Research Paper

Modeling temperature in chocolate mass to predict temperingquality

Delphine Bettens1, Yves Kegelaers2, Benoit Haut1, Véronique Halloin1 and Frédéric Debaste1

1 Chemical Engineering Unit, Free University of Brussels, Brussels, Belgium2 Puratos Group, Groot-Bijgaarden, Belgium

A computational model is developed to predict the temperature in chocolate as a function of space andtime during a new tempering process where pellets of seed crystals of form V are added to the meltedchocolate. The model, a refined version of a previous one (Debaste et al., J Food Eng. 2008, 88, 568–575)including a new expression for the heat sink, is based on the energy equation, simplified by the use of aneffective thermal conductivity. A shrinking core model is used to express the heat sink term associated withthe melting of pellets. The width of the limit layer around the pellets is the fitting parameter. The model isvalidated on experimental temperature profiles in bowls and compared to a previous kinetic model. Thequality of tempering is then assessed by differential scanning calorimetry. Combining this informationwith the model, the quality of tempering based on the initial experimental conditions is predicted.

Keywords: Chocolate tempering / Differential scanning calorimetry / Heat transfer

Received: December 12, 2008; accepted: February 16, 2009

DOI 10.1002/ejlt.200800290

Eur. J. Lipid Sci. Technol. 2009, 111, 273–279 273

1 Introduction

Cocoa butter, the main ingredient of chocolate, crystallizes insix different polymorphs, each with specific properties. Onlyone crystalline form, form V, is preferred in the final productas its melting point gives a firm and glossy chocolate at roomtemperature and at the same time a chocolate that melts in themouth. Chocolate with cocoa butter crystallized in form V hasa higher capacity to resist to “fat bloom”, which gives a non-attractive aspect of chocolate, and also leads to a high volu-metric contraction during solidification, which facilitates thedemolding stage [1]. Tempering of chocolate is thereforeneeded to crystallize cocoa butter in the right crystalline form.

The objective of the tempering process is to bring seedcrystals of form V into the melted chocolate to secure crystal-lization of chocolate in the preferred form. The commontempering process submits chocolate to a complex tempera-ture profile over time [2]. A new patented tempering processon pre-crystallization for use by professional pastry chefs hasbeen developed by Belcolade [3]. It is based on the introduc-

tion of solid pellets of cocoa butter into melted chocolate at atemperature such that the melting of the pellets results in seedcrystals. The pellets are added to the melted chocolate at atemperature low enough not to melt completely (leading tountempered chocolate), yet high enough to generate seedcrystals with appropriate dimensions allowing induction ofcrystal growth (leading to well-tempered chocolate after soli-dification). The advantages of this new process are ease of useand saved time.

This process has been developed based on experimentalobservations. It is therefore useful to develop a mathematicalmodel in order to enhance the understanding and control ofthe process. Such a model is used to make the process lessempirical and to determine the temperature of the meltedchocolate at which the pellets have to be introduced to lead to awell-tempered product.

The developed model is based on the numerical resolutionof a heat transport equation in the bulk of the melted choco-late. It is based on a previously published model by Debaste etal. [4] studying the tempering process in a stirred bowl using atwo-dimensional axisymmetric geometry. In this first paper,the heat sink produced by the melting of the cocoa butter pel-lets was modeled by an experimentally fitted kinetics. Al-though the temperature profile of the chocolate during most ofthe experiment was correctly predicted by this model, thetemperature profile in the chocolate in the first minutes after

Correspondence: Frédéric Debaste, Chemical Engineering Unit, FreeUniversity of Brussels, Avenue F.D. Roosevelt 50 CP 165/67, 1050 Brus-sels, Belgium.Fax: 132 2 6502910E-mail: fdebaste@ulb.ac.be

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274 D. Bettens et al. Eur. J. Lipid Sci. Technol. 2009, 111, 273–279

the seeding was significantly different. As this initial periodmight play a key role for the tempering quality prediction, amore detailed model had to be developed. In this paper, themelting kinetics is expressed as a shrinking core model of themelting particle. Consequently, the resulting kinetics has astrong physical meaning. Moreover, the obtained curves arecorrelated with the quality of the tempering as measured bydifferential scanning calorimetry (DSC) measurements.Therefore, the final model predicts the quality of temperingbased on the initial experimental conditions.

2 Materials and methods

2.1 Tempering experiment

The tempering experiments were done under the conditionsof the new patented process. A mass of chocolate (1, 2 or3 kg) was heated to 323 K. The bowl containing this cho-colate was then placed in an air-conditioned room and themelted chocolate was cooled by manual mixing. At a giventemperature, solid seeds of cocoa butter (5–10 mm diameterspherical pellets) were added to the chocolate. The bowlused for experiments was made of plastic, axisymmetric,shaped as a flat-bottom half sphere, with a depth at thecenter of 7 cm and a diameter at the upper surface of20 cm. Two type K thermocouples, located at the center ofthe chocolate at a depth of 3 cm, were connected to a dataacquisition system from Agilent Technology (Diegem, Bel-gium) recording the temperature every second. All theexperimental results presented were obtained using darkchocolate (composed of 50% cocoa mass and 50% sugar,with a total fat content of 27–28%) produced by Belcolade(Erembodegem, Belgium) and using spherical cocoa butterparticles as solid pellets (obtained from tempered PurePrime Pressed cocoa butter deposited on a moving belt andsolidified in a cooling tunnel).

2.2 Quality of the tempering analysis

A differential scanning calorimeter (DSC821e) from Mett-ler-Toledo (Zaventem, Belgium) was used to analyze thetempering of the chocolate during the experiments. Samplesof chocolate (between 5 and 25 mg) were taken from thebowl at different times during an experiment and placed in aDSC pan (40 mL in aluminum). The pans were rapidly ana-lyzed on the differential scanning calorimeter with the fol-lowing short program: starting at 298 K for 1 min, heating to313 K at 5 K/min, isothermal period of 1 min at 313 K. Themelting curves recorded give the fusion enthalpy that is pro-portional to the amount of crystals present in the chocolate.In that way it was possible to evaluate the quality and thespeed of crystallization of chocolate as a function of theconditions of tempering.

2.3 Mathematical model

A mathematical model of the temperature fields in the choco-late mass during a tempering experiment was developed. Adetailed description of the heat transfer equation can be foundin Debaste et al. [4]. The idea was to include the contributionof conductive and convective terms on heat transfer in onesingle term, expressed as a conductive term with an effectivethermal conductivity coefficient. The assumption made is thatconvective terms can be expressed as conductive terms. Thisapproach simplifies the transient heat balance equation to thefollowing expression [5]:

rlCpqTqt� keffr2T ¼ �Q (1)

where rl is the volumetric mass of the liquid chocolate (kg/m3), Cp is the heat capacity (J kg21 K21), T is the temperature(K), t is the time (s), keff is the effective thermal conductivity(W m21 K–1) and Q is the volumetric rate of heat taken by themelting of seeds (W/m3).

Equation (1), considering a known sink term, can besolved in combination with boundary conditions for con-ductive and radiative heat transfer from the surrounding air[6]:

�keffqTqx

����xL

¼ h T � Textð Þ þ seS T4 � T4ext

� �

(2)

where h is a heat transfer coefficient for the combination of theinner wall of the bowl, its outer wall and the conductiveresistance (W m–2 K–1), s is the Planck constant (W m–2 K–4),es is the bowl emissivity and Text is the ambient temperature.Details on the values of the coefficients for the boundaryconditions can be found in Debaste et al. [4].

In the previous study, the sink term had the form of akinetic reaction; its parameters were identified from an adia-batic melting experiment. In the present work, a model basedon mass and energy balances was set up to express the heatsink term.

It is assumed that the pellets are uniformly dispersed in thewhole chocolate mass due to the stirring. The heat needed tobring the pellets of seed crystals from their initial temperatureto their melting temperature is assumed to be transferredinstantaneously to the pellets. Upon pellets introduction intothe chocolate mass, the chocolate temperature is uniformlylowered to compensate this transfer of energy. The solving of acomplementary transport equation for the pellets temperatureis therefore avoided. Analytical calculations using a conduc-tion equation in a pellet submitted to an abrupt change oftemperature can assess the validity of this hypothesis [6]. Atime of 9 s is needed to reduce the temperature gradient in thepellet to 4% of its initial value. This time is of smaller order ofmagnitude than the characteristic time for the pellets melting,which is a few minutes.

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Eur. J. Lipid Sci. Technol. 2009, 111, 273–279 Modeling temperature in chocolate 275

The heat sink term can therefore be expressed as theproduct of a melting kinetics and the latent heat of fusion:

Q ¼ Ldcdt

(3)

where L is the latent heat of fusion (J/kg) and c is the con-centration of solid pellets (kg/m3).

The kinetics of melting of solid pellets is built on mass andconductive and convective energy balance equations on a solidpellet and on the molten chocolate surrounding it. The pelletsare assumed to be spherical. Figure 1 presents a schematicview of a pellet of radius R surrounded by a layer film ofthickness DR. The detailed derivation of the heat sink term isgiven in Appendix A. The final expression is given by

Q ¼ � 3c0kL

R20rS

1þ RDR

� �

T � Tmð Þ� � ffiffiffiffi

cc0

3

r

(4)

where rs is the volumetric mass of the solid pellets (kg/m3), c isthe concentration of solid pellets (kg/m3), c0 is the initial con-centration of pellets (kg/m3), R0 is the initial radius of a solidpellet (m) and Tm is the melting temperature of the solid pel-lets (K).

Equations (5) and (6) form a system of non-linear partialdifferential equations:

rlCpqTqt¼ keff DT � 3c0k

R20rs

1þ 1f

� �

T � Tmð Þ� � ffiffiffiffi

cc0

3

r

(5)

dcdt¼ � 3c0k

R20LrS

1þ 1f

� �

T � Tmð Þ� � ffiffiffiffi

cc0

3

r

(6)

where f is the ratio of the layer thickness DR and the radius ofthe pellet.

Simulations are started when solid pellets are added to thechocolate. The initial condition for the simulations is a uni-form temperature considering the cooling induced by the factthat pellets are not added at their melting temperature. Math-ematically, this is expressed as follows:

T t ¼ 0ð Þ ¼ Tci �Cppc0

rlCpTm � Tbið Þ (7)

where Tci is the temperature of chocolate at which solid pelletsare added (K), Tbi is the initial temperature of the solid pellets(K), Cpp is the specific heat of the pellets (J kg–1 K–1).

The above system of non-linear partial differential equa-tions with boundary conditions is solved by the finite elementsmethod using the commercial package COMSOL® 3.4(Comsol, Sweden). In this method, the studied geometry ismeshed, i.e. divided into polygonal elements. For each ele-ment, the solution of the equations is approached by a poly-nomial function. The computational domain for the simula-tions includes the chocolate mass only. The domain is dividedinto triangular elements. Figure 2 shows the geometry and thecorresponding mesh, composed of 1440 elements. This cor-responds to the coarsest mesh where the simulation results do

Figure 1. Schematic drawing of a spherical pellet of radius Rmelting surrounded by a heat transfer layer zone of DR thickness. ris the radial coordinate. The surface of the pellet is assumed to beat the melting temperature Tm while the bulk temperature is T.

Figure 2. Axisymmetric section of the bowl with mesh, imple-mented in COMSOL, for 2 kg of chocolate.

not depend on elements size and number. Evolution of thetemperature over 200 s has been simulated using the time-de-pendent iterative non-linear solver already implemented in theChemical Engineering Module of COMSOL.

3 Results

3.1 Measured and modeled temperature evolution inthe bowl

A comparison between experimentally measured values oftemperature and calculated values obtained from the modelwas made to validate the computational model.

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276 D. Bettens et al. Eur. J. Lipid Sci. Technol. 2009, 111, 273–279

Figure 3. Modeled and measuredtemperature in the bulk of chocolateas a function of time for three differ-ent ambient temperatures. 2 kg ofchocolate were used. Text is theambient temperature.

Figure 3 shows the evolution of the temperature of thechocolate obtained from experiments and from the modelwith 2 kg of dark chocolate at three different ambient tem-peratures. The model parameters are summarized in Table 1.The value of the parameter f = 1.3 has been determined sothat the simulated temperature evolution during time in thetempering process at a temperature of 295 K fits in sense ofleast squares to the experimental data based on the first 200 s.This short period is crucial for melting of the seeds. Moreover,the previously developed model, presented by the dashed line,based on an empirical kinetic equation for the sink term, pre-sented a too fast cooling during this initial period. Enhance-ment due to the new expression of the heat sink can thereforebe highlighted in this initial period. After 200 ms, the model isthe same as the one validated by Debaste et al. [4].

The evolution of the chocolate temperature during thetempering process can be explained by the physical phenom-ena happening. When the pellets are introduced into the mol-ten chocolate, the melting of the pellets, an endothermic phe-nomenon, induces a rapid decrease of the chocolate tempera-ture. The melting ends when the temperature of the chocolateis lower than the melting temperature of the pellets. If thechocolate is well tempered, there should remain some seeds ofpellets in the chocolate. Then, only the exchange of heat withambient air still occurs and leads to further cooling of thechocolate. This cooling is slower. The slope of the evolution oftemperature along time changes at around 100 s. Before thischange of slope, experimental fluctuations of temperature areobserved. Small non-uniformities of temperature exist in thestirred chocolate, for example, when a melting solid seedcomes close to a thermocouple. Except for these oscillations,the model shows a fairly good match with the experiments.The physical interpretation of the value of f = 1.3 is that, atany time, the order of width of the heat limit layer is the

Table 1. Values of parameters used in the model and sources.(*) corresponds to data from this work. (**) corresponds to datafrom Debaste et al. [4]. (***) stands for values measured at thebeginning of the experiment. (****) are adjusted values.

Parameter Value Source

Cp 1580 J kg–1 K–1 (*)Cpp 2010 J kg–1 K–1 (*)c0 129 kg/m3 (***)es 0.93 (**)f 1.3 (****)hout 6 W m–2 K–1 (**)k 0.2 W m–1 K–1 (**)keff 10.7 W m–1 K–1 (**)L 157000 J/kg (**)R0 0.0015 m (*)rl 1290 kg/m3 (**)rs 990 kg/m3 (**)Tbi 277 K (***)Tci 315.5 K (***)Text 295 K (***)Tm 306 K (*)

same as the radius of the grain. While such a value does notseem to be unrealistic, we cannot assure its physical validity.However, a parametric study, such as presented in Fig. 4,highlights a low influence of f on the temperature evolution.The low sensitivity of the equation system to the value of f ispartially mathematically explained by the fact that f appears in

Eqs. (5) and (6) in a term having the form of 1þ 1f

� �

. For

f ..1 this term reduces to 1 and is therefore independentof f. The dotted curve, for f = 3.3 is therefore really similar tothose obtained for any upper value.

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Eur. J. Lipid Sci. Technol. 2009, 111, 273–279 Modeling temperature in chocolate 277

Figure 4. Sensitivity analysis on the value of f. Tci is the initialtemperature of the chocolate.

3.2 DSC measurement

Two peaks are observed in a typical DSC curve of the melt-ing of a sample of chocolate during a tempering experiment(Fig. 5). The first one around 300.8 6 0.8 K corresponds tothe melting of form IV crystals, the second one, around304.2 6 0.4 K, corresponds to the melting of form V crys-tals [7]. In order to assess the quality of the tempering andthe amount of crystals present during the process, the areaof the second peak is plotted as a function of time during thetempering. Figure 6 presents results for five differentexperiments. Two distinct behaviors are observed:

(1) Form V appears since the beginning of the experi-ment. The amount of crystals grows fast and viscosity risesrapidly. The chocolate is over-tempered and cannot be fur-

ther used. This is due to a too large quantity of form V seeds.The pellets are not completely melted.

(2) An induction period of 50 min appears beforegrowth of the form V crystals. The chocolate is well tem-pered.

A third behavior, not plotted, corresponds to untem-pered chocolate, without form V crystals due to a completemelting of the seeds. Figure 6 also shows that very smallvariations of conditions can modify the tempering quality.

3.3 Tempering quality prediction

The high sensitivity of the result to the ambient conditionscomplicates the systematic determination of conditionsleading to a well-tempered chocolate. To limit the number ofexperiments required, it is proposed to define a criterion ofgood tempering based on the measured temperature in thechocolate. Such a criterion is based on the temperaturereached shortly after the end of the melting of seeds. At thattime, the temperature is characteristic of the melting and notof recrystallization. This criterion is a small (,1 K) intervalof temperature after a fixed given time.

It is then possible to use the presented model to predictthe tempering quality. This was done for three chocolates ofBelcolade. For each of these chocolates, an interval of tem-perature leading to well-tempered chocolate was identifiedcombining tempering experiments and DSC. Using themodel, it was possible to predict the temperature of additionof the pellets to have a well-tempered chocolate for a widerange of room temperatures (from 291 to 301 K). Thesimulations were realized for various masses of chocolate inthe bowl (from 1 to 3 kg).

4 Discussion

Combination of the model with this criterion represents apowerful tool, as it can be used to determine the initial tem-

Figure 5. Typical DSC curve.

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278 D. Bettens et al. Eur. J. Lipid Sci. Technol. 2009, 111, 273–279

Figure 6. Evolution of the form V heat of fusion as afunction of time for five different experiments. Text andTci are the ambient temperature and the initial temper-ature of the chocolate, respectively.

perature of melted chocolate at which pellets of seed crystalhave to be introduced to give a well-tempered chocolate over awide range of operating conditions, mass of chocolate andbowl geometries. In this way, it is possible to minimizeexpensive and time-consuming pilot tests and to have goodindications on the quality of the final products.

The presented model is a refined version of a previous one[4]; the heat sink term is changed to explicitly consider thetransport phenomena. This improvement clearly enhances thefitting quality of the model for the initial period where theseeds are melting. As this period is the key one for the tem-pering criterion, using this new sink term offers a significantimprovement to predict the tempering quality.

The model can be improved in particular by looking for amore reliable determination of heat transfer layer thickness. Inthis model, the ratio between the layer thickness and the pelletradius is kept to the constant value f. Using a more complexrelationship between layer thickness and pellet radius, forexample using classical correlation or computational fluiddynamics around a pellet, might improve the model. However,the main challenge is the change of the criterion of good tem-pering from a temperature determined empirically to a criter-ion relying on the kinetics of crystallization of the form V.

Although this model was developed for a particularpatented process, the principle of the modeling of the meltingcould be used in any other process where the heat transportphenomenon is limiting the melting.

Appendix A: Derivation of the sink term Eq. (4)

Firstly, mass balance applied to a solid pellet expresses thebalance between the solid mass lost during melting and theliquid mass formed during melting.

ddt

rs43pR3

� �

¼ �4pR2J (8)

where J is the melting rate per surface unit (kg m22 s–1).Mass balance on a solid pellet leads to

J ¼ �rsdRdt

(9)

Secondly, energy balance applied to the melting front of asolid pellet states that heat transported by conduction frommolten chocolate to the external surface of the pellet is usedintegrally to melt the solid pellet:

4pR2kdTdr

����R¼ 4pR2JL (10)

where r is the radial position (m).Thirdly, with the assumption of quasi stationary condi-

tions, mass balance applied on a liquid volume of chocolateleads to

4pvðrÞr2 ¼ 4pR2Jrl

� 4pR2Jrs (11)

where v(r) is the radial velocity of liquid chocolate (m/s) and rl

is the volumetric mass of liquid chocolate (kg/m3). Equa-tion (11) expresses the mass flux of liquid resulting from themelting of the chocolate due to a difference of volumetric massbetween the solid and the liquid. As the liquid is denser thanthe solid, the melting of the pellets leads to a local mass flow inthe direction of the pellets. This movement generates a con-vective heat transfer.

Fourthly, a steady-state energy balance applied to a liquidvolume (thus without sink term) included between position rand position r 1 dr, considering conductive heat transfer (left-

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Eur. J. Lipid Sci. Technol. 2009, 111, 273–279 Modeling temperature in chocolate 279

hand side) and local convective terms due to the liquid velocity(right-hand side), leads to

kddr

r2 dTdr

� �

¼ rlCpr2vdTdr

(12)

Combining Eqs. (7) and (8) gives

kddr

r2 dTdr

� �

¼ rlCpR2J1rl� 1rs

� �dTdr

(13)

The solution of this equation at r = R is

dTdr

����R¼ 1

R1fðT � TmÞ þ T � Tm

� �

(14)

Combining Eqs. (9), (10) and (14) gives

dRdt¼ � k

RLrs

1fðT � TmÞ þ T � Tm

� �

(15)

Finally, it can be shown that the radius of solid pellets is asso-ciated with the concentration of solid pellets in the chocolatemass. This concentration can be expressed as the ratio of themass of pellets added to the molten chocolate and the choco-late volume.

cðtÞ ¼ NpMpðtÞV

and c0 ¼NpMpð0Þ

V(16)

where Np is the number of pellets, Mp is the mass of one pellet(kg) and V is the volume of chocolate (m3).

Moreover, the mass of one pellet is expressed as a functionof its radius through

MpðtÞ ¼ rs43pRðtÞ3 (17)

Therefore, Eqs. (16) and (17) lead to

cðtÞc0¼ RðtÞ

R0

� �3

(18)

Differentiation of Eq. (18) leads to

dcdt¼ 3c0R2

R30

dRdt

(19)

Equations (3), (15) and (19) lead to the final expression of theheat sink term

Q ¼ � 3c0kLR2

0rs1þ R

DR

� ��

ðT � TmÞ� ffiffiffiffi

cc0

3

r

(20)

Acknowledgments

This research was conducted as part of a project funded by Belco-lade manufacture (Belgium). The authors wish to thank PierreDescamps (Belcolade, Belgium) for useful discussions. F.D. andB.H. acknowledge financial support from the Fonds National de laRecherche Scientifique, Belgium. The authors also wish toacknowledge the ARCHIMEDES project for its financial support.

Conflict of interest statement

The authors have declared no conflict of interest.

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