Characterization of a thermochemical storage material I.M. van de Voort WET 2007.04 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
Eindhoven University of TechnologyDepartment of Mechanical EngineeringDivision Thermo Fluids EngineeringEnergy Technology group
2
Abstract
Thermochemical materials are a promising new alternative for long term heat storage. Theprocess concerned is based on a reversible chemical reaction, which is energy demandingin one direction and energy yielding in the reverse direction. Preliminary research markedmagnesium sulfate hepta-hydrate (MgSO4 · 7H2O) as a specifically suitable material. Thematerial dehydrates when heated, forming MgSO4, this can be regarded as ’charging’ of thematerial. The reverse reaction is initiated when the anhydrous component is exposed to watervapor, producing heat.
To this point, literature is insufficient and inconsistent regarding the material properties ofmagnesium sulfate hepta-hydrate. Experiments are performed in order to characterize thematerial and its properties. Furthermore experiments are executed to check the applicabilityof MgSO4 · 7H2O in a seasonal storage system. Utilizing these results a 2D macromolecularnumerical model is proposed, describing the chemical conversion of the material. The modelis validated by comparing experimental and numerical results for some base case problems.
Two applications of the model can be identified. Firstly, other thermochemical materials canbe easily implemented in order to check their fitness as a storage material. Secondly, thewhite 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 warmteopslag. Het proces is gebaseerd op een reversibele chemische reactie die energievragend isin een richting en energieleverend in de andere richting. Voorgaand onderzoek heeft uit-gewezen dat magnesium sulfaat hepta-hydraat (MgSO4 · 7H2O) een bijzonder geschikt ma-teriaal zou kunnen zijn. Het material dehydrateert wanneer het verwarmd wordt, waarbijMgSO4 gevormd wordt. Dit wordt ook wel aangeduid als het ’opladen’ van het materiaal.De tegengestelde reactie wordt genitieerd door het gedehydrateerde materiaal bloot te stellenaan waterdamp. Bij deze reactie komt warmte vrij.
De beschikbare literatuur is onvolledig en inconsistent wat betreft de materiaal eigenschappenvan magnesium sulfaat hepta-hydraat. Daarom zijn experimenten uitgevoerd waarmee hetmateriaal en het functioneren van het materiaal als een medium voor lange termijn warmteopslag gekarakteriseerd kunnen worden. Aan de hand van de resultaten van deze experimentenis een 2D macromoleculair numeriek model opgesteld dat de chemische omzetting beschrijftdie het materiaal ondergaat. Het model is gevalideerd door experimentele en numeriekeresultaten voor enkele typische situaties te vergelijken.
Het model heeft twee belangrijke toepassingen. In de eerste plaats kunnen andere thermo-chemische materialen eenvoudig worden geımplementeerd om zodoende de geschiktheid voorgebruik in een lange termijn warmte opslag te bepalen. Ten tweede geeft het verkregenparametrische model beter inzicht in de processen die plaats vinden tijdens deze reversibelechemische reactie.
A (Arrhenius) frequency factor [s−1]Bi Biot number [−]C (phase rule) number of indep.
chemical comp. [−]cp heat capacity [J kg−1 K−1]D mass diffusion coefficient [m2 s−1]d (Bragg) interplanar distance [m]E energy [J]E (Arrhenius) activation energy
[J mol−1]f volume force [N]Fo Fourier number [−]G Gibbs free-energy [J]G concentration gas phase [mol m−3]H enthalpy [J mol−1]h heat transfer coefficient [W m−2 K]k reaction rate [min−1]L length [m]Le Lewis number [−]M molar mass [g mol−1]m mass [kg]
m′′′
mass source term [kg s−1 m−3]n integer number [−]N mass flux vector [mol m−2 s]n outward facing normal [−]p pressure [Pa]P (phase rule) number of phases [−]Q heat [J]Q source term [W m−3]q heat flux vector [W m−2]R gas constant [8.314 J K−1 mol−1]r radius [m]r interface velocity [m s−1]
Rp (Thiele) typical length scale [m]S entropy [J mol−1 K−1]S1 concentration solid 1 [mol m−3]S2 concentration solid 2 [mol m−3]T temperature [K]t time [s]Th Thiele modulus [−]V volume [m3]V (phase rule) degrees of freedom [−]v velocity vector [m s−1]W work [J]w uptake of vapor [kg kg−1]x problem domain direction [−]Y concentration [−]y problem domain direction [−]z problem domain direction [−]
The energy supply of most countries is presently based on fossil fuels. In the past decennia theawareness of the drawbacks of this energy supply is raised. A large drawback of consumingfossil 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 isdepleting. There are two main solutions to the problem; new energy resources have to beresearched and more efficient ways of using the current limited energy resources have to beexplored.
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 storedfor domestic use in winter, for instance for hot tap water. A current technology is storage ofwarm 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 of25 to 100 meter, the top and bottom of which are impermeable to water. Another technologyis the use of phase change materials (PCM). All these systems are relatively easy to install atreasonable cycle efficiency. Due to the relatively low energy density and large requirements ofinsulation however, these systems are not very cost efficient. A new way of long term storageof solar energy, without the necessity for thermal insulation, is by means of chemical energyin so-called thermochemical materials (TCM’s). Thermochemical materials can undergo re-versible chemical reactions, which are energy consuming in one direction and energy yieldingin the reverse direction. Because the reaction temperature of this process is relatively high(exceeding well over 100oC), 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 hasbeen conducted by ECN and Utrecht University. The emphasis of this research was placedon an extensive literature survey, to which a footnote was added that literature values onthe properties of materials and their chemical reactions are often incomplete and sometimesunreliable. In order to apply these materials in an energy storage system more insight shouldbe gained into characteristics of such materials.
The goal of the current research is to characterize a thermochemical storage material experi-mentally and use the properties obtained in formulating a macromolecular numerical model
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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 chang-ing 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 athand.
In Chapter 2 theoretical background is discussed on the processes and principles concerningthe use of thermochemical materials in a seasonal storage. Chapter 3 treats analysis methodswhich 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 resultsare presented which determine the applicability of the thermochemical material in a storagesystem. In Chapter 5 a finite element model is developed in which a porous layer of salt-hydrate is converted chemically. The simulation results are then compared with experimentalresults. Conclusions and recommendations for future research are formulated in the finalchapter.
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. Thereare a number of materials and reactions that conform to this requirement, as will be discussedin this chapter. The group of salt-hydrates in general and magnesium sulfate hepta-hydrate(MgSO4 · 7H2O) specifically, are considered to be suitable materials [Visscher, 2004]. Thischapter provides background on chemical processes and principles concerning the use of ther-mochemical materials in a seasonal storage. Thermochemistry is not a discipline on its own, itis a combination of thermodynamics, structural chemistry, catalysis and physical chemistry. Itis defined roughly as the application of heat effects due to a chemical reaction between two ormore species. First some chemical processes and definitions will be elucidated. Subsequentlya 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 seasonalstorage 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 systemis constant. The total energy change ∆E of a chemical system is represented by the sum ofthe 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, denotedby ∆H.
Q = ∆H = Hproducts − Hreactants (2.3)
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4 Thermochemical storage materials
If the products have more enthalpy than the reactants, then heat has flown into the systemfrom the surroundings and ∆H thus has a positive sign. Such a reaction is said to beendothermic. If the products have less enthalpy than the reactants, heat has flown out of thesystem 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 statefunction, its value does not depend on the path followed between two states. Thus, the sumof the enthalpy changes for the individual reactions in a sequence must equal the enthalpychange 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 moleculardisorder or randomness in a system. The difference in entropy between two states of the systemis given by dS = dQ/T and has units [ J K−1]. Gases, for example have more randomnessand 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. Aspontaneous process is characterized by a decrease in enthalpy and an increase in entropywhereas 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 representedby 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 atequilibrium and if ∆G is positive the process is non-spontaneous. The relation can also begiven for standard conditions, the terms G, H and S then get a zero in the suffix. Anotherexpression for the Gibbs free-energy is:
∆G = ∆G0 − RTln(p
p0) (2.5)
In which R is the gas constant, p is the vapor pressure and p0 the standard pressure. For anequilibrium reaction, the free energy G is zero and Equation 2.4 and Equation 2.5 can thenbe combined to give:
ln(p
p0) = −
∆H0
RT+
∆S0
R(2.6)
This equation is also known as the Clausius Clapeyron equation and relates the equilibriumvapor pressure to the thermodynamic parameters.
For a system that is in equilibrium the phase rule, stated by Gibbs, relates the numberof components (substances) and phases to the degree of freedom of the system. It readsV = C − P + 2, in which V is the number of degrees of freedom, i.e. the number of variablesthat must be arbitrarily fixed to establish the state of the system, C the number of independentchemical components and P the number of phases. The equilibrium reaction of a salt-hydratecan be represented as follows:
Salt · x H2O(s) + Heat ⇀↽ Salt · (x − y)H2O(s) + y H2O(g)
In such a system, there are two independent components, C = 2; three phases (2 independentsolids and 1 vapor phase), P = 3; and thus the degree of freedom is V = 1. Such a systemis 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 acomparable process, however water is bonded in a different way so that only one independentsolid is present in the system. The phase rule then states that the system is bi-variant, whichmeans 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 character-istics of the reactants and products and the conditions under which the reaction is takingplace. The rate law is an equation expressing the instantaneous reaction rate in terms of theconcentrations of the substances taking part in the reaction at that instant. A decompositionreaction, in which one reactant decomposes into products (e.g. a dehydration step of a salt-hydrate) is a typical first order reaction. This means that the reaction rate is proportional tothe concentration of the reactant. The reverse reaction in which the anhydrous salt togetherwith water vapor form a hydrated salt is a typical second order reaction, which means that thereaction 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, whichreads:
k = A exp(
−E
RT
)
(2.7)
in which k is the reaction rate, A a so-called frequency factor, E the activation energy, R thegas constant and T temperature. This relation can be plotted in an Arrhenius plot, which isa graph of ln(k) vs 1/T . When points measured in experiments make a straight line in thiscoordinate system the reaction rate obeys the Arrhenius law. Reactions with large activationenergies result in steep lines and can therefore be concluded to have rates that depend stronglyon temperature.
2.2 Crystalline salt-hydrates
Crystalline salt-hydrates (or shorter; hydrates) form a class of materials, in which water (oranother inert molecule) can be incorporated into the crystal lattice. The amount of watercontained within the lattice depends on both temperature and (vapor) pressure. In generalit is not possible for neutral atoms or molecules to be bound in ionic crystal structures. Themolecule 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 typicalspatial 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 inwhich 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 positiveregion 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 arebound chemically to the salt, therefore this is also denoted as a chemisorption process. Thewater molecules in such crystals are essential in the stability of the structure; i.e. removalof 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 ofstructures in which the cations are not directly coordinated by the water molecules. Theseare said to contain structural water. The structure of the material is very open and watermolecules merely occupy intersticial voids in the structure. Here they can add to the electro-static energy without upsetting the balance of charge, the bonding forces involved are calledVan der Waals forces. This type of bonding is also denoted physisorption and the energyinvolved is typically low, i.e. lower than 40 kJ mol−1. In some hydrate structures water isfound in both a coordinating and a structural capacity.
Table 2.1: The classification of hydrate structures
Containing only Containing only Containing both coordinatingcoordinating water structural water and structural water
Salt-hydrates may be represented by the general formula AaBb · cH2O, where A is the cationand B is the anion, such as Cl−, CO2−
3 , SO2−4 . The structure adopted will depend on the rela-
tive number of water molecules and cations present. The number of atoms or ligands directlybonded to an atom is called the coordination number. This number implies the geometricalarrangement 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 ratioc/a exceeds the coordination number of the cation, more water molecules are available thanare 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 struc-ture is shown, the framework is composed of siliciumoxide tetrahedra and water molecules can be incorporatedin the pores. Figure 2.1(b) represents a gas hydrate; a crystalline solid in which a gas molecule is surroundedby 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 whichcontain water in a coordinating role. Some examples are shown in Figure 2.1. Here a briefreview of the renowned example of zeolites will be given. Zeolites are framework silicatesin which (Si, Al)O4 tetrahedra are linked together into a three dimensional network. Thisnetwork is very open and can readily accommodate water molecules in its pores. The trivialfunction of water in zeolites is emphasized by the fact that it can be expelled from the crystallattice without destruction (or deterioration) of the structure and can even be replaced byother neutral molecules, such as ammonia. Much research has been done in the past yearson energy storage systems employing zeolites, which can be useful for this research to someextent. Zeolites are also widely used as molecular sieves.
Many hydrated salts contain an odd number of water molecules, of which some are present ina coordinating role and some occur in a structural capacity. The hydrates MgSO4·7H2O andCuSO4·5H2O may be quoted as examples. In the structure of CuSO4·5H2O, four moleculesof coordinating water coordinate the copper atom, which is also coordinated by two oxygenatoms of two SO4 groups and is therefore octahedrally surrounded by neighbors, as can beseen in Figure 2.2. The fifth water molecule is only coordinated by other water molecules andoxygen atoms.
Figure 2.2: Structure of CuSO4 · 5H2O. Each copper atom is coordinated by two sulfate groups and four watermolecules. The fifth water molecule is not coordinated by the cation, but by other water molecules and oxygenatoms 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 waterinto the crystal lattice, while remaining crystalline. When a hydrated salt is heated, the crystalwater is driven off. In a seasonal storage system solar heat can be employed to dehydrate thesalt-hydrate in summer. Subsequently the anhydrous salt is stored until winter. In winterthis salt is exposed to water vapor, initiating the reverse reaction that yields energy in theform 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 seasonalstorage was performed by [Visscher, 2004]. Requirements for candidate materials embody ahigh energy storage density, low corrosivity, reasonable cost and limited toxicity. The estima-tion 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 150oC. Anumber of salt-hydrates was investigated [Visscher, 2004] of which magnesium sulfate hepta-hydrate had the most promising properties. An overview of candidate materials is shown inTable 2.2. Most of the data presented here originates from [Wagman, 1982]. A number ofproperties for reactions of the form A ⇀↽ B +C are given, namely the reaction temperature ofboth the association and dissociation reaction, the accompanied reaction entropy change, itsenthalpy 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 B C Reac. temp. ∆S ∆H Energy dens.dis/as [oC] [J/mol/K] [kJ/mol] [GJ/m3]
MgSO4 · 7H2O MgSO4 7H2O 200/122 1041 411 2.8
FeCO3 FeO CO2 -/180 178 81 2.6
MgSO4 · 7H2O MgSO4·H2O 6H2O 150/105 887 336 2.3
Fe(OH)2 FeO H2O 150/150 137 58 2.2
CaSO4 · 2H2O CaSO4 2H2O -/89 290 105 1.4
MgSO4 · 1H2O MgSO4 1H2O 200/216 154 75 1.3
CaCl2 · 2H2O CaCl2·H2O H2O -/174 104 47 0.6
for descending energy storage density. The full dehydration of MgSO4 ·7H2O to the anhydrouscomponent has the highest energy storage density and an association reaction temperaturewhich is high enough for the purpose of heating tap water. Thereby it is marked as the mostpromising material. A drawback of the use of FeCO3 is that the reaction product CO2 ismore toxic than water. The dehydration of MgSO4 · 7H2O into the mono-hydrate still has ahigh energy storage density, while the reaction temperatures of the reactions are lower. Thedissociation reaction temperature corresponds better to the maximum yield temperature ofcurrent solar vacuum collectors and the association temperature is still high enough for tapwater 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 acombination of temperature and vapor pressure. In literature, some data is found on theintermediate crystalline phases that occur in the dehydration of magnesium sulfate hepta-hydrate. However some bias is found. In [Wagman, 1982] thermochemical data is reportedon the phases: MgSO4 · 7H2O (hepta-hydrate), MgSO4 · 6H2O (hexa-hydrate), MgSO4 · 4H2O(tetra-hydrate), MgSO4 ·2H2O (di-hydrate), MgSO4·H2O (mono-hydrate) and the anhydrouscomponent MgSO4. These intermediate phases are often summarized as MgSO4 · xH2O,where x = 7, 6, 4, 2, 1, 0. In [Paulik, 1981] it is stated that during dehydration MgSO4 · 7H2Odecomposes successively into the 3, 5
3 , 1 and the anhydrous salt. Some years later new datais reported by Paulik, together with three others in [Emons, 1990]. Here the phases MgSO4 ·xH2O, where x = 7, 3, 2, 1, 0 are described to appear during the decomposition. MgSO4 ·3H2Ois stated to be formed at a temperature of 105oC. At a temperature of 120oC the transitionto MgSO4 · 2H2O occurs. The transition to the monohydrate is reported to take place in atemperature range of 150 − 200oC and finally the anhydrous component is formed at 340oC.
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 measurementsthat combine a special crucible with a high resolution measurement [Paulik, 1981]. The 4 and2 hydrate are reported in [Vaniman, 2004] to be metastable, and do not occur in nature. Thedehydration behavior of MgSO4 · 7H2O is not reported unambiguously, therefore dehydrationexperiments will be performed, which are described in Chapter 4.
For MgSO4·7H2O both the enthalpy and entropy change per expelled molecule of water is moreor 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−1K−1 respectively. Themolar weight of magnesium sulfate is 120.37 gram and the molar weight of each water groupis 18.02 gram. The molar weight of magnesium sulfate hepta-hydrate is 246.48 gram and eachwater group that is expelled from the molecule coincides with a decrease in mass of about7.3%. MgSO4 · 7H2O 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 characterizingthermochemical materials. These methods can be utilized to quantify material parametersor system parameters. The term thermal analysis denotes a variety of measuring methods,which involve a change in the temperature of the sample investigated. Typical temperatureprograms involve linear increase of temperature with time, from now on referred to as constantheating rate, or periods where the temperature is kept constant for some time, from now onreferred to as isothermal platforms.
3.1 Thermogravimetric Analysis
By a technique called Thermogravimetric Analysis (TGA) the change in sample mass isanalyzed 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 whicha gas is liberated from the sample. This reduction in mass is registered as a function oftemperature and thus time. A known mass of sample material is placed in a small opencrucible, so that the sample mass is in the order of milligrams. Standard TGA crucibles aremade of platinum or alumina (Al2O3) and have a volume of about 70 mm3. The sample isthen placed in a furnace on a very accurate balance, where it is subjected to a temperatureprogram. In the case of salt-hydrates only water vapor is released from the sample. Thereforethe measured mass decrease can be directly converted into the number of moles of waterexpelled. The ambient within the system is continuously flushed by a so-called purge gas, inmost cases nitrogen. Typical purge flows are 0 − 200 ml min−1. There is a second gas flowpresent in the apparatus, that is called protective gas flow. The protective gas flows throughthe balance housing, thereby prohibiting potentially corrosive gas from the sample chamberto flow into the balance. The protective gas is typically a dry, inert gas, with a flow of about20 ml min−1. Due to the gas flows, the released reaction products are continuously carriedoff.
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 versustemperature and hence time.
the actual temperature program, the temperature range is set and the temperature program isdetermined by a control loop. When no mass reduction is registered, the furnace temperatureis increased at a constant rate. When a differential mass signal is registered however, thetemperature is maintained constant, until the reaction has completed. In this way transitiontemperatures can be determined very accurately, without lengthy measurements.
3.2 Differential Thermal Analysis
In Differential Thermal Analysis (DTA) the temperature difference between a sample andreference sample is measured while the samples are subjected to a temperature program. Thesamples are placed in separate crucibles in the same furnace. Generally an empty crucibleis used as the reference. A typical DTA setup can be seen in Figure 3.2. The output signalfrom 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 DTAsignal 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 theenthalpy change of the reaction. DTA measurements can be employed for temperatures upto 1600oC.
A DTA module can be extended with a TGA module, the system is then referred to as aDTA-TG or STA (Simultaneous Thermal Analysis) device and such a system is shown inFigure 3.2(b). The system now enables the user to perform a simultaneous measurement ofdifferential mass and differential temperature. This makes it easy for the user to distinguishbetween chemical reaction with change in mass, for instance a dehydration reaction, or withouta change in mass, such as melting of the sample.
3.3. DIFFERENTIAL SCANNING CALORIMETRY 13
(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 givenas output signal. In Figure 3.2(b) a schematic overview is given of the Netzsch STA 409 Luxx, which is acombined DTA-TG device.
3.3 Differential Scanning Calorimetry
The difference in heat flow to a sample and reference sample as a function of temperatureis measured by Differential Scanning Calorimetry (DSC) [Dean, 1995]. Both the sample andreference are maintained at nearly the same temperature. The reference sample should havea well-defined heat capacity over the range of temperatures to be scanned. When a thermaltransition occurs in the sample, more (or less) heat will need to flow to it than to the referenceto maintain both at the same temperature. The difference in heat flow is equal to the amountof 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
Constantanheating disk
Sample Ref.
Chromel wafer
Chromel wire
Alumel wireThermocouple junctions
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 additionthe disk serves as part of the temperature sensing unit. The samples are placed on raisedplatforms on the disk. Under each platform there is a chromel (another alloy) wafer. Thejunction 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 isattached to monitor the temperature. Samples are placed in disposable aluminum sample pansof 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 minimizesthermal gradients and maximizes temperature accuracy and resolution. DSC measurementscan be performed for temperatures up to approximately 700oC.
3.4 X-ray crystallography
X-ray crystallography (also called X-ray diffraction or XRD) is a technique that leads to anunderstanding of the material and crystal structure of a substance. In X-ray crystallographyX-rays are passed through a crystal and then caught on a photographic plate resulting ina 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 calledthe 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 thatstrike an atom are reflected off at the same angle θ. Because the crystal has multiple layersthe rays can reach multiple depths, thus resulting in different ray-travel distances. By usingtrigonometry it can be shown that the extra distance traveled between two consecutive layersis equal to twice the distance between atomic layers times the sine of the angle θ. The key tothe Bragg analysis is the realization that the rays striking the two layers of atoms are initiallyin phase, but can only be in phase after reflection if the extra traveled distance is equal toan integer number of wavelengths nλ (n = 1, 2, 3...). If the extra traveled distance is not aninteger number of wavelengths, then the reflected rays will be out phase and will cancel eachother. Imposing this requirement on the previously mentioned relationship gives the Braggequation:
d =nλ
2 sin θ(3.1)
X-ray powder diffraction is a specific example of X-ray crystallography, that is specificallyuseful for structural characterization of powdered materials. The output of the measurementare rings of diffracted intensity as a function of reciprocal lattice units. The method canbe used to quickly identify materials, by comparing the found pattern of powder diffractionpeaks with data from the International Centre for Diffraction Data pattern database. In acrystallographic study of the dehydration reaction of a salt-hydrate, a sample is placed ina camera and then subjected to variations in temperature and pressure in order to inducethe desired reactions. When the diffraction pattern is recorded continuously, while eitherchanging sample temperature or water vapor pressure, phase changes in the material can bedetermined.
3.5. PHASE DIAGRAM 15
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) denotesan example of an X-ray powder diffractogram. The peaks represent positions where the X-ray beam has beendiffracted by the crystal lattice.
3.5 Phase diagram
Phase diagrams are useful for identifying the equilibrium conditions between the thermody-namically distinct phases. Phase diagrams for a single substance, such as water, representthe lines of equilibrium between the three phases (i.e. solid, liquid and gas) spanned bytemperature and pressure. When a boundary line is crossed by changing either temperatureor pressure, a phase change occurs. Phase diagrams can also be constructed for a system inwhich more than one component is present. The composition of the mixture then becomesan important variable. A binary phase diagram represents the relative concentrations of twosubstances against temperature. For salt-hydrates two independent components are present;the crystalline salt and water. For such materials a composition versus temperature diagramfor a constant pressure (mostly 1 atm.), is a very insightful tool. An example is shown inFigure 3.6. A vertical line drawn in a phase diagram is called an isopleth (Greek: equal abun-dance), which is a change to the system that does not affect the overall composition. Whena phase diagram is not at hand in literature, it is possible to construct one from experimen-tal 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 thestable states of the material are known, the melting temperatures of each state should bedetermined 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 bedetermined by using a pressure sensor that is connected to a sealed glass container filled witha sample of the salt-hydrate. The container is placed in a temperature-controlled bath andthe sample temperature is monitored, for instance with a thermocouple. The temperature ofthe bath is raised 10 K after which the sample temperature is left to stabilize. This takesapproximately 15 minutes. When the system is thermally stabilized, the equilibrium vapor
16 Analysis methods
Figure 3.5: Phase diagram of the Na2S·H2O system and measured transition temperatures. Phase denomina-tions: V: vapor, L: H2O based solution, C1, C2 and C3 are the crystalline phases Na2S·9H2O, Na2S·5H2O andNa2S·2H2O respectively. Graph obtained from [De Boer, 2002].
pressure is read. In this way a temperature range can be scanned and pressure readings arerecorded for both heating and cooling. The measurement should be performed for each stablestate of the material. The results of the pressure measurements can be used to calculate thechange in enthalpy and entropy of the dehydration reactions, by using Equation 2.6. A vaporpressure 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 forwater, Na2S·9H2O, Na2S·5H2O and Na2S·2H2O. Graph originates from [De Boer, 2002].
3.7. SCANNING ELECTRON MICROSCOPY 17
3.7 Scanning Electron Microscopy
The scanning electron microscope is designed for the visualization of surfaces of solid objectsat very high resolution. An electron gun produces electrons and accelerates then to an energybetween about 2 and 40 kVe [Goodhew, 1988]. Two or three condensor lenses then focusthe electron beam into a very fine focal spot. The diameter of which can be as small as2 − 10 nm. The beam passes through pairs of scanning coils in the objective lens, whichdeflect 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 twodimensional intensity distribution. The construction of this 2D image depends on the raster-scanned primary beam. The brightness of the signal depends on the number of secondaryelectrons 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 surfaceof the specimen, a higher intensity of secondary electrons is detected. Therefore edges andsteep surfaces are brighter than flat surfaces, resulting in images with a well-defined, threedimensional 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 electronmicroscope over an optical microscope is the large depth of field. For typical operatingconditions, for a magnification of 1000×, the depth of field for an electron microscope and anoptical microscope are 40 and 1µm respectively. A downside of using SEM to produce imagesof salt-hydrates, is the fact that the images are made under vacuum. During the evacuationof the apparatus, the salt-hydrate expels its water, therefore only images of the anhydrouscomponent can be made in this setup. There is a technology called environmental SEM orESEM 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 thecurrent research.
18 Analysis methods
Chapter 4
Experimental results
A number of the analysis methods that are described in Chapter 3 are used for characteriza-tion 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 thateach have their own significance. First some experimental results are discussed that character-ize the material itself, such as the dehydration behavior of the salt-hydrate, the determinationof material parameters and the crystal structure, this part is called material characterization.Secondly the focus lies on experiments which determine parameters that check the applica-bility of magnesium sulfate hepta-hydrate as a thermochemical storage material, this sectionis called system characterization. An extensive list of the applied experimental conditions isgiven in Appendix B.
4.1 Material characterization
The goal of the first set of experiments is to obtain the parameters that determine the spe-cific thermodynamical behavior of the material. These parameters are used as input in thenumerical model. In chemical conversion there are three mechanisms that can affect the prop-agation 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. Forthe purpose of characterizing the material properties, preferably all individual grains of asample are submitted to the same thermodynamic constraints, as they then react at the samerate. Therefore small samples should be employed, in the order of 10 milligram or less. Careshould be given however to the trade-off between signal strength that is required for highexperimental resolution and the desire to employ small samples and low heating rates. Forquantitative studies in the measurement devices which were used for this research a sampleof at least 1 mg is required.
19
20 Experimental results
4.1.1 Stable hydrates
As described before in Section 2.2, literature data on the stable hydrates that appear in thedehydration of MgSO4 ·7H2O is not consistent. Therefore some experiments are performed todetermine the dehydration behavior of the salt-hydrate and the stable phases that are formedduring dehydration. As stated in Section 2.2 each dehydration step corresponds to a decreasein mass of about 7.3%. It should be noted that the mass decrease that is measured in anexperiment only relates to a mean decrease in the sample; it is possible that only part of thesample transformed and part of the sample remains unchanged. The actual composition ofa sample can best be checked by means of a XRD measurement, in the present research thismeasurement 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 sampleof 12 milligram was placed in an open crucible and the purge gas was saturated with watervapor at room temperature (i.e. a vapor pressure of approximately 30 mbar). The results canbe found in Figure 4.1, in which the upper panel shows the sample mass versus temperatureand the lower panel depicts the derived mass versus temperature. In this experiment it is
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s [%
]
50 100 150 200 250 300 350 4000
1
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Der
ived
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/ o C]
Temperature [oC]
Figure 4.1: Dehydration behavior of MgSO4 · 7H2O measured by high resolution measurement on a TGAapparatus. The upper panel shows the mass signal as a function of temperature and in the lower panel thetemperature derivative of the mass signal [ % oC−1] is represented as a function of temperature.
seen that MgSO4 ·7H2O first loses approximately 3.5 water groups at a temperature of about64oC and subsequently gradually expels about 2.5 groups. A last transition is seen at atemperature of about 270oC where 0.5 group is released. The constant decrease in mass inthe temperature span of 70 − 270oC 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 decreasein 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, whichin thermal analysis experiments means employing high heating rates. A large drawback toemploying high heating rates however is the increased chance of local melt. Salt-hydrates with
4.1. MATERIAL CHARACTERIZATION 21
a large fraction of bound water are characterized by low melting temperatures. Dehydrationreactions take time, due to the previously mentioned phenomena of reaction kinetics andlimited mass and heat transfer. In this way, if a sample is heated at too high rate, thedehydration reaction is not completed before the melting temperature is reached. Meltingis an undesired effect because it slows down the reaction, destroys the crystal structure andreinforces 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-TGmeasurement as an endothermic peak in differential temperature signal, without a change inmass. 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 thedecomposition can alter, as can be seen in Figure 4.2. The experiments from Figure 4.2(a)
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(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 theTGA 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. Anauto-sampler is a carousel device in which up to 50 samples can be placed. Samples forsuccessive measurements are prepared at once and the auto-sampler then allows around-the-clock sample analysis, without supervision of an experimentator. The samples which are inqueue for measurement are in open contact with the ambient. The experiments from Figure4.2(b) are performed on the TGA device at ECN. All measurements are performed on samplesof about 10 mg and under a purge flow with a vapor pressure of 30 mbar. When comparingthe results for a heating rate of 1 K min−1 (i.e. the solid line of Figure 4.2(a) and the greysolid line of Figure 4.2(b)) the first observation is that the measurement that is performed atECN displays a transition around T = 40oC which is not found in the measurement whichwas performed with the auto-sampler, where it was the last in queue. Moreover, the samplemass at the end of the measurement with auto-sampler is higher than expected. This leadsto conclude that the initial composition of the material was not the hepta-hydrate, but thehexa-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. Theexperimental result for this heating rate should be scaled and shifted downward in order tobe able to compare it with the other measurements. The end composition is then found to bethe same for all the measurements of Figure 4.2(a), namely the anhydrous component. Theresults for a heating rate of 10 K min−1 are consistent and the overall trend of the curves asa 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 mate-rial. These parameters, such as the thermal conductivity λ, the heat capacity cp and densityρ, should be obtained, so that they can be used in as input for the model. Most materialparameters can be obtained from literature. Where bias in data is found, it is advisable todetermine 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 forMgSO4 ·7H2O to 2660 kg m−3 for MgSO4. These values and those of the intermediate phasesare given in Table 4.1. The density of the salt-hydrate is found to increase when the amountof water, and therefore the molar mass, decreases. The dehydration of the hepta-hydrate tothe anhydrous component corresponds to a decrease in mass of 51% and a density increaseof 58%. Combination of these numbers, yields that the material volume after dehydrationis 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-hydrateis lower than the material density, due to the void fraction between the individual grainsof the material. This void fraction is also denoted macro-porosity and can be determinedby filling a cup with a known volume with the salt-hydrate, then measuring the mass andequating the material density with the actual density. A crucible of 70 mm3 was completelyfilled with 58 mg of MgSO4 · 7H2O, which has a reported density of 1680 kg m−3, yielding amacro-porosity of 0.5.
Table 4.1: Material parameters from literature
M ρ cp λ H[ g mol−1] [ kg m−3] [ J kg−1K−1] [ Wm−1K−1] [ J mol−1]
MgSO4 · 7H2O 246.48 1680 1546 0.48 3364
MgSO4 · 6H2O 228.46 1750 1525 3062
MgSO4 · 4H2O 192.43 2010 1305 2471
MgSO4 · 2H2O 156.40 1124 1871
MgSO4 · 1H2O 138.38 2570 1047 1575
MgSO4 120.37 2660 800 1260
Little information was found on the thermal conductivity of the salt-hydrate. A value of0.48 Wm−1K−1 was found for MgSO4 · 7H2O in [Washburn, 1930]. Because there was nomeasurement device available to determine the thermal conductivity, this value is used in the
4.1. MATERIAL CHARACTERIZATION 23
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−1K−1 for MgSO4 · 7H2O and 1 J g−1K−1 for MgSO4·H2O. Theheat capacity of the hepta-hydrate is also determined experimentally by means of a DSCexperiment, in which a small sample was placed in a closed aluminum sample pan. Thesample is then heated at a heating rate of 5 K min−1. When no reaction takes place in thesample and it is merely heating up, the heat flow to the sample that is measured at thattime, correlates with the heat capacity of the material. In this particular experiment a heatflow of 0.24 mW is measured and a sample mass of 1.876 mg was used. The heat flow permilligram is then determined and divided through the heating rate per second. This yield aheat capacity of 1.5 J g−1K−1, which corresponds with the values found in literature.
The melting temperature of a material can be determined by a coupled DTA-TG measure-ment. It is measured as an endothermic peak in heat flow, that is not accompanied by adecrease in mass. This measurement was performed on a sample of 10 mg which was heatedat a rate of 20 K min−1 over a temperature range of 25 to 325 oC. The result, shown inFigure 4.3(a), displays quite a large peak in DSC signal around T = 60 oC that is not accom-panied by a differential mass signal. The exact melting point is then determined with a DSC
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Figure 4.3: Results from the melting experiment. Figure 4.3(a) shows the DTA-TG result, where the solid linerepresents the DSC signal (which is enlarged two times for scaling) and the dashed line the TG signal. Figure4.3(b) shows the result from the DSC measurement.
measurement where a sample is placed in a sealed aluminum pan, containing little emptyspace. In this way the melting point of MgSO4 · 7H2O was determined to be 52.5 oC, whichis shown in Figure 4.3(b). After the measurement it was determined that no mass had leftthe sample pan, which confirms that no dehydration took place and the determined meltingtemperature is indeed that of the hepta-hydrate.
24 Experimental results
4.1.3 Surface structure
By means of SEM the surface of grains of the salt-hydrate are studied and the effect of thermaldehydration 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 (magnificationapproximately 150×). The sample on the left was not treated before, the sample in Figure 4.4(b) was thermallydehydrated.
depicted. The sample on the right hand side was thermally dehydrated before, the sample onthe left did not undergo previous thermal treatment, but expelled its water in the electronmicroscope under influence of the deep vacuum. The surface of the thermally dehydratedgrains 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. Inorder to validate this statement, the grain surface was also studied for a sample that wasmelted in a DSC measurement, as is already elucidated in Subsection 4.1.2. The results areshown 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 ofgrains with a diameter of 200 − 500 µm.
grains. The surface of the material has changed dramatically, and the structures that areformed resemble the irregularities which were found in Figure 4.4(b). This leads to believe
4.1. MATERIAL CHARACTERIZATION 25
that to some extent melting of the sample, visualized in Figure 4.4(b), has occurred duringthe thermal dehydration of the sample.
4.1.4 Reaction Kinetics
The kinetics of first order chemical reactions are described by the Arrhenius equation. Theconstants, i.e. the frequency factor A and activation energy E, of this equation can bededuced from experimental measurements. The calculation method is based on the factthat the maximum yield of products from a conversion reaction occurs at the peak of thedifferential mass signal. The corresponding temperature is called Tmax. For higher heatingrates, the conversion peak shifts to higher Tmax values. This shift can then be used to calculatethe Arrhenius parameters. An explanation of the exact calculation method can be found inAppendix A.
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Temperature [deg C]
H = 5 K/minH = 10 K/minH = 20 K/min
Figure 4.6: Differential mass signal versus temperature for three different heating rates. The peak temperatureshifts to a higher value for higher heating rate.
Dehydration experiments are performed for three different constant heating rates. The re-sults 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 curvesrepresent the differential mass signal as a function of temperature. The peak shift to highertemperatures for higher heating rate that was expected, is found in the experiment. For thelowest heating rate a small clear peak at low temperature is found, that is hardly distinguish-able in the higher heating rate experiments. The peak shape of the large peak differs for thetwo higher heating rates in comparison to the smooth shape of the lowest heating rate. Thehighest point of each peak is used here to determine the kinetic parameters. Furthermore anadditional peak is found at a temperature of about 150oC for the two high heating rates.
The small first peak, that appears around 50oC yields kinetic values that are not plausible;a frequency factor of 1 · 1024 s−1 is found. The second and larger peak however providesgood results. For the lowest heating rate, the maximum yield for this peak is found at
26 Experimental results
Tmax = 90oC. For a heating rate of 10 K min−1 it is found at Tmax = 113oC and for a heatingrate of 20 K min−1 one finds Tmax = 132oC. The Arrhenius parameters are then calculatedand lead to an activation energy of approximately 55 kJ mol−1 and a frequency factor of about1.67 · 105 s−1. In [Ruiz, 2007] kinetic parameters for the dehydration of magnesium sulfatehepta-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 valuespresented here.
4.2 System characterization
When the material itself is characterized, the influence of different operating conditions onthe progress of the reactions is investigated and these results are compared with the resultsobtained from the numerical continuum model. Then it is possible to set boundaries foremployment of salt-hydrates in a heat storage system. For some of these experiments largersamples are employed, so that transient effects in heat and mass transfer over the layer ofmaterial 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 thesample.
4.2.1 Grain size
For small grains the temperature distribution within each individual grain is stated to beisothermal. When larger grains are used, the isothermal assumption is no longer valid andmass transport may also become a limiting factor. A difference is expected that case betweenexperimental and numerical results, because the numerical model is a continuum model, andtherefore transient effects on the smaller scale are not taken into account. The grain size isexpected to influence the progress of the reaction. The original sample material incorporatesa wide distribution of grain sizes. By means of sieving it is possible to isolate smaller rangesof grain size and by means of grinding smaller grain sizes can be attained. Sieving showedthat the largest fraction of sample had a grain size of 200−500µm. Only a very small fractionof grain size 500 − 600µm was found and a substantial fraction of the sample consisted of agrain size smaller than 200µm. An experiment was performed where a sample of about 12 mgwas heated at a constant rate of 10 K min−1 in a temperature range of 25 − 325 oC. Themeasurement is performed for equal sample mass on two different grain sizes: 200 − 500 µmand 38− 100 µm, the results of which are depicted in Figure 4.7. It is seen that the influenceof grain size on the propagation of the reaction is strongest in the temperature range of50−200 oC. The end product for both experiments is equal, but the timescale of the reactionis larger for larger grain size.
4.2.2 Layer thickness
For increased layer thickness, temperature and mass transport through the adsorbent bed maybecome a problem and transient effects become significant. In order to research this effect an
4.2. SYSTEM CHARACTERIZATION 27
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]
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38−100 mu200−500 mu
Figure 4.7: Dehydration behavior for different grain size, here represented as mass versus temperature. Thestraight 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 isothermalplatforms, the first at 45 oC and the second at 150 oC. Between the platforms a high heatingrate was used of 30 K min−1. This situation was chosen because it simulates actual operatingconditions, 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. Bothsamples that are used have a grain size of 38 − 100 µm, the mass of the small sample was11.4 mg and the large sample was 38.2 mg. In Figure 4.8 the sample mass is presented versustime. The graph shows a clear influence of layer thickness on the propagation of the reaction.
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Figure 4.8: The sample mass as a function of time for different layer thickness; the large sample, representedby 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. Thesmall sample reaches its end product approximately within 5 minutes, whereas the largersample 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 betweenthe conversion curves increases. Overall the result implies that for very large layer thicknessthe propagation of the reaction decelerates significantly, which sets limits to the maximumadsorbent 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 combinationof temperature and vapor pressure. For increased vapor pressure in the ambient, transitiontemperatures are stated to shift to higher values. The purge gas that is used in the experi-ments can be saturated with water vapor by bubbling the gas through water at a constanttemperature. The applied vapor pressure can be calculated from the Clausius Clapeyronequation (Equation 2.6). For water of 24oC a water vapor pressure of 30 mbar is applied tothe gas. The flows of both purge and protective gas through the apparatus can be adjusted inorder 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. Thepurge 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 vaporcontent of 2.5 · 10−5 to 2 · 10−4 mol min−1 at a vapor pressure of respectively 15 to 26 mbaris realized.
An experiment is performed on samples of 10 mg, which are heated at a constant rate of5 K min−1 for three different purge flows. In Figure 4.9 the result of the experiment is
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Figure 4.9: Dehydration experiment for three different purge flow vapor pressures, shown here for the differ-ential sample mass versus time. The solid line represents a vapor pressure of 15 mbar, the dashed line a vaporpressure 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 dehydrationreactions take place at lower temperatures and the first dehydration peak is significantlylarger. The water vapor content of the ambient is shown to limit the propagation of theconversion reaction. For a future system it is therefore important to realize a continuous
4.2. SYSTEM CHARACTERIZATION 29
removal of water vapor from the ambient, for instance by a continuous gas flow with lowvapor 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 thereaction is reversible and can be performed a number of times, this property is called cyclabil-ity. In Figure 4.10 the result of a cycling experiment is presented. A sample of 10.7 mg is first
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Figure 4.10: Dehydration experiment performed three times successively on one sample, displayed here is thesample mass versus time.
heated to 45oC at a heating rate of 10 K min−1, then kept isothermal at that temperaturefor 5 minutes and then heated to 150oC at a heating rate of 30 K min−1 and kept isother-mal at that temperature for 15 minutes. After the measurement, the sample was allowed torehydrate under atmospheric conditions for 24 hours. The sample mass was constant for thelast 3 hours, which indicates that the rehydration process came to an end. This cycle wasperformed three times on the same sample. In the first run the sample mass decreased from10.7 mg to 6.2 mg, which is a reduction of approximately 42%. The material then does notrehydrate fully, but stabilizes at a sample mass of 9.6 mg (i.e. approximately 90% of theoriginal sample mass). After the second dehydration, the sample mass is 6.3 mg, which isalmost equal to the sample mass after the first dehydration. In the following rehydration lesswater 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 hasa mass of 6.3 mg, which is again almost equal to the sample mass after the first dehydra-tion. This result implies that the cyclability of the material is poor, however more researchis necessary to justify this first conclusion.
Another cyclability experiment is performed in a TGA setup under gas with a water vaporpressure of 24 mbar. A sample of 9.6 mg is first heated from 25− 200oC at a heating rate of1 K min−1, and subsequently cooled down at the same rate to 25oC. Finally the temperaturewas kept constant at 25oC for 200 minutes. The result of this experiment is depicted in
30 Experimental results
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Figure 4.11: Dehydration- rehydration experiment performed in the presence of water vapor. The sample massis represented versus time.
Figure 4.11. It shows that first a dehydration reaction takes place, yielding a mass reductionof about 47%. Over a large part of the cooling period no change in mass is found, untilat sufficiently low temperature the rehydration reaction is initiated. During the isothermalperiod a steady mass increase is observed. In the total time of 200 minutes, 25% of mass istaken 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 wasregistered. This method gives more insight into the rehydration kinetics than the previouslymentioned experiment. However, this last measurement is very time consuming, which cancause problems regarding measurement device occupation.
4.3 Experimental resolution
Before processing experimental results it is important to check the resolution of the measure-ment methods that were used to obtain the results. Experimental measurements can be verysensitive to operating conditions and may inhibit large errors. Therefore some research wasdone into the quality of the measurements that are performed.
For thermal analysis measurements a temperature trajectory is given as input. For the TGAand DTA-TG setup a check was performed between the prescribed temperature and the tem-perature realized in the experiment. The result is shown in Figure 4.12. Both measurementsare performed with an empty alumina pan. The temperature program for the TGA mea-surement started with heating from 25 − 40 oC at a heating rate of 40 K min−1, followedby an isothermal platform at T = 40 oC for 5 minutes. Then the sample was heated from40 − 270 oC at a heating rate of 40 K min−1 and kept isothermal at the end temperaturefor 15 minutes. The temperature program for the DTA-TG device started with heating from25−45 oC, followed by an isothermal part for 5 minutes. The sample was subsequently heatedfrom 45−150 oC at a heating rate of 30 K min−1 and kept isothermal at the end temperaturefor 15 minutes. The realized temperature which is shown in Figure 4.12 is measured by a
4.3. EXPERIMENTAL RESOLUTION 31
0 5 10 15 20 25 300
50
100
150
200
250
300TGA
precribedrealized
0 5 10 15 20 25 300
50
100
150
200DTA
Time [min]
Tem
pera
ture
[o C]
Figure 4.12: Comparison between prescribed temperature and realized temperature. Figure (a) is a measure-ment 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 realizedtemperature, in both cases an overshoot in temperature is realized when an isothermal plat-form is prescribed. At higher temperatures the TGA setup clearly performs better than theDTA-TG setup. An overshoot in temperature can cause problems if an isothermal platformis desired close to a melting temperature. When processing experimental data, one shouldbare in mind that the sample temperature can differ significantly from the prescribed tem-perature. Results should therefore be plotted for sample temperature rather than prescribedtemperature.
For TGA the repeatability of the measurements was also verified. A constant heating rateexperiment was performed five times on samples of approximately equal size. The result ofthese measurements is given in Figure 4.13. A sample of about 10 mg is heated at a constantheating rate of 10 K min−1 from 25− 325oC. In the graph the sample mass is plotted versustemperature. The black line is the calculated mean sample mass of the five experimentsand the grey dotted line represents the standard deviation of the experimental data. Themeasurement is characterized by small standard deviation, however locally a larger deviationis found around 120oC. The occurrence of this increased deviation coincides with a largedecrease in sample mass. Overall it is concluded that the repeatability of the measurementsis high.
32 Experimental results
50 100 150 200 250 30050
55
60
65
70
75
80
85
90
95
100
Temperature [oC]
Mas
s [%
]
Figure 4.13: Reproducability measurement performed on a Mettler Toledo TGA apparatus. Depicted here isthe sample mass versus temperature. The black line represents the mean value of the five measurements, thegrey dotted line is the standard deviation.
Chapter 5
Modeling
In order to check the viability of the use of thermochemical materials in an energy storagesystem, a model has been made. This model gives a better phenomenological understandingof the processes present and makes it possible to assign the most important mechanisms andphenomena. Model parameters can be obtained from literature or by performing simple ex-periments. 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 conditionsfor a future operational system, regarding both design and operating conditions. The pos-sibility for prediction of system behavior at different design and operating conditions, willsubstantially 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 formass, momentum and heat. The equations are presented here in general form and are obtainedfrom [Schram, 1994]. The first conservation equation is that of mass, also called the continuityequation:
∂ρ
∂t+ ∇ · (ρv) = 0 (5.1)
where ρ is the density and v the velocity vector. The conservation of momentum for incom-pressible media is written:
∂v
∂t+ (v · ∇)v = −
1
ρ∇p + ν∇2v + f (5.2)
Here p is the pressure, ν the kinematic viscosity and f a volume force, due to an externallyapplied force, e.g. the gravitational acceleration. The conservation of energy is given here inthe so-called temperature notation:
∂
∂t(ρcpT ) + ∇ · (ρcpvT ) = ∇ · (λ ∇T ) + Q (5.3)
where T is the temperature, cp the heat capacity, λ the thermal conductivity and Q a sourceterm. 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 achemical system is described that consists of multiple components, the conservation of eachcomponent can also be stated. The total of balance equations can then be extended with anequation for the conservation of chemical species:
∂Yi
∂t+ v · ∇Yi = ∇ · (D ∇Yi) +
1
ρm′′′
i (5.4)
where Yi is the concentration of each species i, D represents the diffusion coefficient and m′′′
i
is a mass source term related to chemical reaction.
In the conservation equations of momentum, energy and species, the first term on the left-hand side is the change in time and the second term is the convective term (transport due tomacroscopic movement). On the right-hand side the first term is a diffusion term (transportdue to microscopic movement). The remaining terms are different source terms, except forthe momentum equation, in which the pressure gradient also appears.
5.2 Grain models
Before looking into system-behavior of systems employing thermochemical reactions, it isuseful to study the reactions and their kinetics on particle scale first. The reactions that occurin the dehydration or rehydration of crystalline salt-hydrates can be more generally calledsolid-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 stepwood is converted into char, gas and sometimes tar. Also the combustion process of a charparticle in oxygen is an example of a solid-gas reaction. These reaction processes have beenstudied 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 bedescribed by the spherical shrinking core reaction model. The spherical shrinking core modelassumes that in a reacting spherical particle of radius r0, there exists an unreacted shrinking
5.2. GRAIN MODELS 35
core of decreasing radius r surrounded by a growing layer of products. In a reacting particlethree 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 potas-sium carbonate hydrate (K2CO3·1.5H2O). Through experiments and analysis he showed thatthe spherical shrinking core model can be applied to this salt-hydrate. Experiments wereperformed both in the presence of water vapor as well as in vacuum. He showed that thetemperature gradient over the product layer of a particle with an initial radius smaller than 1mm is about 0.008 K and thus negligible. The pseudo-steady state assumption, which impliesthat temperature transients are rapid and heat transport is considered the only source of en-ergy for dehydration, is therefore valid. His study also showed mass diffusional resistances tobe negligible for slow dehydrations of small particles (125−149 µm), but significant for largerparticles or high dehydration rates. The interface velocity of the propagating front r ≡ dr/dtwas 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 statesthat biomass pyrolysis can be described through a primary and a secondary stage. Wood
Wood
k1
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, charand tar. Tar may undergo secondary reactions to char and gas. To make a comparisonwith the current research it suffices to take into account the primary conversion of woodinto char and gas. The formation of tar is thus discarded. As the relative amount of endproducts may vary with conversion parameters, two kinetic parameters are used to describethe conversion; one for each component. [Di Blasi, 1996] introduces a one dimensional modelof a large (τ = 0.025 m) single particle, which is regarded as a slab. The particle is subjectedto a radiative heat flux and degrades thermally. Conservation of mass is stated for eachcomponent. 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 valuefor the species formed. For the solid-phase species wood it reads:
∂ρw
∂t= −(k1 + k3)ρw (5.5)
The parameters k1 and k3 denote the chemical reaction rate and are deducted from an Arrhe-nius 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 thereaction and the chemical reaction source term.
∂
∂t(Tρcp) +
∂T
∂x(ρgcpgu) = λ
(∂2T
∂x2
)
+ (k1 + k3)ρw∆H −ρcpT
V
∂V
∂t(5.6)
The x direction corresponds to the thickness of the slab of biomass. The convective velocityof the gas phase is calculated by the Darcy equation, which reads:
u = −κ
µ
∂p
∂x(5.7)
The pressure term that appears in the Darcy equation is calculated from the ideal gas law inthe following way:
p =ρgRT
Mg(5.8)
where Mg is the molar mass of the gas and R the gas constant. The expression ρcp is the totalheat 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 experimentallythat wood particles experience reductions in particle size up to 60%. The volume occupiedby the solid is assumed to decrease linearly with the wood mass and increase with the charmass, by a chosen shrinkage factor, α, as devolitalization takes place:
VS
Vw0=
mw
mw0+
αmc
mw0(5.9)
Where Vw0 is the initial effective solid volume. It was found that dehydration of MgSO4 ·7H2Oinduces volumetric reduction. This was not included in the current model however. DiBlasiperformed numerical simulations with the described model, in order to simulate the effectsof the shrinkage level, the intensity of the heat flux and the effect of different sets of kineticdata. A qualitative agreement between model predictions and experimental data was found.However, the properties of char and partially charred wood are very poorly known, yieldingpoor quantitative results. The conclusion of the research is that further study is needed toestimate reliable kinetic data (on semi-global scale) and to investigate the dependence ofphysical properties on temperature and solid composition. The uncertainty on the physicalproperties in the current research is smaller, therefore the method described by DiBlasi isconsidered to be valuable.
5.3 Layer models
Reversible solid-gas reactions can be applied well in adsorption heat pumps and storage sys-tems. These reversible reactions take place in fixed-bed reactors, effected by constraints oftemperature and pressure. A lot of modeling work has been performed on adsorption heatpump systems, mostly with the working pair of zeolite and water (vapor) or silicagel and wa-ter. These models are very insightful as the function of the systems described is similar to thecurrent situation, although the principles behind the reactions at hand differ. The availablemodels represent bi-variant systems and differ on that facet from the currently researchedsystem. 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 canbe 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 throughthe gas path, where it is adsorbed while generating heat. The heat that is produced is carriedoff by the heat transfer fluid and available for use. When the adsorbent bed is saturated withvapor, the system is operated in opposite direction; heat is supplied to the adsorbent bed bythe heat exchanger. The increased temperature of the bed effects the desorption reaction andthe water vapor is carried off through the gas path. This cycle can be repeated numeroustimes. The system consists of three main elements, relevant for the mathematical model; theheat transfer fluid, the metal tube and the adsorbent material. The latter includes a poroussolid and water vapor both in gaseous and adsorbate phase. The modeling of such systems isbased 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 andmass. Here only the adsorbent bed is discussed. In the following equation, the subscript gdenotes 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
∂t+
1
r
∂(rvrρg)
∂r+
∂(vzρg)
∂z+ (1 − εt)ρs
∂w
∂t= 0 (5.10)
where v represents the velocity vector, which is determined by Darcy’s equation for transportin porous media; v = −κ
µgrad(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 porosityof 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:
εt∂(ρgcpgTs)
∂t+ (1 − εt) ρs
∂
∂t
[
(1 + w) cpeqTs
]
+1
r
∂
∂r
[
(ρgcpgTs + p) rvr
]
+∂
∂z
[
(ρgcpgTs + p) vz
]
= λeq
[1
r
∂
∂r
(
r∂Ts
∂r
)
+∂2Ts
∂z2
]
+ ρs(1 − εt) |∆H|∂w
∂t(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 porouslayer. Subsequently the vapor uptake distribution, the amount of heat exchanged and theperformance of a given adsorption system can be determined. A base-case assembly wasformulated in which the system performs exactly one cycle. This base-case was used inperforming a sensitivity analysis for a number of parameters. This sensitivity analysis revealedthat increased layer thickness results in increased cycle duration; it was 460 s for a layerthickness of 3 mm, 926 s for 5 mm thick and 1547 s for a thickness of 7 mm. Furthermorethe adsorbent bed configuration was found to have a large impact on the performance of thesystem, 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 EngineeringModule. In this model a layer of hydrated magnesium sulfate is considered as a homogeneouscontinuum and the heat and mass transport through this porous layer is calculated. Thematerial then experiences a single-step chemical conversion, that is based on a transition stepthat was recorded in experiments, as is shown in Figure 5.4(a). The conversion reaction willbe described as a solid S1, that is converted into a solid S2 and a gas phase G, constitutedby water vapor. The reaction kinetics were determined previously in Subsection 4.1.4 for thelarge transition, i.e. the large peak in differential weight loss. The initiation of this reactionis well defined and is found around t = 400 s. The endpoint of the conversion however is lesstrivial. Besides that, the exact composition of the material at the starting point and at theend is unknown. The first model is based on the assumption that the conversion is limited tothe symmetric part of differential mass signal. The transition then occurs between t = 400and 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 massdecrease of 7.3%, as stated before in Section 2.3, in this case 4 water groups are expelled. This
5.4. CURRENT MODEL 39
0 200 400 600 800 1000 1200 1400 1600 1800 200050
55
60
65
70
75
80
85
90
95
100
105
Time [s]
Mas
s [%
]
ExperimentDiff. massModel 2Model 1
(a)
0 200 400 600 800 1000 1200 1400 1600 180055
60
65
70
75
80
85
90
95
100
Time [s]
Mas
s [%
]
experimental resultnumerical model 1numerical model 2
(b)
Figure 5.4: In Figure 5.4(a) an experimentally determined dehydration curve for MgSO4 ·7H2O is shown. Thesolid line is the sample mass and the dashed line is the scaled differential weight loss. The part of the solid linethat is marked in grey is the conversion that is used in numerical model1, the black marked part of the lineis the conversion which is used in numerical model2. In Figure 5.4(b) the output of the model is compared toan experimental result.
observation was compared with literature data on stable states of the salt-hydrate and ledto the conclusion that the corresponding reaction is the dehydration from the hexa-hydrateto the di-hydrate. The numerical results showed poor resemblance with experimental resultshowever, which is seen in Figure 5.4(b). A second model was then formulated, which includeda 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 = 0was not the hepta-hydrate, but a slightly lower hydrate. A consequence of this assumptionis 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. Thecomposition of the material at t = 400 was found to be the hexa-hydrate and the end productthe mono-hydrate, or written: MgSO4 · 6H2O ⇀↽ MgSO4·H2O + 5 H2O.
The numerical model is based on the assumptions that the system is in local thermodynamicalequilibrium, which means that locally the temperature of each phase is equal. The vaporvelocity is governed by Darcy’s law for flow in porous media, the gaseous phase behaves asan ideal gas, the chemical reaction is of first order and therefore the reaction kinetics can bedescribed by an Arrhenius-type equation. Volumetric changes are neglected and the porosityof the material is considered to be constant.
The energy balance equation for a closed system was already stated in Equation 5.3. The termρcp is now composed of the contributions of all phases present, i.e. ρs1cps1 + ρs2cps2 + ρgcpg.Furthermore the densities are stated as a relative density in the domain, that is written asthe concentration of a species times its molar mass; e.g. ρS1 = S1 ·MS1. The energy balanceequation then reads:
where M is the molar mass and the subscript denotes the species concerned. The convectiveterm 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. Inthe case of an endothermic process, this parameter has a negative value, as it then representsa sink of energy.
For a chemically reacting system, the change in mass of each phase is coupled to the chemicalkinetics and stoichiometry. It was chosen to state conservation of concentrations of speciesrather than actual mass as the dependent variable. The mass balance equation for the chem-ically reacting system then reads:
MS1∂S1
∂t+ MS2
∂S2
∂t+ MG
∂G
∂t+ MGv · ∇G = 0 (5.13)
The chemical kinetics for a first order reaction is described by:
∂S1
dt= −k1S1 (5.14)
The reaction rate ki is a function of temperature, in the form of an Arrhenius equation:
ki(T ) = Ai exp(−Ei
RT
)
(5.15)
the values of the constants Ai and Ei were determined experimentally, as described in Sub-section 4.1.4.
The convective velocity that appears in Equation 5.12 and 5.13 is described by Darcy’s lawfor flow in porous media. Darcy’s law states that the velocity vector is determined by thepressure gradient, the fluid viscosity and the structure of the porous medium in the followingway:
v = −κ
µ· ∇p (5.16)
As the only gaseous product is water vapor and the concentration of which is already calcu-lated, the pressure term in the Darcy equation can be rewritten by making use of the idealgas law, it then looks as follows:
v = −κ
µR · ∇(TG) (5.17)
The calculated convective velocity of the gas phase can now be introduced into Equation 5.12and Equation 5.13.
The model considers a layer of porous material, that is represented as a rectangular domain, asis shown in Figure 5.5. The heat and mass transfer through this layer are calculated while thematerial is converted chemically. An unstructured grid of 1040 triangular quadratic elementsis applied to the domain, with 30 elements in x-direction and 8 elements in y-direction. Thewidth of the geometry is 4.8 · 10−3 m and the height is 1.3 · 10−3 m. The equations aresolved 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. Forsystems that incorporate transient effects a small time step is essential, but for a quasi-steadystate effects small time steps are unnecessary and very time consuming. The current systemexperiences large transitions in the beginning, but stabilizes at later times.
5.5. MODEL VERIFICATION 41
H
x
y
1
2
3
4
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 arestated. The initial concentration S1 is calculated from the known density and macro-porosityof 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 boundariesare impermeable. Heat is added to the domain from the bottom side at boundary 2 andthe domain is in convective contact with the ambient at boundary 3. The boundary con-ditions for heat and mass are summarized in Table 5.1, where the heat flux vector readsq = −λ∇T + ρcpT v and the mass flux vector reads Ni = −Di∇ci + civ. In Figure 5.5 aschematic representation of the numerical domain is given.
Table 5.1: Boundary conditions
B1 B2 B3 B4
Heat q · n = 0 Tprescribed −λ∇T · n = 0 q · n = 0q · n = ρcpT v · n
Mass Ni · n = 0 Ni · n = 0 Ni · n = civ · n Ni · n = 0
Initially the domain is at a temperature of 293 K. At t > 0 a temperature boundary conditionis given for boundary 2, which is either a linearly increasing temperature (so-called constantheating 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 testingthe correctness of a model. The second step is called validation, which is meant to assure thatthe model represents the real system to a sufficient level of accuracy. Model verification isperformed by comparison with analytical solutions and by calculation of some dimensionlessnumbers, which give direct insight into the ruling processes in a system.
Conduction is the determining heat transfer mode within the porous layer of material. Intransient 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. Beforetaking the transient effects into account, the temperature distribution over the layer for thesteady-state conduction case is calculated.
When a boundary of the domain is in thermal contact with an ambient, heat is transferred byconvection. When the convection coefficient is very high, the surface temperature becomesapproximately equal to the ambient temperature. A very low convection coefficient resultsin a large temperature difference between the surface and the ambient, which can also bereferred to as a high thermal surface resistance. The Biot number (Bi) is a dimensionlessnumber that represents the ratio of internal to surface transport resistance. For Bi < 0.1the lumped-capacitance assumption is valid [Janna, 2000], which means that the temperatureprofile at any time is (nearly) constant within the domain. Typical values of the convectiveheat transfer coefficient for air in natural convection is 5 − 25 Wm−2K. For a heat transfercoefficient h = 25 Wm−2K and a typical length of L = 1.3 · 10−3 m the Biot number is:
Bi =hL
λ= 0.0677 (5.18)
When T = 393 K at the heated wall, in the current model given by boundary 2, and forthe ambient T∞ = 293 K, then the temperature at the surface, here boundary 3, will theo-retically settle to become 390 K. In the current model this was found to be 389.8 K, whichis a reasonable approximation. In [Janna, 2000] the analytical solutions for the temperaturedistribution 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−2Kand changing the height of the domain to 4.8·10−3 m. The analytical solution states a dimen-sionless temperature difference of 0.67 for this situation, which results in a surface temperatureof T = 360 K. The model calculates a surface temperature of 360.2 K, which is again in goodagreement with the analytical solution. A temperature difference of this quantity has a signif-icant impact on the conversion reaction. Large concentration gradients over the layer emergeand completion of the reaction takes substantially longer time. The Biot number was thenraised to a value of 10. In order to do so, the convection coefficient was set to 300 Wm−2K,which is the reported maximum for a forced convection air flow. The height of the domainwas increased to 1.6 · 10−2m. A surface temperature of 307 K is then predicted and in thesimulation it was calculated to be 309.7 K which is a bit too high. The situation that wassimulated however is not a situation that would ever be operated in reality.
Subsequently the unsteady state conduction problem is considered. The timescale of theproblem can be deducted from the dimensionless temperature difference and the Biot andFourier number:
T − T∞
Tw − T∞
= exp[−(Bi)(Fo)] (5.19)
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 heatconduction 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 reasonsof numerical stability the temperature of boundary 2 is not instantly set to the high value
5.5. MODEL VERIFICATION 43
of 393 K but it is increased gradually over a time period of 10 seconds. When the sink ofenergy, that accounts for the endothermic reaction energy, is set to zero, the final temperatureat boundary 3 is reached 3 seconds after boundary 2 has reached its final temperature, whichis also shown in Figure 5.6(a). When the sink term is active, the temperature at boundary3 is 386 K after 5 seconds. At t = 150 the final boundary temperature is reached, this isdepicted in Figure 5.6(b).
0 10 20 30 40 50 60 70 80 90 100280
300
320
340
360
380
400
Time [s]
Tem
pera
ture
[K]
TwallTsurf
(a)
0 20 40 60 80 100 120 140 160 180 200280
300
320
340
360
380
400
Time [s]
Tem
pera
ture
[K]
TwallTsurf
(b)
Figure 5.6: Two simulation results of the transient temperature over the layer of porous material. In Figure5.6(a) the result is given for the case that no sink-term is present in the domain. In Figure 5.6(b) the sinkterm, related to chemical conversion is active in the domain.
The Thiele modulus determines whether the reaction is controlled by internal diffusion D orthe reaction rate k. It reads Th = Rp
√
k/D, where Rp is a characteristic length scale. Itis found that for the current system Th < 1 which leads to conclude that the conversion iscontrolled 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 currentsystem 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 Kon the domain which is presented in Figure 5.5. The concentrations of all phases and theconcentration distribution over the porous layer are monitored and shown in Figure 5.7. Aconversion of hexa-hydrate to mono-hydrate was considered here. Figure 5.7(a) shows thatthe concentration S1 monotonously decays while forming S2 and G. Stoichiometry of thereaction states that each mole of hexahydrate yields 5 moles of gas phase. The concentrationG becomes larger than the initial concentration S1, however it does not become 5 times aslarge, due to the convective transport to the ambient. The same geometry is also subjectedto a constant heating rate of 20 K min−1 over the same temperature range. The conversionreaction is then completed 300 seconds faster and the maximum concentration G reaches ahigher value, due to the shorter time frame in which the reaction takes place. For the standardgeometry only small differences in concentrations are found over the layer thickness. The
44 Modeling
0 200 400 600 800 1000 12000
500
1000
1500
2000
2500
3000
3500
4000
4500
Time [s]
Con
cent
ratio
n [m
ol m
−3 ]
S1GS2
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−3
0
500
1000
1500
2000
2500
3000
3500
Height [m]
Con
cent
ratio
n [m
ol m
−3 ]
t = 0t = 300t = 500t = 600t = 700t = 800
(b)
Figure 5.7: Two concentration distribution plots calculated by the numerical model. In Figure 5.7(a) theconcentrations 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 thelayer now become visible, as is shown in Figure 5.7(b). Here the concentration course ofspecies S1 over the height of the geometry is shown for a number of time steps. The heightof 3.9 · 10−3 m coincides with the upper boundary 3 of the domain. From this graph it isseen that in the first 300 seconds the concentration S1 reduces slightly, subsequently it fallsrapidly and finally the conversion is almost completed at t = 800 s. The concentration atboundary 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 differentoperating conditions. The first case which is considered, is that of a constant heating rateexperiment. 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. Model2predicts the course of the conversion path very well over a large part of the time range, whichconfirms that the reaction kinetics are described correctly. Around t = 900 however, thereaction tends to slow down in the experiment, whereas it remains constant for some timelonger in the simulation. The reason for this anomalous behavior lies in the fact that thelast water groups that are present in the molecule are strongly held in the lattice, due to thelarge 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
5.6. MODEL VALIDATION 45
0 200 400 600 800 1000 1200 1400 1600 180055
60
65
70
75
80
85
90
95
100
Time [s]
Mas
s [%
]
experimental resultnumerical model 1numerical model 2
(a)
200 400 600 800 1000 120055
60
65
70
75
80
85
90
95
100
Time [s]
Mas
s [%
]
exp 38 muexp 200 munum model2
(b)
Figure 5.8: Two validation results for constant heating rate. Figure 5.8(a) is the result for a heating rateof 5 K min−1, in which the solid line represents the experimental values, the dash-dotted line the result fornumerical model1 and the dashed line for numerical model2. Figure 5.8(b) shows the results for a heating rateof 10 K min−1 two different grain sizes. The solid black line represents the experimental result for a grain sizeof 38 − 100 µm, the grey solid line is the result for a grain size of 200 − 500 µm and the dashed line is thenumerical result for model2.
was also simulated for a constant heating rate of 10 K min−1. The numerical results arecompared with an experiment that is performed at the same constant heating rate, for twodifferent grain sizes. In Figure 5.8(b) the results are shown, where the solid lines represent theexperimental results and the dashed line represents the numerical result. In experiments thegrain size was found to influence the propagation of the dehydration reaction. From Figure5.8(b) it is clear to see that model2 describes the conversion process of the small grain sizesample better than the large grain size sample. This was already expected as the formulatedmodel is a continuum model and therefore does not take transient effects on the smaller scaleinto account. At t = 600 a decrease in reaction rate is found in the experiment, similar tothat found in the previously discussed experiment. Again it results in a difference betweenthe experimental and numerical result.
A numerical run for the step-shaped temperature experiment was performed as well. Onlythe second part of the heating experiment, the heating at high rate from 45 to 150oC andisothermal part afterwards is simulated. The results are presented in Figure 5.9(a), wherethe experimental result is depicted by a solid line and the numerical result is denoted with adashed line. The total conversion is predicted well by the numerical model. Again a deviationis 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 alayer thickness that is three times as large as the original layer. The experimental sample sizewas approximately 38mg. The solid line represents the experimental data and the dashedline 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 andover a larger range of the conversion. The deviations are not alarming in size though and theoverall conversion is described satisfactorily.
46 Modeling
200 400 600 800 1000 1200 1400
60
65
70
75
80
85
90
95
100
Time [s]
Mas
s [%
]
numerical resultexperimental result
(a)
200 400 600 800 1000 1200 1400
60
65
70
75
80
85
90
95
100
Time [s]
Mas
s [%
]
exp largenum large
(b)
Figure 5.9: Two validation results for stepwise heating of a layer of product. Figure 5.9(a) shows the resultsfor a stepwise heating experiment, where the solid line represents the experimental result and the dashed lineis 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 themoment 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 ratiobetween the equilibrium pressure and the current pressure. Including this factor in the kineticsmay yield a better simulation of the conversion, especially at the point where the model resultnow 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 thepredictive capacity of the model. In this way parameters for a future operating system canbe determined. In a future operating system, the reactions are expected to take place ina fixed bed reactor. It was seen before that the height of the layer of product has a largeinfluence on the propagation of the reaction. Therefore the height of the domain was increasedsignificantly. 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 poorheat transport over the layer. When the thermal conductivity of the solids is raised to a valueof 4.8 W m−2K the heat transport over the layer is improved significantly. The conversionreaction for the entire layer is then completed in 2000 seconds. A storage material with a highthermal 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 optionsto enhance the thermal conductivity such as the use of an inert binder. In [Stitou, 1997] thethermal conductivity is increased by using a graphite type binder; the so-called expandednatural graphite (ENG). ENG has a high intrinsic conductivity of 200 W m−1K−1. Thegraphite binder is added to the salt by physical mixing, after which the mixture is compressedand confined within the reactor.
Chapter 6
Conclusions and Recommendations
The goal of this research was to characterize a thermochemical storage material in both ex-perimental and numerical way, in order to determine its applicability in a seasonal storagesystem. Hereto one thermochemical material was selected from a list of promising candidatematerials, which was the outcome of previous literature research performed by ECN. Magne-sium sulfate hepta-hydrate was found to match the stated requirements for a thermochemicalstorage material well.
A number of experiments are performed, consisting of a group of experiments that is per-formed for material characterization and a group to determine the behavior of a heat storagesystem employing a thermochemical material. The experimentally determined properties areused in formulating a 2D numerical model, which describes the dehydration of a porous layerof salt-hydrate.
From experiments it has become clear that the dehydration behavior and the intermediatelyformed phases of MgSO4 · 7H2O did not agree satisfactorily with the data found in literature.The stepwise dehydration which is stated in literature was found in experiments to be moregradual in character and a distinction between the individual steps was fuzzy. The reactionkinetics of a large part of the conversion is described by an Arrhenius type equation, theparameters 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 andthus material deterioration is determined. Water vapor pressure is shown to influence thepropagation of the reactions in the small range of water vapor pressures investigated. Forhigher ambient water vapor pressure, the reactions shift to higher temperatures. Both in-creased layer thickness and increased grain size result in a larger timescale of the conversionreaction, however the end product of the reaction remains the same. A key requirement forthe use of magnesium sulfate in a seasonal storage is that the reaction is reversible and can beperformed a number of times. This property is called cyclability. The results of experimentsperformed imply that the cyclability of the material is poor; the amount of water that is takenup during hydration reduces significantly in every cycle and the timescale of the hydrationprocess is too large to ever be useful as a heat source.
From dimensionless numbers, it is found that for the current setup the temperature differenceover 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 inertbinder, such as expanded natural graphite, increases the thermal conductivity of the salt-hydrate without obstructing mass transport. The numerically retrieved results display goodagreement with the experimental results over a large part of the conversion. The final partof the conversion propagates faster in the numerical model than in reality. Two reasons forthis anomalous behavior can be assigned. Firstly the last water molecules experience a largertransport resistance and are held strongly in the lattice, due to the large amount of freeligands. Secondly, the influence of water vapor pressure on the reaction is not included in thecurrent model.
A number of recommendations are formulated for future research. It is advisable to include astep in the experimental protocol to verify the exact composition of the starting material of anexperiment. This can be done with XRD or by allowing the sample material to settle for sometime at a low temperature in an ambient that is saturated with water vapor. Furthermorea 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 intermediatephases are well-defined, such as copper sulfate penta-hydrate (CuSO4 · 5H2O). Concerningthe cyclability of MgSO4 ·7H2O, more research is necessary in order to validate the presentedresults. Currently an experimental cycling setup is being built at ECN in order to performthese measurements. Reaction kinetics of the hydration reaction should be obtained in orderto extend the numerical model. This enables the user to simulate a full operating cycle andgives insight into system performance.
Bibliography
[Bunyakiat, 2002] Bunyakiat, K., Thanasirsuk, C., Pengpanich, S. Global mass losskinetic studies of Thai coal by TGA Chulalongkorn University, Thailand
[Dean, 1995] Dean, J.A. Analytical chemistry Handbook McGraw- Hill, 1995, pp 15.1-
15.5, London
[De Boer, 2002] De Boer, R., Haije, W.G., Veldhuis, J.B.J. Determination of struc-tural, thermodynamic and phase properties in the Na2S-H2O system forapplication in a chemical heat pump Thermochimica Acta, Vol.395, pp
3-19
[Di Blasi, 1996] Di Blasi, C. Heat, momentum and mass transport through a shrinkingbiomass particle exposed to thermal radiation Chemical Engineering Sci-
ence, Vol.51, pp 1121-1132
[Emons, 1990] Emons, H.-H., Ziegenbalg, G., Naumann, R., Paulik, F. Thermaldecomposition of the magnesium sulphate hydrates under quasi-isothermaland quasi-isobaric conditions Journal of Thermal Analysis, Vol. 36, pp
1265-1279
[Evans, 1966] Evans, R.C. An Introduction to Crystal Chemistry, 2nd ed., repr. withcorrections Cambridge University Press, Cambridge, 1966
[Gallagher, 1998] Gallagher, P.K. Handbook of thermal analysis and calorimetry, volume1 Principles and practice Elsevier Science B.V., The Netherlands, 1998
[Goodhew, 1988] Goodhew, P.J., Humpreys, F.J. Electron Microscopy and Analysis -2nd ed. Taylor & Francis Ltd, London, 1988
[Janna, 2000] Janna, W.S. Engineering heat transfer - 2nd. ed CRC Press LLC, Florida,
2000
[Maggio, 2005] Maggio, G., Freni, A., Restuccia, G. A dynamic model of heat andmass transfer in a double-bed adsorption machine with internal heat re-covery Internation journal of Refrigiration, article in press
[Marletta, 2002] Marletta, L., Maggio, G., Freni, A., Ingrasciotta, M., Restuc-
cia, G. A non-uniform temperature, non-uniform pressure dynamic modelof heat and mass transfer in compact adsorbent beds International Journal
of Heat and Mass Transfer, Vol.45, pp 3321-3330
49
50 BIBLIOGRAPHY
[McMurry, 1995] McMurry, J., Fay, R.C. Chemistry Prentice hall Inc., New Jersey, 1995
[Paulik, 1981] Paulik, J., Paulik, F., Arnold, M. Dehydration of magnesium sul-phate heptahydrate investigated by quasi isothermal - quasi isobaric TGThermochimica Acta, 50, pp 105-110
[Ruiz, 2007] Ruiz-Agudo, E., Martin-Ramos, J.D., Rodriguez-Navarro, C.
Mechanism and kinetics of dehydration of epsomite crystals formed in thepresence of organic additives Journal of Physical Chemistry B, Vol.111, pp
41-52
[Schram, 1994] Schram, P.P.J.M., Van Heijst, G.J.F., Van Dongen, M.E.H. Fy-sische transportverschijnselen voor W Dictaatnummer 3498, Technische
[Stitou, 1997] Stitou, D., Goetz, V., Spinner, B. A new analytical model for solid-gas thermochemical reactors based on thermophysical properties of thereactive medium Chemical Engineering and Processing, Vol.36, pp 29-43
[Thunman, 2002] Thunman, H. Combustion of biomass Nordic Course Lyngby, Danmark,
J.W., Feldman, W.C. Magnesium sulphate salts and the history of wateron 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 ECN-
C–04-074, 2004
[Wagman, 1982] Wagman, D.D, Evans, W.H., Parker, V.B. et al. The NBS tables ofchemical thermodynamic properties: selected values for inorganic and C1and 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 chemicalconversion processes, the so-called Tmax method can be applied. A typical Arrhenius reactionequation has the form:
ki = Aiexp(−Ei
RT) (A.1)
In which ki is the reaction rate [ s−1], Ai the so-called frequency factor [ s−1], Ei the activationenergy [ J/mol], R the gas constant and T the temperature [ K]. The Tmax method isdescribed by [Bunyakiat, 2002]. From TGA measurements a plot can be constructed of thedifferential 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 whichthis maximum yield occurs is denoted by Tmax. The rate of product release from a first-orderchemical reaction i is described as:
dVi
dt= ki(V
∗
i − Vi) (A.2)
51
52 Calculation of kinetic parameters
Where Vi is the accumulated amount of evolved volatile from the reaction i up to time t andV ∗
i the total yield of volatile for reaction i. A constant heating rate implies a linear correlationbetween time and temperature, then Equation A.2 can be expressed in the following form:
dVi
dT=
ki(V∗
i − Vi)
H(A.3)
Where H denotes the heating rate [ K/sec]. At the temperature where the maximum yieldof products is reached, the temperature derivative of the reaction rate is zero.
d2Vi
dT 2=
ki
H
−dVi
dT+
(V ∗
i − Vi)
H
dki
dT= 0 (A.4)
The temperature derivative of the theoretical Arrhenius equation (Equation A.1) reads:
dki
dT= Ai
( Ei
RT 2
)
exp(−Ei
RT
)
(A.5)
Then substituting Equation A.3 and A.5 into Equation A.4 yields:
(ki
H
)(
−ki(V
∗
i − Vi)
H
)
+((V ∗
i − Vi)
H
)
Ai
( Ei
RT 2max
)
exp( −Ei
RTmax
)
= 0 (A.6)
Filling in ki and rearranging gives:
−A2
i
H2exp
(
−2Ei
RTmax
)
= −Ai
H
Ei
RT 2max
exp(
−Ei
RTmax
)
(A.7)
Which can be written in its final form:
ln( H
T 2max
)
= ln(AiR
Ei
)
−Ei
RTmax(A.8)
A series of TGA experiments is then performed for different constant heating rates. TheTmax is determined for each measurement. These points are depicted in a plot of ln( H
Tmax)
vs ( 1Tmax
) and connected with a linear fit. Then the kinetic parameters Ei and Ai can bedetermined from the slope and intercept in the following way:
Ei = −R · slope (A.9)
Ai = exp(intercept) · slope (A.10)
This method can also be applied to determine the kinetic information from a high resolutionmeasurement, despite the fact that a variable heating rate is employed. This is due to thefact that when maximum yield of products is realized, the heating rate in a high resolutionmeasurement is minimal by definition. The actual heating rate at Tmax can be obtained fromthe output data.
Appendix B
Experimental background
B.1 Experimental setups
TGA experiments are performed on two systems. In each TGA device two gas flows arepresent; a purge and a protective flow. The protective flow was set to 20 ml min−1 of drynitrogen. The purge flow was set to 80 ml min−1 of dry nitrogen. This latter gas flowwas wetted for some experiments. Hereto the gas flow was led through two consecutive gasbubbling 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 gasbubbles through the column and is carried off through the tube on the right hand side. Whenthe bubbling length through the bottle is large enough, the gas flow is saturated with watervapor. The water vapor pressure that is applied in this way depends on temperature, followingthe Clausius Clapeyron equation. For a water temperature of 25oC 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 TAInstruments TGA Q500 device is present, which is shown in Figure B.2(a). This system isequipped with an auto-sampler, which is a carousel device that can hold up to 50 samplesfor successive measurements. This setup was used to perform a high resolution measurementas well as some constant heating rate experiments. The purge flow in this setup alwaysconsisted of wetted nitrogen at a flow rate of 80 ml min−1. The device uses platinum samplepans 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 MettlerToledo TGA STARe device is present. This setup was used to perform extensive thermo-gravimetric analysis for a range of operating conditions. The purge flow in the setup wasvaried from 20 − 160 ml min−1. The system uses alumina sample pans, that can be appliedwith 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 whichis used to determine the composition of a gaseous sample, by generating a mass spectrumrepresenting the masses of sample components. This is a specifically useful instrument whenthe reaction products are not exactly known.
B.2. EXPERIMENTAL SETTINGS 55
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. Theprotective flow was fixed at a flow of 20 ml min−1 of dry nitrogen. The same alumina samplepans are used as those used in the Mettler Toledo TGA STARe system. Two schematicrepresentations 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 enterfrom below. The heating of the furnace is provided at the wall, while the temperature programis prescribed for the sample temperature. In Figure B.4(b) a close up is given of the sampleholder. From the four sample holders that are shown on the right, the second setup was usedfor the current measurements. The device was installed recently and experimental resolutiondetermination revealed that a large difference was present between the prescribed and realizedtemperature. Either by extensive calibration or through contact with the manufacturer thismay 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 wereused for measurements. The last column denotes in which figure(s) the experimental resultis shown in this report.
Figure B.5: Overview of experimental settings used in measurements.
Appendix C
Modeling background
For the numerical model a total of 6 application modes is used; one for each phase (threephases are present) and one for each conserved quantity (here conservation of mass and heatis stated). All modes are stated for transient analysis. As the system is considered to be inlocal thermodynamical equilibrium, only one dependent variable, T , is used for all thermalmodes. 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 andthe Conduction application mode for heat transfer. For G mass transfer is described withthe Convection and Diffusion application mode and heat transfer with the Convection andConduction 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 thenumerical model.
In addition a number of expressions are formulated. Scalar expressions are formulated forcalculation of the reaction kinetics and the material densities. This leads to the followingequations:
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:
Px = diff(T · G, x) (C.5)
Py = diff(T · G, y) (C.6)
The constants that appear in the Darcy equation are submitted in the input field for thevelocity in the subdomain settings of the gaseous phase. In this way the total Darcy equationis included in the system of equations without the necessity of a separate application mode.
