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Flow Behavior of Milk Chocolate Meltand the Application to Coating FlowSSSSSUKANYUKANYUKANYUKANYUKANYAAAAA WWWWWICHCHUKITICHCHUKITICHCHUKITICHCHUKITICHCHUKIT, M, M, M, M, MICHAELICHAELICHAELICHAELICHAEL J. M J. M J. M J. M J. MCCCCCCCCCCARARARARARTHYTHYTHYTHYTHY, , , , , ANDANDANDANDAND K K K K KAAAAATHRTHRTHRTHRTHRYNYNYNYNYN L. M L. M L. M L. M L. MCCCCCCCCCCARARARARARTHYTHYTHYTHYTHY

ABSTRACTABSTRACTABSTRACTABSTRACTABSTRACT: The rheological properties of chocolate, especially shear viscosity and yield stress, are important: The rheological properties of chocolate, especially shear viscosity and yield stress, are important: The rheological properties of chocolate, especially shear viscosity and yield stress, are important: The rheological properties of chocolate, especially shear viscosity and yield stress, are important: The rheological properties of chocolate, especially shear viscosity and yield stress, are importantcontrol parameters for enrobing processes in confectionery manufacture. The rheological parameters of moltencontrol parameters for enrobing processes in confectionery manufacture. The rheological parameters of moltencontrol parameters for enrobing processes in confectionery manufacture. The rheological parameters of moltencontrol parameters for enrobing processes in confectionery manufacture. The rheological parameters of moltencontrol parameters for enrobing processes in confectionery manufacture. The rheological parameters of moltenmilk chocolate were measured at 42°C during steady pipe flow using a magnetic resonance imaging (MRI)milk chocolate were measured at 42°C during steady pipe flow using a magnetic resonance imaging (MRI)milk chocolate were measured at 42°C during steady pipe flow using a magnetic resonance imaging (MRI)milk chocolate were measured at 42°C during steady pipe flow using a magnetic resonance imaging (MRI)milk chocolate were measured at 42°C during steady pipe flow using a magnetic resonance imaging (MRI)viscometric method. The experimental method combines shear rate values obtained from an MR velocity imageviscometric method. The experimental method combines shear rate values obtained from an MR velocity imageviscometric method. The experimental method combines shear rate values obtained from an MR velocity imageviscometric method. The experimental method combines shear rate values obtained from an MR velocity imageviscometric method. The experimental method combines shear rate values obtained from an MR velocity imageand shear stress values obtained from an independent pressure drop measurement. The experimental factorsand shear stress values obtained from an independent pressure drop measurement. The experimental factorsand shear stress values obtained from an independent pressure drop measurement. The experimental factorsand shear stress values obtained from an independent pressure drop measurement. The experimental factorsand shear stress values obtained from an independent pressure drop measurement. The experimental factorswere emulsifier type and emulsifier level. The rheogram data were fit by the Casson model to yield the Cassonwere emulsifier type and emulsifier level. The rheogram data were fit by the Casson model to yield the Cassonwere emulsifier type and emulsifier level. The rheogram data were fit by the Casson model to yield the Cassonwere emulsifier type and emulsifier level. The rheogram data were fit by the Casson model to yield the Cassonwere emulsifier type and emulsifier level. The rheogram data were fit by the Casson model to yield the Cassonyield stryield stryield stryield stryield stress and plastic viscosityess and plastic viscosityess and plastic viscosityess and plastic viscosityess and plastic viscosity. . . . . The CThe CThe CThe CThe Casson yield strasson yield strasson yield strasson yield strasson yield stress ress ress ress ress ranged franged franged franged franged from 1.9 to 15.0 Pom 1.9 to 15.0 Pom 1.9 to 15.0 Pom 1.9 to 15.0 Pom 1.9 to 15.0 Pa; the Ca; the Ca; the Ca; the Ca; the Casson viscosity rasson viscosity rasson viscosity rasson viscosity rasson viscosity rangedangedangedangedangedfrom 6.0 to 14.6 Pa s as a function of emulsifier content. The rheological parameters were incorporated into afrom 6.0 to 14.6 Pa s as a function of emulsifier content. The rheological parameters were incorporated into afrom 6.0 to 14.6 Pa s as a function of emulsifier content. The rheological parameters were incorporated into afrom 6.0 to 14.6 Pa s as a function of emulsifier content. The rheological parameters were incorporated into afrom 6.0 to 14.6 Pa s as a function of emulsifier content. The rheological parameters were incorporated into adrainage theory model to predict coating thicknesses in the enrobing process. The model was solved numericallydrainage theory model to predict coating thicknesses in the enrobing process. The model was solved numericallydrainage theory model to predict coating thicknesses in the enrobing process. The model was solved numericallydrainage theory model to predict coating thicknesses in the enrobing process. The model was solved numericallydrainage theory model to predict coating thicknesses in the enrobing process. The model was solved numericallyand yielded good approximations to the experimental values that were between 1.1 to 2.7 mm.and yielded good approximations to the experimental values that were between 1.1 to 2.7 mm.and yielded good approximations to the experimental values that were between 1.1 to 2.7 mm.and yielded good approximations to the experimental values that were between 1.1 to 2.7 mm.and yielded good approximations to the experimental values that were between 1.1 to 2.7 mm.

KKKKKeyworeyworeyworeyworeywords: chocolateds: chocolateds: chocolateds: chocolateds: chocolate, viscosity, viscosity, viscosity, viscosity, viscosity, yield str, yield str, yield str, yield str, yield stressessessessess, enr, enr, enr, enr, enrobing, magnetic robing, magnetic robing, magnetic robing, magnetic robing, magnetic resonance imaging (MRI)esonance imaging (MRI)esonance imaging (MRI)esonance imaging (MRI)esonance imaging (MRI)

Introduction

The use of enrobing technology is prevalent in the chocolatemanufacturing industry. The molten chocolate or chocolate

coating flows in a sheet above a moving belt. The product is coat-ed as it moves through the sheet. For good quality and accurateweight control, the chocolate must have the correct viscosity (Ae-schlimann and Beckett 2000).

Extensive rheological studies of molten chocolate have beenperformed using rotational viscometry to describe flow properties.Prior to 2000, the Casson model was recommended as a standardmethod by the Intl. Office of Cocoa, Chocolate, and Sugar Confec-tionery (IOCCC) to quantify rheological properties of the fluid choc-olate. The Casson parameters are the Casson plastic viscosity andthe Casson yield stress; these values range from 1 to 20 Pa s and 10to 200 Pa, respectively (Chevalley 1975, 1991, 1999).

In a multi-laboratory study, Aeschlimann and Beckett (2000)discuss a sequence of tests and recommendations to improve thestandard method for measuring chocolate viscosity. These research-ers recommend reporting shear stress values at specified shear rates,for example, 2, 5, 10, 20, and 50 s–1. In addition, the chocolate mustbe completely melted prior to viscosity measurements, that is, 40°C.The IOCCC standard for the measurement of viscosity of chocolateproducts continues to recommend rotational viscometry using aconcentric cylinder probe (IOCCC 2000). The Deutsches Instite fürNormung (DIN, standard developed by the German Institute forStandardization) bob is recommended to minimize instrument er-rors. Other instrument errors, such as wall slip, are recognized po-tential reasons for inconsistent measurements, especially at lowshear rates (Beckett 2001; IOCCC 2000).

The success of the enrobing process is dependent both on therheological characterization of the chocolate and the physics of coat-

ing flow. A considerable body of literature published in the mid-1960s and early 1970s discusses liquid entrained on a flat surfaceas the surface is withdrawn vertically from a bath of liquid. Thistype of physical situation is important to coating, cleaning, drain-ing, lubrication, and viscosity measurement. Tallmadge and Gut-finger (1967) cite early applications and present an approach thatincorporates viscous, gravitational, and surface tension forces inthe different regimes. Representative work includes Van Rossum(1958), Gutfinger and Tallmadge (1965), and Groenveld (1970).

Drainage was considered a special case of the withdrawal pro-cess. The unsteady state flow process of a flat plate that has beentotally withdrawn from a liquid bath is governed by only viscous andgravitational forces. Film thickness as a function of position andtime was presented in a straightforward manner for Newtonian flu-ids by Bird and others (2002) as fluid draining from tank sides.Gutfinger and Tallmadge (1965) presented the draining analysis fornon-Newtonian fluids. Specifically, the 3-parameter Ellis model wasevaluated. Depending on the values of the parameters, this mod-el reduced to the Newtonian model, the power law model, or theBingham plastic model. Insightful comments were made regardingthe behavior of power law fluids as compared with Newtonian flu-ids. A change in the power law consistency index, K, has a morepronounced influence on the film thickness than the change ofviscosity in the Newtonian case. In addition, the change in filmthickness with time and position is more pronounced for a Newto-nian fluid than a power law fluid. This difference is primarily dueto the power law fluid being more uniform in thickness than theNewtonian one. This effect becomes more pronounced with de-creasing values of the flow behavior index, n.

This work utilized a magnetic resonance imaging (MRI) techniqueas an integral part of tube viscometry and extends the techniquethat has been reported by McCarthy (1994), Powell and others(1994), Arola and others (1997), and Yoon and McCarthy (2002,2003). The objectives of this study were to evaluate the rheologicalproperties of molten milk chocolate and to incorporate the measuredrheological properties into an unsteady state draining analysis topredict and control coating thickness during the enrobing process.

MS 20040648. Submitted 9/23/04, Revised 11/8/04, Accepted 11/20/04. AuthorWichchukit is with Dept. of Biological and Agricultural Engineering. Au-thors McCarthy M.J. and McCarthy K.L. are with Dept. of Food Science andTechnology and Dept. of Biological and Agricultural EngineeringOne ShieldsAve., Univ. of California, Davis, CA 95616. Direct inquiries to authorMcCarthy (E-mail: klmccarthy@ucdavis.edu).

Flow behavior of milk chocolate melt . . .

E: Food Engineering & Physical Properties

Materials and Methods

TTTTTest samplesest samplesest samplesest samplesest samplesThe test material was milk chocolate obtained directly from the

manufacturer in 5 gallon containers (Hershey Foods Corp., Hershey,Pa., U.S.A.). Although chemical analysis was not performed for theseparticular samples, typical composition by percent for this productis cocoa mass 11.8%, whole milk powder 19.1%, sugar 48.7%, addedcocoa butter 20%, for a total fat content of 31.5% (Jackson 1999). Themilk chocolate was manufactured without added emulsifier. Fivechocolate formulations were used: milk chocolate with no addedemulsifier (the control), milk chocolate with 0.1% soy lecithin, milkchocolate with 0.3% soy lecithin, milk chocolate with 0.1% syntheticlecithin (YN), and milk chocolate with 0.3% YN. The emulsifiers wereadded by weight during the melting and mixing process. The densityof the milk chocolate melt was 1270 kg/m3.

MMMMMagnetic ragnetic ragnetic ragnetic ragnetic resonance imaging viscometresonance imaging viscometresonance imaging viscometresonance imaging viscometresonance imaging viscometryyyyyThe experimental apparatus consisted of a temperature-con-

trolled capillary viscometer connected to a 1818-mL stainless steelsample reservoir (Figure 1). During each experimental run, the testfluid from the reservoir was pressure driven through a straightlength of 9.8 mm inner dimater glass tubing (Pyrex®, Corning, Inc.,Big Flats, N.Y., U.S.A.) by the downward motion of a piston movingat 0.2 mm/s. The cross-sectional area of the piston was 1.04 × 104

mm2 and spanned the cross-section of the fluid reservoir. The con-tact area between the piston and the reservoir surface was a lubri-cated O-ring. The piston was connected to an actuator/control sys-tem fabricated to provide a 0.30-m stroke length and rated for 2200N (Electric Cylinder Model EC2, IDC, Salem, N.H., U.S.A.).

A pressure measurement was acquired upstream of the imagingregion using a pressure transducer (Model PX771-100WD1, Ome-ga Engineering Inc., Stamford, Conn., U.S.A.). The pressure differ-ence was determined by the difference between measurement andambient, characterizing the pressure drop over the length, L, of 1.25m (Figure 1).

The chocolate was melted and held at 42°C in a chocolate-temper-ing system (Revolation X3210, ChocoVision, Poughkeepsie, N.Y.,U.S.A.) to ensure that fat was in the liquid phase and well mixed withthe milk chocolate solids. The sample temperature was maintainedat 42°C throughout testing by jacketing the reservoir and flow tube;the heating medium was warm air (42°C). Although the target tem-perature was 40°C, there was a 2°C offset in the temperature controlof the experimental system. The offset was consistent; off-line mea-surements were therefore performed at the same temperature, 42°C.

For each flow rate and pressure drop measurement, an MR veloc-

ity image was acquired at a position 0.50 m downstream from thereservoir, which ensured fully developed flow. The MR velocityimages were obtained using a SMIS NMR spectrometer (SurreyMedical Imaging Systems, Surrey, U.K.) connected to a 0.1 Teslaelectromagnet, corresponding to 4.2 MHz for 1H resonance frequen-cy. Unshielded gradient coils produced orthogonal gradients withmaximum gradient amplitudes of � 2 Gauss/cm. A pulsed gradientspin-echo pulse sequence was applied to acquire the velocity pro-files (McCarthy 1994; Arola and others 1997). The apparatus geom-etry and MRI experimental parameters are given in Table 1. For theMR image, 32 phase encoding steps were used to characterize thez component of the velocity and 64 frequency encoding steps alongthe x direction read out the radial position (r) of the volume ele-ments. The resulting matrix was zero filled to 64 × 128 and a two-dimensional Fourier transformwas performed. The measurementtime was on the order of 1 min and reflects multiple data acquisi-tions. Data analysis of the MR velocity image was performed tocharacterize flow behavior using MatLab7.0.0 (R14) software (TheMathWorks Inc., Natick, Mass., U.S.A.). Fluid velocity at each radialposition was determined by the position of the maximum signalintensity in each row of the image matrix.

To evaluate shear rate from the velocity profile, the MRI velocitydata were fit with an even-order polynomial; the global fitting errorfor the polynomial was less than 2%. This analytic expression wasthen differentiated and evaluated at each radial position to yieldthe local shear rate

(1)

where v(r) is the axial velocity of the molten chocolate during pipeflow (Arola and others 1997). The range of shear rates evaluatedwas from 1 to 8 s–1; this range corresponds to shear rates relevant tothe enrobing process (Steffe 1996; Aeschlimann and Beckett 2000).

Local shear stress data, to correspond to the shear rate data,were obtained using the force balance

(2)

where � is the shear stress, and �P is the pressure difference acrossthe pipe length, L. The shear rate and shear stress values were plotas a rheogram. The steady shear rheological data for milk chocolatemelt was best described by the Casson model based on goodnessof fit and the coefficient of determination. The expression for theCasson model is

(3)

Table 1—Apparatus geometry and MRI experimental parameters

Apparatus geometry MRI parameters

Tube radius, R (mm) 4.9 Pulse repetition time, TR (ms) 400Tube length, L (m) 1.25 Echo time, TE (ms) 63Reservoir radius (mm) 57.5 Number of phase encoding 32Reservoir height (mm) 130 Number of frequency encoding 64

Number of averages 16Field of view (mm) 60Velocity sweep width (mm/s) 188.9

Figure 1—Schematic diagram of the experimental ap-paratus

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wide by 50.8 mm height, on each side. The mass of chocolate was al-lowed to solidify and was then scraped and weighed to determine anaverage film thickness on each side of the plate.

Soy lecithin was added to the milk chocolate to a level of 0.1%.The dipping procedure was repeated. The 0.1% lecithin mixture wasbrought to a level of 0.3%; the dipping procedure was repeated.

The sample preparation, dipping, and weighing procedure werefollowed for the synthetic lecithin YN as well, starting with emulsi-fier-free milk chocolate.

Results and Discussion

Figure 2 illustrates representative magnetic resonance phase en-coded velocity images of molten milk chocolate at 42°C. Figure

2a represents the MR reference location of fluid at rest (v = 0); Fig-ure 2b illustrates a velocity profile at a volumetric flow rate of 2.4 mL/s. Figure 3 illustrates velocity profiles of the molten milk chocolateunder the experimental conditions. Typical of these velocity pro-files, the velocity is maximum at the center of the pipe and de-creases toward the pipe wall. The maximum velocity value in-creased as the amount of the emulsifier increased from 0% to 0.1%to 0.3%. The velocity values are 38 mm/s, 40 mm/s, and 41 mm/s,respectively, and did not differ significantly with type of emulsifier.Wall slip is also observed as the velocity is nonzero at the pipe wall.

For comparison purposes, the apparent viscosity values of testfluids were determined using rotational viscometry data and MRIviscometry data. The apparent viscosity (�) was calculated by

(5)

Figure 4 illustrates apparent viscosity compared with shear rateof milk chocolate melt for MRI viscometry and rotational viscome-try in the shear rate range of 1 to 8 s–1. Overall, the apparent viscos-ity ranged from 7 to 48 Pa s. Shear-thinning behavior is observed asthe apparent viscosity decreases with increasing shear rate. Forboth emulsifiers, the apparent viscosity decreased as the amountof the emulsifier increased. As the emulsifier level increased to0.3%, the Casson viscosity values decreased by more than a factorof two. Likewise, the Casson yield stress decreased significantlywith increasing emulsifier level (Table 2).

Wall slip, in part, plays a role in the discrepancy between theCasson parameters determined for the MRI data and the rotationalviscometry data (Table 2). Wall slip occurs when a layer of fluid that

where, is the shear rate, �CA is the Casson viscosity, and ��,CA is theCasson yield stress. The yield stress, �o, was also determined directlyfrom the MR image with the plug radius, Ro, and is designated as

(4)

The image analysis procedure was developed and documentedby Sadikin (1999) and Choi (2003) and utilized the graphic userinterface feature of MatLab.

In addition to the MR viscosity measurement, the flow behaviorwas measured off-line by rotational viscometry (CVO, Bohlin Instru-ments, Gloucestershire, U.K.) utilizing the concentric cylinder ge-ometry and the vane method. The chocolate melt temperature wascontrolled at 42°C by a circulating water bath (RTE-111, NESLABInstruments, Inc., Newington, N.H., U.S.A.). A DIN bob with a diam-eter of 25 mm and a cup with a diameter of 27.5 mm operated underthe controlled rate mode over a shear rate range of 1 to 8 s–1. A 4-blade vane with a diameter of 25 mm and the same cup were usedover a shear rate range of 0.05 to 10 s–1 in the controlled rate vanemethod to measure the yield stress (Steffe 1996).

EEEEEnrnrnrnrnrobingobingobingobingobingThe emulsifier-free milk chocolate, described above, was stored

for several days before the experiment to ensure constant temper-ature and mixed thoroughly before removing 1 kg samples. The 1-kg milk chocolate samples were prepared in the tempering bowl ofthe Revolation X3210, chocolate tempering system (ChocoVision,Poughkeepsie, N.Y., U.S.A.). The chocolate was maintained be-tween 40 � 0.5°C.

Flat plastic plates with dimensions of 101.6 mm × 101.6 mm (4 × 4inch) were prepared for dipping into the chocolate. The plastic plateswere paper covered; the adhered paper was removed from the bottom50.8 mm on each side of the plate using an Exacto knife. Three plateswere dipped into the molten milk chocolate for each trial and thengently shook for 20 s over the tempering bowl. The plates were thenhung in a controlled environment chamber, which was maintained at40 � 1°C by two 1500W heaters (Steinel, Model HG3002LCD, HowardElectric Instruments Inc., El Dorado, Kans., U.S.A.). After 30 min, theplates were removed from the chamber and any excess chocolate wasscraped from the side and bottom edges of the plate, the protectivepaper was removed from both sides of the top portion of the plate.The chocolate remaining on the plate had dimensions of 101.6 mm

Figure 2—RepresentativeMR image of milk chocolateflow at 42°C: (1) at rest and(2) at a volumetric flow rate

�P� =

2LR ��

Flow behavior of milk chocolate melt . . .

E: Food Engineering & Physical Properties

has lower viscosity than the main fluid forms at the walls of the vis-cometer and has been observed in many food suspensions andhigh fat products. The lower apparent viscosity values of the rota-tional viscometer compared with the MRI technique are consistentwith wall slip in the concentric cylinder geometry. In addition, thevalues of wall slip can be determined directly from the MR image.The values of slip velocity for the lecithin trials ranged from 17.7 to19.0 mm/s. Similar slip velocities were observed for the syntheticlecithin (YN), though accurate wall slip velocity could not be deter-mined due to lower signal-to-noise ratio at the tube wall.

Table 3 shows reasonable agreement between the yield valuesobtained from the vane method, and the values based on the plugdimension obtained from the MRI viscometry (Eq. 4). However,these yield values are considerably higher that the Casson modelyield values and may well represent static yield stress due to theconditions required to initiate shearing flow. It is not unusual thatyield stress values obtained by 1 method are different than thoseobtained by a different method (Steffe 1996).

Application of rheological parametersApplication of rheological parametersApplication of rheological parametersApplication of rheological parametersApplication of rheological parametersto the enrobing processto the enrobing processto the enrobing processto the enrobing processto the enrobing process

Much of the focus of process rheology is the application of theinformation to real-time decision-making. For instance, will thisparticular milk chocolate provide adequate coverage on a waferduring the enrobing process? Figure 5 illustrates a typical enrobing(or coating) process with the chocolate flowing in a sheet from a binabove a moving belt. As the chocolate covers the wafer, excess drainsfrom the vertical sides of a wafer due to the force of gravity. Choc-olate is a special case of coating because a phase change occursduring the process. The following analysis is relevant for the timeinterval immediately following the enrobing of the product whenthe chocolate is in the molten state prior to solidification of the fat.

For fluids with a yield stress, like chocolate, flow ceases when theyield stress is greater than or equal to the shear stress. Therefore thecoating thickness, Ho, supported by the yield stress is (Steffe 1996;Lang and Rha 1981)

(6)

For a milk chocolate with a Casson yield value of 15 Pa, the fluidwould ultimately drain to a final coating thickness of 1.2 mm,whereas a Newtonian fluid would theoretically drain to an infinite-ly thin layer.

Equation 6 is most appropriately used for the evaluation of astatic film or coat. However, the enrobing process is a transientprocessing as chocolate drains away from vertical surfaces. The firststep of extending the drainage flow analysis to Casson fluids is toevaluate steady state drainage from a vertical surface. The flow offluid on the vertical surface is controlled by the balance of gravita-tional and viscous forces (Bird and others 2002). Schematically, theflow is illustrated in Figure 6a, where x is the vertical downwarddirection and y is the horizontal axis.

Mathematically, the problem is formulated as

(7)

where the left hand side of the equation is the gravitational forces(g is the gravitational acceleration and � is the fluid density) and theright hand side of the equation is the viscous force. For a Newtonianfluid, the relationship between the shear stress and shear rate interms of the velocity gradient is given by

Figure 3—Representative velocity profiles of milk choco-late obtained from the MRI viscometry. �, 0% soy lecithin;�, 0.1% soy lecithin; and �, 0.3% soy lecithin.

Figure 4—Apparent viscosity of milk chocolate meltfor the 2 emulsifiers: (a) soy lecithin and (b) syntheticlecithin. In each figure, circles represent 0% emul-sifier, squares, 0.1% emulsifier, and triangles, 0.3%emulsifier. The filled marks are MRI data; the openmarks are CVO concentric cylinder data.

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(8)

where u is the downward velocity of the fluid and � is the Newto-nian viscosity. This expression is substituted into Eq. 7. Equation7 is integrated to yield the fluid velocity as a function of position (y-direction) and the steady-state fluid thickness (H).

(9)

By integrating over the velocity profile with respect to y, the av-erage velocity (<u>) under steady state conditions is

(10)

This expression is useful because it gives the relationship be-tween the drainage velocity and fluid properties, the more densethe chocolate, the greater the drainage velocity; the higher the vis-cosity, the lower the drainage velocity.

The next step is to consider the unsteady state nature of thechocolate enrobing process. An unsteady state mass balance overa volume of fluid between x and x + �x yields

(11)

which is written in terms of the local fluid thickness, h, as a functionof time (t) and vertical position (x) (Figure 6b). This approach isbased on the draining of a viscous fluid down a tank wall (Bird andothers 2002). The solution to Eq. 11 for fluid thickness of a Newto-nian fluid is

(12)

This approach assumes that the average velocity at a given ver-tical position can be approximated by Eq. 10, with the fluid thick-ness H replaced by h(x,t) (for example, a typical lubrication approx-imation). Equation 12 is useful because the average fluid thickness(<h>) at a time t is determined by integrating the expression over theheight of the region draining for

(13)

As an example, if the height of coverage is L = 5 cm, the averagefluid thickness at 20 s is 2 mm for a Newtonian fluid with a viscosityof 60 Pa s and density of 1270 kg/m3. In concept, this approach wasused by Cisneros-Zevallos and Krochta (2003) to evaluate coatingthickness applied to fruits and vegetables. The limitation of thisapproach is that the solution is valid only for Newtonian fluids andfor finite values of x/t.

In contrast to Newtonian fluids, the unsteady state mass bal-ance (Eq. 11) cannot be solved analytically for Casson fluids (Eq. 2).However, Eq. 11 can be numerically evaluated using the analyticalexpression for the average velocity of the Casson fluid. The Cassonmodel, with the shear rate expressed in terms of velocity gradient,was substituted into Eq. 7. Equation 7 was integrated to yield thefluid velocity as a function of position

(14)

Figure 6—Downward flow of fluid from a vertical surface: (a)steady state and (b) unsteady state

Table 2—Casson parameters from the MRI-based viscometry and rotational viscometry, T = 42 °C

Sample MRI-based viscometry Rotational viscometry

Casson Casson R2 Casson Casson R2

viscosity yield stress viscosity yield stress(Pa s) (Pa) (Pa s) (Pa)

0% 14.6 15.0 0.995 6.7 16.4 0.9860.1% lecithin 12.5 3.1 0.996 4.2 12.7 0.9940.3% lecithin 6.0 2.6 0.977 2.1 10.4 0.999

0.1% YN 12.7 3.9 0.980 4.3 12.1 0.9940.3% YN 6.6 1.9 0.978 2.6 9.6 0.998

Figure 5—Schematic of the enrobing process

Flow behavior of milk chocolate melt . . .

E: Food Engineering & Physical Properties

where Ho = �o,CA/(�g), and y� is 0 at the film surface and y� = H at thesolid vertical surface. Equation 14 is valid in the shearing regionfrom Ho to the solid surface at y� = H. The value of the velocity fromthe film surface to Ho is the maximum velocity of

(15)

The velocity profile of milk chocolate with no added emulsifier(Table 2) is illustrated in Figure 7. The plug region, with fluid trav-eling at the maximum velocity, is at the surface of the film. To rein-force the difference in flow behavior of Casson fluids relative toNewtonian fluids, the velocity profile of a Newtonian fluid is alsoshown. The Newtonian fluid has a viscosity equivalent to the ap-parent viscosity of the milk chocolate at a shear rate of 1 s–1(59.2 Pas). The average velocity over the film, for Eq. 11, is calculated by in-tegrating Eq. 14 and 15 over the relevant region and dividing by thefilm thickness. In Figure 7, the Newtonian fluid has an average ve-locity of 18 mm/s; the average velocity of the Casson fluid is 13mm/s, which is 28% lower due to the yield stress.

The final step in extending the drainage flow analysis to Cassonfluids is to determine the film thickness during the enrobing pro-cess. The local film thickness, h(x,t) is substituted for H in Eq. 14and 15 for the unsteady state mass balance (Eq. 11). This partial

differential equation (Eq. 11) was solved numerically for Cassonfluids with a MatLab finite difference program using central differ-ence for the time derivative and central difference for the spatialderivative (Haberman 2004). The finite difference program wasvalidated by limiting cases (for example, Newtonian fluid) and bydecreasing �t/�x to ensure numerical stability.

The average film thickness over a 20-s time interval is illustrated

Table 4—Average film thickness measured experimentallyand estimated theoretically based on the MRI-based Cassonparameters given in Table 2

Experimental Theoreticalfilm thickness film thickness

����� SD (mm) (mm)

Soy lecithin 0% 2.3 � 0.3 3.00.1% 1.2 � 0.1 1.70.3% 1.1 � 0.1 1.2

Synthetic lecithin 0% 2.7 � 0.2 3.00.1% 1.6 � 0.1 1.80.3% 1.2 � 0.1 1.2

Table 3—Yield stress values of milk chocolate using MRI-based viscometry and the vane method, T = 42°C

MRI Vane

Sample with Yield radius Yield stress Yield stressemulsifier (Ro) (mm) (Pa) (Pa)

0% 0.98 51 540.1% soy lecithin 0.98 37 290.1% synthetic lecithin 0.98 38 300.3% soy lecithin 1.00 18 190.3% synthetic lecithin 0.75 19 20

Figure 8—Average film thickness as a function of time for(a) milk chocolate with soy lecithin and (b) milk chocolatewith synthetic lecithin, YN. The solid line is 0% emulsifier,the dashed line is 0.1% emulsifier, and the dotted line is0.3% emulsifier.

Figure 7—Steady state velocity for a film thickness of 5mm for milk chocolate melt with a plastic viscosity of 14.6Pa s and Casson yield stress of 15.0 Pa (solid line) and fora Newtonian fluid with the same apparent viscosity (59.2Pa s) at 1 s–1

.

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in Figure 8 for each of the milk chocolate samples using the Cassonparameters determined by MRI-based viscometry (Table 2). Thehigher viscosity fluids have greater film thickness. As the emulsifierlevel increases, the film thickness decreases. The predicted averagefilm thicknesses at t = 20 s are given in Table 4, with the experimentalaverage film thicknesses (� 1 standard deviation). The unsteadystate mass balance (Eq. 11) yields a good approximation to the ex-perimental values. Experimentally, the film thicknesses are lowerdue to the draining that took place during the 30 min in the con-trolled temperature chamber. The experimental film thicknesseswere approximately 30% lower than the film thicknesses would havebeen had they been measured at 20 s. This approximation is basedon weighing the chocolate mass that collected in the tray beneaththe draining plates. Specifically, 62 g chocolate was collected afterthe synthetic lecithin trials; a total of 217 g had adhered to the 9plates.

Conclusions

The flow behavior of molten milk chocolate was evaluated ex-perimentally using an MRI-based tube viscometer. The choc-

olate was best characterized by the Casson model; as the emulsifierlevel increased from 0% to 0.3%, the Casson yield stress values de-creased from 15 to 1.9 Pa and Casson viscosity values decreasedfrom 14.6 to 6.0 Pa s. Deviation of the Casson parameters betweenthe MRI-based method and rotational viscometry was due to uncor-rected wall slip during the rotational measurements. The MRI-based viscometer Casson parameters were incorporated into anunsteady state mass balance to predict the average film thicknessof the milk chocolate during enrobing. As the emulsifier level in-creased from 0% to 0.3%, the predicted enrobing thicknesses de-creased by 60%. Experimentally, this trend was verified.

AcknowledgmentWe appreciate the donation of chocolate from Hershey Foods Corp.

Notationg gravitational constant, 9.8 m/s2

H steady state film thickness, mm or mHo film thickness supported by the yield stress, mm or mh unsteady state film thickness, mm or mL pipe length, mP pressure, Par radial directionR pipe radius, mm or mt time, sTE echo time, msTR pulse repetition time, msu film velocity, mm/s or m/sv pipe velocity, mm/s or m/sx flow direction of film, read directiony direction perpendicular to flow direction of film

y’ H-yz axial directionGreek letters

pressure drop, Pashear rate, s–1

� apparent viscosity, Pa s�CA Casson viscosity, Pa s� viscosity, Pa s� melt density, kg/m3

� shear stress, Pa�� yield stress, Pa��, CA Casson yield stress, Pa

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