MICROSCALE SIMULATIONS OF CONDUCTIVE / RADIATIVE HEAT TRANSFERS IN POROUS MEDIA

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1 MICROCALE IMULAION OF CONDUCIVE / RADIAIVE HEA RANFER IN POROU MEDIA J.-F. hovert, V.V. Mourzenko, C. Roudani Institut PPRIME-CNR

only between neighboring grains we expet merely an additional equivalent

2 Context, motivation moldering in porous media 400K (measured) Mirosale simulations ignifiant radiative transfers no long-range tranfers, exhanges only between neighboring grains we expet merely an additional equivalent ondutivity + rue? How muh? ( 3),, r Heat ondution onvetion/diffusion no radiation

pratial range oure power [W] Bakground temperature [K] Grain size (R, length unit)

3 Referene situation Reation front Heat soure in grains urrently burning Continuous heat soure in a single grain, within the paking Parameters: all set aording to pratial range oure power [W] Bakground temperature [K] Grain size (R, length unit) olid thermal ondutivity s [W/mK] Bed porosity ε, etive ondutivity, s Oven at bakground temperature

4 Numerial model Condution: onduting solid, Laplae equation no onvetion nor ondution in the gas time-expliit, finite-volume formulation disretization by a 3 ubi volume elements (a R/5) solid onduting opaque blak body Radiative transfers: gas vauum transparent gas phase opaque solid, blak body the solid surfaes absorb all the inoming radiative flux the solid surfaes emit a flux with - an isotropi Lambert "osθ" orientation distribution - a rate given by tefan-boltzmann law E σ 4 [W/m 2 ] Monte-Carlo simulation Initial and boundary onditions: he bed is initially at bakground temperature External boundaries at onstant bakground temperature Constant ontinuous heat supply in the soure

5 Numerial simulations imulation management: quasi-ontinous time sheme radiated energy quantum: q r [J] set dynamially aording to a ost/nr ompromise (orresponding to at most δ q r /ρ p a ( ) in a volume element) time step δt [s] set dynamially so eah surfae element emits (i.e. at most one quantum during δt) a 2 E δt q r during a time step: eah surfae element emits 0 or quantum (with a probability a 2 E δt / q r ) eah quantum propagates till it hits a solid surfae where it is absorbed the solid temperature is updated (-/+) in real time periodially, ondution is aounted for by an expliit finite-volume step We monitor: the soure temperature the ondutive and radiative and total outgoing fluxes... until a steady regime is reahed

refine q r to improve NR in steady state Outgoing fluxes δ

6 Numerial simulations Example: Mean soure temperature (K) In pratie: - start with a oarse q r (fast), and then - refine q r to improve NR in steady state Outgoing fluxes δ (W) /8 K otal t (s) Condution δ ½ K δ ¼ K Radiation 6/32 t (s)

7 Phenomenology: (r) - in 3 ases with the same parameters: 2.28W, 700K 80 K Referene ase: ondution in plain solid 400 K oure in a grain paking (ondution only) 580 K oure in a grain paking (ondution + radiation)

8 Phenomenology: Radial temperature distribution (r) - Referene ase: ondution in plain solid r/r 4π s r r ( r) +, 0.979W/mK s Analytial solution in a spherial domain. Here, r is an apparent distane for the appliation of the boundary ondition, the solution applies only up to some distane to the oven walls (r 8R)

Phenomenology: Radial temperature distribution (r) - 80 K Referene ase: ondution in plain solid r/r 4π s r r ( r) +, 0.979W/mK s 400 K oure in a grain paking (ondution only) 4π, r r ( r) +, 0.

9 Phenomenology: Radial temperature distribution (r) - 80 K Referene ase: ondution in plain solid r/r 4π s r r ( r) +, 0.979W/mK s 400 K oure in a grain paking (ondution only) 4π, r r ( r) +, 0.73W/mK, Mean (r) Fit Poor thermal Behavior ontat of an equivalent between the soure ontinuous material and the paking he sharp fit applies dropfar enough from the soure and the oven walls (4R r 8R) R/r 8 R/r4

10 Phenomenology: Radial temperature distribution (r) - 80 K Referene ase: ondution in plain solid r/r 4π s r r ( r) +, 0.979W/mK s 400 K oure in a grain paking (ondution only) 4π, r r ( r) +, 0.73W/mK, 580 K oure in a grain paking (ond.+radiation) By analogy, Improved enhaned (radiative) apparent transfer ondutivity (i.e. Rosseland from the approximation) soure to the paking redued temperature drop ( r) +, 0.295W/mK 4π r r

11 emi-loal analysis <> mean temperature in onentri spherial shells of thikness R/5? ( ( ) ) 2 d 2 4 π r 4π r, +, dr r d dr R R mm, 20.5 W mm, 9.86 W <> Condution Radiative ontribution R 3 mm, 20.5 W R mm, 2.28 W R /3 mm, 0.25 W R mm, 0.53 W R mm, 0.25 W 00 K 700 K R no radiation 485 K r/r

12 emi-loal analysis loal etive ondutivity in onentri spherial shells of thikness R/5 4 π r 2 d dr R R mm, 20.5 W mm, 9.86 W [W/mK] Condution (plain solid) R R 3 mm, 20.5 W mm, 2.28 W R /3 mm, 0.25 W R mm, 0.53 W R mm, 0.25 W 480, 700 or 00 K Condution (paking) r/r

13 emi-loal analysis radiative ontribution to the etive ondutivity ( ( ) ) 2 d 2 4 π r 4π r, +, dr r d dr R mm, 20.5 W ( ) 3 R, R mm, 9.86 W, r onstant 3 R R 3 mm, 20.5 W mm, 2.28 W R /3 mm, 0.25 W R mm, 0.53 W R mm, 0.25 W, r ( ) 2 3 σ 8 480, 700 or 00 K r/r

14 emi-loal analysis radiative ontribution to the etive ondutivity, r,, r ( ) 2 3 σ 8 Why this form? 8 ( ) 2 3 σ 8 ( ) 2 3 σ volumetri area alulated independently 8 ( ) 2 3 σ

15 Generalization: reonstruted media hresholded Gaussian fields: l 2 l 3 l 4 porosity orrelation length ε 0.25 ε l (and more...) ε 0.40 ε 0.60 ε 0.80

Comparison Paking / Reonstruted Paking: ε 0.4 0.8 2.

16 Comparison Paking / Reonstruted Paking: ε W 700K Reonstruted: ε W 700K Better ontat soure/surrounding

17 Comparison Without / With radiation 2.28 W 700 K 360 K 320 K ε K ε K ε K ε K ε 0.80 ε 0.80

18 emi-loal analysis radiative ontribution to the etive ondutivity, r,, r ( ) 2 3 σ 8 Why this form? 8 ( ) 2 3 σ 8 ( ) 2 3 σ volumetri area alulated independently 8 ( ) 2 3 σ

19 Conlusion Radiation may ontribute signifiantly to heat transfers in the target appliations. he homogenizable part of their ontribution is well desribed in a wide range of strutures and temperatures by Rosseland approximation, with his model involves only intrinsi dimensionless geometrial parameters: ondutivity oiient, volumetri area, and a fairly onstant shape fator ω ω 0.40 for unonsolidated grain pakings ( ) ω ω 0.47 for onsolidated reonstruted media (ε ) Further work is desirable - to atalogue ω for other strutures (e.g., foams) - to theoretially justify/improve the form of the heuristi formula - to address semi-transparent solid materials, σ 3

20 Interpretation model: periodi vauolar medium, W W w s W w L l L +, r s 4 σ 3 L

21 Interpretation model: intrinsi formulation + 3, 4 l L w W L s r σ Condution in the solid slab Radiation in the void layer s, WL w x 2 In terms of the intrinsi dimensionless geometrial parameters ε ( ) ( ) 3 2 2, 8 x s r σ ε + ( ) ( ) 3 2 2, 8 s r σ ω ε + Not yet fully general: x is an ad-ho volumetri area, for a transfer along x. Introdution of a shape fator ω, multiplied by the whole volumetri area, Paking Reonstruted

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