© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PAMM · Proc. Appl. Math. Mech. 7, 2040041–2040042 (2007) / DOI 10.1002/pamm.200700576

A Stefan problem related to a non-equilibrium shrinking unreacted coremodel with fast reaction for the sphere

Adrian Muntean 1,∗1 CASA - Center for Analysis, Scientific computing and Applications, Department of Mathematics and Computer Science,

TU Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands.

The problem studied in this note refers to a substantial part of a larger system of partial differential equations modelingdiffusion and fast reaction of a gaseous species A in a reactive spherical porous region. We report on the local existence anduniqueness of weak solutions to the corresponding moving-boundary system containing two coupled mass-balance equationsin a moving annulus. Since the system incorporates an explicit description of the velocity of the moving boundary, ourformulation resembles to the one-phase Stefan-like problem with kinetic condition.

1 Introduction. Statement of the problem

In this note, we model a shrinking unreacted core situation (for details, see [4]) by means of a moving-boundary system(say MBP) with kinetic condition modeling the driving force by fast reaction1. We employ this MBP to study the evolutionof a sharp-reaction interface inside a sphere made of partially wet porous material. The model consists of a set of semi-linear mass-balance equations coupled with a non-linear (non-local) ordinary differential equation, which describes the speedof the interface. We refer to this differential equation driving the sharp-reaction interface into the material as the kinetic(non-equilibrium) condition. The model equations are non-linearly coupled by the a priori unknown position of the moving(reaction) interface and non-linearity in the production term by reaction.

We consider the basic geometry illustrated in in Fig. 1: A spherical heterogeneous material element is exposed to an A-richatmosphere. The situation we are looking at is the following: A penetrates Ω1(t) by molecular diffusion in the gaseous phase.Then it rapidly dissolves in the water phase and becomes B. In the water phase, a fast reaction, which has B as a reactant,takes place if and only if some concentration h (the second reactant) is available (via dissolution-type mechanisms that weneglect here) at the reaction locus (i.e. at Γ(t)). This work is inspired by our investigation via 1D MBP approaches of theconcrete carbonation problem initiated in [9]. The challenge is to deal with 2D cases (the spherical case being the first step).

Fig. 1 Left: Basic geometry of the region which our model refers to. Right: Definition of the domains Ω1(t) and Ω2(t).The fast reaction is assumed to take place at the interface Γ(t) separating the two regions. The vector n points out the normaldirection and Γ(t) is expected to move inwards. Ω1(t) denotes the annulus {x : L > |x| = r > R(t)}, L ∈]R0,∞[ for allt ∈ Sθ :=]0, θ[ for a fixed given finite time θ > 0, while |x| is the Euclidian distance between x and the origin.

In this paper, we discuss the system (1)–(7) described in what follows:

at − D1arr =D1(n − 1)

rar − P1(a − Q1b) in Ω1(t) × Sθ (Mass balance of A) (1)

bt − D2brr =D2(n − 1)

rbr + P2(a − Q2b) in Ω1(t) × Sθ (Mass balance of B) (2)

∗ Corresponding author E-mail: a.muntean@tue.nl, Phone: +31 40 247 2702, Fax: +31 40 244 21501 Many reaction-diffusion scenarios taking place in unsaturated reactive porous materials involve the formation and propagation of moving-sharp inter-

faces, where fast chemical reactions are assumed to be concentrated. When spatially separated reactants meet, the separation boundary may be assumed assharp provided that the characteristic time scale of reaction is much smaller than that of transport.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

where Di, Pi, Qi ∈ R+ (i ∈ {1, 2}), n ∈ {2, 3}, and

Ω1(t) × Sθ := {(x, t) ∈ Rn+1 : x ∈ Ω1(t) and t ∈ Sθ}.At the moving boundary Γ(t) := {|x| = R(t), t ∈ Sθ}, we have the following conditions:

−D1ar(R(t), t) = (a(R(t), t) − h)R′(t) − η(a(R(t), t), Λ) for t ∈ Sθ, with given Λ and h > 0, (3)

−D2br(R(t), t) = 0, b(R(t), t) = 0 for t ∈ Sθ, (4)

and

R′(t) = ψ(R(t), a(R(t), t)) for t ∈ Sθ. (Kinetic condition) (5)

The boundary condition at x = L is given by

a(·, t) = a(t), b(·, t) = b(t) for t ∈ Sθ. (6)

We complete the problem with the initial conditions

a(r, 0) = a0(r), b(r, 0) = b0(r) for r > R0, and R(0) = R0. (7)

We assume that the concentration h, which is living in Ω2(t), is strictly positive and constant.

1.1 On the conditions (3)– (5) at the moving interface

(3) and (4) are Stefan-like conditions [3]; they simply express the conservation of the mass of A and B at the interface Γ(t).(5) is our kinetic condition. This can be derived via first principles (cf. sect. 2.3.1 in [9]). As opposed to the use of kineticcondition (see e.g. [13]) at the moving interface, a different attempt is use the continuity of the concentrations when crossingΓ(t) (like in [6, 12], e.g.) or to introduce the Gibbs-Thompson condition a(R(t), t) = − 1

R(t) ; see for instance [11] or, for aderivation, [3]. (5) is in some sense conceptually close to the corresponding relation used in [1].

2 Main results

It is convenient to immobilize the moving boundary by making the substitution y := r−R(t)L−R(t) together with [a(r, t), (b(r, t)]t :=

[u(y, t) + a(t), v(y, t) + b(t)]t, t ∈ Sθ. In this manner, the original problem defined in the non-cylindrical domain Ω1(t)×Sθ

is now mapped into the cylindrical domain ]0, 1[×Sθ, where the weak formulation can be easily written [10] and variousenergy-type estimates in weighted Sobolev spaces [7] can be obtained. See [10], and also [2, 5, 8, 9] for hints on the analysisof a couple of one-dimensional settings.

The assumptions on the reaction rates and material parameters are the following: (H1) Fix the set of reaction parametersΛ ∈ MΛ ⊂ R3 (MΛ compact). Let η(v,Λ) > 0, if v > 0 and h > 0, and η(v,Λ) = 0, otherwise. Assume η(·) to be boundedfor any fixed v > 0; (H2) The reaction rate η : R×MΛ → R+ is locally Lipschitz; (H3) P1Q1k2 ≤ P1k1; P2k1 ≤ P2Q2k2,where k1 and k2 are L∞-bounds on u and v; (H4) The speed ψ has the same structural properties (H1) and (H2) as the reactionrate η. Note that the role of (H1)–(H4) is to ensure the positivity and the existence of L∞-bounds for the concentrations u andv and the interface position R.

Our main result is:

Theorem 2.1 (Local Existence and Uniqueness) Assume the hypotheses (H1)–(H4) as well as some regularity assump-tions on a, b, a0 and b0 be satisfied. Then the following assertions hold:(a) There exists a δ ∈]0, T [ such that the problem (1)–(7) admits a unique local weak solution on Sδ;(b) 0 ≤ u(y, t) + a(t) ≤ k1, 0 ≤ v(y, t) + b(t) ≤ k2, a.e. y ∈ [0, 1] for all t ∈ Sδ . Moreover, R ∈ W 1,∞(Sδ).

Acknowledgements We thank DFG for partial financial support through the grant SPP1122.

References

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[10] A. Muntean, Shrinking spheres by fast reaction: analysis of a moving-boundary system (in preparation).[11] O. A. Oleinik, M. Primicerio, E. V. Radkevitch, Meccanica 28, 129-143 (1993).[12] G. Sestini, Ann. Mat. Pura Appl. 56, 193-207 (1961).[13] A. Visintin, Ann. Mat. Pura Appl. 146, (5) 97-122 (1987).

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