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Thermal analysis for transient radiative cooling of a conducting semitransparent layer of ceramic in high-temperature applications
Infrared Physics & Technology 47 (2006) 278–285 www.elsevier.com/locate/infrared
Thermal analysis for transient radiative cooling of a conducting semitransparent layer of ceramic in high-temperature applications Parham Sadooghi *, Cyrus Aghanajafi K.N.T. University of Technology, Tehran, Iran Received 13 January 2005 Available online 22 July 2005
Abstract A method is developed for obtaining transient temperature distribution in a cooling semitransparent layer of ceramic. The layer is emitting, absorbing, isotropically scattering and heat conducting with a refractive index ranging from 1 to 2. The solution involves solving simultaneously the energy equation and the integral equation for the radiative flux gradient. The energy equation is solved using an implicit finite volume scheme and the integral equation of radiative heat transfer is solved using the singularity technique and Gaussian integration. The effects of scattering are investigated. It is shown that scattering has a significant effect on the transient temperature distribution and the transient mean temperature of the layer. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Fiber reinforced composite materials are now an important class of engineering materials. Because of their useful properties, they are widely used in various fields of engineering. They offer outstanding mechanical properties, unique flexibility in design capabilities, and ease of fabrication. Additional advantages include lightweight, *
Corresponding author. E-mail address: [email protected] (P. Sadooghi).
corrosion resistance, impact resistance, and excellent fatigue strength. Today, fiber composites are routinely used in such diverse applications as automobiles, aircraft, space vehicles, off-shore structures, containers and piping, sporting goods, electronics and appliances. The need for high-temperature reinforcing fibers has led to the development of ceramic fibers. An important factor that limits the performance (efficiency and emission) of current gas turbines is the temperature capability (strength and durability) of the metallic structural components in the engine hot section (blades, nozzles, vanes and combustor liners). It
1350-4495/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2005.04.007
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
279
Nomenclature D thickness of the medium, m E1, E2, E3 the exponential integral functions, R1 En ðxÞ ¼ 0 ln2 expðx=lÞ dl k thermal conductivity of the medium, W/m K N conduction–radiation parameter, k=4rT 4i D n refractive index of the medium Q incidence radiation, W/m2 q1, q2 radiosity from the interior of the two boundaries, W/m2 q1 ; q2 dimensionless radiosities, q1 =rT 4i ; q2 =rT 4i T absolute temperature, K t dimensionless mean temperature, T/Ti tm dimensionless temperature, Tm/Ti Ti initial temperature, K Te temperature of the surroundings, K
is generally agreed that the temperature capabilities of metals have reached their ceiling. Ceramic fibers such as alumina fibers and silicon carbide fibers exhibit superior durability and combine high strength and elastic modulus with high temperature capability, which implies their potential to revolutionize gas-turbine technology. Accurate prediction of temperature distributions in ceramic and other fibrous and semitransparent materials (STM) at high temperature, are essential during various fabricating operations. Within a semitransparent medium, the temperature and heat flux distributions are affected by radiation in addition to heat conduction. The transient coupled radiative and conductive heat transfer in a semitransparent medium is one of the pervasive processes in engineering applications, such as heat dissipating materials [1], tempered glass and the application of its products [2–4], thermal property analysis for ceramic parts [5], manufacture and application of optical fiber and its products [6], melting and removal of ice layers [7], transient responses to volumetrically scattering heat shields [8,9], ignition and flame spread for translucent plastics and solid fuels, and so on. The radiation effects become more important when the STM is
Tm X x a b h jD q qCp r s / x
mean temperature, K dimensionless coordinate, x/D coordinate in direction across the layer, m absorption coefficient of the layer, m1 extinction coefficient of the layer, m1 time, s optical thickness of the layer, bD interface reflectivity product of density and specific heat of the layer, q/m3 K Stefan–Boltzmann constant dimensionless time, ð4rT 4i =qC p DÞh dimensionless incidence radiation, Q=rT 4i scattering albedo
at elevated temperatures, in high temperature surroundings, or subjected to large incident radiation, and the radiation fluxes depend strongly on the temperature level. In semitransparent materials where thermal radiation can affect internal temperature distributions, transient behavior has been studied much less than steady state. To obtain transient solutions numerical procedures such as finite difference and finite element methods, discrete ordinates method, have been used to solve the radiative transfer relations coupled with the transient energy equation. The transient thermal behavior of a single and multiple layers of semitransparent materials have been studied for variety of cases, as reviewed recently by Siegel [10]. Earlier studies on transient coupled radiative– conductive heat transfer in STM were carried out mainly for the boundary conditions of the first kind, with prescribed temperatures at the frontiers. They were summarized by Viskanta and Anderson [11]. Much research has been directed toward the combined effects of conduction and radiation in glass. Chu and Gardon [12] under took the study of combined transfer in gray glasses. Su and Sutton [13] predicted temperatures and heat fluxes with 16 spectral bands in the silicate glass plate
280
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
externally heated by a constant heat input at one boundary for a time interval of 5 s. Siegel [14], and more recently, I and my colleague, [15], used finite difference procedure to predict transient temperature 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 reflections. Hahn and Raether [16] examined the transient heat transfer in a layer of ceramic powder during laser-flash measurements of thermal diffusivity. A three-flux method was used to solve the equation of radiative transfer. However, no results were presented on the effect of scattering and refractive indices. The solution of the exact radiative equation is rather complicated, particularly when the scattering is present; hence a common approach is to use approximate methods such as the two-flux method and diffusion and differential approximations. Unfortunately, approximate methods are usually subject to certain constraints. Under certain conditions, the methods may not be valid. This paper presents a direct numerical procedure for obtaining accurate transient temperature distribution in a semitransparent layer of ceramic fiber. The solution includes the integral equation for the radiative flux gradient, coupled to a transient energy equation that contains both radiative and conductive terms. Solutions are given to demonstrate the effect of isotropic scattering on the radiative cooling of the layer.
2. Analysis
The transient energy equation in dimensionless form is [17]: ot o2 t ¼N RðtÞ; os oX 2
where R(t) is the gradient of the radiative flux under the assumption of isotropic scattering and is a function of X and s ð2Þ RðtÞ ¼ jD ð1 xÞ n2 t4 ð/=4Þ ; where / is the local incident function. Considering the isotropic scattering, / is given by [17]: /ðX Þ ¼ 2qi E2 ðjD X Þ þ 2q2 E2 ½jD ð1 X Þ Z 1 þ 2pjD SðX 0 ÞE1 ðjD jX X 0 jÞ dX 0 ;
where the source function is written as follows: SðX Þ ¼ ð1 xÞðn2 t2n Þ þ ðx=4pÞ/.
ð4Þ
And q1 and q2 are dimensionless diffuse fluxes at the boundaries at X = 0 and X = 1, respectively. For the symmetrical case and under the assumption of diffuse reflection [18], by using of the Fresnel reflection relations q1 and q2 are given by q1 ¼ q2 ¼ 2pqjD
Z
1
1
½SðX 0 ÞE2 ðjD X 0 Þ dX ½2qE3 ðjD Þ .
0
ð5Þ Inserting Eqs. (4) and (5) into Eq. (3), leads to Z 1 /ðX Þ ¼ n2 F ðX ; X 0 Þt4 ðX 0 Þ dX 0
The 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 Ti. It is then placed in a much colder surrounding.
X
ð3Þ
0
0
Te << T (X)
ð1Þ
2
þ 2n jD ð1 xÞ
Z
1
t4 ðX 0 ÞE1 ðjD jX X 0 jÞ dX 0 0
Z 1 x F ðX ; X 0 Þ/ðX 0 Þ dX 4ð1 xÞ 0 Z 1 x þ jD /ðX 0 ÞE1 ðjD jX X 0 jÞ dX 0 ; 2 0
þ
ð6Þ
where T(X)
D
Fig. 1. Geometry and nomenclature of the plane layer.
F ðX ; X 0 Þ ¼ 4
qjD ð1 xÞ fE2 ðjD X Þ 1 2qE3 ðjD Þ þ E2 ½jD ð1 X ÞgE2 ðjD X 0 Þ.
ð7Þ
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
3. Boundary conditions Since convective heat transfer is negligible, boundary conditions are required for radiation and for heat conduction only. Radiation passes out of the layer from within the material; there is 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 the preceding energy equation: otð0; sÞ otð0; sÞ ¼ ¼ 0. ð8Þ oX oX The initial temperature of the layer is uniform, so the non-dimensionalized temperature begins from 1, tðX ; 0Þ ¼ 1.
ð9Þ
4. Numerical solution procedure The discretization of the energy equation is derived using the implicit finite volume scheme [19]: dRðt Þ ðt t Þ dt dRðt Þ dRðt Þ t þ t; ¼ Rðt Þ dt dt
RðtÞ ¼ Rðt Þ þ
ð10Þ
where t is temperature from previous iteration. From Eq. (2), the time derivative of R(t) in Eq. (10) is dRðt Þ 1 d/ 2 3 ¼ jD ð1 xÞ 4n t . ð11Þ dt 4 dt And Eq. (6) gives, Z 1 d/ ðX Þ 3 2 ¼ 4n F ðX ;X 0 Þt ðX 0 ÞdX 0 þ 8ð1 xÞn2 jD dt 0 Z 1 x 3 t ðX 0 ÞE1 ðjD jX X 0 jÞdX 0 þ 4ð1 xÞ 0 Z 1 d/ ðX Þ 0 xjD dX þ F ðX ; X 0 Þ dt 2 0 Z 1 d/ ðX Þ E1 ðjD jX X 0 jÞdX 0 . ð12Þ dt 0
281
By solving Eqs. (11) and (12), dRðt Þ=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 quadrature [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 temperature distribution. For a solution at each time increment, an initial guess of the temperature distribution across the layer must be provided. Typically in this calculation, the temperature distribution at a previous time level is used for the first guess. The values of temperature at the Gaussian abscissas are obtained using cubic spline interpolation [20], with temperature specified, 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 temperature distribution is obtained by the solution of Eq. (1) to have ahead one time increment. The new temperature 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 higher order Gaussian quadratures are needed because of steep variations of the radiative flux at the boundaries.
5. Results and discussion The effect of variations of the influence parameters, such as refractive indices, optical thicknesses and scattering albedos on the temperature distributions are shown in Figs. 2–7. 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. Results are given at three instances during the transient.
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55
a
= 0.11
= 1.5 =0 = 0.3 = 0.6 = 0.9
n = 1, kD = 5, N = 0.1
0
0.1
0.2
0.3
0.4
0.5
= 0.11 = 0.45
t =T/Ti
=1.5 =0
= 0.3 = 0.6 = 0.9
n = 1, kD = 5, N = 0.0
0
0.1
0.2
0.3
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 n = 1, k 15, N = 0.1 D 0.55 0 0.1 0.2 0.3
0.4
0.5
X= x/D
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45
= 0.66 = 2.2 =0
= 0.3 = 0.6 = 0.9 0.4 0.5
= 0.14 = 0.66 = 2.2 = 0
= 0.3 = 0.6 = 0.9
n = 1, kD = 15, N = 0.0
0
d
= 0.14
X= x/D
b
X= x/D
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55
c
t =T/Ti
= 0.45
t =T/Ti
t =T/Ti
282
0.1
0.2
0.3
0.4
0.5
X= x/D
Fig. 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 layer of n = 1 and Fig. 3 for n = 1.5 with x = 0.3, 0.6 and 0.9 for all figures. Different optical thicknesses and radiation–conduction parameters are considered. The results shows that heat conduction serves to equalize temperature across the layer and increased optical thickness makes the layer optically thick and gives a steeper temperature distribution near the boundaries. An important effect of a larger refractive index is that internal reflections promote the distribution of radiative energy within the layer, this makes the transient temperature distributions more uniform. Figs. 2 and 3 show the effect of scattering when the optical thickness is held fix, in creasing scattering gives a move uniform temperature distribution 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 radiative heat loss of the layer depends strongly upon the mag-
nitude of aD which is equal to (1 x)jD. In this case, the effect aD of is somewhat similar to that of jD on a non-scattering layer. In the limiting case when x = 1, the layer remains at the initial temperature because there is no emitting and absorption and, hence no heat loses from the layer. The effect of increasing scattering albedo is significant for lower optical thickness. Comparisons 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 effect of scattering. The reason is that a larger optical thickness and, thus, a larger absorption thickness, means more scattered energy being absorbed, which offsets the effect 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 kept constant as in parts (a) and (b) of Figs. 2 and 3, increasing scattering, decreasing the value of aD, slows down the decrease of temperatures, particularly at larger scattering albedos.
t =T/Ti
a
= 1.0 = 2.5 =0 = 0.3 = 0.6 = 0.9
N = 0.1
0.2
0.3
0.4
0.5
b
X= x/D 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 n =1.5, k 0.5 0 0.1
= 0.25
= 0.25
= 1.0
N = 0.1
n =1.5, k
0
0.1
= 2.5 =0 = 0.3 = 0.6 = 0.9
0.2
0.3
0.4
0.5
X= x/D
c
283
= 0.25 = 1.0
t =T/Ti
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 n =1.5, k 0.5 0 0.1
t =T/Ti
t =T/Ti
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
= 2.5 =0 = 0.3 = 0.6 = 0.9
N = 0.1
0.2
0.3
0.4
0.5
X= x/D 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 n =1.5, k 0.5 0 0.1
d
= 0.25 = 1.0 = 2.5 =0 = 0.3 = 0.6 = 0.9
N = 0.1
0.2
0.3
0.4
0.5
X= x/D
Fig. 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, N = 0, jD = 5; (d) n = 1.5, N = 0, jD = 20.
1
k =5 D kD = 15
0.9 0.85 0.8
N = 0.1 n =1
0.75 0.7
0.9 0
0.6
0.3
0.65
=0
0.3 0.6
0.6
Mean temperature, Tm/Ti
Mean temperature,Tm/Ti
1 0.95
k =5 D k = 20
0.95
D
0.9 0.85 0.8
0.9
0.75
N = 0.1 n = 1.5
0.7 0.65
0.3
0.3
0.6 0.6 0
=0
0.6
0
0.5 1 1.5 Dimensionless time,
2
2.5
0
0.5
1 1.5 Dimensionless time,
2
2.5
Fig. 4. Transient mean temperature of the layer of n = 1 with N = 0.1, jD = 5 and jD = 15.
Fig. 5. Transient mean temperature of the layer of n = 1.5 with N = 0.1, jD = 5 and jD = 20.
The fact that a larger optical thickness weakens the effect of scattering can be shown from Fig. 4, the difference between the curve corresponding to x = 0 and the curve 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 the
curve 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 various optical thickness is different from that at small scattering albedos. The effect of large scattering is
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
Mean temperature,Tm/Ti
284 1 0.95 0.9 0.85
κD = 2
N = 0.1 n =1 ω = 0.9
0.8 0.75
5
50 20
1.5
2
10
0.7 0.65 0
0.5
1
2.5
Dimensionless time, τ Fig. 6. Transient mean temperature of the layer of various jD at x = 0.9 for n = 1, N = 0.1.
shown in Fig. 6. The scattering albedo is held constant at x = 0.9 but the optical thickness of the layer is varied from 2 to 50. At small values of optical thickness jD < 10, the transient mean temperature decreases with an increasing optical thickness; however at larger jD, this trend is reversed. This is related to the magnitude of aD, when aD is small, the temperature distribution is relatively uniform and the radiation rays pass out of the layer because of relatively small absorption. Increasing aD enhances the radiation, and hence the cooling rate. When jD is large, the temperature near the boundaries becomes increasingly cooler than the interior. This explains the special case appearing in Fig. 4 that was mentioned earlier. At such small aD, the cooling effectiveness increases with increasing aD.
κD = 1, ω = 0 κD = 10, ω = 0.9 κD = 100, ω = 0.99
0.98 0.96
t=T/Ti
0.94
1
n=1
Mean temperature, Tm/Ti
1
n=2 n=1
αD = 1 τ = 0.45
0.92 0.9
n=2 n=1 n=2
0.88 0.86
In Fig. 7 the effect of scattering is shown in a different way, aD is held constant, while jD is increased such that the effect of additional scattering can be shown. The temperature distribution at s = 0.45 is shown in Fig. 7(a). It is seen that increasing the scattering leads to steeper temperature distribution near the boundaries, an observation different from that shown in Figs. 2 and 3. Note that in the case of Fig. 7, jD is increased much faster than the increase of x, thus the layer quickly becomes increasingly opaque. In contrast, in Figs. 2 and 3 jD is held constant when effect of increasing scattering is examined. Fig. 7(b) reflects that the cooling rate of the layer decreases with increasing x and jD, with aD being fixed. This is caused by the temperature distribution depicted in Fig. 7(a). It is interesting to note the effect of refractive indices on the transient process at large scattering albedos. In Figs. 7, 2 and 3, when x = 0, a larger n gives more uniform temperatures and that at N = 0.1 and with jD ranging from 2 to 20, there is a decrease in the cooling rate when n is increased. However the effect of n is quite different at large x as is shown in Fig. 7. When x = 0.9 and 0.99, the cooling rate with n = 2 is considerably faster than that with n = 1, and at jD = 100, x = 0.99, the temperature distribution for n = 2 is steeper than that for n = 1. This is most likely a result of enhanced cooling at positions near the boundary by a combined effect of strong scattering and interface reflection.
0.84 0
a
0.1
0.2
0.3
x/D
0.4
αD = 1 N = 0.1
0.95 0.9
n=1
0.85
n=1
0.8 0.75
n=2
b
n=1
κ = 1, ω = 0 D n=2 κ = 10, ω = 0.9 D κD = 100, ω = 0.99
0.7 0.65 0.6 0
0.5
n=2
0.5
1
1.5
2
2.5
Dimensionless time, τ
Fig. 7. Effect of increasing scattering with fixed aD = 1 for n = 1 to 2 and N = 0.1: (a) temperature distribution at s = 0.45 and (b) transient mean temperature of the layer.
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
6. Conclusions Transient solutions were obtained for an emitting, absorbing, isotropically scattering, and heat-conducting layer of ceramic. The solution procedure includes simultaneously solving the transient energy equation using an implicit finite volume scheme and the integral equation for the radiative heat flux using the singularity subtraction technique and Gaussian numerical quadrature the effect of influence parameters, optical thicknesses, scattering albedos, and refractive index of the layer are considered carefully. Scattering is found to have a significant effect on the transient temperature distributions of a layer subject to radiative cooling.
References [1] P. Sadooghi, C. Aghanajafi, Coating effects on transient cooling a hot spherical body, Journal of Fusion Energy 22 (2004) 59–65. [2] T. Kunc, M. Lallemand, J.B. Saulnier, Some new developments on coupled radiative–conductive heat transfer in glasses—Experiments and modeling, International Journal of Heat and Mass Transfer 27 (1984) 2307–2319. [3] R.E. Field, R. Viskanta, Measurement and prediction of the dynamic temperature distributions in soda lime glass plates, Journal of the American Ceramic Society 73 (1990) 2047–2053. [4] R. Ducharme, P. Kapadia, F. Scarfe, J. Dowden, A mathematical model of glass flow and heat transfer in a platinum downspout, International Journal of Heat and Mass Transfer 36 (1993) 1789–1797. [5] J.R. Thomas, Coupled radiation–conduction heat transfer in ceramic liners for diesel engines, Numerical Heat Transfer 21 (1992) 109–120. [6] Z. Yin, Y. Jaluria, Zonal method to model radiative transport in an optical fiber drawing furnace, Journal of Heat Transfer 119 (1997) 597–603.
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[7] B. Song, R. Viskanta, Deicing of solids using radiant heating, Journal of the Thermophysics and Heat Transfer 4 (1993) 311–317. [8] J.T. Howe, L. Yang, Earth atmosphere entry thermal protection by radiation backscattering ablating materials, Journal of Thermophysics and Heat Transfer 7 (1993) 74–81. [9] C.J. Cornelison, J.T. Howe, Analytic solution of the transient behavior of radiation backscattering heat shields, Journal of Thermophysics and Heat Transfer 6 (1992) 612–617. [10] R. Siegel, Transient thermal effects of radiant energy in translucent materials, Journal of Heat Transfer 120 (1998) 4–20. [11] R. Viskanta, E.E. Anderson, Heat transfer in semitransparent solids. Advances in Heat Transfer, vol. 11, Academic Press, New York, 1975, pp. 317–441. [12] G.K. Chui, R. Gardon, Interaction of radiation and conduction in glass, Journal of the American Ceramic Society 52 (1969) 548–553. [13] M.H. Su, W.H. Sutton, Transient conductive and radiative heat transfer in a silica window, Journal of Thermophysics and Heat Transfer 9 (1995) 370–373. [14] R. Siegel, Refractive index effects on transient cooling of a semitransparent radiating layer, Journal of Thermophysics and Heat Transfer 9 (1995) 55–62. [15] P. Sadooghi, C. Aghanajafi, Radiation effects on a ceramic layer, Journal of Radiation Effects and Defects in Solids 159 (2004) 61–71. [16] O. Hahn, F. Raether, Transient coupled conductive– radiative heat transfer in absorbing, emitting and scattering media, International Journal of Heat and Mass Transfer 40 (1997) 689–697. [17] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, third ed., Hemisphere, Washington, DC, 1992 (Chapter 14). [18] C.M. Spuckler, R. Siegel, Refractive index and scattering effects on radiative behavior of a semitransparent layer, Journal of Thermophysics and Heat Transfer 7 (1993) 303– 310. [19] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1992. [20] W.H. Press, S.A. Teukolsky, T.W. Vetterling, B.P. Flannery, Numerical Recipes in Fortran, second ed., Cambridge University Press, New York, 1992.
Thermal analysis for transient radiative cooling of a conducting semitransparent layer of ceramic in high-temperature applications Parham Sadooghi *, Cyrus Aghanajafi K.N.T. University of Technology, Tehran, Iran Received 13 January 2005 Available online 22 July 2005
Abstract A method is developed for obtaining transient temperature distribution in a cooling semitransparent layer of ceramic. The layer is emitting, absorbing, isotropically scattering and heat conducting with a refractive index ranging from 1 to 2. The solution involves solving simultaneously the energy equation and the integral equation for the radiative flux gradient. The energy equation is solved using an implicit finite volume scheme and the integral equation of radiative heat transfer is solved using the singularity technique and Gaussian integration. The effects of scattering are investigated. It is shown that scattering has a significant effect on the transient temperature distribution and the transient mean temperature of the layer. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Fiber reinforced composite materials are now an important class of engineering materials. Because of their useful properties, they are widely used in various fields of engineering. They offer outstanding mechanical properties, unique flexibility in design capabilities, and ease of fabrication. Additional advantages include lightweight, *
Corresponding author. E-mail address: [email protected] (P. Sadooghi).
corrosion resistance, impact resistance, and excellent fatigue strength. Today, fiber composites are routinely used in such diverse applications as automobiles, aircraft, space vehicles, off-shore structures, containers and piping, sporting goods, electronics and appliances. The need for high-temperature reinforcing fibers has led to the development of ceramic fibers. An important factor that limits the performance (efficiency and emission) of current gas turbines is the temperature capability (strength and durability) of the metallic structural components in the engine hot section (blades, nozzles, vanes and combustor liners). It
1350-4495/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.infrared.2005.04.007
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
279
Nomenclature D thickness of the medium, m E1, E2, E3 the exponential integral functions, R1 En ðxÞ ¼ 0 ln2 expðx=lÞ dl k thermal conductivity of the medium, W/m K N conduction–radiation parameter, k=4rT 4i D n refractive index of the medium Q incidence radiation, W/m2 q1, q2 radiosity from the interior of the two boundaries, W/m2 q1 ; q2 dimensionless radiosities, q1 =rT 4i ; q2 =rT 4i T absolute temperature, K t dimensionless mean temperature, T/Ti tm dimensionless temperature, Tm/Ti Ti initial temperature, K Te temperature of the surroundings, K
is generally agreed that the temperature capabilities of metals have reached their ceiling. Ceramic fibers such as alumina fibers and silicon carbide fibers exhibit superior durability and combine high strength and elastic modulus with high temperature capability, which implies their potential to revolutionize gas-turbine technology. Accurate prediction of temperature distributions in ceramic and other fibrous and semitransparent materials (STM) at high temperature, are essential during various fabricating operations. Within a semitransparent medium, the temperature and heat flux distributions are affected by radiation in addition to heat conduction. The transient coupled radiative and conductive heat transfer in a semitransparent medium is one of the pervasive processes in engineering applications, such as heat dissipating materials [1], tempered glass and the application of its products [2–4], thermal property analysis for ceramic parts [5], manufacture and application of optical fiber and its products [6], melting and removal of ice layers [7], transient responses to volumetrically scattering heat shields [8,9], ignition and flame spread for translucent plastics and solid fuels, and so on. The radiation effects become more important when the STM is
Tm X x a b h jD q qCp r s / x
mean temperature, K dimensionless coordinate, x/D coordinate in direction across the layer, m absorption coefficient of the layer, m1 extinction coefficient of the layer, m1 time, s optical thickness of the layer, bD interface reflectivity product of density and specific heat of the layer, q/m3 K Stefan–Boltzmann constant dimensionless time, ð4rT 4i =qC p DÞh dimensionless incidence radiation, Q=rT 4i scattering albedo
at elevated temperatures, in high temperature surroundings, or subjected to large incident radiation, and the radiation fluxes depend strongly on the temperature level. In semitransparent materials where thermal radiation can affect internal temperature distributions, transient behavior has been studied much less than steady state. To obtain transient solutions numerical procedures such as finite difference and finite element methods, discrete ordinates method, have been used to solve the radiative transfer relations coupled with the transient energy equation. The transient thermal behavior of a single and multiple layers of semitransparent materials have been studied for variety of cases, as reviewed recently by Siegel [10]. Earlier studies on transient coupled radiative– conductive heat transfer in STM were carried out mainly for the boundary conditions of the first kind, with prescribed temperatures at the frontiers. They were summarized by Viskanta and Anderson [11]. Much research has been directed toward the combined effects of conduction and radiation in glass. Chu and Gardon [12] under took the study of combined transfer in gray glasses. Su and Sutton [13] predicted temperatures and heat fluxes with 16 spectral bands in the silicate glass plate
280
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externally heated by a constant heat input at one boundary for a time interval of 5 s. Siegel [14], and more recently, I and my colleague, [15], used finite difference procedure to predict transient temperature 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 reflections. Hahn and Raether [16] examined the transient heat transfer in a layer of ceramic powder during laser-flash measurements of thermal diffusivity. A three-flux method was used to solve the equation of radiative transfer. However, no results were presented on the effect of scattering and refractive indices. The solution of the exact radiative equation is rather complicated, particularly when the scattering is present; hence a common approach is to use approximate methods such as the two-flux method and diffusion and differential approximations. Unfortunately, approximate methods are usually subject to certain constraints. Under certain conditions, the methods may not be valid. This paper presents a direct numerical procedure for obtaining accurate transient temperature distribution in a semitransparent layer of ceramic fiber. The solution includes the integral equation for the radiative flux gradient, coupled to a transient energy equation that contains both radiative and conductive terms. Solutions are given to demonstrate the effect of isotropic scattering on the radiative cooling of the layer.
2. Analysis
The transient energy equation in dimensionless form is [17]: ot o2 t ¼N RðtÞ; os oX 2
where R(t) is the gradient of the radiative flux under the assumption of isotropic scattering and is a function of X and s ð2Þ RðtÞ ¼ jD ð1 xÞ n2 t4 ð/=4Þ ; where / is the local incident function. Considering the isotropic scattering, / is given by [17]: /ðX Þ ¼ 2qi E2 ðjD X Þ þ 2q2 E2 ½jD ð1 X Þ Z 1 þ 2pjD SðX 0 ÞE1 ðjD jX X 0 jÞ dX 0 ;
where the source function is written as follows: SðX Þ ¼ ð1 xÞðn2 t2n Þ þ ðx=4pÞ/.
ð4Þ
And q1 and q2 are dimensionless diffuse fluxes at the boundaries at X = 0 and X = 1, respectively. For the symmetrical case and under the assumption of diffuse reflection [18], by using of the Fresnel reflection relations q1 and q2 are given by q1 ¼ q2 ¼ 2pqjD
Z
1
1
½SðX 0 ÞE2 ðjD X 0 Þ dX ½2qE3 ðjD Þ .
0
ð5Þ Inserting Eqs. (4) and (5) into Eq. (3), leads to Z 1 /ðX Þ ¼ n2 F ðX ; X 0 Þt4 ðX 0 Þ dX 0
The 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 Ti. It is then placed in a much colder surrounding.
X
ð3Þ
0
0
Te << T (X)
ð1Þ
2
þ 2n jD ð1 xÞ
Z
1
t4 ðX 0 ÞE1 ðjD jX X 0 jÞ dX 0 0
Z 1 x F ðX ; X 0 Þ/ðX 0 Þ dX 4ð1 xÞ 0 Z 1 x þ jD /ðX 0 ÞE1 ðjD jX X 0 jÞ dX 0 ; 2 0
þ
ð6Þ
where T(X)
D
Fig. 1. Geometry and nomenclature of the plane layer.
F ðX ; X 0 Þ ¼ 4
qjD ð1 xÞ fE2 ðjD X Þ 1 2qE3 ðjD Þ þ E2 ½jD ð1 X ÞgE2 ðjD X 0 Þ.
ð7Þ
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
3. Boundary conditions Since convective heat transfer is negligible, boundary conditions are required for radiation and for heat conduction only. Radiation passes out of the layer from within the material; there is 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 the preceding energy equation: otð0; sÞ otð0; sÞ ¼ ¼ 0. ð8Þ oX oX The initial temperature of the layer is uniform, so the non-dimensionalized temperature begins from 1, tðX ; 0Þ ¼ 1.
ð9Þ
4. Numerical solution procedure The discretization of the energy equation is derived using the implicit finite volume scheme [19]: dRðt Þ ðt t Þ dt dRðt Þ dRðt Þ t þ t; ¼ Rðt Þ dt dt
RðtÞ ¼ Rðt Þ þ
ð10Þ
where t is temperature from previous iteration. From Eq. (2), the time derivative of R(t) in Eq. (10) is dRðt Þ 1 d/ 2 3 ¼ jD ð1 xÞ 4n t . ð11Þ dt 4 dt And Eq. (6) gives, Z 1 d/ ðX Þ 3 2 ¼ 4n F ðX ;X 0 Þt ðX 0 ÞdX 0 þ 8ð1 xÞn2 jD dt 0 Z 1 x 3 t ðX 0 ÞE1 ðjD jX X 0 jÞdX 0 þ 4ð1 xÞ 0 Z 1 d/ ðX Þ 0 xjD dX þ F ðX ; X 0 Þ dt 2 0 Z 1 d/ ðX Þ E1 ðjD jX X 0 jÞdX 0 . ð12Þ dt 0
281
By solving Eqs. (11) and (12), dRðt Þ=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 quadrature [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 temperature distribution. For a solution at each time increment, an initial guess of the temperature distribution across the layer must be provided. Typically in this calculation, the temperature distribution at a previous time level is used for the first guess. The values of temperature at the Gaussian abscissas are obtained using cubic spline interpolation [20], with temperature specified, 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 temperature distribution is obtained by the solution of Eq. (1) to have ahead one time increment. The new temperature 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 higher order Gaussian quadratures are needed because of steep variations of the radiative flux at the boundaries.
5. Results and discussion The effect of variations of the influence parameters, such as refractive indices, optical thicknesses and scattering albedos on the temperature distributions are shown in Figs. 2–7. 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. Results are given at three instances during the transient.
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55
a
= 0.11
= 1.5 =0 = 0.3 = 0.6 = 0.9
n = 1, kD = 5, N = 0.1
0
0.1
0.2
0.3
0.4
0.5
= 0.11 = 0.45
t =T/Ti
=1.5 =0
= 0.3 = 0.6 = 0.9
n = 1, kD = 5, N = 0.0
0
0.1
0.2
0.3
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 n = 1, k 15, N = 0.1 D 0.55 0 0.1 0.2 0.3
0.4
0.5
X= x/D
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45
= 0.66 = 2.2 =0
= 0.3 = 0.6 = 0.9 0.4 0.5
= 0.14 = 0.66 = 2.2 = 0
= 0.3 = 0.6 = 0.9
n = 1, kD = 15, N = 0.0
0
d
= 0.14
X= x/D
b
X= x/D
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55
c
t =T/Ti
= 0.45
t =T/Ti
t =T/Ti
282
0.1
0.2
0.3
0.4
0.5
X= x/D
Fig. 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 layer of n = 1 and Fig. 3 for n = 1.5 with x = 0.3, 0.6 and 0.9 for all figures. Different optical thicknesses and radiation–conduction parameters are considered. The results shows that heat conduction serves to equalize temperature across the layer and increased optical thickness makes the layer optically thick and gives a steeper temperature distribution near the boundaries. An important effect of a larger refractive index is that internal reflections promote the distribution of radiative energy within the layer, this makes the transient temperature distributions more uniform. Figs. 2 and 3 show the effect of scattering when the optical thickness is held fix, in creasing scattering gives a move uniform temperature distribution 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 radiative heat loss of the layer depends strongly upon the mag-
nitude of aD which is equal to (1 x)jD. In this case, the effect aD of is somewhat similar to that of jD on a non-scattering layer. In the limiting case when x = 1, the layer remains at the initial temperature because there is no emitting and absorption and, hence no heat loses from the layer. The effect of increasing scattering albedo is significant for lower optical thickness. Comparisons 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 effect of scattering. The reason is that a larger optical thickness and, thus, a larger absorption thickness, means more scattered energy being absorbed, which offsets the effect 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 kept constant as in parts (a) and (b) of Figs. 2 and 3, increasing scattering, decreasing the value of aD, slows down the decrease of temperatures, particularly at larger scattering albedos.
t =T/Ti
a
= 1.0 = 2.5 =0 = 0.3 = 0.6 = 0.9
N = 0.1
0.2
0.3
0.4
0.5
b
X= x/D 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 n =1.5, k 0.5 0 0.1
= 0.25
= 0.25
= 1.0
N = 0.1
n =1.5, k
0
0.1
= 2.5 =0 = 0.3 = 0.6 = 0.9
0.2
0.3
0.4
0.5
X= x/D
c
283
= 0.25 = 1.0
t =T/Ti
1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 n =1.5, k 0.5 0 0.1
t =T/Ti
t =T/Ti
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
= 2.5 =0 = 0.3 = 0.6 = 0.9
N = 0.1
0.2
0.3
0.4
0.5
X= x/D 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 n =1.5, k 0.5 0 0.1
d
= 0.25 = 1.0 = 2.5 =0 = 0.3 = 0.6 = 0.9
N = 0.1
0.2
0.3
0.4
0.5
X= x/D
Fig. 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, N = 0, jD = 5; (d) n = 1.5, N = 0, jD = 20.
1
k =5 D kD = 15
0.9 0.85 0.8
N = 0.1 n =1
0.75 0.7
0.9 0
0.6
0.3
0.65
=0
0.3 0.6
0.6
Mean temperature, Tm/Ti
Mean temperature,Tm/Ti
1 0.95
k =5 D k = 20
0.95
D
0.9 0.85 0.8
0.9
0.75
N = 0.1 n = 1.5
0.7 0.65
0.3
0.3
0.6 0.6 0
=0
0.6
0
0.5 1 1.5 Dimensionless time,
2
2.5
0
0.5
1 1.5 Dimensionless time,
2
2.5
Fig. 4. Transient mean temperature of the layer of n = 1 with N = 0.1, jD = 5 and jD = 15.
Fig. 5. Transient mean temperature of the layer of n = 1.5 with N = 0.1, jD = 5 and jD = 20.
The fact that a larger optical thickness weakens the effect of scattering can be shown from Fig. 4, the difference between the curve corresponding to x = 0 and the curve 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 the
curve 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 various optical thickness is different from that at small scattering albedos. The effect of large scattering is
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
Mean temperature,Tm/Ti
284 1 0.95 0.9 0.85
κD = 2
N = 0.1 n =1 ω = 0.9
0.8 0.75
5
50 20
1.5
2
10
0.7 0.65 0
0.5
1
2.5
Dimensionless time, τ Fig. 6. Transient mean temperature of the layer of various jD at x = 0.9 for n = 1, N = 0.1.
shown in Fig. 6. The scattering albedo is held constant at x = 0.9 but the optical thickness of the layer is varied from 2 to 50. At small values of optical thickness jD < 10, the transient mean temperature decreases with an increasing optical thickness; however at larger jD, this trend is reversed. This is related to the magnitude of aD, when aD is small, the temperature distribution is relatively uniform and the radiation rays pass out of the layer because of relatively small absorption. Increasing aD enhances the radiation, and hence the cooling rate. When jD is large, the temperature near the boundaries becomes increasingly cooler than the interior. This explains the special case appearing in Fig. 4 that was mentioned earlier. At such small aD, the cooling effectiveness increases with increasing aD.
κD = 1, ω = 0 κD = 10, ω = 0.9 κD = 100, ω = 0.99
0.98 0.96
t=T/Ti
0.94
1
n=1
Mean temperature, Tm/Ti
1
n=2 n=1
αD = 1 τ = 0.45
0.92 0.9
n=2 n=1 n=2
0.88 0.86
In Fig. 7 the effect of scattering is shown in a different way, aD is held constant, while jD is increased such that the effect of additional scattering can be shown. The temperature distribution at s = 0.45 is shown in Fig. 7(a). It is seen that increasing the scattering leads to steeper temperature distribution near the boundaries, an observation different from that shown in Figs. 2 and 3. Note that in the case of Fig. 7, jD is increased much faster than the increase of x, thus the layer quickly becomes increasingly opaque. In contrast, in Figs. 2 and 3 jD is held constant when effect of increasing scattering is examined. Fig. 7(b) reflects that the cooling rate of the layer decreases with increasing x and jD, with aD being fixed. This is caused by the temperature distribution depicted in Fig. 7(a). It is interesting to note the effect of refractive indices on the transient process at large scattering albedos. In Figs. 7, 2 and 3, when x = 0, a larger n gives more uniform temperatures and that at N = 0.1 and with jD ranging from 2 to 20, there is a decrease in the cooling rate when n is increased. However the effect of n is quite different at large x as is shown in Fig. 7. When x = 0.9 and 0.99, the cooling rate with n = 2 is considerably faster than that with n = 1, and at jD = 100, x = 0.99, the temperature distribution for n = 2 is steeper than that for n = 1. This is most likely a result of enhanced cooling at positions near the boundary by a combined effect of strong scattering and interface reflection.
0.84 0
a
0.1
0.2
0.3
x/D
0.4
αD = 1 N = 0.1
0.95 0.9
n=1
0.85
n=1
0.8 0.75
n=2
b
n=1
κ = 1, ω = 0 D n=2 κ = 10, ω = 0.9 D κD = 100, ω = 0.99
0.7 0.65 0.6 0
0.5
n=2
0.5
1
1.5
2
2.5
Dimensionless time, τ
Fig. 7. Effect of increasing scattering with fixed aD = 1 for n = 1 to 2 and N = 0.1: (a) temperature distribution at s = 0.45 and (b) transient mean temperature of the layer.
P. Sadooghi, C. Aghanajafi / Infrared Physics & Technology 47 (2006) 278–285
6. Conclusions Transient solutions were obtained for an emitting, absorbing, isotropically scattering, and heat-conducting layer of ceramic. The solution procedure includes simultaneously solving the transient energy equation using an implicit finite volume scheme and the integral equation for the radiative heat flux using the singularity subtraction technique and Gaussian numerical quadrature the effect of influence parameters, optical thicknesses, scattering albedos, and refractive index of the layer are considered carefully. Scattering is found to have a significant effect on the transient temperature distributions of a layer subject to radiative cooling.
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