A Area (m2) Vamp Volume flow (m3s) cp Specific heat (Jkg K) Wamp Power (W) COP Coefficient of performance w Solute mass fraction Dtube Tube diameter (m) Dp Equivalent particle diameter (m) Greek Dscr Scraper diameter (m) α Heat transfer coefficient (Wm2K) G Growth rate (ms) δ Thickness (m) g Gravity (ms2) ε Bed voidage I Investment costs (keuro) η Efficiency k Mass transfer coefficient (ms) λ Thermal conductivity (Wm K) L Length (m) micro Viscosity (Pa s) N Scraper passes per second (1s) ρ Density (kgm3) Nuscr Liquid Nusselt number α Dscrλliq Nuh Hydraulic Nusselt number Subscripts αi Diinnerλliq add Additional p Pressure (Pa) comp Compressor ∆p Pressure drop (Pa) crys Crystallizer Prliq Liquid Prandtl number cpliq microliqλliq cool Coolant ampQ Heat (W) eq Equilibrium
R Removal rate (ms) FBHE Fluidized bed heat exchanger Reliq Liquid Reynolds number he Heat exchanger ρliq u Diinnermicroliq ice Ice Reh Hydraulic Reynolds number in Inlet ρliq u Diinnermicroliq is Ice slurry Ret Liquid Reynolds number KNO3 Potassium nitrate ρliq N Diinnermicroliq liq Liquid T Temperature (degC) min Minimum Tfr Initial freezing temperature (degC) motor Motor ∆Tln Logarithmic mean temperature net Net difference (K) out Outlet ∆Ttrans Transition temperature difference p Particle (K) pd Pressure drop t Time (s) plate Plate U Overall heat transfer coefficient pump Pump (Wm2K) scr Scrapers us Superficial velocity (ms) SSHE Scraped surface heat exchanger STHE Shell-and-tube heat exchanger
Bel O Lallemand A 1999 Etude drsquoun fluide frigoporteur diphasique ndash 2 Analyse expeacuterimentale du comportement thermique et rheacuteologique International Journal of Refrigeration vol22 pp175-187
Bellas I Tassou SA 2005 Present and future applications of ice slurries International Journal of Refrigeration vol28 pp115-121
Comparison between Fluidized Bed and Scraped Surface Ice Slurry Generators
135
Ben Lakhdar M Cerecero R Alvarez G Guilpart J Flick D Lallemand A 2005 Heat transfer with freezing in a scraped surface heat exchanger Applied Thermal Engineering vol25 pp45-60
Chhabra RP 1995 Wall effects on free-settling velocity of non-spherical particles in viscous media in cylindrical tubes Powder Technology vol85 pp83-90
Chhabra RP Agarwal L Sinha NK 1999 Drag on non-spherical particles An evaluation of available methods Powder Technology vol101 pp288-295
Drewett EM Hartel RW 2006 Ice crystallization in a scraped surface freezer Journal of Food Engineering in press
EPS Ltd 2006 Orbital Rod Evaporator Capacity Curves httpwwwepsltdcouk
Field BS Kauffeld M Madsen K 2003 Use of ice slurry in a supermarket display cabinet In Proceedings of 21st IIR International Congress of Refrigeration 17-23 August 2003 Washington DC (USA) Paris International Institute of Refrigeration
Gladis SP Marciniak MJ OHanlon JB Yundt B 1996 Ice crystal slurry TES system using orbital rod evaporator In Conference Proceedings of the EPRI International Conference on Sustainable Thermal Energy Storage 7-9 August 1996 Bloomington (USA)
Gladis S 1997 Ice slurry thermal energy storage for cheese process cooling ASHRAE Transactions vol103 part 2 pp725-729
GMF 1992 Personal communication Goudsche Machine Fabriek BV Gouda (The Netherlands)
Goede R de Jong EJ de 1993 Heat transfer properties of a scraped-surface heat exchanger in the turbulent flow regime Chemical Engineering Science vol48 pp1393-1404
Haid M Martin H Muumlller-Steinhagen H 1994 Heat transfer to liquid-solid fluidized beds Chemical Engineering and Processing vol33 pp211-225
Haid M 1997 Correlations for the prediction of heat transfer to liquid-solid fluidized beds Chemical Engineering and Processing vol36 pp143-147
Jellema P Nijdam JL 2005 Ice slurry production under vacuum In Proceedings of the 6th IIR Workshop on Ice Slurries 15-17 June 2005 Yverdon-les-Bains (Switzerland) Paris International Institute of Refrigeration pp74-78
Katz T 1997 Auslegung und Betrieb von Wirbelschichtwaumlrmeaustauscher PhD Thesis RWTH Aachen (Germany)
Kauffeld M Kawaji M Egolf PW 2005 Handbook on Ice Slurries Fundamentals and Engineering Paris International Institute of Refrigeration
Kiatsiriroat T Vithayasai S Vorayos N Nuntaphan A Vorayos N 2003 Heat transfer prediction for a direct contact ice thermal energy storage Energy Conversion and Management vol44 pp497-508
Chapter 6
136
Kollbach JS 1987 Entwicklung eines Verdampfungsverfahrens met Wirbelschicht-Waumlrmeaustauscher zum Eindampfen krustenbildender Abwaumlsser PhD Thesis RWTH Aachen (Germany)
Kurihara T Kawashima M 2001 Dynamic ice storage system using super cooled water In Proceedings of the 4th IIR Workshop on Ice Slurries 12-13 November 2001 Osaka (Japan) Paris International Institute of Refrigeration pp61-69
Meewisse JW 2004 Fluidized Bed Ice Slurry Generator for Enhanced Secondary Cooling Systems PhD Thesis Delft University of Technology (The Netherlands)
Meewisse JW Infante Ferreira CA 2003 Validation of the use of heat transfer models in liquidsolid fluidized beds for ice slurry generation International Journal of Heat and Mass Transfer vol46 pp3683-3695
Mil PJJM van Bouman S 1990 Freeze concentration of dairy products Netherlands Milk Dairy Journal vol44 pp21-31
Mito D Mikami Y Tanino M Kozawa Y 2002 A new ice-slurry generator by using actively thermal-hydraulic controlling both supercooling and releasing of water In Proceedings of the 5th IIR Workshop on Ice Slurries 30-31 May 2002 Stockholm (Sweden) Paris International Institute of Refrigeration pp185-196
Nagato H 2001 A dynamic ice storage system with a closed ice-making device using supercooled water In Proceedings of the 4th IIR Workshop on Ice Slurries 12-13 November 2001 Osaka (Japan) Paris International Institute of Refrigeration pp97-103
Nelson KP Pippin J Dunlap J 1999 University ice slurry system In 12th Annual IDEA College-University Conference 10-12 February 1999 New Orleans (USA) Westborough International District Energy Association
Nelson KP 1998 Ice slurry generator In 89th Annual IDEA Conference 13-16 June 1998 San Antonio (USA) Westborough International District Energy Association
Ophir A Koren A 1999 Vacuum freezing vapor compression process (VFVC) for mine cooling In Proceedings of the 20th IIR International Congress of Refrigeration Sydney (Australia) Paris International Institute of Refrigeration
Patience DB Rawlings JB Mohameed HA 2001 Crystallization of para-xylene in scraped-surface crystallizers AIChE Journal vol47 pp2441-2451
Paul J 1996 Compressors for refrigerating plants and ice makers with water as refrigerant In Applications for Natural Refrigerants 3-6 September 1996 Aarhus (Denmark) Paris International Institute of Refrigeration pp577-584
Paul J Jahn E Lausen D Schmidt K-P 1999 Chillers and ice machines with ldquowater as refrigerantrdquo In Proceedings of 20th IIR International Congress of Refrigeration 19-24 September 1999 Sydney (Australia) Paris International Institute of Refrigeration
Comparison between Fluidized Bed and Scraped Surface Ice Slurry Generators
137
Pronk P Meewisse JW Infante Ferreira CA 2005 Erratum to Validation of the use of heat transfer models in liquidsolid fluidized beds for ice slurry generation [International Journal of Heat and Mass Transfer 46 (2003) 3683-3695] International Journal of Heat and Mass Transfer vol48 pp3478-3483
Qin FGF Chen XD Ramachandra S Free K 2006 Heat transfer and power consumption in a scraped-surface heat exchanger while freezing aqueous solutions Separation and Purification Technology vol48 pp150ndash158
Qin FGF Chen XD Russell AB 2003 Heat transfer at the subcooled-scraped surface withwithout phase change AIChE Journal vol49 pp1947-1955
Rautenbach R Katz T 1996 Survey of long time behavior and costs of industrial fluidized bed heat exchangers Desalination vol108 pp335-344
Richardson JF Zaki WN 1954 Sedimentation and fluidization Transactions of the Institute of Chemical Engineers vol32 pp35-53
Roos AC Verschuur RJ Schreurs B Scholz R Jansens PJ 2003 Development of a vacuum crystallizer for the freeze concentration of industrial waste water Chemical Engineering Research and Design vol81 part A pp881ndash892
Sari O Egolf PW Ata-Caesar D Brulhart J Vuarnoz D Lugo R Fournaison L 2005 Direct contact evaporation applied to the generation of ice slurries modelling and experimental results In Proceedings of the 6th IIR Workshop on Ice Slurries 15-17 June 2005 Yverdon-les-Bains (Switzerland) Paris International Institute of Refrigeration pp57-72
Sheer TJ Butterworth MD Ramsden R 2001 Ice as a coolant for deep mines In Proceedings of the 7th International Mine Ventilation Congress 17-22 June 2001 Krakow (Poland) pp355-361
Soe L Hansen T Lundsteen BE 2004 Instant milk cooling system utilising propane and either ice slurry or traditional ice bank In Proceedings of the 6th IIR Gustav Lorentzen Conference on Natural Working Fluids 29 August-1 September 2004 Glasgow (UK) Paris International Institute of Refrigeration
Stamatiou E 2003 Experimental Study of the Ice Slurry Thermal-Hydraulic Characteristics in Compact Plate Heat Exchangers PhD thesis University of Toronto (Canada)
Stamatiou E Kawaji M 2003 Heat transfer characteristics in compact scraped surface ice slurry generators In Proceedings of 21st IIR International Congress of Refrigeration 17-23 August 2003 Washington DC (USA) Paris International Institute of Refrigeration
Stamatiou E Meewisse JW Kawaji M 2005 Ice slurry generation involving moving parts International Journal of Refrigeration vol28 pp60-72
Tanino M Kozawa Y Mito D Inada T 2000 Development of active control method for supercooling releasing of water In Proceedings of the 2nd IIR Workshop on Ice Slurries 25-26 May 2000 Paris (France) Paris International Institute of Refrigeration pp127-139
Chapter 6
138
Trommelen AM Beek WJ Westelaken HC van de 1971 A mechanism for heat transfer in a Votator-type scraped-surface heat exchanger Chemical Engineering Science vol26 pp1987-2001
Vaessen RJC 2003 Development of Scraped Eutectic Crystallizers PhD thesis Delft University of Technology (The Netherlands)
Vaessen RJC Himawan C Witkamp GJ 2002 Scale formation of ice from electrolyte solutions Journal of Crystal Growth vol237-239 pp2172-2177
Vaessen RJC Seckler MM Witkamp GJ 2004 Heat transfer in scraped eutectic crystallizers International Journal of Heat and Mass Transfer vol47 pp717-728
Vuarnoz D Sletta J Sari O Egolf PW 2004 Direct injection ice slurry generator In Proceedings of the 6th IIR Gustav Lorentzen Conference on Natural Working Fluids 29 August-1 September 2004 Glasgow (UK) Paris International Institute of Refrigeration
Wakamoto S Nakao K Tanaka N Kimura H 1996 Study of the stability of supercooled water in an ice generator ASHRAE Transactions vol102 part 2 pp142-150
Wang MJ Kusumoto N 2001 Ice slurry based thermal storage in multifunctional buildings Heat and Mass Transfer vol37 pp597-604
Wang MJ Lopez G Goldstein V 2002 Ice slurry for shrimp farming and processing In Proceedings of the 5th IIR Workshop on Ice Slurries 30-31 May 2002 Stockholm (Sweden) Paris International Institute of Refrigeration pp161-168
Wijeysundera NE Hawlader MNA Andy CWB Hossain MK 2004 Ice-slurry production using direct contact heat transfer International Journal of Refrigeration vol27 pp511-519
Zakeri GR 1997 Vacuum freeze refrigerated circuit (VFRC) a new system design for energy effective heat pumping applications In Proceedings of the IIRIIF Linz lsquo97 Conference Heat Pump Systems Energy Efficiency and Global Warming 28 September-1 October 1997 Linz (Austria) Paris International Institute of Refrigeration pp182-190
Zwieg T Cucarella V Worch H 2002 Novel bio-mimetically based ice-nucleating coatings for ice generation In Proceedings of the 5th IIR Workshop on Ice Slurries 30-31 May 2002 Stockholm (Sweden) Paris International Institute of Refrigeration pp142-150
139
7 Long-term Ice Slurry Storage
71 Introduction
Ice slurries are interesting secondary refrigerants compared to single-phase fluids since they use the latent heat of ice resulting in high heat capacities An important advantage of this high heat capacity is the possibility of cold storage where ice slurry is produced during off-peak hours and is stored in insulated tanks for later use Cold storage with ice slurry can lead to economic and environmental benefits such as reduced installed refrigeration power lower average electricity tariffs and lower energy consumption due to lower condensing temperatures in the primary refrigeration cycle during nighttime operation (see Chapter 1)
Ice slurry can be stored as a homogeneous or heterogeneous suspension In case of homogeneous storage (see Figure 71) a stirring device keeps the ice crystals in suspension which is achievable for ice fraction up to 35 wt (Christensen and Kauffeld 1998) In case of heterogeneous storage (see Figure 72) the tank does not contain a stirring device and as a result the ice crystals float to the top of the tank and the lower part of the tank contains only liquid (Kozawa et al 2005)
Figure 71 Homogeneous ice slurry storage
(Egolf et al 2001) Figure 72 Heterogeneous ice slurry storage
(adapted from Kozawa et al 2005)
An advantage of homogeneous storage compared to heterogeneous storage is the possibility to pump ice crystals to the application heat exchangers which is beneficial since the high heat capacity of ice is then also applied in these heat exchangers and in the piping network A disadvantage of homogeneous storage is however the required mixing power to maintain a homogeneous suspension According to Christensen and Kauffeld (1998) approximately 70 Wm3 mixing power is required to keep an ice slurry homogeneously mixed In their experiments Christensen and Kauffeld used ice slurries made of a 10 wt ethanol solution with an density of approximately 980 kgm3 The density difference between the solution and the ice crystals (917 kgm3) was therefore relatively resulting in a relatively low required mixing power It is expected that the required mixing power is higher in aqueous solutions with higher densities which is the case for most other solutions discussed in this thesis Parts of this chapter have been published in the International Journal of Refrigeration vol28 pp27-36 2005 and in the Journal of Crystal Growth vol275 ppe1361-e1367 2005
Chapter 7
140
Egolf et al (2001) proposed to combine the advantages of both storage methods Their proposal consisted of a storage tank without mixing device from which ice slurry with a high ice fraction is pumped from the top and is mixed with liquid from the bottom (see Figure 73) In this way every desired ice fraction can be achieved It is also possible to operate with an intermittent mixing device that is switched off when no cooling load is applied Meili et al (2001) showed that stratified slurries with all ice crystals floating on the top can easily be turned into an homogeneously suspension by mixing even after 15 hours without mixing
Figure 73 Heterogeneous ice slurry storage with mixing device (Egolf et al 2001)
If heat uptake from the surroundings and mixing power are neglected storage of ice slurry can be considered as an adiabatic process with a virtually constant temperature and ice fraction Although the amount of ice hardly changes during storage the size and shape of crystals may alter due to recrystallization mechanisms Changes in size and shape are supposed to have significant influences on other components of an ice slurry system For example Kitanovski and Poredoš (2002) showed that an increased average crystal size has an effect on the rheological behavior of ice slurry in pipes Analogously Frei and Egolf (2000) measured different pressure drop values for freshly produced ice slurry and for the same ice slurry after storage probably caused by difference in crystal size Furthermore it is shown in Chapter 8 that the average ice crystal size influences the performance of heat exchangers Large crystals with a relatively small surface-to-volume ratio will cause higher superheating values at heat exchanger outlets resulting in reduced cooling capacities Finally crystal characteristics might also have an effect on pump performance and on the minimum required power to keep the ice slurry homogeneously mixed in a storage tank
Next to ice slurry systems for refrigeration recrystallization of ice crystals during storage is also interesting for other industrial processes such as freeze concentration and long-term storage of ice cream In freeze concentration processes ice crystals are stored for a certain period in order to increase the average crystal size which enables a more efficient washing of the crystals in wash columns (Huige and Thijssen 1972 Verschuur et al 2002) In case of ice cream storage the ice crystal size strongly determines the product quality and therefore several investigations have been carried out on the role of storage conditions on crystal sizes (Adapa et al 2000 Donhowe and Hartel 1996 Hagiwari and Hartel 1996)
The objective of this chapter is to give more insight in the physical phenomena that alter ice crystals during adiabatic storage The development of the ice crystals size distribution during adiabatic storage is experimentally studied for ice crystals stored in different solutions Subsequently the obtained experimental results and results from other researchers are used to develop a computer-based dynamic model of ice crystals in aqueous solutions placed in
Long-term Ice Slurry Storage
141
adiabatic storage tanks Finally this model is used to simulate the development of the ice crystal size distribution in time and is validated with the experimental results
72 Recrystallization Mechanisms
When ice crystals are stored in a saturated aqueous solution three mechanisms are distinguished that may alter its size and shape namely attrition agglomeration and Ostwald ripening These three mechanisms are separately discussed in this section
721 Attrition
In mechanically agitated vessels crystals can be damaged by collisions with solids such as the stirrer the walls or other crystals All these kinds of damaging mechanisms are called attrition In general two types of attrition can be distinguished namely breakage and abrasion (Mazzarotta 1992) In case of breakage the collision energy is relatively high and the collision subdivides the initial crystal into a number of fragments with a wide spectrum of sizes When the collision energy is not high enough to break the crystal into numerous pieces it may occur that only a small part of the crystal will be pulled off This phenomenon is called abrasion The fragments produced by abrasion are in most cases much smaller than the parent crystals In order to study abrasion Biscans et al (1996) carried out experiments with a suspension of sodium chloride crystals and acetone which is an anti-solvent for these crystals In these experiments the size of the initial crystals ranged from 100 to 500 microm while the fragments produced by abrasion ranged from 5 to 50 microm (see Figure 74) Besides the production of small fragments abrasion will round off the parent crystals
00
01
02
03
04
05
1 10 100 1000Crystal size (microm)
Mas
s fr
actio
n (-
)
(a)
00
01
02
03
04
05
1 10 100 1000Crystal size (microm)
Mas
s fr
actio
n (-
)
(b)
Figure 74 Crystal size distributions before (a) and after (b) an attrition experiment of 12 hours (Biscans et al 1996)
In literature no information is available for attrition effects on ice crystals but the attrition behavior of ice crystals can be deduced from a comparison with other crystals Gahn and Mersmann (1995) carried out experiments to study the attrition behavior of several kinds of crystals From these experiments it was concluded that the crystals with high hardness values are more sensitive for attrition than softer crystals The hardness of ice strongly depends on the temperature and varies between 0 to ndash15degC from 10 to 100 MPa which are rather low values compared to other crystals (Barnes et al 1971) As far as the hardness is concerned ice
Chapter 7
142
crystals can be compared with sodium chloride crystals and potassium chloride crystals with hardness values of 166 MPa and 91 MPa respectively (Gahn and Mersmann 1995) In spite of the low hardness sodium chloride crystals are affected by abrasion as was shown by Biscans et al (1996) Since hardness values are comparable it is therefore expected that ice crystals are also affected by abrasion It is however not expected that breakage of crystals will occur in ice slurry systems
722 Agglomeration
Agglomeration or accretion is the adherence of two small crystals resulting in one large polycrystalline particle In case of strong agglomeration the average diameter of crystals increases seriously while the number of crystals decreases According to Kasza and Hayashi (1999) ice crystals have a strong tendency to agglomerate in storage tanks and it is therefore important to study this phenomenon
In order to study agglomeration of ice crystals in solution Shirai et al (1987) carried out experiments in which ice crystals were produced and stored in lactose and glucose solutions with different concentrations Microscopic pictures of ice crystals after storage clearly indicated whether agglomeration had occurred since agglomerated crystals could clearly be distinguished from mono-crystalline crystals During the experiments with lactose solutions agglomeration was only observed at concentrations of 10 wt lactose (Tfr=ndash06degC) and not in solutions of 15 wt (Tfr=ndash10degC) A similar phenomenon was observed for glucose solutions in which agglomeration took place in solutions of 5 75 and 10 wt with freezing temperatures of ndash06 ndash09 and ndash12degC respectively but not in a 15 wt solution with a freezing temperature of ndash19degC Kobayshi and Shirai (1996) experimentally confirmed the strong influence of solutes on agglomeration During storage experiments with glucose solutions extensive agglomeration did only occur at glucose concentrations of 10 wt (Tfr=ndash12degC) and lower but not with concentrations of 20 wt (Tfr=ndash27degC) and 30 wt (Tfr=ndash47degC) In experiments with agglomeration the average ice crystal size increased from 100 to 500 microm at the start to 1 to 3 mm after two hours of storage Finally Hayashi and Kasza (2000) observed similar trends during storage experiments with ethylene glycol solutions during which agglomeration only occurred at concentrations below 04 wt (Tfr=ndash01degC)
723 Ostwald Ripening
Ice slurries normally consist of a spectrum of crystal sizes both large and small Due to surface energy contributions small ice crystals have a lower equilibrium temperature than larger ones During isothermal storage of ice slurries these differences in equilibrium temperature result in the growth of large ice crystals and the melting of small ones This phenomenon is called Ostwald ripening or migratory recrystallization and provides an increase in average crystal size over relatively long periods
Theory
The difference in equilibrium temperature between differently sized crystals is deduced from the free Gibbsrsquo energy of a single crystal with respect to the liquid phase (Nielsen 1964)
32V
Am
n micro microB LG A B LV
γ γ∆ = ∆ + = ∆ + (723)
Long-term Ice Slurry Storage
143
In this equation n represents the number of moles in the crystal γ is the surface tension between the crystal and the liquid A is the total surface of the crystal Vm is the molar volume of the solid state BV is the volume shape factor and BA is the surface shape factor Parameter ∆micro is the difference in chemical potential of water between the solid and the liquid state
liq sol smicro micro ( ) -micro ( )T w T∆ = (724)
Equation 72 can be rewritten into the following equation since the chemical potential of the solid state equals the chemical potential of the liquid at equilibrium conditions of a crystal with infinite dimensions
( ) ( ) liq sol liq sol s s liq sol smicro micro ( ) -micro ( ) - micro ( ) -micro ( ) since micro ( ) micro ( )T w T w T T T w Tinfin infin infin infin∆ = = (725)
The chemical potential of the liquid state can be split up into a concentration dependant and independent contribution
( ) ( ) sol
0liq 0liq s ssol
( )micro ln micro ( ) -micro ( ) - micro ( ) -micro ( )( )T wRT T T T TT w
ψψ infin infin
infin
∆ = +
(726)
Since the activity coefficient Ψ is only a weak function of temperature the ratio of the activity coefficients in the first term is close to unity as a result of which the contribution of the first term can be neglected Subsequently the differences in chemical potential of the liquid and the solid state at different temperatures can be calculated with the integral over the entropy
( ) ( )
fliq s f f fmicro - - - - since
T T
T T
hs dT s dT s T T T T h T sT
infin infin
infin infin infininfin
∆∆ = + = ∆ = ∆ = ∆int int (727)
Equations 723 and 727 can be combined into Equation 728
( )3
2V fA
m
-B L hG T T B LV T
γinfininfin
∆∆ = + (728)
A crystal with size L is in equilibrium with the surrounding liquid when its free Gibbsrsquo energy reaches its minimum
( ) 0d G
dL∆
= (729)
Applying Equation 77 to Equation 728 results in the equilibrium temperature of a crystal with size L
A
V ice f
2( ) 1-3
BT L TB h L
γρinfin
= ∆
(730)
Chapter 7
144
Previous Studies on Ostwald Ripening in Ice Slurries
A number of experimental studies have been carried out on Ostwald ripening in ice slurries during isothermal storage In several of these studies ripening experiments were performed with ice crystals in thin liquid films placed under a microscope (Savory et al 2002 Sutton et al 1994 Sutton et al 1996 Williamson et al 2001) Since convection did not occur in the films the location of ice crystals did not change during storage and the evaluation of individual crystals could be observed Microscopic pictures were taken at regular intervals to analyze the growth or dissolution of ice crystals All these studies suggest that Ostwald ripening is the main recrystallization mechanism for the tested conditions since small crystals became smaller and larger ones grew However during some experiments agglomeration of small crystals was also observed Analyses of the shapes of crystal size distributions after storage and the developments of the average crystal size in time indicated that the crystallization kinetics of Ostwald ripening can be considered diffusion controlled for the tested conditions
In other experimental studies Ostwald ripening of ice slurries was investigated during isothermal storage in mixed or unmixed tanks The operating conditions of these experiments were much closer to the storage conditions of ice slurries for refrigeration applications as discussed in Chapter 1 Because of this agreement these experimental studies are discussed in more detail below
Huige and Thijssen (1972) proposed using Ostwald ripening to increase the average ice crystal size of ice slurries produced from sucrose solutions (see also Huige 1972) Ice slurry was produced in a scraped-surface crystallizer with a mean residence time of only five seconds resulting in ice crystal sizes of about 10 to 20 microm These small nuclei were added to a recrystallization tank which contained larger crystals whose residence time was varied at values of 115 20 and 32 hours Since neither cooling nor heating was applied to the slurry in the recrystallization tank the temperature in the tank was between the equilibrium temperature of large crystals and the equilibrium temperature of the small crystals (see Equation 730) As a result the smaller crystals dissolved and the larger grew even larger resulting in an increase of the average crystals size in time Through their experimental study Huige and Thijssen showed that applied method can be used to produce large ice crystals which is very useful in freeze concentration processes where the efficiency of wash columns increases with the average ice crystal size
Smith and Schwartzberg (1985) studied Ostwald ripening of ice crystals in aqueous solutions in more detail (see also Smith 1984) In their experiments they produced ice slurry from aqueous sucrose solutions with different concentrations varying from 9 to 36 wt After production up to ice fractions of 9 to 16 wt ice crystals with an average diameter of about 100 microm were stored in an insulated homogeneously mixed tank of 10 liter The experimental results displayed in Figure 75 clearly show that Ostwald ripening changes the crystal size distribution in a sucrose solution with an initial concentration of 9 wt and an ice fraction of 16 wt The average crystal size increased from 90 to 250 microm within five hours of storage Storage experiments with different sucrose solutions showed that the ripening rate decreases strongly with increasing sucrose concentrations (see Figure 76) Smith and Schwartzberg explained the slower ripening process at higher concentrations of sucrose by the lower mass transfer rate of crystal growth and dissolution at higher concentrations
Long-term Ice Slurry Storage
145
00020406081012141618
0 50 100 150 200 250 300 350Crystal diameter (microm)
Num
ber
dens
ity (1
06 1
m) Initial
After 1 hr
After 2 hrs
After 3 hrs
0
50
100
150
200
250
0 1 2 3 4 5 6Storage time (hours)
Ave
rage
dia
met
er (micro
m)
10
15
223342
Figure 75 Development of ice crystal size distribution during Ostwald ripening in a 9 wt sucrose solution (adapted from Smith
and Schwartzberg 1985)
Figure 76 Ostwald ripening of ice crystals in different sucrose solutions (adapted from
Smith and Schwartzberg 1985)
In some of their storage experiments Smith and Schwartzberg (1985) added relatively small amounts of gelatin to a sucrose solution of 10 wt The experiments showed that gelatin concentrations of 001 to 005 seriously slowed down Ostwald ripening to rates comparable with the ripening rate in sucrose solutions of 22 to 44 wt Smith and Schwartzberg ascribed this phenomenon to a strong reduction of mass transfer coefficients by gelatin
Ice slurry storage experiments in a continuously mixed 6-liter tank with solutions of sucrose and betaine by Louhi-Kultanen (1996) confirmed the strong influence of the solute concentration on Ostwald ripening described above For both sucrose and betaine solutions the ripening rate was significantly lower at solute concentrations of 15 wt compared to 8 wt
Hansen et al (2003) performed ice storage experiments with ethanol and propylene glycol solutions with initial concentrations of 10 (Tfr=ndash43degC) and 15 wt (Tfr=ndash51degC) respectively (see also Hansen et al 2002) Ice slurries with ice fractions of 10 and 30 wt were homogeneously stored in a 1000 liter tank and ice slurries with ice fractions of 30 and 46 wt were heterogeneously stored in a 285 liter tank For both storage methods ice crystals were isothermally stored for about 90 hours The crystal size distribution of the stored ice crystals was determined by analyzing microscopic pictures of ice crystals after 0 20 40 and 90 hours of storage The results displayed in Figure 77 show that the average crystal size increased during all experiments as a result of Ostwald ripening For one experiment the average crystal size even increased from 100 microm to more than 500 microm after 90 hours of storage In general it was concluded from all experiments that the ripening rate was higher during experiments with lower ice fractions Furthermore it appeared that the ripening rate was higher during homogeneous storage than during heterogeneous storage Finally Ostwald ripening in the 10 wt ethanol solution was faster compared to the 15 wt propylene glycol solution
Chapter 7
146
0
100
200
300
400
500
600
0 10 20 30 40 50 60 70 80 90 100Storage time (hours)
Ave
rage
cry
stal
siz
e (micro
m)
pg-m-30 eth-m-10 eth-m-30 pg-t-30 pg-t-46 eth-t-30 eth-t-46
15 wt PG homo φ =01015 wt PG homo φ =03010 wt EtOH homo φ =01010 wt EtOH homo φ =03015 wt PG hetero φ =03015 wt PG hetero φ =04610 wt EtOH hetero φ =03010 wt EtOH homo φ =046
Figure 77 Development of average crystal size during homogeneous and heterogeneously storage at different ice fraction in two different solutions (adapted from Hansen et al 2003)
Besides the described experiments Hansen et al (2002) experimentally studied the influence of air access and a surfactant (015 wt polyoxyethylensorbitan-trioleate) on the ripening process However during the experiments no significant influence on crystal size distributions was observed of neither air access nor the surfactant
724 Conclusions
From the analysis of the three recrystallization mechanisms described in this section can be concluded that Ostwald ripening is likely the most important mechanism altering the crystal size distribution during ice slurry storage for thermal storage applications The average crystal size can increase seriously as a result of Ostwald ripening in isothermally stored ice slurries Attrition might occur in storage tanks by contacts with the mixer the walls or other ice crystals In this respect it is most likely that only abrasion takes place and no breakage which means that only small fragments are pulled off larger crystals These small fragments will however dissolve as a result of Ostwald ripening and the crystal mass will subsequently be attached to the larger crystals in the storage tank Agglomeration might occur in ice slurry tanks for thermal storage especially when low amounts of freezing point depressants are applied However in most applications with ice slurry temperatures below ndash1degC agglomeration plays a minor role
73 Experiments on Ice Slurry Storage
The literature review in the previous section revealed that Ostwald ripening is likely the main recrystallization mechanism during ice slurry storage for thermal storage applications In order to obtain more knowledge on parameters that determine the rate of Ostwald ripening in different ice slurries experiments were performed with different solutions of both sodium chloride and ethylene glycol Only homogeneous storage was studied experimentally and in this respect the influence of the mixing rate was investigated
Long-term Ice Slurry Storage
147
731 Experimental Set-up
For the ice slurry storage experiments an experimental setup as shown in Figure 78 was applied
Figure 78 Experimental setup for ice slurry storage experiments
The main part of the experimental setup is a stirred crystallizer which consists of a double-wall cylindrical glass tank The inner diameter of the tank is 125 mm and its inner height is 130 mm The outside of the tank is insulated to minimize heat uptake The fluid in the vessel can be mixed with a circulator impeller with 3 blades and has a diameter of 47 mm A variable speed motor drives the impeller with a controllable frequency between 40 and 2000 RPM The temperature of the fluid inside the tank is measured with a PT-100 element which is connected to an ASL F250 temperature measurement set This combination enables temperature measurements within an accuracy of 001 K
During the experiments the tank was filled with 10 kg of aqueous solution of sodium chloride (NaCl) or ethylene glycol (EG) of the desired concentration Pure ethylene glycol was pumped through the annular space between the two glass walls to control the temperature inside the tank A low-temperature thermostat controlled the temperature of ethylene glycol
732 Experimental Procedure
The experimental procedure is described here by means of Figure 79 At the start of an experiment the temperature of ethylene glycol in the thermostat was set at a value of 30 K below the initial freezing temperature of the solution inside the vessel In case of the lowest NaCl concentration (26 wt) this value was only 15 K to avoid ice scaling on the wall The initial number of revolutions of the impeller was set at 400 RPM At a certain degree of supercooling (∆Tmaxsuper) initial nucleation of crystals was forced by putting a small steel rod in contact with the impeller After formation of the first crystals crystallization was continued until an ice fraction of approximately 14 wt was reached At this moment the temperature of ethylene glycol was increased in order to provide global thermal equilibrium in the tank and to keep the ice fraction constant The temperature difference for equilibrium was deduced from another experiment in which the temperature of ethylene glycol was constant and the temperature of the solution inside the vessel was measured after a long time After the increase of the ethylene glycol temperature ice slurry was stored for at least 22 hours with a constant mixing rate temperature and ice fraction
Chapter 7
148
-70
-60
-50
-40
-30
-20
-10
00
-2 -1 0 1 2Time (hours)
Tem
pera
ture
(degC
)
Slurry in tankInlet ethylene glycol
CrystallizationCooling Storage
Seeding
∆T maxsuper
24
Figure 79 Temperature profiles during experiment 2 (see Table 71)
Just after the onset of storage a sample of produced ice slurry was taken from the tank The ice crystals in this sample were photographed with a microscope and a CCD camera The microscope was equipped with a thermostatic glass which ensured that the crystals of the sample would neither melt nor grow during the observation After 2 6 and 22 hours this procedure was repeated in order to investigate the development of crystal size and shape in time The ice crystals on the 2-D photographs were analyzed by measuring both the projected area (Ap) and the perimeter (P) At least 80 crystals were measured from each ice slurry sample
For each single crystal the projected area and the perimeter were used to calculate two characteristics parameters The first parameter is the Feret diameter and is defined as the diameter of a circle with the same area as the projection of the crystal
p
Feret
4 AD
π= (731)
The second parameter is the roundness and is defined as the ratio between the perimeter of a circle with the same area as the crystal and the crystal perimeter
FeretDΓP
π= (732)
The roundness varies between 0 and 1 If the roundness is close to 1 the crystal is almost circular As the ratio decreases from 1 the object departs from a circular form
733 Results
During this study five experiments with different conditions were carried out An overview of the experimental conditions is given in Table 71
Long-term Ice Slurry Storage
149
Table 71 Experimental series of ice slurry storage experiments No Solute type Solute
concentration Initial freezing
temperature Temperature
during storageIce fraction
during storage Mixing rate
during storage (wt) (degC) (degC) (wt) (RPM)
1 NaCl 26 ndash15 ndash18 15 400 2 NaCl 49 ndash30 ndash35 15 400 3 NaCl 92 ndash60 ndash71 14 400 4 NaCl 49 ndash30 ndash35 15 750 5 EG 166 ndash62 ndash77 15 400
Four typical microscopic photographs of experiment 2 are shown in Figure 710 The photos have the same scale and were taken after 0 2 6 and 22 hours of storage as described above The four photos clearly show that the crystal size increases in time It is supposed that the main cause for this increase is Ostwald ripening and that agglomeration plays a minor role Two typical examples of agglomeration can be seen in Figure 710c and Figure 710d in which it is obvious that two crystals are cemented together and became one crystal However this cementing behavior is only observed for the minority of the crystals It seems that some ice crystals in Figure 710a and Figure 710b are also agglomerated but in reality they are only overlapping each other forming flocks
a b
c d
Figure 710 Microscopic photographs of experiment 2 (a) 0 hours (b) 2 hours (c) 6 hours and (d) 22 hours of storage
From crystal measurements crystal size distributions were constructed and characteristic parameters such as Feret diameter and roundness were calculated Figure 711 shows the
Chapter 7
150
crystals size distributions at the four sampling moments during experiment 2 In these crystal size distributions the crystals are divided into classes of 100 microm Because the change in crystal size is not quite clear in this figure the development of crystal sizes is also shown in Figure 712 by means of cumulative crystal size distributions From this figure it is more obvious that the average crystal size increases in time
00
01
02
03
04
0 250 500 750 1000 1250Feret diameter (microm)
Num
ber
frac
tion
(10 4 1
m) 0 h
2 h6 h22 h
00
02
04
06
08
10
0 250 500 750 1000 1250Feret diameter (microm)
Cum
ulat
ive
num
ber
frac
tion
(-)
0 h2 h6 h22 h
Figure 711 Development of crystal size distributions in time for experiment 2
Figure 712 Development of cumulative crystal size distributions in time for
experiment 2
Effect of Solute Type and Concentration
Figure 713 shows that the increase of the Feret diameter in time is smaller in solutions with higher sodium chloride concentrations This observation is in accordance with the studies mentioned in Section 723 The figure also shows that Ostwald ripening was slower in a ethylene glycol solution than in a sodium chloride solution with approximately the same freezing temperature (92 wt NaCl and 166 wt EG)
0
100
200
300
400
500
0 4 8 12 16 20 24Time (hours)
Ave
rage
Fer
et d
iam
eter
(microm
)
26 wt NaCl49 wt NaCl92 wt NaCl166 wt EG
Figure 713 Development of average Feret diameter for different types of ice slurry
Figure 714 shows that the roundness of crystals slightly increases during storage for all types of ice slurries Besides it can be concluded that ice slurries with lower solute concentrations resulted in rounder crystals
Long-term Ice Slurry Storage
151
080
085
090
095
100
0 4 8 12 16 20 24Time (hours)
Ave
rage
cry
stal
rou
ndne
ss (-
)26 wt NaCl49 wt NaCl92 wt NaCl166 wt EG
Figure 714 Development of crystal roundness for different types of ice slurry
Effect of Stirring Rate
The effect of the stirring rate was studied by storing the same type of ice crystals at stirring rates of 400 and 750 RPM The results shown in Figure 715 and Figure 716 indicate that the difference in mixing rate has only a marginal effect on the average crystal size However the average crystal roundness increases faster for the storage experiment with the higher stirring rate
0
100
200
300
400
500
0 4 8 12 16 20 24Time (hours)
Ave
rage
Fer
et d
iam
eter
(microm
)
400 rpm750 rpm
080
085
090
095
100
0 4 8 12 16 20 24Time (hours)
Ave
rgae
cry
stal
rou
ndne
ss (-
)
400 rpm750 rpm
Figure 715 Development of average Feret diameter at different stirring rates
Figure 716 Development of average crystal roundness at different stirring rates
A possible explanation for this behavior is that abrasion rounds off the parent crystals and produces new relatively small crystals (see Section 721) These small crystals will melt because of their relatively low equilibrium temperature and their crystal mass subsequently attaches to larger crystals Due to this phenomenon crystals become rounder but the average Feret diameter follows the same trend as with intermediate mixing
734 Comparison of Results with Results from Literature
Both the results reported in literature and the experimental results obtained in this study clearly show that the average crystal size increases during isothermal storage which can be
Chapter 7
152
ascribed to Ostwald ripening The results from different researchers show some interesting similarities
Influence of Solute Type and Concentration
Figure 717 shows that the type of solute strongly influences the ripening rate for example ripening of ice crystals in a 10 wt sucrose solution was much faster than in a 10 wt ethanol solution Besides the type of solute also its concentration appears to be very important since the ripening rate increases with decreasing sodium chloride concentration
0
100
200
300
400
500
0 10 20 30 40 50Time (hours)
Ave
rage
Fer
et d
iam
eter
(microm
)
10 wt sucrose (Smith et al 1985)10 wt EtOH (Hansen et al 2002)15 wt PG (Hansen et al 2002)166 wt EG (present study)26 wt NaCl (present study)49 wt NaCl (present study)92 wt NaCl (present study)
Figure 717 Average Feret diameter during homogeneous storage in different aqueous
solutions
Separate experiments with constant initial concentrations of ethanol or propylene glycol and different ice fractions showed that the increase of the average Feret diameter was smaller at higher ice fractions At higher ice fractions the actual solute concentration is higher and it is likely that this higher solute concentration decreases the ripening rate in case of higher ice fractions
Influence of Mixing Regime
Experimental results by Hansen et al (2002) shown in Figure 718 demonstrate that ripening is faster during homogeneous than during heterogeneous storage in a 10 wt ethanol solution since the slope of the curve for homogeneous storage is steeper than the curve for heterogeneous storage for the same average crystal size However experiments with different mixing rates namely 400 and 750 RPM in a 49 wt NaCl solution show that the increase in crystal size is hardly influenced by the mixing rate
Long-term Ice Slurry Storage
153
0
100
200
300
400
500
0 20 40 60 80 100Time (hours)
Ave
rage
Fer
et d
iam
eter
(microm
)10 wt EtOH homogeneous(Hansen et al 2002)10 wt EtOH heterogeneous(Hansen et al 2002)
49 wt NaCl homogeneous400 RPM (Present study)49 wt NaCl homogeneous750 RPM (Present study)
Figure 718 Average Feret diameters during experiments with heterogeneous and
homogeneous storage
735 Discussion
Since the crystal size distribution is an important parameter of ice slurries for several applications it would be useful to be able to predict the development of this crystal size distribution in time Existing models for Ostwald ripening are based on a theoretical description of the asymptotic increase of the crystal size and have the following form (Lifshitz and Slyozov 1961 Wagner 1961)
2avg avginit 1= + CL L C t (733)
Since storage conditions in practice do rarely correspond with the assumptions of this theoretical model this equation is mostly used as empirical correlation Because of its empirical character the correlation is not applicable to explain differences in ripening rates at different conditions In this section the crystallization kinetics of ice crystals are studied in order to explain different ripening rates in different ice slurries
Ice crystal growth can be divided into three stages namely heat transport from the crystal surface due to the release of the heat of fusion diffusion of solute from the surface towards the bulk and integration of molecules into the crystal lattice During dissolution of crystals the opposite of these three processes occur where the detachment of molecules from the lattice is assumed to be infinitely fast
A schematic representation of temperature and concentration profiles near a growing ice crystal is shown in Figure 719
Chapter 7
154
Figure 719 Temperature and concentration profiles near a growing ice crystal
The temperature difference in the boundary layer TindashTb can be calculated from the heat balance (Mersmann 2001)
( )Ai b
V ice f
-3
=∆
BG T TB h
αρ
(734)
The heat transfer coefficient α for spherical particles is mostly calculated from a correlation proposed by Brian et al (1969)
0 173 4 3 4liq equiv liq equiv0 25 6
3 3liq liq
2 1 3 for lt10
D DNu Pr
ξ ρ ξ ρmicro micro
= +
(735)
Analogously the mass balance gives the concentration difference in the boundary layer
( )2 2
liqAdsi H Ob H Oi
V ice
-3BG k w wB
ρρ
= (736)
In this equation parameter kdsi represents the mass transfer coefficient to a semi-permeable interface which can be deduced from the normal mass transfer coefficient (Mersmann 2001)
2
ddsi
H Ob1-kk
w= (737)
A correlation by Levins and Glastonbury (1972) can be used to determine this mass transfer coefficient for small spherical particles in agitated tanks
0207 0173 4liq equiv 036 mix
3liq tank
2 047D DSh Sc
Dξ ρ
micro
= + (738)
The concentration difference in Equation 736 can be transformed into a temperature difference which enables a comparison with the heat transfer growth rate
Long-term Ice Slurry Storage
155
( )2 2
2
b i H Ob H OiH O
- - dTT T w wdw
= (739)
Finally the difference between the temperature at the crystal interface and the equilibrium temperature can be calculated from the integration speed of crystals into the crystal lattice
( )int i i-=
rG k T T (740)
Huige and Thijssen (1969) proposed the following fitted correlation for the growth rate of ice crystals when the integration stage is limiting
( )155-3 i i27 10 -=G T T (741)
This correlation was deduced from experimental results with pure water It is possible that the growth rate decreases with an increasing concentration of solute because of adsorption of solute molecules on the crystal surface
Combining Equations 734 to 741 results in the following equation from which the total growth rate can be determined by iteration at a given supercooling Tb
-Tb by
2
2
155
H Ob-3 V ice ice fb b
A d liq H O
1-327 10wB hdTG T T G
B k dwρ ρρ α
∆ = minus minus +
(742)
In order to analyze which crystallization stage controls Ostwald ripening of ice crystals and what the effect of solute is on ripening a spherical ice crystal of 300 microm in diameter is considered here which is surrounded by liquid with a concentration wsoluteb and a bulk temperature Tb which is equal to the equilibrium temperature of a crystal of 200 microm The overall temperature driving force because of Ostwald ripening is about 12 10-4 K
Figure 720 shows the three isolated growth rates and the total growth rate without mixing as a function of the sodium chloride concentration for the considered crystal The isolated growth rates have been calculated by neglecting the growth resistance of the other two processes The figure shows that the isolated mass transfer growth rate highly depends on the solute concentration while the heat transfer and the integration growth rate only slightly decrease with increasing solute concentration Furthermore it can be seen that at low solute concentrations integration and heat transfer mainly determine the total growth rate while at higher concentrations mass transfer is the limiting stage The fact that the calculated total growth rate decreases with increasing solute concentration explains the observations shown in Figure 717 that higher solute concentrations decrease the ripening rate
Chapter 7
156
10E-11
10E-10
10E-09
10E-08
0 5 10 15 20w soluteb (wt)
G (m
s)
Heat transferMass transferSurface integrationTotal
Figure 720 Isolated growth rates and total growth rate of a 300 microm crystal at the considered
conditions without mixing in sodium chloride solutions
A surfactant can slow down the integration stage and it is therefore plausible that a surfactant only influences the ripening speed if the integration stage is limiting In cases where the diffusive or convective resistance is limiting a surfactant has a minor effect explaining the observations discussed in Section 723
Figure 721 shows the total growth rate for the considered crystal for different mixing regimes The figure shows that mixing increases the total growth rate but that increasing the mixing input has a minor effect This explains the results displayed in Figure 718 showing higher ripening rates during homogeneous storage compared to heterogeneous storage but that the mixing rate hardly influences the ripening process However higher mixing rates probably lead to more abrasion reducing the effect of increased mass and heat transfer coefficients
10E-11
10E-10
10E-09
10E-08
0 5 10 15 20w soluteb (wt)
G (m
s)
00 Wkg01 Wkg10 Wkg
Figure 721 Total growth rate of a 300 microm crystal at the considered conditions with different
values for mixing input in sodium chloride solutions
Figure 722 shows the total growth rates of the considered ice crystal during ripening without mixing for different aqueous solutions In most liquids the mass transfer stage determines the
Long-term Ice Slurry Storage
157
ripening rate at solute concentration above 5 wt which implies that the ripening rate highly depends on the solute concentration This is in correspondence with the experiments presented in Figure 717 The growth rate of ice crystals in sucrose solutions is higher than in the other solutions at the same concentration which is not expected on the first sight since diffusion coefficients of sucrose solutions are quite low However the derivative of the freezing line dTdwH2O has a relatively small value and therefore the growth rate is high with respect to the growth rate in other solutions
10E-12
10E-11
10E-10
10E-09
10E-08
0 5 10 15 20 25 30w soluteb (wt)
G (m
s)
NaClEthylene glycolEthanolPropylene glycolSucrose
Figure 722 Total growth rate of considered crystal as a function of concentration without
stirring
Figure 723 shows the total growth rates of different aqueous solutions as a function of the freezing temperature and can be used to compare different solutes for ice slurry applications Ostwald ripening in solutions of sodium chloride appears to be faster than in other solutions at the same freezing temperature At a freezing temperature of ndash5degC for example the growth rate of the considered crystal in a sodium chloride solution is approximately twice the growth rate in an ethanol solution Furthermore the figures show that fast ripening mainly occurs in slurries with high freezing temperatures
The preceding analysis on crystallization kinetics shows that the differences in observed ripening rates can be explained by theory on mass transfer heat transfer and surface integration kinetics A qualitative comparison of the ripening rate of two different ice slurries can be made with Figure 722 or Figure 723 However a quantitative prediction of the increase of the average crystal size in time is not possible with this analysis and therefore a dynamic model is developed for this purpose in the next section
Chapter 7
158
10E-12
10E-11
10E-10
10E-09
10E-08
-120 -100 -80 -60 -40 -20 00T freeze (degC)
G (m
s)
NaClEthylene glycolEthanolPropylene glycolSucrose
Figure 723 Total growth rate of considered crystal as a function of freezing temperature
without stirring
736 Conclusions
The rate of Ostwald ripening decreases with increasing solute concentration which is caused by the fact that crystal growth at solute concentrations above 5 wt is mainly determined by mass transfer resistance At low solute concentrations integration of molecules into the crystal lattice and heat transfer resistance play a major role The mass transfer growth resistance increases as the solute concentration increases while the heat transfer and surface integration resistance are hardly influenced by the solute concentration Mixing increases the ripening rate which can be explained by the fact that mass transfer coefficients are enhanced Increasing the mixing rate however hardly influences the ripening rate
74 Dynamic Modeling of Ostwald Ripening
The previous analysis showed that Ostwald ripening is the dominant recrystallization mechanism during storage of ice slurry for thermal storage applications This section presents a dynamic model to predict the development of the crystal size distribution based on these conclusions
741 Model Development
A dynamic model of an ice slurry storage tank has been developed in which ice slurry is assumed to be a perfectly mixed suspension During isothermal storage small crystals melt and large crystals grow as a result of Ostwald ripening The storage tank is considered as a closed and insulated system without transport of mass or energy across its boundaries However it is possible to adapt the model to include these fluxes in future versions
The most important equations of the dynamic model are the population the total mass the solute mass and the energy balance as shown in Table 72
Long-term Ice Slurry Storage
159
Table 72 The balance equations Population balance ( )( ) ( )( ) -
G L t n L tn L tt L
partpart=
part part (743)
with boundary conditions (0 ) 0n t = and (744) init( 0) ( )n L n L= (745) Total mass balance ( )( )( )liq ice1- 0V
tρ φ φ ρpart
+ =part
(746)
Solute mass balance ( )( )sol liq 1- 0V wt
ρ φpart=
part (747)
Energy balance ( ) ( )( )( )liq pliq ice pice f icetot1- 0V c T c T h At
ρ φ ρ φ γpart+ + ∆ + =
part (748)
with Mass fraction of ice
3ice V
tot 0
( )L
L
B n L L dLm
ρφ=infin
=
= int (749)
Total surface of crystals
2icetot A
0
( )L
L
A B n L L dL=infin
=
= int (750)
Because the storage tank is considered as a closed system neither inlet nor outlet mass flows can be found in the presented set of equations Next the storage tank is considered adiabatic which means that both heat transport from the surroundings and heat input by a mixing are neglected It is supposed that the size of ice crystals can only change as a result of Ostwald ripening and that other recrystallization mechanisms such as attrition and agglomeration can be neglected During Ostwald ripening the equilibrium temperature of the smallest crystals is below the actual slurry temperature and it can therefore be assumed that nucleation does not occur Since the formation of ice crystals from aqueous solutions is a very selective process (Vaessen 2002) it is assumed in the model that ice crystals do not contain any solute
In order to solve the equations shown in Table 72 the right-hand side of the population balance is discretized for the crystal size into a finite number of intervals resulting in a set of differential equations (Heijden and Rosmalen 1994) The width of each interval is chosen to be 5 microm and the maximum crystal size is set at 2000 microm The time integration of the total set of equations is performed in MATLAB using a differential equation solver based on an implicit Runge-Kutta formula (MATLAB 2002)
The crystallization kinetics of the considered ice crystals are given by the growth rate which is determined by transport phenomena and the temperature driving force given by the Gibbs-Thomson equation (see Table 73)
Chapter 7
160
Table 73 Crystallization kinetics used in simulations Growth rate ( )A
ice f ice solV
liq d sol
1 13
BG T Th w dTB
k dwρ ρ
α ρ
= minus ∆ +
(751)
with Equilibrium temperature
A
V ice lat
21-3
BT TB h L
γρinfin
= ∆
(730)
Heat transfer (Brian et al 1969)
0173 4liq equiv 025
3liq
2 13D
Nu Prξ ρ
micro
= +
(735)
Mass transfer (Levins and Glastonbury 1972)
0207 0253 4liq equiv 036 mix
3liq tank
2 047D DSh Sc
Dξ ρ
micro
= + (738)
The ice crystal shape and the surface tension between aqueous solution and ice are important parameters for Ostwald ripening but unequivocal values are lacking in literature Hillig (1998) has reviewed literature on determination of the surface tension with different kind of measurement techniques reporting values between 20 and 44 mJm2 Experiments by Hillig discussed in the same work give a value of 317plusmn27 mJm2 In the model presented here a constant value of 30 mJm2 is used for the surface tension and it is assumed that this value is not influenced by the solute type or concentration Literature references on the geometry of bulk ice crystals report disc-shaped ice crystals with height-to-diameter ratios varying from 01 to 05 (Margolis et al 1971 Huige 1972 Swenne 1983 Shirai et al 1985) During the experiments used for the validation of the model only two-dimensional pictures of ice crystals were analyzed and therefore it was not possible to determine their three dimensional shape Therefore ice crystals were modeled as circular discs with the disc diameter as characteristic crystal size L and a constant height-to-diameter ratio of 025 which was taken as an average value from the mentioned references
Heat transfer mass transfer and integration kinetics determine the growth rate of ice crystals while the former two transfer resistances determine the melting rate In the model heat and mass transfer correlations by Brian et al (1969) and Levins and Glastonbury (1972) have been applied in which an equivalent diameter Dequiv of the disc-shaped crystals is used as characteristic length An unequivocal model for surface integration kinetics is lacking in literature and therefore a model by Huige (1972) is considered here
( )155-3 27 10 -G T T= (741)
Figure 724 shows the isolated growth rates of mass transfer heat transfer and integration kinetics as a function of the crystal size in a 49 wt sodium chloride solution that is in equilibrium with a crystal of 200 microm in size which is called the neutral diameter Crystals smaller than this neutral diameter show negative growth rates which means that these crystals are melting The dominant transport resistance for a specific crystal size can be determined from the smallest isolated growth or melting rate for this crystal size The figure reveals that mass transfer resistance is the dominant stage for most crystal sizes and that the integration kinetics are only of importance for crystal sizes slightly larger than the neutral diameter Because of the latter conclusion the model assumes that crystal growth kinetics are controlled
Long-term Ice Slurry Storage
161
by heat and mass transfer resistances while the resistance of integrating water molecules into the lattice of a growing ice crystal is neglected
-15
-10
-05
00
05
0 200 400 600 800 1000Crystal size (microm)
Cry
stal
gro
wth
rat
e (1
0 -7 m
s)
G heat
G surf G mass
Figure 724 Isolated growth rates for mass transfer heat transfer and surface integration
kinetics for a crystal in a 49 wt NaCl solution in equilibrium with 200 microm crystals
742 Validation Conditions
The developed model has been validated with experimental results for Ostwald ripening in homogeneously mixed tanks described in Sections 72 and 73 of this thesis (see Table 74)
Table 74 Parameters of experimental studies used for model validation Solute Reference Solute conc
(wt) Ice fraction (wt)
Tank volume (l)
Time (h)
Betaine Louhi-Kultanen (1996) 8 and 15 - 6 2 Ethylene glycol This thesis 166 15 1 25 Sodium chloride This thesis 26 49 and 92 15 1 25 Sucrose Louhi-Kultanen (1996) 8 and 15 - 6 2 Smith and Schwartzberg (1985) 9 to 38 9 to 16 10 5 Different types of experimental facilities were used to carry out the ripening experiments listed in Table 74 During the experiments by Louhi-Kultanen (1996) and the experiments carried out during the present study stirred tank crystallizers of 6 and 1 liter were used The crystallizers were equipped with a 3-blade propeller and cooling jacket enabling to operate at adiabatic conditions Hansen et al (2002) applied an insulated storage tank of 1000 liter equipped with two 3-blade propellers In order to compensate for heat penetration from the surroundings and to keep a constant ice fraction in the tank a continuous flow of ice slurry was pumped through an ice slurry generator Smith and Schwartzberg (1985) used a 10-liter insulated flask equipped with an auger-type impeller which was placed in a cold room to achieve adiabatic conditions The ratio between the diameter of the mixer and the tank was 06 for the experiments performed by Louhi-Kultanen (1996) and about 04 for the other experiments The mixing power per unit of mass was mentioned by none of the researchers but it was assumed that ice slurry was gently mixed and therefore an estimated value of 02 Wkg was used Measured initial crystal size distributions of experiments were transformed into Rosin-Rammler distributions and used as initial size distributions for simulations
Chapter 7
162
743 Validation Results
After simulation the development of the average crystal size in time was deduced from the changing crystal size distribution during simulation and compared to experimental results First the validation with experimental results obtained in closed adiabatic storage tanks is discussed followed by the validation with results obtained from a storage tank combined with an ice slurry generator
Figure 725 shows the comparison for ripening in sodium chloride and ethylene glycol solutions Both simulation and experimental results show that Ostwald ripening is slower at higher solute concentrations which can be attributed to the increased mass transfer resistance of transporting solute from or to the ice crystal surface in case of growing or melting respectively The model seems to be able to predict average crystal sizes after one day of storage fairly although the real process seems to be faster in the early stage of ripening than the model predicts The figure also shows that the developed model confirms the experimental conclusion that the mixing rate does hardly have any influence on the ripening rate for mixing rates of 400 and 750 rpm corresponding with 02 and 13 Wkg respectively
0
100
200
300
400
500
600
0 5 10 15 20 25Time (hours)
Ave
rage
cry
stal
siz
e (micro
m)
26 wt NaCl 400 rpm 49 wt NaCl 400 rpm 49 wt NaCl 750 rpm 92 wt NaCl 400 rpm 166 wt EG 400 rpm
Figure 725 Development of average ice crystal size obtained from model (lines) and
experiments (points) for solutions of sodium chloride (NaCl) and ethylene glycol (EG) for different mixing rates
The validation for ripening in sucrose solutions shown in Figure 726 reveals that the development of the crystal size during simulation is qualitatively in accordance with the experiments since higher solute concentrations show lower ripening rates However quantitative errors of predicted average crystal size after five hours of storage show values up to 40 microm The initial average crystal sizes of simulations shown in this figure slightly differ from the experimental values since experimental distributions could not exactly be represented by Rosin-Rammler distributions
Long-term Ice Slurry Storage
163
0
50
100
150
200
250
300
00 10 20 30 40 50 60Time (hours)
Ave
rage
cry
stal
siz
e (micro
m)
8 wt φ =1
86 wt φ =0142
130 wt φ =0142
15 wt φ =1
190 wt φ =0132
369 wt φ =0122
Figure 726 Development of average crystal size obtained from model (lines) and
experiments (points) by 1Louhi-Kultanen (1996) and 2Smith and Schwartzberg (1985) for aqueous sucrose solutions
Besides the average crystals size the crystal size distribution is also an important parameter of ice slurry Figure 727 and Figure 728 show the validation of crystal size distributions before and after ripening in aqueous solutions of 8 wt betaine and 26 wt sodium chloride respectively The crystal size distributions obtained from simulations generally resemble the experimental ones but latter distributions seem to have a longer tail from which is concluded that the dynamic model underestimates the fraction of relatively large crystals
00102030405060708090
100
0 100 200 300 400 500Crystal size (microm)
Num
ber
frac
tion
(10 3 1
m) 5
After 0 hours of storage After 2 hours of storage
00
10
20
30
40
50
0 200 400 600 800 1000 1200Crystal size (microm)
Num
ber
frac
tion
(10 3 1
m) After 0 hours of storage
After 22 hours of storage
Figure 727 Comparison of crystal size distributions obtained from model (lines) and experiments (points) before and after
storage for solutions of 8 wt betaine
Figure 728 Comparison of crystal size distributions obtained from model (lines) and experiments (points) before and after storage for solutions of 26 wt sodium
chloride
Figure 729 displays the validation of the model with the experimental results of ethanol and propylene glycol solutions obtained by Hansen et al (2002) who used an ice slurry generator to keep a constant ice fraction in the storage tank The figure clearly shows that the increase of the average crystal size is faster during these experiments than in simulations The fact that
Chapter 7
164
the storage tank was not closed and adiabatic during the experiments might be an explanation for these deviations
0
100
200
300
400
500
600
700
0 20 40 60 80 100Time (hours)
Ave
rage
cry
stal
siz
e (micro
m)
10 wt EtOH φ =010 10 wt EtOH φ =030 15 wt PG φ =010 15 wt PG φ =030
Figure 729 Development of average crystal size obtained from model (lines) and
experiments (points) for solutions of ethanol (EtOH) and propylene glycol (PG) in a storage tank combined with ice slurry generator
744 Discussion
The figures discussed in the previous section showed that the developed dynamic model is able to predict the development of the average crystals size in aqueous sodium chloride ethylene glycol and sucrose solutions at different concentrations fairly but that there are also relatively small deviations between simulations and experiments
A first cause for these deviations can be revealed by means of the development of crystals size distributions shown in Figure 727 and Figure 728 in which the right-hand side of the experimental distributions after several hours of storage is longer and flatter compared to the ones obtained from simulations Limited agglomeration of ice crystals during experiments could be a cause for this observation which is supported by the fact that some agglomerated ice crystals were identified during experiments Although Ostwald ripening is believed to be the main cause for the increase of ice crystals during adiabatic storage limited agglomeration can influence the development of the average crystals size
Another justification for deviations is the fact that the dynamic model assumed crystals to be circular discs with a constant height-to-diameter ratio while the experiments did not show perfect circular discs and experimental height-to-diameter ratios might differ from the constant value taken from literature Simulations with other height-to-diameter ratios have shown that a smaller ratio results in faster Ostwald ripening (see Figure 730) Furthermore experiments showed that the crystal discs become rounder during the first hours of storage which might explain the faster development of the crystal size in the early stages of ripening observed during experiments shown in Figure 725
Long-term Ice Slurry Storage
165
0
100
200
300
400
500
600
700
0 5 10 15 20 25Time (hours)
Ave
rage
cry
stal
siz
e (m
m)
ExperimenthD=015hD=020hD=025hD=030
HD =015 HD =020 HD =025HD =030
Figure 730 Development of average crystal size obtained from model with different height-
to-diameter ratios (HD) and experiment for an aqueous solution of 26 wt sodium chloride
A third explanation for differences between simulations and experiments is the error introduced by the method of modeling crystallization kinetics namely the neglect of the integration kinetics and the implicit errors introduced by the applied heat and mass transfer models The error of these models might be increased by the fact that they have been used for circular discs while they were originally proposed for spherical particles
The final explanation for deviations is the uncertainty in the surface tension between liquid and ice for which a constant value has been taken from literature For this surface tension exact values or models are not available while it might also depend on the solute type and concentration
Figure 730 demonstrates that the developed model for Ostwald ripening in closed adiabatic storage tanks is not applicable for ripening of ice crystals in storage tanks that are connected to an ice slurry generator that compensates for heat penetration In order to simulate the development of the crystal size in this type of storage tanks the model can be extended with the ice slurry generator This extended system is both closed and adiabatic since there is no transport of mass over the system boundaries and the heat that is added to the storage tank equals the heat that is removed by the ice slurry generator For these reasons heat and mass balances are not modified and only the population balance (see Equation 743) is extended with a crystal size distribution flowing to (nin) and from (n) the storage tank respectively
( ) ( )isgin
tot
( ) ( )( ) ( ) ( )mG Lt n Ltn Lt - n Lt - n Lt
t L mpartpart
= +part part
amp (752)
It is assumed that the crystal size distribution n(Lt) coming from the storage tank is subject to growth in the ice slurry generator resulting in crystal population with a larger average crystal size and a higher ice fraction flowing back to the storage tank The growth rate in the ice slurry generator can be approached to be independent of the crystal size and is just enough to compensate for heat penetration Nucleation is neglected in the ice slurry generator and the mass flow through the ice slurry generator has no influence on simulation results and is therefore arbitrarily chosen The results of the extended model shown in Figure 731 demonstrate that the extended model is able to simulate ripening and that the cold loss
Chapter 7
166
compensation by the ice slurry generator accelerates the ripening process For these simulations the value for heat penetration has been tuned at 925 W which represents thermal convection to the storage tank and piping and heat input by the circulation pump
0
100
200
300
400
500
600
700
0 20 40 60 80 100Time (hours)
Ave
rage
cry
stal
siz
e (micro
m)
10 wt EtOH φ =010 10 wt EtOH φ =030 15 wt PG φ =015 15 wt PG φ =030
Figure 731 Development of average crystal size obtained from extended model (lines) and experiments (points) by Hansen et al (2002) for solutions of ethanol (EtOH) and propylene
glycol (PG)
745 Conclusions
A dynamic model for Ostwald ripening of bulk ice crystals during adiabatic storage has been developed Validation of the developed model with experimental data has shown that the model is able to predict the development of the ice crystal size in time fairly In analogy with the considered experiments the simulations showed that mass transfer is the limiting transport mechanism for the considered ice suspensions Deviations between model and experiments are believed to be mainly the result of limited agglomeration and differences in crystal shape
75 Conclusions
Ostwald ripening is the most important recrystallization mechanism during isothermal storage of ice slurry for thermal storage applications During storage small crystals dissolve while larger crystals grow resulting in an increase of the average crystal size The rate of Ostwald ripening decreases with increasing solute concentration which is caused by the fact that crystal growth is mainly determined by mass transfer resistance Mixing increases the ripening rate which can be explained by the fact that mass transfer coefficients are enhanced Increasing the mixing rate however hardly influences the ripening rate A developed dynamic model enables to predict the development of the bulk ice crystals size distribution by Ostwald ripening in time fairly
Long-term Ice Slurry Storage
167
Nomenclature
A Surface area of crystal (m2) V Volume of crystal (m3) Ap Projected area of crystal (m2) Vm Molar volume (m3mol) BA Surface shape factor equal to A L-2 w Mass fraction BV Volume shape factor equal to V L-3 c1 Ripening constant in Eq 733 Greek c2 Ripening exponent in Eq 733 α Heat transfer coefficient (Wm2K) cp Specific heat (Jkg K) γ Surface tension between ice and D Diameter of disc (m) liquid (Jm2) DFeret Feret diameter defined in Eq 731 Γ Roundness defined by Eq 710 (m) δ Boundary layer thickness (m) Dmix Mixer diameter (m) λ Heat conductivity (Wm K) Dtank Tank diameter (m) micro Viscosity (Pa s) D Mutual diffusion coefficient (m2s) micro Chemical potential (Jmol) G Crystal growth rate (ms) ∆micro Chemical potential difference ∆G Free Gibbsrsquo energy (J) (Jmol) ∆hf Heat of fusion (Jkg) ξ Power input by mixer (Wkg)
fh∆ Heat of fusion (Jmol) ρ Density (kgm3) H Height of disc (m) φ Ice mass fraction kd Mass transfer coefficient (ms) Ψ Activity coefficient kdsi Mass transfer coefficient to a semi- permeable interface (ms) Subscripts kint Integration kinetics constant avg Average (ms Kr) b Bulk L Characteristic crystal size (m) equiv Equivalent m Mass H2O Water mamp Mass flow (kgs) heat Heat transfer n Number of moles i Interface n Number of crystals ice Ice Nu Particle Nusselt number α Lλ init Initial P Perimeter in Inlet Pr Prandtl number cp microλ int Surface integration r Order of crystal growth isg Ice slurry generator s Entropy (Jmol) liq Liquid
fs∆ Entropy of fusion (Jmol) m Mass transfer Sc Schmidt number microρliq D mass Mass transfer Sh Sherwood number kd LD max Maximum T Temperature (K or degC) s Solid Tfr Freezing temperature (degC) solute Solute
T Equilibrium temperature (K) super Supercooling Tinfin
Equilibrium temperature of infinite surf Surface integration crystal (K) th Thermal ∆T Temperature difference (K) tot Total t Time (hours)
Chapter 7
168
Abbreviations
EG Ethylene glycol NaCl Sodium chloride EtOH Ethanol PG Propylene glycol
References
Adapa S Schmidt KA Jeon IJ Herald TJ Flores RA 2000 Mechanisms of ice crystallization and recrystallization in ice cream A review Food Reviews International vol16 pp259-271
Barnes P Tabor D Walker FRS Walker JCF 1971 The friction and creep of polycrystalline ice Proceedings of the Royal Society of London Series A vol324 pp127-155
Biscans B Guiraud P Lagueacuterie C Massarelli A Mazzarotta B 1996 Abrasion and breakage phenomena in mechanically stirred crystallizers The Chemical Engineering Journal vol63 pp85-91
Brian PLT Hales HB Sherwood TK 1969 Transport of heat and mass between liquids and spherical particles in an agitated tank AIChE Journal vol15 pp727-733
Christensen KG Kauffeld M 1998 Ice slurry accumulation In Proceedings of the Oslo Conference IIR commission B1B2E1E2 Paris International Institute of Refrigeration pp701-711
Donhowe DP Hartel RW 1996 Recrystallization of ice during bulk storage of ice cream International Dairy Journal vol6 pp1209-1221
Egolf PW Vuarnoz D Sari O 2001 A model to calculate dynamical and steady-state behaviour of ice particles in ice slurry storage tanks In Proceedings of the 4th IIR Workshop on Ice Slurries 12-13 November 2001 Osaka (Japan) Paris International Institute of Refrigeration pp25-39
Frei B Egolf PW 2000 Viscometry applied to the Bingham substance ice slurry In Proceedings of the 2nd IIR Workshop on Ice Slurries 25-26 May 2000 Paris (France) Paris International Institute of Refrigeration pp48-60
Gahn C Mersmann A 1995 The brittleness of substances crystallized in industrial processes Powder Technology vol85 pp71-81
Hagiwari T Hartel RW 1996 Effect of sweetener stabilizer and storage temperature on ice recrystallization in ice cream Journal of Dairy Science vol79 pp735-744
Hansen TM Radošević M Kauffeld M 2002 Behavior of Ice Slurry in Thermal Storage systems ASHRAE Research project ndash RP 1166
Hansen TM Radošević M Kauffeld M Zwieg T 2003 Investigation of ice crystal growth and geometrical characteristics in ice slurry ASHRAE HVACampR Research Journal vol9 pp9-32
Long-term Ice Slurry Storage
169
Hayashi K Kasza KE 2000 A method for measuring ice slurry particle agglomeration in storage tanks ASHRAE Transactions vol106 pp117-123
Heijden AEDM van der Rosmalen GM van 1994 Industrial mass crystallization In Hurle (Ed) Handbook of Crystal Growth Part 2A ndash Bulk Crystal Growth Basic Principles pp372-377
Hillig WB 1998 Measurement of interfacial free energy for icewater system Journal of Crystal Growth vol183 pp463-468
Huige NJJ Thijssen HAC 1969 Rate controlling factors of ice crystal growth from supercooled water glucose solutions In Industrial Crystallization Proceedings of a Symposium on Industrial Crystallization April 15-16 London (Great Britain) London The Institution of Chemical Engineers pp69-86
Huige NJJ 1972 Nucleation and Growth of Ice Crystals from Water and Sugar Solutions in Continuous Stirred Tank Crystallizers PhD thesis Eindhoven University of Technology (The Netherlands)
Huige NJJ Thijssen HAC 1972 Production of large crystals by continuous ripening in a stirrer tank Journal of Crystal Growth vol1314 pp483-487
Kasza KE Hayashi K 1999 Ice slurry cooling research storage tank ice agglomeration and extraction ASHRAE Transactions vol105 pp260-266
Kitanovski A Poredoš A 2002 Concentration distribution and viscosity of ice-slurry in heterogeneous flow International Journal of Refrigeration vol 25 pp827-835
Kobayashi A Shirai Y 1996 A method for making large agglomerated ice crystals for freeze concentration Journal of Food Engineering vol27 pp1-15
Kozawa Y Aizawa N Tanino M 2005 Study on ice storing characteristics in dynamic-type ice storage system by using supercooled water Effects of the supplying conditions of ice-slurry at deployment to district heating and cooling system International Journal of Refrigeration vol28 pp73-82
Levins BE Glastonbury JR 1972 Particle-liquid hydrodynamics and mass transfer in a stirred vessel Part II ndash Mass transfer Transactions of the Institution of Chemical Engineers vol50 pp132-146
Lifshitz IM Slyozov VV 1961 The kinetics of precipitation from supersaturated solid solutions Journal of Physics and Chemistry of Solids vol19 pp35-50
Louhi-Kultanen M 1996 Concentration and Purification by Crystallization PhD thesis Lappeenranta University of Technology (Finland)
MATLAB 2002 Version 6 Mathwork Inc Natwick
Margolis G Sherwood TK Brian PLT Sarofim AF 1971 The performance of a continuous well stirred ice crystallizer Industrial and Engineering Chemistry Fundamentals vol10 pp439-452
Chapter 7
170
Mazzarotta B 1992 Abrasion and breakage phenomena in agitated crystal suspensions Chemical Engineering Science vol47 pp3105-3111
Meili F Sari O Vuarnoz D Egolf PW 2001 Storage and mixing of ice slurries in tanks In Proceedings of the 3rd IIR Workshop on Ice Slurries 16-18 May 2001 Lucerne (Switzerland) Paris International Institute of Refrigeration pp97-104
Mersmann A 2001 Crystallization Technology Handbook Second edition New York Marcel Dekker Inc
Nielsen AE 1964 Kinetics of Precipitation Oxford Pergamon Press
Savory RM Hounslow MJ Williamson AM 2002 Isothermal coarsening anisotropic ice crystals In Proceedings of the 15th International Symposium on Industrial Crystallization September 15-18 Sorrento (Italy)
Shirai Y Nakanishi K Matsuno R Kamikubo T 1985 Effects of polymers on secondary nucleation of ice crystals Journal of Food Science vol50 pp401-406
Shirai Y Sugimoto T Hashimoto M Nakanishi K Matsuno R 1987 Mechanism of ice growth in a batch crystallizer with an external cooler for freeze concentration Agricultural and Biological Chemistry vol51 pp2359-2366
Sutton RL Evans ID Crilly JF 1994 Modeling ice crystal coarsening in concentrated disperse food systems Journal of Food Science vol59 pp1227-1233
Sutton RL Lips A Piccirillo G Sztehlo A 1996 Kinetics of ice recrystallization in aqueous fructose solutions Journal of Food Science vol61 pp741-745
Smith CE 1984 Ice Crystal Growth Rates during the Ripening Stage of Freeze Concentration (Mass-transfer Sequential Analysis Neutral Diameter) PhD thesis University of Massachusetts (USA)
Smith CE Schwartzberg HG 1985 Ice crystal size changes during ripening in freeze concentration Biotechnology Progress vol1 pp111-120
Swenne DA 1983 The Eutectic Crystallization of NaClmiddot2H2O and Ice PhD thesis Eindhoven University of Technology (The Netherlands)
Verschuur RJ Scholz R Nistelrooij N van Schreurs B 2002 Innovations in freeze concentration technology In Proceedings of the 15th International Symposium on Industrial Crystallization September 15-18 Sorrento (Italy)
Wagner C 1961 Theorie der Alterung von Niederschlaumlgen durch Umloumlsen (Ostwald-Reifung) Zeitschrift fuumlr Elektrochemie vol65 pp581-591
Williamson A Lips A Clark A Hall D 2001 Ripening of faceted ice crystals Powder Technology vol121 pp74-80
171
8 Melting of Ice Slurry in Heat Exchangers
81 Introduction
After production and storage ice slurry is transported to applications where it provides cooling to rooms products or processes (see Chapter 1) Due to the absorption of heat the ice slurry temperature increases and ice crystals melt The melting process is expected to be strongly influenced by the properties of ice slurry such as the ice fraction and the average ice crystal size Since these properties are mainly determined during the production and storage stage it is important to know their influences on the melting process Furthermore knowledge on heat and mass transfer processes during melting may improve the knowledge of ice slurry production processes or vice versa
In general two different methods of ice slurry melting can be distinguished The first method is called direct contact melting and is mainly applied in food industry for cooling of fish fruit and vegetables (Fikiin et al 2005 Torres-de Mariacutea et al 2005) In this method ice slurry is poured directly onto fresh harvested products resulting in high cooling rates which ensure a high product quality In the second method ice slurry is pumped through a regular heat exchanger absorbing heat from air or another fluid This method is frequently applied in refrigerated display cabinets for supermarkets and in air conditioning systems for buildings
This chapter focuses on the melting process of ice slurries in heat exchangers First a literature review on hydrodynamics and heat transfer aspects of melting ice slurries is presented to investigate which aspects of melting ice slurries are not fully understood yet The second part consists of an experimental study of ice slurry melting in a tube-in-tube heat transfer coil which aims to give a contribution to the understanding of these aspects
82 Literature Review on Ice Slurry Melting in Heat Exchangers
The performance of ice slurry as secondary refrigerant is partly determined by its performance during melting in application heat exchangers Important design aspects in this respect are the heat transfer coefficient between the melting ice slurry and the heat exchanger wall and the pressure drop of the ice slurry flow between inlet and outlet Both heat transfer and pressure drop are influenced by the flow pattern and rheology of the flowing ice slurry Another aspect that plays a role during melting is superheating of ice slurry which can seriously reduce the heat transfer capacity of a heat exchanger
This section gives a brief literature review on these various aspects of ice slurry melting in heat exchangers More extensive reviews on this subject have been presented by Ayel et al (2003) Egolf et al (2005) and Kitanovski et al (2005)
821 Flow Patterns
According to Kitanovski et al (2002) three different patterns can be distinguished for ice slurry flows in horizontal tubes namely moving bed flow heterogeneous flow and homogeneous flow (see Figure 81) In moving bed flow ice crystals accumulate in the upper part of the tube forming a crystal bed while the liquid flows underneath it The velocity of the
Chapter 8
172
crystal bed is lower than the liquid velocity In heterogeneous flows the crystals are suspended over the entire cross section of the tube but their concentration is higher in the upper part of the tube than in the lower part In case of a homogeneous flow ice crystals are randomly distributed and the crystal concentration is therefore constant over the entire cross section
Figure 81 Flow patterns for ice slurry flow in horizontal tubes
The boundaries between the different flow patterns are mainly determined by the ice slurry velocity the average crystal size the density ratio between ice and solution and the ice fraction In case of low velocities large crystals or high density ratios between liquid and ice the ice crystals have the tendency to float to the top of the tube forming a moving bed flow As the velocity increases the ice crystals are smaller or the liquid density is closer to the density of ice the flow pattern turns initially to heterogeneous flow and finally to homogeneous flow According to Lee et al (2002) the flow pattern tends also more towards the homogeneous flow regime when the ice fraction increases Validated correlations to predict flow patterns for ice slurry are lacking in literature General correlations to predict flow patterns of suspension flows are given by Wasp et al (1977) Shook and Roco (1991) and Darby (1986)
Kitanovski et al (2002) presented experiments to determine flow patterns of ice slurry with ice crystals of 01 to 03 mm in 10 wt ethanol solutions The transition from moving bed flow to heterogeneous flow occurred at velocities between 01 to 03 ms At velocities above 02 to 05 ms the ice slurry flow became homogeneous
822 Rheology
Several researchers have studied the rheological behavior of homogeneous ice slurry flows They generally agree that ice slurry with ice fractions up to 15 wt can be considered as Newtonian which means that the shear rate is proportional to the yield stress (Ayel et al 2003 Meewisse 2004 Kitanovski et al 2005) For ice fractions above 15 wt two different types of rheology have been proposed namely pseudo-plastic (Guilpart et al 1999) and Bingham type of flow behavior (Doetsch 2001 Frei and Egolf 2000 Niezgoda-Żelasko and Zalewski 2006) Doetsch (2002) proposed to use the Casson model which combines Newtonian behavior at low ice fractions with Bingham behavior at higher ice fractions
823 Pressure Drop
Most experimental studies on pressure drop of ice slurries were performed with horizontal tubes In general these studies report an increase in pressure drop with increasing ice fraction especially at low ice slurry velocities (Christensen and Kauffeld 1997 Jensen et al 2000
Melting of Ice Slurry in Heat Exchangers
173
Bedecarrats et al 2003 Lee et al 2006 Niezgoda-Żelasko and Zalewski 2006) Bedecarrats et al (2003) for example measured pressure drop values for a velocity of 05 ms and an ice fraction of 20 wt that were a factor of six higher than for the case without ice crystals At higher ice slurry velocities pressure drop values also increased with increasing ice fraction but to a lower extent
At high velocities of about 1 to 2 ms and ice fractions of about 20 wt Bedecarrats et al (2003) and Niezgoda-Żelasko and Zalewski (2006) observed a sudden decrease in pressure drop with increasing ice fraction When the ice fraction was further increased the pressure drop restarted to rise resulting in a local minimum in pressure drop Niezgoda-Żelasko and Zalewski (2006) attribute this sudden decrease in pressure drop to a laminarization of the ice slurry flow at high ice fractions This explanation can also be used for the experimental results of Knodel et al (2000) which showed a decrease in pressure drop of 8 as the ice fraction increased from 0 to 10 wt
Experiments with ice slurry in plate heat exchangers were performed by Bellas et al (2002) Frei and Boyman (2003) and Noslashrgaard et al (2005) All three studies showed an increase in pressure drop with increasing ice fraction especially at low ice slurry velocities Frei and Boyman (2003) reported that the pressure drop for low velocities increased with 44 as the ice fraction increased from 0 to 29 wt At higher velocities this increase was 32
824 Heat Transfer Coefficients
Experimental results for wall-to-slurry heat transfer coefficients in horizontal tubes showed approximately the same results as the pressure drop measurements discussed above Christensen and Kauffeld (1997) Jensen et al (2000) and Lee et al (2006) found that heat transfer coefficients increased up to a factor of three with increasing ice fraction The highest relative increase was measured for low ice slurry velocities up to 1 ms while the enhancement at high ice slurry velocities of about 3 to 4 ms was only small Bedecarrats et al (2003) and Niezgoda-Żelasko (2006) measured approximately the same trends but at high velocities and ice fractions around 20 wt they also observed a decrease in heat transfer coefficient as the ice fraction increased This decrease was attributed by Niezgoda-Żelasko to a laminarization of the ice slurry flow Knodel et al (2000) used the same explanation for their experimental results which showed a continuous decrease of the heat transfer coefficient as the ice fraction increased from 0 to 10 wt at high ice slurry velocities of about 5 ms
The influence of the heat flux on wall-to-slurry heat transfer coefficients was investigated by several researchers (Christensen and Kauffeld 1997 Jensen et al 2000 Lee et al 2006 Niezgoda-Żelasko 2006) All these experimental studies demonstrated that the heat flux has no effect on heat transfer coefficients of melting ice slurry
Ice slurry melting experiments with plate heat exchangers by Noslashrgaard et al (2005) Frei and Boyman (2003) and Stamatiou and Kawaji (2005) showed increasing heat transfer coefficients with increasing ice fractions In accordance with the experiments with horizontal tubes the relative increase of the heat transfer coefficient was especially high at low mass flow rates Experiments with a plate heat exchanger by Bellas et al (2002) showed different trends since the results indicated that ice fractions up to 22 did not have any influence on heat transfer
In none of the experimental studies on heat transfer coefficients of melting ice slurry the influence of the ice crystal size has been studied
Chapter 8
174
825 Superheating
Ice slurry is called superheated when its liquid temperature is higher than its equilibrium temperature Superheating can be explained by considering the melting process of ice slurry in heat exchangers as a two-stage process First the heat exchanger wall heats the liquid and consequently the superheated liquid melts the ice crystals The relation between the rates of both processes determines the degree of superheating For example when crystal-to-liquid heat and mass transfer processes are relatively slow compared to the wall-to-liquid heat transfer process then the degree of superheating is high Superheating always occurs in melting heat exchangers but its degree depends on the operating conditions
Due to superheating the average ice slurry temperature in the heat exchanger is higher than is expected from equilibrium calculations and as a result the heat exchanger capacity is lower Figure 82 shows an example to explain the effect of superheating on the heat exchanger capacity The figure represents the temperature of an ice slurry on its path from the storage tank via the pump to the heat exchanger and back to the storage tank The ice slurry that enters the heat exchanger is in equilibrium and has an ice fraction of 10 wt and a temperature of ndash50degC The ice slurry is heated by a heat source of 20degC and as a result the ice crystals melt According to equilibrium calculations the ice fraction at the outlet is zero and the temperature of the solution equals its freezing temperature of ndash45degC However the real temperature of the ice slurry in the heat exchanger is higher and the slurry leaves the heat exchanger with a temperature of ndash15degC A fraction of the ice crystals is still present at the outlet of the heat exchanger and the melting process continues in the tubing between the heat exchanger and the tank resulting in a decrease of the slurry temperature The figure clearly shows that the real temperature difference between the slurry and the heat source is smaller than is expected from equilibrium calculations As a result of this smaller temperature difference the heat exchanger capacity is also significantly lower
Figure 82 Example of real and equilibrium temperature profiles of ice slurry in a melting
loop
Up to now superheated ice slurry at the outlet of melting heat exchangers has been observed by Hansen et al (2003) Kitanovski et al (2003) and Frei and Boyman (2003) Only the latter
Melting of Ice Slurry in Heat Exchangers
175
researchers reported superheating values indicating that superheating especially occurs at low ice fractions This trend is explained by the reduced crystal surface at low ice fractions which slows down the crystal-to-liquid heat and mass transfer processes
826 Outlook for Experiments
The preceding literature review has shown that the influences of ice slurry velocity and ice fraction on pressure drop and heat transfer coefficients have extensively been studied by various researchers Although the reported pressure drop and heat transfer data show approximately the same trends in the various studies more experiments are required to fully understand the role of all parameters In particular the role of the average ice crystal size and the crystal size distribution on the heat transfer process needs attention in this respect because these aspects have not been considered in any experimental study up to now
Another issue that has only attained little attention is superheating of ice slurry in heat exchangers Superheating can seriously reduce the capacity of melting heat exchangers and it is therefore important to investigate the physical phenomena behind it In this respect it is interesting to study the influences of ice slurry velocity and heat flux on superheating Furthermore the average crystal size and the ice fraction are expected to have a strong effect on the degree of superheating since they determine the available crystal surface for the crystal-to-liquid process The melting of ice crystals may be limited by mass transfer and in that case the solute concentration also has a strong influence on the degree of superheating
In the next sections an experimental study on melting of ice slurry in a heat exchanger is presented This study gives a contribution to the knowledge on ice slurry melting especially on the subjects that have been mentioned above
83 Experimental Method
831 Experimental Set-up
The experiments on melting of ice slurry in a heat exchanger were performed with the experimental set-up shown in Figure 83 A fluidized bed heat exchanger as described in Section 22 was used to produce ice slurry from aqueous sodium chloride solutions The produced ice slurry was stored in an insulated tank that was equipped with a mixing device to keep the ice slurry homogeneous The tank could easily be disconnected from the set-up and be placed in a cold room After production and eventually isothermal storage in the cold room a visualization section consisting of a flow cell and a microscope was applied to analyze the produced ice crystals (see Figure 511)
The ice slurry was subsequently pumped through the inner tube of a tube-in-tube heat transfer coil which had an internal diameter of 70 mm an outside diameter of 95 mm and a total external heat-exchanging surface of 0181 m2
A 20 wt ethylene glycol solution which was extracted from a thermostatic bath flowed counter currently through the annulus and heated the ice slurry in the inner tube The hydraulic diameter of this annulus measured 62 mm The melting process was continued until all ice crystals had melted and the tank contained only liquid
Chapter 8
176
Figure 83 Schematic overview of the experimental set-up
PT-100 elements with an accuracy of 001 K measured the temperatures of the ice slurry and the ethylene glycol solution at the inlets and outlets of the heat exchanger A pressure difference sensor was used to measure the pressure drop of ice slurry The mass flow of ice slurry was measured using a coriolis mass flow meter and a magnetic flow meter measured the flow rate of ethylene glycol solution The coriolis mass flow meter was also able to measure the temperature of ice slurry downstream of the heat exchanger All flow rates and temperatures were automatically measured every ten seconds with the exception of the temperature measured in the coriolis mass flow meter which was manually read
832 Experimental Conditions
This chapter presents a series of ten melting experiments In this experimental series the operating conditions were systematically varied as shown in Table 81 in order to study their effect on superheating heat transfer coefficients and pressure drop
Table 81 Experimental conditions of melting experiments Exp w0 Tfr uis TEGin τstor DFeret ininitφ no (wt) (degC) (ms) (degC) (h) (microm) (wt) 1 66 -41 10 30 0 2491 17 2 66 -41 15 30 0 249 18 3 66 -41 20 30 0 2491 18 4 66 -41 25 30 0 2491 16 5 66 -41 15 30 16 283 16 6 35 -21 15 52 0 338 14 7 110 -74 15 -07 0 133 17 8 71 -44 18 26 0 148 10 9 70 -44 17 26 15 277 9
10 71 -44 18 00 0 1482 10 1Assumed equal as in experiment 2 2Assumed equal as in experiment 8
Melting of Ice Slurry in Heat Exchangers
177
The varied operating conditions were the ice slurry velocity the heat flux the ice crystal size and the sodium chloride concentration The ice slurry velocity was varied by controlling the gear pump to the desired mass flow rate The heat flux was adjusted by varying the inlet temperature of the aqueous ethylene glycol solution In most experiments the difference between the initial freezing temperature of the aqueous solution and the inlet temperature of the ethylene glycol solution was 71plusmn01 K except for experiment 10 in which this temperature difference was only 44 K For the latter experiment the heat flux varied from 4 to 7 kWm2 while the heat flux in the other experiments was 7 to 13 kWm2 The average crystal size was determined by analyzing the crystals with the visualization section In this respect the Feret diameter was used as characteristic crystal size which is defined as the diameter of a circle with the same area as the projection of the crystal (see Section 732) Ice crystals produced from aqueous solutions with equal solute concentration and equal production procedure appeared to have approximately the same average crystal size The average crystal sizes at the start of experiments 1 3 and 4 were therefore assumed equal to the average crystal size determined at the start of experiment 2 The same assumption was made for the crystal sizes of experiment 8 and 10 Ice crystals produced from aqueous solutions with higher solute concentrations appeared to have smaller crystals In order to vary the average crystal size for a certain solute concentration ice slurry was isothermally stored in the cold room During isothermal storage the average crystal size increased as a result of Ostwald ripening (see Chapter 7)
833 Data Reduction
The total heat flux in the heat exchanger was determined from the flow rate and the inlet and outlet temperatures of the ethylene glycol solution This total heat flux was used to calculate the overall heat transfer coefficient Uo
he o o lnQ U A T= ∆amp (81)
The logarithmic temperature difference in Equation 81 was calculated from the measured temperatures at the inlets and outlets of the heat exchanger The use of the logarithmic temperature difference to determine the overall heat transfer coefficient is only valid when the specific heat of both fluids in the heat exchanger is constant In case the ice slurry is superheated in the heat exchanger this condition is not completely fulfilled However the errors introduced by this method are expected to be small and a more sophisticated method is not available Therefore the overall heat transfer coefficient is based here on the logarithmic temperature difference as is shown in Equation 81
The overall heat transfer resistance (1Uo) consists of three parts the annular side heat transfer resistance the heat transfer resistance of the tube wall and the wall-to-liquid heat transfer resistance
oinner oinner oinner
o o w iinner i iinner
1 1 1ln2
D D DU D Dα λ α
= + +
(82)
The Wilson plot calibration technique was used to formulate single-phase heat transfer correlations for both sides of the heat exchanger (see Appendix C3) The correlation for the annular side was used to calculate the heat transfer coefficient of the ethylene glycol flow αo Subsequently this heat transfer coefficient was used to determine the heat transfer coefficient for the ice slurry flow αi from Equation 82 Finally this experimentally determined heat
Chapter 8
178
transfer coefficient for ice slurry flow was compared with the heat transfer coefficient predicted from the correlation for single-phase flow The measurement accuracies of the physical parameters obtained with the melting heat exchanger are given in Appendix D2
84 Results and Discussion on Superheating
In general the ten melting experiments listed in Table 81 showed mutually the same trends on superheating The observed phenomena are therefore initially discussed for one experiment only namely experiment 1 Subsequently results of the different experiments are compared
841 Analysis of a Single Experiment
The ice slurry temperatures measured at the inlet and outlet of the heat exchanger during melting experiment 1 are shown in Figure 84 At the start of the experiment the ice fraction at the inlet was 17 wt at a temperature of ndash50degC According to the heat balance the reduction in ice fraction was initially approximately 9 wt per pass which resulted in an expected outlet ice fraction of about 8 wt Since ice crystals were present at the outlet the equilibrium temperature at this location was at least lower than the initial freezing temperature of -41degC However the measured outlet temperature exceeded this initial freezing temperature with about 1 K which means that the ice slurry at the outlet was superheated The temperature measured in the coriolis mass flow meter was below the temperature measured at the outlet of the heat exchanger This decrease in temperature is attributed to the release of superheating downstream of the heat exchanger
-60
-50
-40
-30
-20
-10
00
10
20
0 600 1200 1800 2400 3000 3600Time (s)
Tem
pera
ture
(degC
)
00
50
100
150
200
250
300
350
400
Ice
frac
tion
(wt
)
Tfr
T inmeas
T outmeas
T coriomeas
ineqinφ
Figure 84 Measured ice slurry temperatures and ice fraction at the inlet based on
equilibrium during melting experiment 1
If it assumed that ice slurry entering the heat exchanger is in equilibrium than the ice fraction at the inlet is calculated by
ineqin 0ineqin
ineqin
w ww
φminus
= (83)
Melting of Ice Slurry in Heat Exchangers
179
In Equation 83 the mass fraction of solute in the solution wineqin is determined from the measured inlet temperature assuming equilibrium (see Equation A2 in Appendix A11)
ineqin eq inmeasw w T= (84)
The development of this inlet ice fraction φineqin in Figure 84 suggests that all ice crystals had melted at t=2400 s However by that time ice crystals were still observed in the ice suspension tank Another indication that ice crystals were still present in the system is that the slope of the measured inlet temperature does not change significantly at t=2400 s A considerable change of this slope is however observed at t=3200 s indicating that all ice crystals had melted by that time
The described observations indicate that ice slurry is also not in equilibrium at the inlet of the heat exchanger at least during the final stage of the experiment In order to quantify superheating of ice slurry at the inlet and outlet of the heat exchanger the enthalpy of ice slurry at both locations is considered
Enthalpy at Inlet
First the enthalpy of ice slurry at the inlet is considered for the assumption of equilibrium at this location
( ) isineqin ineqin liq ineqin inmeas ineqin ice inmeas1 h h w T h Tφ φ= minus + for inmeas frT Tle (85)
isineqin liq 0 inmeash h w T= for inmeas frT Tgt (86)
For temperatures above the freezing temperature the enthalpy simply equals the enthalpy of the aqueous solution (see Equation A18 in Appendix A14) At temperatures below the freezing temperature the enthalpy of ice slurry is the weighed average of the enthalpy of the solution and the enthalpy of ice (see Equation B8 in Appendix B23)
The enthalpy based on equilibrium calculations can be compared with the enthalpy based on the cumulative heat input which consists of the heat transferred in the heat exchanger and the heat input by other components such as the pump and the mixing device in the tank
( )he rest
isinreal isineqinis0
0t Q Q
h t h t dtm
+= = + int
amp amp (87)
The enthalpy at t=0 is determined by assuming that the ice slurry is in equilibrium at the beginning of the experiment The integral in Equation 87 is rewritten into a summation in order to apply it to the measured data
( )he rest
isinreal isineqin0 is
0tn t
n
Q Q th t h t
m
= ∆
=
+ ∆= = + sum
amp amp (88)
Initially the heat input by other components is set equal to zero Now both the enthalpy based on equilibrium at the inlet and the enthalpy based on the cumulative heat input are compared in Figure 85
Chapter 8
180
-800
-700
-600
-500
-400
-300
-200
-100
00
0 600 1200 1800 2400 3000 3600Time (s)
Ent
halp
y (k
Jkg
)
h isineqin
h isinreal
h isinreal rest( 170 W)Q =amp
rest
( 0 W)Q =amp
Figure 85 Ice slurry enthalpies at the inlet during melting experiment 1
At the end of the experiment (t=3400 s) the tank contained only liquid and the enthalpy based on equilibrium is supposed to represent the correct enthalpy for this time The difference between this enthalpy and the enthalpy based on the cumulative input is attributed to the heat input by the other components In order to estimate this heat input both enthalpies are equated for the final measurement of the experiment
isineqin isinreal endforh h t t= = (89)
If the heat input from the other components is assumed constant then substitution of Equation 88 in 89 gives a correlation for this heat input
end
is herest isineqin end isineqin
0end is
0tn t
n
m Q tQ h t t h tt m
= ∆
=
∆ = = minus = +
sum
ampamp (810)
Application of Equation 810 for melting experiment 1 results in a heat input by the other components of 170 W (see also Figure 85) Heat input values calculated for the other melting experiments showed similar numbers
Enthalpy at Outlet
Now the real enthalpy at the inlet of the heat exchanger is known from Equation 88 the enthalpy at the outlet can be calculated by
is
isoutreal isinrealis
Qh hm
= +amp
amp (811)
Melting of Ice Slurry in Heat Exchangers
181
Ice Fractions and Equilibrium Temperatures at Inlet and Outlet
The foregoing analysis clearly indicates that ice slurry is neither in equilibrium at the inlet nor at the outlet of the heat exchanger The ice fraction can therefore not be calculated by using the initial solute concentration and the measured temperature only as is shown in Equations 83 and 84 The non-equilibrium state requires a third thermodynamic property to calculate the ice fraction for example the enthalpy
The enthalpy of ice slurry which is not in equilibrium is given by
( ) isreal real liq real meas real ice eq real1 h h w T h T wφ φ= minus + (812)
At the inlet and outlet of the heat exchanger the enthalpies are known from Equations 88 and 811 and the temperature is known from measurements Equation 812 contains therefore only two unknown variables namely the solute concentration in the solution wreal and the ice fraction φreal Since ice slurry is homogenously mixed in the tank it is assumed that the solute concentration in the slurry always equals the initial solute concentration w0 The ice fraction φreal is therefore directly related to the solute concentration in the solution wreal by means of the solute mass balance
( )0 real real1w wφ= minus (813)
The ice fraction φreal and the solute concentration wreal can now be solved iteratively from Equations 812 and 813 The ice fractions at the inlet and outlet calculated with this method are shown for experiment 1 in Figure 86
-60
-50
-40
-30
-20
-10
00
10
20
0 600 1200 1800 2400 3000 3600Time (s)
Tem
pera
ture
(degC
)
0
5
10
15
20
25
30
35
40Ic
e fr
actio
n (w
t)
T inmeas
T outmeas
inreal
outreal
φφ
Figure 86 Measured ice slurry temperatures and calculated ice fractions during melting
experiment 1
The solute concentration in the liquid wreal is now used to calculate the equilibrium temperature at the inlet and outlet of the heat exchanger
eq eq realT T w= (814)
Chapter 8
182
Figure 87 shows that the measured outlet temperatures exceed the calculated equilibrium temperatures at the outlet indicating that the ice slurry is significantly superheated
-60
-50
-40
-30
-20
-10
00
10
20
0 600 1200 1800 2400 3000 3600Time (s)
Tem
pera
ture
(degC
)
0
5
10
15
20
25
30
35
40
Ice
frac
tion
(wt
)
T outmeas
T outeq
outrealφ
∆Tsh
Figure 87 Measured ice slurry temperature and calculated ice fractions and equilibrium
temperatures at the outlet of the heat exchanger during melting experiment 1
Superheating Definition
In order to quantify superheating at the outlet of the heat exchanger the degree of superheating ∆Tsh is defined as the difference between the measured temperature and the equilibrium temperature
sh liqmeas eq realT T T w∆ = minus (815)
Melting of ice slurry in a heat exchanger can be considered as a process consisting of two stages as shown in Figure 88 (see also Section 825) The first stage consists of the heat transfer process from the wall to the liquid The driving force of this process is the temperature difference between the wall and the liquid phase The second stage is the actual melting of the ice crystals where the difference between the liquid temperature and the equilibrium temperature hence the degree of superheating is the driving force
Figure 88 Schematic representation of temperatures during melting of ice slurry in a heat
exchanger
Melting of Ice Slurry in Heat Exchangers
183
The degree of superheating can be seen as a fraction of the total driving force of the melting process
meas eqsh
w-liq sh w eq
T TTT T T T
ζminus∆
= =∆ + ∆ minus
(816)
This relative superheating ζ enables to compare superheating results from experiments with different mass flow rates and different heat fluxes
For the analysis of superheating it is necessary to calculate the wall temperature at the outlet of the ice slurry flow Here the ratio of heat transfer coefficients the ice slurry temperature and the temperature of the ethylene glycol solution are used to determine this temperature
( )( )
oinnerw is o
EG is i iinner
DT T UT T Dα
minus=
minus (817)
842 Influence of Ice Fraction and Ice Slurry Velocity
The superheating results for different ice slurry velocities in Figure 89 clearly show that the degree of superheating increases as the ice fraction decreases The figure also shows that for ice fractions higher than 5 wt the degree of superheating is higher in the experiments with low ice slurry velocities This higher degree of superheating is mainly the result of the higher wall temperature caused by the lower wall-to-liquid heat transfer coefficient at low slurry velocities The results for the relative superheating ζ in Figure 810 take these different wall temperatures into account This figure shows that the relative superheating of the experiments with slurry velocities of 10 15 and 20 ms are very similar but that the relative superheating at a velocity of 25 ms is slightly lower
00
10
20
30
40
50
00 50 100 150Outlet ice fraction (wt)
Deg
ree
of s
uper
heat
ing
(K)
10 ms15 ms20 ms25 ms
00
02
04
06
08
10
00 50 100 150Outlet ice fraction (wt)
Rel
ativ
e su
perh
eatin
g ζ
10 ms15 ms20 ms25 ms
Figure 89 Degree of superheating at the outlet for various ice slurry velocities
(Experiments 1 2 3 and 4)
Figure 810 Relative superheating ζ at the outlet for various ice slurry velocities
(Experiments 1 2 3 and 4)
Chapter 8
184
843 Influence of Heat Flux
The results from the experiments with different ethylene glycol solution inlet temperatures in Figure 811 show that the degree of superheating increases as the heat flux increases However the relative superheating ζ is similar for different heat fluxes as is shown in Figure 812
00
10
20
30
40
50
00 20 40 60 80 100Outlet ice fraction (wt)
Deg
ree
of s
uper
heat
ing
(K)
26degC 148 microm00degC 148 microm26degC 277 microm
00
02
04
06
08
00 20 40 60 80 100Outlet ice fraction (wt)
Rel
ativ
e su
perh
eatin
g ζ
26degC 148 microm00degC 148 microm26degC 277 microm
Figure 811 Degree of superheating at the outlet for various heat fluxes (TEG) and crystal sizes (Experiments 8 9 and 10)
Figure 812 Relative superheating ζ at the outlet for various heat fluxes (TEG) and crystal sizes (Experiments 8 9 and 10)
844 Influence of Crystal Size
The results of experiments 8 and 10 in Figure 811 indicate that ice slurries consisting of larger crystals exhibit higher degrees of superheating Accordingly the relative superheating also increases as the average ice crystal size increases (see Figure 812) A comparison of the superheating results of experiments 2 and 5 in which the crystal size was also the only varied variable gives the same conclusion
845 Influence of Solute Concentration
The superheating results of the experiments with different solute concentrations are shown in Figures 813 and 814 The two figures indicate that both the degree of superheating and the relative superheating are higher in liquids with low solute concentration However it is difficult to compare the presented results because not only the solute concentration was different in these experiments but also the average crystal size As is shown above the average crystal size influences superheating significantly A more comprehensive analysis is therefore presented in the next subsection to unravel the influence of the solute concentration on superheating
Melting of Ice Slurry in Heat Exchangers
185
00
10
20
30
40
50
00 50 100 150Outlet ice fraction (wt)
Deg
ree
of s
uper
heat
ing
(K)
35 wt 338 microm66 wt 249 microm110 wt 133 microm
00
02
04
06
08
10
00 50 100 150Outlet ice fraction (wt)
Rel
ativ
e su
perh
eatin
g ζ
35 wt 338 microm66 wt 249 microm110 wt 133 microm
Figure 813 Degree of superheating at the outlet for various solute concentrations
(Experiments 2 6 and 7)
Figure 814 Relative superheating ζ at the outlet for various solute concentrations
(Experiments 2 6 and 7)
846 Discussion
The presented results for superheating at the outlet of the heat exchanger can be explained by a model of the melting process This model is based on the heat and mass transfer processes in a control volume of the heat exchanger as shown in Figure 815 It is assumed that the control volume is ideally mixed which means that ice slurry is homogeneously distributed and that the liquid temperature is constant in the entire control volume
Figure 815 Schematic representation of melting process in a control volume
The control volume is considered as a steady state system and the heat balance is therefore
( )is isout isinQ m h h= minusamp amp (818)
The heat transferred from the wall to liquid in the control volume is given by
( )i i w liqQ A T Tα= minusamp with i iinnerA D xπ= ∆ (819)
Chapter 8
186
The increase of the enthalpy of ice slurry in Equation 818 is represented by
( )( ) ( )( )isout isin out liqout out iceout in liqin in icein1 1h h h h h hφ φ φ φminus = minus + minus minus + (820)
The ice fraction at the outlet in Equation 820 can be replaced by
out inφ φ φ= minus ∆ (821)
Combining Equations 820 and 821 gives the following expression for the change in enthalpy
( ) ( )( ) ( )isout isin liqout iceout in liqout liqin in iceout icein1h h h h h h h hφ φ φminus = ∆ minus + minus minus + minus (822)
The change in liquid enthalpy is approximated by the product of the temperature increase and the specific heat of the liquid It is assumed here that the heat of mixing can be neglected and that specific heats are constant for small temperature changes With these assumptions Equation 822 becomes
( )( )isout isin f in pliq in pice1h h h T c cφ φ φminus asymp ∆ ∆ + ∆ minus + (823)
Equation 823 shows that the increase of the enthalpy consists of a latent heat contribution represented by a decrease of the ice fraction and a sensible heat contribution represented by an increase of the temperature During the initial stage of the melting experiments the sensible heat contribution was 20 of the total enthalpy increase on average For simplicity the sensible heat contribution is neglected in this analysis and the enthalpy difference is assumed equal to the product of the change in ice fraction and the latent heat of fusion
isout isin fh h hφminus asymp ∆ ∆ (824)
Combining Equations 818 819 and 824 leads to the following heat balance for the control volume
( )i iinner w liq is fD x T T m hα π φ∆ minus = ∆ ∆amp (825)
The decrease of the ice fraction is caused by the melting of individual ice crystals The mass reduction of ice in the control volume is proportional to the total surface of ice crystals Aice and the negative growth rate G
ice ice icem A Gρ∆ = minusamp (826)
The decrease of the ice fraction is now calculated as the ratio between the reduction of the ice mass and the mass flow rate of ice slurry
ice ice ice
is is
m A Gm m
ρφ ∆∆ = = minus
amp
amp amp (827)
Melting of Ice Slurry in Heat Exchangers
187
The total available crystal surface Aice for the melting process is proportional to the number of crystals in the control volume and the characteristic crystal size squared
2ice 1 FeretA c N D= (828)
It is assumed here that both the shape of the individual crystals as well as the shape of the crystal size distribution were the same in the various experiments The number of crystals N in Equation 828 is deduced from the total mass of ice in the control volume with the same assumptions
ice3
2 ice Feret
mNc Dρ
= with 2ice is is iinner4
m m D xπφ φρ= = ∆ (829)
The negative crystal growth rate G in Equation 827 is determined by mass and heat transfer between the crystal surface and the liquid phase of the slurry
eq liqA
eqice V f
liq cr
3T TBG
dTB hwk dw
ρρ α
minus=
∆minus +
(830)
Rearranging of Equation 830 shows explicitly the ratio between the heat and mass transfer coefficient
eq liqA
ice V eqcrf
cr f liq
31
T TBGB dTh w
k h dwρ α
α ρ
minus=
∆minus + ∆
(831)
This ratio of the coefficients is determined from the analogy between heat and mass transfer close to the crystal surface
1 1 1 23 3 3 3
1 23 3
cr liq liq pliq liq liqcr
cr D D DNu Pr c
k Sh Scλ λ ρ λα
= = = (832)
The expression of Equation 832 is substituted in Equation 831 resulting in a new expression for the crystal growth rate
1 2
3 3
2 23 3
eq liqcrA
ice V f pliq liq eq
f liq
31
D
T TBGB h c w dT
dwh
αρ λ
ρ
minus=
∆ minus + ∆
(833)
Equation 833 shows that both heat and mass transfer resistances determine the total resistance for melting However the ratio of these contributions strongly depends on the solute concentration as is shown in Figure 816 At low solute concentration of 35 wt for example the crystal growth rate is determined by equal contributions of heat and mass transfer resistance while at high concentrations the growth rate is almost completely determined by mass transfer
Chapter 8
188
00
10
20
30
40
50
60
70
00 20 40 60 80 100 120 140NaCl concentration in the liquid (wt)
Con
trib
utio
n to
cry
stal
gro
wth
re
sist
ance
rel
ativ
e to
hea
t tra
nsfe
r
Total
Mass transfer
Heat transfer
Figure 816 Contributions to crystal growth resistance relative to heat transfer resistance
Equations 828 829 and 833 are now substituted in Equation 827
1 2
3 3
2 23 3
2liq eqiinneris cr1 A
2 V ice Feret is f pliq liq eq
f liq
121
D
T TD xc Bc B D m h c w dT
dwh
φρ απφρ λ
ρ
minus∆∆ =
∆ minus + ∆
amp
(834)
Substitution of Equation 834 in the heat balance of Equation 825 finally gives an expression for the degree of superheating
( )
1 23 3
2 23 3
pliq liq eqV ice Feret2 ish liq eq w liq
1 A is iinner cr f liq
12 1 1Dc w dTB DcT T T T T
c B D dwhλρ α
ρ φ α ρ
∆ = minus = minus + minus ∆
(835)
Equation 835 shows that the degree of superheating is higher for slurries with large crystals which is in accordance with the experiments Ice slurries with large crystals have a relatively small crystal surface resulting in a slow melting process and exhibit therefore high degrees of superheating Ice slurries with low ice fractions have also relatively little crystal surface and exhibit therefore also high degrees of superheating This phenomenon is represented in Equation 835 by the ice fraction in the denominator
In correspondence with the experiments Equation 835 shows that the degree of superheating increases with increasing heat flux which is represented here by the temperature difference between wall and liquid However the ratio between the driving forces of the two stages of melting is not influenced by the heat flux Therefore the relative superheating does not depend on the heat flux which is in accordance with the experiments (see Figure 812)
The experiments showed that the relative superheating is hardly influenced by the ice slurry velocity This observation can also be explained by Equation 835 A higher ice slurry velocity results first of all in a higher heat transfer coefficient between wall and liquid However the heat and mass transfer coefficients between crystals and liquid also increase It is expected that the relative increases of all these coefficients are approximately similar as the
Melting of Ice Slurry in Heat Exchangers
189
velocity increases and that therefore the relative superheating is almost independent of the ice slurry velocity
According to Equation 835 the degree of superheating is higher in aqueous solutions with higher solute concentrations This trend can not directly be confirmed by the experiments because the experiments with different solute concentrations also had different average crystals sizes In order to confirm the influence of the solute concentration all variables that have been varied in the experiments have been considered simultaneously For this purpose all experimental constants of Equation 835 are combined in one constant c3
( )
1 23 3
2 23 3
pliq liq eqice Feretsh 3 w liq
is f liq
1Dc w dTDT c T T
dwhλρ
ρ φ ρ
∆ = minus + minus ∆
with V2 i3
1 A iinner cr
12Bccc B D
αα
= (836)
The ratio of the heat transfer coefficients in the expression for c3 is assumed constant here The experiments with different ice slurry velocities showed similar relative superheating values indicating that this assumption is reasonable
The experimental variables at the right-hand side of Equation 836 are considered at the start of each experiment This analysis is limited to the initial phase of the experiments since the average ice crystal size was only determined prior to each experiment It is expected that the average crystal size decreases in the course of an experiment but this was not quantified
The results of this analysis for all ten melting experiments shown in Figure 817 confirm proportionality between the variables and the degree of superheating stated in Equation 836
00
05
10
15
20
000 001 002 003 004 005
∆T
shm
eas (
K) 1
2 34
56
7
8
9
10
-25
+25
(m K)( )1 2
3 3
2 23 3
pliq liq eqice Feretw liq
is f liq
1Dc w dTD T T
dwhλρ
ρ φ ρ
minus + minus ∆
Figure 817 Relation between variables at right-hand side of Equation 836 and measured
degrees of superheating the numbers in the figure represent the experiment number as listed in Table 81
Chapter 8
190
The expression in Equation 835 shows that the degree of superheating also depends on the tube diameter According to the expression the degree of superheating decreases with increasing tube diameter Since the diameter of the heat exchanger tube was not varied in the experiments this influence can not be confirmed
847 Conclusions
The degree of superheating at the outlet of melting heat exchangers is proportional to the average ice crystal size and the heat flux It is furthermore inversely proportional to the ice fraction and therefore superheating increases as the ice fraction decreases The negative growth rate of melting ice crystals is determined by both mass and heat transfer The degree of superheating increases therefore with increasing solute concentration especially at higher concentrations at which the heat transfer resistance plays a minor role Finally the degree of superheating is expected to be higher in heat exchangers with small hydraulic diameters but does hardly depend on the ice slurry velocity
85 Results and Discussion on Heat Transfer Coefficients
851 Influence of Ice Fraction and Ice Slurry Velocity
The experimental results of all ten melting experiments show that the wall-to-liquid heat transfer coefficient increases with increasing ice fraction as is shown for four experiments in Figure 818 The figure also shows that the relative increase of the heat transfer coefficient is especially high at low ice slurry velocities For an ice slurry velocity of 10 ms for example the heat transfer coefficient at an ice fraction of 13 wt is approximately 50 higher than for the case that all crystals have melted The relative increase of the heat transfer coefficient at a velocity of 25 ms for the same ice fractions is rather small The relatively high increase at low velocities and the limited increase at higher velocities is in accordance with the results in literature discussed in Section 824
0
1000
2000
3000
4000
5000
6000
00 50 100 150 200Average ice fraction (wt)
Hea
t tra
nsfe
r co
effic
ient
(Wm
2 K)
10 ms15 ms20 ms25 ms
Figure 818 Wall-to-slurry heat transfer coefficients versus average ice fraction for different
ice slurry velocities (Experiments 1 2 3 and 4)
Melting of Ice Slurry in Heat Exchangers
191
It is interesting to compare the measured heat transfer coefficients of Figure 818 with the values predicted by a heat transfer correlation for single-phase flow Such a heat transfer correlation has been formulated on the basis of calibration experiments with aqueous solutions and the Wilson plot calibration technique The entire procedure is described in Appendix C3 The heat transfer correlation for the inner tube is based on the Reynolds Nusselt and Prandtl number
3 0 903 0 33liq liq liq7 36 10 Nu Re Prminus= sdot for liq 6700Re le (837)
2 0 687 0 33liq liq liq5 06 10 Nu Re Prminus= sdot for liq 6700Re gt (838)
When this correlation is applied for ice slurry flow the dimensionless numbers can either be based on the thermophysical properties of the two-phase mixture or on the thermophysical properties of the liquid phase only The dynamic viscosity of a slurry is for example always higher than the viscosity of the liquid phase and the thermal conductivities of slurry and liquid can also strongly deviate (see also Appendix B2)
In order to compare measured heat transfer coefficients with heat transfer coefficients predicted on the basis of thermophysical liquid properties the heat transfer factor based on liquid properties is defined as
measliq
predliq
αα
Ψ = with predliqα based on liq liqNu Re and liqPr (839)
In analogy the heat transfer factor based on slurry properties is defined as
measis
predis
αα
Ψ = with predisα based on is isNu Re and isPr (840)
Figure 819 shows heat transfer factors based on liquid properties for the four experiments with different velocities while Figure 820 shows heat transfer factors based on slurry properties For high velocities the heat transfer factor based on liquid properties is close to unity for all ice fractions This means that the heat transfer coefficient of ice slurry for these velocities can be predicted within 10 by the heat transfer correlation in Equations 837 and 838 in combination with the thermophysical liquid properties For low velocities however real heat transfer coefficients are up to 50 higher than calculated by the heat transfer correlation using liquid properties The same is valid for heat transfer coefficients calculated on the basis of ice slurry properties For high ice fractions the heat transfer factors show values much higher than unity (see Figure 820) Real heat transfer coefficients are in fact up to 75 higher than expected from the heat transfer correlation based on slurry properties
A possible explanation for the relative steep increase in heat transfer as a function of the ice fraction at low velocities is that the ice crystals are not homogeneously distributed over the cross section of the tube Due to low turbulence levels at low velocities ice crystals float to the top of the tube It is plausible that these ice crystals touch the tube wall and disturb the thermal boundary layer which enhances the heat transfer coefficient At higher velocities the ice slurry flow shows a more homogeneous flow pattern For these conditions ice crystals are hardly present in the relatively hot vicinity of the tube wall It is therefore expected that the thermal boundary layer mainly consists of liquid and that the heat transfer coefficient can be predicted by the heat transfer correlation based on liquid properties
Chapter 8
192
08
10
12
14
16
18
20
00 50 100 150Average ice fraction (wt)
10 ms15 ms20 ms25 ms
Hea
t tra
nsfe
r fa
ctor
Ψliq
08
10
12
14
16
18
20
00 50 100 150Average ice fraction (wt)
10 ms15 ms20 ms25 ms
Hea
t tra
nsfe
r fa
ctor
Ψis
Figure 819 Heat transfer factors based on liquid properties for various ice slurry velocities (Experiments 1 2 3 and 4)
Figure 820 Heat transfer factors based on slurry properties for various ice slurry velocities (Experiments 1 2 3 and 4)
In the rest of this section measured heat transfer coefficients are only compared to values calculated on the basis of slurry properties
852 Influence of Heat Flux and Ice Crystal Size
The experimental results in Figure 821 indicate that neither the heat flux nor the average ice crystal size influence the heat transfer coefficient This negligible influence of the heat flux is in accordance with several experimental studies reported in literature (see Section 824) The effect of the ice crystal size on heat transfer coefficients has not been studied before but this effect seems to be small according to the presented results It is however possible that even larger crystals do influence the heat transfer coefficient Larger crystals have namely a stronger tendency to float to the top of the tube and may therefore enhance heat transfer coefficients
853 Influence of Solute Concentration
Figure 822 shows that the enhancement of the heat transfer coefficient with increasing ice fraction is stronger at higher solute concentrations This phenomenon may be explained by the higher density difference between the liquid phase and the ice crystals At an ice fraction of 10 wt the density difference between liquid and solid phase is 113 kgm3 for the slurry with an initial solute concentration of 35 wt while this value is 179 kgm3 for the slurry with an initial solute concentration of 110 wt This higher density difference increases the buoyancy force on the ice crystals and therefore more ice crystals are located in the upper part of the tube It is expected that these crystals are in touch with the tube wall increasing heat transfer coefficients analogously to the heat transfer enhancement at low velocities
Melting of Ice Slurry in Heat Exchangers
193
08
10
12
14
16
18
20
00 20 40 60 80 100Average ice fraction (wt)
Hea
t tra
nsfe
r fa
ctor
Ψis
26degC 148 microm00degC 148 microm26degC 277 microm
08
10
12
14
16
18
20
00 50 100 150Average ice fraction (wt)
Hea
t tra
nsfe
r fa
ctor
Ψis
35 wt 338 microm66 wt 249 microm110 wt 133 microm
Figure 821 Heat transfer factor based on slurry properties for various heat fluxes (TEG)
and crystal sizes (Experiments 8 9 and 10)
Figure 822 Heat transfer factors based on slurry properties for various solute
concentrations (Experiments 2 6 and 7)
854 Conclusions
Wall-to-liquid heat transfer coefficients of ice slurry during melting increase with increasing ice fraction This enhancement is especially high at low slurry velocities and for high density differences between liquid and ice For the studied operation conditions the heat flux and the average ice crystal size have no influence on the heat transfer coefficient
86 Results and Discussion on Pressure Drop
861 Influence of Ice Fraction and Ice Slurry Velocity
Figure 823 shows the pressure drop results as a function of the average ice fraction for the four experiments with different ice slurry velocities The figure shows that the pressure drop increases as the ice fraction increases which was observed for all ten melting experiments The measured pressure drop values can be compared with values predicted by the pressure drop model formulated in Appendix C34 According to this model the pressure drop of a single-phase flow in the inner tube of the heat exchanger can be predicted by
0 404liq liq2
pred 0 118liq liqiinner
1 42 for 67001 with 0 112 for 67002
f Re ReLp f uf Re ReD
ρminus
minus
= lt∆ = = ge (841)
Analogously to the prediction of heat transfer coefficients the pressure drop of solid-liquid flows can be predicted either on the basis of the thermophysical properties of the liquid phase or on the basis of the properties of the slurry The main difference in thermophysical properties with respect to pressure drop is the dynamic viscosity The viscosity of an ice slurry with an ice fraction of 10 wt is namely 45 higher than the viscosity of the liquid phase only This higher viscosity leads to a lower Reynolds number and therefore to a higher friction factor f
Chapter 8
194
000
020
040
060
080
100
120
140
00 50 100 150 200Average ice fraction (wt)
Pres
sure
dro
p (b
ar)
10 ms15 ms20 ms25 ms
Figure 823 Pressure drop versus average ice fraction for different ice slurry velocities
(Experiments 1 2 and 3)
The pressure drop factor based on liquid properties compares measured pressure drop values with values predicted on the basis of liquid properties and is defined as
measliq
predliq
pp
∆Π =
∆ with predliqp∆ based on liqRe (842)
In the same way the pressure drop factor based on slurry properties is defined as
measis
predis
pp
∆Π =
∆ with predisp∆ based on isRe (843)
Figures 824 and 825 show the pressure drop factors for the experiments with different ice slurry velocities The pressure drop factor based on liquid properties in Figure 824 increases up to values of 13 as the ice fraction increases from 0 to 15 wt This means that the application of liquid properties in the pressure drop model of Equation 841 leads to underestimation of real pressure drop values for ice slurry flow However the pressure drop factor based on slurry properties in Figure 825 shows values close to unity for all tested ice fractions and velocities The pressure drop of ice slurry with ice fractions up to 15 wt can thus be predicted by the model of Equation 841 using slurry properties
862 Influence of Heat Flux Ice Crystal Size and Solute Concentration
The results of the experiments with different heat fluxes and different average ice crystal sizes show the same relation between ice fraction and pressure drop which means that the pressure drop is not influenced by any of these parameters Figure 826 confirms this observation by showing pressure drop factors close to unity for all different conditions
Melting of Ice Slurry in Heat Exchangers
195
08
10
12
14
16
00 50 100 150Average ice fraction (wt)
10 ms15 ms20 ms25 ms
Pres
sure
dro
p fa
ctor
Πliq
08
10
12
14
16
00 50 100 150Average ice fraction (wt)
10 ms15 ms20 ms25 ms
Pres
sure
dro
p fa
ctor
Πis
Figure 824 Pressure drop factor based on liquid properties for various ice slurry velocities (Experiments 1 2 3 and 4)
Figure 825 Pressure drop factor based on slurry properties for various ice slurry velocities (Experiments 1 2 3 and 4)
The pressure drop results for the experiments with different solute concentrations show approximately the same results In accordance with the other experiments the pressure drop factor for the experiments with initial solute concentrations of 35 and 66 wt is also close to unity However the experiment with the highest solute concentration of 110 wt shows a slightly increasing pressure drop factor as the ice fraction increases (see Figure 827) This behavior may be caused by the relative high density difference between the liquid and the solid phase at high solute concentration As a result the buoyancy force on the crystals is stronger and the flow pattern may change from homogeneous to heterogeneous flow or even moving bed flow This changing flow pattern may be the cause for the 10 difference between the measured and the predicted pressure drop value
08
10
12
14
16
00 20 40 60 80 100Average ice fraction (wt)
Pres
sure
dro
p fa
ctor
Πis
26degC 148 microm00degC 148 microm26degC 277 microm
08
10
12
14
16
00 50 100 150Average ice fraction (wt)
Pres
sure
dro
p fa
ctor
Πis
35 wt 338 microm66 wt 249 microm110 wt 133 microm
Figure 826 Pressure drop factor based on slurry properties for various heat fluxes (TEG)
and crystal sizes (Experiments 8 9 and 10)
Figure 827 Pressure drop factor based on slurry properties for various solute
concentrations (Experiments 2 6 and 7)
Chapter 8
196
863 Conclusions
Pressure drop values of ice slurry flows with ice fractions up to 15 wt can be predicted by using pressure drop correlations for single-phase flow The application of the thermophysical properties of the slurry in these correlations leads to absolute errors of 10 and smaller
87 Conclusions
The liquid temperature of ice slurry in melting heat exchangers can be significantly higher than the equilibrium temperature This phenomenon is referred to as superheating and can lead to a serious reduction of heat exchanger capacities The degree of superheating at the outlet of heat exchangers is proportional to the average ice crystal size and the heat flux It is furthermore inversely proportional to the ice fraction and therefore superheating increases as the ice fraction decreases The negative growth rate of melting ice crystals is determined by both heat and mass transfer The degree of superheating increases therefore with increasing solute concentration especially at higher concentrations at which the heat transfer resistance plays a minor role Finally the degree of superheating is expected to be higher in heat exchangers with small hydraulic diameters
Wall-to-liquid heat transfer coefficients and pressure drop values increase with increasing ice fraction The heat transfer enhancement is especially high at low slurry velocities or high density differences between liquid and ice Pressure drop values for ice fractions up to 15 wt can be predicted within 10 by applying a single-phase flow pressure drop correlation in which the slurry properties are used Both heat flux and average ice crystal size do neither influence the heat transfer coefficient nor the pressure drop
Melting of Ice Slurry in Heat Exchangers
197
Nomenclature
A Area (m2) Greek BA Area shape factor α Heat transfer coefficient (Wm2K) BV Volume shape factor δ Boundary layer thickness (m) c13 Constants ζ Relative superheating defined in cp Specific heat (Jkg K) Eq 816 DFeret Average crystal Feret diameter (m) λ Thermal conductivity (Wm K) D Tube diameter (m) micro Viscosity (Pa s) D Diffusion coefficient (m2s) Πliq Pressure drop factor based on liquid f Friction factor properties defined in Eq 842 G Growth rate (ms) Πis Pressure drop factor based on slurry h Enthalpy (Jkg) properties defined in Eq 843 ∆hf Latent heat of fusion of ice (Jkg) ρ Density (kgm3) k Mass transfer coefficient (ms) τ Period (h) L Tube length (m) φ Ice mass fraction m Mass (kg) Ψliq Heat transfer factor based on liquid ampm Mass flow (kgs) properties defined in Eq 839
N Number of crystals Ψis Heat transfer factor based on slurry n Number of measurements properties defined in Eq 840 Nucr Liquid Nusselt number αcr DFeretλliq Nuliq Liquid Nusselt number αi Diinnerλliq Subscripts Nuis Slurry Nusselt number αi Diinnerλis corio Coriolis mass flow meter Prliq Liquid Prandtl number cpliq microliqλliq cr Crystal Pris Slurry Prandtl number cpsensis microisλis EG Ethylene glycol solution ∆p Pressure drop (Pa) end End of experiment ampQ Heat (W) eq Equilibrium
Reliq Liquid Reynolds number eqin Equilibrium assumed at inlet ρliq u Diinnermicroliq fr Freezing point Reis Slurry Reynolds number ρis u Dmicrois he Heat exchanger Sc Schmidt number microliq(ρliq D) i Inside Shcr Crystal Sherwood number k DFeretD ice Ice T Temperature (degC) in Inlet heat exchanger Tfr Initial freezing temperature (degC) init Initial ∆Tln Logarithmic mean temperature inner Inner difference (K) is Ice slurry ∆Tsh Degree of superheating (K) defined liq Liquid in Eq 815 meas Measured t Time (s) o Outside ∆t Measurement interval (s) out Outlet heat exchanger U Overall heat transfer coefficient pred Predicted (Wm2K) real Real u Velocity (ms) rest Other components Vamp Volume flow (m3s) sens Sensible w Solute mass fraction src Source w0 Initial solute mass fraction in liquid stor Storage ∆x Length of control volume (m) w Wall
Chapter 8
198
References
Ayel V Lottin O Peerhossaini H 2003 Rheology flow behaviour and heat transfer of ice slurries a review of the state of the art International Journal of Refrigeration vol26 pp95-107
Bedecarrats J Strub F Peuvrel C Dumas J 2003 Heat transfer and pressure drop of ice slurry in a heat exchanger In Proceedings of 21st IIR International Congress of Refrigeration 17-23 August 2003 Washington DC (USA) Paris International Institute of Refrigeration
Bellas J Chaer I Tassou SA 2002 Heat transfer and pressure drop of ice slurries in plate heat exchangers Applied Thermal Engineering vol22 pp721-732
Christensen K Kauffeld M 1997 Heat transfer measurements with ice slurry In International ConferencendashHeat Transfer Issues in Natural Refrigerants Paris International Institute of Refrigeration pp127ndash141
Darby R 1986 Hydrodynamics of slurries and suspensions In Cheremisinoff NP (Ed) Encyclopedia of fluid mechanics ndash Volume 5 Slurry Flow Technology Houston Gulf pp49-92
Doetsch C 2001 Pressure drop and flow pattern In Proceedings of the 3rd IIR Workshop on Ice Slurries 16-18 May 2001 Lucerne (Switzerland) Paris International Institute of Refrigeration pp53-68
Doetsch C 2002 Pressure drop calculation of ice slurries using the Casson model In Proceedings of the 5th IIR Workshop on Ice Slurries 30-31 May 2002 Stockholm (Sweden) Paris International Institute of Refrigeration pp15-21
Egolf PW Kitanovski A Ata-Caesar D Stamatiou E Kawaji M Bedecarrats JP Strub F 2005 Thermodynamics and heat transfer of ice slurries International Journal of Refrigeration vol28 pp51-59
Fikiin K Wang M-J Kauffeld M Hansen TM 2005 Direct contact chilling and freezing of foods in ice slurries In Kauffeld M Kawaji M Egolf PW (Eds) Handbook on Ice Slurries Fundamentals and Engineering Paris International Institute of Refrigeration pp251-271
Frei B Boyman T 2003 Plate heat exchanger operating with ice slurry In Proceedings of the 1st International Conference and Business Forum on Phase Change Materials and Phase Change Slurries 23-26 April 2003 Yverdon-les-Bains (Switzerland)
Frei B Egolf PW 2000 Viscometry applied to the Bingham substance ice slurry In Proceedings of the 2nd IIR Workshop on Ice Slurries 25-26 May 2000 Paris (France) Paris International Institute of Refrigeration pp48-60
Guilpart J Fournaison L Ben-Lakhdar MA Flick D Lallemand A 1999 Experimental study and calculation method of transport characteristics of ice slurries In Proceedings of the 1st IIR Workshop on Ice Slurries 27-28 May 1999 Yverdon-les-Bains (Switzerland) Paris International Institute of Refrigeration pp74-82
Melting of Ice Slurry in Heat Exchangers
199
Hansen TM Radošević M Kauffeld M Zwieg T 2003 Investigation of ice crystal growth and geometrical characteristics in ice slurry International Journal of HVACampR Research vol9 pp9-32
Jensen E Christensen K Hansen T Schneider P Kauffeld M 2000 Pressure drop and heat transfer with ice slurry In Proceedings of the 4th IIR Gustav Lorentzen Conference on Natural Working Fluids 25-28 July 2000 Purdue (USA) Paris International Institute of Refrigeration pp521ndash529
Kitanovski A Poredoš A Reghem P Stutz B Dumas JP Vuarnoz D Sari O Egolf PW Hansen TM 2002 Flow patterns of ice slurry flows In Proceedings of the 5th IIR Workshop on Ice Slurries 30-31 May 2002 Stockholm (Sweden) Paris International Institute of Refrigeration pp36-46
Kitanovski A Sarlah A Poredoš A Egolf PW Sari O Vuarnoz D Sletta JP 2003 Thermodynamics and fluid dynamics of phase change slurries in rectangular channels In Proceedings of the 21st IIR International Congress of Refrigeration 17-223 August 2003 Washington DC (USA) Paris International Institute of Refrigeration
Kitanovski A Vuarnoz D Ata-Caesar D Egolf PW Hansen TM Doetsch C 2005 The fluid dynamics of ice slurry International Journal of Refrigeration vol28 pp37-50
Knodel BD France DM Choi U Wambsganss M 2000 Heat transfer and pressure drop in ice-water slurries Applied Thermal Engineering vol20 pp671ndash685
Lee DW Yoon CI Yoon ES Joo MC 2002 Experimental study on flow and pressure drop of ice slurry for various pipes In Proceedings of the 5th IIR Workshop on Ice Slurries 30-31 May 2002 Stockholm (Sweden) Paris International Institute of Refrigeration pp22-29
Lee DW Yoon ES Joo MC Sharma A 2006 Heat transfer characteristics of the ice slurry at melting process in a tube flow International Journal of Refrigeration vol29 pp451-455
Meewisse JW 2004 Fluidized Bed Ice Slurry Generator for Enhanced Secondary Cooling Systems PhD thesis Delft University of Technology (The Netherlands)
Niezgoda-Żelasko B 2006 Heat transfer of ice slurry flows in tubes International Journal of Refrigeration vol29 pp437-450
Niezgoda-Żelasko B Zalewski W 2006 Momentum transfer of ice slurry flows in tubes experimental investigations International Journal of Refrigeration vol29 pp418-428
Noslashrgaard E Soslashrensen TA Hansen TM Kauffeld M 2005 Performance of components of ice slurry systems pumps plate heat exchangers and fittings International Journal of Refrigeration vol28 pp83-91
Shook CA Roco MC 1991 Slurry Flow Principles and Practice Boston Butterworth-Heinemann
Chapter 8
200
Stamatiou E Kawaji M 2005 Thermal and flow behavior of ice slurries in a vertical rectangular channel - Part II Forced convective melting heat transfer International Journal of Heat and Mass Transfer vol48 pp3544-3559
Torres-de Mariacutea G Abril J Casp A 2005 Coefficients deacutechanges superficiels pour la reacutefrigeacuteration et la congeacutelation daliments immergeacutes dans un coulis de glace International Journal of Refrigeration vol28 pp1040-1047
Wasp EJ Kenny JP Gandhi RL 1977 Solid-liquid Flow Slurry Pipeline Transportation Clausthal Trans Tech
201
9 Conclusions
Ice Scaling Prevention in Fluidized Bed Heat Exchangers
Ice scaling during ice crystallization from aqueous solutions in liquid-solid fluidized bed heat exchangers can only be prevented when a certain difference between the wall temperature and the equilibrium temperature of the solution is not exceeded This so-called transition temperature difference depends on operating parameters such as fluidized bed parameters and liquid properties The explanation for this phenomenon is that ice scaling is only successfully prevented when the removal rate induced by fluidized particles exceeds the growth rate of ice crystals attached to the cooled wall
The scale removal rate in stationary fluidized beds is proportional to the impulse exerted by particles-wall collisions The transition temperature difference increases therefore as the bed voidage decreases or the particle size increases Of all tested fluidized bed conditions the highest transition temperature difference was achieved for a fluidized bed with a bed voidage 81 consisting of 4 mm stainless steel particles In circulating fluidized beds the scale removal rate is determined by both particle-wall collisions and liquid pressures fronts induced by particle-particle collisions The scale removal rate by liquid pressure fronts is also proportional to the impulse they exert on the wall but with a lower proportionality constant At higher circulation rates both the frequency and the average maximum pressure of liquid pressure fronts increases resulting in a higher total exerted impulse on the wall and enhanced removal rates Due to this enhancement the transition temperature difference for ice scaling increases as the circulation rate increases A disadvantage of circulating fluidized beds for ice slurry production is the high risk of blockages in the downcomer tube
The growth rate of ice crystals attached to a cooled wall in an aqueous solution is determined by mass transfer The ice crystals that grow on the wall absorb only water molecules and therefore dissolved solute molecules or ions accumulate near the ice interface and slow down the crystal growth rate The growth rate of ice crystals on the wall is therefore inversely proportional to the solute concentration and increases with increasing diffusion coefficient Besides the growth rate is proportional to the difference between the wall temperature and the equilibrium temperature of the solution Due to these two effects the transition temperature difference for ice scaling is proportional to the solute concentration and is higher in aqueous solutions with low diffusion coefficients A model based on these physical mechanisms has been proposed to predict ice scaling in fluidized bed heat exchangers A validation with experimental results demonstrates that the model is applicable for a wide variety of solutes and concentrations showing an average absolute error of 144
Fluidized Bed Heat Exchangers for other Crystallization Processes
Besides ice crystallization processes fluidized bed heat exchangers are also attractive for other industrial processes that suffer from severe crystallization fouling such as cooling crystallization of salts and eutectic freeze crystallization In case of cooling crystallization of salts fluidized bed heat exchangers are able to prevent crystallization fouling of KNO3 and MgSO47H2O for heat fluxes up to 17 kWm2 Crystallization fouling during eutectic freeze crystallization from binary solutions is however not prevented by fluidized particles which can be explained by an extension of the ice scaling model It is supposed that salt crystallization during eutectic freeze crystallization takes up the salt ions that accumulate near
Chapter 9
202
the ice interface during ice growth The crystallizing ions therefore do not hinder the growth of ice crystals on the wall resulting in higher growth rates and more severe ice scaling The addition of a non-crystallizing solute considerably reduces fouling and achieves that eutectic freeze crystallization can be operated at heat fluxes of 10 kWm2 or higher From these results can be concluded that the ice growth rate and thus ice scaling is only determined by the non-crystallizing solutes
Comparison between Fluidized Bed and Scraped Surface Heat Exchangers
The transition temperature difference for ice scaling in a scraped surface heat exchanger is about 75 times higher than in a stationary fluidized bed heat exchanger with stainless steel particles of 4 mm in size operated at a bed voidage of 81 The heat flux at which ice scaling occurs is more than four times higher than in the fluidized bed heat exchanger The transition temperature difference in scraped surface heat exchangers increases with decreasing ice slurry temperature or with increasing solute concentration which is in correspondence with fluidized bed ice slurry generators The heat transfer performance of both ice slurry generators is comparable
The investment costs of fluidized bed heat exchangers per square meter of heat transfer surface are relatively low compared to the costs of scraped surface heat exchangers Fluidized bed ice slurry generators should be operated at ice slurry temperatures of about ndash5degC with a heat flux of approximately 10 kWm2 while scraped surface ice slurry generators are normally operated at ndash2degC with a heat flux of 20 kWm2 A comparison between these two crystallizers for installations of 100 kW and higher shows that the investment costs of fluidized bed ice slurry generators are about 30 tot 60 lower than of commercially available scraped surface ice slurry generators In addition the energy consumption of systems using fluidized bed ice slurry generators is about 5 to 21 lower It can therefore be concluded that the fluidized bed ice slurry generator is an attractive ice crystallizer concerning both investment costs and energy consumption
Ice Crystallization Phenomena during Storage and Melting of Ice Slurry
Besides the ice slurry production stage ice crystals are also subject to crystallization phenomena in other components of ice slurry systems such as storage tanks and melting heat exchangers During isothermal storage in tanks the crystal size distribution alters by means of recrystallization mechanisms of which Ostwald ripening is most important Due to surface energy contributions small crystals dissolve while larger crystals grow resulting in an increase of the average crystal size The rate of Ostwald ripening decreases with increasing solute concentration and depends furthermore on the solute type and initial average ice crystal size In melting heat exchangers ice slurry can seriously be superheated which means that the liquid temperature is significantly higher than the equilibrium temperature This phenomenon may result in reduced heat exchanger capacities The degree of superheating at the outlet of heat exchangers is proportional to the average ice crystal size and the heat flux It is furthermore inversely proportional to the ice fraction and depends also on the solute concentration
The described phenomena in storage tanks and melting heat exchangers can be explained by crystallization kinetics The growth and melting rates of suspended ice crystals are mainly determined by heat and mass transfer resistances while surface integration plays a minor role during growth The mass transfer resistance is proportional to the solute concentration while the heat transfer resistance hardly depends on the solute At low solute concentrations
Conclusions
203
corresponding to equilibrium temperatures of about ndash2degC and higher the crystallization kinetics are therefore dominated by heat transfer while mass transfer dominates at higher concentrations These crystallization kinetics result in slow crystal growth and melting processes at high solute concentrations or for solutes with relatively small diffusion coefficients Slow growth and melting processes lead to low Ostwald ripening rates in storage tanks and high degrees of superheating at the outlet of melting heat exchangers The latter effect is also achieved when the available ice crystal surface is small which occurs at low ice fractions or for relatively large ice crystals
Overall Conclusions
Fluidized bed heat exchangers are attractive crystallizers for ice slurry production in indirect refrigeration systems Installations using fluidized bed ice slurry generators have lower investment costs and lower energy consumptions compared to systems that use scraped surface heat exchangers Besides ice slurry production fluidized bed heat exchangers are also attractive for other industrial crystallization processes that suffer from severe crystallization fouling such as cooling crystallization and eutectic freeze crystallization
Several phenomena in ice slurry systems can be explained by the crystallization kinetics of ice crystals in aqueous solutions which are determined by heat transfer mass transfer and surface integration Ice scaling during ice slurry production in fluidized bed heat exchangers is for example only prevented when the growth rate of ice crystals attached to the heat exchanger wall does not exceed the removal rate induced by fluidized particles This ice growth rate is mainly determined by mass transfer and is therefore lower in solutions with higher solute concentrations and with lower diffusion coefficients resulting in less severe ice scaling The crystallization kinetics of suspended ice crystals in storage tanks and melting heat exchangers are also strongly influenced by mass transfer although heat transfer also plays a role at low solute concentrations In these components the crystallization kinetics determine the rate of Ostwald ripening during storage and the degree of superheating during melting in heat exchangers
Chapter 9
204
205
Appendix A Properties of Aqueous Solutions
This appendix describes models to predict properties of the aqueous solutions used in this thesis The solution properties that are discussed are phase equilibrium data density specific heat thermal conductivity dynamic viscosity enthalpy and diffusion coefficient The first part of this appendix gives the general models to calculate these properties the second part contains coefficients for these models for each specific aqueous solution Some of the coefficients were directly taken from literature sources while other coefficients were fit with help of measurement data points from literature or were determined with models that had been expressed in a different form
A1 Model Description
A11 Phase Equilibrium Data
Figure A1 shows a characteristic phase diagram of a binary water-solute system The ice line represents the relation between the solute concentration and the temperature at which solution and ice crystals are in equilibrium The solubility line analogously represents the equilibrium between the solution and the solid phase of the solute Some of the electrolyte solutes used have more than one solid state since they can form different hydrates In these cases also more than one solubility line exists
Solute concentration
Tem
pera
ture
Aqueous solution
Eutectic point
Solid solute +aqueous solution
Ice +aqueous solution
Ice + solid solute
Ice line
Solubility line
0degC
T eut
0 wt w eut
Figure A1 Typical binary phase diagram of water-solute system
The point where the ice line intersects the solubility line is called the eutectic point At this temperature (Teut) and solute concentration (weut) solid solute ice and aqueous solution are in equilibrium and can exist simultaneously
In this appendix phase equilibrium lines are represented by polynomials as shown in Equations A1 and A2
5i
eq ii 0
T C w=
= sum with T in (degC) and w in (wt) (A1)
Appendix A
206
5i
eq ii 0
w C T=
= sum with w in (wt) and T in (degC) (A2)
The coefficients Ci for the different solutes used are given in the second part of this appendix Coefficients that are not given are equal to zero
A12 Density Specific Heat and Thermal Conductivity
Density specific heat and thermal conductivity are a function of both the solution temperature and the solute concentration Equation A3 presents the general expression that is used to calculate these three properties for different solutions (Melinder 1997)
( ) ( )( )
5 3i j
ij m mi 0 j 0
f C w w T T= =
= sdot minus sdot minussumsum with w in (wt) and T in (degC) (A3)
The function f in this expression represents the density ρ in (kgm3) the specific heat cp in (Jkg K) or the thermal conductivity λ in (Wm K) Coefficients Cij and constants wm and Tm for different solutions are listed in the second part of this appendix
A13 Dynamic Viscosity
In analogously with the previous properties the dynamic viscosity micro in (Pa s) can be calculated with Equation A4 (Melinder 1997)
( ) ( ) ( )( )5 3
i j3ij m m
i 0 j 0
ln 10 C w w T Tmicro= =
sdot = sdot minus sdot minussumsum with T in (degC) and w in (wt) (A4)
Coefficients Cij and constants wm and Tm for different solutions are listed in the second part of this appendix
A14 Enthalpy
In this thesis the enthalpy of water in a liquid state at 0degC and the enthalpy of solute in its normal state at 0degC are defined as zero
2H O 0degC 0h equiv (A5)
solute 0degC 0h equiv (A6)
With this definition it is possible to formulate the enthalpy of an aqueous solution
2
diss
sol solute diss H O diss diss diss psol 1 100 100
T
T
w wh w T h T h T h w T c w T dT = + minus + ∆ + int (A7)
The enthalpy of an aqueous solution firstly consist of the partial sensible heat contributions to heat both solute and water from 0degC to the temperature Tdiss at which the heat of dissolution ∆hdiss is defined For the case where the solute is mixed with water instead of dissolved the enthalpy of dissolution should be replaced by the enthalpy of mixing ∆hmix and the dissolution temperature by the mixing temperature Tmix The second contribution is the heat of dissolution
Properties of Aqueous Solutions
207
or mixing itself and the last contribution is sensible heat contribution of the solution Below all three contributions will be discussed in more detail
The sensible heat contribution of the solute is estimated by
solute psolute psolute
0degC
T
h T c T dT c T= asympint with T in (degC) (A8)
The sensible heat contribution of water is estimated by an expression which is deduced from specific heat measurements cited by Dorsey (1940) and which is valid between 0 and 30degC
2 2
2 2 3H O pH O
0degC
42163 1495 1925 10T
h T c T dT T T Tminus= asymp minus + sdotint with T in (degC) (A9)
The heat of dissolution or mixing is a function of both the solute concentration and the temperature However most literature sources provide only data on heats of dissolution or mixing at a specific temperature Tdiss or Tmix The data for different solutions found in literature have all been transformed into the following form
3i
diss diss ii 1
h w T C w=
∆ = sum with w in (wt) (A10)
The sensible heat contribution of the solution can be split up into two parts
m
diss diss m
psol psol psol TT T
T T T
c w T dT c w T dT c w T dT= +int int int with w in (wt) (A11)
The first part of the right-hand side of Equation A11 can be simplified by using the expression for specific heat given in Equation A3
m
diss
psol T
T
c w T dTint ( )diss
0
psol m mmT T
c w T T d T Tminus
= minus minusint (A12)
( ) ( )( ) ( )
diss 5 3i j
ij m m mi 0 j 00
mT T
C w w T T d T Tminus
= =
= minus sdot minus sdot minus minussumsumint (A13)
( ) ( )
5 3i j+1
ij m diss mi 0 j 0
1j+1
C w w T T= =
= sdot minus sdot minus
sumsum (A14)
( )( )5
ii ij diss m m
i 0 c j C T T w w
=
= sdot minussum (A15)
Appendix A
208
Analogously the second part of the right-hand side of Equation A11 can be simplified by the same method as shown above
m
psol T
T
c w T dTint ( ) ( )( ) ( )m 5 3
i jij m m m
i 0 j 00
T T
C w w T T d T Tminus
= =
= sdot minus sdot minus minussumsumint (A16)
( ) ( )
5 3i j+1
ij m mi 0 j 0
1j+1
C w w T T= =
= sdot minus sdot minus
sumsum (A17)
Equations A8 A9 A10 A15 and A17 can be combined into Equation A7 as a result of which one general enthalpy model can be derived for a specific aqueous solution The model can be transformed into the same form as the expressions for density specific heat and thermal conductivity as proposed by Melinder (1997)
( ) ( )( )
5 4i k
ik m mi 0 k 0
h C w w T T= =
= sdot minus sdot minussumsum with h in (Jkg) w in (wt) and T in (degC) (A18)
The coefficients Cij and constants wm and Tm for calculating the enthalpy of aqueous solutions can be found in the second part of this appendix
A15 Diffusion Coefficient
The binary diffusion coefficient of an aqueous solution depends on the solute concentration and the temperature (Cussler 1997) Calculation models for binary diffusion coefficients of aqueous solutions that are suitable over a large temperature range are not available in literature Therefore an expression for the diffusion coefficient at the lowest reported temperature T0 is deduced from measured data available in literature
5i
0 ii 1
D wT C w=
= sum with D in (m2s) w in (wt) and T in (degC) (A19)
According to Reid et al (1987) and Cussler (1997) the product of the diffusion coefficient and the dynamic viscosity divided by the temperature in Kelvin does hardly depend on the temperature
D constant273 15T
micro=
+ (A20)
This statement was experimentally confirmed for aqueous solutions by Garner and Marchant (1961) for a temperature range from 15 to 40degC and by Byers and King (1966) for a temperature range from 20 to 70degC In this thesis binary diffusion coefficients of aqueous solutions are therefore estimated by using Equation A20 in which the diffusion coefficient at T0 is estimated from Equation A19
0
00
273 15D D273 15
wT T wT wTwT T
micromicro
+= +
(A21)
Properties of Aqueous Solutions
209
A2 Organic Aqueous Solutions
A21 D-glucose (C6H12O6)
Other names Dextrose grape sugar
CAS number 50-99-7
Molecular mass 18016 gmol
State at 0degC Solid
-100
-50
00
50
100
150
200
250
00 50 100 150 200 250 300 350 400 450 500Dextrose concentration (wt)
Tem
pera
ture
(degC
)
Aqueous solution
Ice + aqueous solution
Ice linedextrose
α-monohydrate + aqueous solution
Solubility line
Figure A2 Phase diagram of the water-dextrose system
Table A1 Ice line of water-dextrose system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -5hellip0degC -9291 -08127 -4617E-02 -1389E-03 -1666E-05Teqw1 0hellip31 wt -01217 1179E-03 -1185E-04 1832E-06 -1811E-08
1Deduced from Young (1957)
Table A2 Solubility line of C6H12O6middotH2O in water (see Equations A1 and A2) Function Domain C0 C1 C2 C3 C4 weqT1 -5hellip52degC 3378 06215 3080E-03 -2310E-05 - Teqw2 31hellip71 wt -6929 2632 -2097E-02 1116E-04 -
1Young (1957) 2Deduced from Young (1957)
Appendix A
210
Density specific heat thermal conductivity dynamic viscosity and diffusion coefficient data of aqueous dextrose solutions are only available at the ice line Because of this the properties of dextrose solutions are not presented here in the standard form as described in Section A1 Below expressions for the mentioned thermophysical properties at the ice line are given as a function of the freezing temperature Teq in (degC) for the range from ndash7 to 0degC (Huige 1972)
Density in (kgm3) 2eq eq1000 0 3606 2 266 T Tρ = minus minus (A22)
Specific heat in (Jkg K) 2p eq eq4216 244 3 15 77c T T= minus + (A23)
Thermal conductivity (Wm K) 2 4 2eq eq0 5576 2 307 10 9 595 10 T Tλ minus minus= + sdot + sdot (A24)
Dynamic viscosity (Pa s) ( )3 4 2eq eqln 10 0 5179 0 3208 9 793 10 T Tmicro minussdot = minus minus sdot (A25)
Diffusion coefficient (m2s) 10 11 12 2eq eqD 3 541 10 6 683 10 4 322 10 T Tminus minus minus= sdot + sdot + sdot (A26)
Properties of Aqueous Solutions
211
A22 Ethanol (C2H6O)
Other names Ethyl alcohol alcohol
CAS number 64-17-5
Molecular mass 4607 gmol
State at 0degC Liquid
Specific heat 2438 Jkg K at 25degC (Lide 1995)
-250
-200
-150
-100
-50
00
50
00 50 100 150 200 250 300 350 400Ethanol concentration (wt)
Tem
pera
ture
(degC
) Aqueous solution
Ice + aqueous solution
Ice line
Figure A3 Phase diagram of water-ethanol system
Table A3 Ice line of water-ethanol system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -30hellip0degC -2635 -8340E-02 -1583E-03 -7171E-06 - Teqw1 0hellip40 wt -04268 3709E-03 -4336E-04 -1806E-06 1347E-07
1Deduced from Flick (1998)
Table A4 Heat of mixing of water-ethanol system (see Equation A10) Function Domain Tmix C1 C2 C3 ∆hmixw1 0hellip40 wt 00degC -3394E+03 5666 1055E-02
1Deduced from Beggerow (1976)
Appendix A
212
Table A5 Coefficients of water-ethanol system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat1 Thermal conductivity1
Dynamic viscosity1 Enthalpy
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 11hellip60 wt 11hellip60 wt 11hellip60 wt 11hellip60 wt 11hellip40 wt T Teqhellip20degC Teqhellip20degC Teqhellip20degC Teqhellip20degC Teqhellip20degC
wm 389250 389250 389250 389250 389250 Tm -49038 -49038 -49038 -49038 -49038 C00 9544E+02 3925E+03 03545 2214 -6381E+04 C01 -06416 3876 4421E-04 -5710E-02 3925E+03 C02 -2495E-03 2300E-04 -2942E-07 4679E-04 1938 C03 1729E-05 1322E-05 -1115E-08 -1374E-06 7667E-05 C04 - - - - 3305E-06 C10 -1729 -2795 -4334E-03 8025E-04 1316E+03 C11 -1824E-02 01773 -2021E-05 2618E-04 -2795 C12 3116E-04 4769E-05 -4865E-09 -8472E-06 8865E-02 C13 -6425E-07 3008E-06 2972E-10 1478E-07 1590E-05 C14 - - - - 7520E-07 C20 -2193E-02 -9620E-02 3021E-05 -7330E-04 5844 C21 5847E-04 -3908E-03 4239E-07 7056E-06 -9620E-02 C22 -2517E-06 1951E-05 1007E-09 2473E-07 -1954E-03 C23 -2875E-08 3366E-08 -7325E-12 -1329E-08 6503E-06 C24 - - - - 8415E-09 C30 6217E-04 7580E-03 6904E-07 4285E-07 -2686E-02 C31 4208E-06 2283E-05 -3203E-09 3239E-07 7580E-03 C32 -3460E-07 -9149E-07 -1439E-11 -1234E-08 1142E-05 C33 - - - - -3050E-07 C40 2288E-06 -1213E-04 -1512E-08 4313E-08 5642E-04 C41 -4141E-07 2545E-06 -3486E-10 8582E-09 -1213E-04 C42 - - - - 1273E-06 C50 -6412E-07 2235E-07 -1012E-09 7654E-09 -1096E-06 C51 - - - - 2235E-07
1Melinder (1997)
Table A6 Diffusion coefficient of water-ethanol system (see Equation A19) Function Domain T0 C0 C1 C2 C3 C4 Dw 1 0hellip35 wt 250degC 1240E+09 3237E+11 2835E-13 - -
1Deduced from Hammond and Stokes (1953)
Properties of Aqueous Solutions
213
A23 Ethylene Glycol (C2H6O2)
Other names 12-ethanediol ethylene alcohol
CAS number 107-21-1
Molecular mass 6207 gmol
State at 0degC Liquid
Specific heat 2350 Jkg K on average between 0 and 25degC (Holman 1997)
-250
-200
-150
-100
-50
00
50
00 50 100 150 200 250 300 350 400Ethylene glycol concentration (wt)
Tem
pera
ture
(degC
) Aqueous solution
Ice + aqueous solution
Ice line
Figure A4 Phase diagram of water-ethylene glycol system
Table A7 Ice line of water-ethylene glycol system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -40hellip0degC - 3409 -01429 -4401E-03 -7259E-05 -4809E-07Teqw1 0hellip54 wt -02869 -5450E-03 1230E-04 -8090E-06 8911E-08
1Deduced from Melinder (1997)
Table A8 Heat of mixing of water-ethylene glycol system (see Equation A10) Function Domain Tmix C1 C2 C3 ∆hmixw1 0hellip100 wt 250degC -10865 12534 -1675E-02
1Deduced from Beggerow (1976)
Appendix A
214
Table A9 Coefficients of water-ethylene glycol system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat1 Thermal conductivity1
Dynamic viscosity1 Enthalpy
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 0hellip56 wt 0hellip56 wt 0hellip56 wt 0hellip56 wt 0hellip56 wt T Teqhellip40degC Teqhellip40degC Teqhellip40degC Teqhellip40degC Teqhellip40degC
wm 381615 381615 381615 381615 381615 Tm 63333 63333 63333 63333 63333 C00 1056E+03 3501E+03 04211 1453 -2971E+03 C01 -03987 3954 7995E-04 -3747E-02 3501E+03 C02 -3068E-03 6065E-05 -5509E-08 2842E-04 1977 C03 1233E-05 -5979E-06 -1460E-08 -8025E-07 2022E-05 C04 - - - - -1495E-06 C10 1505 -2419 -3694E-03 2920E-02 -2300E+02 C11 -8953E-03 01031 -1751E-05 -1131E-04 -2419 C12 6378E-05 4312E-05 6656E-08 1729E-06 5155E-02 C13 -1152E-07 5168E-06 2017E-09 -5073E-08 1437E-05 C14 - - - - 1292E-06 C20 -1634E-03 4613E-03 2095E-05 1264E-04 1052 C21 1541E-04 -6595E-05 2078E-07 6785E-09 4613E-03 C22 -1874E-06 1620E-05 -2394E-09 -1685E-08 -3298E-05 C23 -9809E-09 -3250E-07 -6772E-11 -1082E-09 5400E-06 C24 - - - - -8125E-08 C30 -2317E-04 6028E-03 3663E-07 4386E-06 -01374 C31 2549E-06 5642E-05 -5272E-09 -2191E-07 6028E-03 C32 -5523E-08 -7777E-07 -1126E-10 -9117E-11 2821E-05 C33 - - - - -2592E-07 C40 -8510E-06 -7977E-05 -6389E-09 -9223E-08 1399E-03 C41 -3848E-08 5190E-07 -1112E-10 -4294E-09 -7977E-05 C42 - - - - 2595E-07 C50 -1128E-07 -3380E-06 -1820E-10 -3655E-09 6309E-05 C51 - - - - -3380E-06
1Melinder (1997)
Table A10 Diffusion coefficient of water-ethylene glycol system (see Equation A19) Function Domain T0 C0 C1 C2 C3 C4 Dw1 0hellip100 wt 250degC 1189E-09 1152E-11 2837E-14 -5773E-17 -
1Fernaacutendez-Sempere et al (1996)
Properties of Aqueous Solutions
215
A24 Propylene Glycol (C3H8O2)
Other names 12-propanediol
CAS number 57-55-6
Molecular mass 7609 gmol
State at 0degC Liquid
Specific heat 2481 Jkg K at 20degC (Bosen et al 2000)
-250
-200
-150
-100
-50
00
50
00 50 100 150 200 250 300 350 400 450Propylene glycol concentration (wt)
Tem
pera
ture
(degC
) Aqueous solution
Ice + aqueous solution
Ice line
Figure A5 Phase diagram of water-propylene glycol system
Table A11 Ice line of water-propylene glycol system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -35hellip0degC -3465 -01190 -2696E-03 -2669E-05 - Teqw1 0hellip51 wt -01617 -1592E-02 3924E-04 -5471E-06 -
1Deduced from Melinder (1997)
Table A12 Heat of mixing of water-ethylene glycol system (see Equation A10) Function Domain Tmix C1 C2 C3 ∆hmixw1 0hellip60 wt 250degC -1300 1100 5974E-02
1Deduced from Christensen et al (1984)
Appendix A
216
Table A13 Coefficients of water-propylene glycol system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat1 Thermal conductivity1
Dynamic viscosity1 Enthalpy
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 15hellip56 wt 15hellip56 wt 15hellip56 wt 15hellip56 wt 15hellip56 wt T Teqhellip40degC Teqhellip40degC Teqhellip40degC Teqhellip40degC Teqhellip40degC
wm 427686 427686 427686 427686 427686 Tm 53571 53571 53571 53571 53571 C00 1042E+03 3679E+03 3806E-01 2274E+00 -1692E+04 C01 -4907E-01 1571E+00 5765E-04 -5342E-02 3679E+03 C02 -2819E-03 1331E-02 -3477E-07 5372E-04 07855 C03 -5895E-07 1975E-07 -6041E-09 -4955E-06 4437E-03 C04 - - - - 4938E-08 C10 8081E-01 -1933E+01 -3815E-03 4500E-02 -9801 C11 -9652E-03 1118E-01 -1423E-05 -5488E-04 -1933 C12 7168E-05 -1108E-03 -1203E-08 1845E-06 5590E-02 C13 2404E-07 4924E-06 -5854E-10 1192E-07 -3693E-04 C14 - - - - 1231E-06 C20 -7156E-03 -4879E-02 8420E-06 -7808E-05 1961 C21 1088E-04 -2338E-04 1081E-07 1453E-06 -4879E-02 C22 -3328E-06 2753E-05 1959E-09 -2816E-07 -1169E-04 C23 1153E-07 -3148E-07 1271E-10 8562E-09 9177E-06 C24 - - - - -7870E-08 C30 1190E-04 4749E-03 -1110E-06 6565E-06 -3174E-02 C31 -6226E-06 -2621E-05 -1612E-09 -4032E-07 4749E-03 C32 -3026E-08 1286E-06 3005E-10 -1212E-09 -1311E-05 C33 - - - - 4287E-07 C40 -1170E-05 -2871E-04 5503E-09 6441E-07 5657E-03 C41 -2915E-07 -9050E-08 1437E-10 -1430E-08 -2871E-04 C42 - - - - -4525E-08 C50 -6033E-07 -1068E-05 1290E-09 1092E-08 2098E-04 C51 - - - - -1068E-05
1Melinder (1997)
Table A14 Diffusion coefficient of water-propylene glycol system (see Equation A19) Function Domain T0 C0 C1 C2 C3 C4 Dw 1 0hellip45 wt 20degC 9004E-10 -1477E-11 -1420E-13 3892E-15 -
1Deduced from Garner and Marchant (1961)
Properties of Aqueous Solutions
217
A3 Inorganic Aqueous Solutions
A31 Magnesium Sulfate (MgSO4)
Other name Epsom salt
CAS number 7487-88-9
Molecular mass 12037 gmol
State at 0degC Solid
Specific heat 800 JkgK (Seeger et al 2000)
-100
-50
00
50
100
150
200
250
00 50 100 150 200 250 300MgSO4 concentration (wt)
Tem
pera
ture
(degC
)
Aqueous solution
Ice + aqueous solutionIce line
Ice + MgSO412H2O
aqueous solution + MgSO412H2O
aqueoussolution +
MgSO47H2O
Solubility lineMgSO 4 7H 2 O
Eutectic point
Solubility lineMgSO 4 12H 2 O
Figure A6 Phase diagram of water-MgSO4 system
Table A15 Ice line of water-MgSO4 system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -39hellip0degC -6733 -06153 -2952E-02 - - Teqw1 0hellip18 wt - 01293 -3892E-03 -2725E-05 - -
1Deduced from Gmelin (1952)
Table A16 Solubility line of MgSO4middot12H2O in water (see Equations A1 and A2) Function Domain C0 C1 C2 C3 C4 weqT1 -39hellip27degC 203 0594 - - - Teqw1 18hellip22 wt -342 1684 - - -
1Deduced from Gmelin (1952)
Table A17 Solubility line of MgSO4middot7H2O in water (see Equations A1 and A2) Function Domain C0 C1 C2 C3 C4 weqT1 27hellip25degC 213 0206 833E-04 - - Teqw1 22hellip27 wt -1339 7759 -694E-02 - -
1Deduced from Gmelin (1952)
Appendix A
218
Table A18 Heat of dissolution of MgSO4 in water (see Equation A10) Function Domain Tmix C1 C2 C3 ∆hdissw1 0hellip25 wt 18degC -7161E+03 - -
1Deduced from Beggerow (1976)
Table A19 Coefficients of water-MgSO4 system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat1 Thermal conductivity1
Dynamic viscosity2 Enthalpy
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 0hellip30 wt 0hellip40 wt 0hellip30 wt 0hellip24 wt 0hellip40 wt T Teqhellip30degC 0hellip23degC3 Teqhellip40degC 15hellip55degC4 0hellip30degC3
wm 0 0 0 0 0 Tm 0 0 0 0 0 C00 10004E+03 4216E+03 05607 05743 3230 C01 2045E-02 -2990 2027E-03 -3278E-02 4216E+03 C02 -5390E-03 5775E-02 -6852E-06 2355E-04 -1495 C03 - - - -1009E-06 1925E-02 C04 - - - - - C10 1021 -5046 -6369E-04 5200E-02 -6940E+03 C11 -2381E-02 3611E-02 -2302E-06 5234E-05 -5046 C12 2644E-04 -6974E-04 7784E-09 -6310E-07 1806E-02 C13 - - - - 1204E-02 C14 - - - - - C20 5561E-02 03493 - 8370E-04 -6257 C21 4243E-04 -2477E-04 - -1974E-05 03493 C22 -5402E-06 4785E-06 - 1256E-07 -1239E-04 C23 - - - - 1595E-06 C30 - - - 2496E-05 -
1Deduced from Gmelin (1952) 2Deduced from Lobo (1989) 3Extrapolated values are used below 0degC 4Extrapolated values are used below 15degC Viscosity measurements have shown that errors of this extrapolation are below 20
Table A20 Diffusion coefficient of water-MgSO4 system (see Equation A19) Function Domain T0 C0 C1 C2 C3 C4 Dw1 0hellip35 wt 181degC 5725E-10 -8984E-12 1112E-13 - -
1Deduced from Gmelin (1952)
Properties of Aqueous Solutions
219
A32 Potassium Chloride (KCl)
Other name -
CAS number 7447-40-7
Molecular mass 7455 gmol
State at 0degC Solid
Specific heat 694 Jkg K (Schultz et al 2000)
-150
-100-50
0050
100150
200250
300
00 50 100 150 200 250 300KCl concentration (wt)
Tem
pera
ture
(degC
) Aqueous solution
Ice + aqueous solution
Ice line
Ice + KClH2O
aqueous solution + KCl
Eutectic point
Solubility line
aq sol + KClH2O
Figure A7 Phase diagram of water-KCl system
Table A21 Ice line of water-KCl system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -106hellip0degC -2245 -3454E-02 3300E-04 - - Teqw1 0hellip197 wt - 04502 -1680E-03 -1553E-04 - -
1Deduced from Gmelin (1952)
Table A22 Solubility line of KCl in water (see Equations A1 and A2) Function Domain C0 C1 C2 C3 C4 weqT1 -66hellip80degC 2193 01929 -4398E-04 -6186E-06 5677E-08 Teqw1 207hellip34 wt -4685E+02 6124 -3234 7982E-02 6957E-04
1Deduced from Gmelin (1952)
Table A23 Solubility line of KClmiddotH2O in water (see Equations A1 and A2) Function Domain C0 C1 C2 C3 C4 weqT1 -106hellip-66degC 2428 07524 3063E-02 - - Teqw1 197207 wt -9896E+02 9301 -2199 - -
1Deduced from Gmelin (1952)
Appendix A
220
Table A24 Heat of dissolution of KCl in water (see Equation A10) Function Domain Tmix C1 C2 C3 ∆hdissw1 0hellip23 wt 00degC 2973E+03 -25575 -
1Deduced from Gmelin (1952)
Table A25 Coefficients of water-KCl system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat1 Thermal conductivity1
Dynamic viscosity1 Enthalpy
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 0hellip30 wt 0hellip40 wt 0hellip25 wt 0hellip25 wt 0hellip23 wt T Teqhellip40degC Teqhellip40degC Teqhellip25degC 0hellip85degC2 Teqhellip40degC
wm 0 0 0 0 0 Tm 0 0 0 0 0 C00 10000 41772 05607 05767 0000 C01 2674E-02 -028626 2027E-03 -3267E-02 4177E+03 C02 -5394E-03 - -6852E-06 2214E-04 -01431 C03 - - - -8117E-07 - C10 6647 -3172 -2243E-03 -1175E-02 2973E+03 C11 -2446E-02 -7126E-02 -8106E-06 4996E-04 -3172 C12 2401E-04 - 2741E-08 -3400E-06 -3563E-02 C13 - - - 6261E-09 - C20 1711E-02 -01368 - 2675E-04 -2558 C21 4005E-04 6843E-03 - -6164E-06 -01368 C22 -4094E-06 - - 3697E-08 3421E-03 C30 - - - -2262E-06 - C31 - - - -1389E-08 - C40 - - - 1607E-07 -
1Deduced from Gmelin (1952) 2Extrapolated values are used below 0degC Viscosity measurements have shown that errors of this extrapolation are below 20
Table A26 Diffusion coefficient of water-KCl system (see Equation A19) Function Domain T0 C0 C1 C2 C3 C4 Dw 1 05hellip23 wt 18degC 1566E-09 -1269E-11 3542E-12 -1897E-13 3712E-15
1Deduced from Lobo (1989)
Properties of Aqueous Solutions
221
A33 Potassium Formate (KCOOH or KFo)
Other name -
CAS number 590-29-4
Molecular mass 8412 gmol
State at 0degC Solid (Aittomaumlki 1997)
-450-400-350-300-250-200-150-100-500050
100
00 50 100 150 200 250 300 350 400 450 500KCOOH concentration (wt)
Tem
pera
ture
(degC
) Aqueous solution
Ice + aqueous solution
Ice line
Figure A8 Phase diagram of water-KCOOH system
Table A27 Ice line of water-KCOOH system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -45hellip0degC -2150 -4183E-02 -3696E-04 - - Teqw1 0hellip45 wt -04658 -1151E-03 -2261E-04 - -
1Deduced from Melinder (1997)
Data on the heat of dissolution of potassium formate in water has not been found in literature
Appendix A
222
Table A28 Coefficients of water-KCOOH system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat1 Thermal conductivity1
Dynamic viscosity1 Enthalpy2
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 19hellip50 wt 19hellip50 wt 19hellip50 wt 19hellip50 wt 19hellip50 wt T Teqhellip40degC Teqhellip40degC Teqhellip40degC Teqhellip40degC Teqhellip40degC
wm 25 25 25 25 25 Tm 0 0 0 0 0 C00 1156E+03 3314E+03 05111 08142 0000 C01 -04035 1520 1292E-03 -2982E-02 3314E+03 C02 1054E-04 1757E-03 2949E-06 1849E-04 7600E-01 C03 - - - - 5857E-04 C10 6691 -2982 -1584E-03 1486E-02 0000 C11 5108E-04 7153E-02 -6271E-06 -1751E-04 -2982E+01 C12 -1724E-05 -1737E-04 -2135E-07 5847E-06 3577E-02 C13 - - - - -5790E-05 C20 3977E-02 01262 8820E-06 5258E-04 0000 C21 -1549E-05 -2274E-04 -1852E-07 3712E-06 1262E-01 C22 - - - - -1137E-04 C30 5434E-07 3619E-06 4430E-09 -9631E-08 0000 C31 - - - - 3619E-06
1Deduced from Melinder (1997) 2The enthalpy function does not contain the heat of dissolution The function can therefore only be applied to calculate enthalpy differences at a constant solute concentration
Data on the diffusion coefficient of aqueous potassium formate solution has not been found in literature
Properties of Aqueous Solutions
223
A34 Potassium Nitrate (KNO3)
Other name -
CAS number 7757-79-1
Molecular mass 1011 gmol
State at 0degC Solid
Specific heat 953 Jkg K at 25degC (Laue et al 2000)
-100
-50
00
50
100
150
200
250
300
00 50 100 150 200 250 300KNO3 concentration (wt)
Tem
pera
ture
(degC
)
Aqueous solution
Ice + aqueous solution
Ice line
Ice + KNO3
aqueous solution + KNO3
Eutectic point
Solubility line
Figure A9 Phase diagram of water-KNO3 system
Table A29 Ice line of the water-KNO3 system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -29hellip0degC -3026 01498 -8989E-03 - - Teqw1 0hellip10 wt -03304 5361E-03 -7069E-05 - -
1Deduced from Gmelin (1952)
Table A30 Solubility line of KNO3 in water (see Equations A1 and A2) Function Domain C0 C1 C2 C3 C4 weqT1 -29hellip115degC 1182 04489 1077E-02 -1494E-04 5578E-07 Teqw1 10hellip75 wt -2837 2918 -5053E-02 5777E-04 -1208E-06
1Deduced from Gmelin (1952)
Table A31 Heat of dissolution of KNO3 in water (see Equation A10) Function Domain Tdiss C1 C2 C3 ∆hdissw1 0hellip15 wt 147degC 36034 -36091 -
1Deduced from Gmelin (1952)
Appendix A
224
Table A32 Coefficients of water-KNO3 system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat1 Thermal conductivity2
Dynamic viscosity2 Enthalpy
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 0hellip24 wt 0hellip20 wt 0hellip24 wt 0hellip32 wt 0hellip20 wt T Teqhellip30degC Teqhellip30degC Teqhellip30degC Teqhellip30degC Teqhellip30degC
wm 0 0 0 0 0 Tm 0 0 0 0 0 C00 9999E+02 4216E+03 056848 05379 4559E-02 C01 4285E-02 -2990 1616E-03 -3603E-02 4216E+03 C02 -6099E-03 5775E-02 -4309E-06 4062E-04 -1495 C03 - - - -3693E-06 1925E-02 C04 - - -1653E-03 - - C10 6630 -4745 -4698E-06 7153E-03 3746E+03 C11 -2521E-02 3365E-02 1253E-08 -1901E-06 -4745 C12 2187E-04 3365E-02 - 1445E-06 1683E-02 C13 - - -465E-12 5716E-08 -2166E-04 C14 - - 744E-14 - - C20 1728E-02 04911 - -1166E-03 -4328 C21 1775E-04 -3482E-04 - 1768E-05 04911 C22 - 6726E-06 109E-13 -3043E-07 -1741E-04 C23 - - - - 2242E-06 C30 2098E-04 - - 5434E-05 - C31 - - - -1422E-07 - C40 - - - -9349E-07 -
1Deduced from Gmelin (1952) 2Deduced from Vaessen (2003)
Table A33 Diffusion coefficient of water-KNO3 system (see Equation A19) Function Domain T0 C0 C1 C2 C3 C4 Dw 1 0hellip15 wt 185degC 1460E-09 -3275E-11 9739E-13 - -
1Deduced from Gmelin (1952)
Properties of Aqueous Solutions
225
A35 Sodium Chloride (NaCl)
Other name Table salt
CAS number 7647-14-5
Molecular mass 5844 gmol
State at 0degC Solid
Specific heat 850 Jkg K at 25degC (Westphal et al 2000)
-250
-200
-150
-100
-50
00
50
00 50 100 150 200 250NaCl concentration (wt)
Tem
pera
ture
(degC
)
Aqueous solution
Ice + aqueous solution
Ice line
Ice + NaCl2H2O
aqueous solution + NaCl2H2O
Eutectic point
Solubility lineNaCl 2H 2 O
Figure A10 Phase diagram of water-NaCl system
Table A34 Ice line of water-NaCl system (see Equations A1 and A2) Function Domain C1 C2 C3 C4 C5 weqT1 -211hellip0degC -1758 -3830E-02 -3147E-04 6977E-07 - Teqw1 0hellip232 wt - 05615 -1057E-02 3132E-04 -2202E-05 -
1Deduced from Lide (2004)
Table A35 Solubility line of NaClmiddot2H2O in water (see Equations A1 and A2) Function Domain C0 C1 C2 C3 C4 weqT1 -211hellip01degC 26086 01409 - - - Teqw1 232hellip261 wt - 1851 7097 - - -
1Deduced from Gmelin (1952)
Table A36 Heat of dissolution of NaCl in water (see Equation A10) Function Domain Tmix C1 C2 C3 ∆hdissw1 0hellip25 wt 200degC 90224 -3522 04973
1Deduced from Beggerow (1976)
Appendix A
226
Table A37 Coefficients of water-NaCl system for calculating thermophysical properties (see Equations A3 A4 and A18)
Property Density1 Specific heat2 Thermal conductivity2
Dynamic viscosity2 Enthalpy
Symbol ρ cp λ micro h Unit kgm3 Jkg K Wm K Pa s Jkg
w 0hellip23 wt 0hellip23 wt 0hellip23 wt 0hellip23 wt 0hellip23 wt T Teqhellip30degC Teqhellip30degC Teqhellip30degC Teqhellip30degC Teqhellip30degC
wm 0 123539 123539 123539 123539 Tm 0 92581 92581 92581 92581 C00 10002 3619E+03 5692E-01 4951E-01 4334E+04 C01 4487E-02 1893 1677E-03 -2743E-02 3619E+03 C02 -6919E-03 -2804E-04 -2661E-06 2397E-04 09465 C03 1657E-05 - - - -9347E-05 C10 7767 -3384 -8528E-04 2277E-02 -4872 C11 -3773E-02 6473E-02 -1519E-05 -9952E-06 -3384 C12 5316E-04 -1467E-03 3244E-07 4419E-06 3237E-02 C13 - - - - -4890E-04 C20 -1174E-02 07992 -9082E-06 4907E-04 -2453 C21 6761E-04 -1458E-02 -4241E-08 -9974E-06 07992 C22 -1318E-05 - - - -7290E-03 C30 7610E-04 -1959E-02 -3147E-07 -2524E-06 07077 C31 - - - - -1959E-02
1Deduced from Lobo (1989) 2Melinder (1997)
Table A38 Diffusion coefficient of water-NaCl system (see Equation A19) Function Domain T0 C0 C1 C2 C3 C4 Dw 1 0hellip15 wt 185degC 1259E-09 -4266E-11 1094E-11 -8930E-13 2567E-14
1Deduced from Gmelin (1952)
Properties of Aqueous Solutions
227
Nomenclature
cp Specific heat (Jkg K) micro Dynamic viscosity (Pa s) C Constant ρ Density (kgm3) D Diffusion coefficient (m2s) f Function Subscripts h Enthalpy (Jkg) diss Dissolution T Temperature (K or degC) eut Eutectic Tm Constant in Eqs A3 and A4 (degC) eq Equilibrium w Solute concentration (wt) mix Mixing wm Constant in Eqs A3 and A4 (wt) sol Solution solute Solute Greek λ Thermal conductivity (Wm K)
References
Aittomaumlki A Lahti A 1997 Potassium formate as a secondary refrigerant International Journal of Refrigeration vol20 pp276-282
Beggerow G 1976 Heats of mixing and solution In Landolt-Boumlrnstein Numerical Data and Functional Relationships in Science and Technology - New Series Group 4 Physical Chemistry Band 2 Berlin Springer
Bosen SF Bowles WA Ford EA Perlson BD 2000 Antifreezes In Ullmanns encyclopedia of industrial chemistry 6th edition Electronic Release 2000 Weinheim Wiley-VCH
Christensen C Gmehling J Rasmussen P Weidlich U 1984 Heats of mixing data collection Part 1 Binary systems Frankfurt am Main DECHEMA (Deutsche Gesellschaft fuumlr Chemisches Apparatewesen)
Cussler EL 1997 Diffusion Mass Transfer in Fluid Systems 2nd edition Cambridge Cambridge University Press
Dorsey NE 1940 Properties of ordinary water-substance in all its phases Water-vapor water and all the ices New York Reinhold Publishing Corporation
Fernaacutendez-Sempere J Ruiz-Beviaacute Colom-Valiente J Maacutes-Peacuterez F 1996 Determination of diffusion coefficients of glycols Journal of Chemical and Engineering Data vol41 pp47-48
Flick EW 1998 Industrial Solvents Handbook 5th edition Westwood Noyes
Garner FH Marchant PJM 1961 Diffusivities of associated compounds in water Transactions of the Institution of Chemical Engineers vol 39 pp397-408
Gmelin 1952 Gmelins Handbuch der Anorganischen Chemie 8th edition Deutsche Chemische Gesellschaft Weinheim Verlag Chemie
Appendix A
228
Hammond BR Stokes RH 1953 Diffusion in binary liquid mixtures Transactions of the Faraday Society vol49 pp890-895
Holman JP 1997 Heat Transfer 8th edition New York McGraw-Hill Inc
Huige NJJ 1972 Nucleation and Growth of Ice Crystals from Water and Sugar Solutions in Continuous Stirred Tank Crystallizers PhD thesis Eindhoven University of Technology (The Netherlands)
Kemira Chemicals 2003 Product Brochure Freezium Kemira Chemicals BV Europoort-Rotterdam (The Netherlands)
Laue W Thiemann Scheibler E Wiegand KW 2000 Nitrates and nitrites In Ullmanns Encyclopedia of Industrial Chemistry 6th edition Electronic Release 2000 Weinheim Wiley-VCH
Lide DR 1995 Handbook of Organic Solvents Boca Raton CRC Press
Lide DR 2004 CRC Handbook of Chemistry and Physics A Ready-reference Book of Chemical and Physical Data 84th edition Boca Raton CRC Press
Lobo VMM 1989 Handbook of Electrolyte Solutions Amsterdam Elsevier
Melinder Ǻ 1997 Thermophysical Properties of Liquid Secondary Refrigerants Charts and Tables Paris International Institute of Refrigeration
Plessen H von 2000 Sodium sulfate In Ullmanns Encyclopedia of Industrial Chemistry 6th edition Electronic Release 2000 Weinheim Wiley-VCH
Reid RC Prausnitz JM Poling BE 1987 The Properties of Gases and Liquids 4th edition New York McGraw-Hill Inc
Schultz H Bauer G Schachl E Hagedorn F Schmittinger P 2000 Potassium compounds In Ullmanns Encyclopedia of Industrial Chemistry 6th edition Electronic Release 2000 Weinheim Wiley-VCH
Seeger M Otto W Flick W Bickelhaupt F Akkerman OS 2000 Magnesium compounds In Ullmanns Encyclopedia of Industrial Chemistry 6th edition Electronic Release 2000 Weinheim Wiley-VCH
Vaessen RJC 2003 Development of Scraped Eutectic Crystallizers PhD thesis Delft University of Technology (The Netherlands)
Westphal G et al 2000 Sodium chloride In Ullmanns Encyclopedia of Industrial Chemistry 6th edition Electronic Release 2000 Weinheim Wiley-VCH
Young FE 1957 D-Glucose-water phase diagram Journal of Physical Chemistry vol61 pp616-619
229
Appendix B Properties of Ice and Ice Slurries
B1 Properties of Ice
B11 Density
The density of ice between ndash100 and 001degC is given by the following expression deduced from an expression for the specific volume of ice by Hyland and Wexler (1983)
4 2ice 916 67 0 15 3 0 10 T Tρ minus= minus sdot + sdot sdot with ρ in (kgm3) and T in (degC) (B1)
B12 Thermal Conductivity
An expression for the thermal conductivity of ice between ndash100 and 001degC is given by the following expression deduced from data points given by Dorsey (1940)
3 5 2ice 2 23 9 7 10 4 7 10 T Tλ minus minus= minus sdot sdot + sdot sdot with λ in (Wm K) and T in (degC) (B2)
B13 Enthalpy
The enthalpy of ice between ndash100 and 001degC is given by an expression deduced from Hyland and Wexler (1983) in which the enthalpy of water in liquid state at 00degC equals zero
2 3 3ice 333430 2106 9 3 7991 1 0876 10h T T Tminus= + sdot + sdot + sdot sdot with h in (Jkg)
and T in (degC)(B3)
B14 Specific Heat
The specific heat of ice between ndash100 and 001degC has been deduced by taking the derivative of the expression for the enthalpy of ice stated in Equation B4
3 2pice 2106 9 7 5982 3 2628 10c T Tminus= + sdot + sdot sdot with cp in (Jkg K) and T in (degC) (B4)
B2 Properties of Ice Slurries
The properties of ice slurries are both influenced by the properties of ice and the liquid properties In this section models are presented to determine density thermal conductivity enthalpy specific heat and dynamic viscosity of ice slurry
B21 Density
The specific volume of ice slurry is given by the weighted average of the specific volumes of both phases
( )is ice liq1-v v vφ φ= + (B5)
Appendix B
230
The specific volumes in Equation B5 can be replaced by the reciprocal value of the density which gives a relation for the density of ice slurry
( )is
ice liq
11-
ρφ ρ φ ρ
=+
(B6)
Values for the density of ice are given in Section B1 and values for the density of several aqueous solutions can be found in Appendix A
B22 Thermal Conductivity
The thermal conductivity of ice slurry can be calculated with a model proposed by Tareef (1940) for liquid-solid mixtures
( )( )
liq ice liq iceis liq
liq ice liq ice
2 2
2
λ λ ξ λ λλ λ
λ λ ξ λ λ
+ minus minus = + + minus
(B7)
Bel and Lallemand (1999) proposed to use the model presented by Jeffrey (1973) to calculate the thermal conductivity of ice slurries However the differences between the results of Jeffreyrsquos and Tareefrsquos model applied to ice slurries appear to be smaller than 05 for ice fractions up to 40 vol
Values for the thermal conductivity of ice and aqueous solutions can be found in Section B12 and Appendix A respectively
B23 Enthalpy
The enthalpy of an ice slurry can simply be deduced from the weighted average of the enthalpy of the liquid phase and the enthalpy of ice
( )is ice liq1h h hφ φ= + minus (B8)
Values for the enthalpies of ice and aqueous solutions can be found in Section B13 and Appendix A respectively
B24 Specific Heat
The specific heat cp is defined as the temperature derivative of the enthalpy
p
hcT
part=
part (B9)
The formula for the enthalpy of ice slurry in Equation B9 can be written more explicitly
( ) is ice pice liq pliq
0degC 0degC
0degC 1 0degCT T
h h c dT h c dTφ φ
= + + minus +
int int (B10)
The first term in Equation B10 represents the enthalpy contribution of the ice phase with the latent heat at 0degC and the sensible heat respectively The second part of Equation B10
Properties of Ice and Ice Slurries
231
represents the enthalpy contribution of the liquid phase constructed of the enthalpy of the liquid at 0degC and a sensible heat contribution
The derivative of Equation B10 is shown in Equation B11
( ) is
ice pice pice liq pliq0degC 0degC
10degC 0degC
T Th h c dT c h c dTT T T
φφ φ part minuspart part
= + + + + + part part part int int
( ) pliq1 cφ+ minus
(B11)
The infinitesimal temperature change partT causes a infinitesimal change of ice fraction and with that also a change of the solute concentration in the liquid phase The effect of this change on the liquid enthalpy at 0degC is neglected in this analysis
Rearranging Equation B11 leads to Equation B12 in which the right-hand side shows a clear separation between latent and sensible heat contributions to the specific heat
( ) ( )is
ice pice pliq liq pice pliq0degC
0degC 0degC 1Th h c c dT h c c
T Tφ φ φ
part part= + minus minus + + minus part part
int (B12)
If both latent and sensible contributions are taken into account than the derivative of the enthalpy is called apparent specific heat cpappis
( ) ( )ice
pappis ice pice pliq liq pice pliq0degC
0degC 0degC 1Twc h c c dT h c c
Tφ φ
part= + minus minus + + minus part
int (B13)
If the latent heat is neglected and only sensible contributions are used than the derivative of the enthalpy is called sensible specific heat cpsensis
( )psensis pice pliq1c c cφ φ= + minus (B14)
B25 Dynamic Viscosity
The dynamic viscosity of ice slurry increases with the ice fraction In most publications on ice slurries a viscosity model by Thomas (1965) is used to estimated the viscosity of the ice slurry from the dynamic viscosity of the liquid and the volumetric ice fraction
( )2 3 16 6is liq 1 2 5 10 05 2 73 10 e ξmicro micro ξ ξ minus= + + + sdot (B15)
Values for the viscosity of aqueous solution used in this thesis can be found in Appendix A
Experimental validation of Thomasrsquo model for ice slurry by Kauffeld et al (1999) has shown that the model is able to predict viscosities of ice slurries reasonably well below volumetric ice fractions of 020 At higher ice fractions considerable errors have been found which is ascribed to the fact that the ice slurry behaves no longer as a Newtonian but as a non-Newtonian fluid at higher fractions A study by Kitanovski and Poredoš (2002) has shown that the average ice crystal size and the velocity also influence the viscosity of ice slurries
Appendix B
232
Nomenclature
cp Specific heat (Jkg K) ρ Density (kgm3) h Enthalpy (Jkg) φ Ice mass fraction T Temperature (K or degC) v Specific volume (m3kg) Subscripts app Apparent Greek ice Ice λ Thermal conductivity (Wm K) is Ice slurry micro Dynamic viscosity (Pa s) liq Liquid ξ Ice volume fraction sens Sensible
References
Bel O Lallemand A 1999 Etude drsquoun frigoporteur diphasique 1 Caracteacuteristiques thermophysiques intrinsegraveques drsquoun coulis de glace International Journal of Refrigeration vol22 pp164-174
Dorsey NE 1940 Properties of Ordinary Water-substance in all its Phases Water-vapor Water and all the Ices New York Reinhold Publishing Corporation
Jeffrey DJ 1973 Conduction through a random suspension of spheres Proceedings of the Royal Society London volA335 pp355-367
Kauffeld M Christensen KG Lund S Hansen TM 1999 Experience with ice slurry In Proceedings of the 1st IIR Workshop on Ice Slurries 27-28 May 1999 Yverdon-les-Bains (Switzerland) Paris International Institute of Refrigeration pp42-73
Kitanovski A Poredoš A 2002 Concentration distribution and viscosity of ice-slurry in heterogeneous flow International Journal of Refrigeration vol 25 pp827-835
Hyland W Wexler A 1983 Formulations for the thermodynamic properties of the saturated phases of H2O from 17315 K to 47315 K ASHRAE Transactions vol89 (2A) pp500-519
Tareef BM 1940 Colloidal Journal USSR vol6 p545
Thomas DG 1965 Transport characteristics of suspension VIII A note on the viscosity of Newtonian suspensions of uniform spherical particles Journal of Colloid Science vol20 pp267-277
233
Appendix C Calibration of Heat Exchangers
In this thesis four different heat exchangers have been used to study ice crystallization phenomena The first two are vertical tube-in-tube heat exchangers that have been used for fluidized bed experiments The third one is a tube-in-tube heat transfer coil that has been applied for the ice slurry melting experiments described in Chapter 8 The final heat exchanger is a scraped surface heat exchanger that consisted of a crystallization tank with a scraped cooled bottom plate
In all four heat exchangers temperature and flow rate measurements have been used to determine characteristic parameters such as heat flux heat transfer coefficients and wall temperatures To be able to calculate these parameters heat uptake from the surroundings friction losses and heat transfer models were verified or determined during sets of calibration experiments For the inner tube of the tube-in-tube heat transfer coil also a pressure drop model was determined
This appendix describes the four heat exchangers used in this thesis in detail and presents the calibration methods and results
C1 Small Fluidized Bed Heat Exchanger
C11 Dimensions
The small fluidized bed heat exchanger consists of two identical tube-in-tube heat exchangers made of stainless steel with a transparent section in between (see Figure C1) A 34 wt potassium formate solution flows through the annuli of the heat exchanger and is able to cool the inner tube in which the fluidized bed is located The outer tube is well insulated to reduce heat uptake from the surroundings
The most important dimensions of the small fluidized bed heat exchanger are listed in Table C1
Table C1 Dimensions of the small fluidized bed heat exchanger Dimension Value Inside diameter of inner tube Diinner (m) 427 10-3 Outside diameter of inner tube Doinner (m) 483 10-3 Thickness inner tube δinner (m) 277 10-3 Inside diameter of outer tube Doouter (m) 548 10-3 Outside diameter of outer tube Diouter (m) 603 10-3 Thickness outer tube δouter (m) 277 10-3 Hydraulic diameter annulus Dhyd (m) 653 10-3
Heat transfer length per part L1 (m) 214 Length of one part L2 (m) 234 Length of transparent section L3 (m) 020 Total outside heat transfer surface inner tubes (m2) 0649
Figure C1 Schematic overview of small fluidized
bed heat exchanger