels on slab and bloom casters. Both mold fluxes (used with refractory shroud pouring) and mold powders result in greater heat transfer.The powders serve not only as lubricants but also provide other functions:∙Enhanced heat transfer at the strand-mold interface.∙Protection of the liquid metal surface in the mold from reoxidation by surrounding air.∙Thermal insulation of the liquid metal surface to prevent unwanted solidification, particularly at the wall-meniscus interface and at the submerged shroud.∙Absorb non-metallic inclusions that float to the liquid surface.Mold powder is added to the surface of the liquid steel shortly after the start of casting either manually by rakes or by mechanical feeders. Powder in contact with the liquid steel melts forming a liquid slag that then infiltrates between the mold wall and surface of the solidifying steel. Additional powder is added continually to replace that removed on the surface of the cast section. Lubrication by mold powders is a complex phenomenon and depends not only on flux properties such as viscosity, but also on the operating conditions, such as steel grade, casting speed, and oscillation condition.In addition to viscosity, which is dependent on the silica and alumina content of the powder, the melting point or crystallization temperature characteristics of the powder are also important. Very "fluid" slugs with low viscosities and low crystallization temperatures tend to provide the most effective heat transfer in the mold.Additional characteristics affecting the other functional requirements of powders include: a minimal iron oxide content, for example, to protect the liquid steel surface from reoxidation; and a low density which, together with graphitic carbon to retard sintering, fusibility and melting, enhances the thermal insulation capabilities.Mold powders consist of a mixture of materials of which Si02-CaO-AI203-Na2O-CaF2 is the basic component with varying amounts of carbon and other coinpounds. They can be broadly divided into:∙fly-ash based powders∙synthetic powders∙prefused, fritted or granulated powdersReferences[1] The making, Shaping and Treating of Steel, 1985, US Steel.[2] The making, Shaping and Treating of Steel, 2002, AISE Steel Foundation.。

Heat Transfer热传作业3题目

Heat Transfer热传作业3题目

2015/03/12, M3510A, Heat Transfer, Homework III, Due 2015/03/19Problem 2.5 Assume steady-state, one-dimensionalheat conduction through the symmetric shape shown.Assuming that there is no internal heat generation,derive an expression for the thermal conductivity()k x for these conditions: ()()1A x x =-,()()330012T x x x =--, and q = 6000 W, where A is in square meters, T in kelvins, and x in meters.Problem 2.16 Steady-state, one dimensionalconduction occurs in a rod of constant thermalconductivity k and variable cross-sectional area()0ax x A x A e =, where 0A and a are constants. Thelateral surface of the rod is well insulated. (a) Write anexpression for the conduction heat rate, ()x q x . Usethis expression to determine the temperaturedistribution ()T x . (b) Now consider conditions for which thermal energy is generated in therod at a volumetric rate ()0exp q q ax ''''''=-, where 0q ''' is a constant. Obtain an expression for ()x q x when the left face (x = 0) is well insulated.Problem 2.28 Uniform internal heat generation at 73610W m q '''=⨯ is occurring in a cylindrical nuclear reactor fuel rod of 60-mm diameter, and under steady-state conditions the temperature distribution is of the form ()52900 5.2610T r r =-⨯⨯, where T is in degrees Celsius and r is in meters. The fuel rod properties are 30W m K k =⋅, 31100kg m ρ=, and 800J kg K P c =⋅. (a) What is the rate of heat transfer per unit length of the rod at r = 0 (the centerline) and at r = 30 mm (the surface)? (b) If the reactor power level is suddenly increased to 83210W m q '''=, what is the initial time rate of temperature change at r = 0 and r = 30 mm?Problem 2.40 The steady-state temperature distribution in a one-dimensional wall of thermal conductivity k and thickness L is of the form 32T ax bx cx d =+++. Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces (x = 0, L).Problem 2.43 The cylindrical systemillustrated has negligible variation oftemperature in the r- and z-directions.Assume that o i r r r ∆=- is small comparedto i r , and denote the length in thez-direction, normal to the page, as L . (a)Beginning with a properly defined controlvolume and considering energy generationand storage effects, derive the differentialequation that prescribes the variation in temperature with the angular coordinate φ. (b) For steady-state conditions with no internal heat generation and constant properties, determine the temperature distribution ()T φ in terms of the constants 1T , 2T , i r , and o r . (c) For the conditions of part (b) write the expression for the heat rate q φ.Problem 2.51 A Spherical shell of inner and outer radii i r and o r , respectively, contains heat-dissipating components, and at a particular instant the temperature distribution in the shell is known to be of the form ()12C T r C r=+. Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius?。

裴攀-翻译中文

裴攀-翻译中文

第6章光源和放大器在光纤系统,光纤光源产生的光束携带的信息。

激光二极管和发光二极管是两种最常见的来源。

他们的微小尺寸与小直径的光纤兼容,其坚固的结构和低功耗要求与现代的固态电子兼容。

在以下几个GHz的工作系统,大部分(或数Gb /秒),信息贴到光束通过调节输入电流源。

外部调制(在第4、10章讨论)被认为是当这些率超标。

我们二极管LED和激光研究,包括操作方法,转移特性和调制。

我们计划以获得其他好的或理念的差异的两个来源,什么情况下调用。

当纤维损失导致信号功率低于要求的水平,光放大器都需要增强信号到有效的水平。

通过他们的使用,光纤链路可以延长。

因为光源和光放大器,如此多的共同点,他们都是在这一章处理。

1.发光二极管一个发光二极管[1,2]是一个PN结的半导体发光时正向偏置。

图6.1显示的连接器件、电路符号,能量块和二极管关联。

能带理论提供了对一个)简单的解释半导体发射器(和探测器)。

允许能带通过的是工作组,其显示的宽度能在图中,相隔一禁止区域(带隙)。

在上层能带称为导带,电子不一定要到移动单个原子都是免费的。

洞中有一个正电荷。

它们存在于原子电子的地点已经从一个中立带走,留下的电荷原子与净正。

自由电子与空穴重新结合可以,返回的中性原子状态。

能量被释放时,发生这种情况。

一个n -型半导体拥有自由电子数,如图图英寸6.1。

p型半导体有孔数自由。

当一种P型和一种N型材料费米能级(WF)的P和N的材料一致,并外加电压上作用时,产生的能垒如显示的数字所示。

重参杂材料,这种情况提供许多电子传到和过程中需要排放的孔。

在图中,电子能量增加垂直向上,能增加洞垂直向下。

因此,在N地区的自由电子没有足够的能量去穿越阻碍而移动到P区。

同样,空穴缺乏足够的能量克服障碍而移动进入n区。

当没有外加电压时,由于两种材料不同的费米能级产生的的能量阻碍,就不能自由移动。

外加电压通过升高的N端势能,降低一侧的P端势能,从而是阻碍减小。

如果供电电压(电子伏特)与能级(工作组)相同,自由电子和自由空穴就有足够的能量移动到交界区,如底部的数字显示,当一个自由电子在交界区遇到了一个空穴,电子可以下降到价带,并与空穴重组。

Heat transfer by convection, conduction and radiation

Heat transfer by convection, conduction and radiation
* Corresponding author. Tel.: +1 514 340 4711x4579; fax: +1 514 340 5917. E-mail address: bilgen@polymtl.ca (E. Bilgen).
1 Deceased author.
0142-727X/$ - see front matter Ó 2008 Published by Elsevier Inc. doi:10.1016/j.ijheatfluidflow.2008.08.006
article info
Article history: Received 3 April 2008 Received in revised form 20 August 2008 Accepted 22 August 2008 Available online 28 November 2008
Keywords: Solar chimney Conjugate heat transfer Natural convection Radiation Conduction Numerical simulation
Solar chimney may be described as a channel with two walls where an air movement is generated by buoyancy force generated by solar energy. The basic problem without considering conduction and surface radiation was studied in (Aung et al., 1972; Rheault and Bilgen, 1988). Both were on developing free convection in vertical channels, with asymmetric heating by Aung et al. (1972) and with non-isothermal vertical plates by Rheault and Bilgen (1988). In later numerical works, the conjugate heat transfer by free convection and conduction in vertical channels was studied by Kim et al. (1990), Burch et al. (1985) and Bilgen and Yamane (2004). Conjugate heat transfer by convection and conduction in systems having a glazing, solid wall and ventilation ports was numerically studied to determine their thermal performance in heating and ventilating of buildings by solar energy (e.g. Ben Yedder and Bilgen (1991) and references therein). Conjugate heat transfer by convection and surface radiation was studied numerically by Moshfegh and Sandberg (1996). They considered a vertical channel heated from one side with constant heat flux and insulated on the other side, and with various aspect ratios. They studied heat transfer by turbulent natural convection and surface radiation and revealed the importance of surface radiation on fluid flow in the channel. Bouali and Mezrhab (2006) studied heat transfer by laminar free convection and surface radiation in a divided vertical channel with

电池单元的安排和传热流体的寄生功率消耗和混合动力电动汽车的电池的温度分布的影响

电池单元的安排和传热流体的寄生功率消耗和混合动力电动汽车的电池的温度分布的影响

An observer looks at the cell temperature in automotive battery packsMaxime Debert a,Guillaume Colin b,n,Gérard Bloch c,Yann Chamaillard ba Renault SAS,Franceb Laboratoire PRISME,Universitéd'Orléans,Francec Centre de Recherche en Automatique de Nancy(CRAN),Universitéde Lorraine,CNRS,Francea r t i c l e i n f oArticle history:Received19September2012Accepted4March2013Keywords:Li-ion batteryElectric Vehicle(EV)Cell temperature estimationInternal resistance estimationLinear Parameter-Varying(LPV)systemPolytopic observera b s t r a c tThe internal temperature of Li-ion batteries for electric or hybrid vehicles is an important factorinfluencing their ageing.Generally not measured,it can be reconstructed from an external measurementand a model.This paper presents the simplified modelling of heat transfers in a battery module,leadingto a Linear Parameter-Varying(LPV)model.Then,a polytopic observer is proposed to estimate the celltemperature and internal resistance,ensuring a tradeoff between the convergence speed and the noise ofthe estimated states.Experimental results show the good quality of the estimation and the diagnosispotential offered by internal resistance reconstruction.&2013Elsevier Ltd.All rights reserved.1.IntroductionThe battery is the keystone of Electric Vehicles(EV)and Plug-inHybrid Electric Vehicles(PHEV),which require electric traction overextended periods in charge-depleting mode.It is a heavy,bulky,andcostly device,which constitutes the main obstacle to the massdistribution of such vehicles.Moreover,the lifetime of a currentbattery is shorter than that of the car.Improving the battery ageing iscertainly the key to the durability of the whole vehicle.The ageing process of Li-ion batteries is highly complex andaffected by many factors:internal temperature,use,calendar life(even unused,a battery loses little by little its capacity over time,Ratnakumar,Smart,Blosiu,&Surampudi,1999),state of charge(SOC),depth of discharge/charge(DOD/DOC),etc.Amongst theparameters influencing the performance degradation of suchbatteries,the internal temperature is known to be one of themost important(Broussely et al.,2005;Vetter et al.,2005).However,in general,due to the cost and manufacturing process,the internal temperature cannot be measured,neither at the coreof the reactions nor close to the battery cells.The Li-ion battery pack studied in this work is made up ofnumerous modules,and these modules are in turn composed ofseveral prismatic cells.The battery temperature is measured in thevehicles by a thermistor mounted in an interstice of an equippedmodule.The measurement gives therefore an image of thetemperature of the air trapped in the module,and the informationabout the cell temperature is smoothed and delayed.The reduc-tion can reach tens of degrees,with very long time delays,leadingto very conservative strategies of power limitation.Moreover,thismeans that the internal parameters,which vary according to theinternal temperature,cannot be properly estimated.It is therefore necessary to estimate the temperature as close aspossible to the cell.To this end,the temperature and internalresistance of the cell can be reconstructed from the externalmeasurement given at the module level,by using a thermal model.A similar approach was proposed to estimate the SOC in Hu,Yurkovich,Guezennec,and Yurkovich(2009)and Hu andYurkovich(2012).Lin et al.(in press)propose an adaptive observerof the core temperature of a cylindrical lithium ion battery whichpermits to deduce the internal resistance.In particular,monitoringthe internal resistance of the battery would make it possible todevelop diagnosis functions(Hall,Schoen,Powers,Liu,&Kirby,2005).In this paper the estimation of internal resistance is used formonitoring the battery safety by detecting abnormal conditions.The rest of the paper is organized as follows.Section2presentsthe control-oriented modelling of heat transfers in a batterymodule,leading to a Linear Parameter-Varying(LPV)model.Then,a so-called polytopic observer,with guaranteed convergence,isproposed in Section3to estimate the cell temperature andinternal resistance.An approach to limit the amplification of themeasurement noise for high observer gains is developed.Someresults obtained from a specially instrumented module are illu-strated in Section4,showing the good quality of the estimationand the ease of use for real-time implementation.The main contributions of the paper are twofold:a genericapproach for the design of an observer for LPV systems ensuringContents lists available at SciVerse ScienceDirectjournal homepage:/locate/conengpracControl Engineering Practice0967-0661/$-see front matter&2013Elsevier Ltd.All rights reserved./10.1016/j.conengprac.2013.03.001n Corresponding author.Tel.:+33238492420;fax:+33238417383.E-mail addresses:maxime.debert@(M.Debert),guillaume.colin@univ-orleans.fr(G.Colin),gerard.bloch@univ-lorraine.fr(G.Bloch),yann.chamaillard@univ-orleans.fr(Y.Chamaillard).Control Engineering Practice21(2013)1035–1042both the convergence of the estimation and a tradeoff between the convergence speed and the noise of the estimated states;the estimation of Li-ion battery internal resistance and temperature,simple to implement and built for on board diagnosis of batteries for electric or hybrid vehicles.2.Thermal model 2.1.Physical modelFor the design of the cell temperature observer,a control-oriented but representative model has to be built.In our case,the battery module contains several prismatic cells and the model differs from the one of Lin et al.(in press),which considers a cylindrical battery.The main assumptions of the proposed zero-dimensional model are that the cells have the same uniform temperature (this has been checked from measurements)and exchange heat with the air trapped in the module by natural convection.A thermistor,used in mass-produced vehicles,mea-sures this air temperature.The module is cooled down by convection,forced or not,depending on the cooling mode of the battery pack.The thermal model of the battery module is based on the physical equations of heat flow balance using equivalent thermal resistances (Debert,Colin,Mensler,Chamaillard,&Guzzella,2008;Forgez,Do,Friedrich,Morcrette,&Delacourt,2010;Guzzella &Sciarretta,2007;Lin et al.,2012;Muratori,Canova,Guezennec,&Rizzoni,2010;Pesaran,2002).The heat transfers are presented in Fig.1,where four temperatures are considered:the cell tempera-ture T cell ,the one given by the sensor T sens ,the temperature of the cooled wall (casing)T cas and the temperature of the air cooling the module T air .The difference between the cell edge and core temperatures does not justify another state (this is con firmed by the fact that the Biot number is small).The temperatures are in K,heat flows in W ¼J/s and thermal resistances in K/W.At the first node,the temperature T cell is givenby integrating the difference between the heat flows produced _Q 1and lost _Q 2:T cell ¼Zt0_Q 1−_Q 2C V 1dt ;ð1Þwhere C V 1the heat capacity at constant volume (J/K)of the cell.The heat flow _Q 1is produced by the Joule effect and chemical reaction:_Q 1¼RI 2batþT cell ΔS 1;ð2Þwith R the cell internal resistance (Ω),I bat the charge/dischargecurrent of the battery (A),ΔS the entropy change of the cell reaction,n the number of electrons exchanged and F Faraday 's constant.However,as shown in Williford,Viswanathan,and Zhang (2009)and con firmed by our experiments,the chemical contribu-tion ΔS can be neglected for the considered chemistry,so that _Q 1≈RI 2bat:ð3ÞThe heat flow _Q 2lost by the cell is caused by the convection of the air trapped inside the module:_Q 2¼T cell −T sens R eff 1;ð4Þwhere R eff 1¼1=h 1A 1is a thermal resistance,with A 1the area (m 2)swept by the cooling air and h 1the heat transfer coef ficient (W/m 2/K).At the second node,the temperature measured by the sensor T sens is in turn modelled byT sens ¼Z t 0_Q 2−_Q 3C V 2dt ;ð5Þwith C V 2the heat capacity at constant volume (J/K)of the airtrapped inside the module,and where again the heat flow _Q 3is expressed from a thermal resistance:_Q 3¼T sens −T cas R eff 2:ð6ÞFinally at the third node,the casing temperature T cas is obtained with the same approach:T cas ¼Z t 0_Q 3−_Q 4C p 3dt ;ð7Þwhere C p 3is the heat capacity at constant pressure (J/K)of theexternal air,and where _Q 4¼T cas −T air R eff 3:ð8ÞThe temperature of the air cooling the module T air is given by the cooling system and the thermal resistance R eff 3differs depending on the cooling mode (convection forced or not).2.2.State space modelIt is known that the internal resistance is in fluenced by many factors as the SOC,temperature or sign of the current.For the sake of estimator design,it is assumed that battery internal resistance variations are rather slow for a wide range of battery operating conditions,because SOC and battery temperature are typically slowly changing as well.When the battery is nearly fully charged or deeply discharged,these variations tend to be much more emphasized.However,these operating conditions are not likely to occur,because the battery management system needs to prevent them beforehand.On the other hand,detecting abnormal battery state from the estimated internal resistance is also aimed.There-fore,in order to avoid the non-detection of an abnormal situation due to an incorrect value of a parameter like SOC,temperature,cell voltage,the cell internal resistance R is simply assumed to vary slowly.This simpli fication has been also proposed in Plett (2004)for estimating the State Of Health (SOH)with a Kalman filter.However,if necessary,a more complex model could be used for the internal resistance.Thus,finally the thermal model canbeFig.1.Heat flow transfers and equivalent resistance modelling.M.Debert et al./Control Engineering Practice 21(2013)1035–10421036written as a state space model _T cell ¼k 1ðRI 2bat −k 4ðT cell −T sens ÞÞ_T sens ¼k 2ðk 4ðT cell −T sens Þ−k 5ðT sens −T cas ÞÞ_T cas ¼k 3ðk 5ðT sens −T cas Þ−k 6ðT cas −T air ÞÞ_R ¼0;8>>>><>>>>:ð9Þwith the parameters k 1¼1C V 1;k 2¼1C V 2;k 3¼1C p 3;k 4¼1R eff 1;k 5¼1R eff 2;k 6¼1R eff 3:8>>><>>>:ð10ÞThese parameters,assumed to be constant because of the tem-perature variation considered,can be theoretically calculated,e.g.from Nusselt numbers (Cheron,1999).They will be identi fied here from experiments,as presented below in Section 2.3.De fining the system state vector,input and output respectively as x ¼½T cell T sens T cas R T ,u ¼T air and y ¼T cell ,a Linear Parameter-Varying (LPV)model in continuous time t :_xðt Þ¼A c ðρðt ÞÞx ðt Þþb c u ðt Þy ðt Þ¼cx ðt Þ(ð11Þcan be written,withA c ðρðt ÞÞ¼−k 1k 4k 1k 40k 1I 2bat ðt Þk 2k 4−k 2ðk 4þk 5Þk 2k 500k 3k 5−k 3ðk 5þk 6Þ000266664377775;ð12Þb c ¼00k 3k 602666437775;c ¼½0100 ;ð13Þwhere the varying parameter ρðt Þ¼I 2bat ðt Þis available because the current of battery charge or discharge I bat is measured.Approximating _xðt Þby ðx k þ1−x k Þ=t s ,with the sampling period t s ,a discrete-time model is finally obtained:x k þ1¼A ðρk Þx k þb u ky k ¼cx k ;(ð14Þwith A ¼I þt s A c and b ¼t s b c .2.3.Parameter identi ficationFor the parameter identi fication of the proposed thermal model (14),an experimental module was introduced in a battery pack.This module,shown in Fig.2,is instrumented with a thermocouple placed between the most internal cells (in red,giving T cell ),the mass production temperature sensor (in green,giving T sens ),and a thermocouple placed on the casing (in blue,giving T cas ).A cycle of battery use was carried out,with a charge followed by a typical Electric Vehicle (EV)charge depleting.The gathered data have been divided into two parts,an identi fication set only used for parameter estimation and a validation set for model testing.The parameters were identi fied by a Nelder –Mead simplex (direct search)algorithm minimizing the Root Mean Squared Errors (RMSE)between the predicted and measured values,giving k 1¼0:0008,k 2¼0:4657,k 3¼0:0041,k 4¼1:1461,k 5¼3:8434,k 6¼0:3801.Several initializations have been performed to ensure satisfactory convergence.For a sampling period t s ¼0:1s,thisleads for system(14)to the following matrices:A ¼0:9999089:169e −408e −4I 2bat ðk Þ0:5337380:7676381:78987000:015750:9982681266664377775;ð15Þb ¼001:5584e −402666437775;c ¼½0100 :ð16ÞData and estimation results are shown in Fig.3.The model is then simulated on the validation data as shown in Fig.4.It is worth noting that the simulation is an N-step ahead prediction based only on the battery current,included in the time-varying parameter,and constant air external temperature T air .Table 1shows the Root Mean Squared Error and the correlation coef ficient for the estimation of the three temperatures T cell ,T sens and T cas on the identi fication data and on the validation data.ThethermistorThermocoupleFig.2.Specially equipped module for parameter identi fication.(For interpretationof the references to color in this figure caption,the reader is referred to the web version of this article.)50010001500200025003000350025262728293031323334T e m p e r a t u r e (°C)500100015002000250030003500−0.4−0.200.2E r r o r (°C )−200−1000100C u r r e n t (A )Time (s)Fig.3.Identi fication of the thermal model.M.Debert et al./Control Engineering Practice 21(2013)1035–10421037model gives quite good results on the identi fication data.On the validation data,the errors are bigger but the dynamics are respected.Therefore an (closed loop)observer is necessary to compensate these estimation errors and accommodate the ageing and parameter dispersion of batteries.3.Observer for LPV systemThis section is devoted to the design of an observer for system (14),with guaranteed convergence while limiting the effect of the noise measurement on the reconstructed state.3.1.PrincipleConsider the discrete time Linear Parameter-Varying (LPV)system subject to measurement noise x k þ1¼A ðρk Þx k þBu ky k ¼Cx k þDv k (ð17Þwhere x k ∈R n ,u k ∈R p ,y k ∈R m are respectively the state,input and output vectors of the system,v k ∈R q the zero-mean measurement noise,and ρk a measurable time varying parameter vector.As ρk is bounded,it evolves in a compact set and can always be included in a convex polytope D ρ.Hence,ρk admits the polytopic decomposition ρk ¼∑Ni ¼1ξi k θi ;ð18Þwhere the vector ξk ¼½ξ1k …ξN kTbelongs to the convex set S ¼f ξk ∈R N ;ξi k ≥0∀i ;∑Ni ¼1ξi k ¼1g .The constant vectors θi ;…;θN are the N vertices of the convex polytope D ρ.Note that,at each time k ,ξk depends on the available time-varying parameter ρk .It will be noted ξk ðρk Þto recall this dependence.Then,as the state matrix A ðρk Þfor LPV systems depends linearly on ρk ,it can be rewritten in a polytopic form (Millérioux,Anstett,&Bloch,2005;Halimi,Millerioux,&Daafouz,2013)asA ðρk Þ¼∑Ni ¼1ξi k ðρk ÞA i :ð19ÞThe constant matrices A i ,i ¼1;…;N ,are the vertices of the convex hull D A including the compact set where A ðρk Þevolves.The state reconstruction for system (17)can be achieved by a so-called polytopic observer ^xk þ1¼A ðρk Þ^x k þBu k þK ðρk Þðy k −^y k Þ^y k ¼C ^x k ;(ð20Þwhere the observer gain can be time varying,i.e.K ðρk Þ¼∑N i ¼1ξi k ðρk ÞK i ,or constant,i.e.K ðρk Þ¼K ,which is simpler for further use,but will constrain the LMIs to be solved.From (17),(20)and (19),it can be shown that the statereconstruction error ϵk ¼x k −^x k is governed by ϵk þ1¼A ðρk Þϵk −K ðρk ÞDv k ;ð21Þwhere A ðρk Þ¼∑N i ¼1ξik ðρk ÞðA i −K i C Þ,with K i ¼K for constant gain.Consider first the noise free case,i.e.v k ¼0andϵk þ1¼A ðρk Þϵk :ð22ÞThe conditions of global convergence toward zero of the state reconstruction error (22)are ensured from the following theorem (Millérioux,Rosier,Bloch,&Daafouz,2004).Theorem 1.The global convergence of (22)is achieved whenever the following set of Linear Matrix Inequalities :P i A T i G T i −C T F TiG i A i −F i C G i þG Ti −P j 243540ð23Þwhere the positive-de finite matrices P i ,the matrices F i and G i are unknown ,is feasible for all ði ;j Þ∈f 1;…;N g Âf 1;…;N g .The matrices F i and G i are determined when solving (23)by convex optimization.Consider now noisy measurements,i.e.v k ≠0.Although the stability of the state reconstruction error is ensured by Theorem 1,high gain values which can be obtained by solving the LMIs (23)can lead to unusable estimated states.Indeed,as can be seen in (21),the observer gain K also acts on the measurement noise.Thus,there is a need to balance the speed of convergence and the in fluence of measurement noise.To this end,let us de fine the upper bound s of the L 2gain as a scalar verifying sup ∥v ∥2≠0∥z ∥2∥v ∥2o s ð24Þwhere z k ¼~C ϵk .Considering z k instead of ϵkdirectly allows for instance to select particular reconstructed states through ~C.The following Theorem 2can thus be stated,with a proof,based on Millérioux and Daafouz (2006),given in the Appendix.Theorem 2.The global convergence of (21),with v k ≠0and the constraint (24),is ensured if there exist symmetric matrices P i and matrices G i and F i such that ,for ði ;j Þ∈f 1…N g Âf 1…N g ,the followingT e m p e r a t u r e (°C )−202E r r o r (°C )C u r r e n t (A )Time (s)Fig.4.Validation of the thermal model.Table 1Root Mean Squared Error (RMSE)and correlation coef ficient with the thermal model for the estimation of the three temperatures T cell ,T sens and T cas .DataTemperature RMSE (1C)Correlation coef ficient Identi ficationT cell 0.170.998T sens 0.100.999T cas 0.100.999ValidationT cell 1.60.959T sens 1.30.961T cas1.20.957M.Debert et al./Control Engineering Practice 21(2013)1035–10421038matrix inequalities are feasible :P i0ðG i A i −F i C ÞT ~C T 0s 1ð−F i D ÞT 0G i A i −F iC −F iD G i þG T i −P j0~Cs 126666437777540:ð25ÞFor both noise free and noisy cases,the time varying observer gain is given byK ðρk Þ¼∑N i ¼1ξik K iwith K i ¼G −1i F ið26Þand,replacing the matrices F i by F and G i by G in (23)or (25),theconstant gain K simply by K ¼G −1F :ð27Þ3.2.Limiting the noise effectFrom the thermal system (14)with (15)and (16),the detri-mental effect of the measurement noise on the estimated cell temperature and the interest of limiting this effect are illustrated in simulation.Note that,near zero current,the observability matrix is not of full rank so that the system is not observable.Thus for j I bat j o 0:5A (maximum considered sensor error),the observer is not active and only the open loop model (14)is run.Another solution could be to estimate only a part of the states and so to consider a constant internal resistance at low load.For a battery current switching from 0.5A to 100A at 100s during 100s and a temperature of the air cooling the module T air ¼301C,the observers are simulated when a white Gaussian measurement noise,with a rather weak standard deviation of 0.11C,is added to the air temperature T sens given by the thermis-tor.A wide range of variation 0:5A ≤j I bat j ≤400A is considered for the charge/discharge current.Since the time varying parameter ρk ¼I 2bat ðk Þis scalar,the polytopic decomposition (18)reduces to elementary computations.Indeed,ρk evolves between the minimal ρmin and maximal ρmax values,which are the N ¼2vertices θ1and θ2of the convex polytope D ρ.Thus,(18)turns into ρk ¼ξ1k ρmin þξ2k ρmax .Since ∑Ni ¼1ξi k ¼1,ρk ¼ξ1k ρmin þð1−ξ1k Þρmax ,and (19)isreduced to A ðρk Þ¼ξ1k A 1þð1−ξ1k ÞA 2,where A 1and A 2are obtainedfrom A (15)by replacing I 2bat ðk Þby ρmin ¼0:52and ρmax ¼4002.By applying Theorem 1ignoring the measurement noise and solving the LMIs (23)with the toolbox Yalmip,1a time varying gain observer gain K ðρk Þ(26)is obtained by linear interpolation between the gains K 1¼½16:31391:6241−0:07380780:528547 T and K 2¼½23:22291:65147−0:07005010:519493 T .Nevertheless,as illustrated in Fig.5,the observer yields noisy estimates,particu-larly for T cell .By using now Theorem 2with ~Cð1;1Þ¼1,~C ði ;j Þ¼0,i ;j ≠1,in order to limit the noise effect on the reconstruction of T cell ,the gains K 1and K 2are found to be similar.Then Theorem 2is applied to find a constant observer gain (27).Solving the corresponding LMIs with,for (24),s ¼7:8then yields K ¼½0:418830:122810:035940:00010075 T .The results given in Fig.6show that the observer with a constant gain,easier to implement in real time,is suitable for the considered application,and is robust to noise.Recall that the high level bound s (24)can be tuned in order to decrease the noise effect.4.Experimental resultsThe observer proposed in Theorem 2,with a constant gain (27),is now applied on two real data sets (a normal use and a battery overcharge)for the reconstruction of the cell temperature and internal resistance,still based on model (14)with (15)and (16).4.1.Normal useThe first real cycle,shown in Fig.7,concatenates a typical EV charge depleting and a complete charge of the battery.Note that this State Of Charge (SOC )estimation is not used by the observer that allows the battery monitoring unit to be independent of this estimated variable.Again,for the state reconstruction,a wide range of variation 0:5A ≤j I bat j ≤400A is considered for the charge/discharge current.With s ¼7:8,a constant observer gain K ¼½0:418830:122810:035940:00010075 T is obtained.Fig.8shows that the tempera-tures T sens and T cas are very well estimated,and that the cell temperature T cell is estimated with a good accuracy,at least suf ficient for real time reconstruction and monitoring of the internal battery temperature.The corresponding estimated time varying cell internal resistance is shown in Fig.9compared with the estimation ^R¼f ðT sens ;SOC Þð28Þ05001000150029.53030.53131.53232.5Time (s)T e m p e r a t u r e s (°C)Fig.5.Simulation results with noise using Theorem 1.29.83030.230.430.630.83131.231.431.631.8Time (s)T e m p e r a t u r e s (°C )Fig.6.Simulation with noise using Theorem 2(constant gain,s ¼7:8).1http://users.isy.liu.se/johanl/yalmipM.Debert et al./Control Engineering Practice 21(2013)1035–10421039where f is commonly given by a look-up table.This table isillustrated in Fig.10and the estimation ^Ris calculated from data shown in Fig.8for T sens ,and Fig.7for SOC .The trends of both the estimations are similar.The differences can be attributed to the fact that the look-up table was built from a cell at the beginning of its life with a static identi fication of the internal resistance whereas the module used for the observer design of internal cell temperature and resistance was equipped with cells in the middle of their lifecycle.Moreover,the observed internal resistance represents the overall impedance of the cell which is not the same as the static resistance (Jossen,2006).Table 2shows the Root Mean Squared Error and the correlation coef ficient for the observation of the three temperatures T cell ,T sensand T cas on the validation data of Section 2.3.The observer improves the previous “open loop ”estimation shown in Table 1.Note that it gives relevant results even with a reduced thermal model and the assumption of slowly varying internal resistance.4.2.OverchargeThe observer is now applied in a second real test,a battery overcharge with a constant current of 8A.The real and estimated temperatures are shown in Fig.11and the corresponding esti-mated cell internal resistance in Fig.12.Because the look-up table does not include overcharge conditions,no comparison is given.However,it is known that in the case of overcharge the internal resistance increases (Hall et al.,2005).Hence,the simultaneous50100Time (s)S t a t e O f C h a r g e (\%)C u r r e n t (A )Fig.7.Battery current (A)(solid line)and State Of Charge (%)(dashed line)versus time (s)for the first experimental test.Time (s)T e m p e r a t u r e s (°C)Fig.8.Real and estimated temperatures (constant gain,s ¼7:8).0.20.250.30.350.40.450.5Time (s)N o r m a l i z e d i n t e r n a l r e s i s t a n c eFig.9.Estimated (normalized w.r.t.minimum and maximum)internal resistances:proposed observer (solid line)and look-up-table (28)(dashline).Temperature (°C)S O C (%)Normalized resistance102030405060708090100Fig.10.Look-up table of the internal resistance (normalized w.r.t.minimum and maximum)w.r.t.temperature and State Of Charge.Table 2Root Mean Squared Error (RMSE)and correlation coef ficient with the polytopic observer for the estimation of the three temperatures T cell ,T sens and T cas on the validation data.Temperature RMSE (1C)Correlation coef ficient T cell 0.380.997T sens 0.140.998T cas0.140.998010002000300040005000600070008000900014161820222426Time (s)T e m p e r a t u r e s (°C )Fig.11.Real and estimated temperatures during a battery overcharge.M.Debert et al./Control Engineering Practice 21(2013)1035–10421040reconstruction of the cell temperature and internal resistance makes it possible to detect overcharges by checking a threshold crossing of the estimated resistance.For example,as the resistance is normalized,it must be always less than 1,which is not the case in Fig.12due to the overcharge.The proposed overcharge detec-tion from the estimation of internal resistance increases the battery safety since overcharge can be monitored by two inde-pendent modules of the Battery Management System (BMS),the standard one using a cell voltage sensor and the proposed one based on current and temperature sensors.5.ConclusionFor the simultaneous estimation of the cell temperature andinternal resistance of a Li-ion battery pack,a control-oriented thermal modelling has been carried out.This led to a LPV model from which an observer,with guaranteed convergence,can be designed.A generic approach for LPV systems limiting the in flu-ence of measurement noise on the estimates has been proposed.Considering the quality of the estimation which has been demon-strated on real data,the proposed observer could be included in the battery management systems of electric vehicles to increase availability,longevity and safety of the batteries.Indeed,the simultaneous reconstruction of the cell temperature and internal resistance makes it possible to develop diagnosis functions in order for example to detect overcharges increasing the safety during the battery charge procedure.Appendix A.Proof of Theorem 2Proof.Let us de fine matrices G i ,P i and M i :G i ¼G i 00s 1 ;P i ¼P i 00s 1 ;M i ¼A i −F i C−F i D s −1~C 0ð29ÞEq.(25)can be rewritten as P iðG i M i ÞT G i M i G i þG T i −P j "#40:ð30ÞAs P i is strictly positive,thenG i P −1j G T i ≥G i þG T i −P jð31Þand thus P i ðG i M i ÞT G i M iG i P −1j G Ti"#40:ð32ÞThis inequality is identical to100G i P −1j "#P iðP j M i ÞT P j M i P j "#100ðG i P −1j ÞT"#40ð33Þand thusP i ðP j M i ÞT P j M iP j"#40ð34Þbecause G i and P j are full rank matrices.Finally,for each i ¼1…N ,the j ¼1…N inequalities (34)aremultiplied by ξj k þ1and summed.Then,the i ¼1…N inequalities are multiplied by ξik and summed again.Then P kðP k þ1M ÞT P k þ1M P k þ1"#40ð35Þwith P k ¼∑N i ¼1ξi k P i and P k þ1¼∑N i ¼1ξi k þ1P i .With the Schur com-plement,one has directlyM T P k þ1M −P k o 0ð36Þwith P k þ1¼P k þ100s 1;P k ¼P k 00s 1;M ¼A B s −1~Cð37Þand finally A ¼∑N i ¼1ξi k ðA i −F i C Þand B ¼∑Ni ¼1ξi k ð−F i D Þ.Eq.(36)can be then rewritten as A T P k þ1A þs −1~C T ~C −P kA T P k þ1B 0B T P k þ1AB T P k þ1B −s 1000−s 1264375o 0:ð38ÞMultiplying,respectively left and right,by ½ϵk v k and its transpose,leads toV ðϵk þ1;ξk þ1Þ−V ðϵk ;ξk Þþs −1ð~C ϵk ÞT ð~C ϵk Þ−s v T k v ko 0:ð39ÞConsidering this equation for k ¼0…N and summing leads to V ðϵN þ1;ξN þ1Þþs −1∑Nk ¼0ð~C ϵk ÞT ð~C ϵk Þ−s ∑Nk ¼0v T k v ko 0:ð40ÞBecause V ðϵN þ1;ξN þ1Þ¼ϵT N þ1P k ϵN þ140,then s −1∑Nk ¼0ð~C ϵk ÞT ð~C ϵk Þo s ∑Nk ¼0v T k v kð41Þand,when N -∞,this expression is equivalent to (24).□ReferencesBroussely,M.,Biensan,P.,Bonhomme,F.,Blanchard,P.,Herreyre,S.,Nechev,K.,&Staniewicz,R.(2005).Main aging mechanisms in Li ion batteries.Journal of Power Sources ,146(1–2),90–96.Cheron,B.(1999).Transferts thermiques,Ellipses ..Debert,M.,Colin,G.,Mensler,M.,Chamaillard,Y.,&Guzzella,L.(2008).Li-ionbattery models for HEV simulator.In:IFP (Ed.),Advances in hybrid powertrains,Rueil-Malmaison,France ..Forgez,C.,Do,D.V.,Friedrich,G.,Morcrette,M.,&Delacourt,C.(2010).Thermalmodeling of a cylindrical LiFePO 4/graphite lithium-ion battery.Journal of Power Sources ,195(9),2961–2968.Guzzella,L.,&Sciarretta, A.(2007).Vehicle propulsion systems.Introduction tomodeling and optimization (2nd ed.).Springer.Halimi,Meriem Millerioux,Gilles Daafouz,Jamal 2013.Polytopic Observers for LPVDiscrete-Time Systems.In:Sename,Olivier Gaspar,Peter Bokor,József (Eds.),Robust Control and Linear Parameter Varying Approaches.Lecture Notes in Control and Information Sciences,vol.437,pp.97-124.Doi:10.1007/978-3-642-36110-4_5,ISBN:978-3-642-36109-8.01000200030004000500060007000800090000.40.60.811.21.41.61.8Time (s)N o r m a l i z e d i n t e r n a l r e s i s t a n c eFig.12.Estimated cell internal resistance (normalized)during a batteryovercharge.M.Debert et al./Control Engineering Practice 21(2013)1035–10421041。

热学技术 HEAT TECHNOLOGY

热学技术 HEAT TECHNOLOGY

Radiation- Surface radiation
The radiation is emitted and absorbed by a surface depends on the material, the surface structure, and the temperature.
Black Body

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For a forced convection flow the Reynolds number Re is : vL Re
Convection
For the free convection, the Grashof number
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Coefficience h is not a material property, but depends on: ※ the geometry of wall(horizontal, vertical, straight, curved) ※ the flow (velocity, laminar or turbulent, free convection or forced convection) ※ the material properties of the gas or the liquid (thermal conductivity, density, heat capacity, viscosity)
Surface radiation
The spectral emissivity of several materials often depends to a large extent on the wavelength.

已定型的含阿拉伯数字的词语

已定型的含阿拉伯数字的词语

114青岛大学学报(工程技术版)第33卷[14]C h e n S C, W a n C C,W a n g Y Y. T h e r m a l Analysis of L i t h i u m-I o n Batteries[J]. Journal of P o w e r Sources, 2005,140(1):111-124.[15]李聪聪,陈强,喻凡,等.起停系统电池温度建模及试验研究)].汽车工程,2013,35(1):60 - 65.[16]胡明辉,秦大同,石万凯,等.混合动力汽车镍氢电池组温度场研究)].汽车工程,2007,29(1):37 -40.[17]杨世铭,陶文铨.传热学[M].4版.北京:高等教育出版社,2006.[18]段瑶娟.动力锂离子电池包热场分析与散热优化研究[D].北京:北京交通大学,2015.[19]余志生.汽车理论[M].5版.北京:机械工业出版社,2009.)0]喻凡,林逸.汽车系统动力学[M].北京:机械工业出版社,2005.Study on Temperature Characteristics of High-PowerLi-Ion Cells under NEDCL I Y u a n y u a n,H U O W e i,L I Z h i m i n g,F A N H u a m i n,H U A N G F u c h u a n g (School of Electrom echanic E n gin eerin g,Qingdao U n iv rrs ity,Qingdao 266071, C hina)Abstract:In this p a p rr, 18650-Li-ionbatteries are research ed,to explore the therm al characteristics of the high-power Li-ionbattery undrr the N E D C.A ccording to O C V curve of Li-ion battery and theJoule's law of internalre sista n t,the m athem atical model of heating characteristics of high-pow er Li-ionbattery based on N E D C is tablish ed.Based on Newton's cooling form ula and Stephen-Boltzm ann's la w,a m athem atical model of heat d isii-pation during battery N E D C operation is established.T hen therm al sim ulation of the battery is perform ed w ith-M atlab.T h e experim ental m ethod of therm al characteristics of high-pow er Li-ionbattery based on N E D C is ex­p lo red,and the circuit platform of the experim entis b uilt.T h e output power of the high-pow er Li-ioncell iscontrolled by the duty cycle of PW M w ave based on N E D C.D uring Experim ental p ro c ess,the experim ental da­ta im port into the hard disk of a P C that connected w ith an o scilloscop e,the device m easures experim ental vari­ables.T h e results show that the PW M w ave m ethod w hich is equivalent to N E D C output power cou m ission of m easurem ent during lithium b attery discharge in a high-pow er and the end of the second cy c le,the-high-pow er Li-ioncell is a bout to e x h a u st,and its internal resistance rose sh arp ly.T akin g into account the safe­ty problem at this tim e thehigh-pow er Li-ioncell should no longer be used.Keywords:N E D C#tem p eratu re#M a tla b#lithium-ion c e ll#single-chip摘要的写作要求摘要应按照四要素编写,即:目的(本人针对什么问题,或为解决什么问题)、方法(解决该问题采用 了那些方法,包括:所用的原理、理论、条件、对象、材料、工艺、结构、手段、装备、程序等)、结果(应陈述主要研究成果,包括所获得的实验结果。

APS 传热学 Types of Heat transfer

APS  传热学 Types of Heat transfer

Types of Heat transferThere are three modes of Heat transfer:1. Conduction2. Convection3. RadiationConduction:takes place between the two solid bodies in contact, two regions of the same solid body. This will happen because of the hot, vibrating, and rapidly moving molecules transfer the heat to their neighboring atoms.Convection:takes place through a medium and the medium may be liquid or the gas. The heat transfer takes place by the movement of fluid from one place to another. The heat transfer here is due to the bulk motion of the fluid.Convection is described by the Newton’s law of cooling; the law states that the rate of heat loosed by a body is proportional to the difference in temperatures between the body and its surroundings.Radiation:The Third mode of energy transfer is Radiation Heat Transfer. The transfer of heat from hot body to a cold body with any material medium for propagation.Thermal Radiation:The waves emitted are of different frequency, the variation in the frequency is because of the difference in the degree of hotness of the material. Sun also emits thermal radiation due to the extremely hot gases present in it, and this radiation heats the Earth. This thermal radiation is of great importance in the Thermodynamics. It is just a kind of conversion of heat energy in the form of electromagnetic waves. Warmer body emits more heat than the colder body. Stefan Boltzmann LawStefan Boltzmann Law states that the emissive power of a (black) body is proportional to the forth power of the absolute temperature.Mathematically, it is given asp = σAεT4where ,Ρ= Power radiated in watts by the substance,σ = Stefan-Boltzmann constant,A = surface area of the substance in meters,ε= emissivity constant, 0<= ε<=1T = Surface temperature in KThe value of ε depends on the properties of the substance surface from which the radiations are to be transmitted.A heat exchanger isa piece of equipment built for efficient heat transfer from one medium to another.A Shell and Tube heat exchangerShell and tube heat exchanger, single pass (1–1 parallel flow)Countercurrent (A) and parallel (B) flows热电偶Thermocouples are a widely used type of temperature sensor for measurement and control, [ 1 ] and can also convert a temperature gradient into electricity.。

heat-transfer

heat-transfer
set up.
The third method of heat transfer
How does heat energy get from the Sun to the Earth?
There are no particles between the Sun and the Earth so it CANNOT travel by conduction or by convection.
Water movement
Cools at the surface
Cooler water sinks
Convection current
Hot water rises
Why is it windy at the seaside?
Cold air sinks
Where is the freezer
RADIATION
?
Radiation
Radiation travels in straight lines True/False
Radiation can travel through a vacuum True/False
Understanding Heat Transfer, Conduction, Convection and
Radiation
Heat Transfer
• Heat always moves from a warmer place to a cooler place.
• Hot objects in a cooler room will cool to room temperature.
Insulators, such as wo_o_d_ and pl_a_st_i_c , do not have this ‘sea of electrons’ which is why they do not conduct heat as well as metals.

铁素体转变开始温度的英语

铁素体转变开始温度的英语

铁素体转变开始温度的英语The Transformation of Ferrite to Austenite Start TemperatureThe transformation of ferrite to austenite is a critical phase change that occurs in the heat treatment of steel. This transformation plays a crucial role in determining the final microstructure and properties of the steel, and understanding the factors that influence the start temperature of this transformation is essential for optimizing the heat treatment process.The start temperature of the ferrite to austenite transformation is influenced by a variety of factors including the chemical composition of the steel, the heating rate, and the prior microstructure of the material. The chemical composition of the steel, particularly the content of alloying elements such as carbon, chromium, and nickel, can have a significant impact on the transformation start temperature.Carbon is one of the most important alloying elements in steel and has a significant influence on the transformation start temperature. Increasing the carbon content of the steel generally lowers the transformation start temperature. This is because carbon atoms canstabilize the austenite phase, making it more difficult for the ferrite phase to form. As a result, the transformation to austenite requires higher temperatures to occur.Other alloying elements can also affect the transformation start temperature. Chromium, for example, tends to increase the transformation start temperature by promoting the formation of ferrite. Nickel, on the other hand, can lower the transformation start temperature by stabilizing the austenite phase.The heating rate is another important factor that can influence the transformation start temperature. Faster heating rates can result in a higher transformation start temperature, as the steel does not have as much time to undergo the necessary atomic rearrangements required for the transformation to occur. Slower heating rates, on the other hand, allow more time for the transformation to occur at lower temperatures.The prior microstructure of the steel can also play a role in the transformation start temperature. For example, the presence of residual stresses or defects in the microstructure can affect the ease with which the transformation can occur, leading to changes in the transformation start temperature.Understanding the factors that influence the transformation starttemperature is crucial for optimizing the heat treatment of steel. By carefully controlling the chemical composition, heating rate, and prior microstructure of the steel, it is possible to precisely control the transformation start temperature and achieve the desired microstructure and properties.One important application of this knowledge is in the heat treatment of high-strength steels used in the automotive industry. These steels often undergo complex heat treatment processes involving quenching and tempering to achieve the desired combination of strength and ductility. The transformation start temperature is a critical parameter in these processes, as it determines the temperature at which the austenite phase begins to form during heating.By understanding the factors that influence the transformation start temperature, engineers can optimize the heat treatment process to ensure that the desired microstructure is achieved. This can lead to improvements in the mechanical properties of the steel, such as increased strength and toughness, which are essential for the safety and performance of automotive components.In addition to the automotive industry, the transformation start temperature is also an important consideration in the heat treatment of other types of steel, such as those used in construction, aerospace,and energy applications. By understanding and controlling this parameter, manufacturers can ensure that their steel products meet the required specifications and perform reliably in service.In conclusion, the transformation of ferrite to austenite is a critical phase change in the heat treatment of steel, and the start temperature of this transformation is influenced by a variety of factors. By understanding and controlling these factors, engineers and manufacturers can optimize the heat treatment process and produce steel products with the desired microstructure and properties.。

食品工程原理chapter-5 Evaporation

食品工程原理chapter-5 Evaporation
Δ= ΔtT- Δt
Δt=Ts-t ΔtT =Ts-T
Δt——传热的有效温度差, ℃
ΔtT ——理论上的传热温度差, ℃ t —— 溶液的沸点, ℃ T——纯水在操作沸点, ℃ Ts——加热蒸气的温度, ℃
Example 8.1.:用476kN/m2(绝压)的水蒸气作为加热蒸汽 (Ts=150 ℃),蒸发室内压力为1atm,蒸发30%的NaOH溶液, 沸点为t=115 ℃,其最大传热温度差,用ΔtT来表示:
4、mix the foam:a mount of foam mixed with the vapor, which will lead to material losses and condenser pollution. 5、energy reclaim:the utilization of vapors is one of the most important problems in the evaporation operation.
Problems in evaporation involve primarily heat transfer and material and energy balances
Energy of an evaporator is supplied by vapor/steam, The hot steam is called raw steam, while the vapor produced during Evaporation is called vapor, flash steam or secondary steam.
The pressure exerted by a column of liquid of height, h, and density, ρ, is

High pressure freezing and thawing

High pressure freezing and thawing

ReviewHigh pressure freezing and thawing of foods:a reviewA.LeBail a,*,D.Chevalier a ,D.M.Mussa a ,M.Ghoul baGEPEA-ENITIAA,Rue de la Ge´raudie `re,BP 82225,F-44322Nantes cedex,3France bENSAIA,2avenue de la foreˆt de Haye,F-54500Vandoeuvre les Nancy,France Received 1October 1999;received in revised form 2March 2001;accepted 13March 2001AbstractThe phase diagram of water shows that the melting temperature of water decreases with pressure down to À21 C at210MPa while the opposite effect is observed above this pressure.This phenomenon allows the achievement of rapid freezing and thawing of foods that mainly contain water.In addition,pressure-assisted thawing has the advantage of inducing a reduction in drip loss which tends to be a function of process parameters and nature of the product.Con-cerning pressure shift freezing,this process permits the significant preservation of the microstructure of biological substances.The current status of high pressure freezing and thawing applications in foods is reviewed in this paper.Concepts and principles underlying the application of these technologies are firstly developed.Available literature on the applications of high pressure freezing and thawing is then presented and discussed.Finally,the modelling aspects of theses processes are dealt with.#2002Elsevier Science Ltd and IIR.All rights reserved.Keywords:Food;Frozen food;Freezing;Thawing;Slurry;Very high pressure;Survey0140-7007/02/$22.00#2002Elsevier Science Ltd and IIR.All rights reserved.P I I :S 0140-7007(01)00030-5International Journal of Refrigeration 25(2002)504–513/locate/ijrefrigContents1.Introduction...........................................................................................................................................................5052.Freezing and thawing processes based on phase transition under pressure...........................................................5063.Applications of high pressure freezing processes....................................................................................................5073.1.The earliest studies........................................................................................................................................5073.2.Applications of high pressure freezing in microscopic preparation...............................................................5073.3.Applications of high pressure freezing on foods ...........................................................................................5084.Applications of high pressure thawing processes...................................................................................................5105.Modelling of high pressure freezing and thawing processes...................................................................................5116.Conclusion .............................................................................................................................................................512Acknowledgements......................................................................................................................................................512References . (512)*Corresponding author.Tel.:+33-2-51-78-54-73;fax:+33-2-51-78-54-67.E-mail address:lebail@enitiaa-nantes.fr (A.Le Bail).Conge lation et de conge lation de produits alimentaires soushaute pressionRe sumeLe diagramme de phase d’eau montre que la tempe´rature de fusion d’eau diminue lorsqu’on augmente la pression, atteignantÀ21 C pour une pression de210MPa;au-dessus de cette pression,la tendance contraire est observe´e.Ce phe´nome`ne permet la conge´lation et la de´conge´lation rapides de produits alimentaires compose´s avant tout d’eau.En outre, la de´conge´lation sous pression diminue les pertes dues a`l’exsudat,qui ont tendance a`eˆtre fonction des parame`tres de traitement et de la nature du conge´lation sous pression variable permet une meilleure conservation de la microstructure des substances biologiques.Cette communication donne une vue d’ensemble de l’e´tat de l’art actuel des applications de la conge´lation et de la de´conge´lation sous haute pression.Les concepts et les principes gouvernant l’appli-cation de cette technologie sont de´veloppe´s.Les re´fe´rences aux applications des la conge´lation et la de´conge´lation sous haute pression sont pre´sente´es et commente´es.Enfin,on pre´sente la mode´lisation de ces proce´de´s.#2002Elsevier Science Ltd and IIR.All rights reserved.Mots cle´s:Produit alimentaire;Produit congele;Conge lation;De conge lation;Tre s haute pression(hyperpression);Enque te1.IntroductionHigh pressure(HP)has appeared as a new parameter in food processing for about10years.Three domains of food processing are of potential interest in HP including food texturation,preservation and phase change[1–3]. HP freezing and thawing processes are related to these three applications.Since1912it has been known that water undergoes different phases when submitted to HP at different temperatures[4].As shown in Fig.1,the phase change temperature of water decreases with pressure from 0 C at0.1MPa toÀ21 C at210Mpa and the opposite effect is observed above this pressure level.This phe-nomenon allows to carry out different high pressure processes of freezing and thawing of foods.Further-more,high pressure treatments from100MPa prevent microbial growth[5].Nevertheless,protein modifica-tions may take place from100to200MPa[6].HP freezing and thawing processes have recently attracted research and development interests to the food industry in order to improve the quality of frozen and thawed foods.This new interest in high pressure phase change processes dates from the beginning of the eighties and is particularly related to the commercialisation ofhigh Fig.1.Possibilities of high pressure freezing and thawing processes based on the phase diagram of water(ABCD:pressure-assistedfreezing,DCBA:pressure-assisted thawing,ABEFG:pressure shift freezing,GFEBA:pressure-induced thawing,ABEFHI:freezing toice III,IHFEBA:thawing to ice III).Fig.1.Possibilite´s des proce´de´s de conge´lation et de de´conge´lation fonde´s sur le diagramme de phase d’eau(ABCD:conge´lation sous haute pression;DCBA:de´conge´lation sous pression;ABEFG:conge´lation sous pression variable;ABEFHI:conge´lation formant de la glace III,IHFEBA:de´conge´lation formant de la glace III).A.LeBail et al./International Journal of Refrigeration25(2002)504–513505pressure processed products in Japan and more recently in USA and Europe[1,7].This paper reviews the literature on high pressure freezing and thawing processes applied to biological sub-stances and food,particularly during the last10years.The subject is discussed in three parts,withfirstly a description of the different possibilities of high pressure freezing and thawing processes.Then,the food applications of these processes are described andfinally,a discussion on the modelling aspect of each process is proposed.2.Freezing and thawing processes based on phase transition under pressureSeveral processes of high pressure freezing and thaw-ing can be carried out thanks to the phase diagram of water.They were enumerated by[8]who proposed6 different high pressure freezing and thawing processes as shown in Fig.1.The pressure-assisted freezing(PAF),ABCD in Fig.1, consists in cooling a sample under pressure up to its phase change temperature at the applied pressure.In this way,the freezing takes place under a constant pressure.Reciprocally,pressure-assisted thawing(PAT)corre-sponds to a thawing under a constant pressure(DCBA in Fig.1).The decrease of the melting point of ice until 210MPa allows the increase of the temperature gap between the heat source and the phase change front and thus enhances the heatflux rate in the case of PAT.The reduction of the thawing time during PAT is directly related to this relative increase of the temperature gap. Temperature and pressure evolutions during PAT are presented in Fig.2for a pure water sample.The pressure transmissionfluid is maintained above0 C.During the pressure increase(stage1,Fig.2),the temperature of the sample starts to decrease when the corresponding phase change pressure is reached.Thawing occurs under pres-sure during stage2(Fig.2).Pressure is released when the sample temperature is a few degrees above the initial freezing point of the sample.Indeed,cooling occurs during depressurisation(around5 C for200MPa dur-ing stage3,Fig.2).If thefinal temperature reaches the initial freezing point before the atmospheric pressure, partial freezing will occur.Pressure shift freezing(i.e.PSF,ABEFG in Fig.1)is detailed in Fig.3.This freezing process permits particu-larly an increase in the ice-nucleation rate.It consists in placing a food in a HP vessel whose temperature is regu-lated above the initial freezing temperature of the food. The vessel is closed and pressure increased(stage1,Fig.3) up to a pressure for which freezing is not possible[3]. During thisfirst step,water remains in a liquid phase. Holding the pressure and temperature at these condi-tions for the time required(stage2,Fig.3)causes a drop in sample temperature until the regulated temperature of the media is reached.During the pressure release(stage3, Fig.3),the temperature of the sample isfirst depressed due to depressurisation.When the difference between the sample temperature and that corresponding to the cur-rent pressure becomes too large a suddentemperature Fig. 2.Sample temperature and medium pressure changes during the different stages of pressure-assisted thawing for pure water sample.Fig.2.Tempe´rature de l’e´chantillon et modifications moyennes de la pression lors de plusieurs stades de de´conge´lation sous haute pression pour un e´chantillon d’eaupure.Fig. 3.Sample temperature and medium pressure changes during the different stages of pressure shift freezing process for pure water sample.Fig.3.Tempe´rature de l’e´chantillon et modifications moyennes de la pression lors de plusieurs stades de conge´lation sous pression variable pour un e´chantillon d’eau pure.506 A.LeBail et al./International Journal of Refrigeration25(2002)504–513rise is observed when nucleation occurs.This tempera-ture difference is called supercooling.It induces the nucleation phase at an elevated rate.Thereafter,the temperature increases up to the atmospheric freezing temperature according to the phase change pressure tem-perature relationship.During this second phase of the pressure release step,ice crystals grow while nucleation might still be occurring due to the rapid evolution.As a result,the formation of homogenous ice nucleation at the core is induced.Numerous andfine ice nuclei obtained during the stage3(Fig.3)grow into a massive number of small ice crystals during stage4(Fig.3)at atmospheric pressure.This has a significant implication in food quality since smaller ice crystals are less dama-ging to tissues and cell structures than large ice crystals. Tempering is completed during the stage5(Fig.3)at atmospheric pressure.One of the limitations of the pro-cess comes from the limited effectiveness in cooling the sample under pressure and during thefinal stage of freezing at atmospheric pressure.The mass ratio of ice to water during adiabatic expansion may reach up to 0.36kg of ice per kg of water for a pressure release from 210MPa[9].Apparently,the mass ratio to be formed during PSF of real foods would be more complicated to establish since these products contain dissolved sub-stances,fat and dry matters.The reversion process of PSF is pressure-induced thawing(i.e.PIT,GFEBA in Fig.1).This thawing pro-cess corresponds to a phase transition initiated with a pressure change and continued at a constant pressure. Freezing to ice III(ABEFHI in Fig.1)and thawing to ice III(IHFEBA in Fig.1)are processes using the ice III form because it has a higher density than liquid water and ice I.Nevertheless this ice form is only stable under pressure that limits the application of these processes.3.Applications of high pressure freezing processes3.1.The earliest studiesThe earliest studies of high pressure freezing were carried out mainly in the medical sector in the sixties to test cell survival as well as microbial destruction.Taylor [10]compared the survival of human conjunctiva and chicken embryo cells that were frozen,either con-ventionally(using coolants)or using pressure-assisted freezing.Several series of experiments involving cooling of living cells at temperatures ranging fromÀ5to À120 C for different time intervals,followed by count-ing the survivors were carried out.A rapid cooling of cells through the crystallisation zone or afinal tempera-ture belowÀ20 C were found to be the best operations. It was therefore hoped that the application of high pressure could accelerate the phase transition within the zone of crystallisation.Living cells were then subjected to240MPa atÀ22 C in order to achieve rapid freezing. However,the survival obtained from this PAF did not show apparent advantage of this technology over con-ventional freezing.This failure was attributed to insuffi-cient ice formation that would be required to pass through the safety zone of crystallisation or to the high velocity of the freezing that could have contributed to damage the cells[10].The question of freezing rate was ruled out later[11]when it was found out that90%of Euglena cells could tolerate PAF at210MPa and À130 C when freezing time is maintained at0.1s.It may be argued besides that cell survival depends on species.However,such rapid freezing rates usually result in vitrification rather than formation of large crystals of ice I or ice III which are more stressful for cells.Although the exact path followed on the pressure-temperature phase diagram of water is unknown,it is possible that pressure and temperature selected by Tay-lor[10]have induced formation of ice III or ice V that affected the cells survival.It should also be noted that pressure shifting between ices I and III at160–460MPa atÀ25 C has been reported as being an effective means of killing micro-organisms[11]and thus it may be useful to reduce the micro-organisms in foods.3.2.Applications of high pressure freezing in microscopic preparationHigh pressure freezing has been widely used in the ultrastructure analysis of micro-organisms,animals as well as plants tissues and has permitted the freezing of specimens as thick as500m m.The HP freezing appara-tus using liquid nitrogen and pressure shift from200 MPa was developed in order to obtain vitrification of the sample in a superficial zone of the sample(50m m thickness)[12].Freezing rate was estimated at104K sÀ1. This experiment showed formation of vitrification at the surface of samples while ice crystals were obtained at the core.In addition,larger samples pressurised at100MPa atÀ20 C(PAF)yielded larger ice crystals that damaged food structures.A few years later,new equipment was developed that was able to pressurise samples to210 MPa with simultaneously cooling toÀ100 C in about 30ms[13].This equipment was used to evaluate effects of PAF on stabilisation of nerve samples at low tem-peratures.The results showed that in these conditions pressure-assisted frozen nerves were comparable to those frozen conventionally for cryo-fixation of thin section.High pressure freezing technology was never-theless more preferable because it worked without che-mical treatments and permittedfixation of thicker specimens.A commercial high pressure unit(HPM010, Balzers Union,FL)was used to study the effect of PAF at250MPa on various organs of rat,plant tissues as well as a variety of micro-organisms[14,15].The animal samples were frozen successfully under pressureA.LeBail et al./International Journal of Refrigeration25(2002)504–513507throughout the entire thickness of400m m.Membranes in samples treated with PAF showed smooth contours unlike undulating appearance observed in samples sub-jected to chemicalfixation.The yield of well frozen cells was very high and the freezing quality was the same throughout the entire cells for animal and microbial cells.However,more variability was reported in quality of pressure-assisted frozen botanical samples compared with animal tissues and it depended mainly on the spe-cimen properties.3.3.Applications of high pressure freezing on foods From1992,studies of PAF and PSF on food systems have mainly focused on the effects of these technologies on the food quality as shown in Table1.For example, changes in food texture or colour following PAF or PSF have been widely studied due to the importance of these quality parameters on consumer acceptability of frozen foods.Freezing under a gaseous pressure(nitrogen pressure of0.3–10MPa atÀ20toÀ25 C)resulted in an improvement in the quality of subsequently air-dried fruit,vegetables and meat products compared with samples frozen atÀ20 C without pressure[16].Freezing under gaseous pressure resulted in reduced textural damage and shrivelling,due to tiny gas bubbles being trapped in tissues.Pressure-frozen fruits possessed a softer texture and could be eaten without thawing. Kanda et al.[17]compared the quality of tofu sub-jected to PSF(200MPa,À18 C)and air-blast freezing (À10andÀ18 C).Tofu frozen by pressure release was reported to maintain its initial shape and texture and to have no drip loss.These were attributed to the smaller ice crystals that were formed in PSF tofu,whereas much larger ice crystals were observed in air-blast frozen sys-tem.Further study on tofu has also reported similar findings[18].In this study,the authors maintained tofu atÀ19 C for45or90min and observed that no ice was formed at200MPa,while ices I,III,V and VI were formed at100,340,400–600and700MPa,respectively. During the pressure release,the samples compressed to 200MPa underwent nucleation and freezing.In addi-tion,some samples were air-blast frozen at atmospheric pressure atÀ20,À30orÀ80 C or pressurised without freezing.A significance difference was observed between samples after thawing at+20 C.Samples frozen at340 MPa(PAF)or by pressure release from200MPa(PSF) showed almost the same textural characteristics as the untreated samples.The ice crystals cross sectionsTable1High pressure freezing studies available in literature(PAF:pressure-assisted freezing,PSF:pressure shift freezing)Tableau1Etudes sur la conge´lation sous haute pression publie´es dans la litte´rature(PAF:conge´lation sous haute pression,PSF:conge´lation sous pression variable)Food system Process Analysis Reference Tofu PSF Microstructure[17]Tofu FUP/PSF Microstructure[18] Carrot FUP/PSF TextureMicrostructure[19]Pectin contentCarrot FUP/PSF Microstructure[20] Potato PSF Texture,colour,Microstructure[22]TextureChinese cabbage FUP/PSF Microstructure[21]Pectin contentEggplant PSF TextureMicrostructure[23]Drip lossesPork PSF Microstructure[26] Protein gels FUP/PSF TextureMicrostructure[24]Drip lossesNorway lobsters PSF Protein stabilityMicrostructure [27]508 A.LeBail et al./International Journal of Refrigeration25(2002)504–513obtained at200MPa(PSF)were smaller than at100MPa (PAT)(0.85Â10À3vs.8.5Â10À3m m2at the surface and 1.7Â10À3vs.20Â10À3m m2at the centre).The size of ice crystals increased at500and700MPa(12.5.10À3m m2at the surface,and28.10À3m m2at the centre).Samples subjected to a pressure level of100MPa atÀ20 C (PAF)were reported to yield larger ice crystals that damaged food structures.The texture of raw or thermally blanched carrots subjected to high pressure freezing in a prechilled toÀ18 $À20 C system at100–700MPa was compared[19].A freezing process of45min was quite sufficient except for samples pressure shift frozen at200MPa which required a storage in a freezer atÀ30 C and atmospheric pres-sure to complete the freezing process.The texture of the carrots blanched for3minutes then frozen by pressure release at200or under340and400MPa was the same as for non frozen blanched carrots.The carrots frozen at100or500MPa(PAF)showed a decrease in the firmness and an increase in strain.Such textural mod-ifications were attributed to the formation of ices I and III for the samples frozen at100and500MPa,respec-tively[20].The texture and microstructure studies in Chinese cabbage subjected to high pressure freezing between100 and700MPa were also carried out[21].Again in these experiments,the high pressure equipment was prechilled toÀ18$À20 C.The freezing process lasted for45min followed by the thawing of the samples at+20 C.Con-trary to the results obtained in carrots[20],there was a significance difference between the texture of raw and pressured treated Chinese cabbages.Raw midribs were firmer compared to non-frozen samples pressurised at 200–400MPa.PAF at100and700MPa was found to be destructive by inducing excessive softness to the midribs as reported later in carrots[20].These results of high pressure freezing of carrots and Chinese cabbage mid-ribs showed that conditions for which ice crystals were formed before depressurisation have critical impact on final texture.Koch et al.[22]compared the quality changes on potatoes subjected to PSF(400MPa,À15 C,20min) with samples pressurised without phase change(400 MPa,15 C,20min)and air-blast frozen samples (À30 C).The results showed that PSF preserved potato texture while air-blast freezing resulted in a reduction of the rupture strength.Changes in colour parameters of potatoes were also evaluated.The lightness(L*)values decreased during freezing and were more obvious for air-blast freezing than for PSF whereas a*values(red values)increased only slightly in PSF samples compared to air-blast freezing.There was no significant difference observed in b*values(yellow values)for samples sub-jected to PSF or air-blast freezing.In addition,a higher concentration in dissolved substances in drip from potatoes subjected to air-blast freezing than those pres-sure shift frozen was observed.Concerning drip thaw-ing,PSF of eggplants[23]showed that PSF allowed a decrease in the drip volume by20%compared with air freezing.Changes on the structure,hydration and mechanical properties of a heat-induced gel ofß-lactoglobulin sub-jected to various cooling system were evaluated[24]. Three different freezing processes were carried out in a high pressure vessel maintained atÀ33 C by a propa-nediol and water mixture:freezing at atmospheric pressure,freezing under pressure at200MPa(PAF)and PSF induced by a pressure release from207MPa at À20 C.These processes were besides compared with freezing at atmospheric pressure using still air atÀ43 C and gaseous nitrogen atÀ80 C.PSF(207MPa,À20 C) was reported to be faster than PAF and freezing under air and liquid N2.Concerning PSF,the release of pres-sure from207induced a massive ice nucleation resulting in warming of gel toÀ1 C due to the latent heat of crys-tallisation.The growth of ice crystals took place while the latent heat of crystallisation was progressively removed by the coolant until the temperature at the centre of the gel reachedÀ20 C.The texture of the pressure shift fro-zen gels was found to be similar to that of non-pres-surised non-frozen control gels.Both gels were smooth, white,translucent and glossy except that the gel matrix was sectioned by a pattern of small ice containing cracks evenly distributed throughout the gel formed by an unknown mechanism.On the other hand,gels frozen in air or liquid N2differed significantly from control gels by being less smooth,spongy and with dull and granular appearance.In addition,the ice matrix contained numerous round orflattened alveolus of varying size due to temperature gradient during freezing.Unfortunately, PAF induced more gel drip and higher gel rigidity after thawing than freezing in the same cooling medium at atmospheric pressure.The residual water holding capa-city,the elasticity and the relaxation time of PSF gels remained close to those of untreated gels.Levy et al.[25]studied the influence of PSF(207 MPa,À18 C)compared to classical freezing methods and PAF(100MPa,À9 C)on oil-in-water emulsions.It appeared that PSF induced higher supercooling and more nucleation than conventional freezing and PAF and besides did not modified the size distribution of oil droplets.Histological studies on pork samples[26]have shown that PSF(200MPa,À20 C)caused less damage in terms of structural preservation compared to classical methods such as air-blast and cryogenicfluid freezing. This was mainly due to the fact that,unlike PSF,clas-sical methods induced thermal gradient and,therefore,a non-uniform ice crystal distribution that was harmful to tissue structures.As a result,the PSF process is appro-priate to freeze large pieces of food when uniform ice crystal size are required.A.LeBail et al./International Journal of Refrigeration25(2002)504–513509PSF(200MPa,À18 C)of Norway lobsters was compared with pressurisation without freezing(200 MPa,+4 C)and air-blast freezing(À30 C)[27].PSF and pressurisation without freezing induced a significant increase in the toughness of the Norway lobster tails compared to the untreated and air-blast frozen samples. These changes of texture were attributed to protein denaturation induced by high pressure.In addition,a reduction of the ice crystal size in case of PSF compared to air-blast freezing was showed by scanning electron microscopy.A study on turbotfillets[28]showed in a first time that a pressure level of140MPa at+4 C allowed to minimise the effect of pressure on colour, lipid and protein stability’s.Secondly,PSF at140MPa atÀ14 C of turbotfillets showed that this freezing pro-cess improved the quality of thawedfillets compared to a conventional process(air-blast freezing atÀ20 C). Thus,the thawing and cooking drips were reduced by PSF and ice crystals formed by this process were sig-nificantly smaller than those obtained by air-blast freezing.The ice crystal size distribution was studied in cylind-rical gelatin gel frozen either by air-blast or in brine or by pressure release from100,150and200MPa[29]. These authors observed that the pressure shift frozen samples presented smaller and more homogeneous ice crystals compared to those obtained by conventional freezing processes.In addition,the mean representative ice crystal diameter decreased as the pressure level increased in the case of PSF samples.This has been associated with the degree of supercooling achieved during PSF and experimentally evaluated in this study.It appeared that enzymes were particularly resistant to high pressure freezing technique although the resis-tance varied with enzyme nature.This was demonstrated by freezing selected enzymes at0toÀ22 C under high pressure up to400MPa and comparing their activity with that of atmospheric pressure frozen samples[30].The results showed that peroxidase,polyphenoloxidase and Bacillus subtilis alpha-amylase expressed no irreversible inactivation after PAF,whereas pectinmethylesterase was only slightly irreversibly inactivated.Lipoxygenase, on the other hand,expressed higher inactivation but up to55%of the activity was retained.In summary,many studies of high pressure freezing showed that PSF process results in a much preserved texture of the foods.Reports of microscopic observa-tions have shown preserved food microstructures as long as no ice is formed before depressurisation.At the oppo-site,texture measurements have shown that denaturation might occur under pressure,resulting in a modification of the texture of food in case of high protein contain samples.Drip volume reduction is variable from one product to the other while no specific studies were done on potential microbial destruction of micro-organisms in real foods.4.Applications of high pressure thawing processes The application of high pressure to thaw frozen bio-logical materials has been studied since the early sixties by Taylor[10]who observed that a slow freezing rate combined to PAT at225MPa permitted a significant increase in the survival ratio of human conjunctiva cells. The advantages of PAT applied to food systems were later tested in tuna meat and surimi[31].Tuna meat samples(44cm3,35g)and surimi blocks(62cm3,60g) were reported to demand shorter thawing time under pressure than at atmospheric pressure.In addition,the temperature of these samples could be maintained at a lower temperature than5 C throughout the thawing process.Unfortunately,this process,as PSF,induced colour changes in the meat due to protein denaturation under pressure.In addition,the texture of the surimi and tuna meat became harder than the raw samples.Mur-akami et al.[32]also observed a colour change(increase of the lightness)in tuna meat when subjected to PAT. However,a substantial reduction of the drip volume of tuna meat was noted while the microbial population was kept at a constant level.Deuchi and Hayashi[33]presented PAT experiments of ice blocks and beef.The ice blocks were previously frozen atÀ10,À15,À20,andÀ30 C.Pressure-assisted thawing at50–200MPa and5 C for30min was com-pared with thawing at atmospheric pressure at5 C for 30min.The results showed that ice blocks subjected to PAT were completely thawed while those left to stand at atmospheric pressure were only partially thawed.Beef samples,previously frozen atÀ20 C,were thawed at 50–200MPa at20 C for30min.It appeared that pres-sure around50MPa was reported to be the optimal for beef with respect to drip volume and colour change whereas higher pressure levels induced protein dena-turation resulting in whitening of meat.Eshtiaghi and Knorr[34]studied PAT(600MPa,15 min,50 C)of whole or sliced frozen strawberries vacuum packaged and heat-sealed in polyethylene bags. After the high pressure treatment,the strawberries with the released juice were immersed in a20%sugar solution (60 Brix)and stirred at20 C for60min or at90 C for20 min at atmospheric pressure.The results concluded that pressure-assisted thawed samples subsequently immersed in a sugar solution at room temperature had similar Brix values as the thermally treated samples.High pressure treatment followed by thermal treatment for20min at 90 C resulted in an additional increase in sugar content. For the whole intact strawberries,the overall sugar uptake was less but the impact of high pressure thawing prior to the thermal processing was substantially more dramatic than for the strawberry slices.This improve-ment of sugar uptake post to PAT was due to pressure-induced permeabilisation of the cell membrane as well as increased mass transfer due to degassing of the tissue510 A.LeBail et al./International Journal of Refrigeration25(2002)504–513。

Modeling of Heat Transfer in Nanoporous Silica Influence of Moisture

Modeling of Heat Transfer in Nanoporous Silica Influence of Moisture

Modeling of Heat Transfer in Nanoporous Silica - Influence of MoistureR. Coquard1, D. Quenard21. Centre Thermique de Lyon (CETHIL), UMR CNRS 5008, Domaine Scientifique de la Doua, INSA de Lyon, Bâtiment Sadi Carnot, 9 rue de la physique, 69621 Villeurbanne CEDEX, France. Phone: 04 72 43 84 74 Email: remi.coquard1@insa-lyon.fr2. Centre Scientifique et Technique du Bâtiment (CSTB), 24 rue Joseph FOURIER, 38400 Saint Martin d’Hères, France. Phone : 04 76 76 25 46 Email : quenard@cstb.frAbstractThis work is interested in the modelling of the thermal behaviour of nanoporous silica. These highly porous materials are suitable for the manufacturing of Vacuum Insulation Panels given that the exceptionally small size of their pores permit to reduce drastically the conductive heat transfer when a primary vacuum is applied. Nanoporous silicas have been widely studied experimentally but only few studies have already dealt with the theoretical modelling of their thermal characteristics from their nanometric structure. That is the reason why we have developed a model of heat transfer based on a simplified representation of their structural morphology which permits us to analyse the contributions of each heat transfer modes (gaseous conduction, solid conduction, radiation) to the total heat transfer.The coupled radiation-conduction heat transfer is treated using a simultaneous resolution of the Energy Equation and the Radiative Transfer Equation. The effective conductivity of the medium is computed by means of a Finite Element Method while its monochromatic radiative properties are calculated assuming that only silica particles are present and that they scatter radiation independently. The model can take into account the presence of condensed water at the surface of silica particles as well as the possible contact area between neighbouring particles. The results of the model permits us to investigate the evolution of the equivalent thermal conductivity of the nanoporous silica with different parameters related to their structural characteristics (density, particle size, contact area between particles …) or to the external conditions (pressure, water content …). They notably indicate in which conditions the silica should be used in order to maximize their insulating performances. The results of the model are compared with experimental results previously published and show good agreement. In particular, the bad influence of the presence of condensed water and of the contact area between touching particles on the conductive heat transfer are clearly confirmed theoretically. This demonstrates the important role played by the adsorbed water on the thermal performances of this very absorbent material. The study also reveals that, for materials with common densities containing silica particles alone, the radiative heat transfer is significant and noticeably degrades the insulating performances of the material. Thus, the addition of opaque particles to the solid matrix is indispensable.1Nomenclaturea : constant parameter proper to the gas consideredA cube : characteristic length of the cubic lattice in mc0 : light speed in vacuum (≈ 2.99776.108 m/s)C ext, C sca et C abs : extinction, scattering and absorbing cross sections in m²d c : contact diameter between neighboring grains in md eau : diameter of the water meniscus in md grain : diameter of the silica grains in mh : Planck constant (≈ 6.626.10-34 J.s)Iλ : Monochromatic radiant intensity (W/m²/Str/µm)I0λ : Monochromatic radiant intensity of the black-body at temperature T (W/m²/Str/µm) k : thermal conductivity in W/m/KK : Boltzmann constant (≈ 1.3805.10-23 J/K)L g : mean free path of the gas molecules (m)~: complex refractive index=ikn−nn~: equivalent refractive indexeqN grain :Number of grains per volume unit (m-3)P : Pressure in PaP max : Maximal strain on the material in PaPλ(θ) : Monochromatic scattering phase functionq c: Conductive heat flux density (W/m²)q r : Radiative heat flux density (W/m²)Q k : thermal power passing through the horizontal slab z k in WR : Radius of the particles in mt p : Characteristic size of the pores in mT : temperature in KT cold, T hot : temperature of the hot and cold boundaries in KTh water : thickness of the condensed water on the sides of the silica grains in mx, y, z : cartesian coordinatesGreek symbolsβλ, σλ, et κλ : monochromatic extinction, scattering and absorption coefficient in m-1γ : convergence parameterε : porosityλ : radiation wavelength in µmµ =cosθ : directing cosine of the directionρ : density in kg/m3σSB: Stefan-Boltzmann constant (≈ 5.67.10-8 W/m²/K4)θ : angle between incident and scattering directions in radωλ = σλ/βλ : monochromatic scattering albedo (-)ξ : convergence criterionSubscriptsair : of airch : relative to the chain of silica grainside: relative to the lateral boundaries of the computation volumewater : of watereff: effectiveeq : equivalentg : gasbound. : relative to the horizontal hot and cold boundaries of the computation volume i,j,k : relative to the discretised nodes of coordinates x i, y j, z kfree : free gassi : of silicax, y, z : relative to the point of coordinates x, y, zλ : at radiation wavelength λ21 IntroductionUsual building thermal insulators (cellular or fibrous materials) have now reached a limit in their thermal performances and it appears that an equivalent thermal conductivity of 0.03 W/m/K is an almost incompressible value. Thus, the improvement of the thermal insulation required to reduce the expenditure of energy of the building sector could only be achieved by the development of new thermal insulators able to reach lower equivalent thermal conductivities. One of the most promising materials are the Vacuum Insulating Panels made of nanoporous media under air vacuum. The principle of action of these new insulators is based on the “Knudsen effect”. By this technique, it is theoretically possible to reach equivalent conductivities lower than the thermal conductivity of free air. The Knudsen effect is already used in several products commercialized by industrial groups such as POREXTHERM or CABOT-CORP.Several studies have already been interested in the thermal properties of this kind of materials. Rochais, Domingues and Enguehard [1] have developed a model enabling to study the transient or steady-state conductive heat transfer through a fractal representation of a nanoporous material in order to compute the thermal conductivity and diffusivity of these materials. At the same time, Enguehard [2] have modeled the radiative heat transfer in these highly porous materials by developing a simplified model based on the Rosseland approximation and the Mie Theory under the assumption of independent scattering. The author pointed out the fact that, for the density considered, thermal radiation contributes significantly to the total heat transfer. Then, he suggested to introduce micrometric particles into the nanometric network in order to reduce the radiative contribution. The theoretical results showed that a substantial decrease of this contribution could be achieved by introducing a relatively small amount of particle with suitable sizes (diameter close to 1µm).Quenard and Sallée [3] have also conducted a detailed experimental study on the thermophysical properties of these materials. The samples tested, with porosities greater than 90%, are essentially composed of fumed silica associated with a very small amount of fibers insuring the mechanical consistency. Some opaque particles are also added to the material to make it more opaque to thermal radiation. The authors measured the porosities, the specific areas and the pore size distribution. They have also been interested in the evolution of their equivalent thermal conductivity with pressure, temperature and water content using the hot-wire method which enable a simple, fast and non-intrusive measurement in a wide range of experimental conditions. The experimental results showed that, at room temperature and atmospheric pressure, the equivalent conductivity of these materials is close to 20 mW/m/K. This conductivity could be reduced noticeably to approximately 6 mW/m/K by applying a primary vacuum (P=1-10 hPa). Moreover, the measurements reveal that the equivalent conductivity increases significantly with the temperature which certainly implies that a non negligible part of the heat transfer is due to thermal radiation. Finally, Quenard and Sallee clearly pointed out the very bad influence of water adsorption on the thermal performances of nanoporous silica whose thermal conductivity increases linearly with the moisture content at a rate of approximately 1.5 to 2 mW/m/K per mass percentage of adsorbed waterResearchers from the Bavarian Center for Energy (ZAE Bayern) [4, 5, 6, 7] have also looked into the characterization of opacified nanoporous medium under partial air vacuum. They have shown that for the opacified silica aerogel, equivalent conductivities of only several mW/m/K could be reached at ambient temperature [4]. For fumed, precipitated or pyrogenic silica powders [5, 6], the addition of opacifiant particles allows to reach equivalent conductivities close to 3 mW/m/K under partial air vacuum. However, they also remarked that the mechanical strains undergone by the material reduces the insulating abilities of the material significantly. Finally, the ZAE Bayern also studied the influence of condensed water on the thermal properties of fumed silica [7] by measuring their conductivities for different moisture content using a guarded hot-plate apparatus. The increase of the thermal conductivity is noticeable and is almost proportional to the water content.One can notice that, at the moment, the thermal properties of nanoporous materials have mainly been studied from an experimental point of view. The relation between their nanometric structure and the evolution of their thermal characteristics with the different parameters (density, pressure, water content …) remain relatively badly known. Thus, we have developed a simple model representing the silica grain network constituting the porous structure which allowed us to estimate the thermal conductivity of these insulators from their structural characteristics (density, solid conductivity, grains diameter, contact area between grains …). We assumed that the macroscopic heat conduction law are still valid at the nano-grain scale. Moreover, we have also modeled the presence of condensed water to predict the influence on the equivalent conductivity. Finally, given that radiative heat transfer could be significant, we also computed the radiative properties of non-opacified34 nanoporous silica. The knowledge of these properties and of the effective conductivity of the nanoporous silica permits to compute their equivalent conductivity.2 Structure of the nanoporous silicaThe use of nanoporous materials proves to be particularly convenient for the manufacturing of V.I.P. given that the very small size of the pores allows to benefit from the Knudsen effect without requiring high vacuum. Silica aerogels, the most common nanoporous material, are constituted of pyrogenic or precipitated silicium micrograins ( amorphous SiO 2). These micro-grains are almost spherical and their characteristic size is approximately 10 nm. Fumed silica could also be considered for this application. In this case, the structure is still constituted of silica grains with very small diameters but they are obtained from the combustion of silica tetrachlorure (SiCl 4) in aqueous atmosphere. For these two kinds of nanoporous silica, the structure of the grain network is random and the chains have random orientations. Thus, the properties of the material could be considered as isotropic. The scheme and the photograph of Fig. 1 illustrate the structure of the network obtained by considering that all the grains have a uniform size.In order to simplify the theoretical morphology of nanoporous silica, We have modeled the grain network by considering that all the identical grains are spread on the arêtes of cubes abutted sides by sides. The grains are mechanically linked to their neighbors. The illustration of this cubical network is given on Fig. 2.The porosity ε and the average size of the pores t p constituting the nanoporous silica only depend on the diameter of the silica grains d grain (radius r grain ) and on the length of the arête A cube of the cubical network. We have computed these two parameters for different grain diameter and different length of arête by generating the entire grain network from a unique representative cube incorporating several grains on its arêtes. The porosity of the material is estimated by generating a great number of points randomly distributed in the elementary cube and by calculating the proportion of points belonging to an air pore. To match the porosity of usual nanoporous silicas which is comprised between 0.9 and 0.95, our representation corresponds to a ratio grain cube d A / comprised between 7 and 10.As regards the average size of the pores, it is very difficult to define it given that the nanometric structure does not form closed pores. However, we managed to express a characteristic length of the pores. Actually, this characteristic size represents the average distance that an air molecule have to travel after impacting a silica grain to reach the silica matrix again. To compute this average distance, we randomly generate a point into the air pores and a direction of propagation of the molecule. Then, a ray-tracing procedure determines the distance between the two silica grains encountered in this direction on both sides of the point. For a great number of rays, it is possible to estimate the average size of the pores. According to our geometric model, for grain diameters close to 10 nm, the characteristic size of the pores is several hundreds of nanometers for usual silica (porosity between 0.9 and 0.95). This average size is the same order of magnitude as the mean free path of air molecules and allows to benefit from the KNUDSEN effect which reduces the thermal conductivity of air. Indeed, according to Griesinger et al. [8], the conductivity k g of the gas can be estimated by :Figure 1: Simplified representation of the random structure of nanoporous silicas5p g libre g g t L a k k /...21+= (1)For air, under atmospheric pressure: k air,free = 0.0262 W/m/K ; a = 1.5 and L g ≈ 67 nm.Thus, according to our model, when the diameter of the grain is about 10 nm, the decrease of thermal conductivity of air due to the Knudsen effect for usual silicas (ε between 0.9 and 0.95) is comprised between 12 and 15 mW/m/K at ambient pressure and is already very interesting.3 Modelling of the heat transfer in nanoporous silicaThe principle of action of all traditional thermal insulators is to prevent the convective heat transfer due to the temperature gradient and to limit the conductive heat transfer through the solid matrix. Thus, the heat transfer is primary due to the conduction through the gas (generally air). In nanoporous materials, the thermal conductivity of air could even be reduced by the Knudsen effect. Actually, when the insulators is highly porous, another heat transfer mode due to the propagation of thermal radiation occurs. In the case of 1-D steady-state heat transfer between two horizontal boundaries at different temperatures, the total heat flux can be computed by solving the Energy Equation (EE) and the Radiative Transfer Equation (RTE). To solve these equations, it is necessary to know the effective thermal conductivity and the infrared radiative properties of the nanoporous silica.3.1Modelling of the conductive heat transfer 3.1.1 Thermal conductivity without condensed waterIn order to compute the effective conductivity of dry nanoporous silica composed of a solid and a gas phase, we used a model based on the Finite Element Method (FEM) applied to the grain network. This model assumes that the heat transfer by conduction at the grain scale follow the macroscopic laws represented by the Fourier equation :T k q c ∂−= (2)Although the grain diameter is not vastly greater than the size of the silica molecules, we will consider, in the next, that the Fourier law is suitable. From this hypothesis, we have modeled the thermal conductivity of a chain of silica grain in punctual contact, noted k ch . This conductivity can be calculated by analyzing the temperature field in a representative cube containing a grain subjected to a temperature gradient. It is necessary to solve the differential equation of heat conduction in steady-state by the Finite Element Method. The FEM has already been used by Druma et al. [9], Quintard et al. [10] and Bakker [11] for conductive heat transfer calculations in materials made of several phases. The principle is to simulate an experimental steady-state apparatus measuring the conductivity and to solve the steady-state energy equation 0=∇∇).(T K r r r (3)Figure 2 : Theoretical representation of the cubical grain network proposed for the modelling of thethermal properties of nanoporous silicaElementarycube dgrain6 The resolution is based on a spatial discretization in cubical elements of the volume modeled. Each volume is made of silica or air and a node is placed at its center. The thermal conductivity of the solid node is k Si = 1.4 W/m/K and that of air volume is set to k air = 0.0001 W/m/K. We voluntarily set the thermal conductivity of air to a very small value given that air is confined in a very narrow space comprised between two neighboring grains and is subjected to a strong Knudsen effectOnce the effective conductivity of the chain of grain has been computed, the conductivity of the entire network forming the nanoporous silica is calculated assuming that it is actually made of two continuous phases with different conductivities : one phase corresponding to the grain in contact forming the skeleton of the network and one phase corresponding to air. We used a numeric model based on the FEM similar to the previous model to compute the effective conductivity of the entire network of silica grain. The model used only differs from the previous one by the geometry of the discretised volume which now corresponds to a cubical volume whose arête are occupied by the chains of grain. The thermal conductivity of the silica chain used for the computation has been computed in the previous step: k ch . On the other hand, the thermal conductivity of air used is not the same as that of the previous computation given that air is no more confined between the silica grains but in the entire network. Then, the mean free path of the air molecules is noticeably greater than previously and the thermal conductivity of air could no more be neglected although the Knudsen effect is present. The thermal conductivity used is given by Eq. (1)3.1.2 Conductivity in the presence of condensed water and contact between grains Due to their high specific area and the hydrophilic character of silica, the risk of water condensation in nanoporous silica is major. Thus, it is necessary to take into account the influence of the presence of condensed water on its thermal properties. In addition, the contact between neighboring grains is not punctual since the mechanical strain could lead to a deformation of the grains. A model enabling the simulation of these two phenomenon has been developed.The ideal structure of the nanoporous silica is then modified to take into account the condensed water and the contact diameter d c between two touching grains. We assumed that water can condense according to two means :- by forming a meniscus with diameter d water located at the junction between two grains,- by forming a continuous water layer with thickness Th water at the surface of the grainsThe cube representing the chain of grain composing the nanoporous silica could be illustrated by the scheme of Fig. 3.The effective conductivity of the chain of grain in the presence of condensed water could be computed using a Finite Element Method similar to that used previously but in which, the spatial discretization is modified. In the new configuration, three different phases are considered : a solid phase (conductivity k Si ), a liquid phase (conductivity k water = 0.6 W/m/K) and a gas phase ( k air = 0.0001 W/m/K). The proportions of each phase depend on the values of d grain , d water , Th water and d c .3.1.3Evolution of the effective conductivity with the different parametersFigure 3 : Illustration of the cube representing the chain of silica grains in the presence ofcondensed water and of a non-punctual contactRepresentative cube containing a silica graincondensed water7Influence of the densityThe density of nanoporous silica only depends on their porosity : 2).1(SiO ρερ−= (4) with ρSiO2 = 2500 kg/m 3 the density of silicaIn order to analyze the relative influence of these two effects, we have conducted the computations without the presence of condensed water and assuming punctual contacts between grains fro different densities and different grain diameters. Two air pressures have been modeled : atmospheric pressure (1000 hPa) and a primary vacuum (P=10 hPa). The results are depicted on Fig. 4 where we have also represented the equivalent thermal conductivities measured by Quenard et al.[3] on opacified fumed silicas and by Caps et al. [5] on opacified pyrogenic silicas. In both cases, theauthors have shown that due to the presence of opacifiers, the radiative heat transfer at ambient temperature could be neglected and that the equivalent conductivity is sensibly equal to the effective conductivity.The results of Fig. 4 show that, for relatively weak densities, the theoretical effective conductivity decreases with the density when the silica grain are sufficiently small. This can be explained by the fact that, in this range of densities, the diminution of the conductivity of air due to the Knudsen effect when the silica gets denser (decrease of the average size of the pores) is more important than the increase of the thermal conduction through the solid phase. On the other hand, when the silica grains are relatively large (> several tens of nm), the decline of thermal conductivity of air is practically imperceptible and the augmentation of solid conduction due to the densification is predominant. This causes an increase of the effective conductivity with the density.The evolution observed when the grains are sufficiently small is quite noteworthy given that for the other porous materials classically used for thermal insulation, the effective conductivity generally tends to increase with the density.One can also notice that for common densities and grain diameters of approximately 10 nm, the effective conductivity at atmospheric pressure is lower than the thermal conductivity of free air. The interest of the Knudsen effect is then clearly apparent even without partial vacuum. At the same time, when a partial air vacuum is applied, the theoretical effective conductivity reaches particularly low values lower than 10 mW/m/K.All these theoretical conclusions agree well with the experimental results of Quenard and Sallee [3] who measured an equivalent thermal conductivity at atmospheric pressure slightly lower than the thermal conductivity of free air. However, it seems that our theoretical results slightly overestimates the conductivity of nanoporous silica for the two pressures considered.Influence of the grain diameterThe size of the grain forming the nanoporous matrix affects directly the average pore size t p . Then, very small grains are required in order to maximize the Knudsen effect. The influence of the parameter d grain on the effective conductivity is illustrated on Fig. 4. One can notice that silica grains100 200300400500600 700density in kg/m 3 k eff inW/m/K Figure 4 : Theoretical evolution of the effective conductivity of nanoporous silicas with their densities at atmospheric pressure and under a partial air vacuum (P=10 hPa) for different grain diameters8 with diameters lower than several tens of nm are required so that the Knudsen effect is perceptible at atmospheric pressure. Above this limit, microporous silicas are not suitable to reach very low thermal conductivities. However, for lower pressures, we notice that when the grains are small enough, their size no more affects the effective conductivity since the conductivity of confined air is then extremely low and almost negligible when compared with the conductivity of the silica skeleton.Influence of air pressureWe have seen that the air pressure have an influence on the mean free path of air molecules and thus on the strength of the Knudsen effect. To illustrate this more clearly, we have depicted on Fig. 5 the variations of the effective conductivity with pressure for different porosities and assuming that the grain diameter is d grain = 10 nm. We have also illustrated on this figure the equivalent conductivities measured at different air pressures by the CSTB (Quenard et Sallée [3]) and the ZAE Bayern (Caps et al. [5]) on opacified fumed silicas.The influence of air pressure is significant. Then, when an advanced air vacuum in the order of 10-100 hPa is applied in common nanoporous silicas (porosities between 0.9 and 0.95, d grain ≈ 10 nm), very weak thermal conductivities lower than 10 mW/m/K can be obtained theoretically. But, if we reduce the pressure under 10 hPa, the improvement of the conductivity is less and less pronounced. Very low pressures are not necessarily interesting particularly if we consider that the mechanical strains on the structure are then increased. This results are in very good agreement with the measurements given by Quenard and Sallee [3] and Caps et al. [5].Influence of the contact between grainsOwing to the mechanical strain undergone by the nanoporous silicas during the manufacturing process or utilization, the contact between grains is not punctual. Actually, there is a contact area between the two touching grains that is likely to enhance the conductive heat transfer in the solid phase. This effect has been underlined by Caps and Fricke [6] who proposed a law of variation of the conductivity in the form max P A k c =.We have evaluated from a theoretical point of view the influence of this non-punctual contact on the effective conductivity of the nanoporous silica by assuming that contact area is circular. We varied the contact diameter d c . The computations were conducted at atmospheric pressure for nanoporous silica with densities 0.9455 and 0.968 with a grain diameter d grain = 10 nm. We specify that, in each case considered, the porosity of the silica remains unchanged when d c varies. The variations observed are thus entirely due to the variation of the contact. They are illustrated on Fig. 6As expected, the contact area between the grains leads to an increase of k eff . The conduction of heat by solid-solid contact is enhanced. However, one can notice that this influence is relatively limited as long as the deformation of the grains is weak compared with the grain diameter (d c /d grain < 0.2). But, it becomes significant when the deformation is more important.Pressure in hPak eff inW/m/KFigure 5 : Theoretical evolution of the effective conductivity of nanoporous silica with air pressure for different densities and d grain = 10 nm0,010,02 0,03 0,040,1 1 10101001000。

aspen10燕化高级培训班英文培训资料

aspen10燕化高级培训班英文培训资料

Course Agenda (2)
5. Sequential Modular (SM) Simulation – Run flowsheets in SM mode 6. Introduction to Equation Oriented (EO) Modeling – Compare EO and SM solution strategies 7. Basic EO Functionality – Demonstrate the basic features of EO modeling, including Spec groups and EO input
3. Unit Operations – Review enhancements to columns, heat exchangers, and reactors; plus new manipulator blocks
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Thermodynamics and Heat Transfer

Thermodynamics and Heat Transfer

Thermodynamics and Heat Transfer Thermodynamics and heat transfer are two concepts that are closely related to each other. Thermodynamics is the study of the relationship between heat, energy, and work, while heat transfer is the movement of heat from one object to another. Both concepts are essential in understanding how energy is transferred and transformed in various systems. In this essay, we will explore the significance of thermodynamics and heat transfer in various fields, including engineering, physics, and chemistry.Engineering is one of the fields where thermodynamics and heat transfer play a crucial role. Engineers use these concepts to design and optimize various systems, such as engines, refrigeration systems, and power plants. For example, in the design of an engine, engineers need to understand the thermodynamics of the combustion process to ensure that the engine operates efficiently. They also need to consider heat transfer to prevent overheating and damage to the engine components. In the design of a refrigeration system, engineers need to understand the thermodynamics of the refrigerant and the heat transfer between therefrigerant and the surrounding environment to ensure that the system operates efficiently.Physics is another field where thermodynamics and heat transfer are essential. Physicists use these concepts to understand the behavior of matter and energy in various systems, such as the universe, stars, and planets. For example, in astrophysics, thermodynamics and heat transfer are used to understand the behavior of stars and their energy output. In planetary science, these concepts are used to understand the energy balance of planets and their atmospheres. In condensedmatter physics, thermodynamics and heat transfer are used to understand the behavior of materials at different temperatures and pressures.Chemistry is also a field where thermodynamics and heat transfer play acrucial role. Chemists use these concepts to understand the behavior of chemical reactions and the energy changes that occur during these reactions. For example,in the design of a chemical reactor, chemists need to understand the thermodynamics of the reaction to ensure that the reaction proceeds efficiently. They also need to consider heat transfer to prevent overheating and damage to thereactor components. In the study of materials science, thermodynamics and heat transfer are used to understand the behavior of materials at different temperatures and pressures.Thermodynamics and heat transfer are also essential in everyday life. For example, in the design of buildings, architects and engineers need to consider heat transfer to ensure that the building is well-insulated and energy-efficient. In the design of consumer products, manufacturers need to consider thermodynamics and heat transfer to ensure that the products operate efficiently and safely. In cooking, thermodynamics and heat transfer are used to understand how heat is transferred from the stove to the food and how different cooking methods affect the texture and flavor of the food.In conclusion, thermodynamics and heat transfer are essential concepts in various fields, including engineering, physics, and chemistry. These concepts are used to understand the behavior of matter and energy in various systems, from engines and refrigeration systems to stars and planets. They are also essential in everyday life, from the design of buildings and consumer products to cooking. Understanding these concepts is crucial for developing efficient and sustainable technologies and for improving our understanding of the natural world.。

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Modeling of heat transfer during controlled cooling in hot rod rolling of carbon steelsA.H.Nobari,S.Serajzadeh *Department of Materials Science and Engineering,Sharif University of Technology,Azadi Ave.,PO BOX 11155-9466,Irana r t i c l e i n f oArticle history:Received 11May 2010Accepted 2October 2010Available online 23October 2010Keywords:Hot rolling Carbon steelMathematical modeling Controlled coolinga b s t r a c tA mathematical model is developed to predict temperature variations and austenite phase trans-formation kinetics in steel rods and wires during controlled cooling after hot rolling.Two-dimensional finite element analysis is employed to solve the governing heat conduction equation with non-uniform boundary conditions as it is subjected in practical cooling layout.The kinetics of austenite transformation is also determined using the cooling curves achieved from the thermal analysis together with the additivity rule.To verify the model results,temperature pro files in steel rods cooled under different conditions are measured experimentally in laboratory and also,temperature history during controlled cooling of hot rolled wire rods are recorded using infrared pyrometer.The comparison between the predicted and the experimental results shows a reasonable agreement.Ó2010Elsevier Ltd.All rights reserved.1.IntroductionCooling rates and subsequent austenite phase transformation after hot rolling of steels is an important task in designing a proper cooling layout regarding the final microstructures and mechanical properties.In hot rolling of steels,the bar exiting the last stand is often in the austenitic condition and as the steel is cooled on the run-out table,austenite phase transformation occurs continuously as the hot rolled product is being cooled.In this regard different layout including water spray and air fans may be employed for controlled cooling of the hot rolled steel [1]and as a result,different cooling conditions are provided on upside and downside of the rolled metal.Investigation into the thermal behaviors of steels under such conditions has been considered in several works.For instance,Izzo [2]have employed a one-dimensional thermal model to predict and to control temperature variations after hot strip rolling process.Packo et al.[3]have investigated thermal response of steel strips subjected to water-cooling conditions.Monsalve and Celentano [4]have presented a mathematical model to assess thermal and microstructural behaviors during steel sheet coiling.Sun et al.[5]have developed a mathematical model based on the finite element method to evaluate coupled thermal e metallurgical behavior of steel strip on the run-out table.Han et al.[6]have developed a two-dimensional model to assess thermo-mechanical response of steels on the run-out table in hot strip rolling opera-tions.Zhou et al.[7]have used a finite element code,ABAQUS,todetermine temperature pro file and residual stresses after cooling on the run-out table in hot strip rolling process.Serajzadeh [8]has utilized a two-dimensional model to determine temperature distribution and kinetics of austenite transformation on the run-out table in hot strip rolling process.Phadke et al.[9]have utilized a code,ROLPAS,to determine temperature variations and final microstructures after hot rolling.Lindemann and Schmidt [10]have developed a mathematical model including the geometrical char-acterization of wire rod to predict thermal behavior of hot rolled wire rod.Zhang et al.[11]have used a mathematical model for prediction of temperature history and ferrite grain size after hot rolling of plain carbon steels.In this work,a two-dimensional thermal model is developed to predict temperature pro file and kinetics of austenite decomposi-tion during controlled cooling after hot rod rolling of steels.The non-uniform thermal boundary conditions at lower and upper sides are taken into account and the additivity rule has been coupled with the thermal model to include the effect of heat of transformation on the temperature variations of steel being cooled.In addition,the effect of decarburizing in austenite is also predicted and implemented in the thermal model using the Fick ’s equation and a coupled finite element analysis.2.Mathematical modelTo derive the governing heat transfer equation,heat conduction along the longitude axis (z -axis)is neglected owing to the length of hot rolled rods and wires.So,the heat conduction equation may be described in two dimensions as follows:*Corresponding author.Tel.:þ9821661615218;fax:þ982166005717.E-mail address:serajzadeh@ (S.Serajzadeh).Contents lists available at ScienceDirectApplied Thermal Engineeringjournal ho mepage:www.elsevier.co m/locate/apthermeng1359-4311/$e see front matter Ó2010Elsevier Ltd.All rights reserved.doi:10.1016/j.applthermaleng.2010.10.003Applied Thermal Engineering 31(2011)487e 492v v x k v T v x þv v y k v T v y þ_q¼r c p v Tv t(1)where T and t are temperature and time,respectively,_qis the rate of heat of transformation.k ,c p ,r are thermal conductivity,speci fic heat,and density of the metal,respectively.Both radiation and convection occurs on the surface at high temperatures while convection heat transfer is the dominant heat transfer process at lower temperatures i.e.less than 600 C.In this regard,the thermal boundary conditions on the surface can be de fined as below:Àk v T v n¼h ðT ÀT C Þþ3s T 4ÀT 4C(2)where “h ”is the convection heat transfer coef ficient which may be different for the upper and lower sides of the rod,T c is surrounding temperature varies from one zone to the other zone i.e.air cooled region or forced cooled region by fans,s is the Stefan e Boltzman constant,and 3is the emissivity that is assumed to be a temperature dependent factor [12]while the effect of radiation may be ignored at temperatures below 600 C where the convection is the main heat transfer mechanism at the surface boundary.In addition,heat of transformation may be released during cooling while the rate of heat of transformation can be determined as [13]:_q¼r D H ðT Þd X(3)where D H (T )is heat of transformation assuming to be temperature dependent and X is transformed fraction.To determine the starting time of austenite decomposition,the additivity rule is utilized as follows [14]:Xt t ¼0D t is iðd g =d g TTT Þ¼1(4)where D t i is the time step at the temperature T i ,and s i is incubation time and temperature T i .d g and d g TTT are the grain size before transformation and the grain size used to construct TTT diagrams,respectively.To predict austenite decomposition kinetics under continuous cooling conditions,Avrami equation together with theadditivity rule may be employed.Accordingly,the progress of phase change at a given temperature can be described using Avrami ’s equation as below [15]:X ¼1Àexp ðÀbt n Þ(5)where,b and n are the material parameters that are obtained from the steel TTT diagram.To include the effect of temperature varia-tions on the above equation,the cooling time is divided into small time steps assuming constant temperature during each of them.Then,the Avrami equation can be utilized to predict amount of phase change,however,for the next time step,the effect of previous transformation is included in Eq.(5)by de fining a new elapsed time as t *i as denoted below:t *¼264ln 11ÀX i À1 b T i3751(6)X i ¼1Àexp Àb T i t nit i ¼D t i þt *(7)here X i À1is the amount of phase transformation until the (i À1)th step and D t i is the time step and T i is temperature at i th time step.It is worth noting that to determine volume fraction of martensite during quenching of austenite,the following equation may be utilized [16].X ¼1Àexp ½À0:0110ðMs ÀT Þ (8)here Ms is the temperature of beginning of martensite trans-formation.The above equation can be employed together with the Avrami ’s equation to predict the kinetics of microstructural changes under high-rate cooling conditions like the one that occurs during water quenching.Now,the above problem may be solved utilizing the finite element method.For doing so,the cross-section of rod is divided into four-node elements as shown in Fig.1.Then,the equation (1)is converted to its corresponding weak from using Galerkin scheme [17]as presented below:Zv N T v x k v N v x d A þZ v N T v y k v Nv yd A þZN T hN d c þZN T s3ðT +Þ3N d c !a e þ Z N T r c p N d A _a e ¼ZN T q +d A þZN ThT c d c þZN T s3T 4c d c(9)Fig.1.The mesh system used in the model.Edge RegionFig.2.Schematic illustration of wire rod on the cooling table.Table 1Chemical composition of the steel used in this work.ElementC Si Mn SPWeight percent0.670.280.600.0140.020A.H.Nobari,S.Serajzadeh /Applied Thermal Engineering 31(2011)487e 492488Here,“c ”denotes the element boundary side,N shape functionmatrix,T +is an initial guess to linearize the nonlinear boundary conditions due to radiation,a e is the nodal temperature vector and _ae is the nodal temperature rate vector.The above equation can be rewritten in matrix form as below:Ka þC _a¼f (10)where “K ”is the stiffness matrix,“C ”the capacitance matrix and f is nodal force vector.This differential equation may now be solved by Euler Backward difference scheme in time domain,as follows:a n þ1¼D tC À1Àf n ÀD tK n Áþa n(11)Here,a n and a n þ1is the nodal temperature vectors at the nth and (n þ1)th time-steps,respectively.It is worth noting that the above scheme is conditionally stable and thus in the model small time steps of the order of 10À2s were employed particularly at the early stage of cooling where rapid changing in wire temperature occurs.It should be noted that decarburizing in austenite might also beoccurred before during cooling period particularly in the early stages of cooling before austenite decomposition begins.Therefore,to predict the distribution of carbon content within the rod being cooled in austenite region,the following equation is taken into account.1v v r Dr v C v r ¼v Cv t(12)here C and D are the carbon concentration and diffusion coef ficient,respectively.The later may be determined at different temperatures as below:D ¼ð0:05þ0:09C 0Þexp ðÀQ =RT Þ(13)where,Q and R are the activation energy (132.1kJ/mol)and the ideal gas constant,respectively and C 0is initial carbon content of the employed steel.The boundary conditions at the surface were assumed as follows:C j r ¼r 0¼0(14)here “r 0”represents rod radius.Note that two-dimensional finite element analysis together with four-node isoparametric elements are employed in the model and a code was written in MATLAB.In addition,the above problem is solved iteratively at each time-step owing to heat of transformation.Accordingly,the solution proce-dure can be described as follows:1-The distributions of temperature and carbon content alongradius direction are predicted before staring the austenite transformation as a sequentially coupled problem.First temperature distribution is calculated and then the distri-bution of carbon content is determined in each time step.2-The appropriate boundary conditions are determined basedon the position of the wire on the run-out table i.e.the edge region or the central part as shown in Fig.2.3-After staring austenite transformation,in each time steptemperature distribution is first calculated and then the fraction of transformation during the time step is determined using additivity rule.Subsequently,in the same time step temperature field is recalculated considering the effect of heat of transformation.4-The error norm for two successive solutions is determined bythe following criterion.k T i þ1ÀT i kk T i k<3;where k T k ¼T T $T1=2(15)where k T k represents Euclidean norm of nodal temperature vectors and superscript “i ”describes the iteration number.If the above error value is small enough i.e.less than 10À2then the above step is followed for the next time step and if not the calculations are repeated in the same time step.Table 2Thermo-physical properties used in the thermal model.Speci fic heat of pearlitec P p ðJ =Kg C Þ¼480:57þ0:1226T þ8Â10À4T 2À5Â10À7T À3Speci fic heat of ferrite in high temperatures c F p ¼148822:82e 495:64T þ0:55237T 2À2:0495Â10À4T 3ðT >760 C ÞSpeci fic heat of ferrite in low temperatures c F p ¼521:89þ11:303exp T À192:36135:42ðT 760 C ÞSpeci fic heat of austenite c A p ¼466:45þ0:1561T þ2Â10À5T2ðT >610 C ÞThermal conductivityk (W/m C)¼59.92e 0.0221T À5.4Â10À5T 2þ4.3Â10À8T 3Heat of pearlite transformation D H p (J/kg)¼120848þ52.417T À0.158T 2Heat of ferrite transformation D H F ¼207898À15.632T À0.246T 2Emissivity factor3¼T 1000ð0:125T 1000À0:38Þþ1:1Time (s)51015202530T e m p e r a t u r e (oC )0150300450600750900aTime (s)50100150200250300T e m p e r a t u r e (oC )400500600700800900bparison between the recorded and the predicted cooling curves for the rod pre-heated at 870 C for 10min a)quenched in water,b)air cooled.A.H.Nobari,S.Serajzadeh /Applied Thermal Engineering 31(2011)487e 492489It should be noted that Four-node isoparametric elements are employed the thermal analysis and two-node elements are employed in determination of carbon content.In the thermal model,288node and 280elements are used while 18elements are employed in the decarburizing model.In order to obtain the optimum con figuration of the mesh system providing both shorter computational time and high accuracy,different mesh systems with different element sizes were tested and finally the above-mentioned con figuration was found to be suitable while the corresponding computational time is about 870s.3.Materials and experimentsPlain carbon steel with the chemical composition given in Table 1was studied in this work while the parameters used in the Avrami equation,n and b ,were calculated using the TTT diagram of the employed steel.It is worth nothing that the above parameters were also determined for the carbon steel having the carbon content ranging 0.6e 0.1.This is particularly important for the sample subjected to decarburizing leading to different chemical composition at surface region.To validate the model predictions,experiments have been carried out on steel rods.Specimens with 16.8mm diameter and 180mm length were austenitized at 870 C for 10min.Then,K-type thermocouples were embedded in the drilled holes to record time e temperature history while the ther-mocouples were connected to data recorder apparatus saving 10data per second.Finally,the steel rods were subjected to different cooling media and the temperature pro files in the center of the rod were recorded.The austenite initial grain sizes were also deter-mined by metallographic method on quenched samples using a solution of saturated picric acid at 35 C.Note that the thermo-physical properties of the employed steel are assumed to be temperature-dependent using data in Refs.[18,19]and Table 2shows different thermo-physical properties used in the model.In addition,temperatures at different position during controlled cooling after hot rod rolling were measured using an infrared pyrometer.Two positions of the rod including edge and centralregions were considered in practical measurement as schemati-cally illustrated in Fig.2.4.Model resultsFig.3shows the measured and the predicted time e temperature variations at the center of the specimen austenitized at 870 C for 10min with initial austenite mean grain size of 47m m and then cooled in water and still air.The convection heat transfer coef ficientTable 3Heat convection coef ficients used in the model.For forced air cooling on ROT “h ”at lower surface (W/m 2 C)“h ”at upper surface (W/m 2 C)Cooling conditions 2fans 1fan closed 2fans 1fan closed Center of the table 207.1159.213.9155.3119.413.9Edge of the table162.7125.412.8122.094.012.8Position(m)481216202428323640T e m p e r a t u r e (o C )0100200300400500600700800900parison between the measured temperatures and the predicted coolingcurves for the wire rod air cooled after hot rolling.Position (m)2468T e m p e r a t u r e (o C )500600700800900abPosition (m)02468T e m p e r a t u r e (o C )400500600700800900Fig.6.Temperature pro files in the upper and lower surfaces of cooling wire locating in the center of cooling table for the wire coil velocity of 0.3m/s,a)when three first fans are open,b)when the second and the third fans are closed.Fig.4.Predicted cooling curves for the steel rod quenched in water.A.H.Nobari,S.Serajzadeh /Applied Thermal Engineering 31(2011)487e 492490in the air was taken as 12W/m 2K and the data in Ref.[20]was used for determination of this factor for water cooling.It is seen that there is a good agreement between the measured and the predicted cooling curves.The proposed model is capable of determining temperature pro files under different cooling practice.For example,Fig.4shows the time e temperature curves for a sample pre-heated at 870 C for 30min with austenite mean grain size of 55m m and then quenched in water.As it is expected,surface regions experi-ence severe temperature variations while the temperature pro file in central zone is smoother.However,after 20s the temperature is distributed homogenously.Thermal behavior during controlled cooling after hot wire roll-ing (Stelmor)may also be considered by the model.The cooling table in this research consists of two air conveyor cooling sections having 27air fans.The different combinations of cooling conditions are then possible to impose to the wire.The wire diameter is 5.5mm and its temperature is 870 C when enters the first air conveyor.The initial austenite grain size was taken as 49m m and wire coil velocity of 0.23m/s.It was assumed that the effect of work hardening was removed by static recrystallization just after the last rolling stand thus the effect of work hardening is not included in the phase transformation model.The data published in Ref.[3]have been employed for evaluation of the convection heat transfer coef ficients at different positions and temperatures.For instance,Table 3shows the employed heat transfer coef ficients for different positions and cooling conditions.Fig.5compares predicted and measured temperatures pro files in different regions of wire coil being cooled.It is seen that there is a reasonable consistency between the experimental and the predicted results by the model.Note that the experimental temperature measurement is also hard to perform and the mentioned experimental data is based on threesets of measurements while the relative error is calculated based on the population standard deviation as 9%.Fig.6shows the temperature variations in upper and lower sides of steel wire coil locating in the center of cooling table under different cooling conditions e.g.when the first three fans are completely open as the first cooling condition and the other one when the second and the third fans are closed.It is found that the difference between the upper and lower surfaces for both cooling conditions are trivial which it may be attributed to the small diameter of wire rod,5.5mm,considered in this research.However,the difference may be larger in rods with greater diameters.Fig.7shows time e temperature curves and austenite decom-position progress when the first two air fans are completely closedTime (s)4681012T e m p e r a t u r e (oC )200300400500600700800P e a r l i t e f r a c t i o n0.00.20.40.60.81.0aPosition (m)24681012T e m p e r a t u r e (oC )200300400500600700800P h a s e f r a c t i o n0.00.20.40.60.81.0bFig.7.Predicted temperature pro file and phase transformation on the cooling table a)Center of the wire rod,b)Surface of the wire rod.Time (s)246810T e m p e r a t u r e (oC )200300400500600700800900P e a r l i t e f r a c t i o n0.00.20.40.60.81.0abTime (s)246810T e m p e r a t u r e (oC )200300400500600700800900P e a r l i t e f r a c t i o n0.00.20.40.60.81.0Fig.8.Predicted temperature pro file and phase transformation at the center of wire rod for wire coil velocity of 0.15m/s,a)Program 1,b)Program 2.Time (s)246810T e m p e r a t u r e (oC )100200300400500600700800900P e a r l i t e f r a c t i o n0.00.20.40.60.81.0Fig.9.Effect of velocity on temperature pro file and kinetics of phase transformation at the center of cooling rod under cooling program 1,2C3C.A.H.Nobari,S.Serajzadeh /Applied Thermal Engineering 31(2011)487e 492491and the wire coil velocity is0.23m/s.It is seen that at the cooling profile for both central and edge positions are almost similar. However,the ferrite is formed at the upper surface region,within the surface elements,where decarburizing occurs and as a result the carbon concentration decreases to about0.4wt%while at the central part the main structure is just pearlite.As displayed in Fig.6,it is expected that with changing the cooling practice the thermal response and kinetics of austenite decomposi-tion alter accordingly.For example,Fig.8displays the effect of cooling layout on temperature profile at the central region of wire rod and kinetics of austenite to pearlite decomposition.The cooling layouts are considered as program#1when the second and the thirds fans are closed,2C3C,and program#2when thefirst two fans are closed 1C2C.As it is seen both rate of cooling and positions of starting and finishing of austenite decomposition change.Therefore,the model makes it possible to trace the progress austenite decomposition and may be used as predictive tool to more accurately control austenite transformation and as a result to produce the wire rod with the required mechanical properties.Wire coil speed or bar velocity is also an important parameter that controls the thermal and metallurgical events on the cooling table.Fig.9illustrates the effect of conveyor speed for the cooling program2C3C,on the temperature profile and progress of austenite transformation.It is seen that by increasing the speed from0.15m/s to0.30m/s the time for starting the austenite decomposition increases as well.5.ConclusionsIn this work,the kinetics of austenite decomposition and thermal response during controlled cooling during hot wire rod rolling process are predicted employing two-dimensionalfinite element model together with the additivity rule.The model considers the effects of process parameters including the wire coil velocity or bar velocity along the cooling table as well as different cooling layouts. 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