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Accepted Manuscript

Rheological characterization of sorbet using pipe rheometry during the freezing

process

Marcela Arellano, Denis Flick, Graciela Alvarez

PII: S0260-8774(13)00252-5

DOI: http://dx.doi.org/10.1016/j.jfoodeng.2013.05.017

Reference: JFOE 7391

To appear in: Journal of Food Engineering

Received Date: 16 January 2013

Revised Date: 6 May 2013Accepted Date: 16 May 2013

Please cite this article as: Arellano, M., Flick, D., Alvarez, G., Rheological characterization of sorbet using pipe

rheometry during the freezing process, Journal of Food Engineering (2013), doi: http://dx.doi.org/10.1016/

j.jfoodeng.2013.05.017

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Rheological characterization of sorbet using pipe rheometry1

during the freezing process.2

Marcela Arellanoa,b,c,d, Denis Flickb,c,d, Graciela Alvareza,1.3

aIrstea. UR Gnie des Procds Frigorifiques. 1 rue Pierre-Gilles de Gennes CS 10030, 92761 Antony Cedex, France4bAgroParisTech. UMR 1145 Ingnierie Procds Aliments. 16 rue Claude Bernard, 75231 Paris Cedex 05, France5

cINRA.UMR 1145 Ingnierie Procds Aliments. 1 avenue des Olympiades, 91744 Massy Cedex, France6dCNAM. UMR 1145 Ingnierie Procds Aliments. 292 rue Saint-Martin, 75141 Paris Cedex 03, France7

8

Abstract9

Sorbet produced without aeration is a dispersion of ice crystals distributed randomly in a freeze-10

concentrated liquid phase. The rheological properties of this suspension will be affected by the viscosity of the11

continuous liquid phase and the volume fraction of ice crystals. The knowledge of the viscosity of sorbet is12

essential for the improvement of product quality, the selection of process equipment, and for the optimal design13

of piping systems. This work aimed firstly, at studying the influence of the ice volume fraction (determined by14

the product temperature) on the apparent viscosity of a commercial sorbet, and secondly, to propose a15

rheological model that describes the evolution of the viscosity of the product as a function of the ice volume16

fraction. The rheology of sorbet was measured in situ by means of a pipe rheometer connected at the outlet of a17

continuous scraped surface heat exchanger (SSHE). The pipe rheometer was composed of a series of pipes in18

PVC of different diameters, making it possible to apply a range of apparent shear rate from 4-430 s -1. The flow19

behaviour index of sorbet decreased as the temperature of the product decreased, the effect of which indicates20

that the product becomes more shear thinning as the freezing of sorbet occurs. The consistency coefficient and21

therefore the magnitude of the apparent viscosity of sorbet increased with the decrease in product temperature22

and with the increase of the ice volume fraction. Results also showed that the rheological model described the23

experimental data within a 20% error.24

25

Key words: Apparent viscosity; Pipe rheometry; Draw temperature; Ice volume fraction; Freezing; Scraped surface heat exchanger.26

1 Corresponding author. Tel.: +33 140 96 60 17; Fax: +33 140 96 60 75.E-mail address: [email protected]


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1. Introduction27

The characterisation of the rheological properties of sorbet has significant applications28

throughout the manufacturing process of frozen desserts. The understanding of the influence29

of changes in product microstructure on its rheological properties is necessary for the30

improvement of the freezing process and the quality of the product. The knowledge of the31

viscosity of the product is also essential for the selection of process equipment, and for the32

optimal design of piping systems. The freezing of sorbet is carried out in a continuous scraped33

surface heat exchanger (SSHE) or freezer. Once the freezing of sorbet starts and ice crystals34

are being formed, the liquid sorbet mix starts to freeze concentrate and the viscosity of this35

continuous liquid phase increases (Burns and Russell, 1999; Goff et al., 1995).36

Simultaneously, the ice crystals are dispersed in the liquid sorbet mix by the rotation of the37

scraping blades, modifying the fluid flow field and increasing the viscosity of sorbet. Sorbet38

exiting from the freezer at a draw temperature between -4 to -6 C, contains roughly 20-40%39

of the total amount of water in the form of ice crystals, which are suspended in a viscous40

liquid phase composed of water, sugar, stabilizers (polysaccharides) and salts. At this point,41

the product must have an adequate viscosity to be pumped for moulding and packaging.42

Further on, the product is hardened in a blast freezer to attain a core temperature of -18 C (Cook43

and Hartel, 2010), where roughly 80% of the amount of water is frozen (Marshall et al., 2003).44

The measurement of the viscosity of sorbet and ice cream is highly complex, because45

the product is temperature sensitive and it behaves as a non-Newtonian shear-thinning fluid46

(Burns and Russell, 1999; Haddad, 2009). A number of studies in the literature, based on47

oscillatory thermo-rheometry for the analysis of the rheology of ice cream, have reported48

viscoelastic behaviour which is strongly related to the ice crystal microstructure (Goff et al.,49

1995; Granger et al., 2004; Wildmoser et al., 2004). In order to improve the online control of50


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product quality, it is necessary to investigate further the rheology of frozen desserts flowing51

directly from the SSHE.52

Pipe rheometers have been used to measure the apparent viscosity of ice cream and53

sorbet in situ and online during the freezing process (Cerecero, 2003; Martin et al. 2008;54

Elhweg et al., 2009). Pipe rheometers are generally composed of a set of pipes of different55

diameters, through which the product flows under pressure. The relationship between the56

shear rate and the shear stress is determined from volumetric flow rate and pressure drop57

measurements. The challenge of pipe rheometry measurements lies on the difficulties of58

controlling a steady temperature and flow conditions. Furthermore, the effects of wall slip59

behaviour and viscous dissipation must be evaluated so as to ensure accurate rheological60

measurements.61

Apparent wall slip behaviour occurs in multi-phase systems due to the displacement of62

the disperse phase away from solid interfaces. This creates a layer of fluid near the wall63

region that has a lower viscosity and a higher velocity gradient as compared to the bulk of the64

product, forming a layer of high shear (Martin and Wilson, 2005). This apparent wall slip65

modifies the flow velocity profile and the shear rate gradient, the effect of which leads to66

inaccurate rheological measurements. Mooney (1931) proposed a method to identify the slip67

wall behaviour. This technique consists on tracing the flow behaviour curves (shear stress68

versus shear rate) for different pipe diameters and different flow rates. In the absence of wall69

slip, these curves overlap. However, a significant separation of these curves reveals the70

existence of wall slip. Apparent wall slip behaviour has been observed in multi-phase food71

products such as fruit purees (Balmforth et al., 2007), tomato ketchup (Adhikari and Jindal,72

2001) and coarse food suspensions of CMC-green pea solutions. More recently Martin et al.73

(2008) and Elhweg et al. (2009) reported some evidence of apparent wall slip in ice cream,74


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but wall slip effects were neglected due to insufficient data and control of pressure to discern75

clear trends.76

The phenomenon of viscous dissipation refers to the mechanical energy dissipated77

during the flow of the fluid through the pipe which is converted into internal energy, increasing78

the temperature of the product along the pipe axis (Winter, 1977). Thus, due to the high shear79

rates obtained near the pipe wall, the temperature of sorbet will increase near the wall region,80

leading to the decrease in the viscosity of the product, increasing the fluid flow velocity and81

consequently leading to a higher wall shear rate. The impact of viscous dissipation can be82

assessed by evaluating the Nahme dimensionless number (Na ), which indicates the degree at83

which the temperature rise will affect the viscosity of the product (Macosko, 1994). The effect84

of viscous dissipation becomes significant when Na > 1 (Macosko, 1994). Elhweg et al.85

(2009) reported that the phenomenon of viscous dissipation in ice cream was significant for a86

certain range of product temperatures (-6 to -12 C) and shear rates (0.3 to 360 s-1).87

Sorbet produced without aeration is a dispersion of ice crystals distributed randomly in88

a freeze concentrated liquid phase. The flow of this suspension will be affected by the89

viscosity of the continuous liquid phase, the volume fraction ( ) of ice crystals, crystal-90

crystal interactions and ice crystal shape. A number of theoretical and empirical equations91

have been developed to describe the viscosity of Newtonian suspensions (Einstein, 1906;92

Mooney, 1951; Krieger and Dougherty, 1959; Thomas, 1965; Batchelor, 1977). A summary93

of the models available in the literature is shown in Table 1.94

Most of these models are extended versions of the expression developed by Einstein95

(1906) to predict the evolution of the viscosity of a Newtonian suspension of rigid spheres96

( < 0.02), as a function of the volume fraction of the suspended spheres and of the97

viscosity of the continuous phase l, written as:98


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( )5.21= l (1)

This model takes only into account the Brownian movement of the spheres, neglects particle-99

particle interactions, and is only valid in the case of dilute solutions.100

For higher particle concentrations ( < 0.625) and a range of particle size between101

0.099 to 435 m, Thomas (1965) proposed a semi-empirical expression which predicts the102

viscosity of Newtonian suspensions as a function of the viscosity of the continuous phase l103

and the volume fraction of the suspended rigid spheres, expressed as:104

( )( ) 6.16exp00273.005.105.21 2 +++= l (2)

In this model, the first three terms inside the parentheses account for the effect of the105

hydrodynamic interactions of spheres and particle-particle interactions, whereas the106

exponential term considers the rearrangement of particles as the suspension is sheared107

(Thomas, 1965). This model has been widely used to predict the viscosity of ice slurries (Ayel108

et al., 2003; Hansel, 2000), but has been reported to overestimate the viscosity of ice slurries109

when the ice concentration exceeds > 0.15 (Hansel, 2000). Haddad (2009) compared110

experimental viscosity data obtained in a scraped rheometer, during the batch freezing of a111

30% sucrose solution with the predicted values by Thomas equation (Eq. 2). Results showed112

that the Thomas equation underestimates by 60% the viscosity of non-Newtonian shear-113

thinning suspensions of ice crystals.114

The aims of the present work are, firstly, to study the influence of the temperature of115

the product and thus of the ice volume fraction (icev.

), on the apparent viscosity (app

) of a116

commercial sorbet at different stages of the freezing process (i.e. during the flow of the117

product through the SSHE, through a pipe, and through a product filling machine). Secondly,118

to propose a rheological empirical model to describe the evolution of the viscosity of the119

product as a function of the ice volume fraction, that may help ice cream and sorbet120


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manufacturers to improve the online control of product quality. The presence or absence of121

wall slip and viscous dissipation were also assessed in order to assure the reliability of the122

experimental data used to fit the rheological model.123

124

2. Materials and methods125

126

2.1. Sorbet freezing and operating conditions127

The working fluid used in these experiments was an ultra high temperature pasteurized128

lemon sorbet mix (14.6% w/w sucrose, 8% w/w fructose, 0.09% w/w dextrose, 3% w/w129

lemon juice concentrate 60 Brix, 0.5% w/w locust bean gum / guar gum / hypromellose130

stabiliser blend). The mix was stored at 5 C for 24 h prior to use. Freezing of the mix was131

carried out in a laboratory scale continuous pilot SSHE (WCB Model MF 50). A schematic132

representation of the experimental platform is shown in Fig. 1. Product flow rate was adjusted133

within a range of 0.007 to 0.014 kg.s-1 (25 to 50 kg.h-1). The mix flow rate was determined by134

weighing the product exit stream during a given period of time. The accuracy of this135

measurement was determined to be 9.2x10-5 kg.s-1. The dasher speed of the SSHE was fixed136

at 78.5 rad.s-1 (750 rpm). The dasher speed was measured by means of a photoelectric137

tachometer (Ahlborn, type FUA9192) with an accuracy of 0.105 rad.s-1 (1 rpm). The138

temperature of sorbet was varied by adjusting the refrigerant fluid temperature (r22,139

chlorodifluoromethane) evaporating in the cooling jacket of the SSHE. A calibrated type T140

(copper - constantan) thermocouple with an accuracy of 0.2 C was fixed with conductive141

aluminium tape on the external surface wall of the cooling jacket, so as to measure the142

evaporation temperature of the refrigerant fluid. The exterior of the exchanger jacket was143

insulated with 2 cm thick polystyrene foam in order to prevent heat transfer with the144

environment.145


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The operating conditions under which the rheological measurements were carried out are146

given in Table 2. No aeration was employed for any of the rheological measurements. For the147

mix flow rate of 0.007 kg.s-1 (25 kg.h-1), 4 replicates were performed for each ice volume148

fraction. During each of these replicates, 4 readings of pressure drop and flow rate were149

performed, yielding a total of 16 experimental points for each pipe diameter. These150

experimental points were taken into account for the fitting of the rheological model. For the151

mix flow rates of 0.010 kg.s-1 (35 kg.h-1) and 0.014 kg.s-1 (50 kg.h-1), 2 replicates were152

performed for each ice volume fraction. During each of these replicates, 4 readings of153

pressure drop and flow rate were performed, yielding a total of 8 experimental points for each154

pipe diameter. These experimental points were performed to verify the presence or absence of155

wall slip behaviour, but were not taken into account for the fitting of the rheological model.156

157

2.2. Pipe rheometry measurements158

Frozen sorbet was pumped from the SSHE, first, into a contraction/enlargement pipe,159

and then into an instrumented pipe rheometer (cf. Fig. 1). The contraction/enlargement pipe160

was composed of 5 pipes in copper plumbing of 0.10 m length each and of different internal161

diameters (d1c/e=0.025m; d2c/e=0.0157m; d3c/e=0.0094m; d4c/e=0.0157m; d5c/e=0.025m). This162

contraction/enlargement pipe was used to pre-shear the sorbet before it entered into the pipe163

rheometer, so as to prevent any thixotropic behaviour and to obtain repeatable measurements.164

The pipe rheometer was composed of sets of 4 pipes in clear polyvinyl chloride (PVC) of165

different internal diameters (d1=0.0272m, d2=0.0212m, d3=0.0167m, d4=0.013m, d5=0.01m,166

d6=0.0058m) connected in series, making it possible to apply an apparent shear rate range of 4167

< w& < 430 s-1. All pipes were insulated with polystyrene foam of 2 cm thickness in order to168

reduce heat gain.169


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The power requirements for the piston pump of the SSHE to keep the flow of sorbet170

depend on the viscosity of the product, the flow rate, and the pipe length and diameter. For a171

given flow rate and pipe diameter, the increase of the ice volume fraction ( ..icev ) led to the172

increase of the apparent viscosity of the product, and therefore to an increase in the power173

requirements of the piston pump. For this reason, the use of all the pipes was not possible for174

all experimental conditions. Therefore, the pipes of diameters d1 to d6 were used to measure175

the rheology of the liquid sorbet mix and the frozen sorbets with ..icev of 0.058 and 0.11,176

diameters d1 to d5 for frozen sorbet with ..icev of 0.17, diameters d1 to d4 for ..icev of 0.23 and177

0.31, and d1 to d3 for ..icev of 0.39.178

Two piezometric rings, located at the measuring points of each pipe, made it possible179

to measure the pressure drop within each pipe over a length of 0.5 m. The piezometric ring180

was composed of small holes (2mm of internal diameter) that formed a concentric circle181

around the centre of each pipe. Each set of holes were connected together in an annular space182

to give the average pressure of the fluid as it passed through the measuring points of each183

pipe. For these measurements the pressure drop was assumed to be linear throughout each184pipe. Pressure drop measurements were performed by liquid column manometers with an185

accuracy of 2% of the measured value. Pressure drop and flow rate were used to calculate186

the shear stress and the shear rate by Eq. 5 and 6, respectively, as shown in the following187

section.188

The draw temperature of sorbet was measured as it flowed through the pipes of the189

rheometer by means of calibrated Pt100 probes (accuracy of 0.1 C) located in the centre of190

the inlet and outlet pipes of the rheometer (cf. Fig. 1). The thermal steady state in the pipe191

rheometer was achieved by pumping sorbet into it, until the measured temperatures at the inlet192

and outlet of the rheometer were brought to the desired experimental draw temperature.193

194


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2.2.1. Apparent viscosity calculations195

The theory of the pipe rheometer is based on the Rabinowitsch-Mooney equation (Steffe,196

1996) which considers the following assumptions: (1) the fluid flow is laminar and steady, (2)197

there is no slip at the wall of the pipe, (3) the fluid properties are independent of pressure and198

time, and (4) the temperature is constant throughout the whole system. The general Rabinowitsch-199

Mooney equation relating shear rate ( )w& and shear stress ( )w at the wall of the pipe is given by:200

( )( )( )

( )( )

+

==

w

3

w3ww lnd

R/Vlnd

R

V3f

&&

& (3)

where V& is the volumetric fluid flow rate,R the inner radius of the pipe and w the shear stress201

given by:202

( )L

RPw 2

= (4)

where P is the pressure drop measured over a fixed length L of a horizontal pipe.203

Eq. (3) can be solved and simplified in terms of the definition of a power law fluid204

( ( )nww k &= ), then the apparent shear rate w& may be determined by:205

+=

3wR

V4

n4

1n3

&

&

(5)

The plot of the power law relation between ( )wln and ( )wln & fits a straight line with a206

point slope ( )( ) ( )( )3RV4lndL2RPlndn &= and intercept ( )kln .207

For each draw temperature obtained at a given product flow rate and rotational speed,208

n is calculated by taking into account data from all replicates performed at the same209

operating condition.210

A regression analysis of ( )w versus ( )w& to fit a power law model was used to211

characterize the apparent viscosity of non-Newtonians shear-thinning fluids, such as sorbet:212


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1= nwapp k & (6)

where n is the flow behaviour index, w& the shear rate and k is the consistency index. The213

consistency index k represents the apparent viscosity of the product at a shear rate of 1s-1.214

In order to understand the changes of the apparent viscosity of sorbet at different215

stages of the freezing process, we can express the viscosity of the product at the shear rates at216

which the sorbet is submitted during processing. The average shear rate & within the SSHE217

can be calculated with the model proposed by Leuliet et al., (1986) written as follows:218

)44.2345.110213.3( 1754.003.07115.04 RnL NnVVn += &&& (7)

where nL is the number of scraping blades andR

N the rotational speed of the dasher. The shear219

rate within the outlet pipe of the SSHE and within a sorbet cup filling machine were determined by220

Eq. 5 considering a pipe of diameter doutlet.pipe= 0.0225 m and dfilling.pipe = 0.03 m, respectively.221

222

2.3. Ice volume fraction calculations223

Assuming the thermodynamic equilibrium between ice and the solution of solute224

(sweeteners content), the ice mass fraction in sorbet ..icem (kg of ice / kg of sorbet) can be225

determined from the freezing point curve of sorbet mix, which is a function of the mass226

fraction of solute msw in the residual liquid phase. This curve was previously determined by227

means of differential scanning calorimetric (DSC) measurements as reported by Gonzalez228

(2012). The expression that characterizes the freezing point curve and links the solute mass229

fraction at the equilibrium temperature is given by the following equation:230

( )432 0000529.000167.00202.0137.0 TTTTwms = , with T in C (8)

According to a mass balance of solute (sweetener content), the solute is present in231

sorbet mix at a certain initial mass fraction ( imsw . ). Then the solute is concentrated as the232

freezing of sorbet occurs, until it reaches a final mass fraction ( fmsw . ) in the liquid phase. The233


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final liquid phase represents only a fraction ( )..1 icem of sorbet, hence the ice mass fraction234

can be calculated by the following equation:235

( ) ( )

( )Tw

wTww

fms

imsicemfmsicemims

.

....... 11 == (9)

The ice volume fraction (m3 of ice / m3 of sorbet) is then determined as follows:236

ice.m

i

sice.v

= (10)

where s is the density of sorbet and i the density of ice.237

When the experimental conditions of the SSHE were set, the time necessary for the238

whole system to achieve thermal steady state varied between 20 to 30 min. Once the thermal239

steady state was attained, the draw temperature of the product and ice volume fraction data240

were recorded and calculated every 5 seconds for 10 min, by using a program written in241

LabVIEW. Simultaneously, the 4 readings of pressure drop and flow rate were taken within242

a time interval of 10 min.243

244

2.4. Viscous dissipation245

The effect of viscous dissipation on the apparent viscosity of the product app at a246

given apparent shear rate w& , can be assessed by the Nahme number, Na , expressed for the247

pipe flow of a power law fluid as:248

sorbet

21n

w

4

RkNa

+=

& (11)

where is the temperature sensitivity of viscosity defined as ( )( )Tappapp /1 , k and n 249

the consistency and flow behaviour indices, respectively, sorbet the thermal conductivity of250

sorbet, and R the pipe radius (Judy et al., 2002; Elhweg et al., 2009).251


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The thermal conductivity of sorbet is given by the following expression reported by252

Levy, (1981):253

( )

( )

+++

=icewicevicew

icewicevicewwsorbet

.

.

2

22 (12)

where ice is the evolution of the thermal conductivity of ice as a function of the temperature254

of the product and calculated by the following expression (Levy, 1982):255

( ) 156.1310975.524.2 Tice += , with T in C (13)

w is the thermal conductivity of the solute solution, determined by the Baloh relation (1967):256

( ) ( )

( )263263

103.2108.0261.0

10847.710976.1563.01

TTw

TTw

ms

msw

++

+=

, with T in C (14)

with T being the draw temperature of the product and msw the mass fraction of solute in the257

residual liquid phase of the sorbet.258

259

3. Results and discussion260

3.1. Laminar flow regime, wall slip behaviour and viscous dissipation261

The Reynolds numbers (Re) calculated for each of the operating conditions262

investigated are shown in Table 3. As we can see from this data all the apparent viscosity263

measurements were performed under laminar flow conditions (Re < 2300).264

The flow behaviour curves of sorbet for the three different tested product flow rates, at265

different product temperatures and therefore at different ice volume fractions are presented in266

Fig. 2. We can observe from these results that the power law model adequately described the267

shear stress and apparent shear rate data for all experimental conditions (R2 > 0.92). It can268

also be seen that the different flow curves measured with different pipe diameters overlap and269

do not separate significantly, demonstrating thus the absence of wall slip. Therefore we have270

neglected the wall slip behaviour for the analysis of our experimental data. Although Martin271


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lower product temperatures, the effect of which is likely due to the higher temperature297

difference between the environment and the product, leading to an increase of heat gain.298

The rheological properties of liquid sorbet mix and frozen sorbet measured at different299

operating conditions are shown in Table 5 (see also Fig. 5). These results confirm the shear-300

thinning behaviour for the liquid sorbet mix and for the frozen sorbet (flow behaviour index n301

< 1). It can also be seen that the flow behaviour index n decreases from n = 0.55 for the302

liquid sorbet mix at 5.03 C to n = 0.52 for frozen sorbet at -2.89 C (increase in ice volume303

fraction of 0.058). Then, there is a further decrease in the flow behaviour index from n = 0.52304

to n = 0.41 between the product temperatures of -2.89 to -3.11 C (ice volume fraction305

increase from 0.058 to 0.17). Furthermore, from -3.11 C to -5.68 C there is an establishment306

of a plateau at roughly n = 0.41 with the further decrease in the temperature of the product.307

This effect indicates that sorbet becomes more shear-thinning as the temperature of the308

product decreases and as the ice volume fraction increases. Cerecero (2003) also reported a309

decrease in the flow behaviour index and the establishment of a plateau with a decrease in the310

product temperature during the freezing of sucrose-water solutions (30% w/w).311

It can also be seen from these results (cf. Table 5), that a decrease in the product312

temperature and an increase in the ice volume fraction, lead to an increase in the consistency313

index of sorbet and therefore to an increase in the magnitude of the apparent viscosity of the314

product. In view of these results it is our opinion that this effect is due to two simultaneous315

phenomena occurring during the freezing of sorbet: firstly, there is a freeze concentration in316

the content of polysaccharides and sweeteners within the unfrozen phase, increasing the317

colloidal interparticle forces and leading to the increase in the apparent viscosity of this liquid318

phase. Secondly, the increase in the volume fraction of the ice crystals suspended in the liquid319

phase, leads to the increase in hydrodynamic perturbations in the flow field and increases the320

viscosity of the suspension. Similarly, Cerecero (2003) reported an increase in the viscosity of321


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the product with the increase in the ice volume fraction during the freezing of sucrose-water322

solutions (30% w/w). Goff et al., (1995) as well as Martin et al. (2008) and Elhweg et al.323

(2009) reported an increase in the apparent viscosity of ice cream with the decrease in product324

temperature and the increase in the concentration of ice crystals.325

The apparent viscosity of the product as a function of the shear rate at different draw326

temperatures and ice volume fractions is shown in Fig. 3. As previously mentioned, these327

results can be useful to understand the behaviour of the apparent viscosity of sorbet as it328

passes through the SSHE, through the outlet pipe of the SSHE and through a cup filling329

machine. The average shear rate within the SSHE was calculated by Eq. 7 (Leuliet et al.,330

1986) for a mix flow rate of 0.007 kg.s-1 (25 kg.h-1) and dasher speed of 78.5 rad.s-1 (750331

rpm), and determined to be roughly 360 s-1. For this given product flow rate, the shear rate332

applied within a standard outlet pipe of the SSHE was determined to be roughly 10 s-1, and333

the shear rate within the cup filling machine was roughly 4s-1. Taking into account this334

information, it can be seen from Fig. 3, that the product exhibited the lowest apparent335

viscosity when it is submitted to higher shear rates, as those applied within the SSHE. Then,336

when the product flows through the outlet pipe the viscosity of the product is increased. And337

finally, the highest value of apparent viscosity is reached when the product flows through the338

cup filling machine, where the product is submitted to the lowest shear rate. These changes in339

the apparent viscosity throughout the different processing stages are obviously due to the340

shear-thinning behaviour of sorbet.341

342

3.3. Experimental uncertainty343

The estimation of the apparent viscosity is affected by the uncertainties of the344

parameters R , P and V& ( dR 0.02 mm, dP= 2% of the measured P, Vd& 8.36x10-8345


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m3.s-1) which are involved in the relation between the shear stress and the shear rate and346

determine the apparent viscosity of the product, expressed as:347

( )

+==

3

w

w

app

R

V4

n4

1n3

L4

DP

&&

(16)

The differential of this equation makes it possible to account the propagation of the348

uncertainties of the measured variables R , P and V& into the uncertainty of the estimation of349

the apparent viscosity app of the product.350

The uncertainty of the estimation of the ice volume fraction icevd . is principally due to351

the uncertainty of the temperature of the product T( dT 0.1 C). Therefore, the differential352

of Eq. 8, makes it possible to account the propagation of the uncertainty of the measured353

temperature T into the uncertainty of the estimation of the ice volume fraction icevd . .354

The apparent viscosity as a function of the ice volume fraction inside the 5 different355

pipes used in the rheometer for a mix flow rate of 0.007 kg.s-1 (25kg.h-1) is shown in Fig. 4.356

The points represent the mean values for the different replicates. The vertical error bars in Fig.357

4 represent the standard deviation of the apparent viscosity due to the variability of the358

pressure drop measurements between the different replicates. This measure variation of the359

apparent viscosity is always higher than the calculated uncertainty (values not shown); this360

result is likely due to the use of a volumetric piston pump which supplies a jerky flow and361

affects the reading of the pressure drop within the liquid column manometers, causing thus a362

higher measurement error than the accuracy reported by the supplier of the manometers. The363horizontal error bars in Fig. 4 represent the calculated uncertainty in the estimation of the ice364

volume fraction. This uncertainty is higher as the concentration of ice crystals approaches to365

zero, and thus as the temperature of the product approaches the initial freezing temperature of366

sorbet at -2.63 C. This effect is likely due to the accuracy of 0.1 C of the calibrated Pt100367


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17

probe that was used to measured the product temperature, that is to say, when the temperature368

of the product decreases from -2.63 to -2.73 C, the concentration of ice crystals increases369

from 0 to 0.028, thus the propagation of the uncertainty of the product temperature370

measurement into the calculation of the ice volume fraction is not negligible.371

372

4. Rheological model373

As previously mentioned, the rheological properties of sorbet can be adjusted to an374

empirical expression that can be used to describe the apparent viscosity of the product as a375

function of the ice volume fraction. In this section, a rheological model is presented which376

was inspired from the Thomas equation (Eq. 2) and describes the apparent viscosity of sorbet.377

The parameters of the rheological model were identified by taking into account only the378

apparent viscosity measurements performed at 0.007 kg.s-1 (25kg.h-1).379

380

4.1. Model description381

The apparent viscosity of sorbet is affected by the viscosity of the freeze concentrated382

liquid phase and by the concentration in volume of the ice crystals suspended in the liquid383

phase. The shape of the ice crystals may also affect the viscosity of sorbet, but its effect will384

be neglected in the rheological model presented in this work. Therefore, the rheological model385

presented in this section takes into account the evolution of the viscosity of the liquid phase as386

the temperature of the liquid mix decreases and as the concentration of solute increases; and387

the evolution of the relative viscosity of the sorbet as the ice volume fraction increases.388

The apparent viscosity of the liquid sorbet mix used in this work was previously389

studied by Gonzalez (2012). This author expressed the consistency coefficient of the liquid390

mix as a function of the temperature T of the liquid mix (ranging from -2 to 5 C,391


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18

extrapolation to a product temperature of -5.67 C for this work), and of the concentration of392

solids msw (ranging from 0.25 to 0.53), as shown by Eq. 19:393

557.29

15.273

38.2242exp1002.39 msmix w

Tk

+

= (19)

The rheological model must consider as well the evolution of the flow behavior index,394

from the value of the liquid sorbet mix 55.0nmix = , as its decreases slightly at the beginning395

of the freezing process, and passes through the establishment of a plateau at n = 0.41 with the396

further decrease in the temperature of the product and the increase in the ice volume fraction397

icev. . The equation that describes the evolution of the flow behaviour index is written as398

follows:399

( )

+=

icevmixnn

.exp1 (20)

where the coefficients and were identified to be 0.29 and 0.11, respectively, by400

minimizing the sum of squared errors (SSE) defined in equation 21:401

( )2

1

1

=

M

predictedmeasured

nnM

SSE (21)

where M is the number of experimental points, measuredn is the measured mean value of the402

flow behaviour index and predictedn the predicted value of the flow behaviour index at a given403

ice volume fraction.404

The comparison between the experimental and predicted values of the flow behaviour405

index as a function of the ice volume fraction is presented in Fig. 5. It can be seen that the406

predicted values by Eq. 20 show a reasonably close fit to the experimental means values of407

the flow behaviour index.408

Therefore, the rheological model was constructed by taking into account the evolution409

of the consistency coefficient of the liquid phase mixk (Eq. 19), the evolution of the flow410


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19

behaviour index n (Eq. 20, which also considers the flow behaviour index of the liquid phase411

mixn ), and the effect of the increase in the ice volume fraction icev. on the apparent viscosity412

of sorbet expressed by a modified Thomas equation, as shown in Eq. 22:413

}( ) 1.2.. exp05.105.21 +++= nicevicevicevmixapp k & (22)

where the coefficients and were determined to be 3.32 and 3.42, respectively, by414

minimizing the sum of squared errors (SSE) defined in Eq. 23:415

( )2

1..

1 =M

predictedappmeasuredappM

SSE (23)

where M is the number of experimental points, measuredapp. is the experimental apparent416

viscosity and predictedapp. the predicted value of the apparent viscosity for sorbet.417

The comparison between the calculated apparent viscosity values by the rheological418

model and the apparent viscosity values measured at different product temperatures, different419

ice volume fractions and different shear rates, is shown in Fig 6. We can observe from these420

results that rheological model describes reasonably well the apparent viscosity of the product,421

although the model described the apparent viscosity data at icev. = 0.058 with an error higher422

to 20%.423

424

5. Conclusions425

This work studied the influence of the temperature of the product and thus of the ice426

volume fraction on the apparent viscosity of sorbet by using pipe rheometry measurements.427

Experimental data showed that there was no evidence of wall slip behaviour and therefore428

wall slip effects were neglected. Results demonstrated as well that for the set of experimental429

runs performed, viscous dissipation was negligible.430

Experimental results demonstrated that the value of the flow behaviour index431

decreased from the beginning of the freezing of sorbet until the product reached an ice volume432


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20

fraction about 17%; subsequently the value of the flow behaviour index remained almost433

constant forming a plateau with the further decrease of the temperature of the product. The434

consistency coefficient and therefore the magnitude of the apparent viscosity of sorbet435

increased with the decrease in the temperature of the product and with the increase in the ice436

volume fraction. This effect can be explained by two phenomena occurring during the437

freezing of sorbet: firstly, below the freezing point of sorbet mix, a decrease in the438

temperature of the product leads to the freeze-concentration of the content of polysaccharides439

and sweeteners within the liquid phase, the effect of which increased the colloidal forces440

between particles and increases the apparent viscosity of the unfrozen liquid phase. Secondly,441

an increase in the ice volume fraction leads to an increase in the hydrodynamic forces442

between the ice crystals and the surrounding fluid, increasing the apparent viscosity of the443

suspension of ice crystals.444

An empirical rheological model to describe the apparent viscosity of sorbet was445

presented, the empirical correlation was constructed by taking into the evolution of the446

apparent viscosity of the unfrozen liquid phase as a function of the product temperature and447

solute concentration, the evolution of the flow behaviour index as a function of the ice volume448

fraction, and the effect of the ice volume fraction on the apparent viscosity of the product. The449

rheological model described reasonably well the apparent viscosity of the product.450

The results obtained in this work can be useful to improve the control of the quality of451

frozen desserts and for the improvement of the design of piping systems and for the452

mathematical modelling of the freezing process in SSHEs.453

454

455

456

457


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21

Acknowledgments458

The authors gratefully acknowledge the financial support granted by the European459

Community Seventh Framework through the CAF project (Computer Aided Food processes460

for control Engineering) Project number 212754.461


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22

Abbreviations462

DRS Dasher Rotational Speed463

MFR Mix Flow Rate464

SSE Sum of Squared Errors465

SSHE Scraped Surface Heat Exchanger466

TR22 Evaporation Temperature of r22467

468


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23

Nomenclature469

dc/e internal diameter of contraction/enlargement pipe470

d internal diameter of pipe rheometer471

k consistency coefficient of sorbet, Pa.sn

472

mixk consistency coefficient for sorbet mix, Pa.sn473

n flow behaviour index index, -474

nL number of scraping blades, -475

L pipe length, m476

R pipe radius, m477

RN rotational speed of scraping blades, rad.s-1478

P pressure drop, Pa479

V& volumetric flow rate, m3.s-1480

T temperature of the product, C481

msw initial mass fraction of solute (sweetener content), -482

imsw . initial mass fraction of solute (sweetener content), -483

fmsw . final mass fraction of solute (sweetener content), -484

485

Greek symbols486

& shear rate, s-1487

w& wall shear rate, s-1488

w wall shear stress, Pa489

w thermal conductivity of the solute solution, W.m-1.K-1490

ice thermal conductivity of ice, W.m-1.K-1491

sorbet thermal conductivity of sorbet, W.m-1.K-1492


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24

app apparent viscosity, Pa.s493

viscosity of a Newtonian suspension, Pa.s494

l viscosity of the liquid phase of a suspension, Pa.s495

i ice density, kg.m-3, -496

s sorbet density, kg.m-3, -497

particle concentration, -498

icem. ice mass fraction, -499

icev. ice volume fraction, -500

thermal sensibility, K-1501


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25

References502

Adhikari, B., Jindal V. (2001). Fluid flow characterization with tube viscometer data.Journal503

of Food Engineering. 50. 229-234.504

Ayel, V., Lottin, O., Peerhossaini, H. (2003). Rheology, flow behaviour and heat transfer of505

ice slurries: a review of the state of the art. International Journal of Refrigeration.506

26: 95107.507

Balmforth, N., Craster, R., Perona, P., Rust, A., Sassi, R. (2007). Viscoplastic dam breaks and508

the Bostwick consistometer.Journal of Non-Newtonian Fluid Mechanics.142:63-78.509

Baloh T. (1967). Some diagrams for aqueous sugar solutions.Zucker. 20: 668-679.510

Batchelor, G. K. (1977). The effect of Brownian motion on the bulk stress in a suspension of511

spherical particles.Journal of Fluid Mechanics. 83: 97-117.512

Burns, I. and Russell A. (1999). Process rheology of ice-cream. In ZDS Inter-Ice, Solingen.513

Cerecero, R. (2003).Etude des coulements et de transferts thermiques lors de la fabrication514

dun sorbet lchelle du pilote et du laboratoire. PhD Thesis. INA-PG, Paris, France.515

Cook, K. L. K. and Hartel, R. W. (2010). Mechanisms of Ice Crystallization in Ice Cream516

Production. Comprehensive Reviews in Food Science and Food Safety, 9 (2), 213-222.517

Einstein, A. (1906). Investigations on the theory of Brownian motion. Annalen der Physik.518

19: 289-306.519

Elhweg, B., Burns, I., Chew, Y., Martin, P., Russell, A., Wilson, D. (2009). Viscous520

dissipation and apparent wall slip in capillary rheometry of ice cream. Food and521

Bioproducts Processing. 87: 266-272.522

Granger C., Langendorff, V., Renouf N., Barey P., and Cansell M. (2004). Impact of523

Formulation on Ice cream Microstructures: An Oscillation Thermo-Rheometry Study.524

Journal of Dairy Science, 87: 810-812.525

526


27/29

26

Goff, H. D., Freslon, B., Sahagian, M. E., Hauber, T. D., Stone, A. P., Stanley, D. W. (1995).527

Structural development in ice cream - Dynamic rheological measurements. Journal of528

Texture Studies, 26 (5), 517-536.529

Gonzalez, E. (2012). Contribution au contrle par la modlisation dun procd de530

cristallisation en continu. PhD Thesis. AgroParisTech, Paris, France.531

Haddad, A. (2009). Couplage entre coulements, transferts thermiques et transformation lors532

du changement de phase dun produit alimentaire liquide complexe - Application la533

matrise de la texture. PhD Thesis. AgroParisTech, Paris, France.534

Hansen TM, Kauffeld M. (2000). Viscosity of ice-slurry. 2nd Workshop on ice-slurries,535

International Institute of Refrigeration.536

Judy, J., Maynes, D., Webb B.W. (2002). Characterization of frictional pressure drop for537

liquid flows through microchannels.International Journal of Heat and Mass Transfer,538

45: 34773489.539

Krieger, I.M. and Dougherty, T.J. (1959). A mechanism for non-Newtonian flow in540

suspension of rigid spheres. Transactions of the Society of Rheology. 3: 137-152.541

Leuillet, J. C., Maingonnait, J. F., Corrieu, G. (1986). Etude de la perte de charge dans un542

changeur de chaleur a surface racle traitant des produits newtoniens et non-543

newtoniens.Journal of Food Engineering. 5: 153-176.544

Levy, F. L. (1981). A modified Maxwell-Euken equation for calculating the thermal545

conductivity of two-component solutions or mixtures. Internation Journal of546

Refrigeration. 4 (4): 223-225.547

Levy, F. L. (1982). Calculating the thermal conductivity of meat and fish in the freezing548

range.Internation Journal of Refrigeration. 5 (3): 49154.549

Macosko, C.W. (1994). Rheology, Principles, Measurements and Applications. New York:550

VCH Publishers Inc. 550 p.551


28/29

27

Marshall, R. T., Goff, H. D., Hartel R. W. (2003). Ice cream. 6th Ed. New York: Klumer552

Academic/Plenum Publishers. 371 p.553

Martin, P., Wilson, D. (2005). A critical assessment of the Jastrzebski interface conditions for554

the capillary flow of pastes, foams and polymers. Chemical Engineering Science.555

60: 493-502.556

Martin, P., Odic, K., Russell, A., Burns, I., Wilson, D. (2008). Rheology of commercial and557

model ice creams. Applied Rheology. 18(1), 12913-1:11558

Mooney M. (1931). Explicit formulas for slip and fluidity.Journal of Rheology.2: 210-222.559

Mooney M. (1951). The viscosity of a concentrated suspension of spherical particles. Journal560

of Colloid Science. 6: 162-170.561

Steffe, J. F. (1996). Tube Viscometry. Rheological methods in Food Process Engineering.562

East lansing, MI, USA: F. Press. p.94 -156.563

Thomas, D. G. (1965). Transport characteristics of suspension: VIII. A note on the viscosity564

of newtonian suspensions of uniform spherical particles. Journal of Colloid Science.565

20: 267-277.566

Wildmoser, H., Scheiwiller, J., Windhab, E.J., 2004, Impact of disperse microstructure on567

rheology and quality aspects of ice cream.Lebensmittel-Wissenschaft & Technologie,568

37(8): 881891.569

Winter, H. (1977). Viscous dissipation in shear flows of molten polymers.Advances in Heat570

Transfer. 13:205-267571


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Highlights572573574

- Rheological characterization of sorbet at different ice fractions and shear rates575576

- Rheological model to predict the viscosity of sorbet as a function of ice fraction577578 - The experimental tendencies are represented satisfactorily within a 20% error limit579580581582583

13 - saldarriaga - rheological characterization of sorbet using pipe rheometry during the freezing...

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