Strong well-posedness of a model for an ionic exchange process [Elektronische Ressource] / von Matthias Kotschote

Sujets

• Physik

Informations

• Publié par: martin-luther-universitat_halle-wittenberg
• Publié le: 01 janvier 2003
• Nombre de lectures: 27
• Langue: English

Extrait

Strong Well-Posedness of a Model
for an Ionic Exchange Process
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium (Dr. rer. nat.)
vorgelegt der
Mathematischen-Naturwissenschaftlichen-Technischen Fakultat
(mathematisch-naturwissenschaftlicher Bereich)
der Martin-Luther-Universitat Halle-Wittenberg
von
Herrn Dipl.-Math. Matthias Kotschote
geb. am 21.10.1973 in: Pritzwalk
Gutachter:
1. Prof. Dr. Jan Pruss (Halle)
2. PD Dr. Dieter Bothe (Paderborn)
3. Prof. Dr. Matthias Hieber (Darmstadt)
Halle (Saale), 28. November 2003 (Tag der Verteidigung)
urn:nbn:de:gbv:3-000006407
[http://nbn-resolving.de/urn/resolver.pl?urn=nbn%3Ade%3Agbv%3A3-000006407]To My ParentsContents
Introduction ii
1 The Model 1
1.1 Regeneration of Ionic Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modelling of an Ionic Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Solution Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 The Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Preliminaries 14
2.1 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
12.2 The classesS(X),BIP (X) andH (X) . . . . . . . . . . . . . . . . . . . . . 15
2.3 Operator-Valued Fourier Multipliers andR-Bounded
Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Multiplication Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Maximal L -Regularity for the Linear Problem 32p
3.1 A Full Space Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 A Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 A Two Phase Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 The linear problem in domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.1 Localisation Techniques for Bounded Domains . . . . . . . . . . . . . 51
3.4.2 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 The Nonlinear Problem 80
4.1 Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bibliography 92
iIntroduction
It is a well known fact that chemically reacting systems can be described by means of systems
of reaction di usion equations on the microscopic scale. These kinds of equations have been
studied in great detail by many authors during the last three decades. Although in chemi-
cal engineering the focus is overriding on the macroscopic scale, leading mostly to ordinary
di eren tial equations, in many problems one has to take into account e ects of di usion, con-
vection and dispersion, or physical e ects caused by electrical charges (e.g. electromigration).
The mass balance equations then become reaction-di usion-con vection equations which are
coupled with equations arising from considered physical processes. This leads to systems of
partial di eren tial equations in three dimensions which can be very complicated. Therefore,
models have been developed on the macroscopic scale which allow for the essential informa-
tion about the physical processes taking place. However, many chemical processes involve two
or more phases, which typically means, the reacting species o w into the Continuously- ow
Stirred Tank Reactor (CSTR) and at least one of these species must be transferred to another
phase through an interface. In such situations it is of importance to take into consideration
mass transport in order to arrive at reliable models. However, this brings about the coupling
between the macroscopic reactor scale and the microscopic processes.
In the last decades, e orts have been made to account for electrical forces between particles.
This approach seems to be reasonable for the reacting species not being electrically neutral,
and particularly if electrical interactions can not be neglected in the chemical process. This
applies in case that forces are of the same magnitude as the other driving forces, e.g.
di usion or convection. However this involves a new unknown quantity, namely, the so-called
electrical potential which is caused by the charged particles. Including this item leads to a
strong coupling of the equations for the charged species. One possibility for incorporating
these e ects into the model is the assumption of electroneutrality, which demands that the
total charge has to be zero everywhere at any time. This means that, for concentrations ci
of reacting species and corresponding charges z 2 Z the following algebraic constraint musti
hold X
zc (t;x) = 0; t2J; x2
: (1)i i
i
Thus, the reaction di usion equations are augmented with an algebraic equation. The e ect
of electromigration was rst taken into account by Henry and Louro [14]. To all appearances
there are only a very few papers about electrochemical systems in the mathematical literature,
e.g. see [2], [6], [15], [23], [38] and [4]. Therefore, it is this physical feature which is to play a
decisive role in our treatise.
In this thesis we are concerned with a mathematical model resulting from a regenerative
ionic exchanger, see [21] or [5] for more chemical background. The model will describe in
iirdetail the regeneration of the weak acidic cation exchanger-resins Amberlite IRC-86 (called
2+pellets) charged withCu -ions via hydrochloric acidHCl in a well stirred tank (CSTR). In
fact, the pellets are suspended in a liquid bulk phase, where the acid is fed into the reactor
+continuously via a carrying liquid and dissociates intoH andCl . The exchange of cations
+ 2+H andCu is connected with a subsequent reaction of neutralisation between the moving
+protons H arising from the acid and the attached ions COO . The chemical reaction
equation reads as follows:
2+ + 2+R(COO ) Cu + 2H ! 2RCOOH + Cu : (2)2
The model is illustrated schematically by the following gure
Reaction in Pellet: Filmf 2+ + 2+_V R(COO ) Cu +2H !2RCOOH+Cu2
HCl
Cl
2+R(COO ) Cu2
Cl
Pellet
+H
+H
2+CuReaction in Bulk:
+ CuClHCl H + Cl 2
2+CuCl Cu + 2Cl f_2 V
HCl
Figure 1: Processes in the CSTR
As visualised above, the underlying chemical system consists of three phases: the almost
perfectly mixed bulk phase, the porous pellet and the lm. The balance of each phase has to
take into consideration coupling of mass transport for all species and chemical reactions. The
resulting equations yield systems of heterogeneous reaction di usion equations in each phase
which are connected to boundary conditions. In the end, a system of parabolic equations for
concentrations in lm and pellet is obtained, and ordinary di eren tial equations reproduce
the situation in the bulk phase. As mentioned above, the e ect of electromigration caused
by considering charged species is to be involved, which in turn requires the electroneutrality
condition (1).
Now, we shall describe the equations modelling the above situation. Let
be a bounded
3domain in R which decomposes according to
=
[
and their boundaries :=@
P F P P
2and@
= [ , :=@ , areC -smooth with dist ( ; ) > 0. The domain
represents aF P P P
ktypical pellet and
its surrounding liquid lm. For the unknown functionsu : [0;T]
!F k
iiiN b N k
R , k = P;F, u : [0;T]! R and : [0;T]
! R, k = P;F, we are concerned withk
the problem
P P P P P P P P@ u r (D ru ) r (M u
r ) =R (t;x;u ); (t;x)2J
;Pt
F F F F F F F F@ u r (D ru ) r (M u
r ) =R (t;x;u ); (t;x)2J
;Ft
P P P P P F F F F FD @ u +M u @ =D @ u +M u @ ; (t;x)2J ;P
P P P F F Fln( u ) + z = ln( u ) + z; (t;x)2J ;0 0 Pi i i i1iN 1iN
(3)
F bu =u ; (t;x)2J ; Z d 1b f b b b b F F F F Fu = u u +R (u ) a D @ u +M u @ d ; t2J; dt
P P F F b bu (0;x) =u (x); x2
; u (0;x) =u (x); x2
; u (0) =u ;P F0 0 0
and
T k T bz u (t;x) = 0; (t;x)2J
; k =P;F; z u (t) = 0; t2J: (4)k
k kLet us explain the relevant quantities and constants. The functions u and , k = P;F;b
denote the concentration vectors and electrical potentials, respectively, where the superscripts
k kindicate the corresponding phase. The di usion coe cien tsd summarised to the matrixDi
k kare known functions of (t;x), and the quantity m (t;x) := d (t;x)z is the so-called0 ii i
k kelectrochemical mobility. We set M = diag[m ] . Finally, the constant :=F=RT is1iN 0i
positive, where F denotes the Faraday constant, T the absolute temperature and R the gas
constant.
The charge of species i denoted by z is the same in each phase. The rst boundaryi
condition is caused by the continuity of uxes on , whereas the second boundary conditionP
is due to continuity of chemical potentials. At the outer surface of the lm continuity of
concentrations is imposed. The next equation describes the evolution of the bulk concentra-
f btions. The feedsu are time-dependent non-negative functions and the constanta comprisesi
among other things the total number of pellets in the bulk volume. Finally, the functions
kR , k = P;F;b and