On Continuum Models for Heat Transfer in Small Scale Porous Materials Professor Jinliang Yuan

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1 On Continuum Models for Heat Transfer in Small Scale Porous Materials Professor Jinliang Yuan August 30, 2013 Department of Energy Sciences Lund University, Sweden

2 Why porous structures applied? How small scale of porous structures? What are important parameters? Which inds of impacts? l= Kn p = T B l d 2pd 2 g

3 Unit Cell Performance based on the Coupled Interactions of Transport and Reactions

4 Important Interactions and Couplings Performance/Cost Reaction Transport Material/structure/ configuration/process

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6 Four representative SEM image sections of the Ni-YSZ anode separated by 150 nm, illustrating the change in the microstructure with depth along the milling direction. J. R. Wilson, et al. Nature Materials 5, (2006). SOFC was first cut and polished, leaving a cross-sectional surface with the anode, electrolyte, and cathode exposed. The FIB was then used to mill a rectangular trench into this surface in the vicinity of the anode. A series of SEM images was acquired from one of the trench side walls as the FIB 'shaved' away material from this surface.

7 A view of the 3D reconstruction showing the Ni (green), YSZ (translucent/grey), and pore (blue) phases. a. A ey application of 3D reconstruction is to mae quantitative correlations between processing and microstructure, such as phase volume fractions. (The volume percentages of Ni, YSZ and pores calculated from the 3D reconstruction were 25.9%, 54.6% and 19.5%, respectively, i.e., Ni to YSZ volume ratio of 32:68). b. To find the interface/surface areas of each phase in the anode. Pores Ni Catalyst YSZ c. To evaluate the connections between microstructure and electrochemical reactions, based on TPB length. d. Determing the tortuosity of the ionic, electronic, and gas-phase transport pathways within the electrode networ.

8 Porous Materials Applied in Energy Systems Transport Reactions Much more...

9 Multi-functional Porous Materials Heat Generation/ sin I q =- DH M - I h b 2 2 2F HO HO to ensure maximum active surfaces, and to allow the injection of the generated products to the ducts. The mass transfer is dominated by the gas diffusion or/and convection. This is ensured by the open pores of the electrodes, in terms of permeability, porosity/tortuosity. to have a good charge transfer conductivity.

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11 Porous Materials in Fuel Cells Channel: mm to cm Diffusion Layer (GDL): ~0.5 mm Catalyst Layer (CL): ~20 μm Membrane: ~ 200 μm Pores/Particles: nm - μm Multi-functional Components

12 Continuum flow. The intermolecular collisions dominate the flow and the conventional boundary conditions at the interface between fluid and solid body are valid. The velocity and temperature of the fluid are continuous over the interface (the velocity and temperature of the fluid are the same as ones for the solid body at the interface. Transition flow. the intermolecular and with wall collisions are of minor or bigger importance. Knudsen flow or free molecule flow. Where molecules hitting and leaving the solid surface can travel rather far before they collide with some other molecules. Slip flow. The Knudsen number is small but the molecular effects are not possible to be neglected. The velocity and temperature at the interface between fluid and solid are no longer continuous. The velocityslip and the temperature-jump are related to λ and so called accommodations- respective reflection-coefficients have been introduced.

13 Continuum Heat Transfer level 5b 1 5c 5a é v ù 2 r f ê + ( v Ñ ) v = -Ñ p+ mñ ë t ú v û 4a 3 6 æ Tf ö r c T T q f ç + Ñ =Ñ Ñ + & è t ø ( ) v ( ) f f f f Bed Wall 2 4b Gas Flow T r c s =Ñ Ñ T + q& s t ( ) ( ) s s s T = T, Ñ T = Ñ T f G s G f f G s s G

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15 Continuum level Tf e( r c) + ( r c) v Ñ T =Ñ (, Ñ T ) + h ( T - T ) + q& f t f f f eff f v s f f T (1-e)( r c) s =Ñ ( s, t ÑT )-h ( T - T ) + q& hsf Asf hv = DV s eff s v s f s T r c + r c Ñ T =Ñ Ñ Tù+ q eff t eff ë û & ( ) ( ) v é eff

16 General CFD Modeling y ( rf) f + ( ruif) = ( G ) t x x x i i i + S i, j+1 i-1, j i, j i, j-1 i+1, j x

17 CFD Modeling Approches for Porous Materials Currently CFD is often adopted for fuel cell modeling, and a number of empirical parameters are employed with their values being selected not on the solid foundation: Effective properties for the functional materials,, D, τ, ε, K, etc mesh size issue? Surface tension; Reaction inetics; Fabrication process. D eff = e ( + ) =e ( + ) t D D D D im 2 bul bul im ik, im ik,

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19 Universal expression for the apparent conductivity app app = 1 (9g-5) T + T + a ( g+ 1) 2 T w1 w2 1 Kn m app = 1 + 3Kn g Kn < <Kn < 0.1 Kn > 10 app,fm app,tj = app,c = 1 1+ BKn a 1 (9g-5) T + T a ( g+ 1) 2 T m = m w1 w2 g 1+3Kn Kn

20 Nusselt-number vs. Reynolds number for a sphere Ma Nu Nu = (2-as) [ Ma/(RePr) ] Nu a s Ana. solutions at subsonic region (Ma<1) 0 0 Exp. solutions Nu tf becomes smaller!!

21 Typical Knudsen numbers and heat transfer regimes in SOFC and PEMFC electrodes Operating conditions λ (nm) d p (nm) Kn Heat transfer regime Remar 850 o C/1atm 372 (O 2 ) Transition SOFC cathode active layer 1000 o C /1atm 900 (H 2 ) Transition SOFC anode substrate 80 o C/1atm 82 (air) Transition PEMFC CL 80 o C/1atm 85(H 2 ),41(O 2 ), 51(H 2 O) Transition PEMFC MPL 80 o C/1atm 85(H 2 ),41(O 2 ), 51(H 2 O) Slip flow PEMFC GDL Kn = l d p l= p T B 2pd 2 g

22 Structure schematic of: a) parallel model, b) series model, c) Maxwell-Eucen 1 model (blac = continuous phase, white = dispersed phase), d) Maxwell-Eucen 2 model (blac = dispersed phase, white = continuous phase) and e) random model.

23 1 {(3 1) [3(1 ) 1] [(3 1) (3(1 ) 1) ] 2 eff = e- f + -e - s + e- f + -e - s + 8 f s } 4 External porosity region = (1-e) +e eff s f eff / s ε eff = s ( - ) e s f s f ( - ) e s f s f Internal porosity region eff = f ( - )(1-e) f s f s ( f - )(1-e) f s s eff = 1 (1-e)/ +e s f

24 Effective thermal conductivity in fuel cell electrodes. Case Volume Fraction (%) f, (W/m K) s, (W/m K) s,eff, (W/m K) Para. (W/m K) Series (W/m K) ME1 (W/m K) ME2 (W/m K) EMT (W/ m K) Exp. Data (W/m K) Remars SOFC anode 41(pore) /33(YSZ) /26(Ni) (YSZ) /50(Ni) /4.54 /3.27 x/y/z direction [58] PEMFC GDL 71(pore) /20(C) /9(PTFE) (C)/ 11.7 (PTFE) [59, 60]

25 eff Evaluation Method for the Effective Thermal Conductivity = 1 f / + (1- f)/ series parallel f (between 0 and 1) is the weighted geometric or arithmetic means

26 Sigmoidal average of the high and low thermal conductivities s eff s s, =j + (1-j ) If the ratio of the solid and fluid thermal conductivities is about one order of magnitude or higher, such as the case involving the solid particles and the fluid-filled pores in the typical porous materials, the lower bounds will eff = approach a very small value. 3 1-e ( ) 2, 1+ e 2 s eff eff = Multi-component solid phases with different thermal conductivity - e exp( 2 ) 1-e s, eff Non-linear (exponential) relation Case Porosity (%) s,eff eff eff Exp. Data SOFC anode /4.54 /3.27 PEMFC GDL

27 ' s, eff = s, eff t Case Por. (%) Kn f,kn (W/ mk) SOFC anode PEMFC GDL Tortuosity τ /1.96 /2.10(for 8YSZ) s,eff (corrected by tortuosity) 12.91/11.79 /11.0 eff (Linear) 3.73/3.4/ 3.19 eff (Nonlinear) 4.55/4.19 /3.91 eff (EMT) 5.13/4.70/ < 0.1 eff 2 3 = (1- e) s, eff ε>2/3=67%, eff =0 s>>f 1 { [ 2 eff = a+ a + 8 fs } 4 a = (3e- 1) + [3(1-e) -1] a = (3.4e- 1) + [3.4(1-e) -1] f f eff (EMT)=0.48 s s

28 Modeling of Heat Transfer in Nanoporous Silica, 8th Int. Vacuum Insulation Sym., 2007 = (1- e) + e eff s f R themal L L = = (1- e) + e eff s f T1 R 1 T2 T1 T2 R 2 R 2 R 1 eff = 1 (1- e) + s e f ε= ε

29 Final Remars Micro and even nano-scaled pores/particles are often involved in the FC multi-functional porous regions, and the effects of the fluid/solid interactions on the transport processes may be significant; Modeling approaches for various transport phenomena in the porous regions should be selected based on the particle/pore sizes, and operating conditions (Kn number) In general, the continuum models (such as CFD) can be further applied in some cases, subject to the careful evaluation of the effective parameters and reactions. Various models are available for effective thermal conductivity of the porous structures, but not well applied!