Perturbation Solutions for Thermal Process of Honeycomb Regenerator

Available online at www.sciencedirect.comScienceDirectJ0URNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2007, 14(4); 06-10, 48Perturbation Solutions for Thermal Process of Honeycomb RegeneratorAI Yuan-fang，MEI Chi,HUANG Guo dong, JIANG Shao-jian(School of Energy Science and Enginering, Central South Univeriy, Changsha 410083, Hunan, China)Abstract; A parameter perturbation for the unsteadystate heat transfer characteristics of honeycomb regenerator ispresented, It is limited to the cases where the storage matrix has a small wall thickness s0 that no temperature varia-tion in the matrix perpendicular to the flow direction is .considered. Starting from a two phase transient thermal mod-el for the gas and storage matrix, an approximate solution for regenerator heat transfer process is derived using themultiple scale method for the limiting case where the longitudinal heat conduction of solid matrix is far less than theconvective heat transfer between the gas and the solid. The regenerator temperature profiles are expressed as Taylorseries of the cofficient of solid heat conduction item in the model. The analytical validity is shown by comparing theperturbation solution with the experinent and the numerical solution. The results show that it is possible for the per-turbation to improve the effectiveness and economics of thermal research on regenerators.Key words: honeycomb regenerator; thermal process; asymptotic analysis; semi-analytic methodHigh-temperature air combustion[1 used in themal performance. The analytical process withoutforging furnaces, soaking furnaces, continuousconsidering longitudinal solid heat conduction is easy.heating furnaces, ladle bakers, gas-fired radiantbut coarse, Considering the solid heat conductiontubes, and aluminum melting furnaces of the meta!-parallel to flow direction, Klein H[8] derived an ap-lurgical industry can provide high thermal efficien-proximate heat transfer solution using the Taylor' scy，reduced NO: emission, and uniform heat fluxseries expansions as a function of switch time t forfield. Most of these furnaces use the honeycomb re-the limiting case where the period of heating andgenerator-2.8. Its thermal characteristics such as thecooling process became infinitesimally small. Histemperature fluctuation and the temperature effi~conclusions could not be applied to the real situationciency are critical to achieve the optimal thermal per-because he treated“switching rapidly" as“t→0" andformance. Therefore, a more accurate research onassumed a0/at to be differentiable on t=0. In theregenerator heat transfer process is required,present study, the asymptotic solution of heat trans-Numerical simulation[4-61 is a theoretical meth-fer process for weak solid heat conduction is ob-od often used to solve regenerator thermal perform-tained through the perturbation method.ance. The mathematical analysis is an effective andDynamic Energy Balancesconvenient method. Zheng C H7] had carried outseveral investigations on the analytic solutions inThe regenerator temperature changes continu-case that the storage matrix has a small wall thick-ously with its location and time. Temperature varia-ness so that no temperature variation in the matrixtion perpendicular to the flow direction can be usual-perpendicular to the flow direction is considered,ly neglected because regenerators have a compara-but they are limited to the case where the longitudi-tively small wall thickness. Because the residual gasnal heat conduction is neglected. For regenerator,mass (the volume of the gas within the honeycombespecially compact regenerator, operating at a shortmatrix void) is less than 0.1% of the total gas massswitching period, the solid heat conduction in theflow, its effect on heat transfer is negligible'7间. There-length direction may lead to the change in the ther-fore,中国煤化工state and two-phaseFoundatlon Item:Item Sponsoted by High Technology Research and DevelopmentMHC N M H G0lAA514013)Biography: AI Yuan-fang(1968-), Male, Doctor, Associate Professor; E mail: yfai@maiL csu. edu. cn Revised Date; April 10, 2006No.4.Perturbation Solutions for Thermal Process of Honeycomb Regeneratorheat transfer model with energy balance equations(a,L)/(A.p.cp.), X=x/l, A.=(B, -0.,)/(8.o -..),for the gas and solid matrix can be sufficient.Ay=(), -0.)/(6,.o -08.1), the dimensionless constantsConsidering the symmetrical geometry of the几= (a,LI)/(qm,yCp,) and业= (入.A.)/(a,Ll2), the :honeycomb structure, it can be assumed that theabove equations can be rewritten asdistribution of physical parameters is the same forAA./ar,-Q,H*A,/aX2 = =A,-A.(3)each cell of the honeycomb matrix. The region be-AAy/aX= -业,(A,-A.)tween two symmetrical adiabatic surfaces shown inA,(0,F,)=1,JA. (0,r,)/aX=aA,(1,r,)/aX=0 (5)Fig. 1 is taken as the control volume. The flue gasA,(X,0)= f(X)and air are regarded as the ideal gas.Eqn. (3) to Eqn. (6) are the 2nd order linearDuring the flue gas cooling period shown inpartial differential equations with constant coeffi-Fig. 1, the coordinate origin is at the flue gas inlet,cients. Under certain conditions, its solution exists'andx axis is along the flow direction. The energyand is unique, continuous, and differentiable.balance for the element dx at x shown in Fig. 1The subscript “y" and“s" can be neglected toyields the following equations:simplify the expression, Combining Eqn. (3) withA.p.cp..a0./at- A,A.J0,/ax2 =a,L(0,-θ.)(1) Eqn. (4), A, can be eliminated and thereforegqm,yCp,DU,/Ax=-a,L(O, -0,)(2)a.|2A_ -341+2+ 23A- ψ°A=0(7)whereA is area, m2; P is density, (kg ●m~*); cp is”|示ax|taxtaxarspecific heat at constant pressure, (kJ.kg-'●K-);θIntroducing the complex variables s and w(X,s)==is temperature,"C; tis time, s; λ is thermal conduc-L[A(X,r)], and considering the derivative theoremtivity, (kW●m-1● K-); a is surface coefficient ofand Eqn. (6), the following equation can be ob-heat transfer including heat convection and thermaltained.radiation, (kW●m-2●K-1); L is inner perimeterL[aA(X ,r)/ar]= srw(X,s)- f(X)(8)of channel, m; qm is mass flow rate, (kg●s~);Applying the Laplace transformation to Eqn. (7), .subscript“s”stands for solid matrix and“y" standssubstituting Eqn. (8), and thenfor flue gas.2. sw(X,s)-f(X)- 4ψ dw(X,s)]. t dw(X,s). +The boundary conditions are 0,(0,t) = 0,o and" dXa,(0,t)/ax= 20,(l,t)/ax=0, and the initial condi-sw'(X,)- f'(x)- yPu(x,22=0(9)tion is 0,(x,0)= f(x), where l. is channel length,dXm; and f(x) is initial function of solid temperature.Let业= = e2 and introduce the differential symbolTo decrease the error from the large tempera-H(s,Q,e), which is defined byture change of gas or flue gas resulting from the longH(s,Q,e)=e2(d*/dX +0. d*/dX")-switch time, the average temperature between the(1+s)d/dX-s.s(10)inlet and outlet temperature of gas or flue gas isLet F(X)=- L2f(X)-f'(X), and Eqn. (9)taken as the reference temperature.can be rewritten as:For air heating period, the origin is the air out-H(s,Q,e)[w]= F(X)(11)let and at the inlet 0,(I,t)= 0. is satisfied, whereThe Laplace transformation is applied to Eqrthe subscript“a”stands for air.(5), thenw' (0,s) =w'(l,s)=0(12)2 Perturbation ProceduresEqn. (3) is substituted into Eqn. (5), then 1=Afrer introducing the dimensioles variablesr,= [4+3s/ar- -虹As/AX*]1x-oApplying Laplace transformation to the aboveInner surface Symmetricalequation and substituting Eqn. (8) , then the follow-ing equation is obtained.Adiabatic--Adiabatic[(1+s)w- e2w"]|x-o= 1/s+f(0)(13)Fuegas宁Eqn. (12) and Eqn. (13) are the boundary con-dition:中国煤化工AdiabaticEqn. (YHCNMHGin Eqn. (11) to .ceramic regenera-Inner surfacerx+tr Syrminetricaltor, λ,~100-1(W●m~1●K-1), a~10'(W●m-2●Fig1 Physical model and coordinate systemK-), A.~10-6 m, L~10-2 m, and l~10-1 m.Journal of Iron and Steel Research, InternationalVol.14When the equation is degenerated, that is e=0, thewo=A。(6)+B。(6)●e-V+*+C,(E)●e-/TP (26)solution will be transferred to its special form. If theThe reason for stting the solution of Eqn. (24) is .solution of Eqn. (11) to Eqn. (13) w(X,s,e) is re-that when the complex plane s is separated by Im(s)=0garded as the function of ε and it has the necessaryalong the line Re(s)∈(- -∞,1) and√1+s is ana-continuous and differentiable properties, w(X,s,E)lyzed on the separated plane, the main value branchcan be expanded to Taylor series ofe near e=0.Ife=0, the boundary condition Eqn. (12) shouldwill satisfy Re (√1+s) >0; hence, there will bebe discarded. Therefore, there is a boundary layer inno singularities for Eqn. (27) onφ，中++∞.Eqn. (21) is simplified as follows:each end of Eqn. (11). Introducing the variables of mul-Ko[w]=- K[wo]+F(引)(27)tiple scales ξ= ; X, φ= =u(X)/e, and φ=v(X)/e, satisfySubtituting Eqn. (26) and Eqn. (16) into Eqn. (27)u(X)→X when X-→0 and v(X)→1-X when X→1.the following equation can be obtainedLet D= a/dξand E=u' a/aφ+v' a/a p,Ko[w]=(1+s)A'o+s2sAo+ F(引-[2(1+s)B'。+then the regularity of multivariate chain differentia-B。]●e-v1+x-[2(1+s)C。+Cj].e-/I+p (28)tion is obtained as follows:When the righthand side of Eqn. (28) is notd/dX=D+e-'E .equal to zero, there will be such terms as φ●(a+b.d/dX*=D2 +ε-'(DE+ED)+ε-'Ed/dX3 =D* +e-(D*E+ DED+ED*)+e~v1+#) and φ●(c+d●e-JIT印) in the solution,e-"(DE*+ EDE+ED)+e-*Ewherea, b, c, and d are coefficients. These termsThe above regularities are substituted intomake w:/wo infinite when φ, φ→∞,Similar to deduction of Eqn. (26), the homoge-Eqn. (10), thenneous equation of Eqn. (28) can be defined as folH(s,s2,e)=e~'Ko+K1+εK2+e2. Ks(14)lowswhere K,(n=0,1,2,3) is linear differential opera-w1=A()+B,(E).e- /T+*+C.(6)●e-VIFo (29)tor.If the hypothesis that the right-hand side ofKo=E(E-1-s)(15)K = DEr +EDE+E D+nE*-(1+s)D-Qs (16)Eqn. (28) equals zero uniformly holds, it can be ob-tained.Kz= D'E+DED+ ED2 +Q2(DE+ ED)Ks = D'(D+Q)(18)(1+s)A'o+0.s. Ao+F(月)=0(30)2(1+s)B',+B。=0(31)An asymptotic expansion of w(X,s) can be2(1 +s)C'.+C。=032)constructed as follows:Eqn. (30) to Eqn. (32) are the differential equa-w(X,s,e)=. z e*w,(ξ,中,p,s)(19)tions satisfied by Ao, B。and Co.Substituting Eqn. (14) and Eqn. (19) into .Because the term e' can result from the termEqn. (11), the following equation can be obtainede'w" of Eqn. (26)，the 2nd order solution should bestudied to solve the coefficient of e° and e' under the(Ko+eK: +e Kz+e' Ks)[ Ee'rw(ξ,φ,p)]=εF()boundary conditions. The deduction of 2nd order so-After the left-hand side of the above equation is ex-lution is similar to that of Eqn. (26).panded and the coefficient of e" in both sides is equal,wr=A2(日)+B2(4)●e-VTP+Cr(6)●e-VT+ (33)a series of asymptotic equations can be achieved as(1+s)A'1+Q.s. AI=0(34)follows: .B.=[B(0)-mB。(0)E]●e(1([Q(1+0]) .(35)Ko[wo]=0(20) C; =[Cr (0) +mC。(0)]●e(0/C1+)])(36)Ko[w.]+ K[wo]= F(月)(21)where m is constant; B, C1 are function solved byKo[wr]+Kr[wr]+ Kz[wo]=0(22) Eqn. (30) to Eqn. (32); and Br, Cz are functionKo[w3]+ K,[wr]+ K2[wiJ+ Ks[wo]=0(23)solved by Eqn. (34) to Eqn. (36).Combining Eqn. (15), Eqn. (20) can be rewrit-Considering the existent reason of Eqn. (26),ten asthe following equation can be obtained.E(E*-1-s)[w,]=0(24)w2=fE+ge=].Supposeu'=-v'=1, that is to say,中国煤化工行)u(X)=X≥0 v(X)=1- X≥0(25)fYHCNMHG(37)The solution of Eqn. (24) can be constructed aswhere fg, gB, fc, and gc are constant related to s.The 2nd order approximation of w(X,s) can beNo.4Perturbation Solutions for Thermal Process of Honeycomb Regeneratorexpressed asv1+x1- A'.(1)。. n_w(X,e)=wo +ewn +e wr +O(e' )”/1+5exPL(2(1+5)~Substituting Eqn. (26), Eqn. (29), and Eqn. (37)into Eqn. (19), the result can be obtained as fol-y1+s))(1- x)]} +O(e2)(47)lows:To get the analytical expression of A, the inver-w(X)=Ao(X) +e. A:(X) +e'Ar(X)+sion of Laplace transform should be applied to w(X ,p[ -0x_√1+5X] {B。+eB, (0)+e B,(0)+s). Considering the subscript “y”and“s", the re-2(1+s)一 esults can be obtained by using the MATLAB symbol[efg-emB。(0)]X+e'g8X}+{C+e. C1(0)+operation function as follows:e'Cr(0)+[e*fc+e. mC(0)]X+e'gcX"}●A.(X,r,)=1+e~',[h(X)+史, . (X-w)e,".-nXv正(1-x)| +0(e*)(38)| 2(1+5)一一 EdI。(2 /平,wT)dw](48)With the boundary conditions of Eqn. (11) todwEqn. (13), all integral constants in Eqn. (38) can beA.(X,r,)=\,e-,[(f(0)-1)H(X,r,)-solved.f(1)K(X ,r,)+e',M(X,r,)+Considering the existent reason of Eqn. (26),2y;N(X,r,)* f f(1-w)●Re( /1+5)>0 always holds. Thus lime ~"e~ 1T/e=0 whenn=1, 2, 3, ..we ,I,(2 V亚L四dw](49)Substituting Eqn. (38) into Eqn. (12), and the2 v亚,I,wcefficient ofe-',e', ande' in both sides is equal, wherethe following equations can be obtained.(39H(X,I,)= erfc(X _ + sin V2业,XI,B。(0)=C。(0)=02 ar, √0.5π●,XA'。(0)-v1+sB,(0)=0(40)1-X i。sinh√24,(1-X)T,K(X,r,)= eric(,√0.5π. ,(-X)A'。(1)+√1+sCi(0)●e-([2(1+m]=0(41)2 Vn,T,Substituting Eqn. (39) to Eqn. (41) into Eqn.1-x、. sinh√2平,(3- X)Ty(13), and the cefficient ofe~', e", and e' in bothM(X,r)= erfc(2 2,T v0. 5x(3-X)sides of Eqn. (13) is equal, the following equationscan be obtained.P(X,r,)=sinh√24,(1- XF√0.5π●,一对A。(0)=1/[s(1+s)]+f(0)/(1+s)(42)1-XA:(0)=043)N(X,I,)= erfc().[P(x,r,)-~2 V21,Solving Eqn. (30) and Eqn. (42), the followingequation can be obtained.「”' P(X ,w)dw]A(x)=+-+.+46)The perturbation solution is so complex that itSolving Eqn. (34) and Eqn. (43), A(X)=0 iscannot be used directly. After the spatial domain andobtained.the tin中国煤化工T: semi-analytic solu-The 1st order asymptotic solution w(X,s) cantion ine solution processbe written asfor airYHC N M H G are pursued alter-A'。(0)natively. The solid temperature distribution at thew(X,s)=Ao(X) +e{EexPL2(1+5)-.√1+send of one period is used as the initial condition for●10.Journal of Iron and Steel Research, InternationalVol.14the next period. The judgment criterion to stop the near the outlet for flue gas and solid increases gradu-iteration is that the heat amount released from theally as the heat stored in the solid media is enforced.flue gas equals that absorbed by the air.The solid temperature at the outlet [x=400 mm inFig.2 (c)] can never exceed the flue gas tempera-3 Results and Discussionture [x= 400 mm in Fig.2 (a)], which remainsTo prove the perturbation validity, the analyti-physically correct. The closer to the inlet or thecal result is compared with the experiment and the shorter time after the switch, the larger the temper-numerical solution. The same parameters are T=ature change of gas and solid, which agrees well30 s, the regenerator dimension 0.3 mX0.3 mXwith the Hausen regenerative theory. Their differ-0.4 m, L=8.4 mm, 8=1 mm, the velocity uy,o=ence comes from several aspects such as the instabil-0.5 kg/s, u.,=0.43 kg/s, 0,o=950 C, and .1= ity of gas temperature at the inlet of the experiment150 C. As to the analysis, Q,= 12.777, r, =[x= 0 mm in Fig.2 (a)], the unstable combustion10.908, 2=9.977 82, r.=7.183, 虫= 2.363X heat of the fuel, the assumed fixed thermal parame-10-',业=3. 588X10-s, the step Ax=5 mm, andters of gas, and the neglected entrance effect insideOt=0.5 s. Fig. 2 shows the comparison between the the passage in thermal model. Considering the equiva-perturbation solution and the experiment as well aslent heat transfer coefficient between the gas and sol-the comparison between the perturbation solution and id phases, the analysis difference from the heat radi-numerical simulation using the software FLUENT.ation can be decreased.The tendency of perturbation solution ( realThe perturbation result (real lines) in Fig.2 (b)lines) in Fig. 2 (a) and (c) coincides with that of theand (d) agree well with the numerical result (dashedexperiment ( dashed lines), and their difference is lines). Contrary to the numerical calculation, thevery small. During the flue gas cooling period, theconvergent conditions for the iteration generallyflue gas is passing through the regenerator anchold; therefore, the perturbation is more adaptable.cooled down by the solid media. The temperatureOn the other hand, the boundary condition type,1 000一Semi analysis@) |- Semi- analysis叫)308一Experiment例一Numerical result00 t、30s30s600 t205心 20g4006s208is2001000|e)d)|- - Semi-analysis- Serni-analysis800一Experiment图、30S 600、20s20st;心305-6s200L .1003004000中国煤化工学x/mmMYHCNMHGFig.2 Temperature profile of fue gas (a), (b) and sold preheated one (c), (d)(Continued on Page 48)●48.Journal of Iron and Steel Research, InternationalVol. 14slightly higher value of friction consistently at allWear Resistance for Plin Carbon Steels [J]. Mat Sci Tecnol,1998, 14; 776-782.loads than the H steel.，(5) The wear for both the steels is mainly mild[7] Sawa M, Rigney D A. Sliding Behavior of Dual-Phase Steels inVacum and Air []. Wear, 1987, 119; 369-390.oxidative wear and the wear particles are mainly a [8]Clayton P. The Relations Between Wear Behaviour and BasicFe2O3.Material Properties for Pearlitic Steels []. Wear, 1980, 60;75-93.[9] Smith A F. The Friction and Sliding Wear of Unlubricated 316References:Stainless Steel in Air at Room Temperature in the Load Range[1] Davies R G, Magee C L. Structure and Properties of Dua-0.5-90 N[J]. Wear, 1986, 110; 151-168.Phase Steels [A]. Kott R A, Morrise J w, eds. Physical Met-10] Smith A F. The Unlubricated Reciprocating Sliding Wear of aallurgy of Automotive High-Strength Steels [C]. New York;Martensitic Stainless Steel in Air and CO2 Between 20 andTMS AIME, 1979. 1-19.300个[]. Wear, 1986, 123: 313-331.:2] Tyagi R, NathS K, Ray S. Development of Wear Resistant[11] Iwabuchi A, Hori K, Kubosawe H. The Efet of Oxide PerMedium Carbon Dual Phase Steels and Their Mechanical Propticles Supplied at the Interface Before Sliding on the Severeerties [J]. Mat Sci Technol, 2004, 20: 645 652.Mild Wear Transition [J]. Wear, 1988, 128; 123-137.3] JhaA K, Prasad BK, Modi OP, et al Correlating Micro-[12] Quinn TF J, Rowson D M, Sullivan J M. Application of thestructural Features and Mechanicel Properties With AbrasionOxidational Theory of Mild Wear to the Sliding Wear of LowResistance of a High Strength Low Ally Steel [J]. Wear,Alloy Sreel [J]. Wear, 1980, 65; 1-20.2003, 254; 120-128.[13] SulivanJ L, Quinn T FJ, Rowson D M. Developments in theWayne s F, Rice s L. The Role of Microstructure in Wear ofOxidational Theory of Mild Wear [J]. Tribol Int, 1980, 13Selected Steels [J]. Wear, 1983, 85; 93-106.(4); 153-158.5] ModiOP, Prasad BK, Jha AK, etal, Low Stress Wear Be[ 14] Blau P J. Mechanisms for Transitional Friction and Wearhavior of 0. 2%C Steel; Influence of Microstructure and TestBehavior of Sliding Metals U]. Wear, 1981, 72; 5-66.Parameters [J]. Tribol Lett, 2003, 15; 249-255.15] Saka N, Pamies-Teixeria JJ,Suth N P. Wear of Two-Phase6]Basak A, Reddy D C, Kanth D V K. Computer Modeling ofMetals [J]. Wear, 1977, 44: 77-86.(Continued From Page 10)the velocity m agni- [2] ChunX T, Jien DT. Apiacion and Devloment of Ho-eycomb Regenerative Combustion System [J]. Industrial Heat-tude, and its direction alternatively change becauseing, 1998, 35(3); 26-35 (in Japanese).the gas inlet and outlet positions are peioially var [3] u We, Qi Harying, You Chang fo, er aL Numeric Reearchied. The simulation parameters need to be inputon Heat Transfer in Honeycomb Regeneretor [J]. Journal ofmanually and periodically. Hence, the perturbationEngineering Thermophysics, 2001, 22(5); 657-660 (in Chi-nese).is more effective.[4] LI Jin, FU Wei-biao, HOU Ling -yun. Numerical Analysis fora4 ConclusionsRegenerative and Generative Hydrogen Unit [J]. Journal ofCombustion Science and Technology, 2003, 9(3): 261-266 (inBecause the coefficient of heat conduction termChinese),in the energy equation is a small parameter, the pa-[5] Nabil R, Wlodzinierz B. Thermal Performance Analysis on aTwo Composite Material Honeycomb Heat Regenerators Usedrameter perturbation can be used to study the regen-for HTAC Burners []. Applied Thermal Engineering. 2005,erator heat transfer process. The results proved that25(17-18)。2966-2982.the perturbation analysis of regenerative heat ex[6] Zarrinchkafsh/M T, Sadrameli s M, Simulation of Fixed Bedchanger makes its thermal research including tem-Regenerative Heat Exchangers for Flue Gas Heat Recovery[J]. Applied Therinal Engineeing, 2004, 24(2-3); 373-382.perature distribution, maximum temperature effi-[7] Zheng C H, Clements B. The Thermal Performance Characte-ciency, and switch time more economic and effec-istics of Regenerators in HITACG Furnaces [A]. Gaswirmetive.Institute V Essen, eds. 6th International Symposium on HighIt is necessary to carry out further research onTemperature Air Combustion and Gasification [C]. Ruhrgebi-et, Germany; Gas Warme Institut, 2005. A9. 1-A9. 12.mathematic analysis when the heat conduction of stor-[8]Klein H, Eigenberger G. Approximate Solutions for Metllicage matrix along length direction becomes larger.Regenerative Heat Exchangers [J]. International Journal ofHeat and Mass Transfer, 2001, 44(18): 3553-3563.[9]_ LI Mao-de, CHENG Hui-er. Theretical Analysis of Heat1] Hiroshi T, Gupta A, Hasegawa T, et al. High Temperature中国煤化工High Tempernture AirAir Combustion From Energy Conservation to Pollution ReducEThermal Scienceeandtion [M]. New York, USA; CRC Press, 2003.Y片C N M H Gin Chinese).