p9787115013u4204.pdf

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 61, 91–108

Temperature Driven Mass Transport

in Concentrated Saturated Solutions

Antonio Fasano and Mario Primicerio

Abstract. We study the phenomenon of thermally induced mass transport in

partially saturated solutions under a thermal gradient, accompanied by de-

position of the solid segregated phase on the “cold” boundary. We formulate

a one-dimensional model including the displacement of all species (solvent,

solute and segregated phase) and we analyze a typical case establishing exis-

tence and uniqueness.

1. Introduction

It is well known that saturation of a solution of a solute S in a solvent Σ is achieved

at some concentration c S depending on temperature T . Typically c S is a smooth

function of T such that c S ( T ) > 0. Therefore, if one excludes supersaturation, it

is possible to produce the following facts by acting on the thermal ﬁeld:

(i) cooling a solution of concentration c ∗ to a temperature T such that c ∗ >

c S ( T ), segregation of the substance S is produced as a solid phase, typically

in the form of suspended crystals,

(ii) maintaining a thermal gradient in a saturated solution creates a concentration

gradient of the solute inducing diﬀusion.

These phenomena are believed to be the most important origin of the formation

of a deposit of solid wax on the pipe wall during the transportation of mineral oils

with a high content of heavy hydrocarbons (waxy crude oils) in the presence of

signiﬁcant heat loss to the surroundings (see the survey paper [1]).

In the paper [2] we have illustrated some general features of the behavior of

non-isothermal saturated solutions in bounded domains, including the appearance

of an unsaturated region and the deposition of solid matter at the boundary.

Work performed in the framework of the cooperation between Enitecnologie (Milano) and I2T3

(Firenze).

2005 Birkhauser Verlag Basel/Switzerland

A. Fasano and M. Primicerio

The analysis of [2] was based on the following simplifying assumptions:

(a) the three components of the system, namely the solute, the solvent and the

segregated phase, have the same density (supposed constant in the range of

temperature considered),

(b) the concentrations of the solute and of the segregated phase are small in

comparison with the concentration of the solvent.

The consequences of (a) are that gravity has no eﬀect and that the segrega-

tion/dissolution process does not change volume.

The consequences of (b) are that solvent can be considered immobile and

that the presence of a growing solid deposit has a negligible eﬀect on the mass

transport process.

For the speciﬁc application to waxy crude oil assumption (a) is reasonable

on the basis of experimental evidence, but assumption (b) may not be realistic. Of

course eliminating (b) leads to a much more complex situation.

For this reason we want to formulate a new model in which, diﬀerently from

[2], the displacement of all the components is taken into account, as well as the

inﬂuence of the growing deposit on the whole process.

In order to be able to perform some mathematical analysis of the problem and

to obtain some qualitative results we conﬁne our attention to the one-dimensional

case, considering a system conﬁned in the slab 0 <x<L . Of course the results

can be adapted with minor changes to a region bounded by coaxial cylinders (the

geometry of some laboratory device devoted to the measure of thickness of deposit

layers formed under controlled temperature gradients).

The general features of the model are presented in Section 2. In Section 3 we

consider a speciﬁc experimental condition in which we pass through three stages:

at time t = 0 the system is totally saturated with the segregated phase present

everywhere, next a desaturation front appears and eventually the saturated zone

becomes extinct. The rest of the paper is devoted to the study of the three stages,

showing existence and uniqueness and obtaining some qualitative properties.

2. Description of physical system and

the governing diﬀerential equations

During the evolution of the process we are going to study we can ﬁnd a saturated

and an unsaturated region. Supposing that at any point and at any time the segre-

gated phase is in equilibrium with the solution, there will be no solid component in

the unsaturated region. We recall that all the components have the same density

ρ , whose dependence on temperature is neglected.

The saturated region is a two-phase system:

– The solid phase is the segregated material. It is made of suspended particles

(crystals) having some mobility. We denote its concentration by G ( x, t )

– The liquid phase is a saturated solution. Its concentrantion for the whole

system is

Γ( x, t ).

Temperature Driven Mass Transport

In turn, the solution is a two-component system containing

– the solute with concentration c (mass of solute per unit volume of the system)

– the solvent with concentration γ (mass of solvent per unit volume of the

system).

In the sequel we will use the non-dimensional quantities

G = G/ρ, Γ=Γ /ρ, c = c/ρ, γ = γ/ρ.

Clearly

Γ= γ + c (2.1)

G +Γ=1 . (2.2)

We can also introduce the relative non-dimensional concentrations (mass of

solute and of solvent per unit mass of the solution)

c rel = c/ Γ γ rel = γ/ Γ . (2.3)

As we pointed out, saturated region is characterized by the fact that c rel =

c S ( T ) where the latter quantity is the saturation concentration and depends on

the local temperature T only.

On c S ( T ) we make the following assumption:

(H1)

c S ∈

C 3 ,c S > 0

in a temperature interval [ T 1 ,T 2 ].

Displacement of the various components is generated by spatial dishomogeneity.

Let J G ,J Γ be the ﬂuxes of segregated solid and of solution, respectively, in a

saturated region.

Let Q be the mass passing, per unit time and per unit volume (rescaled by

ρ ), from segregated to dissolved phase. Then we have the balance equations

∂G

∂t

+ ∂J G

∂x

−

(2.4)

∂ Γ

∂t

+ ∂J Γ

∂x

= Q.

(2.5)

From (2.2) it follows that

∂

∂x ( J G + J Γ )=0 ,

(2.6)

expressing bulk volume conservation and implying

J G + J Γ =0

(2.7)

if there is no global mass exchange with the exterior, as we suppose.

At this point we do not take the general view point of mixture theory, but

we make the assumption that G is transported by diﬀusion. Thus

J G =

−

D G ∂G

∂x ,

(2.8)

where D G is the diﬀusivity coe cient for the segregated phase.

A. Fasano and M. Primicerio

We notice that (2.8) is consistent with the fact that all the components of

the system have the same density, so that we may say that suspended particles do

not feel internal rearrangements of the solution components.

Next we have to describe the ﬂow of the components in the solution, and

denote by J γ and J c the ﬂux of solvent and of solute, respectively. Of course

J Γ = J γ + J c . (2.9)

Here too we take a simpliﬁcation supposing that the solute ﬂow J c relative

to the solution is of Fickian type, i.e.,

J c =

−

D ∂c rel

∂x

(2.10)

where D>D G is the solute diﬀusivity so that in the saturated region J c is a given

function of the thermal gradient.

The ﬂux J c is the sum of Γ J c and of the convective ﬂux due to the motion

of the solution. Introducing the velocity of the solution

V Γ = J Γ / Γ

(2.11)

we have

J c = cV Γ +Γ J c = c rel J Γ +Γ J c . (2.12)

Consequently we have the following expression for J c for the saturated and

unsaturated case (still retaining the basic assumption of absence of bulk mass

transfer, (2.7))

J c = c S D G ∂G

∂x −

−

G ) D ∂c S

∂x , for the saturated case

(2.13)

D ∂c rel

∂x

D ∂c

J c =

−

∂x , for the unsaturated case ( G =0) .

(2.14)

At this point we can write the balance equation for the solute

∂c

∂t + ∂J c

= Q.

(2.15)

∂x

While for the unsaturated case, (2.15) is nothing but

∂c

∂t −

D ∂ 2 c

∂x 2

= 0

(2.16)

in the saturated case we have c = c S ( T )(1

−

G ) and hence

−

c S ∂G

∂t

∂

∂x {

c S D G ∂G

∂x −

−

G ) D ∂c S

∂x }

= Q,

(2.17)

which provides the expression of Q . Thus (2.4) takes the form

∂G

∂t −

D G ∂ 2 G

∂x 2

c S {

( D G + D ) ∂c S

∂x

∂G

∂x −

−

G ) D ∂ 2 c S

∂x 2 }

=0 . (2.18)

−

It is also convenient to observe that from γ + c + G =1and c = c S (1

−

G )

we obtain

γ =(1 − c S )(1 − G ) ,

(2.19)

Temperature Driven Mass Transport

from which using (2.7), (2.9), (2.13) we deduce the expression

J γ =

−

D G ∂γ

∂x +( D

−

D G )

∂c S

∂x .

(2.20)

−

c S

We are interested in the in which T is a linear function of x independent

of time:

T 1 ) x

T = T 1 +( T 2 −

(2.21)

where the boundary temperatures T 1 ,T 2 (with T 1 <T 2 )aregiveninsuchaway

that a saturation phase is present, at least for some time.

Remark 1. The assumption that temperature has the equilibrium proﬁle (2 . 21) is

acceptable if heat diﬀusivity is much larger than D (which is certainly true), so

that thermal equilibrium is achieved before any signiﬁcant mass transport takes

place, and if we may neglect the amount of heat that is released or absorbed during

the segregation/dissolution process. In the speciﬁc case of waxy crude oils it can be

seen that the latter assumption is fulﬁlled (the inﬂuence of latent heat associated

to deposition is likewise negligible).

3. Modelling a speciﬁc mass transport process with deposition

We restrict our analysis to the following process, easily reproducible in a laboratory

device.

We start with a solution at uniform concentration c ∗ ( <ρ ) and uniform tem-

perature T ∗ with c ∗ below saturation. Then we cool the system rapidly to the

temperature proﬁle (2.21) in such a way that c ∗ = c ∗ /ρ > c S ( T 2 ), so that the

whole system becomes saturated with a (non-dimensional) concentration

G 0 ( x )= c ∗ − c S ( T ( x ))(1 − G 0 ( x ))

(3.1)

of segregated phase, with

c 0 ( x )= c S ( T ( x ))(1 − G 0 ( x ))

(3.2)

being the corresponding concentration of the solute.

These will be our initial conditions. Starting from t = 0 the system will evolve

through the following stages.

Stage 1. G> 0 throughout the system

Themassﬂowtowardsthecoldwall x = 0 produced by the gradient of c S ( T ( x ))

generates various phenomena:

•

the solute mass leaving the warm wall x = L has to be replaced by the

segregated phase,

•

mass exchange occurs between the solid and the liquid phase, as described in

the previous section,

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