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Rheology, flow behaviour and heat transfer of ice slurries:

a review of the state of the art

V. Ayel, O. Lottin*, H. Peerhossaini

Laboratoire de Thermocinetique, CNRS UMR 6607, Ecole Polytechnique de lUniversite de Nantes, BP 50609, 44306 Nantes, France

Received 25 October 2001; received in revised form 23 February 2002; accepted 26 February 2002

Abstract

This paper reviews recent studies on rheology, flow behaviour and heat transfer of two-phase aqueous secondary

refrigerants (ice slurries). The difficulties in measuring their rheological properties for which standard rheometers prove

to be poorly adapted are analysed. Special attention is paid to vane-in-cup rheometers, which can make it possible to

determine the yield stress. Pressure losses in cylindrical tubes have been measured by many authors. Even if the results

generally agree with two-phase flow theory, divergences in sensitivity to the solid fraction can be observed. The strati-

fication observed by some authors for small-Reynolds-number flows and its effects on the pressure drop are addressed.

In a last part, information concerning numerical values of the heat transfer coefficient of ice slurries are summarized.

Heat transfer coefficients depend on many parameters, but are largely influenced by the flow regime. A geometry of

heat exchanger is then proposed, which has already been effective for single-phase flows, and which may enhance heat

transfer in ice slurry flows.# 2002 Elsevier Science Ltd and IIR. All rights reserved.

Keywords: Two-phase secondary refrigerant; Ice slurry; Rheology; Heat transfer; Pressure loss; Survey

Rhe ologie, pertes de charge et transferts de chaleur des coulis

de glace : une revue de le tat de lart

Re sume

Cet article passe en revue des etudes recentes concernant la rheologie, le comportement des ecoulements et le transfert

thermique des frigoporteurs diphasiques aqueux (coulis de glace). Les difficultes de mesure des proprietes rheologiques,

pour lesquelles des rheome`tres courants save`rent mal adaptes, sont analysees. Une attention particulie`re est portee sur lesrheome`tres du type vane-in-cup qui permettent de determiner la contrainte seuil dun materiau. Les pertes de charge

lineaires dans des canalisations cylindriques ont ete mesurees par de nombreux auteurs. Meme si les resultats sont

generalement conformes a` la theorie des ecoulements diphasiques, on peut observer des divergences quant a` la sensibilite a`

la concentration en glace. La stratification, observee par quelques auteurs pour des ecoulements a` faible nombre de Rey-

nolds, et ses effets sur la chute de pression sont etudies. Dans une dernie`re partie, les valeurs numeriques du coefficient de

transfert thermique des coulis de glace sont recapitulees. Les coefficients de transfert thermique dependent de nombreux

0140-7007/03/$20.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved.

P I I : S 0 1 4 0 - 7 0 0 7 ( 0 2 ) 0 0 0 1 6 - 6

International Journal of Refrigeration 26 (2003) 95107

www.elsevier.com/locate/ijrefrig

* Corresponding author. Tel.: +33-240-683-153; fax: +33-240-683-141.

E-mail address: olivier.lottin@polytech.univ-nantes.fr (O. Lottin).

http://www.elsevier.com/locate/ijrefrig/a4.3dmailto:olivier.lottin@polytech.univ-nantes.frmailto:olivier.lottin@polytech.univ-nantes.frhttp://www.elsevier.com/locate/ijrefrig/a4.3d

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parame`tres, mais sont fortement influences par le regime decoulement. Nous proposons enfin un type dechangeur de

chaleur, dont lefficacitea deja` eteremarquee pour des ecoulements monophasiques, et il pourrait saverer tout a` fait adapte

a` des ecoulements de coulis de glace. # 2002 Elsevier Science Ltd and IIR. All rights reserved.

Mots cles : Caloporteur diphasique ; Coulis ; Rhe ologie ; Transfert de chaleur ; Chute de pression ; Base de donne es ; Enquete

1. Introduction

Cold distribution by means of secondary refrigerant is

an alternative solution to direct expansion systems,

which require large quantities of refrigerant. Ice slurries

exploit the latent heat of ice and are much more efficient

heat carriers than single-phase fluids; their high apparent

heat capacity results in lower mass flow rates and tem-

perature changes. Also, they are suitable for cold storage.

However, because of the presence of solid particles within

the carrying fluid, the flow of slurries is highly constrained.

It is thus necessary to determine the rheological nature of

these fluids for a better flow control in distribution plants.

At the present state of the art, the data and opinions ofdifferent authors are often divergent, so that reliable

conclusions on the rheology and flow behaviour of ice

slurries are difficult to draw. However, the technological

advantages of ice slurries and their potential economic

impact are great enough to warrant a critical review of

past work, the suggestion of new research areas, and the

search for new design geometries.

This paper first focuses on the rheology of the liquid

ice mixtures and assesses recent studies on ice slurries.

In the second part we address the flow and pressure

losses of slurries in cylindrical pipes by examining some

previous work on the subject. Then, the stratification

phenomenon observed by some authors for small-Rey-

nolds-number flows and its effects on pressure drops are

treated. In the last part we study the numerical values

and the evolutions of the heat transfer coefficients in

rectilinear cylindrical type geometries and we propose a

promising design geometry that enhances transport

phenomena between the ice slurries and a solid wall.

2. Rheological behaviour of ice slurries

Liquidsolid mixtures are often considered as non-

Newtonian; Table 1 gives examples of non-Newtonian

models that can be used to characterise slurries. Itshould be noted that some of these models could be

satisfactory on a restricted range of shear rates but

completely inappropriate on wider ranges.

Common to most of these models is a yield stress

below which no flow is observed. The value of the yield

stress depends strongly on the solid particle fraction:

Nguyen and Boger [1] suppose that the yield stress is

linked to the interaction and attraction among particles,

giving rise to structures larger than the primary crystals;

the yield stress then represents the breaking limit of these

structures. Similarly, Barnes [2] suggests that the yield

stress is a consequence of microstructural reorganization

Nomenclature

C thermal conductivity (W m1 K)

d particle diameter (m)

D tube diameter (m)

f friction factor

Fr Froude number

Gz Graetz number

h heat transfer coefficient (W m2 K)

H height of the vane (m)

He Hedstrom number

K constant of proportionalityL tube length (m)

n power index

Nu Nusselt number

Pr Prandtl number

Re Reynolds number

T torque (N m)

V velocity (m s1)

xs mass fraction of solids (wt.%)

:

shear stress (s1)

P pressure drop (Pa)

l specific heat (J kg1 K1)

viscosity (Pa s)

density (kg m3)

shear rate (Pa)

0 yield stress (Pa)

s volume fraction of solids (%)

Subscripts

ai antifreeze (initial value)b Bingham

c Casson

D-B DittusBolter

exp experimental

m average (for solidliquid mixture)

MAX maximum

s solid

w water-antifreeze (fluid)

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in the stagnant mixture into a structure allowing flows.

In spite of this, the real definition of the yield stress isstill a subject of discussion. For instance, the Bingham

model can lead to an overestimation of the yield stress if

not properly used. One sees in Fig. 1 that if a nonlinear

shear stress and shear rate behaviour is represented by a

linear function f :

, then the apparent yield stress is

higher than its real value. The yield stress cannot be con-

sidered as a fundamental property of the material as long

as its value depends on the model or on the viscometer [1].

3. Measurement of the rheological properties of ice slurries

Rheological measurements with ice slurries require

significant care, making experimentation difficult. There

are many reasons for these difficulties; some specific to

ice slurries and some linked to the general features of

liquidsolid mixtures:

frictions generated by the apparatus during

rheological measurement will induce warming of

the sample and modification of its ice content;

the low viscosity of the carrying fluid requires

small gap to allow accurate measurements; on

the other hand, the relatively large ice crystals

(from 10 mm to a few hundred mm) require muchlarger gap;

the stratification of solid particles, according to

their shape and size, does not permit to keep

samples homogeneous;

end effects (stresses on surfaces that ideally should

not be considered in measurement) are not easy to

take into account in stress-field calculations;

glides between solid particles and wall, or even

within the tested material, may significantly

complicate measurements.

The carrying fluid in ice slurries can generally beconsidered as Newtonian, as can slurries themselves if

the ice mass fraction remains low. Authors are in good

agreement that the ice mass fraction limit separating

Newtonian and non-Newtonian behaviours is between 6

and 15% (Table 2). Christensen and Kauffeld [5] propose

a correlation for apparent viscosity:

w 1 2:5s 10:052s 0:00273exp16:6s

1

Many authors attempted to reduce or suppress stratifi-

cation in ice slurries during rheological measurements:

pumping systems can remove particles from the upper

parts of the apparatus to inject them in the lower parts;

helical grooves may prevent stratification in rotating

viscometers at higher shear rates. Klein et al. [9] studied

the stratification rate of particles according to the height

of the vessel, considering the intermediate level at which

the particle concentration would remain, for some time,

constant and equal to the initial concentration.

Other authors tried to minimize the end effects. In

rotating viscometers for instance, the higher part of the

bob is not immersed in the sample while the lower part

can be dug so as to insert an air bubble during

measurement, which minimises stress transfer to thebob. This method is effective with viscoplastic fluids that

tend to maintain the bubble but will be delicate to

implement with ice slurries [10].

Barnes [11] studied glide in two-phase liquidsolid

mixtures inside viscometers. He explains that a narrow

layer of the carrying fluid free of particles (depletion

zone), establishes between the wall and the fluid. This

depletion zone acts as a lubricant. According to Barnes,

glides can explain the thixotropy phenomenon men-

tioned by some authors. Nguyen and Boger [12] report

that glide effects are all the more significant when the

solid fraction is high or the particles are flocculated. On

Table 1

Rheological non-Newtonian models that can be used for ice

slurries.

Tableau 1Mode`les rheologiques non-newtoniens pouvant etre utilises pour

les coulis de glace

Models Equations

Ostwald K0: n0

:5 0

Bingham 0 B:

> 0:

0 < 0Herschel-Buckley 0 K

0n0

> 0:

0 < 0

Casson 1=20 c

1=2h i2

> 0

:

0 < 0

Fig. 1. Comparison of yield stress values given by a linear

(Bingham) and a non-linear model.

Fig. 1. Comparaison de la contrainte de cisaillement entre un

mode`le lineaire (Bingham) et un mode`le non lineaire.

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the other hand, glide effects can be reduced by using a

larger gap between the rotating and fixed part of the

viscometer or a larger tube diameter. Barnes [13] esti-

mates that the gap must be at least 20 times greater than

the average particle diameter, provided that this dia-

meter stays below 0.25 mm. This ratio increases for

higher particle diameters.

Nguyen and Boger [12] compared the efficiency of

viscometers used with liquidsolid mixtures. The cone

plate and plateplate viscometers are proscribed because

the distance between the cone or the upper plate and the

lower plate is smaller than the particle size. Further-

more, the liquid will be expelled from the lower plate.

According to Yoshimura and Prudhomme [14], who

carried out experiments in order to compare 3 types ofviscometers (parallel plate, bob-in-cup and vane-in-cup

viscometers), the parallel plate viscometers are the least

appropriate apparatus for the determination of the yield

stress. The capillary viscometer is efficient in the sense that

it simulates the flow of the mixture in a pipe. It is also less

affected by the melting of the ice during the process (the

fluid crosses the tube, unceasingly regenerated at the entry

of the viscometer), but the measurements are distorted on

the one hand by the end effects (pressure drops in the input

and output of the tube) and on the other by the glide

effects, which cause the particles to concentrate in the

center of the tube [12].

Vane-in-cup viscometers are sometimes used for

rheological measurements with liquid-solid mixtures.

Their external cylinders are similar to those of bob-and-

cup viscometers but the bob is replaced by vanes (Fig. 2),

generally four of them. According to Barnes [13], vane-

in-cup viscometers have several advantages:

since the inner cylinder is replaced by some uni-

formly moving fluid, glide effects are minimised;

due to their reduced thickness, the vanes create

little disturbance within the sample;

the gap can be sufficiently wide compared to the

size of the particles.

However, vane-in-cup viscometers are well adaptedfor high-viscosity mixtures, which ice slurries are not

when the ice mass fraction is low. Furthermore, one can

often use them only to measure the yield stress; while

the vane rotates at constant speed, one observes a max-

imum value of the torque that corresponds to the tran-

sition between elastic and plastic behaviour. The

relation between the maximum torque and the yield

stress is:

TMAX D 3

2

H

D

1

m

!0 2

Table 2

Rheological behaviour of ice slurries

Tableau 2Comportements rheologiques des coulis de glace

Authors Descriptions of the

slurries

Viscosimeters Results

Bel and Lallemand [3] xs eau.

Jensen et al. [6] xs

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where H is the immersed height and D the external dia-

meter of the vane; m is a parameter characterising the

end effects, equal to 3 or 6 depending on whether or not

the vane is completely immersed in the sample. The

main design parameters for accurate measurements have

been summarised by Nguyen and Boger [15]. Barois et

al. [16] added a vibrating membrane inside a vane-in-cup viscometer to reduce stratification.

4. Distance flow and pressure drop in cylindrical ducts

According to Darby [17], the flow of suspensions can

be classified in three categories:

in homogeneous flows, the particle fraction is

large and the average particle diameter is much

smaller than the characteristic length of the

pipe; the dispersion can then be considered as auniform single-phase fluid.

In heterogeneous flows, the average particle size

is no longer negligible compared to the charac-

teristic length of the pipe, and the particle

fraction is lower or the density of the solid

phase is very different from that of the carrying

fluid.

In saltated flows, two phases of different den-

sities are clearly separated and the particles

accumulate at the bottom or at the top of the

pipe, thus constituting a bed of suspensions.

Generally, practical flows are often a combination of

homogeneous and heterogeneous flows.

Quemada [18] listed the parameters that differentiate

these three kinds of flows:

the geometric characteristics of the suspensions:

the particle drag coefficient depends on a Rey-

nolds number whose characteristic length may

include one or more ratios describing particleshape and orientation in the flow;

the concentration of particles: interactions

increase with solid fraction, having an effect

starting from 2 to 3% [19];

the density: adding solid particles changes the

density of the mixture, the Reynolds number

and the stratification speed. It has a dominating

influence on the saltation of the particles.

The model suggested by Darby [17] indicates that in a

constant-velocity flow, the pressure drops are always

higher in a two-phase mixture than in the carrying fluidonly. In Fig. 3, the dashed line represents the evolution

of the linear pressure drop according to the velocity in a

pipe filled with the carrying fluid only. There is a cri-

tical deposit rate Vm2 that ensures minimum displace-

ment of the solid particles and can be interpreted as the

transition between saltated and heterogeneous flows.

This transition velocity was observed with most of the

slurries and corresponds to a minimum of the pressure

drop. It may be useful to approach this velocity, but at a

velocity higher than the critical deposit rate, in order to

minimise the pressure drop and to avoid stratification.

Concerning the ice slurries flows, it is noted (Table 3)

that in many cases [3,4,6,2027], the evolution of the

pressure drop corresponds well to the general behaviour

Fig. 2. Schematic diagram of a vane-in-cup viscometer.

Fig. 2. Diagramme dun viscosime`tre de type vane-in-cup.

Fig. 3. Pressure gradient plotted versus mean velocity for the

various flow regimes (xs represents ice fraction) [17].

Fig. 3. Pertes de charge en fonction de la vitesse moyenne pour

differents regimes decoulement xs

represente la fraction de

glace) [17].

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described by Darby [17]. Jensen et al. [6] and Bel and

Lallemand [3] observed no change in the pressure drop

with changing ice mass fraction provided it remained

lower than 1015%. For Winters and Kooy [27], the

pressure drop measured with the slurry is even veryclose to that obtained with the carrying fluid for ice

mass fraction not exceeding 22%. Snoek and Joanis [21]

made the assumption that the ice slurry friction factor

could be determined from the knowledge of the water-

only and waterglycol (liquid only) friction factors, and

developed a correlation for this. On the other hand, in

some works, like these of Liu et al. [28], Knodel and

France [30] and Knodel et al. [31], the pressure drop

decreases with an increase in the ice fraction, which

seems to be contrary to general studies on two-phase

flows. Knodel et al. explain this observation by a

decrease in the flow turbulence caused by interactions

between the fluid and the particles. According to them,

there is a critical ice fraction beyond which the pressure

drop remains constant (30% in this case). Knodel et al.

[31] found that for small tubes, Reynolds numbers ran-

ging from 38,000 to 74,000 and when xs was above 4%,

they could write:

f

0:18Re0:2 0:946 3

Knodel and France [30] observed that when the pipe

diameters are low, the number of interactions betweenthe wall and particles increases, which tends to intensify

friction and to decrease the damping of turbulence. Liu

et al. [28] confirm this hypothesis and argue that the

solid particles bring a coherent structure to the fluid,

thus decreasing the energy dissipated by viscous fric-

tions. They propose a correlation:

P 15:135V1:931m 0:02227s 4

where Vm is the mean velocity and s is the solid

volume fraction.

5. Flow stratification

The forces acting on a single particle in a flow are

(Fig. 4): the gravitational force, which is toward the

bottom, the buoyancy force, and the drag force of thefluid on the particle; the latter is a function of the velo-

city difference between the solid and liquid phases. Its

direction is mainly that of the flow, with an added

component in the direction opposed to that of the par-

ticle sedimentation (if a movement of sedimentation is

induced). The drag force can slow down or even prevent

saltation. Some other forces acting on the particles that

are more complex to determine arise from the inter-

action among particles and between the particles and

the walls. Their direction is random, but it tends to slow

down the saltation rate, and due to the non-uniformity

of the movements it acts on the whole field of saltation.

The stratification velocity thus depends on several

parameters:

The particle size and shape: these parameters act

primarily on the drag coefficient of the fluid on

the ice crystals. The drag force can be reduced to

its simpler expression for rigid spherical particles,

Table 3 (continued)

Authors Description of the slurries Geometries Comments

Knodel et al. [31] s

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but their influence is more complex when their

shape is far from spherical. The effect of the par-

ticle shape on the drag coefficient results in a

modification of the critical velocity for saltation.

Generally, this factor is taken into account in cal-

culations by coefficients of nonsphericity.

The particle weight: the average rate of saltationis proportional to the relative density of the

crystals and of the carrying fluid.

The solid concentration: interactions between

particles and the particles wakes slow down the

saltation rate. On the other hand, a high mass

fraction of ice may induce flocculation of the

crystals. Agglomerations of particles will trap

fluid, and the whole will move at sedimentation

velocities higher than those of the single parti-

cles. These formations are controlled by the

nature of the flow. As long as the relative influ-

ence of flocculation and particle interaction willbe unknown, stratification of ice crystals will

remain difficult to describe.

To approach an optimal flow by minimizing the

pressure drop, one has to be as close as possible to the

laminated flow mode (see Fig. 3). However, this presents

the risk that the flow will be in a laminated or stationary

bed mode, which would increase solid friction between

the particles and the walls. According to Takahashi et

al. [25] and Kawashima et al. [33], pressure drops in ice

slurry flows are higher than those in pure liquid at low

speeds, because ice particles accumulate at the top of the

horizontal pipe, sometimes even forming solid blocks.

These phenomena disappear at higher speeds (Vm> 1.6

m s1). For Inaba [22], the effects of the drag force

depend on the particle size: the smaller particles accu-

mulate at the top of the tube while the larger ones, on

which the drag is more significant, remain below. Snoek

et al. [21] observed that below the velocity Vm2, solid

friction is more significant, and the bypass section of the

carrying fluid (with Vfluid> Vice) is reduced. Pressure

drops thus increase even at reduced speeds. In the same

way, Winters and Kooy [27] studied Vm2, which seems

to be dependent on xs (for xs=7.5%, Vm2=0.12 m s1,

while for xs=12%, Vm2=0.18 m s

1). They alsoobserved contacts between the ice particles and the wall

for flow rates lower than 0.3 m s1.

Sasaki et al. [34] studied the particle-fluid relative

velocity, which tends to increase with increasing average

flow speed, thus preventing particle settlement.

6. Cooling coil performance

Table 4 gathers the informations available on the heat

transfer coefficient h or the Nusselt number Nu

[3,5,6,23,29,31,3540]. As a general rule, the majority of

observations show that h or Nu increase with the flow

rate (velocity, mass flow rate or Reynolds). Only Knodel

et al. [31] an Snoek and Bellamy [35] observed that Nu

decreases with increasing xs. Meewisse and Ferreira [23]

propose a correlation for the calculation of the Nusselt

number in laminar flow of ice slurries in circular tubes:

Nu 38:3Gz0:15x0:52s 5

where Gz RePrD4z

is the Graetz number, and z is the

axial distance along the tube. Re and Pr are respectively

the Reynolds and the Prandtl numbers, calculated with

the average properties of the ice slurry. The viscosity

was evaluated with the correlation (1) of Christensen

and Kauffeld [5]. This equation is valid for 35%:

Nu

Nuw 1 0:103xs 2:003Re

0:192 30xs30 x

0:339 Re10000

s 12

where Re and Nu are calculated with the properties of

the ice slurry, and Nuw is calculated with the properties

of the carrier fluid, using the average velocity. Eq. (1)

was used to evaluate the viscosity. This model estimates

the measured values within 35%, and 75% of the

measurements are estimated within 30%. The validity

of the correlation is limited to tubes with an inner dia-

meter between 18 and 25 mm. Christensen and Kauffeld

[5] suggest that the melting of ice crystals close to the wallreduces the temperature gradient through the boundary

layer, which can explain the increase in the heat transfer

coefficient.

Ben Lakhdar et al. [39] and Inaba [22] studied the

evolution of the Nusselt number with the length covered

in the heat exchanger. The value of Nu decreases slowly

with the progression of the slurry in the heat exchanger,

and tends towards the values of Nu corresponding to

the single phase flow. Kawanami et al. [40] studied the

evolution of the heat transfer coefficient h in a heat

exchanger, formed by a tube bended at 180 in a vertical

plane. At the concave wall, h increases with the velocity

of the flow. For velocities higher than 0.4 m s1, h

decreases according to the angle of the bend, whereas

for velocities close to 0.1 m s1, h decreases then stays

constant. At the convex wall, h decreases according to

the angle of the bend, but for velocities close to 0.1 m

s1, it tends to be in general greater than that for the

other velocity conditions at the area from 0 to about

120. The friction of ice crystals against the wall, caused

by their stratification, could explain these phenomena.

The increase in the velocity of ice slurry flow causes an

increase in the accumulation of the ice particles in the heat

transfer area of the concave wall, as well as the decrease

of the depletion zone thickness, owing to the mixing

effects based on both the centrifugal and the inertial

forces.

Flow geometries that can combine the low-pressurelosses of laminar flows and efficient mixing would be

particularly adapted to flows of ice slurry. An example

of such geometries consists of a series of bent tubes with

planes of curvature at a 90 angle with respect to the

previous tube, thus creating alternating Dean flows

whose chaotic characteristics can enhance mixing and

heat transfer (Fig. 5). The counterrotating secondary

flows called Dean roll-cells appear in curved duct due

to the centrifugal force, that pushes the fluid particles

against the concave wall. If the curvature plane of the

curved ducts maintains the same orientation (as in a

helical coil), the flow has the same structure at the

Table 4 (continued)

Authors Description of the

slurries

Geometries Comments

(ethanol) Di=38 mm

L=0.5 m

h (xs=0%)=500 W/m2C, h (xs=35%)=

2000 W/m2 K.

Ben Lakhdar et al. [39] xs

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entrance and at the exit of the coil, i.e. a regular Dean

flow. On the other hand, if each bend is turned 90

compared to the preceding one, the Dean roll-cells

formed in an elbow are broken by the subsequent one,

which imposes a different orientation on them. In a coil

composed of a series of elbows with alternating curva-

ture planes, the fluid particles follow chaotic trajec-

tories, even for small Reynolds numbers, allowing better

homogenisation of the particle distribution and tem-

peratures within the fluid. This phenomenon increases

heat transfer considerably, provided that the Reynolds

numbers stay low [41]. It is then possible to improve the

efficiency of heat exchangers used with high viscosity

fluids without increasing the velocity and the pressure

losses.

In a numerical study, Mokrani et al [42] used a reduced

model to compute the flow in a helical and a chaotic coil,

and observed the distribution at the exit plane of 40,000

passive tracer injected at a point in the input section. Fig. 6

shows that for a tracer injection at the centre of the input

section and a Reynolds number of 40, the mixing is far

more homogeneous in the chaotic coil than in the helical

one, and this only after 10 bends. At the output of the

chaotic coil the particles are in close to total apparent

disorder, whatever the injection position of the particles.

On the other hand, in the helical coil the particles follow

the trajectories of the regular Dean roll-cells, which do

not evolve in the flow direction.

Mokrani et al. [43] also found experimentally that at

very low Reynolds numbers (Re=63) the chaotic coil

has a heat transfer efficiency 28% greater than the heli-

cal coil. This type of coil could probably be adapted to

the flows of ice slurries, either to allow efficient mixingof ice particles all along the distributing pipes, or to

optimize heat transfer in some heat exchangers. Fig. 7

shows that the temperature profile at the outlet of the

chaotic coil, for a Reynolds number of 63, is flat and does

not exceed a temperature difference of 3 K, while it reaches

12 K for the helical coil under the same conditions.

7. Conclusions

The rheological study of ice slurries suffers from a ser-

ious handicap, since no current apparatus has really

Fig. 6. Images of the injected tracer after passing through dif-

ferent numbers of bends, for both helical and chaotic configur-

ations [42].

Fig. 6. Images du filet de traceur deforme par lecoulement en

fonction du nombre de coudes et de la position dinjection, pour la

configuration helicodale et la configuration chaotique [42].

Fig. 7. Temperature profile at the exit of the two heat exchan-

gers for two measurements, and for Reynolds number 63 [43].

Fig. 7. Profils de temperature a` la sortie des deux e changeurs de

chaleur pour deux mesures et pour un nombre de Reynolds de 63

[43].

Fig. 5. Dean-roll-cells in a curved duct.

Fig. 5. Cellules de Dean dans une conduite incurvee.

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proved satisfactory: vane-in-cup rheometers, which seem

in general well adapted to liquidsolid two-phase systems,

do not allow precise measurement of any rheological

characteristics except the yield stress. These difficulties are

due to the low viscosity of the carrying fluid, to the size of

the crystals and to the stratification problem (which

could be perhaps be solved by small recirculationpumps). The capillary rheometer can be used to measure

the linear pressure drops in a cylindrical pipe according

to the average speed of the mixture and to the ice frac-

tion, as many authors did, but viscosity measurements

will be disturbed by glide effects and end effects. In any

case, opinions seem to converge on the proposition that

the pressure drop increases with average speed and tends

to approach that of the carrying fluid. On the other hand,

different authors do not agree on the behaviour of ice

slurries with ice fraction, probably because of different

flow reorganisation patterns in different regimes. Studies

of the stratification regimes confirm Darbys [17] theoryof a critical deposit rate that increases as the solid

fraction increases.

Although the experimental conditions are different,

the heat transfer coefficients seem to follow the same

behaviour overall, from the point of view of the orders

of magnitude of the numerical values as well as their

evolutions according to various parameters.

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