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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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http://dx.doi.org/10.1016/j.jfoodeng.2013.05.017http://dx.doi.org/http://dx.doi.org/10.1016/j.jfoodeng.2013.05.017http://dx.doi.org/http://dx.doi.org/10.1016/j.jfoodeng.2013.05.017http://dx.doi.org/http://dx.doi.org/10.1016/j.jfoodeng.2013.05.017http://dx.doi.org/http://dx.doi.org/10.1016/j.jfoodeng.2013.05.017http://dx.doi.org/10.1016/j.jfoodeng.2013.05.0177/27/2019 13 - Saldarriaga - Rheological Characterization of Sorbet Using Pipe Rheometry During the Freezing Process 2013
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1
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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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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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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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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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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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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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
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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