March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 1 A GENERAL MODEL FOR WAX DIFFUSION IN CRUDE OILS UNDER THERMAL GRADIENT E. COMPARINI∗ and F. TALAMUCCI Dipartimento di Matematica “U. Dini”, Università di Firenze, Firenze, I50134, Italy ∗ E-mail: elena.comparini@math.unifi.it We consider a general model for the complex phenomenon of wax deposition in crude oils. Wax is present either as dissolved in oil or suspended as a crystallized phase. The solubility of wax decreases very sharply with temperature. The presence of a thermal gradient induces both a dynamics of transfer from dissolved to solid phase and the formation of a gel–like deposit layer at the cold wall. The process is described including different stages of evolution: we start from the fully saturated system, then, after the onset of an unsaturated front we deal with the simultaneous presence of saturated and unsaturated regions up to the complete unsaturation of the system. Keywords: Waxy crude oils; Molecular diffusion; Heat and mass transfer; Wax deposition; Free boundary problem 1. Introduction Waxy crude oils are mixtures of mineral oils, paraffins, aromatics and other impurities. The presence of wax components in oil, under particular thermal conditions, causes problems during transport in subsea pipelines, like wax precipitation with the consequent formation of wax crystals, deposition of the crystals on internal walls of pipeline, increasing of viscosity, diminuition of capacity of crude oil, removal of paraffin deposit. The formation of a solid deposit in pipeline walls is a phenomenon of crucial importance in oil industry because it can cause the blockage of a line, so that this complex phenomenon has been the object of a large number of papers, see the survey paper [7]. We refer to [11], [10] for the description of the mechanism of diffusion in non-isothermal solutions that causes wax deposition. A one-dimensional model describing the phenomenon of thermally induced mass transport in March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 2 partially saturated solutions under thermal gradient, including the displacement of all species (solvent, solute and segregated phase), is formulated in [10], where a mathematical analysis of the problem and some qualitative results have been obtained. In [11] the authors include in their analysis the case in which, because of a sufficiently low temperature, wax crystals aggregate in a gel-like structure that can be considered as immobile and not subject to diffusion. In this paper we deal with a theoretical investigation of the process of thermally induced mass transport in a general situation, taking into account both the molecolar diffusion of wax and the displacement of crystals suspended in oil saturated with wax. The mechanism at the origin of the formation of a deposit at the wall of the pipeline can be summarized remarking that solubility of wax depends on temperature, then dissolved wax under a certain temperature (cloud point) precipitates in form of wax crystals. Thermal gradient induces a concentration gradient that in its turn causes migration of wax towards the cold wall where it precipitates. Wax adheres to the wall forming a gel-like layer of increasing thickness. The mathematical model is based on the assumption that the three components of the system, oil (solvent), dissolved wax (solute) and wax crystals (segregated phase) have the same density (supposed constant in the range of temperature considered). That implies that gravity has no effect and that the segregation/dissolution process does not change volume. We start from a situation of full saturation of the solution in a threedimensional domain assuming no flux condition at the boundaries analyzing the formation of the solid wax deposit at the cold wall and the possible desaturation of the solution starting from the warm wall. 2. Mass balance, mass flux Let be Ω ⊆ R3 a connected domain. We define ̺α as the mass concentration (mass per unit volume of the system) of each component, referring to α = γ for oil, c for dissolved wax and G for segregated wax. It is also convenient to introduce the mass concentration of the liquid part, say Γ, made of dissolved wax and oil: ̺Γ = ̺γ + ̺c . The general conservation principle writes for each component ∂̺α + ∇ · Jα = Iα (1) ∂t where Jα is the mass flux of the species α and Iα is the rate of mass production or loss. March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 3 Mass conservation requires P Iα = 0. Furthermore, by the physics of α=γ,c,G the process is evident that the only mass transfer processes are dissolution or segregation of wax, so that IG = −Ic , Iγ = 0, and then from (1): ∂̺ + ∇ · J = 0, ̺ = ̺γ + ̺c + ̺G , J = Jγ + Jc + JG . (2) ∂t A particular attention must be paid to modelling the mass fluxes Jα = ̺α vα , α = γ, c, G, where vα denotes the velocity of the α species (Jα = ̺α vα ). Following a general scheme encompassing both dilute and concentrated solutions, the mass transfer of each species is due to the additive effects of diffusion and convection. Let us say that VA is some convective reference velocity and write Jα = ̺α (vα − VA ) + ̺α VA (3) so that mass flux of each component is the sum of the convective flux JA = ̺α VA and the diffusive flux J∗α = ̺α (vα − VA ). According to [5], multicomponent diffusion of a N –species system, where the N th species is designated as the solvent, can be described as J∗i = Ci vi − V A
=− N −1 X A,C Di,j ∇Cj , i = 1, . . . , N − 1 (4) j=1 where Ci denotes the i–concentration (not necessarily mass concentration) A,C stress the and the superscripts A, C in the molecular diffusivities Di,j dependence on the choice of the concentrations (bulk density, mass fraction, volume fraction, relative concentration,...) and on the reference velocity. A,C The multicomponent diffusion coefficients Di,j are generally not symmetA,C ric and the diagonal terms Di,i (main terms) are similar in magnitude to A,C binary values, see [5]. Each cross term Di,j , i 6= j is the contribution to the flux of the i solute by the concentration gradient of the j solute. Commonly, off–diagonal diffusion coefficients are much smaller than the main terms diffusion coefficients. We have to remark that (4) demands no forced pressure and no thermal diffusion: actually, the range of temperature of the process we are studying is so restricted that (4) can be assumed with good approximation. Let us examine now mass flux Jα for each specific component. It must be said that (see [5]) there is no evidence for what convective reference velocity VA should be: the choice of the mass average velocity or the velocity of the solvent or other reference velocities depends on the phenomenology of the process. March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 4 As for segregated wax, it seems to be reasonable to assume a process of binary diffusion (N = 2) where G is the solute and the liquid part Γ (oil + dissolved wax) is the solvent. The reference velocity is in that case the mass P ̺α vα /̺ = J/̺ and average velocity defined by (see also (2)) V∗ = α=γ,c,G we write (4) (referring to mass concentrations) simply as J∗G = ̺G (vG − V∗ ) = −DG ∇̺G where DG is the binary diffusivity of wax crystals in the mixture. Therefore, we get from (3) JG = −DG ∇̺G + ̺G J/̺. (5) It is likely to assume that DG depends on ̺G . If the concentration of solid wax is sufficiently high, we may think that DG reduces to zero and wax crystals are simply transported via convection by the average mass flux. On the other hand, if the liquid component is present in eccess, the mass average velocity is approximately the solvent velocity, which can be assumed as the reference velocity VA . Diffusion of dissolved wax c requires a special comment, since two different possibilities can be imagined: A. dissolved wax diffuses in the mixture, B. the process of diffusion of c occurs only in the liquid part whose concentration is ̺Γ = ̺γ + ̺c . In the first case, the situation is identical to what we have just concluded for G in (5), so we write Jc = −Dc ∇̺c + ̺c J/̺, where Dc is the binary diffusivity of dissolved wax in the whole system and with respect to the mass average velocity. In the latter case, we have to imagine a binary diffusion of c in the liquid part Γ only, disregarding wax crystals displacement. Following such a point of view, (4) is used in order to link the relative flux of dissolved wax (i. e. with respect to the liquid part only) with the relative concentration of c in Γ. Hence, we write (4) as e c ∇̺c,rel J∗c,rel = ̺c,rel (vc − VA ) = −D where the appropriate choice of the reference velocity is VA = vΓ = (̺γ vγ + ̺c vc )/̺Γ and the relative concentration has to be ̺c,rel = ̺c /ηΓ with ηΓ ec volumetric fraction of Γ in a unit volume of the mixture. The coefficient D is the binary diffusivity of dissolved wax in the liquid part. We finally get e c ∇ ̺c + ̺c DG ∇̺G + ̺c J. e c ∇ ̺c + ̺c vΓ = −ηΓ D (6) Jc = −ηΓ D ηΓ ηΓ ̺Γ ̺ March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 5 The flux of oil is obtained by Jγ = J − (Jc + JG ). Namely, we obtain in case A and B respectively ̺γ Jγ = DG ∇̺G + Dc ∇̺c + J, (7) ̺ ̺γ e c ∇ ̺c + ̺γ J. DG ∇̺G + ηΓ D (8) Jγ = ̺Γ ηΓ ̺ Obviously, different mechanisms describing the mass transfer process can be introduced, as, for istance, assuming a ternary diffusion of c, G in the solvent γ and the validity of (4) with N = 3, VA = V∗ (in that case, the diffusion coefficients Dc,c , DG,G , Dc,G and DG,c must be specified). One can use a volumetric approach defining the volumetric contents as γ = ̺γ /dγ , c = ̺c /dc , G = ̺G /dG , Γ = γ + c, and the volumetric velocities as Jα /dα = αvα , where dα , α = γ, c, G are the specific densities. If all the components have the same constant specific density d, that is dα = d, α = γ, c, G, and if G + c + γ = 1 (volume saturation), we have    JG = −d DG ∇G + GJ, (9) Jc = −d Dc ∇c + cJ,   J = d (D ∇G + D ∇c) + γJ γ G c e c ∇(c/Γ) + d (c/Γ)DG ∇G + cJ depending on the choice of or Jc = −d ΓD assumption either A or B, respectively, for Jc . The corresponding balance equations for G and c are ∂G − ∇ · [DG ∇G] + ∇G · J/d = IG /d ∂t ∂c − ∇ · [Dc ∇c] + ∇c · J/d = −IG /d case A ∂t ∂c e c ∇(c/Γ)] + ∇ · [(c/Γ)DG ∇G] − ∇ · [ΓD ∂t + ∇ c · J/d = −IG /d case B (10) (11) (12) The species γ is computed simply by means of γ = 1 − c − G. In the following analysis we will consider only case B, probably more adherent with the physical process. In that case, the flux of total wax ctot = c + G is e c ∇c − dD∇G b + ctot J Jctot = −dD (13) b = (1 − c/Γ)DG + (c/Γ)D e c and the mass balance can be written as where D ∂ctot e c ∇c + D∇G) b − ∇ · (D + ∇ctot · J/d = 0. ∂t (14) March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 6 3. Thermal balance, full or partial saturation The dynamics of transition, i. e. the specification of the term IG , depends on the thermodynamical assumptions we make for the model. Indeed, let us assume that the temperature T can be defined in each point of the system (i. e. the components are in thermal equilibrium in a representative elementary volume). It is known that for each value of T the concentration of dissolved wax in the solvent cannot exceed a saturation threshold, say ̺c,sat . Thus, we have ̺c,rel ≤ ̺c,sat (T ). Typically ̺′c,sat (T ) > 0. If the components have the same specific densities we can write c/Γ ≤ cS (T ), cS (T ) = ̺c,sat (T )/d, c′S (T ) > 0. Concerning with the total content of wax ctot = c + G, either dissolved or crystallized, we can adopt two different points of view (see [10]): (E) thermodynamical equilibrium is istantaneously reached between dissolved wax and segregated wax, so that c = ΓcS (T ), G = ctot − ΓcS (T ) in case of full saturation, c = ctot < ΓcS , G = 0, in case of partial saturation, (N E) segregated wax is present even if the solution is partially saturated. In case of full saturation, that is c = ΓcS (T ) each component can be expressed in term of G and T , since c = ΓcS (T ) = (1 − G)cS (T ), γ = [1 − cS (T )](1 − G). Thus, the dynamics of transition is automatically assigned and the mass balance for G is ∂G − ∇ · [(1 − cS (T ))DG ∇G] ∂t ∂T e c (1 − G)c′ (T )∇T ] − ∇ · [D +(1 − G)c′S (T ) S ∂t ′ +[(1 − cS (T ))∇G + (1 − G)cS (T )∇T ] · J/d = 0. (1 − cS (T )) (15) If the total flux J can be in some way specified (as, for istance, whenever it is identically zero), then (15) is in terms of G and T only, since the diffusion coefficients are expected to be functions of G, c, (possibly of the sum ctot ) and of T . In case of partial saturation c/Γ < cS (T ), equation (15) cannot be used. Actually, an additional statement must be added in order to describe the mechanism of transition. In that case, which we refer to as nonequilibrium models (N E), one can directly specify the term IG as a function of c, G, cS and T . A general law could be IG = −Φ(ΓcS − c) Ψ(G) where Φ, Ψ are empirical functions March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 7 satisfying Φ(0) = 0, Φ(y) > 0 if y > 0, Ψ(0) = 0, Ψ(G) > 0 if G > 0 Φ′ (y) > 0, (16) (17) In [9] Φ = βy, with β positive constant, and Ψ = H, Heavyside function, are considered. More generally, it can be imagined that Ψ(G) ∼ 1 starting from a value G0 > 0 close to zero, while for very low concentration of G, corresponding to few isolated wax crystals, dissolution is more rapid, so that Ψ(G) > 1, Ψ′ < 0 for 0 < G < G0 . The heat balance of the system, assuming that all the constituents have locally a common temperature T is (see [12])  X  ∂Eα  ∂T + ∇ · (−k∇T ) + (18) ∂T ∂t α=γ,c,G   T  Z X X  ∂Eα Jα · ∇T + Iα  ηα (y)dy + Eα  = 0, ηα + ∂T K(T ) + α=γ,c,G ̺α α=γ,c,G 0 where K(T ) = ̺ηγ + ̺c ηc + ̺G ηG is the equivalent heat capacity (ηα are the specific heats α = γ, c, G), Eα latent energy per unit mass, α = γ, c, G, k thermal conductivity of the mixture. According to (18), the quasi–steady equation (−k∇T ) = 0 is consistent if one makes the assumptions: ◦ heat released or absorbed during segregation or dissolution process is neglected, ◦ heat flux by conduction is no relevant, ◦ wax diffusivity is much less than the thermal diffusivity, so that the thermal equilibrium is istantaneously reached, ◦ the specific heat ηc and ηG are nearly the same. Nevertheless, it must be remarked that thermal conductivities of different zones (saturated region, deposit of segregated wax,...) may be very different. In several cases, the range (T1 , T2 ) of temperature is quite small, so that a linear solubility curve c′′s (T ) = 0 can be used in order to describe the real situation. March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 8 4. Conditions at the external boundary and at the interfaces At the boundary ∂Ω of Ω we assume that there is no mass exchange with the exterior of the species γ and ctot : Jγ · n = Jctot · n = 0, x ∈ ∂Ω (19) (n outward normal of ∂Ω) and that the temperature is assigned: T |∂Ω1 = T1 T |∂Ω2 = T2 > T1 , (20) where ∂Ω1 (“cold” wall) and ∂Ω2 (“warm” wall) are the subsets of δΩ where the thermal flux outcoming or incoming, respectively (see [9]). A special care must be devoted to writing the conditions on the internal boundaries of the system. Call Σ any interface in Ω (namely any internal ∂g boundary) and assume that +∇·j = 0 holds for a quantity g with current ∂t flow j. Then, the possible discontinuities across the interface Σ follow to the Rankine–Hugoniot condition (see, for istance, [6]) [[g]]V · n = [[j]] · n (21) where V is the velocity of the front Σ, n, [[∗]] stand for the jump [∗]+ −[∗]− , with + and − denoting the two regions separated by Σ and n is the unit normal, say pointing +. Applying (21) for the total wax and for oil and using the volumetric contents one gets [[γ]] V · n = [[Jγ /d]] · n, [[c + G]] V · n = [[(Jc + JG )/d]] · n. (22) We remark that [[J]] · n = 0 and that the two conditions in (22) are non independent, since c + G = 1 − γ, Jc + JG = J − Jγ . Let us apply (22) for the two key situations: 1. Σ = Σ1 is the interface between the deposit of segregated wax, say D, and the rest of the mixture R (let us choose − for D, + for R), 2. Σ = Σ2 is the front between the saturated region S and the unsaturated part U (− for S, + for U). With respect to point 1, we postulate a pure mechanical criterion for the deposit growth, which is engendered by ⊲ capture by adhesion of suspensions of a fraction cD tot of total wax c+ at the front, tot ⊲ contribution of the incoming mass flux, according to a part JD tot of + + + the total flux of wax Jctot = Jc + JG . March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 9 Assuming (by experimental evidence) that everything is at rest in the deposit D, we formulate such as mechanism by stating − D d[cD tot − (1 − γ )]V1 · n = Jctot · n (23) where γ − is the oil content in the deposit. Condition (23) is consistent with the total wax balance (22) only if the complementary fractions, which do not take part to deposition, are arranged in the right way beyond the front, namely D D + d[c+ tot − ctot ]V1 · n = [Jctot − Jctot ] · n. (24) Both (23) and (24) have to be considered at Σ1 in order to formulate the free boundary problem. It may be of use to consider (23) and (22) in place + of (24): as a matter of fact, (22) allows us to relate the velocity vγ+ = J+ γ /γ of the solvent at Σ1 with the front speed: (1 − γ − /γ + )V1 · n = vγ+ · n, γ + = 1 − c+ tot . (25) + + D Remark 4.1. It must be said that if cD tot = ctot and Jctot = Jctot , then condition (23) coincides with the total balance (22) and (24) is of no use. In that case, an additional assumption must be introduced: one possibility is to state that c+ = (1 − G+ )cS (T ). Actually, if all the dissolved wax is captured by the front, we may think that c attains saturation. + Whenever cD tot 6= ctot , we can eliminate V1 · n from (23), (24) and get the following implicit condition at Σ1 : + − D − + [cD tot − (1 − γ )]Jctot · n = [ctot − (1 − γ )]Jctot · n. (26) D In (23) the fractions cD tot and Jctot which build on the deposit must be specified. A simple but reasonable possibility consists in setting + + cD tot = λc c + λG G , + + JD ctot = µc Jc + µG JG (27) where λc , λG , µc , µG are constant values in [0, 1] to be specified. We remark that if one assumes that only the diffusive flux of dissolved + wax contributes to the formation of the deposit, then JD ctot = µJc,dif f = e c ∇(c/Γ), and conditions (23) and (24) become −µd ΓD ( + − d[cD tot − (1 − γ )]V1 · n = µJc,dif f , + D d[c+ tot − ctot ]V1 · n = (1 − µ)Jc,dif f − d(1 − c/Γ)DG ∇G. Let us point out some cases of (27), which have been considered in specific models in literature. March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 10 + 1. λc = λG = 1, that is cD tot = λctot (both dissolved and segregated wax in loco are entrapped in the deposit). Conditions (23), (24) reduce to ( + d[(c + G)+ − (1 − γ − )]V1 · n = [µc J+ c + µG JG ] · n, (28) + [(1 − µc )J+ c + (1 − µG )JG ] · n = 0. In particular, if µc = 1 (i. e. the entire liquid wax flux contributes to the deposit), then (28) is, whenever µG 6= 1 d[(c + G)+ − (1 − γ − )]V1 · n = J+ c · n, J+ G · n = 0, (29) which corresponds to µG = 0 (see [9]). On the contrary, if µG = 1, we run into the case of Remark 4.1. 2. λc = 0, λG = 1, µc = 1, µG = 0, stating that wax entering the deposit is only the total segregated wax and flux flowing into the deposit is only the dissolved wax flux. In this case, (23), (24) are [G+ − (1 − γ − )]V1 · n = J+ c · n, c+ V1 · n = J+ G · n. (30) 3. λc = 0, λG = η ≤ 1, µc = χ ≤ 1, µG = 0, wich corresponds to postulating the deposit of a fraction η of wax crystals and a part χ of dissolved wax flux. This time, (23) and (24) are ( d[ηG+ − (1 − γ − )]V1 · n = [χJ+ c ] · n, + d[c+ + (1 − η)G+ ]V1 · n = [(1 − χ)J+ c + JG ] · n, Case 3 corresponds to the model formulated in [10]. Remark 4.2. Following a different point of view, if one conjectures to + D assign the rates of deposition as cD tot = λ(c, G, T )ctot , 0 < λ ≤ 1, Jctot = + µ(c, G, T )Jctot , 0 < µ ≤ 1, we deduce from (23), (24) that λ = 1 if and only if µ = 1 and λ = µ if and only if they are both 1. Whenever λ 6= µ, condition (26) allows us to calculate the amount of total − wax at the front: c+ tot = 1 − µ(1 − γ )/(λ − µ), which makes sense only if the functions λ, µ are such that λ > µ + (1 − γ − )(1 − µ). Whenever the deposit is contiguous to a fully saturated region, then, assuming (27), condition (23) writes d[λG G + λc (1 − G)cS − (1 − γ − )]V1 · n = e c ∇T, d[µc cS − µG ]DG ∇G − µc (1 − G)c′ (T )D S (we omitted the symbol +, for simplicity) and (24) similarly. March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 11 On the other hand, if R is unsaturated and thermodynamical equilibrium is assumed to hold, then G is identically zero in R. Thus, according to (27), + + one should make the assumption cD JD tot = λc c , ctot = µc Jc . Arguing as in Remark 4.2 and considering that we expect c = cS (T ) at Σ1 , we conclude that it must be λc = µc = 1 and that d(c+ − (1 − γ − ))V1 · n = J+ c · n, c = cS (T ). (31) We finally discuss the conditions on the desaturation front Σ2 , which separates the saturated region S = {x ∈ Ω | c = ΓcS (T )} (say −) from the partially saturated part U = {x ∈ Ω | c < ΓcS (T )} (say +). It makes sense to assume that both c and G are continuous at the desaturation front: [[c]] = 0, [[G]] = 0 x ∈ Σ2 , (32) so that c = (1−G)cS (T ) at the front (we have implicitly assumed [[T ]] = 0). Hence, condition (22) at Σ2 reduces to Jctot · n = 0, which can be written, according to (13) and (32)
e c (∇c)+ − (1 − G)c′S (T )(∇T )− = (33) D + e c (∇G) . −(1 − cS (T ))DG [[∇G]] − cS (T )D Note that the diffusion coefficients are continuous at Σ2 , since they are expected to depend (at most) on c, G and T . If we assume thermodynamical equilibrium (model E), we have G ≡ 0 in U, c|Σ2 = cS , G|Σ2 = 0 and (33) reduces to
e c (∇c)+ − c′S (T )(∇T )− = −(1 − cS (T ))DG (∇G)− . (34) D It is clear that a wide set of situations can be described by varying the initial scenario of the model. Nevertheless, as we stated in Section 1, we focus our attention on the process which is expected to evolve according to the sequential stages: • initial full saturation of Ω, • formation of the solid wax deposit at the cold wall, • desaturation process starting from the warm wall. In order to follow such a scheme, we choose initially a total content of wax ctot (x, 0) = c∗tot > cS (TM ) where TM = max T2 (x, 0) (see (20)). x∈Ω2 Assuming that the thermodynamical equilibrium is instantaneously reached, we have (c/Γ)(x, 0) = cS (T (x, 0)), hence c(x, 0) = (1 − c∗tot ) cS (T (x, 0)) , [1 − cS (T (x, 0))] G(x, 0) = c∗tot − c(x, 0). (35) March 9, 2007 11:36 WSPC - Proceedings Trim Size: 9in x 6in 12pages 12 The initial profile of T is obtained by solving −k∇T (x, 0) = 0 together with (see (20)) T (x, 0)|Ω1 = T1 (x, 0), T (x, 0)|Ω2 = T2 (x, 0). Although we specify particular initial conditions, the problem offers formidable difficulties from the mathematical point of view: as to cite one, even assuming thermodynamical equilibrium at any time t, the onset of Σ2 has to be ascribed to a time t̄ such that G(x, t̄)|x∈Ω2 = 0 and such as check seems to be very difficult in a generic domain Ω. References 1. J. R. Cannon, A. Fasano, Boundary value multidimensional problems in fast chemical reactions, Arch. Rat. Mech. Anal., 53 (1973) 1-13. 2. S. Correra, A. Fasano, L. Fusi, M. Primicerio, Modelling wax diffusion in crude oils: the cold finger device, Appl. Math. Model., (Elsevier 2006), to appear. 3. S. Correra, A. Fasano, L. Fusi, M. Primicerio, F. Rosso, Wax diffusivity under given thermal gradient: a mathematical model. ZAMM (2006), to appear. 4. J. Bear Dynamics of Fluids in Porous Media, Environmental science serier (Elsevier 1972). 5. E. L. Cussler Diffusion: Mass transfer in fluid systems, (Cambridge University Press 1984). 6. Evans Partial differential Equations, AMS, (Providence 1999). 7. A. Fasano, S. Correra, L. Fusi, Mathematical models for waxy crude oils. Meccanica 39, (2004) 441-483. 8. A. Fasano, M. Primicerio, General free-boundary problems for the heat equation, I, J. Math. Anal. Appl. , 57 (1977), 694-723. 9. A. Fasano, M. Primicerio, Heat and mass transport in non-isothermal partially saturated oil-wax solutions. New Trends in Math. Phys., P. Fergola, F. Capone, M. Gentile & G. Guerriero editors, (World Scientific 2005) 10. A. Fasano, M. Primicerio, Temperature driven mass transport in concentrated saturated solutions. Progress in Nonlinear Differential Equations and their Applications 61 (2005), 91-108. 11. A. Fasano, M. Primicerio, Wax deposition in crude oils: a new approach. Rendiconti Lincei, Matematica e Applicazioni, 16 Serie 9, Issue 4, Mathematical Physics (2005) 251-263 12. F. Talamucci, Some problems concerning with mass and heat transfer in a multi–component system, MAT Conferencias, seminarios y trabajos de Matemàtica, serie A 6 (2002) .