thermal analysis for transient radiative cooling of a conducting layer of ceramic in high-temperature applications

1、Infrared Physics & Technology 47 (2006) 278285/locate/infraredThermal analysis for transient radiative cooling of a conducting semitransparent layer of ceramic in high-temperature applicationsParham Sadooghi *, Cyrus AghanajaK.N.T. University of Technology, Tehran, IranReceived 13 Ja

2、nuary 2005Available online 22 July 2005AbstractA method is developed for obtaining transient temperature distribution in a cooling semitransparent layer of cera- mic. The layer is emitting, absorbing, isotropically scattering and heat conducting with a refractive index ranging from 1 to 2. The solut

3、ion involves solving simultaneously the energy equation and the integral equation for the radiative ux gradient. The energy equation is solved using an implicit nite volume scheme and the integral equation of radiative heat transfer is solved using the singularity technique and Gaussian integration.

4、 The eects of scattering are investi- gated. It is shown that scattering has a signicant eect on the transient temperature distribution and the transient mean temperature of the layer.2005 Elsevier B.V. All rights reserved.1. Introductioncorrosion resistance, impact resistance, and excel- lent fatig

5、ue strength. Today, ber composites areFiber reinforced composite materials are now an important class of engineering materials. Be- cause of their useful properties, they are widely used in various elds of engineering. They oer outstanding mechanical properties, unique exi- bility in design capabili

6、ties, and ease of fabrica- tion. Additional advantages include lightweight,routinely useduch diverse applications as auto-mobiles, aircraft, space vehicles, o-shore struc-tures, containers and piping, sporting goods, electronics and appliances. The need for high-tem- perature reinforcing bers has le

7、d to the develop- ment of ceramic bers. An important factor that limits the performance (eciency and emission) of current gas turbines is the temperature capa- bility (strength and durability) of the metallic structural components in the engine hot section (blades, nozzles, vanes and combustor liner

8、s). It* Corresponding author.E-mail address: (P. Sadoo- ghi).1350-4495/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2005.04.007279P. Sadooghi, C. Aghanaja / Infrared Physics & Technology 47 (2006) 278285is generally agreed that the tempe

9、rature capabili- ties of metals have reached their ceiling. Ceramic bers such as alumina bers and silicon carbide - bers exhibit superior durability and combine high strength and elastic modulus with high tempera- ture capability, which implies their potential to revolutionize gas-turbine technology

10、. Accurate prediction of temperature distributions in ceramic and other brous and semitransparent materials (STM) at high temperature, are essential during various fabricating operations. Within a semi- transparent medium, the temperature and heat ux distributions are aected by radiation in addi- ti

11、on to heat conduction. The transient coupled radiative and conductive heat transfer in a semi- transparent medium is one of the pervasive pro- cesses in engineering applications, such as heat dissipating materials 1, tempered glass and the application of its products 24, thermal property analysis fo

12、r ceramic parts 5, manufacture and application of optical ber and its products 6, melting and removal of ice layers 7, transient re- sponses to volumetrically scattering heat shields 8,9, ignition and ame spread for translucent plastics and solid fuels, and so on. The radiation eects become more imp

13、ortant when the STM isat elevated temperatures, in high temperature sur- roundings, or subjected to large incident radiation, and the radiation uxes depend strongly on thetemperature level.emitransparent materialswhere thermal radiation can aect internal temper-ature distributions, transient behavio

14、r has been studied much less than steady state. To obtain transient solutions numerical procedures such as nite dierence and nite element methods, dis- crete ordinates method, have been used to solve the radiative transfer relations coupled with the transient energy equation. The transient thermal b

15、ehavior of a single and multiple layers of semi- transparent materials have been studied for variety of cases, as reviewed recently by Siegel 10.Earlier studies on transient coupled radiative conductive heat transfer TM were carried out mainly for the boundary conditions of the rst kind, with prescr

16、ibed temperatures at the frontiers. They were summarized by Viskanta and Anderson 11. Much research has been directed toward the combined eects of conduction and radiation in glass. Chu and Gardon 12 under took the study of combined transfer in gray glasses. Su and Sut- ton 13 predicted temperatures

17、 and heat uxes with 16 spectral bands in the silicate glass plateNomenclatureDthickness of the medium, mTmmean temperature, KE1, E2, E3 the eRxponential integral functions,Xdimensionless coordinate, x/DEnx 1 ln 2exp x=l dlxcoordinate in direction across the layer,0kthermal conductivity of the medium

18、,mW/m Kaabsorption coecient of the layer, m 1Nconductionradiation parameter,bextinction coecient of the layer, m 1k=4rT 4Dhtime, sinrefractive index of the mediumjDoptical thickness of the layer, bD Qincidence radiation, W/m2qinterface reectivityq1, q2radiosity from the interior of the twoqCpproduct

19、 of density and specic heat of boundaries, W/m2the layer, q/m3 Kq1; q2dimensionlessradiosities, q1=rT 4; q2=rT 4rStefanBoltzmann constantiiTabsolute temperature, Ksdimensionless time, 4rT 4=qCpDhitdimensionless mean temperature, T/Ti/dimensionless incidence radiation,tmdimensionless temperature, Tm/

20、TiQ=rT 4iTiinitial temperature, Kxscattering albedoTetemperature of the surroundings, K280P. Sadooghi, C. Aghanaja / Infrared Physics & Technology 47 (2006) 278285dimensionlessexternally heated by a constant heat input at one boundary for a time interval of 5 s. Siegel 14, and more recently, I and m

21、y colleague, 15, used nite dierence procedure to predict transient tem- perature distribution in a semitransparent material with a refractive index of one. We also solved the problem for refractive indices larger than one, that provides internal reections. Hahn and Raether16 examined the transient h

22、eat transfer in a layer of ceramic powder during laser-ash measure- ments of thermal diusivity. A three-ux method was used to solve the equation of radiative trans- fer. However, no results were presented on the eect of scattering and refractive indices. The solution of the exact radiative equation

23、is rather complicated, particularly when the scattering is present; hence a common approach is to use approximate methods such as the two-ux method and diusion and dierential approximations. Unfortunately, approximate methods are usually subject to certain constraints. Under certain condi- tions, th

24、e methods may not be valid. This paper presents a direct numerical procedure for obtain- ing accurate transient temperature distribution in a semitransparent layer of ceramic ber. The solu- tion includes the integral equation for the radiative ux gradient, coupled to a transient energy equa-The tran

25、sient energy equation in form is 17:oto2tos N oX 2 Rt;where R(t) is the gradient of the1radiative uxunder the assumption of isotropic scattering and is a function of X and sRt jD 1 xn2t4/=4 ;2where / is the local incident function. Considering the isotropic scattering, / is given by 17:/X 2q E2Dj X

26、2q E 2Dj 1X i2Z1 2pjDSX 0E1jDjX0X 0j dX 0;3where the source function is written as follows:SX 1 x n2 2t x=4p/.4nAnd q1 and q2 are dimensionless diuse uxes at the boundaries at X = 0 and X = 1, respectively. For the symmetrical case and under the assump- tion of diuse reection 18, by using of the Fre

27、s- nel reection relations q1 and q2 are given byq1 q2Z1tion that contaboth radiative and conductive1. 2pqjDSX 0E2jDX 0 dX 2qE3jDterms. Solutions are given to demonstrate the ef- fect of isotropic scattering on the radiative cooling of the layer.05erting Eqs. (4) and (5) into Eq. (3), leads toZ1/X n2

28、F X ; X 0t4X 0 dX 002. AnalysisZ1The analysis is for a gray absorbing, emitting, isotropically scattering and heat conducting layer of thickness D, as shown in Fig. 1.Initially the layer is at a uniform temperature 2n2jD1xt4X 0E1jDjX0X 0j dX 0Z1F X ; X 0/X 0 dX x 41x0T . It is then placed in a much

29、colder surrounding.Zi1x 2 jD/X 0E1jDjXX 0j dX 0;60Te T (X)where qjD1x DF X ; X 0 4fE2jDX 12qE3Dj E2jD1X gE2jDX 0.7Fig. 1. Geometry and nomenclature of the plane layer.XT(X)281P. Sadooghi, C. Aghanaja / Infrared Physics & Technology 47 (2006) 2782853. Boundary conditionsBy solving Eqs. (11) and (12),

30、 dRt =dt is obtained.The integral equations (6) and (12) are solved numerically by using the well know Nystrom method 20.The layer is divided into M increments in X with smaller DX near the boundaries. The discretization of the integral equations (6) and (12) is derived by using the Gaussian quadrat

31、ure 20. Variable time increments are used with smaller time steps in the beginning of the process. At each time step Eqs. (1), (2), (6), (11) and (12) are solved simultaneously and iterations are performed to obtain the temper- ature distribution. For a solution at each time increment, an initial gu

32、ess of the temperature distribution across the layer must be provided. Typically in this calculation, the temperature distri- bution at a previous time level is used for the rst guess. The values of temperature at the Gaussian abscissas are obtained using cubic spline interpola- tion 20, with temper

33、ature specied, the values of R and dR/dt are obtained from the solution of Eqs. (2), (3), (11) and (12). The exponential integral functions E3 and values of R and dR/dt are then calculated using the Nystrom interpolation 20. With R and dR/dt evaluated, an updated tempera- ture distribution is obtain

34、ed by the solution of Eq.(1) to have ahead one time increment. The newtemperature distribution is used to reevaluate the values of R and dR/dt. This process is repeated until the desired convergence is achieved. It is found that the larger the optical thickness of the layer, the denser grids and the

35、 higher order Gaussian quadratures are needed because of steep variations of the radiative ux at the boundaries.Since convective heat transfer is negligible, boundary conditions are required for radiationand for heat conductiononly. Radiation passesout of the layer from within the material; there is

36、 no emission at the boundaries, which are planes without volumes. There is no external conduction or convection, such as a device used for dissipating waste heat from equipment operating in the cold vacuum of outer space or on the moon. Therefore the following boundary conditions should be used for

37、the preceding energy equation:ot0; soXot0; soX 0.8The initial temperature of the layer is uniform, so thenon-dimensionalized temperature begtX ; 0 1.from 1,94. Numerical solution procedureThe discretization of the energy equation is de- rived using the implicit nite volume scheme 19:dRtRt Rt t tdt d

38、Rt dRt Rt tt;10dtdtwhere t is temperature from previous iteration.From Eq. (2), the time derivative of R(t) in Eq.(10) isdRt1 d/4 dt 4n2t 311 j 1 x.DdtAnd Eq. (6) gives,5. Results and discussionZ1d/ X 4n2FX ; X 0t 3 X 0dX 0 81 xn2j0DThe eect of variations of the inuence parame- ters, such as refract

39、ive indices, optical thicknesses and scattering albedos on the temperature distri- butions are shown in Figs. 27. The conditions are symmetric on both sides of the layer so the transient temperatures are symmetric and the distributions are given for one-half of the layer.dtZ1t X3 0 x41 xE j jX X 0jd

40、X 0 1 D0Z1F X;X 0d/ X dtxj D2dX 0 0Z1 d/ X dtResults are given at three transient.tances during theE1jDjX X 0jdX 0.120282P. Sadooghi, C. Aghanaja / Infrared Physics & Technology 47 (2006) 27828510.950.90.850.80.750.70.650.60.5510.950.90.850.80.750.70.650.60.55= 0.11= 0.14= 0.45= 0.66= 1.5= 2.2 = 0 =

41、 0= 0.3 = 0.3n = 1, k = 5, N = 0.1 = 0.6= 0.6D n = 1, k 15, N = 0.1 = 0.9= 0.9D 00.40.500.40.5abX= x/DX= x/D10.950.90.850.80.750.70.650.60.5510.950.90.850.80.750.70.650.60.550.50.45= 0.14=0.11= 0.66= 0.45= 2.2=1.5 = 0 = 0 = 0.3 = 0.3= 0.6 = 0.6 = 0.9n = 1, kD = 5, N= 0.0= 0.9n = 1,

42、 k = 15, N= 0.0D00.40.500.40.5cdX= x/DX= x/DFig. 2. Transient temperature distribution under various conditions for a layer with (a) n = 1, N = 0.1, jD = 5; (b) n = 1, N = 0.1,jD = 15; (c) n = 1, N = 0, jD = 5; (d) n = 1, N = 0, jD = 15.Fig. 2 give transient temperatures for a laye

43、r ofn = 1 and Fig. 3 for n = 1.5 with x = 0.3, 0.6 and0.9 for all gures.Dierent optical thicknesses and radiationcon- duction parameters are considered. The results shows that heat conduction serves to equalize tem- perature across the layer and increased optical thickness makes the layer optically

44、thick and gives a steeper temperature distribution near the bound- aries. An important eect of a larger refractive in- dex is that internal reections promote the distribution of radiative energy within the layer, this makes the transient temperature distributions more uniform.Figs. 2 and 3 show the

45、eect of scattering when the optical thickness is held x, in creasing scattering gives a move uniform temperature dis- tribution and slows down the temperature drop. This is because of the relatively reduced emitting ability or absorption thickness of the layer, aD, when x is increased, because the r

46、adiative heat loss of the layer depends strongly upon the mag-nitude of aD which is equal to (1 x)jD. In this case, the eect aD of is somewhat similar to that of jD on a non-scattering layer. In the limiting case when x = 1, the layer remaat the initial temperature because there is no emitting and a

47、bsorption and, hence no heat loses fromthe layer. The eect of increasing scattering albedo is signicant for lower optical thickness. Compar- isons between parts (a) and (b) and between parts(c) and (d) in Figs. 2 and 3, show that a larger optical thickness tends to weaken the eect of scattering. The

48、 reason is that a larger optical thickness and, thus, a larger absorption thickness, means more scattered energy being absorbed, which osets the eect of scattering. The transient mean temperatures are shown in Fig. 4 (for n = 1) and Fig. 5 (for n = 1.5). The result show that with other conditions ke

49、pt constant as in parts (a) and (b) of Figs. 2 and 3, increasing scat- tering, decreasing the value of aD, slows down the decrease of temperatures, particularly at lar- ger scattering albedos.t =T/Tit =T/Tit =T/Tit =T/Ti283P. Sadooghi, C. Aghanaja / Infrared Physics & Technology 47 (2006) 27828510.9

50、50.90.850.80.750.70.650.60.550.510.950.90.850.80.750.70.650.60.550.5= 0.25= 0.25 = 1.0 = 1.0 = 2.5= 2.5 = 0 = 0 = 0.3= 0.3= 0.6N = 0.1 n =1.5,k = 0.6N = 0.1 = 0.9n =1.5,k= 0.900.40.500.40.5aX= x/DbX= x/D10.950.90.850.80.750.70.650.60.550.510.950.90.850.80.750.70.650.60.550.5= 0.25=

51、 0.25= 1.0= 1.0= 2.5= 2.5 = 0= 0.3 = 0= 0.3= 0.6N = 0.1 = 0.6n =1.5,kN = 0.1 n =1.5,k= 0.9 = 0.900.40.500.40.5cX= x/DdX= x/DFig. 3. Transient temperature distribution under various conditions for a layer with (a) n = 1.5, N = 0.1, jD = 5; (b) n = 1.5, N = 0.1,jD = 20; (c) n = 1.5,

52、N = 0, jD = 5; (d) n = 1.5, N = 0, jD = 20.10.950.90.8510.950.90.85= 5k = 5kDk = 15k D = 20DD0.80.750.70.60.9N = 0.1n = 10.750.70.650.60.6N = 0.1n = 1.50 0= 00.6= 000.511.522.500.511.522.5Dimensionless time,Dimensionless time,Fig. 4. Transient mean temperature of the laye

53、r of n = 1 withN = 0.1, jD = 5 and jD = 15.Fig. 5. Transient mean temperature of the layer of n = 1.5 withN = 0.1, jD = 5 and jD = 20.The fact that a larger optical thickness weakens the eect of scattering can be shown from Fig. 4, the dierence between the curve corresponding to x = 0 and the curve

54、corresponding to x = 0.9 for jD = 5 is larger than that for jD = 15, at any time and at any temperature level. Fig. 4 shows that thecurve corresponding to jD = 5, x = 0.9 is above the curve corresponding to jD = 15 at the same scattering level. This is because at large scattering albedos, the behavior of mean temperature at var- ious optical thickness is di