Journal of Thermal Science and Technology

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1 Science and Technology Design Considerations of Porous Gas Enthaly Radiation Converters for Exhaust-Heat Recovery Systems Preecha KHANTIKOMOL and Kouichi KAMIUTO High-Temerature Heat Transfer Laboratory, Deartment of Mechanical and Energy Systems Engineering, Oita University, Oita , Jaan, Phone: ; fax: Abstract A redictive model for describing the gas enthaly-radiation conversion rocess in an oen-cellular orous late is roosed. The ability of the model in redicting temerature distributions within orous converters is examined in comarison with available exerimental data. Extensive numerical simulations for commercially available orous lates made of Ni-Cr and artially-stabilized ZrO 2 are erformed by varying several system arameters such as orous late thickness, inlet gas temerature and mass flow rate in order to evaluate decreases in gas temerature across the orous lates and the conversion efficiency defined by a ratio of recatured radiation to incoming energy and also to give design tis on this kind of orous converter. Key words: Heat Recovery, Energy Saving, Gas Enthaly, Radiation, Porous Converter 1. Introduction Received 8 Jan., 2008 (No [DOI: /jtst.3.319] To recover as much as ossible of the enthaly of exhaust gases from various high-temerature facilities, Echigo [1] resented an innovative idea of lacing high-orosity media normal to the flow direction in an exhaust duct. Due to the resence of the solid hase of orous media, art of the gas enthaly and the radiant energy, mainly, incoming from the region ustream is absorbed and then converted to raise the temerature of the solid hase, which, in turn, emits and anisotroically scatters thermal radiation in both ustream and downstream directions. Through these rocesses, the gas temerature downstream of the orous media becomes much lower than that ustream. In fact, Echigo exerimentally showed that the temerature decrease of a high-temerature combustion gas across a 15 mm thick Ni-Cr-Al oen-cellular orous late reaches at 100~200 K and thereby established a concet for thermal insulation in flow systems [2], which has been successfully alied for imroving energy efficiency of reformers, metallurgical furnaces and ceramic radiant tubes and for lean combustion enhancement [3, ]. However, design tis on effective use of this kind of orous converter have not yet been fully clarified due to lack of a redictive model for orous converter. Under this circumstance, the urose of the resent study is twofold: first to establish a redictive model for gas enthaly-radiation conversion rocesses in oen-cellular orous lates and then to erform extensive numerical simulations for tyical metallic and ceramic orous materials in order to obtain design tis on orous gas enthaly-radiation converters. 319

2 Nomenclature a, b : constants c : secific heat caacity of fluid, J ( kg K c m,1 : mean secific heat caacity of fluid over a temerature range from ( to T f ( 0, J ( kg K c m,2 : mean secific heat caacity of fluid over a temerature range from to T f ( 0, J ( kg K c m,2 : dimensionless mean secific heat caacity of fluid ( = cm,2 c D n : nominal mean cell diameter (= 0.025/PPI, m D : equivalent strut diameter, m f : constant ( = 1 3 G 2 : incident radiation, W m g : scaled asymmetry factor of a scattering hase function 3 h v : volumetric heat transfer coefficient, W(m K k f : thermal conductivity of fluid, W ( m k s : thermal conductivity of solid, W ( m K 3 N R : conduction-radiation arameter ( = kf σ x0 2 Nu : Nusselt number ( = hd v kf PPI : ores er inch Pr : Prandtl number ( = µ f c kf 2 q Rx : net radiative heat flux in the flow direction, W m q R : radiative heat flux vector : Reynolds number ( = ρ fud µ f T bu : equivalent blackbody temerature of radiant energy coming from the region ustream, K T bd : equivalent blackbody temerature of radiant energy coming from the region downstream ( =, K T f : fluid temerature, K : ambient temerature, K T s : solid temerature, K T 0 : inlet gas temerature, K x : coordinate in the flow direction : thickness of a orous late, m,c : critical thickness of a orous late, m u : fluid velocity, ms w : dimensionless width of a strut consisting of a cubic unit cell Greeks β : scaled extinction coefficient, m 1 γ : T 1000, K θ f : decrease in the dimensionless fluid temerature ( = ( (0 ( x0 T0 θ f : decrease in dimensionless fluid temerature across an infinite orous late δ : dimensionless thickness of a orous late ( = x0 D η c : conversion efficiency θ f : dimensionless fluid temerature ( = θ s : dimensionless solid temerature ( = Ts θ 0 : fluid temerature ratio ( = T0 θ r0 : radiation temerature ratio ( = Tbu Tbd λ f : dimensionless thermal conductivity of fluid ( = kf kf λ s : dimensionless thermal conductivity of solid ( = ks kf µ f : viscosity, Pa s ξ : dimensionless coordinate in the flow direction ( = xx0 320

3 π : a ratio of the circumference of a circle to its diameter ρ f : 3 density of fluid, kg m ρ : hemisherical reflectivity of the surface of struts and strut junctures H σ : Stefan-Boltzmann s constant, W ( m 2 K 0 0 τ : otical thickness ( = β x φ : orosity χ : dimensionless incident radiation ( = G σt f ψ : dimensionless net radiative heat flux in the flow direction ω : scaled albedo Subscrit : quantity at the ambient + : forward comonent - : backward comonent f ( = q σt Rx 2. Theoretical Analysis 2.1 Governing equations The hysical model and coordinate system for analyzing combined forced-convection and radiation in an oen-cellular orous late are shown in Fig. 1. A high-temerature exhaust gas, which is air in the resent study, is normally assing through an oen-cellular orous late with the thickness, orosityφ and nominal cell diameter D n. The mass flow rate ρ f u is constant everywhere. Conduction through the fluid and solid hases is taken into account. The fluid is transarent to all radiation, while the orous late can emit, absorb and anisotroically scatter thermal radiation. The external radiations are coming from both the regions ustream and downstream and their intensities can be quantitatively reresented by equivalent blackbody temeratures. The radiative roerties of the orous late are considered to be gray. In addition, the thermal and radiative roerties of the gas and orous lates deend on temerature. Under these assumtions, the law of conservation of energy for each hase yields the following exressions: d d d ρfcu = hv( Ts + φ kf, dx dx dx (1 d dts f ( 1 φ ks + hν ( Ts divq R = 0, dx dx (2 ( div q R = β 1 ω σt s G, (3 where h v is the volumetric heat transfer coefficient between the solid and fluid hases, is the scaled extinction coefficient and ω is the scaled albedo. β I b ( T bu u T 0 Porous late I b ( T bd 0 x Fig.1 The hysical model and coordinate system 321

4 The boundary conditions for Eqs. (1 and (2 are given by dt = s 0 : f = 0, 0, dx = x T T d dts x = x0 : = = 0. dx dx ( The quantity G aearing in Eq.(3 reresents the incident radiation within a orous late. In addition to G, the net radiative heat flux in the flow direction qrx is also needed for evaluating the conversion efficiency defined in Section 2.. To evaluate the radiative transorts within oen-cellular orous materials, the P 1 method has been successfully emloyed in revious studies [5, 6], and thus we adot this method for evaluating G and q Rx. The P 1 equations are written as ( ω β ( G σts ( ω g β qrx dqrx + 1 = 0, (5 dx dg + 31 = 0. (6 dx The boundary conditions for these equations are given by Marshak s ones: = 0 : + 2 Rx = σ bu, = 0 : 2 Rx = σ bd, x G q T x x G q T (7 where Tbu and Tbd denote equivalent blackbody temeratures of thermal radiation coming from the regions ustream and downstream, resectively. From Eq. (6, qrx is related to G as follows: q Rx = 31 1 ( ω g dg. β dx (8 Although Eqs. (1, (2 and (3 cannot be solved analytically, integration along the flow direction is ossible and is instructive to understand the hysical mechanism of the resent energy conversion rocess: ( ( 0 ( ( 0 ( + + q ( q ( q ( x q ( x ρ fcm,1u x0 = qrx + qrx x0 = (9 Rx Rx R R In obtaining Eq. (9, we assumed that d dx = 0 at x = 0 in addition to Eqs. (. If a orous converter is otically thick and the incoming radiation from the region + downstream is small, then both qrx ( x0 and qrx ( x0 can be disregarded and Eq. (9 may be reduced to + ( ( ( ( ( ρ c u T 0 T x q 0 q 0. (10 f m,2 f f 0 Rx Rx This equation means that if a decrease in the enthaly of a flowing gas occurs, it is recovered as radiation, mainly, leaving for the region ustream from the orous late. 322

5 2.2 Physical roerties The volumetric heat transfer coefficient between the fluid and solid hases estimated using the correlation roosed by Kamiuto and San San Yee [7]: h v was Nu 2 hd v = = 0.12 ( Re Pr, (11 k f where D is an equivalent strut diameter defined by D = 2 wd π. (12 n Here, w is the dimensionless width of struts consisting of a cubic unit cell assumed in Dul nev s model [8] for an oen-cellular orous material and is given by 1 1 π w = cos cos ( 2φ (13 Moreover, on the basis of Dul nev s cubic unit cell model, analytical exressions for the radiative roerties of an oen-cellular orous material have been develoed by Kamiuto [9] and are resented as follows: π 23 2 β = ( 6 π w w( 1 w π Dn ( 1 w, + w = ρh, g = 9, (1 where ρ denotes the hemisherical reflectivity of the surface of struts and strut junctures. H 2.3 Numerical analysis For the convenience of numerical analysis, the governing equations and associated boundary conditions are rewritten in dimensionless form by introducing the following quantities: 3 c = c c, NR = kf σ x0, 2 Nu = hvd k f, Pr = µ f c k f, Re = ρ fud µ f, δ = x0 D, θf =, θ f 0 = T0, θr0 = Tbu Tbd, θs = Ts, λf = kf kf, λs = ks kf, ξ = x x0, τ0 = β x0, χ = G σ, ψ = qrx σ. (15 The governing equations are rewritten in dimensionless form and are given by dθ f 2 d dθf δreprc = λf Nu ( θ f θs δ + φ λf, d dθs 2 τ0 1 f ( 1 φ λs + λfδ Nu( θf θs ( 1 ω θs χ = 0. NR (16 323

6 The dimensionless boundary conditions are dθ s = 0: f = 0, = 0, ξ θ θ dθ f dθs ξ = 1: = = 0. (17 The P 1 equations are also rewritten in dimensionless form: ( ω τ0 ( χ θs dψ + 1 = 0, ( ω g dχ + 31 τ 0 ψ = 0. (18 The associated boundary conditions are ξ = 0 : χ + 2ψ = θ r0, ξ = 1 : χ 2ψ =. (19 These equations were solved numerically using an imlicit finite difference method. The orous region was divided into 200 equally saced increments for comutations of θ f and θ s, whereas the otical thickness was divided into 00 equally saced increments and the P 1 equations for χ and ψ were discretized at staggered lattice oints [5]. 2. Conversion efficiency To quantitatively evaluate the efficiency of the gas enthaly-radiation conversion rocesses, the following quantity defined by a ratio of recatured radiation to incoming energy is introduced in accordance with Wang and Tien [2]: ( ( ( ( ηc = qr ρf ucm,2 0 + σtbu + σt bd, = ψ ( 0 δreprnrcm,2 θf ( θr (20 Here, c m,2 is the mean secific heat caacity of gas over a temerature range from to T f ( 0 and qrx ( 0 is the backward radiative heat flux at the inlet of a orous late. The equivalent blackbody temerature of radiant energy coming from the region ustream may change variously, and thus can be greater or smaller than an inlet gas temerature T 0, deending on thermal environment in the region ustream. 3. Results and Discussion 3.1 Comarison with available exerimental data Firstly, to address the validity of the roosed model, theoretical analyses were made under conditions corresonding to Echigo s exeriments, where several kinds of Ni-Cr-Al foam metal were tested, but only the exerimental results for 5 mm and 15 mm thick orous lates are discussed here. In the analyses, we assumed that Tbd = = 300 K and T bu = 850 K and that the hemisherical reflectivity and thermal conductivity of Ni-Cr-Al are given by those of Ni-Cr. Results are shown in Figs. 2, where the exerimental data are indicated by symbols. As seen from these figures, the exit gas temeratures of the orous lates are well redicted, irresective of the inlet gas temerature and the hysical 32

7 characteristics of the examined orous lates, and thereby the validity of the roosed theoretical model is confirmed Ni-Cr-Al # 7 φ = 0.98 T 0 = K = Ni-Cr-Al#3 φ = 0.93 T 0 = K = = (m Ni-Cr-Al #1 φ = 0.92 T 0 = K = T 0 = (K T 0 = (K Ni-Cr-Al #1 φ = 0.92 PPI = 8.5 = (m = T 0 = (K θ f, θ s θ f, θ s 2.5 : θ f Theoretical 2.0 : θ (T s bu = 850 K : θ f (0 Exerimental : θ f (1 (Echigo ξ (a Effects of the kind of orous late : θ f : θ s : θ f (0 : θ f (1 Theoretical (T bu = 850 K Exerimental (Echigo ξ (b Effects of the inlet gas temerature Figs.2 (a(b Predicted temerature rofiles within Ni-Cr-Al orous lates (Comarison with Echigo s exeriments ( Numerical simulations Utilizing the resent model, we erformed extensive numerical simulations as for two kinds of commercially available oen-cellular orous material: Ni-Cr and artially-stabilized ZrO 2. Physical characteristics of these orous materials are summarized in Table 1. The thermal conductivities and hemisherical reflectivities of Ni-Cr [12] and artially-stabilized ZrO 2 [12, 13] are reresented as follows: ( W (m K = γ for Ni-Cr, = ( γ ( γ ( γ (21 1+ γ ( γ ( γ for s ZrO. ρh = γ ( γ γ γ γ for Ni-Cr, (22 = γ γ γ.63 + γ γ for s ZrO. ks ( 2 ( ( ( ( ( ( ( 2 Moreover, the isobaric secific heat caacity and the thermal conductivity of air are reresented by ( ( c kj kg K = γ( γ( ( ( γ( γ( γ. k W m K = γ( γ( f γ( γ. (23 (2 Here, γ is defined by T 1000 (K. Throughout the simulations, we assumed that Tbd = = 300 (K. The relevant system arameters were varied in the following ranges: 500 (K T (K, 300 (K T bu 1250 (K, and 0.005(m 0.015( m. Comuted results are summarized in Figs

8 Table 1. Physical characteristics of the examined oen-cellular orous materials [10, 11]. Material Porosity, φ PPI D (m β (m -1 Ni-Cr # Ni-Cr # Ni-Cr # # # # Figs. 3 (a(b show tyical variations in the dimensionless gas temerature decrease θ f = ( ( 0 ( x0 T 0 against the otical thickness of a orous converter. As seen from these figures, when the ustream radiation temerature is greater than the inlet gas temerature ( Tbu > T0, θ f becomes negative in a certain region of τ 0 : the orous late does not act as a converter of gas-enthaly to radiation and, inversely, acts as a converter of radiation to gas-enthaly. Consequently, if thermal engineers want to use a orous late effectively as a orous gas enthaly-radiation converter, then the orous late should be used under the condition of Tbu T T 0 = 700 K = 1 T bu = 500 K Ni-Cr #1 Ni-Cr #3 T bu = 700 K T 0 = 700 K = 1 T bu = 500 K #10 #20 T bu = 700 K θ f 0.2 θ f T bu = 900 K 0.0 T bu = 900 K -0.2 T bu = 1100 K -0.2 T bu = 1100 K τ 0 (a Case of the Ni-Cr orous lates τ 0 (b Case of the orous lates Figs.3 (a(b Effects of the ustream radiation temerature Tbu on. θ f In the case of Tbu T0, θ f increases with τ (or and is asymtotic to a certain limiting value and is well aroximated by the following exression: θ f b ( 0 θf = θ f 1 ex ax. (25 The least-squares fits of Eq.(25 to numerical results of θ f are shown in Figs. and determined values of θ f, a and b are summarized in Table 2. More comrehensive results as for other cases examined in the resent study will be reorted in Ref.1. Good agreement is achieved. From a view oint of design, it is convenient to indicate an aroximate thickness of the orous converter used under the condition of Tbu = T0. Here, it should be noted that Tbu = T0 is a quite natural assumtion as long as the region ustream is surrounded by well insulated walls and no strong radiation emitters exist there. For this urose, we evaluate a critical thickness of the orous converter,c yielding 0.95 θ f by varying. As seen from Figs.5,,c deends on both and Tbu ( = T

9 Table 2. Values of, a and b ( Re = 1. θ f Material T 0 (K T bu (K θ f a b Ni-Cr# # Numerical results Least-squares fit Numerical results Least-squares fit θ f Ni-Cr # 3 T 0 = 1100 (K = 1 T bu = 500 K T bu = 700 K T bu = 900 K T bu = 1100 K θ f # 20 T 0 = 1100 (K = 1 T bu = 500 K T bu = 700 K T bu = 900 K T bu = 1100 K x (m 0 (a Case of the Ni-Cr #3 orous lates x (m 0 (b Case of the #20 orous lates Figs. (a(b Comarison of numerical results of θ f with least-squares fits.,c (m T bu = T 0 Ni-Cr # 1 Ni-Cr # 3 Ni-Cr # T bu = T 0 #10 #20 #5 T bu = K 900 K 700 K 500 K T bu = 1100 K 900 K 700 K 500 K,c (m (a Case of the Ni-Cr orous lates (b Case of the orous lates θ Figs.5 (a(b Critical thicknesses yielding 0.95 f. Figs. 6 (a(b indicate contour mas for the asymtotic gas temerature decrease θ across a orous late. Here, the vertical axis denotes the inlet gas temerature, whereas the horizontal axis reresents the Reynolds number. For a constant inlet gas temerature, the temerature decrease diminishes as the Reynolds number increases. We recommend from a view oint of engineering design that be taken less than 1, rovided that is controllable as a system arameter. f 327

10 Figs. 7 (a(b indicate the conversion efficiencies η c of Ni-Cr and orous lates evaluated by Eq. (20. The results show that the conversion efficiency decreases as the Reynold number increases because convective heat transfer increases with, but variations in η against are generally dull in the range of less then 1. c T Ni-Cr #1 #10 bu = T0 Tbu = T0 T Ni-Cr #3 bd = = 300 ( K Tbd = = 300 ( K #20 Ni-Cr # θ 1500 s f = θ f = ZrO 2 # T 0 (K T 0 (K (a Case of the Ni-Cr orous lates (b Case of the orous lates Figs. 6 (a(b Contour mas for the asymtotic gas temerature decrease Ni-Cr and orous lates. θ f across Ni-Cr # 7 Ni-Cr # # 10 η c Ni-Cr # 3 T bu = T 0 = 900 (K 0.01 T f = 300 (K = (m = (m = (m η c # 20 T bu = T 0 = 900 (K # T f = 300 (K = (m = (m = (m (a Case of the Ni-Cr orous lates (b Case of the orous lates Figs.7 (a(b The conversion efficiencies of Ni-Cr and orous lates. 5. Conclusions The major conclusions that can be drawn from the resent study are summarized as follows: 1 The roosed theoretical model is accurate in redicting the temerature rofiles within a orous gas enthaly-radiation converter. 2 The orous late acts as a gas enthaly-radiation converter only when Tbu T0. 3 For a constant inlet gas temerature satisfying Tbu T0, the temerature decrease of a gas across a orous gas enthaly-radiation converter increases with a thickness of the converter and decreases with an increase in a mass flow rate. 328

11 The critical thicknesses of the Ni-Cr and orous gas enthaly-radiation converters can be estimated from Figs The conversion efficiency decreases as the Reynold number increases. 6 A mean gas velocity through an exhaust duct should be controlled so as to realize Re < 1. References 1. Echigo, R., Effective Energy Conversion Method between Gas Enthaly and Thermal Radiation and Alication to Industrial Furnaces. Proc.7th Int. Heat Transfer Conf., Vol.6 (1982, Wang, K.Y. and Tien, C.L., Thermal Insulation in Flow Systems : Combined Radiation and Convection through a Porous Segment, J. Heat Transfer, Vol. 106 (199, Echigo, R., High Temerature Heat Transfer Augmentations, in High Temerature Heat Exchangers, Y. Mori, A. E. Sheindlin and N. H. Afgan (ed., Hemishere, (1986, Viskanta, R., Radiative Transfer in Combustion Systems : Fundamentals and Alications, Begel House, (2005, Cha Kamiuto, K., Saitoh, S., and Itoh, K., Numerical Model for Combined Conductive and Radiative Heat Transfer in Annular Packed Beds, Numer. Heat Transfer, Part A, Vol.23 (1993, Kamiuto, K., Combined Conductive and Radiative Heat Transfer through Oen-Cellular Porous Plates, JSME Int. J., Series B, Vol.3 (2000, No.2, Kamiuto, K. and San San Yee., Heat Transfer Correlations for Oen-Cellular Porous Materials, Int. Communi. Heat Mass Transf., vol.32 (2005, No.7, Dul nev, G.N., Heat Transfer through Solid Disersion Systems, Eng. Phys. J.,Vol. 9 (1965, Kamiuto, K., Study of Dul nev s Model for the Thermal and Radiative Proerties of Oen-Cellular Porous Materials, JSME Int. J., Series B., Vol. 0 (1997, Sumitomodenko Co.Ltd., Catalogue for CELLMET, JT-6R5, ( Hendricks, T.J., Ph.D. Dissertation, University of Texas at Austin, ( Toloukian, Y.S., (ed., Thermohysical Proerties of High Temerature Solid Materials. MacMillan, New York, ( Krijijanousky, R.J. and Schuteln, Z. Yu., Thermohysical Proerties of Oxides, Jaan-Soviet Press, Osaka, (1975, Khantikomol, P. and Kamiuto, K., Thermal Characteristics of Porous Gas Enthaly-Radiation Converters for Exhaust-Heat Recovery Systems, Reorts of the Faculty of Engineering, Oita University, No.56, to aear. 329