Characterization of a
thermochemical storage
material
TU/e Master Thesis
March, 2007
Engineering thesis committee
prof.dr.ir. A.A. van Steenhoven (Chairman, TU/e)
dr.ir. C.C.M. Rindt (Coach, TU/e)
dr.ir. A.J.H. Frijns (TU/e)
ir. J.G. Wijers (TU/e)
dr.ir. W.G.J. van Helden (Coach, ECN)
Eindhoven University of Technology
Department of Mechanical Engineering
Division Thermo Fluids Engineering
Energy Technology group
I.M. van de Voort
WET 2007.04
2
Abstract
Thermochemical materials are a promising new alternative for long term heat storage. The
process concerned is based on a reversible chemical reaction, which is energy demanding
in one direction and energy yielding in the reverse direction. Preliminary research marked
magnesium sulfate hepta-hydrate (MgSO4 · 7H2 O) as a specifically suitable material. The
material dehydrates when heated, forming MgSO4 , this can be regarded as ’charging’ of the
material. The reverse reaction is initiated when the anhydrous component is exposed to water
vapor, producing heat.
To this point, literature is insufficient and inconsistent regarding the material properties of
magnesium sulfate hepta-hydrate. Experiments are performed in order to characterize the
material and its properties. Furthermore experiments are executed to check the applicability
of MgSO4 · 7H2 O in a seasonal storage system. Utilizing these results a 2D macromolecular
numerical model is proposed, describing the chemical conversion of the material. The model
is validated by comparing experimental and numerical results for some base case problems.
Two applications of the model can be identified. Firstly, other thermochemical materials can
be easily implemented in order to check their fitness as a storage material. Secondly, the
white box model provides a better phenomenological understanding of the processes at hand.
i
ii
Abstract
Samenvatting
Thermochemische materialen vormen een veelbelovend alternatief voor lange termijn warmte
opslag. Het proces is gebaseerd op een reversibele chemische reactie die energievragend is
in een richting en energieleverend in de andere richting. Voorgaand onderzoek heeft uitgewezen dat magnesium sulfaat hepta-hydraat (MgSO4 · 7H2 O) een bijzonder geschikt materiaal zou kunnen zijn. Het material dehydrateert wanneer het verwarmd wordt, waarbij
MgSO4 gevormd wordt. Dit wordt ook wel aangeduid als het ’opladen’ van het materiaal.
De tegengestelde reactie wordt genitieerd door het gedehydrateerde materiaal bloot te stellen
aan waterdamp. Bij deze reactie komt warmte vrij.
De beschikbare literatuur is onvolledig en inconsistent wat betreft de materiaal eigenschappen
van magnesium sulfaat hepta-hydraat. Daarom zijn experimenten uitgevoerd waarmee het
materiaal en het functioneren van het materiaal als een medium voor lange termijn warmte
opslag gekarakteriseerd kunnen worden. Aan de hand van de resultaten van deze experimenten
is een 2D macromoleculair numeriek model opgesteld dat de chemische omzetting beschrijft
die het materiaal ondergaat. Het model is gevalideerd door experimentele en numerieke
resultaten voor enkele typische situaties te vergelijken.
Het model heeft twee belangrijke toepassingen. In de eerste plaats kunnen andere thermochemische materialen eenvoudig worden geı̈mplementeerd om zodoende de geschiktheid voor
gebruik in een lange termijn warmte opslag te bepalen. Ten tweede geeft het verkregen
parametrische model beter inzicht in de processen die plaats vinden tijdens deze reversibele
chemische reactie.
iii
iv
Samenvatting
Contents
Abstract
i
Samenvatting
iii
Nomenclature
vii
1 Introduction
1
2 Thermochemical storage materials
3
2.1
Chemical energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Crystalline salt-hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Salt-hydrates in a seasonal storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3 Analysis methods
11
3.1
Thermogravimetric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2
Differential Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.3
Differential Scanning Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.4
X-ray crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.5
Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.6
Vapor pressure measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.7
Scanning Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4 Experimental results
4.1
Material characterization
19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.1.1
Stable hydrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.1.2
Material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
v
vi
CONTENTS
4.2
4.3
4.1.3
Surface structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
4.1.4
Reaction Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
System characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2.1
Grain size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2.2
Layer thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2.3
Vapor pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.2.4
Cyclability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Experimental resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
5 Modeling
33
5.1
Introductory theory on heat and mass transfer . . . . . . . . . . . . . . . . . . . . . .
33
5.2
Grain models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5.3
Layer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
5.3.1
Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.4
Current model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.5
Model verification
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
5.6
Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5.7
Model predicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
6 Conclusions and Recommendations
47
Bibliography
49
A Calculation of kinetic parameters
51
B Experimental background
53
B.1 Experimental setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
B.2 Experimental settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
C Modeling background
57
Nomenclature
A
Bi
C
cp
D
d
E
E
f
Fo
G
G
H
h
k
L
Le
M
m
′′′
ṁ
n
N
n
p
P
Q
Q
q
R
r
ṙ
(Arrhenius) frequency factor [s−1 ]
Biot number [−]
(phase rule) number of indep.
chemical comp. [−]
heat capacity [J kg−1 K−1 ]
mass diffusion coefficient [m2 s−1 ]
(Bragg) interplanar distance [m]
energy [J]
(Arrhenius) activation energy
[J mol−1 ]
volume force [N]
Fourier number [−]
Gibbs free-energy [J]
concentration gas phase [mol m−3 ]
enthalpy [J mol−1 ]
heat transfer coefficient [W m−2 K]
reaction rate [min−1 ]
length [m]
Lewis number [−]
molar mass [g mol−1 ]
mass [kg]
mass source term [kg s−1 m−3 ]
integer number [−]
mass flux vector [mol m−2 s]
outward facing normal [−]
pressure [Pa]
(phase rule) number of phases [−]
heat [J]
source term [W m−3 ]
heat flux vector [W m−2 ]
gas constant [8.314 J K−1 mol−1 ]
radius [m]
interface velocity [m s−1 ]
Rp
S
S1
S2
T
t
Th
V
V
v
W
w
x
Y
y
z
(Thiele) typical length scale [m]
entropy [J mol−1 K−1 ]
concentration solid 1 [mol m−3 ]
concentration solid 2 [mol m−3 ]
temperature [K]
time [s]
Thiele modulus [−]
volume [m3 ]
(phase rule) degrees of freedom [−]
velocity vector [m s−1 ]
work [J]
uptake of vapor [kg kg−1 ]
problem domain direction [−]
concentration [−]
problem domain direction [−]
problem domain direction [−]
Greek letters
α
α
∆
ε
θ
κ
λ
λ
µ
ν
ρ
τ
vii
shrinkage factor [−]
thermal diffusivity [m2 K−1 s−1 ]
change [−]
porosity [−]
angle of incidence [deg]
permeability [m2 ]
thermal conductivity [W m−1 K−1 ]
wave length [m]
dynamic viscosity [kg m−1 s−1 ]
kinematic viscosity [m2 s−1 ]
material density [kg m−3 ]
(DiBlasi) half slab thickness [m]
viii
NOMENCLATURE
Super- and subscripts
(g)
(s)
∞
c
eq
g
0
R
S
s
SR
t
w
gas
solid
ambient
char
equilibrium
gas(eous phase)
under standard conditions
reference sample
sample
solid (phase)
sample with respect to reference
sample
total
wood
Chapter 1
Introduction
The energy supply of most countries is presently based on fossil fuels. In the past decennia the
awareness of the drawbacks of this energy supply is raised. A large drawback of consuming
fossil fuels is the emission of carbon-dioxide, which contributes to the greenhouse effect.
Furthermore the global energy demand is still increasing, while the stock of fossil fuels is
depleting. There are two main solutions to the problem; new energy resources have to be
researched and more efficient ways of using the current limited energy resources have to be
explored.
Within the built environment the efficient use of energy has also become a topic of attention.
One of the research activities is focused on the storage of heat over longer periods of time,
also called seasonal storage. The surplus of heat that is available in summer is then stored
for domestic use in winter, for instance for hot tap water. A current technology is storage of
warm water by using either water bags or underground natural buffers, also called aquifers.
These buffers consist of geologically determined water conducting sand layers at a depth of
25 to 100 meter, the top and bottom of which are impermeable to water. Another technology
is the use of phase change materials (PCM). All these systems are relatively easy to install at
reasonable cycle efficiency. Due to the relatively low energy density and large requirements of
insulation however, these systems are not very cost efficient. A new way of long term storage
of solar energy, without the necessity for thermal insulation, is by means of chemical energy
in so-called thermochemical materials (TCM’s). Thermochemical materials can undergo reversible chemical reactions, which are energy consuming in one direction and energy yielding
in the reverse direction. Because the reaction temperature of this process is relatively high
(exceeding well over 100o C), there is no need for auxiliary heating to produce hot tap water.
A preliminary research for candidate materials for a seasonal heat storage based on TCM’s has
been conducted by ECN and Utrecht University. The emphasis of this research was placed
on an extensive literature survey, to which a footnote was added that literature values on
the properties of materials and their chemical reactions are often incomplete and sometimes
unreliable. In order to apply these materials in an energy storage system more insight should
be gained into characteristics of such materials.
The goal of the current research is to characterize a thermochemical storage material experimentally and use the properties obtained in formulating a macromolecular numerical model
1
2
Introduction
that describes the conversion of the material. Two applications of the model can be identified.
Firstly it can be used as a tool to characterize and implement other materials easily, by changing a few parameters, which can be obtained from literature or small experiments. Secondly,
the white box model provides a better phenomenological understanding of the processes at
hand.
In Chapter 2 theoretical background is discussed on the processes and principles concerning
the use of thermochemical materials in a seasonal storage. Chapter 3 treats analysis methods
which can be utilized in order to characterize materials and quantify material parameters.
The experimental results obtained with these analysis methods are presented in Chapter 4.
First some material characterization measurements are discussed and subsequently results
are presented which determine the applicability of the thermochemical material in a storage
system. In Chapter 5 a finite element model is developed in which a porous layer of salthydrate is converted chemically. The simulation results are then compared with experimental
results. Conclusions and recommendations for future research are formulated in the final
chapter.
Chapter 2
Thermochemical storage materials
A thermochemical storage system derives its functioning from a reversible chemical reaction,
that is energy demanding in one direction and energy yielding in the reverse direction. There
are a number of materials and reactions that conform to this requirement, as will be discussed
in this chapter. The group of salt-hydrates in general and magnesium sulfate hepta-hydrate
(MgSO4 · 7H2 O) specifically, are considered to be suitable materials [Visscher, 2004]. This
chapter provides background on chemical processes and principles concerning the use of thermochemical materials in a seasonal storage. Thermochemistry is not a discipline on its own, it
is a combination of thermodynamics, structural chemistry, catalysis and physical chemistry. It
is defined roughly as the application of heat effects due to a chemical reaction between two or
more species. First some chemical processes and definitions will be elucidated. Subsequently
a class of thermochemical materials is discussed, which is called crystalline salt-hydrates.
The chapter is concluded with a review of the use of crystalline salt-hydrates in a seasonal
storage and special attention is given to the material which is characterized in this research:
magnesium sulfate hepta-hydrate.
2.1
Chemical energy
Let’s consider a chemical system in which starting reactants are transferred to final products.
According to the first law of thermodynamics the total internal energy of an isolated system
is constant. The total energy change ∆E of a chemical system is represented by the sum of
the work performed on the system W , in addition to the heat of the system Q.
∆E = Q + W = Q + (−p∆V )
(2.1)
This equation can be rearranged in order to represent the amount of heat transferred, as:
Q = ∆E + p∆V
(2.2)
The heat of a system at constant pressure is also denoted by the change in enthalpy, denoted
by ∆H.
Q = ∆H = Hproducts − Hreactants
(2.3)
3
4
Thermochemical storage materials
If the products have more enthalpy than the reactants, then heat has flown into the system
from the surroundings and ∆H thus has a positive sign. Such a reaction is said to be
endothermic. If the products have less enthalpy than the reactants, heat has flown out of the
system to the surroundings and ∆H has a negative sign. Such reactions are called exothermic.
Examples of exothermic reactions are combustion reactions. As enthalpy change is a state
function, its value does not depend on the path followed between two states. Thus, the sum
of the enthalpy changes for the individual reactions in a sequence must equal the enthalpy
change for the overall reaction. This statement is also known as Hess’s law.
Entropy is a property of a thermodynamic system, that describes the amount of molecular
disorder or randomness in a system. The difference in entropy between two states of the system
is given by dS = dQ/T and has units [ J K−1 ]. Gases, for example have more randomness
and thus higher entropy than liquids and liquids have a higher entropy than solids.
The spontaneity of a process is governed by a change in both enthalpy and entropy. A
spontaneous process is characterized by a decrease in enthalpy and an increase in entropy
whereas a non-spontaneous process exhibits an increase in enthalpy and a decrease in entropy.
A general criterion for the spontaneity of a chemical reaction or physical process is represented
by the Gibbs free-energy change, denoted ∆G:
∆G = ∆H − T ∆S
(2.4)
If ∆G has a negative value the process is spontaneous, if ∆G is zero, the process is at
equilibrium and if ∆G is positive the process is non-spontaneous. The relation can also be
given for standard conditions, the terms G, H and S then get a zero in the suffix. Another
expression for the Gibbs free-energy is:
∆G = ∆G0 − RT ln(
p
)
p0
(2.5)
In which R is the gas constant, p is the vapor pressure and p0 the standard pressure. For an
equilibrium reaction, the free energy G is zero and Equation 2.4 and Equation 2.5 can then
be combined to give:
∆H0 ∆S0
p
+
(2.6)
ln( ) = −
p0
RT
R
This equation is also known as the Clausius Clapeyron equation and relates the equilibrium
vapor pressure to the thermodynamic parameters.
For a system that is in equilibrium the phase rule, stated by Gibbs, relates the number
of components (substances) and phases to the degree of freedom of the system. It reads
V = C − P + 2, in which V is the number of degrees of freedom, i.e. the number of variables
that must be arbitrarily fixed to establish the state of the system, C the number of independent
chemical components and P the number of phases. The equilibrium reaction of a salt-hydrate
can be represented as follows:
Salt · x H2 O(s) + Heat ⇀
↽ Salt · (x − y)H2 O(s) + y H2 O(g)
In such a system, there are two independent components, C = 2; three phases (2 independent
solids and 1 vapor phase), P = 3; and thus the degree of freedom is V = 1. Such a system
is called mono-variant, since for a given temperature there is only one equilibrium pressure
2.2. CRYSTALLINE SALT-HYDRATES
5
and vice versa. The adsorption and desorption of water vapor on silicagel seems to be a
comparable process, however water is bonded in a different way so that only one independent
solid is present in the system. The phase rule then states that the system is bi-variant, which
means that two variables (i.e. the temperature and pressure) can be chosen independently,
and a range of equilibrium states exist.
The speed at which a chemical reaction is completed may vary, depending on the characteristics of the reactants and products and the conditions under which the reaction is taking
place. The rate law is an equation expressing the instantaneous reaction rate in terms of the
concentrations of the substances taking part in the reaction at that instant. A decomposition
reaction, in which one reactant decomposes into products (e.g. a dehydration step of a salthydrate) is a typical first order reaction. This means that the reaction rate is proportional to
the concentration of the reactant. The reverse reaction in which the anhydrous salt together
with water vapor form a hydrated salt is a typical second order reaction, which means that the
reaction rate depends on the concentrations of both substances taking part in the reaction.
First order reaction kinetics are mostly described by the so called Arrhenius equation, which
reads:
E k = A exp −
(2.7)
RT
in which k is the reaction rate, A a so-called frequency factor, E the activation energy, R the
gas constant and T temperature. This relation can be plotted in an Arrhenius plot, which is
a graph of ln(k) vs 1/T . When points measured in experiments make a straight line in this
coordinate system the reaction rate obeys the Arrhenius law. Reactions with large activation
energies result in steep lines and can therefore be concluded to have rates that depend strongly
on temperature.
2.2
Crystalline salt-hydrates
Crystalline salt-hydrates (or shorter; hydrates) form a class of materials, in which water (or
another inert molecule) can be incorporated into the crystal lattice. The amount of water
contained within the lattice depends on both temperature and (vapor) pressure. In general
it is not possible for neutral atoms or molecules to be bound in ionic crystal structures. The
molecule of water and for instance ammonia, are exceptions to this general rule [Evans, 1966].
A reason for this anomalous behavior is their molecular polarity; the molecule has a typical
spatial configuration that results in a region of positive charge and a region of negative charge.
Another reason is the small size of the molecule.
Crystalline salt-hydrates can be classified into two groups, based on the distinctive way in
which water is incorporated in the lattice, also represented in Table 2.1. In the first class,
water molecules are arranged around the positively charged ion of the salt (i.e. the cation).
As the negative region of the water molecules will be directed towards the cation, the positive
region of the molecules will face outward, thereby increasing the effective radius of the cation.
These shall be described as hydrates containing coordinating water. The inert molecules are
bound chemically to the salt, therefore this is also denoted as a chemisorption process. The
water molecules in such crystals are essential in the stability of the structure; i.e. removal
of these molecules leads to a complete structural breakdown. Chemisorption processes are
6
Thermochemical storage materials
characterized by a high enthalpy, i.e. larger than 50 kJ mol−1 . The second class consists of
structures in which the cations are not directly coordinated by the water molecules. These
are said to contain structural water. The structure of the material is very open and water
molecules merely occupy intersticial voids in the structure. Here they can add to the electrostatic energy without upsetting the balance of charge, the bonding forces involved are called
Van der Waals forces. This type of bonding is also denoted physisorption and the energy
involved is typically low, i.e. lower than 40 kJ mol−1 . In some hydrate structures water is
found in both a coordinating and a structural capacity.
Table 2.1: The classification of hydrate structures
Containing only
coordinating water
AlCl3 · 6H2 O
CaSO4 · 2H2 O
Containing only
structural water
Zeolites
Gas hydrates
Containing both coordinating
and structural water
NiSO4 · 7H2 O
CuSO4 · 5H2 O
Salt-hydrates may be represented by the general formula Aa Bb · cH2 O, where A is the cation
and B is the anion, such as Cl− , CO32− , SO42− . The structure adopted will depend on the relative number of water molecules and cations present. The number of atoms or ligands directly
bonded to an atom is called the coordination number. This number implies the geometrical
arrangement of the ligands; a coordination number of six implies a octahedral configuration,
whereas a coordination number of four mostly yields a tetrahedral configuration. If the ratio
c/a exceeds the coordination number of the cation, more water molecules are available than
are required to coordinate that ion, and the excess can be present only in a structural role.
(a)
(b)
Figure 2.1: Structural representation of hydrates with structural water. In figure 2.1(a) a typical zeolite structure is shown, the framework is composed of siliciumoxide tetrahedra and water molecules can be incorporated
in the pores. Figure 2.1(b) represents a gas hydrate; a crystalline solid in which a gas molecule is surrounded
by a cage of water molecules.
2.3. SALT-HYDRATES IN A SEASONAL STORAGE
7
The hydrates containing structural water are more diverse in appearance than those which
contain water in a coordinating role. Some examples are shown in Figure 2.1. Here a brief
review of the renowned example of zeolites will be given. Zeolites are framework silicates
in which (Si, Al)O4 tetrahedra are linked together into a three dimensional network. This
network is very open and can readily accommodate water molecules in its pores. The trivial
function of water in zeolites is emphasized by the fact that it can be expelled from the crystal
lattice without destruction (or deterioration) of the structure and can even be replaced by
other neutral molecules, such as ammonia. Much research has been done in the past years
on energy storage systems employing zeolites, which can be useful for this research to some
extent. Zeolites are also widely used as molecular sieves.
Many hydrated salts contain an odd number of water molecules, of which some are present in
a coordinating role and some occur in a structural capacity. The hydrates MgSO4 ·7H2 O and
CuSO4 ·5H2 O may be quoted as examples. In the structure of CuSO4 ·5H2 O, four molecules
of coordinating water coordinate the copper atom, which is also coordinated by two oxygen
atoms of two SO4 groups and is therefore octahedrally surrounded by neighbors, as can be
seen in Figure 2.2. The fifth water molecule is only coordinated by other water molecules and
oxygen atoms.
Figure 2.2: Structure of CuSO4 · 5H2 O. Each copper atom is coordinated by two sulfate groups and four water
molecules. The fifth water molecule is not coordinated by the cation, but by other water molecules and oxygen
atoms of the sulfate group.
2.3
Salt-hydrates in a seasonal storage
As is discussed in the previous section, salt-hydrates can incorporate large amounts of water
into the crystal lattice, while remaining crystalline. When a hydrated salt is heated, the crystal
water is driven off. In a seasonal storage system solar heat can be employed to dehydrate the
salt-hydrate in summer. Subsequently the anhydrous salt is stored until winter. In winter
this salt is exposed to water vapor, initiating the reverse reaction that yields energy in the
form of heat, which can be used for residential use, such as hot tap water and central heating.
Preliminary literature research into candidate materials for use in a thermochemical seasonal
storage was performed by [Visscher, 2004]. Requirements for candidate materials embody a
high energy storage density, low corrosivity, reasonable cost and limited toxicity. The estimation for the energy which is available for dehydration, is based on the best current available
8
Thermochemical storage materials
solar vacuum collectors, which have a maximum operating temperature of about 150o C. A
number of salt-hydrates was investigated [Visscher, 2004] of which magnesium sulfate heptahydrate had the most promising properties. An overview of candidate materials is shown in
Table 2.2. Most of the data presented here originates from [Wagman, 1982]. A number of
properties for reactions of the form A ⇀
↽ B + C are given, namely the reaction temperature of
both the association and dissociation reaction, the accompanied reaction entropy change, its
enthalpy change and finally its energy storage density. The materials in the table are sorted
Table 2.2: Properties of a number of thermochemical storage materials
A
MgSO4 · 7H2 O
FeCO3
MgSO4 · 7H2 O
Fe(OH)2
CaSO4 · 2H2 O
MgSO4 · 1H2 O
CaCl2 · 2H2 O
B
C
MgSO4
FeO
MgSO4 ·H2 O
FeO
CaSO4
MgSO4
CaCl2 ·H2 O
7H2 O
CO2
6H2 O
H2 O
2H2 O
1H2 O
H2 O
Reac. temp.
dis/as [o C]
200/122
-/180
150/105
150/150
-/89
200/216
-/174
∆S
[J/mol/K]
1041
178
887
137
290
154
104
∆H
[kJ/mol]
411
81
336
58
105
75
47
Energy dens.
[GJ/m3 ]
2.8
2.6
2.3
2.2
1.4
1.3
0.6
for descending energy storage density. The full dehydration of MgSO4 ·7H2 O to the anhydrous
component has the highest energy storage density and an association reaction temperature
which is high enough for the purpose of heating tap water. Thereby it is marked as the most
promising material. A drawback of the use of FeCO3 is that the reaction product CO2 is
more toxic than water. The dehydration of MgSO4 · 7H2 O into the mono-hydrate still has a
high energy storage density, while the reaction temperatures of the reactions are lower. The
dissociation reaction temperature corresponds better to the maximum yield temperature of
current solar vacuum collectors and the association temperature is still high enough for tap
water heating. Due to its good prospects, the current research focuses on magnesium sulfate.
The dehydration of salt-hydrates is reported to occur in discrete steps and a number of
(meta-)stable states often exist. For each transition one equilibrium line exists, that is a
combination of temperature and vapor pressure. In literature, some data is found on the
intermediate crystalline phases that occur in the dehydration of magnesium sulfate heptahydrate. However some bias is found. In [Wagman, 1982] thermochemical data is reported
on the phases: MgSO4 · 7H2 O (hepta-hydrate), MgSO4 · 6H2 O (hexa-hydrate), MgSO4 · 4H2 O
(tetra-hydrate), MgSO4 · 2H2 O (di-hydrate), MgSO4 ·H2 O (mono-hydrate) and the anhydrous
component MgSO4 . These intermediate phases are often summarized as MgSO4 · xH2 O,
where x = 7, 6, 4, 2, 1, 0. In [Paulik, 1981] it is stated that during dehydration MgSO4 · 7H2 O
decomposes successively into the 3, 53 , 1 and the anhydrous salt. Some years later new data
is reported by Paulik, together with three others in [Emons, 1990]. Here the phases MgSO4 ·
xH2 O, where x = 7, 3, 2, 1, 0 are described to appear during the decomposition. MgSO4 ·3H2 O
is stated to be formed at a temperature of 105o C. At a temperature of 120o C the transition
to MgSO4 · 2H2 O occurs. The transition to the monohydrate is reported to take place in a
temperature range of 150 − 200o C and finally the anhydrous component is formed at 340o C.
2.3. SALT-HYDRATES IN A SEASONAL STORAGE
9
The former reported 53 phase is now stated to be a mixture of the 2 and 1 hydrate. These data
were found by the QTG and QDTA method developed by Paulik, which are measurements
that combine a special crucible with a high resolution measurement [Paulik, 1981]. The 4 and
2 hydrate are reported in [Vaniman, 2004] to be metastable, and do not occur in nature. The
dehydration behavior of MgSO4 · 7H2 O is not reported unambiguously, therefore dehydration
experiments will be performed, which are described in Chapter 4.
For MgSO4 ·7H2 O both the enthalpy and entropy change per expelled molecule of water is more
or less constant, except for the transition of the mono-hydrate to the anhydrous component
(which is somewhat higher), and is 55 kJ mol−1 and −160 J mol−1 K−1 respectively. The
molar weight of magnesium sulfate is 120.37 gram and the molar weight of each water group
is 18.02 gram. The molar weight of magnesium sulfate hepta-hydrate is 246.48 gram and each
water group that is expelled from the molecule coincides with a decrease in mass of about
7.3%. MgSO4 · 7H2 O is also known as epsom salt and is readily available at reasonable cost.
10
Thermochemical storage materials
Chapter 3
Analysis methods
In this chapter a number of analysis methods are discussed that are useful in characterizing
thermochemical materials. These methods can be utilized to quantify material parameters
or system parameters. The term thermal analysis denotes a variety of measuring methods,
which involve a change in the temperature of the sample investigated. Typical temperature
programs involve linear increase of temperature with time, from now on referred to as constant
heating rate, or periods where the temperature is kept constant for some time, from now on
referred to as isothermal platforms.
3.1
Thermogravimetric Analysis
By a technique called Thermogravimetric Analysis (TGA) the change in sample mass is
analyzed as a function of temperature. The temperature program is prescribed for the furnace.
A schematic overview is shown in Figure 3.1.
As materials are heated, they can lose weight due to drying or chemical reaction, in which
a gas is liberated from the sample. This reduction in mass is registered as a function of
temperature and thus time. A known mass of sample material is placed in a small open
crucible, so that the sample mass is in the order of milligrams. Standard TGA crucibles are
made of platinum or alumina (Al2 O3 ) and have a volume of about 70 mm3 . The sample is
then placed in a furnace on a very accurate balance, where it is subjected to a temperature
program. In the case of salt-hydrates only water vapor is released from the sample. Therefore
the measured mass decrease can be directly converted into the number of moles of water
expelled. The ambient within the system is continuously flushed by a so-called purge gas, in
most cases nitrogen. Typical purge flows are 0 − 200 ml min−1 . There is a second gas flow
present in the apparatus, that is called protective gas flow. The protective gas flows through
the balance housing, thereby prohibiting potentially corrosive gas from the sample chamber
to flow into the balance. The protective gas is typically a dry, inert gas, with a flow of about
20 ml min−1 . Due to the gas flows, the released reaction products are continuously carried
off.
A new technology within TGA analysis is the high resolution measurement. Instead of defining
11
12
Analysis methods
Figure 3.1: A schematic representation of a TGA measurement system, also known as a thermobalance (From
[Gallagher, 1998]). The output signal of the measurement is the mass of the sample, that can be plotted versus
temperature and hence time.
the actual temperature program, the temperature range is set and the temperature program is
determined by a control loop. When no mass reduction is registered, the furnace temperature
is increased at a constant rate. When a differential mass signal is registered however, the
temperature is maintained constant, until the reaction has completed. In this way transition
temperatures can be determined very accurately, without lengthy measurements.
3.2
Differential Thermal Analysis
In Differential Thermal Analysis (DTA) the temperature difference between a sample and
reference sample is measured while the samples are subjected to a temperature program. The
samples are placed in separate crucibles in the same furnace. Generally an empty crucible
is used as the reference. A typical DTA setup can be seen in Figure 3.2. The output signal
from a DTA measurement is the temperature difference between sample and reference sample,
∆TSR = TS − TR . In case of an endothermic process in the measurement sample, the DTA
signal tends towards negative ∆T values. The conversion process causes an increase in ∆TSR
and results in a peak signal, the area which is enclosed by this peak is a measure for the
enthalpy change of the reaction. DTA measurements can be employed for temperatures up
to 1600o C.
A DTA module can be extended with a TGA module, the system is then referred to as a
DTA-TG or STA (Simultaneous Thermal Analysis) device and such a system is shown in
Figure 3.2(b). The system now enables the user to perform a simultaneous measurement of
differential mass and differential temperature. This makes it easy for the user to distinguish
between chemical reaction with change in mass, for instance a dehydration reaction, or without
a change in mass, such as melting of the sample.
13
3.3. DIFFERENTIAL SCANNING CALORIMETRY
(a)
(b)
Figure 3.2: In Figure 3.2(a) a schematic representation of a DTA block measuring system is given (from
[Gallagher, 1998]). A thermocouple is placed in each sample and the differential temperature ∆TSR is given
as output signal. In Figure 3.2(b) a schematic overview is given of the Netzsch STA 409 Luxx, which is a
combined DTA-TG device.
3.3
Differential Scanning Calorimetry
The difference in heat flow to a sample and reference sample as a function of temperature
is measured by Differential Scanning Calorimetry (DSC) [Dean, 1995]. Both the sample and
reference are maintained at nearly the same temperature. The reference sample should have
a well-defined heat capacity over the range of temperatures to be scanned. When a thermal
transition occurs in the sample, more (or less) heat will need to flow to it than to the reference
to maintain both at the same temperature. The difference in heat flow is equal to the amount
of heat absorbed or released during the transition.
In a heat flux DSC, such as schematically shown in Figure 3.3, heat is transferred to the
Ref.
Sample
Constantan
heating disk
Chromel wafer
Alumel wire
Thermocouple junctions
Chromel wire
Figure 3.3: Disk-type heat flux DSC. DSC measures both temperature and heat of transitions or reactions.
sample and reference through a disk which is made of the alloy constantan. In addition
the disk serves as part of the temperature sensing unit. The samples are placed on raised
platforms on the disk. Under each platform there is a chromel (another alloy) wafer. The
junction between the two alloys forms a thermocouple. The signal from these sensors is used
14
Analysis methods
to measure the differential heat flow. Beneath each chromel wafer a second thermocouple is
attached to monitor the temperature. Samples are placed in disposable aluminum sample pans
of high thermal conductivity which can be sealed and are then weighed on a microbalance.
The sample is placed on the sample holder and an empty sample pan serves as reference.
Sample sizes range from 1 to 100 mg. Thin-layer, large-area sample distribution minimizes
thermal gradients and maximizes temperature accuracy and resolution. DSC measurements
can be performed for temperatures up to approximately 700o C.
3.4
X-ray crystallography
X-ray crystallography (also called X-ray diffraction or XRD) is a technique that leads to an
understanding of the material and crystal structure of a substance. In X-ray crystallography
X-rays are passed through a crystal and then caught on a photographic plate resulting in
a pattern of spots or lines, that originate in X-ray diffraction on the atoms in the crystal.
Bragg was the first to pose an explanation for this phenomenon in 1913, which is also called
the Bragg analysis. The X-rays are diffracted by different layers of atoms in the crystal,
leading to constructive interference in some instances but destructive interference in others.
When incoming x-rays with wavelength λ strike a crystal face at an angle θ, those rays that
strike an atom are reflected off at the same angle θ. Because the crystal has multiple layers
the rays can reach multiple depths, thus resulting in different ray-travel distances. By using
trigonometry it can be shown that the extra distance traveled between two consecutive layers
is equal to twice the distance between atomic layers times the sine of the angle θ. The key to
the Bragg analysis is the realization that the rays striking the two layers of atoms are initially
in phase, but can only be in phase after reflection if the extra traveled distance is equal to
an integer number of wavelengths nλ (n = 1, 2, 3...). If the extra traveled distance is not an
integer number of wavelengths, then the reflected rays will be out phase and will cancel each
other. Imposing this requirement on the previously mentioned relationship gives the Bragg
equation:
d=
nλ
2 sin θ
(3.1)
X-ray powder diffraction is a specific example of X-ray crystallography, that is specifically
useful for structural characterization of powdered materials. The output of the measurement
are rings of diffracted intensity as a function of reciprocal lattice units. The method can
be used to quickly identify materials, by comparing the found pattern of powder diffraction
peaks with data from the International Centre for Diffraction Data pattern database. In a
crystallographic study of the dehydration reaction of a salt-hydrate, a sample is placed in
a camera and then subjected to variations in temperature and pressure in order to induce
the desired reactions. When the diffraction pattern is recorded continuously, while either
changing sample temperature or water vapor pressure, phase changes in the material can be
determined.
15
3.5. PHASE DIAGRAM
q
d
(a)
(b)
Figure 3.4: In Figure 3.4(a) a schematic representation of the Bragg analysis is given. Figure 3.4(b) denotes
an example of an X-ray powder diffractogram. The peaks represent positions where the X-ray beam has been
diffracted by the crystal lattice.
3.5
Phase diagram
Phase diagrams are useful for identifying the equilibrium conditions between the thermodynamically distinct phases. Phase diagrams for a single substance, such as water, represent
the lines of equilibrium between the three phases (i.e. solid, liquid and gas) spanned by
temperature and pressure. When a boundary line is crossed by changing either temperature
or pressure, a phase change occurs. Phase diagrams can also be constructed for a system in
which more than one component is present. The composition of the mixture then becomes
an important variable. A binary phase diagram represents the relative concentrations of two
substances against temperature. For salt-hydrates two independent components are present;
the crystalline salt and water. For such materials a composition versus temperature diagram
for a constant pressure (mostly 1 atm.), is a very insightful tool. An example is shown in
Figure 3.6. A vertical line drawn in a phase diagram is called an isopleth (Greek: equal abundance), which is a change to the system that does not affect the overall composition. When
a phase diagram is not at hand in literature, it is possible to construct one from experimental data. First the transitions within the material should be determined, by means of TGA
(or preferably DTA-TG) measurements in combination with XRD measurements. When the
stable states of the material are known, the melting temperatures of each state should be
determined by using DTA measurements and possibly visual melting point determinations.
3.6
Vapor pressure measurements
The equilibrium water vapor pressure of a salt-hydrate as a function of its temperature can be
determined by using a pressure sensor that is connected to a sealed glass container filled with
a sample of the salt-hydrate. The container is placed in a temperature-controlled bath and
the sample temperature is monitored, for instance with a thermocouple. The temperature of
the bath is raised 10 K after which the sample temperature is left to stabilize. This takes
approximately 15 minutes. When the system is thermally stabilized, the equilibrium vapor
16
Analysis methods
Figure 3.5: Phase diagram of the Na2 S·H2 O system and measured transition temperatures. Phase denominations: V: vapor, L: H2 O based solution, C1, C2 and C3 are the crystalline phases Na2 S·9H2 O, Na2 S·5H2 O and
Na2 S·2H2 O respectively. Graph obtained from [De Boer, 2002].
pressure is read. In this way a temperature range can be scanned and pressure readings are
recorded for both heating and cooling. The measurement should be performed for each stable
state of the material. The results of the pressure measurements can be used to calculate the
change in enthalpy and entropy of the dehydration reactions, by using Equation 2.6. A vapor
pressure measurement system was not available for the current research, an example from
[De Boer, 2002] is shown here.
Figure 3.6: Constructed equilibrium curves of temperature versus water vapor pressure. Curves are given for
water, Na2 S·9H2 O, Na2 S·5H2 O and Na2 S·2H2 O. Graph originates from [De Boer, 2002].
17
3.7. SCANNING ELECTRON MICROSCOPY
3.7
Scanning Electron Microscopy
The scanning electron microscope is designed for the visualization of surfaces of solid objects
at very high resolution. An electron gun produces electrons and accelerates then to an energy
between about 2 and 40 kVe [Goodhew, 1988]. Two or three condensor lenses then focus
the electron beam into a very fine focal spot. The diameter of which can be as small as
2 − 10 nm. The beam passes through pairs of scanning coils in the objective lens, which
deflect the beam in a raster fashion over a rectangular area of the surface of the specimen.
Interactions of the electrons with the surface lead to the subsequent emission of electrons.
Low energy secondary electrons are detected and the resulting signal is rendered into a two
dimensional intensity distribution. The construction of this 2D image depends on the rasterscanned primary beam. The brightness of the signal depends on the number of secondary
electrons reaching the detector. A flat surface results in a uniform intensity distribution.
When the angle of incidence increases, for instance on edges or steep surfaces on the surface
of the specimen, a higher intensity of secondary electrons is detected. Therefore edges and
steep surfaces are brighter than flat surfaces, resulting in images with a well-defined, three
dimensional appearance.
(a)
(b)
Figure 3.7: Comparison between scanning electron microscope image (left) and optical microscope image
(right). The SEM image is in focus over the full depth of the specimen and has high resolution.
Apart from the good spatial resolution, one of the major advantages of the scanning electron
microscope over an optical microscope is the large depth of field. For typical operating
conditions, for a magnification of 1000×, the depth of field for an electron microscope and an
optical microscope are 40 and 1µm respectively. A downside of using SEM to produce images
of salt-hydrates, is the fact that the images are made under vacuum. During the evacuation
of the apparatus, the salt-hydrate expels its water, therefore only images of the anhydrous
component can be made in this setup. There is a technology called environmental SEM or
ESEM which allows the examination of specimens surrounded by a gaseous environment.
This would be a useful tool for studying salt-hydrates, however it was not available for the
current research.
18
Analysis methods
Chapter 4
Experimental results
A number of the analysis methods that are described in Chapter 3 are used for characterization of magnesium sulfate hepta-hydrate. The results presented here are obtained with TGA,
DTA-TG, DSC and SEM. The characterization of the material is divided into two parts that
each have their own significance. First some experimental results are discussed that characterize the material itself, such as the dehydration behavior of the salt-hydrate, the determination
of material parameters and the crystal structure, this part is called material characterization.
Secondly the focus lies on experiments which determine parameters that check the applicability of magnesium sulfate hepta-hydrate as a thermochemical storage material, this section
is called system characterization. An extensive list of the applied experimental conditions is
given in Appendix B.
4.1
Material characterization
The goal of the first set of experiments is to obtain the parameters that determine the specific thermodynamical behavior of the material. These parameters are used as input in the
numerical model. In chemical conversion there are three mechanisms that can affect the propagation of the reaction; the thermal transport, the mass transport and the reaction kinetics.
In Chapter 5 these parameters and their influence will be discussed more thoroughly. For
the purpose of characterizing the material properties, preferably all individual grains of a
sample are submitted to the same thermodynamic constraints, as they then react at the same
rate. Therefore small samples should be employed, in the order of 10 milligram or less. Care
should be given however to the trade-off between signal strength that is required for high
experimental resolution and the desire to employ small samples and low heating rates. For
quantitative studies in the measurement devices which were used for this research a sample
of at least 1 mg is required.
19
20
4.1.1
Experimental results
Stable hydrates
As described before in Section 2.2, literature data on the stable hydrates that appear in the
dehydration of MgSO4 · 7H2 O is not consistent. Therefore some experiments are performed to
determine the dehydration behavior of the salt-hydrate and the stable phases that are formed
during dehydration. As stated in Section 2.2 each dehydration step corresponds to a decrease
in mass of about 7.3%. It should be noted that the mass decrease that is measured in an
experiment only relates to a mean decrease in the sample; it is possible that only part of the
sample transformed and part of the sample remains unchanged. The actual composition of
a sample can best be checked by means of a XRD measurement, in the present research this
measurement method was not available however.
By means of a high resolution thermo-gravimetric measurement (described in Section 3.1)
performed on a TA Instruments TGA Q500 the dehydration behavior was studied. A sample
of 12 milligram was placed in an open crucible and the purge gas was saturated with water
vapor at room temperature (i.e. a vapor pressure of approximately 30 mbar). The results can
be found in Figure 4.1, in which the upper panel shows the sample mass versus temperature
and the lower panel depicts the derived mass versus temperature. In this experiment it is
Mass [%]
100
90
80
70
60
50
50
100
150
50
100
150
200
250
300
350
400
250
300
350
400
o
Derived mass [%/ C]
4
3
2
1
0
200
o
Temperature [ C]
Figure 4.1: Dehydration behavior of MgSO4 · 7H2 O measured by high resolution measurement on a TGA
apparatus. The upper panel shows the mass signal as a function of temperature and in the lower panel the
temperature derivative of the mass signal [ % o C−1 ] is represented as a function of temperature.
seen that MgSO4 · 7H2 O first loses approximately 3.5 water groups at a temperature of about
64o C and subsequently gradually expels about 2.5 groups. A last transition is seen at a
temperature of about 270o C where 0.5 group is released. The constant decrease in mass in
the temperature span of 70 − 270o C is a result of the low relative humidity of the purge gas.
The control loop in a high resolution method is set to a certain sensitivity. The mass decrease
in this temperature span is gradual, and therefore the heating is not paused.
For the experimentator it is desirable to perform experiments as fast as possible, which
in thermal analysis experiments means employing high heating rates. A large drawback to
employing high heating rates however is the increased chance of local melt. Salt-hydrates with
21
4.1. MATERIAL CHARACTERIZATION
a large fraction of bound water are characterized by low melting temperatures. Dehydration
reactions take time, due to the previously mentioned phenomena of reaction kinetics and
limited mass and heat transfer. In this way, if a sample is heated at too high rate, the
dehydration reaction is not completed before the melting temperature is reached. Melting
is an undesired effect because it slows down the reaction, destroys the crystal structure and
reinforces the formation of an amorphous crystal structure, which is unable to bind water,
thereby diminishing the cyclability of the material. Local melt can be detected in a DTA-TG
measurement as an endothermic peak in differential temperature signal, without a change in
mass. An example of such a measurement is shown in Figure 4.3(a).
Dehydration experiments are also performed for constant heating rates, varying from 1 −
50 K min−1 . For higher heating rates the decomposition temperatures shift to higher values,
mainly due to reaction kinetics. Besides that, the components that are formed within the
decomposition can alter, as can be seen in Figure 4.2. The experiments from Figure 4.2(a)
H = 1 K/min
H = 10 K/min
H = 50 K/min
95
90
90
85
85
80
80
75
70
75
70
65
65
60
60
55
55
50
H = 1 K/min
H = 5 K/min
H = 10 K/min
H = 20 K/min
100
95
Mass [%]
Mass [%]
100
50
50
100
150
200
250
Temperature [oC]
(a)
300
350
400
40
60
80
100
120
140
160
180
200
Temperature [oC]
(b)
Figure 4.2: Dehydration behavior for different heating rates, shown here as the mass versus temperature.
Figure 4.2(a) shows results obtained by the TGA device at TU/e, Figure 4.2(b) shows the results from the
TGA device at ECN. The temperature on the left hand side is twice as large as that on the right hand side.
are performed at TU/e on the TGA device that is equipped with an auto-sampler. An
auto-sampler is a carousel device in which up to 50 samples can be placed. Samples for
successive measurements are prepared at once and the auto-sampler then allows around-theclock sample analysis, without supervision of an experimentator. The samples which are in
queue for measurement are in open contact with the ambient. The experiments from Figure
4.2(b) are performed on the TGA device at ECN. All measurements are performed on samples
of about 10 mg and under a purge flow with a vapor pressure of 30 mbar. When comparing
the results for a heating rate of 1 K min−1 (i.e. the solid line of Figure 4.2(a) and the grey
solid line of Figure 4.2(b)) the first observation is that the measurement that is performed at
ECN displays a transition around T = 40o C which is not found in the measurement which
was performed with the auto-sampler, where it was the last in queue. Moreover, the sample
mass at the end of the measurement with auto-sampler is higher than expected. This leads
to conclude that the initial composition of the material was not the hepta-hydrate, but the
hexa-hydrate. The transition of hexa-hydrate to the anhydrous component corresponds to
22
Experimental results
a decrease in mass of 47.3%, which is exactly the reduction found in the experiment. The
experimental result for this heating rate should be scaled and shifted downward in order to
be able to compare it with the other measurements. The end composition is then found to be
the same for all the measurements of Figure 4.2(a), namely the anhydrous component. The
results for a heating rate of 10 K min−1 are consistent and the overall trend of the curves as
a function of heating rate also corresponds well.
4.1.2
Material parameters
There are a number of parameters which determine the thermodynamical behavior of a material. These parameters, such as the thermal conductivity λ, the heat capacity cp and density
ρ, should be obtained, so that they can be used in as input for the model. Most material
parameters can be obtained from literature. Where bias in data is found, it is advisable to
determine parameters experimentally.
Values on material density are consistent in literature; [Ullmann, 2000] and [Washburn, 1930]
report densities for the different states of the salt-hydrate, ranging from 1680 kg m−3 for
MgSO4 · 7H2 O to 2660 kg m−3 for MgSO4 . These values and those of the intermediate phases
are given in Table 4.1. The density of the salt-hydrate is found to increase when the amount
of water, and therefore the molar mass, decreases. The dehydration of the hepta-hydrate to
the anhydrous component corresponds to a decrease in mass of 51% and a density increase
of 58%. Combination of these numbers, yields that the material volume after dehydration
is only a third of the original volume. In experiments a volumetric decrease was observed,
however it was not determined quantitatively. The density of an actual layer of salt-hydrate
is lower than the material density, due to the void fraction between the individual grains
of the material. This void fraction is also denoted macro-porosity and can be determined
by filling a cup with a known volume with the salt-hydrate, then measuring the mass and
equating the material density with the actual density. A crucible of 70 mm3 was completely
filled with 58 mg of MgSO4 · 7H2 O, which has a reported density of 1680 kg m−3 , yielding a
macro-porosity of 0.5.
Table 4.1: Material parameters from literature
MgSO4 · 7H2 O
MgSO4 · 6H2 O
MgSO4 · 4H2 O
MgSO4 · 2H2 O
MgSO4 · 1H2 O
MgSO4
M
[ g mol−1 ]
246.48
228.46
192.43
156.40
138.38
120.37
ρ
[ kg m−3 ]
1680
1750
2010
2570
2660
cp
[ J kg−1 K−1 ]
1546
1525
1305
1124
1047
800
λ
[ Wm−1 K−1 ]
0.48
H
[ J mol−1 ]
3364
3062
2471
1871
1575
1260
Little information was found on the thermal conductivity of the salt-hydrate. A value of
0.48 Wm−1 K−1 was found for MgSO4 · 7H2 O in [Washburn, 1930]. Because there was no
measurement device available to determine the thermal conductivity, this value is used in the
23
4.1. MATERIAL CHARACTERIZATION
model and assumed that it is the same for each phase.
The heat capacity of magnesium sulfate is given in both [Ullmann, 2000] and [Washburn, 1930]
and reported to be 1.5 J g−1 K−1 for MgSO4 · 7H2 O and 1 J g−1 K−1 for MgSO4 ·H2 O. The
heat capacity of the hepta-hydrate is also determined experimentally by means of a DSC
experiment, in which a small sample was placed in a closed aluminum sample pan. The
sample is then heated at a heating rate of 5 K min−1 . When no reaction takes place in the
sample and it is merely heating up, the heat flow to the sample that is measured at that
time, correlates with the heat capacity of the material. In this particular experiment a heat
flow of 0.24 mW is measured and a sample mass of 1.876 mg was used. The heat flow per
milligram is then determined and divided through the heating rate per second. This yield a
heat capacity of 1.5 J g−1 K−1 , which corresponds with the values found in literature.
The melting temperature of a material can be determined by a coupled DTA-TG measurement. It is measured as an endothermic peak in heat flow, that is not accompanied by a
decrease in mass. This measurement was performed on a sample of 10 mg which was heated
at a rate of 20 K min−1 over a temperature range of 25 to 325 o C. The result, shown in
Figure 4.3(a), displays quite a large peak in DSC signal around T = 60 o C that is not accompanied by a differential mass signal. The exact melting point is then determined with a DSC
2
1.8
0
1.6
1.4
Heat flux [ µV/mg ]
−5
−10
1.2
1
0.8
0.6
−15
0.4
DSC−signal
TG−signal
0.2
−20
50
100
150
200
o
Temperature [ C]
(a)
250
300
0
40
45
50
55
60
65
Temperature [ oC ]
(b)
Figure 4.3: Results from the melting experiment. Figure 4.3(a) shows the DTA-TG result, where the solid line
represents the DSC signal (which is enlarged two times for scaling) and the dashed line the TG signal. Figure
4.3(b) shows the result from the DSC measurement.
measurement where a sample is placed in a sealed aluminum pan, containing little empty
space. In this way the melting point of MgSO4 · 7H2 O was determined to be 52.5 o C, which
is shown in Figure 4.3(b). After the measurement it was determined that no mass had left
the sample pan, which confirms that no dehydration took place and the determined melting
temperature is indeed that of the hepta-hydrate.
24
4.1.3
Experimental results
Surface structure
By means of SEM the surface of grains of the salt-hydrate are studied and the effect of thermal
dehydration on the surface of the grains is visualized. In Figure 4.4 two clusters of grains are
(a)
(b)
Figure 4.4: SEM images of a sample of magnesium sulfate, with a grain size of 200 − 500 µm (magnification
approximately 150×). The sample on the left was not treated before, the sample in Figure 4.4(b) was thermally
dehydrated.
depicted. The sample on the right hand side was thermally dehydrated before, the sample on
the left did not undergo previous thermal treatment, but expelled its water in the electron
microscope under influence of the deep vacuum. The surface of the thermally dehydrated
grains has obtained some irregularities, whereas the sample on the left has a smooth surface.
The irregular surface is considered to be a visual indication of material deterioration. In
order to validate this statement, the grain surface was also studied for a sample that was
melted in a DSC measurement, as is already elucidated in Subsection 4.1.2. The results are
shown in Figure 4.5. A first observation is that it is now difficult to distinguish the individual
Figure 4.5: SEM image of a sample that was melted (magnification 131×). The original sample consisted of
grains with a diameter of 200 − 500 µm.
grains. The surface of the material has changed dramatically, and the structures that are
formed resemble the irregularities which were found in Figure 4.4(b). This leads to believe
25
4.1. MATERIAL CHARACTERIZATION
that to some extent melting of the sample, visualized in Figure 4.4(b), has occurred during
the thermal dehydration of the sample.
4.1.4
Reaction Kinetics
The kinetics of first order chemical reactions are described by the Arrhenius equation. The
constants, i.e. the frequency factor A and activation energy E, of this equation can be
deduced from experimental measurements. The calculation method is based on the fact
that the maximum yield of products from a conversion reaction occurs at the peak of the
differential mass signal. The corresponding temperature is called Tmax . For higher heating
rates, the conversion peak shifts to higher Tmax values. This shift can then be used to calculate
the Arrhenius parameters. An explanation of the exact calculation method can be found in
Appendix A.
0.01
H = 5 K/min
H = 10 K/min
H = 20 K/min
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
40
60
80
100
120
140
160
180
200
Temperature [deg C]
Figure 4.6: Differential mass signal versus temperature for three different heating rates. The peak temperature
shifts to a higher value for higher heating rate.
Dehydration experiments are performed for three different constant heating rates. The results are plotted in Figure 4.6, where the solid line represents a heating rate of 5 K min−1 ,
the dashed line represents 10 K min−1 and the dash-dotted line is 20 K min−1 . The curves
represent the differential mass signal as a function of temperature. The peak shift to higher
temperatures for higher heating rate that was expected, is found in the experiment. For the
lowest heating rate a small clear peak at low temperature is found, that is hardly distinguishable in the higher heating rate experiments. The peak shape of the large peak differs for the
two higher heating rates in comparison to the smooth shape of the lowest heating rate. The
highest point of each peak is used here to determine the kinetic parameters. Furthermore an
additional peak is found at a temperature of about 150o C for the two high heating rates.
The small first peak, that appears around 50o C yields kinetic values that are not plausible;
a frequency factor of 1 · 1024 s−1 is found. The second and larger peak however provides
good results. For the lowest heating rate, the maximum yield for this peak is found at
26
Experimental results
Tmax = 90o C. For a heating rate of 10 K min−1 it is found at Tmax = 113o C and for a heating
rate of 20 K min−1 one finds Tmax = 132o C. The Arrhenius parameters are then calculated
and lead to an activation energy of approximately 55 kJ mol−1 and a frequency factor of about
1.67 · 105 s−1 . In [Ruiz, 2007] kinetic parameters for the dehydration of magnesium sulfate
hepta-hydrate are determined. Although the parameters are not regarded to be constant,
over a large part of the conversion they are. The values over this range are equal to the values
presented here.
4.2
System characterization
When the material itself is characterized, the influence of different operating conditions on
the progress of the reactions is investigated and these results are compared with the results
obtained from the numerical continuum model. Then it is possible to set boundaries for
employment of salt-hydrates in a heat storage system. For some of these experiments larger
samples are employed, so that transient effects in heat and mass transfer over the layer of
material become significant. Most of the experiments are performed on a DTA-TG apparatus,
in order to obtain information about both the heat flow to the sample and the mass of the
sample.
4.2.1
Grain size
For small grains the temperature distribution within each individual grain is stated to be
isothermal. When larger grains are used, the isothermal assumption is no longer valid and
mass transport may also become a limiting factor. A difference is expected that case between
experimental and numerical results, because the numerical model is a continuum model, and
therefore transient effects on the smaller scale are not taken into account. The grain size is
expected to influence the progress of the reaction. The original sample material incorporates
a wide distribution of grain sizes. By means of sieving it is possible to isolate smaller ranges
of grain size and by means of grinding smaller grain sizes can be attained. Sieving showed
that the largest fraction of sample had a grain size of 200 − 500µm. Only a very small fraction
of grain size 500 − 600µm was found and a substantial fraction of the sample consisted of a
grain size smaller than 200µm. An experiment was performed where a sample of about 12 mg
was heated at a constant rate of 10 K min−1 in a temperature range of 25 − 325 o C. The
measurement is performed for equal sample mass on two different grain sizes: 200 − 500 µm
and 38 − 100 µm, the results of which are depicted in Figure 4.7. It is seen that the influence
of grain size on the propagation of the reaction is strongest in the temperature range of
50 − 200 o C. The end product for both experiments is equal, but the timescale of the reaction
is larger for larger grain size.
4.2.2
Layer thickness
For increased layer thickness, temperature and mass transport through the adsorbent bed may
become a problem and transient effects become significant. In order to research this effect an
27
4.2. SYSTEM CHARACTERIZATION
38−100 mu
200−500 mu
100
95
90
Mass [%]
85
80
75
70
65
60
55
50
50
100
150
200
250
300
Temperature [oC]
Figure 4.7: Dehydration behavior for different grain size, here represented as mass versus temperature. The
straight line represents a small grain size of 38 − 100 µm and the dashed line is a grain size of 200 − 500 µm.
experiment was performed, in which the sample is subjected to two consecutive isothermal
platforms, the first at 45 o C and the second at 150 o C. Between the platforms a high heating
rate was used of 30 K min−1 . This situation was chosen because it simulates actual operating
conditions, in which a valve is opened instantly and the heat exchanger starts to heat up.
The maximum temperature is set to the yield temperature of a solar vacuum collector. Both
samples that are used have a grain size of 38 − 100 µm, the mass of the small sample was
11.4 mg and the large sample was 38.2 mg. In Figure 4.8 the sample mass is presented versus
time. The graph shows a clear influence of layer thickness on the propagation of the reaction.
small sample
large sample
100
95
90
Mass [%]
85
80
75
70
65
60
55
0
5
10
15
20
25
Time [min]
Figure 4.8: The sample mass as a function of time for different layer thickness; the large sample, represented
by the dashed line is approximately three times larger than the small sample, represented by the solid line.
The end product in both runs is equal, but for the larger sample the timescale increases. The
small sample reaches its end product approximately within 5 minutes, whereas the larger
sample reaches its end composition some minutes later. Towards the end of the conversion,
28
Experimental results
when a reduction of sample mass of approximately 27% is realized, the difference between
the conversion curves increases. Overall the result implies that for very large layer thickness
the propagation of the reaction decelerates significantly, which sets limits to the maximum
adsorbent bed thickness of a future system.
4.2.3
Vapor pressure
As the system is mono-variant, the equilibrium of the reaction is determined by a combination
of temperature and vapor pressure. For increased vapor pressure in the ambient, transition
temperatures are stated to shift to higher values. The purge gas that is used in the experiments can be saturated with water vapor by bubbling the gas through water at a constant
temperature. The applied vapor pressure can be calculated from the Clausius Clapeyron
equation (Equation 2.6). For water of 24o C a water vapor pressure of 30 mbar is applied to
the gas. The flows of both purge and protective gas through the apparatus can be adjusted in
order to vary the water vapor content of the total gas flow over the sample. As stated before,
the protective flow needs to be a dry gas and is set at a constant flow of 20 ml min−1 . The
purge gas is saturated with water vapor at room temperature in the manner discussed above.
The purge flow can be set to values between 20 and 160 ml min−1 . In this way a water vapor
content of 2.5 · 10−5 to 2 · 10−4 mol min−1 at a vapor pressure of respectively 15 to 26 mbar
is realized.
An experiment is performed on samples of 10 mg, which are heated at a constant rate of
5 K min−1 for three different purge flows. In Figure 4.9 the result of the experiment is
0.01
15 mbar
24 mbar
26 mbar
0.009
Differentiel mass [mg/s]
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Time [s]
Figure 4.9: Dehydration experiment for three different purge flow vapor pressures, shown here for the differential sample mass versus time. The solid line represents a vapor pressure of 15 mbar, the dashed line a vapor
pressure of 24 mbar and the dash-dotted line a vapor pressure of 26 mbar.
shown. The graph shows that for lower water vapor content in the purge flow, the dehydration
reactions take place at lower temperatures and the first dehydration peak is significantly
larger. The water vapor content of the ambient is shown to limit the propagation of the
conversion reaction. For a future system it is therefore important to realize a continuous
29
4.2. SYSTEM CHARACTERIZATION
removal of water vapor from the ambient, for instance by a continuous gas flow with low
vapor pressure over the adsorbent bed.
4.2.4
Cyclability
If the material is ever to be used in a heat storage system, it is of vital importance that the
reaction is reversible and can be performed a number of times, this property is called cyclability. In Figure 4.10 the result of a cycling experiment is presented. A sample of 10.7 mg is first
11
run 1
run 2
run 3
10.5
10
Mass [mg]
9.5
9
8.5
8
7.5
7
6.5
6
0
5
10
15
20
25
Time [min]
Figure 4.10: Dehydration experiment performed three times successively on one sample, displayed here is the
sample mass versus time.
heated to 45o C at a heating rate of 10 K min−1 , then kept isothermal at that temperature
for 5 minutes and then heated to 150o C at a heating rate of 30 K min−1 and kept isothermal at that temperature for 15 minutes. After the measurement, the sample was allowed to
rehydrate under atmospheric conditions for 24 hours. The sample mass was constant for the
last 3 hours, which indicates that the rehydration process came to an end. This cycle was
performed three times on the same sample. In the first run the sample mass decreased from
10.7 mg to 6.2 mg, which is a reduction of approximately 42%. The material then does not
rehydrate fully, but stabilizes at a sample mass of 9.6 mg (i.e. approximately 90% of the
original sample mass). After the second dehydration, the sample mass is 6.3 mg, which is
almost equal to the sample mass after the first dehydration. In the following rehydration less
water is taken up than the first time and the sample settles to a sample mass of 7.6 mg (i.e.
approximately 71% of the original sample mass). After the last dehydration the sample has
a mass of 6.3 mg, which is again almost equal to the sample mass after the first dehydration. This result implies that the cyclability of the material is poor, however more research
is necessary to justify this first conclusion.
Another cyclability experiment is performed in a TGA setup under gas with a water vapor
pressure of 24 mbar. A sample of 9.6 mg is first heated from 25 − 200o C at a heating rate of
1 K min−1 , and subsequently cooled down at the same rate to 25o C. Finally the temperature
was kept constant at 25o C for 200 minutes. The result of this experiment is depicted in
30
Experimental results
100
95
90
Mass [%]
85
80
75
70
65
60
55
50
0
100
200
300
400
500
Time [min]
Figure 4.11: Dehydration- rehydration experiment performed in the presence of water vapor. The sample mass
is represented versus time.
Figure 4.11. It shows that first a dehydration reaction takes place, yielding a mass reduction
of about 47%. Over a large part of the cooling period no change in mass is found, until
at sufficiently low temperature the rehydration reaction is initiated. During the isothermal
period a steady mass increase is observed. In the total time of 200 minutes, 25% of mass is
taken up again by the sample, which corresponds with an uptake of approximately 2.4 mg.
As the exothermic hydration reaction propagates slowly, no accompanying heat effect was
registered. This method gives more insight into the rehydration kinetics than the previously
mentioned experiment. However, this last measurement is very time consuming, which can
cause problems regarding measurement device occupation.
4.3
Experimental resolution
Before processing experimental results it is important to check the resolution of the measurement methods that were used to obtain the results. Experimental measurements can be very
sensitive to operating conditions and may inhibit large errors. Therefore some research was
done into the quality of the measurements that are performed.
For thermal analysis measurements a temperature trajectory is given as input. For the TGA
and DTA-TG setup a check was performed between the prescribed temperature and the temperature realized in the experiment. The result is shown in Figure 4.12. Both measurements
are performed with an empty alumina pan. The temperature program for the TGA measurement started with heating from 25 − 40 o C at a heating rate of 40 K min−1 , followed
by an isothermal platform at T = 40 o C for 5 minutes. Then the sample was heated from
40 − 270 o C at a heating rate of 40 K min−1 and kept isothermal at the end temperature
for 15 minutes. The temperature program for the DTA-TG device started with heating from
25−45 o C, followed by an isothermal part for 5 minutes. The sample was subsequently heated
from 45−150 o C at a heating rate of 30 K min−1 and kept isothermal at the end temperature
for 15 minutes. The realized temperature which is shown in Figure 4.12 is measured by a
31
4.3. EXPERIMENTAL RESOLUTION
TGA
300
precribed
realized
250
200
150
o
Temperature [ C]
100
50
0
0
5
10
15
20
25
30
20
25
30
DTA
200
150
100
50
0
0
5
10
15
Time [min]
Figure 4.12: Comparison between prescribed temperature and realized temperature. Figure (a) is a measurement on the TGA apparatus and Figure (b) is a measurement performed on the DTA-TG.
thermocouple that is placed underneath the sample.
For lower temperatures both setups show a large difference between prescribed and realized
temperature, in both cases an overshoot in temperature is realized when an isothermal platform is prescribed. At higher temperatures the TGA setup clearly performs better than the
DTA-TG setup. An overshoot in temperature can cause problems if an isothermal platform
is desired close to a melting temperature. When processing experimental data, one should
bare in mind that the sample temperature can differ significantly from the prescribed temperature. Results should therefore be plotted for sample temperature rather than prescribed
temperature.
For TGA the repeatability of the measurements was also verified. A constant heating rate
experiment was performed five times on samples of approximately equal size. The result of
these measurements is given in Figure 4.13. A sample of about 10 mg is heated at a constant
heating rate of 10 K min−1 from 25 − 325o C. In the graph the sample mass is plotted versus
temperature. The black line is the calculated mean sample mass of the five experiments
and the grey dotted line represents the standard deviation of the experimental data. The
measurement is characterized by small standard deviation, however locally a larger deviation
is found around 120o C. The occurrence of this increased deviation coincides with a large
decrease in sample mass. Overall it is concluded that the repeatability of the measurements
is high.
32
Experimental results
100
95
90
Mass [%]
85
80
75
70
65
60
55
50
50
100
150
200
250
300
Temperature [oC]
Figure 4.13: Reproducability measurement performed on a Mettler Toledo TGA apparatus. Depicted here is
the sample mass versus temperature. The black line represents the mean value of the five measurements, the
grey dotted line is the standard deviation.
Chapter 5
Modeling
In order to check the viability of the use of thermochemical materials in an energy storage
system, a model has been made. This model gives a better phenomenological understanding
of the processes present and makes it possible to assign the most important mechanisms and
phenomena. Model parameters can be obtained from literature or by performing simple experiments. The model will then be validated by performing a number of case-experiments.
The better insight in the process can help detect its drawbacks and set boundary conditions
for a future operational system, regarding both design and operating conditions. The possibility for prediction of system behavior at different design and operating conditions, will
substantially lower the need for experimental trials.
5.1
Introductory theory on heat and mass transfer
A thermodynamic system can be described by the three (local) conservation equations for
mass, momentum and heat. The equations are presented here in general form and are obtained
from [Schram, 1994]. The first conservation equation is that of mass, also called the continuity
equation:
∂ρ
+ ∇ · (ρv̄) = 0
(5.1)
∂t
where ρ is the density and v̄ the velocity vector. The conservation of momentum for incompressible media is written:
1
∂v̄
+ (v̄ · ∇)v̄ = − ∇p + ν∇2 v̄ + f¯
∂t
ρ
(5.2)
Here p is the pressure, ν the kinematic viscosity and f¯ a volume force, due to an externally
applied force, e.g. the gravitational acceleration. The conservation of energy is given here in
the so-called temperature notation:
∂
(ρcp T ) + ∇ · (ρcp v̄T ) = ∇ · (λ ∇T ) + Q
(5.3)
∂t
where T is the temperature, cp the heat capacity, λ the thermal conductivity and Q a source
term. In the case of a chemical reacting system, the source term incorporates an enthalpy
33
34
Modeling
term, representing the energy production or consumption of the chemical reaction. When a
chemical system is described that consists of multiple components, the conservation of each
component can also be stated. The total of balance equations can then be extended with an
equation for the conservation of chemical species:
∂Yi
1
+ v̄ · ∇Yi = ∇ · (D ∇Yi ) + ṁ′′′
∂t
ρ i
(5.4)
where Yi is the concentration of each species i, D represents the diffusion coefficient and ṁ′′′
i
is a mass source term related to chemical reaction.
In the conservation equations of momentum, energy and species, the first term on the lefthand side is the change in time and the second term is the convective term (transport due to
macroscopic movement). On the right-hand side the first term is a diffusion term (transport
due to microscopic movement). The remaining terms are different source terms, except for
the momentum equation, in which the pressure gradient also appears.
5.2
Grain models
Before looking into system-behavior of systems employing thermochemical reactions, it is
useful to study the reactions and their kinetics on particle scale first. The reactions that occur
in the dehydration or rehydration of crystalline salt-hydrates can be more generally called
solid-gas reactions. Similar reaction processes are handled in biomass conversion research.
An example is the devolatilization step in the combustion of wood particles. During this step
wood is converted into char, gas and sometimes tar. Also the combustion process of a char
particle in oxygen is an example of a solid-gas reaction. These reaction processes have been
studied thoroughly over the past years, thus much literature is available. In Figure 5.1 three
Figure 5.1: Overview of simplified particle conversion models.
basic particle conversion models are depicted. Solid-gas reactions on particle level can best be
described by the spherical shrinking core reaction model. The spherical shrinking core model
assumes that in a reacting spherical particle of radius r0 , there exists an unreacted shrinking
35
5.2. GRAIN MODELS
core of decreasing radius r surrounded by a growing layer of products. In a reacting particle
three possible limitations for the progress of the reaction can be assigned:
• Mass transfer through the layer of products
• Heat transfer through the layer of products
• Chemical reaction
[Stanish, 1983] performed pioneering work on the dehydration behavior of crystalline potassium carbonate hydrate (K2 CO3 ·1.5H2 O). Through experiments and analysis he showed that
the spherical shrinking core model can be applied to this salt-hydrate. Experiments were
performed both in the presence of water vapor as well as in vacuum. He showed that the
temperature gradient over the product layer of a particle with an initial radius smaller than 1
mm is about 0.008 K and thus negligible. The pseudo-steady state assumption, which implies
that temperature transients are rapid and heat transport is considered the only source of energy for dehydration, is therefore valid. His study also showed mass diffusional resistances to
be negligible for slow dehydrations of small particles (125 − 149 µm), but significant for larger
particles or high dehydration rates. The interface velocity of the propagating front ṙ ≡ dr/dt
was considered to be constant. The activation energy was calculated to be 91 kJ mol−1 ,
obtained from the slope of a plot of experimental results in Arrhenius coordinates.
Research on the conversion of wood particles was executed by [Di Blasi, 1996]. He states
that biomass pyrolysis can be described through a primary and a secondary stage. Wood
k1
Wood
k2
k3
Gas
Tar
Char
k4
k5
Gas
Char
Figure 5.2: Schematic overview of the wood conversion process.
undergoes thermal degradation according to primary reactions giving as products gas, char
and tar. Tar may undergo secondary reactions to char and gas. To make a comparison
with the current research it suffices to take into account the primary conversion of wood
into char and gas. The formation of tar is thus discarded. As the relative amount of end
products may vary with conversion parameters, two kinetic parameters are used to describe
the conversion; one for each component. [Di Blasi, 1996] introduces a one dimensional model
of a large (τ = 0.025 m) single particle, which is regarded as a slab. The particle is subjected
to a radiative heat flux and degrades thermally. Conservation of mass is stated for each
component. On the right hand side of Equation 5.1 a chemical reaction source term appears,
which has a negative value for the species that is consumed in the reaction and positive value
for the species formed. For the solid-phase species wood it reads:
∂ρw
= −(k1 + k3 )ρw
∂t
(5.5)
The parameters k1 and k3 denote the chemical reaction rate and are deducted from an Arrhenius equation. Convective terms only appear in the conservation equations of the gas phase.
36
Modeling
The source term in the conservation of energy is determined by the enthalpy change of the
reaction and the chemical reaction source term.
∂2T ∂
∂T
ρcp T ∂V
(T ρcp ) +
(ρg cpg u) = λ
+ (k1 + k3 )ρw ∆H −
∂t
∂x
∂x2
V ∂t
(5.6)
The x direction corresponds to the thickness of the slab of biomass. The convective velocity
of the gas phase is calculated by the Darcy equation, which reads:
u=−
κ ∂p
µ ∂x
(5.7)
The pressure term that appears in the Darcy equation is calculated from the ideal gas law in
the following way:
ρg RT
(5.8)
p=
Mg
where Mg is the molar mass of the gas and R the gas constant. The expression ρcp is the total
heat capacity of the control volume and it is composed of three portions; wood, char and gas.
[Di Blasi, 1996] also accounts for shrinkage of the particle, as it was found experimentally
that wood particles experience reductions in particle size up to 60%. The volume occupied
by the solid is assumed to decrease linearly with the wood mass and increase with the char
mass, by a chosen shrinkage factor, α, as devolitalization takes place:
VS
mw
αmc
=
+
Vw0
mw0 mw0
(5.9)
Where Vw0 is the initial effective solid volume. It was found that dehydration of MgSO4 ·7H2 O
induces volumetric reduction. This was not included in the current model however. DiBlasi
performed numerical simulations with the described model, in order to simulate the effects
of the shrinkage level, the intensity of the heat flux and the effect of different sets of kinetic
data. A qualitative agreement between model predictions and experimental data was found.
However, the properties of char and partially charred wood are very poorly known, yielding
poor quantitative results. The conclusion of the research is that further study is needed to
estimate reliable kinetic data (on semi-global scale) and to investigate the dependence of
physical properties on temperature and solid composition. The uncertainty on the physical
properties in the current research is smaller, therefore the method described by DiBlasi is
considered to be valuable.
5.3
Layer models
Reversible solid-gas reactions can be applied well in adsorption heat pumps and storage systems. These reversible reactions take place in fixed-bed reactors, effected by constraints of
temperature and pressure. A lot of modeling work has been performed on adsorption heat
pump systems, mostly with the working pair of zeolite and water (vapor) or silicagel and water. These models are very insightful as the function of the systems described is similar to the
current situation, although the principles behind the reactions at hand differ. The available
models represent bi-variant systems and differ on that facet from the currently researched
system. Heat pump systems mainly display a tubular setup, consisting of a heat exchanger,
5.3. LAYER MODELS
37
Figure 5.3: Schematic of an adsorbent bed tubular setup.
coated by an adsorbent bed, that is surrounded by a gas volume, through which vapor can
be transported. A typical example can be seen in Figure 5.3. This system is described in
[Marletta, 2002] and [Maggio, 2005]. Water vapor can reach the dry adsorbent bed through
the gas path, where it is adsorbed while generating heat. The heat that is produced is carried
off by the heat transfer fluid and available for use. When the adsorbent bed is saturated with
vapor, the system is operated in opposite direction; heat is supplied to the adsorbent bed by
the heat exchanger. The increased temperature of the bed effects the desorption reaction and
the water vapor is carried off through the gas path. This cycle can be repeated numerous
times. The system consists of three main elements, relevant for the mathematical model; the
heat transfer fluid, the metal tube and the adsorbent material. The latter includes a porous
solid and water vapor both in gaseous and adsorbate phase. The modeling of such systems is
based on the following assumptions:
-
The adsorbent bed is homogeneous, with uniform adsorbent porosity and density
The heat and mass transfer resistances on the particle scale are neglected
Thermal gradients in radial direction are neglected for the heat transfer fluid and
metal
The vapor velocity in the adsorbent is determined by Darcy’s equation
The gaseous phase behaves as an ideal gas
The properties of the metal and gaseous phase are assumed to be constant
The properties of the adsorbent, are considered to be temperature dependent
All thermal losses are negligible
5.3.1
Governing equations
The system can be described by governing equations, stating the conservation of energy and
mass. Here only the adsorbent bed is discussed. In the following equation, the subscript g
denotes the gaseous phase, s is the solid phase and w is the uptake of vapor in the adsorbent
38
Modeling
bed. The mass balance for the adsorbent is:
εt
∂ρg
1 ∂(rvr ρg ) ∂(vz ρg )
∂w
+
+
+ (1 − εt )ρs
=0
∂t
r
∂r
∂z
∂t
(5.10)
where v̄ represents the velocity vector, which is determined by Darcy’s equation for transport
in porous media; v̄ = − µκ grad(p). Here κ is the permeability of the medium, µ the viscosity
and p the pressure. The total porosity, εt , is the sum of the micro-porosity, the porosity
of the individual grains; and the macro-porosity, the porosity of the adsorbent bed; εt =
εma + (1 − εma )εmi . The energy balance for the adsorbent is given by:
i 1 ∂ h
i
∂(ρg cpg Ts )
∂h
+ (1 − εt ) ρs
(1 + w) cpeq Ts +
(ρg cpg Ts + p) rvr
∂t
∂t
r ∂r
2
i
h
i
h
∂w
∂Ts
∂ Ts
∂
1 ∂
+ ρs (1 − εt ) |∆H|
(ρg cpg Ts + p) vz = λeq
r
+
+
∂z
r ∂r
∂r
∂z 2
∂t
εt
(5.11)
The equilibrium conditions of an adsorbent-adsorbate pair are represented by ln(p) = A(w) +
B(w)
T , where A(w) and B(w) are cubic polynomials that are determined experimentally and
also given in literature.
The numerical model provides the temperature and pressure distribution over the porous
layer. Subsequently the vapor uptake distribution, the amount of heat exchanged and the
performance of a given adsorption system can be determined. A base-case assembly was
formulated in which the system performs exactly one cycle. This base-case was used in
performing a sensitivity analysis for a number of parameters. This sensitivity analysis revealed
that increased layer thickness results in increased cycle duration; it was 460 s for a layer
thickness of 3 mm, 926 s for 5 mm thick and 1547 s for a thickness of 7 mm. Furthermore
the adsorbent bed configuration was found to have a large impact on the performance of the
system, where a new consolidated bed configuration performed best.
5.4
Current model
A 2D finite element model is built in COMSOL Multiphysics, within the Chemical Engineering
Module. In this model a layer of hydrated magnesium sulfate is considered as a homogeneous
continuum and the heat and mass transport through this porous layer is calculated. The
material then experiences a single-step chemical conversion, that is based on a transition step
that was recorded in experiments, as is shown in Figure 5.4(a). The conversion reaction will
be described as a solid S1 , that is converted into a solid S2 and a gas phase G, constituted
by water vapor. The reaction kinetics were determined previously in Subsection 4.1.4 for the
large transition, i.e. the large peak in differential weight loss. The initiation of this reaction
is well defined and is found around t = 400 s. The endpoint of the conversion however is less
trivial. Besides that, the exact composition of the material at the starting point and at the
end is unknown. The first model is based on the assumption that the conversion is limited to
the symmetric part of differential mass signal. The transition then occurs between t = 400
and t = 1070, as is shown as the grey marked part in Figure 5.4(a). A mass reduction of 30%
is measured over this time interval. When each expelled water group coincides with a mass
decrease of 7.3%, as stated before in Section 2.3, in this case 4 water groups are expelled. This
39
5.4. CURRENT MODEL
105
Experiment
Diff. mass
Model 2
Model 1
100
95
95
90
90
85
85
Mass [%]
Mass [%]
experimental result
numerical model 1
numerical model 2
100
80
75
80
75
70
70
65
65
60
60
55
50
0
200
400
600
800
1000
1200
Time [s]
(a)
1400
1600
1800
2000
55
0
200
400
600
800
1000
1200
1400
1600
1800
Time [s]
(b)
Figure 5.4: In Figure 5.4(a) an experimentally determined dehydration curve for MgSO4 · 7H2 O is shown. The
solid line is the sample mass and the dashed line is the scaled differential weight loss. The part of the solid line
that is marked in grey is the conversion that is used in numerical model1, the black marked part of the line
is the conversion which is used in numerical model2. In Figure 5.4(b) the output of the model is compared to
an experimental result.
observation was compared with literature data on stable states of the salt-hydrate and led
to the conclusion that the corresponding reaction is the dehydration from the hexa-hydrate
to the di-hydrate. The numerical results showed poor resemblance with experimental results
however, which is seen in Figure 5.4(b). A second model was then formulated, which included
a larger part of the conversion, which is shown in Figure 5.4(a) as the black marked part.
Moreover the assumption was made that the initial composition of the material at t = 0
was not the hepta-hydrate, but a slightly lower hydrate. A consequence of this assumption
is that the rejection of one water group no longer coincides with a mass reduction of 7.3%.
A mass reduction of 40% was measured, which was calculated to be 5 water groups. The
composition of the material at t = 400 was found to be the hexa-hydrate and the end product
the mono-hydrate, or written: MgSO4 · 6H2 O ⇀
↽ MgSO4 ·H2 O + 5 H2 O.
The numerical model is based on the assumptions that the system is in local thermodynamical
equilibrium, which means that locally the temperature of each phase is equal. The vapor
velocity is governed by Darcy’s law for flow in porous media, the gaseous phase behaves as
an ideal gas, the chemical reaction is of first order and therefore the reaction kinetics can be
described by an Arrhenius-type equation. Volumetric changes are neglected and the porosity
of the material is considered to be constant.
The energy balance equation for a closed system was already stated in Equation 5.3. The term
ρcp is now composed of the contributions of all phases present, i.e. ρs1 cps1 + ρs2 cps2 + ρg cpg .
Furthermore the densities are stated as a relative density in the domain, that is written as
the concentration of a species times its molar mass; e.g. ρS1 = S1 · MS1 . The energy balance
equation then reads:
∂
(MG GcpG + MS1 S1 cpS1 + MS2 S2 cpS2 )T + ∇(MG GcpG T v̄) = ∇ · (λ∇T ) + k1 MS1 S1 ∆H
∂t
(5.12)
40
Modeling
where M is the molar mass and the subscript denotes the species concerned. The convective
term for the solid phases is zero, so only a convective term for the gaseous phase is present.
The last term on the right-hand side represents a source term due to chemical reaction. In
the case of an endothermic process, this parameter has a negative value, as it then represents
a sink of energy.
For a chemically reacting system, the change in mass of each phase is coupled to the chemical
kinetics and stoichiometry. It was chosen to state conservation of concentrations of species
rather than actual mass as the dependent variable. The mass balance equation for the chemically reacting system then reads:
MS1
∂S1
∂S2
∂G
+ MS2
+ MG
+ MG v̄ · ∇G = 0
∂t
∂t
∂t
(5.13)
The chemical kinetics for a first order reaction is described by:
∂S1
= −k1 S1
dt
(5.14)
The reaction rate ki is a function of temperature, in the form of an Arrhenius equation:
−E ki (T ) = Ai exp
i
RT
(5.15)
the values of the constants Ai and Ei were determined experimentally, as described in Subsection 4.1.4.
The convective velocity that appears in Equation 5.12 and 5.13 is described by Darcy’s law
for flow in porous media. Darcy’s law states that the velocity vector is determined by the
pressure gradient, the fluid viscosity and the structure of the porous medium in the following
way:
κ
v̄ = − · ∇p
(5.16)
µ
As the only gaseous product is water vapor and the concentration of which is already calculated, the pressure term in the Darcy equation can be rewritten by making use of the ideal
gas law, it then looks as follows:
κ
(5.17)
v̄ = − R · ∇(T G)
µ
The calculated convective velocity of the gas phase can now be introduced into Equation 5.12
and Equation 5.13.
The model considers a layer of porous material, that is represented as a rectangular domain, as
is shown in Figure 5.5. The heat and mass transfer through this layer are calculated while the
material is converted chemically. An unstructured grid of 1040 triangular quadratic elements
is applied to the domain, with 30 elements in x-direction and 8 elements in y-direction. The
width of the geometry is 4.8 · 10−3 m and the height is 1.3 · 10−3 m. The equations are
solved by a direct solver that transfers the nonlinear problem into a linear set of equations.
The adaptive time stepping function was used, in order to minimize calculation time. For
systems that incorporate transient effects a small time step is essential, but for a quasi-steady
state effects small time steps are unnecessary and very time consuming. The current system
experiences large transitions in the beginning, but stabilizes at later times.
41
5.5. MODEL VERIFICATION
3
4
1
H
y
2
x
Figure 5.5: A schematic representation of the numerical geometry, in which the boundaries are numbered.
To complete the mathematical formulation of the problem, initial and boundary conditions are
stated. The initial concentration S1 is calculated from the known density and macro-porosity
of the material.
For t = 0
T (x, y) = T0
S1 = 3400
S2 = 0
G=1
The layer of material is open to mass transport at boundary 3, while all other boundaries
are impermeable. Heat is added to the domain from the bottom side at boundary 2 and
the domain is in convective contact with the ambient at boundary 3. The boundary conditions for heat and mass are summarized in Table 5.1, where the heat flux vector reads
q̄ = −λ∇T + ρcp T v̄ and the mass flux vector reads N̄i = −D̄i ∇ci + ci v̄. In Figure 5.5 a
schematic representation of the numerical domain is given.
Table 5.1: Boundary conditions
Heat
B1
q̄ · n̄ = 0
B2
Tprescribed
Mass
Ni · n̄ = 0
Ni · n̄ = 0
B3
−λ∇T · n̄ = 0
q̄ · n̄ = ρcp T v̄ · n̄
Ni · n̄ = ci v̄ · n̄
B4
q̄ · n̄ = 0
Ni · n̄ = 0
Initially the domain is at a temperature of 293 K. At t > 0 a temperature boundary condition
is given for boundary 2, which is either a linearly increasing temperature (so-called constant
heating rate) or a temperature that is increased instantly.
5.5
Model verification
Model verification is used to check whether a model is built right and is the first step in testing
the correctness of a model. The second step is called validation, which is meant to assure that
the model represents the real system to a sufficient level of accuracy. Model verification is
performed by comparison with analytical solutions and by calculation of some dimensionless
numbers, which give direct insight into the ruling processes in a system.
Conduction is the determining heat transfer mode within the porous layer of material. In
transient heat conduction problems, the temperature varies with location within the domain
42
Modeling
and time. These cases are also denoted as unsteady-state heat conduction problems. Before
taking the transient effects into account, the temperature distribution over the layer for the
steady-state conduction case is calculated.
When a boundary of the domain is in thermal contact with an ambient, heat is transferred by
convection. When the convection coefficient is very high, the surface temperature becomes
approximately equal to the ambient temperature. A very low convection coefficient results
in a large temperature difference between the surface and the ambient, which can also be
referred to as a high thermal surface resistance. The Biot number (Bi) is a dimensionless
number that represents the ratio of internal to surface transport resistance. For Bi < 0.1
the lumped-capacitance assumption is valid [Janna, 2000], which means that the temperature
profile at any time is (nearly) constant within the domain. Typical values of the convective
heat transfer coefficient for air in natural convection is 5 − 25 Wm−2 K. For a heat transfer
coefficient h = 25 Wm−2 K and a typical length of L = 1.3 · 10−3 m the Biot number is:
Bi =
hL
= 0.0677
λ
(5.18)
When T = 393 K at the heated wall, in the current model given by boundary 2, and for
the ambient T∞ = 293 K, then the temperature at the surface, here boundary 3, will theoretically settle to become 390 K. In the current model this was found to be 389.8 K, which
is a reasonable approximation. In [Janna, 2000] the analytical solutions for the temperature
distribution over a finite slab for different Biot numbers are given in a series of charts.
The Biot number was then increased to 1 by setting the heat transfer coefficient to 100 Wm−2 K
and changing the height of the domain to 4.8·10−3 m. The analytical solution states a dimensionless temperature difference of 0.67 for this situation, which results in a surface temperature
of T = 360 K. The model calculates a surface temperature of 360.2 K, which is again in good
agreement with the analytical solution. A temperature difference of this quantity has a significant impact on the conversion reaction. Large concentration gradients over the layer emerge
and completion of the reaction takes substantially longer time. The Biot number was then
raised to a value of 10. In order to do so, the convection coefficient was set to 300 Wm−2 K,
which is the reported maximum for a forced convection air flow. The height of the domain
was increased to 1.6 · 10−2 m. A surface temperature of 307 K is then predicted and in the
simulation it was calculated to be 309.7 K which is a bit too high. The situation that was
simulated however is not a situation that would ever be operated in reality.
Subsequently the unsteady state conduction problem is considered. The timescale of the
problem can be deducted from the dimensionless temperature difference and the Biot and
Fourier number:
T − T∞
= exp[−(Bi)(F o)]
(5.19)
Tw − T∞
The dimensionless temperature difference for a Biot number of 0.0677 has a value of 0.97.
This yields a Fourier number of 0.45. The Fourier number is stated as the ratio of heat
conduction to rate of thermal energy storage and reads:
Fo =
αt
L2
(5.20)
Rewriting the equation and solving for t yields a typical timescale of t = 4 s. For reasons
of numerical stability the temperature of boundary 2 is not instantly set to the high value
43
5.5. MODEL VERIFICATION
of 393 K but it is increased gradually over a time period of 10 seconds. When the sink of
energy, that accounts for the endothermic reaction energy, is set to zero, the final temperature
at boundary 3 is reached 3 seconds after boundary 2 has reached its final temperature, which
is also shown in Figure 5.6(a). When the sink term is active, the temperature at boundary
3 is 386 K after 5 seconds. At t = 150 the final boundary temperature is reached, this is
depicted in Figure 5.6(b).
400
400
Twall
Tsurf
380
380
360
360
Temperature [K]
Temperature [K]
Twall
Tsurf
340
340
320
320
300
300
280
0
10
20
30
40
50
Time [s]
(a)
60
70
80
90
100
280
0
20
40
60
80
100
120
140
160
180
200
Time [s]
(b)
Figure 5.6: Two simulation results of the transient temperature over the layer of porous material. In Figure
5.6(a) the result is given for the case that no sink-term is present in the domain. In Figure 5.6(b) the sink
term, related to chemical conversion is active in the domain.
The Thiele modulus determines whetherpthe reaction is controlled by internal diffusion D or
the reaction rate k. It reads T h = Rp k/D, where Rp is a characteristic length scale. It
is found that for the current system T h < 1 which leads to conclude that the conversion is
controlled rather by the chemical reaction rate than internal diffusion.
In [Thunman, 2002] a last dimensionless number is described: the Lewis number Le = α/D,
which represents the ratio between thermal diffusivity and mass diffusivity. For the current
system Le ≈ 1.
Then some first simulations are performed to gain more insight into the numerical model.
A constant heating rate of 10 K min−1 was applied over a temperature range of 20 − 200 K
on the domain which is presented in Figure 5.5. The concentrations of all phases and the
concentration distribution over the porous layer are monitored and shown in Figure 5.7. A
conversion of hexa-hydrate to mono-hydrate was considered here. Figure 5.7(a) shows that
the concentration S1 monotonously decays while forming S2 and G. Stoichiometry of the
reaction states that each mole of hexahydrate yields 5 moles of gas phase. The concentration
G becomes larger than the initial concentration S1 , however it does not become 5 times as
large, due to the convective transport to the ambient. The same geometry is also subjected
to a constant heating rate of 20 K min−1 over the same temperature range. The conversion
reaction is then completed 300 seconds faster and the maximum concentration G reaches a
higher value, due to the shorter time frame in which the reaction takes place. For the standard
geometry only small differences in concentrations are found over the layer thickness. The
44
Modeling
3500
4500
t=0
t = 300
t = 500
t = 600
t = 700
t = 800
S1
G
S2
4000
3000
−3
Concentration [mol m ]
Concentration [mol m−3]
3500
3000
2500
2000
1500
2500
2000
1500
1000
1000
500
500
0
0
200
400
600
800
Time [s]
(a)
1000
1200
0
0
0.5
1
1.5
2
Height [m]
2.5
3
3.5
4
−3
x 10
(b)
Figure 5.7: Two concentration distribution plots calculated by the numerical model. In Figure 5.7(a) the
concentrations of all three phases are shown versus time for the center point of the domain. Figure 5.7(b)
shows the concentration distribution for S1 over the height of the domain for a number of time steps.
geometry was expanded by enlarging the domain in y-direction from 1.3 · 10−3 to 3.9 · 10−3 m.
The experimental equivalent sample mass is 30 mg. The concentration differences over the
layer now become visible, as is shown in Figure 5.7(b). Here the concentration course of
species S1 over the height of the geometry is shown for a number of time steps. The height
of 3.9 · 10−3 m coincides with the upper boundary 3 of the domain. From this graph it is
seen that in the first 300 seconds the concentration S1 reduces slightly, subsequently it falls
rapidly and finally the conversion is almost completed at t = 800 s. The concentration at
boundary 3 at t = 800 s is relatively high compared to the rest of the geometry.
5.6
Model validation
The results from the numerical model are compared to experimental results for different
operating conditions. The first case which is considered, is that of a constant heating rate
experiment. Dehydration behavior for a heating rate of 5 K min−1 was simulated numerically.
The results for both numerical models are compared with the experimentally obtained results,
which is depicted in Figure 5.8(a). In this figure the solid line represents the experiment,
the dash-dotted line is model1 and the dashed line is the result of model2. Figure 5.8(a)
shows that model2 describes the conversion process significantly better than model1. Model2
predicts the course of the conversion path very well over a large part of the time range, which
confirms that the reaction kinetics are described correctly. Around t = 900 however, the
reaction tends to slow down in the experiment, whereas it remains constant for some time
longer in the simulation. The reason for this anomalous behavior lies in the fact that the
last water groups that are present in the molecule are strongly held in the lattice, due to the
large amount of free ligands. This effect is not taken into account in the numerical modeling.
The total mass reduction in the experiment is higher than the mass reduction in model2,
but this is logical as only part of the total conversion is modeled. The dehydration behavior
45
5.6. MODEL VALIDATION
experimental result
numerical model 1
numerical model 2
95
95
90
90
85
85
80
75
80
75
70
70
65
65
60
60
55
0
200
400
600
800
1000
Time [s]
(a)
1200
1400
1600
1800
exp 38 mu
exp 200 mu
num model2
100
Mass [%]
Mass [%]
100
55
200
400
600
800
1000
1200
Time [s]
(b)
Figure 5.8: Two validation results for constant heating rate. Figure 5.8(a) is the result for a heating rate
of 5 K min−1 , in which the solid line represents the experimental values, the dash-dotted line the result for
numerical model1 and the dashed line for numerical model2. Figure 5.8(b) shows the results for a heating rate
of 10 K min−1 two different grain sizes. The solid black line represents the experimental result for a grain size
of 38 − 100 µm, the grey solid line is the result for a grain size of 200 − 500 µm and the dashed line is the
numerical result for model2.
was also simulated for a constant heating rate of 10 K min−1 . The numerical results are
compared with an experiment that is performed at the same constant heating rate, for two
different grain sizes. In Figure 5.8(b) the results are shown, where the solid lines represent the
experimental results and the dashed line represents the numerical result. In experiments the
grain size was found to influence the propagation of the dehydration reaction. From Figure
5.8(b) it is clear to see that model2 describes the conversion process of the small grain size
sample better than the large grain size sample. This was already expected as the formulated
model is a continuum model and therefore does not take transient effects on the smaller scale
into account. At t = 600 a decrease in reaction rate is found in the experiment, similar to
that found in the previously discussed experiment. Again it results in a difference between
the experimental and numerical result.
A numerical run for the step-shaped temperature experiment was performed as well. Only
the second part of the heating experiment, the heating at high rate from 45 to 150o C and
isothermal part afterwards is simulated. The results are presented in Figure 5.9(a), where
the experimental result is depicted by a solid line and the numerical result is denoted with a
dashed line. The total conversion is predicted well by the numerical model. Again a deviation
is found between the experimental and numerical result at a sample mass of about 70%.
Overall however the numerical result corresponds to the experimental result satisfactorily.
Figure 5.9(b) shows the result for the same experiment, now performed on a sample with a
layer thickness that is three times as large as the original layer. The experimental sample size
was approximately 38mg. The solid line represents the experimental data and the dashed
line is the result for numerical model 2. Good resemblance is found between the two curves,
however in contrast to other validation results, deviations are found in both directions and
over a larger range of the conversion. The deviations are not alarming in size though and the
overall conversion is described satisfactorily.
46
Modeling
numerical result
experimental result
95
95
90
90
85
85
80
75
80
75
70
70
65
65
60
60
200
400
600
800
1000
1200
Time [s]
(a)
1400
exp large
num large
100
Mass [%]
Mass [%]
100
200
400
600
800
1000
1200
1400
Time [s]
(b)
Figure 5.9: Two validation results for stepwise heating of a layer of product. Figure 5.9(a) shows the results
for a stepwise heating experiment, where the solid line represents the experimental result and the dashed line
is the numerical result for model2. Figure 5.9(b) shows the result for the same experiment for a larger sample.
The solid line represents the experimental result and the dashed line the result for numerical model2.
Overall one can conclude that the model simulates the different experiments well. At the
moment the influence of vapor pressure on the reaction kinetics was not taken into account.
The effect of vapor pressure on the kinetics can be described as the logarithm of the ratio
between the equilibrium pressure and the current pressure. Including this factor in the kinetics
may yield a better simulation of the conversion, especially at the point where the model result
now deviates from the experimental result.
5.7
Model predicability
When the functionality of the model is checked for correctness, the next step is to use the
predictive capacity of the model. In this way parameters for a future operating system can
be determined. In a future operating system, the reactions are expected to take place in
a fixed bed reactor. It was seen before that the height of the layer of product has a large
influence on the propagation of the reaction. Therefore the height of the domain was increased
significantly. For a layer thickness of 5 centimeter, the conversion reaction propagates slowly.
The total conversion of the layer takes about 12000 seconds. This is mainly due to the poor
heat transport over the layer. When the thermal conductivity of the solids is raised to a value
of 4.8 W m−2 K the heat transport over the layer is improved significantly. The conversion
reaction for the entire layer is then completed in 2000 seconds. A storage material with a high
thermal conductivity enables the use of a fixed bed reactor with substantial layer thickness.
As salt-hydrates in general have a low conductivity it is useful to investigate other options
to enhance the thermal conductivity such as the use of an inert binder. In [Stitou, 1997] the
thermal conductivity is increased by using a graphite type binder; the so-called expanded
natural graphite (ENG). ENG has a high intrinsic conductivity of 200 W m−1 K−1 . The
graphite binder is added to the salt by physical mixing, after which the mixture is compressed
and confined within the reactor.
Chapter 6
Conclusions and Recommendations
The goal of this research was to characterize a thermochemical storage material in both experimental and numerical way, in order to determine its applicability in a seasonal storage
system. Hereto one thermochemical material was selected from a list of promising candidate
materials, which was the outcome of previous literature research performed by ECN. Magnesium sulfate hepta-hydrate was found to match the stated requirements for a thermochemical
storage material well.
A number of experiments are performed, consisting of a group of experiments that is performed for material characterization and a group to determine the behavior of a heat storage
system employing a thermochemical material. The experimentally determined properties are
used in formulating a 2D numerical model, which describes the dehydration of a porous layer
of salt-hydrate.
From experiments it has become clear that the dehydration behavior and the intermediately
formed phases of MgSO4 · 7H2 O did not agree satisfactorily with the data found in literature.
The stepwise dehydration which is stated in literature was found in experiments to be more
gradual in character and a distinction between the individual steps was fuzzy. The reaction
kinetics of a large part of the conversion is described by an Arrhenius type equation, the
parameters of which are determined successfully and are included in the numerical model.
The surface of grains of the salt-hydrate are studied and a visual indication of melting and
thus material deterioration is determined. Water vapor pressure is shown to influence the
propagation of the reactions in the small range of water vapor pressures investigated. For
higher ambient water vapor pressure, the reactions shift to higher temperatures. Both increased layer thickness and increased grain size result in a larger timescale of the conversion
reaction, however the end product of the reaction remains the same. A key requirement for
the use of magnesium sulfate in a seasonal storage is that the reaction is reversible and can be
performed a number of times. This property is called cyclability. The results of experiments
performed imply that the cyclability of the material is poor; the amount of water that is taken
up during hydration reduces significantly in every cycle and the timescale of the hydration
process is too large to ever be useful as a heat source.
From dimensionless numbers, it is found that for the current setup the temperature difference
over a layer of material is small and reaction kinetics is found to be the limiting factor in
47
48
Conclusions and Recommendations
the propagation of the conversion reaction. However at significantly larger layer thickness,
i.e. 10 cm, the heat conduction over the layer becomes insufficient. The use of an inert
binder, such as expanded natural graphite, increases the thermal conductivity of the salthydrate without obstructing mass transport. The numerically retrieved results display good
agreement with the experimental results over a large part of the conversion. The final part
of the conversion propagates faster in the numerical model than in reality. Two reasons for
this anomalous behavior can be assigned. Firstly the last water molecules experience a larger
transport resistance and are held strongly in the lattice, due to the large amount of free
ligands. Secondly, the influence of water vapor pressure on the reaction is not included in the
current model.
A number of recommendations are formulated for future research. It is advisable to include a
step in the experimental protocol to verify the exact composition of the starting material of an
experiment. This can be done with XRD or by allowing the sample material to settle for some
time at a low temperature in an ambient that is saturated with water vapor. Furthermore
a second material should be implemented in the numerical model to validate its functioning.
The best alternative is to select a material of which material parameters and intermediate
phases are well-defined, such as copper sulfate penta-hydrate (CuSO4 · 5H2 O). Concerning
the cyclability of MgSO4 · 7H2 O, more research is necessary in order to validate the presented
results. Currently an experimental cycling setup is being built at ECN in order to perform
these measurements. Reaction kinetics of the hydration reaction should be obtained in order
to extend the numerical model. This enables the user to simulate a full operating cycle and
gives insight into system performance.
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Universiteit Eindhoven, 1994
[Stanish, 1983]
Stanish, M.A., Perlmutter, D.D. Dehydration of crystalline potassium carbonate hydrate AlChE Journal, Vol.29, pp 806-812
[Stitou, 1997]
Stitou, D., Goetz, V., Spinner, B. A new analytical model for solidgas thermochemical reactors based on thermophysical properties of the
reactive medium Chemical Engineering and Processing, Vol.36, pp 29-43
[Thunman, 2002] Thunman, H. Combustion of biomass Nordic Course Lyngby, Danmark,
19-23 August 2002
[Ullmann, 2000]
Ullmann’s Encyclopedia of Industrial Chemistry, Sixth Edition 2000 Electronic Release
[Vaniman, 2004]
Vaniman, D.T., Bish, D.L., Chipera, S.J., Fialips, C.I., Carey,
J.W., Feldman, W.C. Magnesium sulphate salts and the history of water
on Mars Nature, 431, pp 663-665
[Visscher, 2004]
Visscher, K., Veldhuis, J.B.J., Oonk, H.A.J., Van Ekeren, P.J.,
Blok, J.G. Compacte chemische seizoensopslag van zonnewarmte ECNC–04-074, 2004
[Wagman, 1982]
Wagman, D.D, Evans, W.H., Parker, V.B. et al. The NBS tables of
chemical thermodynamic properties: selected values for inorganic and C1
and C2 organic substances in SI units Journal of Physical and Chemcial
Reference Data, 11, 1982
[Washburn, 1930] Washburn, E.W. et.al International critical tables of numerical data,
physics, chemistry and technology McGraw-Hill, London, 1926-1930
Appendix A
Calculation of kinetic parameters
In order to determine the kinetic parameters of Arrhenius type equations concerning chemical
conversion processes, the so-called Tmax method can be applied. A typical Arrhenius reaction
equation has the form:
−Ei
ki = Ai exp(
)
(A.1)
RT
In which ki is the reaction rate [ s−1 ], Ai the so-called frequency factor [ s−1 ], Ei the activation
energy [ J/mol], R the gas constant and T the temperature [ K]. The Tmax method is
described by [Bunyakiat, 2002]. From TGA measurements a plot can be constructed of the
differential sample mass versus temperature.
Figure A.1: Differential mass versus temperature curve for coal pyrolysis.
Each reaction that takes place in the material results in a peak in the differential mass signal,
the top point of which represents the maximum yield of products. The temperature at which
this maximum yield occurs is denoted by Tmax . The rate of product release from a first-order
chemical reaction i is described as:
dVi
= ki (Vi∗ − Vi )
dt
51
(A.2)
52
Calculation of kinetic parameters
Where Vi is the accumulated amount of evolved volatile from the reaction i up to time t and
Vi∗ the total yield of volatile for reaction i. A constant heating rate implies a linear correlation
between time and temperature, then Equation A.2 can be expressed in the following form:
dVi
ki (Vi∗ − Vi )
=
dT
H
(A.3)
Where H denotes the heating rate [ K/sec]. At the temperature where the maximum yield
of products is reached, the temperature derivative of the reaction rate is zero.
ki −dVi (Vi∗ − Vi ) dki
d 2 Vi
=
+
=0
dT 2
H dT
H
dT
(A.4)
The temperature derivative of the theoretical Arrhenius equation (Equation A.1) reads:
E −E dki
i
i
= Ai
exp
dT
RT 2
RT
(A.5)
Then substituting Equation A.3 and A.5 into Equation A.4 yields:
k k (V ∗ − V ) i
i i
i
H
−
H
+
(V ∗ − V ) i
i
H
Ai
−E Ei i
=0
exp
2
RTmax
RTmax
(A.6)
Filling in ki and rearranging gives:
−
Ei Ai Ei
2Ei A2i
exp
−
=
−
exp
−
2
H2
RTmax
H RTmax
RTmax
(A.7)
Which can be written in its final form:
ln
H 2
Tmax
= ln
A R
i
Ei
−
Ei
RTmax
(A.8)
A series of TGA experiments is then performed for different constant heating rates. The
H
)
Tmax is determined for each measurement. These points are depicted in a plot of ln( Tmax
1
vs ( Tmax ) and connected with a linear fit. Then the kinetic parameters Ei and Ai can be
determined from the slope and intercept in the following way:
Ei = −R · slope
Ai = exp(intercept) · slope
(A.9)
(A.10)
This method can also be applied to determine the kinetic information from a high resolution
measurement, despite the fact that a variable heating rate is employed. This is due to the
fact that when maximum yield of products is realized, the heating rate in a high resolution
measurement is minimal by definition. The actual heating rate at Tmax can be obtained from
the output data.
Appendix B
Experimental background
B.1
Experimental setups
TGA experiments are performed on two systems. In each TGA device two gas flows are
present; a purge and a protective flow. The protective flow was set to 20 ml min−1 of dry
nitrogen. The purge flow was set to 80 ml min−1 of dry nitrogen. This latter gas flow
was wetted for some experiments. Hereto the gas flow was led through two consecutive gas
bubbling bottles, an example of which is shown in Figure B.1. The dry gas enters the bottle
Figure B.1: Schematic representation of a gas bubbling bottle.
through the tube on the left hand side. At the bottom of the tube a filter plate is present,
which is a porous piece of material that enhances the formation of small bubbles. The gas
bubbles through the column and is carried off through the tube on the right hand side. When
the bubbling length through the bottle is large enough, the gas flow is saturated with water
vapor. The water vapor pressure that is applied in this way depends on temperature, following
the Clausius Clapeyron equation. For a water temperature of 25o C a water vapor pressure of
53
54
Experimental background
30 mbar is applied to the gas flow.
At the department of Chemical Engineering at Eindhoven University of Technology a TA
Instruments TGA Q500 device is present, which is shown in Figure B.2(a). This system is
equipped with an auto-sampler, which is a carousel device that can hold up to 50 samples
for successive measurements. This setup was used to perform a high resolution measurement
as well as some constant heating rate experiments. The purge flow in this setup always
consisted of wetted nitrogen at a flow rate of 80 ml min−1 . The device uses platinum sample
pans without a lid, some examples of which are shown in Figure B.2(b).
(a)
(b)
Figure B.2: The TA Q500 thermo-gravimetric analyzer and its sample pans.
At the division of Biomass, Coal and Environmental Research at ECN in Petten a Mettler
Toledo TGA STARe device is present. This setup was used to perform extensive thermogravimetric analysis for a range of operating conditions. The purge flow in the setup was
varied from 20 − 160 ml min−1 . The system uses alumina sample pans, that can be applied
with or without a lid with a pinhole. These are depicted in Figure B.3. The gas-outlet of the
Figure B.3: Overview of sample pans available for the Mettler Toledo TGA STARe system.
Mettler Toledo TGA STARe is equipped with a mass spectrometer. This is a device which
is used to determine the composition of a gaseous sample, by generating a mass spectrum
representing the masses of sample components. This is a specifically useful instrument when
the reaction products are not exactly known.
55
B.2. EXPERIMENTAL SETTINGS
For DTA-TG measurements a Netzsch STA 409 PC was used, which was available at ECN.
The purge flow in this setup was fixed and consisted of 60 ml min−1 of dry nitrogen. The
protective flow was fixed at a flow of 20 ml min−1 of dry nitrogen. The same alumina sample
pans are used as those used in the Mettler Toledo TGA STARe system. Two schematic
representations of the system setup are found in Figure B.4. On the left hand side a layout of
(a)
(b)
Figure B.4: Two schematic overviews of the Netzsch STA 409 PC system.
the apparatus is shown. The sample is placed on a long slender rod and the gas flows enter
from below. The heating of the furnace is provided at the wall, while the temperature program
is prescribed for the sample temperature. In Figure B.4(b) a close up is given of the sample
holder. From the four sample holders that are shown on the right, the second setup was used
for the current measurements. The device was installed recently and experimental resolution
determination revealed that a large difference was present between the prescribed and realized
temperature. Either by extensive calibration or through contact with the manufacturer this
may be improved in the future.
DSC measurements are performed on a Netzsch DSC 204 F1 at ECN, using aluminium pans,
which were sealed with a press. No purge gas flow was present during the measurements.
B.2
Experimental settings
In the table given in Figure B.5 an overview is given of the experimental settings that were
used for measurements. The last column denotes in which figure(s) the experimental result
is shown in this report.
56
Experimental background
HiRes_ramp10
mix
12.6
80 ml/min wetted
25-400
STA06111403
STA06112202
STA06112203
mix
200-500
38-100
10.4
9.7
10.4
lid with pinhole
lid with pinhole
lid with pinhole
STA06112703
-
0
60 ml/min dry
STA06112704
STA06112804
STA06112902
200-500
10.7
9.6
7.6
lid with pinhole
STA06112802
38-100
38.2
lid with pinhole
STA06112803
38-100
11.4
lid with pinhole
TGA06101801
mix
9.2
20 ml/min wetted
25-320
25-320
25-320
25-45
45
45-150
150
25-45
45
45-150
150
25-45
45
45-150
150
25-45
45
45-150
150
25-200
200-25
Rate /
Time
2 K/min
1K/min
5 min
Hmax= 10
K/min
20 K/min
20 K/min
20 K/min
10 K/min
5 min
30 K/min
15 min
10 K/min
5 min
30 K/min
15 min
10 K/min
5 min
30 K/min
15 min
10 K/min
5 min
30 K/min
15 min
5 K/min
-5 K/min
TGA06101802
mix
10
80 ml/min wetted
25-200
200-25
5 K/min
-5 K/min
TGA06101803
mix
10.3
160 ml/min wetted
TGA06102501
mix
11.3
80 ml/min wetted
TGA06102506
mix
13.2
80 ml/min wetted
TGA06102507
mix
9.6
80 ml/min wetted
TGA06110801
mix
12.6
80 ml/min wetted
25-200
200-25
25-200
200-25
25-200
200-25
25
25-200
200-25
20
25-325
325
5 K/min
-5 K/min
20 K/min
-20 K/min
10 K/min
-10 K/min
40 min
1 K/min
-1 K/min
200 min
10 K/min
10 min
mix
11.3
12.1
12.2
12.8
80 ml/min wetted
25-325
325
10 K/min
10 min
TGA06110903 200-500
12.2
80 ml/min wetted
TGA06110904
12.3
80 ml/min wetted
0
80 ml/min wetted
11.6
8.0
10.7
80 ml/min wetted
80 ml/min wetted
80 ml/min wetted
25-325
325
25-325
325
25-40
40
40-270
270
25-400
25-400
25-400
10 K/min
5 min
10 K/min
5 min
40 K/min
5 min
40 K/min
15 min
1 K/min
10 K/min
50 K/min
Name
Grain size Sample mass
[um]
[mg]
DSC06112105
TGA06110802
TGA06110803
TGA06110901
TGA06110902
8.5
38-100
TGA06111401
TGAramp1
TGAramp10
TGAramp50
mix
mix
mix
Purge flow
none
Temp. [deg
C]
25-40
40-65
65
Figure B.5: Overview of experimental settings used in measurements.
Ref.
4.3b
4.1
4.3a
5.8b
5.8b
4.12
4.10
4.8
5.9b
4.8
5.9a
4.9
5.8a
4.2b
4.6
4.9
4.9
4.2b
4.6
4.2b
4.6
4.2b
4.11
4.13
4.13
4.7
4.7
4.12
4.2a
4.2a
4.2a
Appendix C
Modeling background
For the numerical model a total of 6 application modes is used; one for each phase (three
phases are present) and one for each conserved quantity (here conservation of mass and heat
is stated). All modes are stated for transient analysis. As the system is considered to be in
local thermodynamical equilibrium, only one dependent variable, T , is used for all thermal
modes. The concentrations of the individual phases are denoted S1, S2 and G respectively.
For S1 and S2 the applied modes are the Diffusion application mode for mass transfer and
the Conduction application mode for heat transfer. For G mass transfer is described with
the Convection and Diffusion application mode and heat transfer with the Convection and
Conduction application mode.
Figure C.1: Screen shot of COMSOL displaying the domain and mesh.
57
58
Modeling background
In Table C.1 an overview is given of the material parameters as they are applied in the
numerical model.
Table C.1: Input parameters for numerical model 2
Parameter
mms1
mms2
mmg
A
E
R
cps1
cps2
cpg
ks1
ks2
kg
dH
mu
kappa
porosity
Value
228.46 · 10−3
138.38 · 10−3
18 · 10−3
1.67 · 105
55 · 103
8.314
1525
1047
1005
0.48
0.48
0.026
−275 · 103
3 · 10−5
1 · 10−15
0.5
Description
molar mass solid 1
molar mass solid 2
molar mass gas
Arrhenius frequency factor
Arrhenius activation energy
ideal gas constant
heat capacity solid 1
heat capacity solid 2
heat capacity gaseous phase
thermal conductivity solid 1
thermal conductivity solid 2
thermal conductivity gaseous phase
reaction enthalpy
viscosity water vapor
permeability
macroporosity
In addition a number of expressions are formulated. Scalar expressions are formulated for
calculation of the reaction kinetics and the material densities. This leads to the following
equations:
k1 = A · exp(−E/R/T )
(C.1)
rhos1 = S1 · mms1
(C.2)
rhos2 = S2 · mms2
(C.3)
rhog = G · mmg
(C.4)
In order to determine the Darcy velocity two subdomain expressions are formulated:
P x = dif f (T · G, x)
(C.5)
P y = dif f (T · G, y)
(C.6)
The constants that appear in the Darcy equation are submitted in the input field for the
velocity in the subdomain settings of the gaseous phase. In this way the total Darcy equation
is included in the system of equations without the necessity of a separate application mode.