Original Research PaperA population balance model for the chocolate roller refining process
Graphical abstract
Fig. 1: (a) schematic passage of chocolate paste through the gap between the set of rollers and (b) Specification of the control volume in the grinding zone.
Introduction
Chocolate is a semi-solid suspension of sugar and cocoa particles in a continuous fat phase [1]. The delicious taste of chocolate is because it evokes a range of stimuli that activates pleasure centers of the brain [2]. The quality of chocolate is attributed mainly to its rheological behavior that is influenced largely by its ingredient composition, particle size distribution (PSD), and processing techniques [3]. Chocolate producers pay special attention to the PSD of the chocolate ingredients as this factor has a critical effect on the mouthfeel of the product [4]. An excessive portion of the particles with a diameter greater than 30 µm is often stated to cause a gritty perceived taste [2]. Therefore, during the chocolate manufacture, the particles are ground to reach an optimal PSD (comminution process).
Ball milling and roller refining are considered as the two traditional processes to reduce the PSD in the chocolate manufacturing process [5]. The comparison between the performances of the ball milling and roller refining has been widely discussed in the literature [6], [7], [8], [9]. The fundamental difference between these processes is the exposure of particles to the grinding forces. The breakage of particles in roller refining is because of the shear stress and pressure caused by the different rotational velocities and narrowing distance of the rolls during the passage, respectively. While in ball milling, the grinding of particles is due to the exposure to short-time, high-impact pulses induced by the contacts of particles with balls/walls [6].
Several modeling approaches have been proposed to improve the understanding of the roller refining process. The main focus of the studies was to simulate the high-pressure grinding roll (HPGR) used in the grinding/compression of minerals/ores [10], [11]. Fuerstenau and co-workers [12], [13] proposed a roller grinding kinetics equation in terms of energy input. They introduced an energy dissipation component to normalize the specific selection function. Austin and co-workers [14], [15] modified the roller crusher model for two dominant breakage mechanisms: contact and particle-bed. Morrell and co-workers [16] developed a modular model considering three grinding zones in an HPGR. They applied the drop-weight test to fit the parameters of their model. The group around Tavares [17], [18], [19] used the discrete element method (DEM) to simulate the breakage of particles in HPGR.
Fig. 1 portrays schematically the three zones that the particles pass through in the space between the two counter-rotating rolls [16]. The feed enters from the acceleration zone and transfers to the grinding zone where the particles break under the effect of the shear stress and pressure caused by the rolls [20], [21].
In the grinding of minerals, the air is the primary/continuous and solid particles are the secondary/disperse phases of the particulate system. As minerals/ores get compressed in the HPGR, the trapped air in the void space between the particles leaves the bed from the top. Therefore, the density of the bulk (air + particles) increases. The compression continues up to the center-center line of the rolls as the width of the grinding zone decreases. The compression zone is represented by the nip angle (Fig. 1). The angle depends on the material properties, the roll surface pattern, and roll rotational speed [10]. The third zone after the compression zone is the relaxation zone where the particles face hardly any pressure [16].
Population balance models (PBM) have been widely employed as a convenient way to model particulate systems [22], [23], [24], [25], [26]. The quality of a PBM depends on the adequacy of the chosen rate terms for growth, breakage, and aggregation [27], [28], [29]. Liu developed the first PBM for the HPGR [30]. They provided two mass and energy balance equations to account for the effect of energy adsorption in different size classes. They used the piston-die tests for mono-sized particles to determine the breakage functions. Torres and Casali [31] developed a modular discretized PBM based on the Morrell-Tondo-Shi model [16]. Torres and Casali [31] considered that there are two types of compression in the grinding zone: single-particle compression (SPC) and particle bed compression (PBC). SPC happens when the diameter of a particle is greater than the local width of the grinding zone. Therefore, the particle breaks to pass through the space between the rolls. While PBC happens when a bed of particles is put under the shear stress/pressure caused by the rolls. However, they assumed that the single-particle compression happens in a specific region between the rolls [31]. They considered the SBC as an event-based phenomenon where the big particles instantly break into particles smaller than a certain size [31]. Moreover, they considered constant axial velocity and breakage function for the bulk passes through the space between the rolls. The study of Torres and Casali [31] can be considered as the benchmark of the HPGR population balance modeling as their methodology was followed in other studies [32], [33], [34], [35], [36], [37].
Based on our literature research, we could not find any study that developed a PBM to simulate the chocolate refining process. As the chocolate roller refinement is in its essence an HPGR, we were mainly inspired by the correspondingly developed PBMs. However, there are fundamental differences between the HPGR and chocolate roller refining. The chocolate paste is composed of cacao butter/soy lecithin fats as primary/continuous, and sugar/cacao particles as secondary/disperse phases [1]. Therefore, unlike the roller grinding of minerals, the density of chocolate paste does not change by the compression. Moreover, in the grinding of minerals in HPGRs, the rotational velocities of a pair of rolls are typically equal. While the rotational velocities of a pair of rolls in the chocolate roller refining are typically different. This emphasizes the importance of shear stress in the roller refining compared to the compression of minerals/ores in HPGRs.
This work aims to develop a population balance model to interpret the evolution of PSD in the chocolate roller refining process. The continuity equation was used to calculate the axial velocity of the chocolate paste as it passes through the space between the rolls. The PBM of Torres and Casali [31] was extended/enhanced to model the possible steady-state SPC/SBC in all points in the space between the rolls. The parameters of the model were adjusted using the experimental (industrial) data provided by the Bühler AG Company (Uzwil, Switzerland). To the best of the authors’ knowledge, this is the first work where the population balance and continuity equations are coupled to model the breakage of sugar particles in the chocolate roller refining process.
Section snippets
Description of the industrial unit
In this section, a brief description of a typical industrial process unit of a 5-roll refiner for chocolate refining is presented (Bühler AG). An illustration of the most common 5-roll refiner with a length () of 180 cm is shown in Fig. 2-a. As the chocolate paste advances in the equipment, it passes through four stages of roller grinding. The schematic passage of the chocolate paste through the gaps between the set of rolls where represents the stages of roller grinding )
Mathematical model
A phenomenological model was developed to analyze the grinding/breakage of sugar particles as they pass through the space between the chocolate roller refiners. To model the system, sugar particles’ continuity and population balance equations for the space between the rolls were coupled. The control volume of the grinding zone used for the model derivation is shown in Fig. 3. In this figure, the positive direction of is towards the center-center line of the rolls. is the height of the
Results and discussion
The values of the estimated parameters for powder and crystalline sugars are presented in Table 2. Here, the parameters were estimated using all of the experimental PSDs of the four stages of roller grinding for the powder and crystalline sugar operational cases (Section 2).
In the formulation of the breakage frequency function (Eq. (17)), represents the specific rate of a reference size in respect to the power needed to rotate the rolls and apply pressure to the chocolate paste [31]. Also,
Conclusion
A study on the grinding of sugar particles in the chocolate roller refining process was presented. As the main contribution of the work, a model based on continuity and population balance equations was developed to explicate the experimental (industrial) data provided by the Bühler AG company (Uzwil, Switzerland). The model proposed equations to calculate the axial velocity, grinding height, nip angle, and sugar PSD of chocolate paste as it passes through the space between the rolls based on
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors would like to acknowledge the Bühler AG company (Uzwil, Switzerland) and German Institute of Food Technology (DIL), Quakenbrück specifically Dr. Ute Bindrich, Dr. Dana Middendorf, Sarah Schroeder, and Dr. Knut Franke for providing valuable information and industrial data. This IGF Project of the FEI is/was supported via AiF within the program for promoting the Industrial Collective Research (IGF) of the German Ministry of Economic Affairs and Energy (BMWi), based on a resolution of
References (47)
- et al.
Factors influencing rheological and textural qualities in chocolate - a review
Trends Food Sci. Technol.
(2007) A first survey of grinding with high-compression roller mills
Int. J. Miner. Process.
(1988)- et al.
Energy consumption and product size distributions in choke-fed, high-compression roll mills
Int. J. Miner. Process.
(1991) - et al.
A preliminary model of high pressure roll grinding using the discrete element method and multi-body dynamics coupling
Int. J. Miner. Process.
(2016) - et al.
Experiences in dry grinding with high compression roller mills for end product quality below 20 microns
Miner. Eng.
(1990) - et al.
Scale-up procedure for continuous grinding mill design using population balance models
Int. J. Miner. Process.
(1980) - et al.
Population balance modeling of non-linear effects in milling processes
Powder Technol.
(2005) Simulation of crystal size and shape by means of a reduced two-dimensional population balance model
Chem. Eng. Sci.
(2006)- et al.
Identification of the breakage rate and distribution parameters in a non-linear population balance model for batch milling
Powder Technol.
(2011) - et al.
Solution of inverse problems in population balances-II. Particle break-up
Comput. Chem. Eng.
(1995)
Solutions of inverse problems in population balances-I. Aggregation kinetics
Comput. Chem. Eng.
Modelling of interparticle breakage
Int. J. Miner. Process.
A novel approach for the modelling of high-pressure grinding rolls
Miner. Eng.
Systems optimization model for energy management of a parallel HPGR crushing process
Appl. Energy.
Robust HPGR model calibration using genetic algorithms
Miner. Eng.
Modeling and energy-based model predictive control of high pressure grinding roll
Miner. Eng.
Science and Technology of Enrobed and Filled Chocolate, Confectionery and Bakery Products
Sci. Technol. Enrobed Fill. Choc. Confect. Bak. Prod.
On the solution of population balance equations by discretization - I. A fixed pivot technique
Chem. Eng. Sci.
The effect of ball size on mill performance
Powder Technol.
The back-calculation of specific rates of breakage and non-normalized breakage distribution parameters from batch grinding data
Int. J. Miner. Process.
Characterization and modeling of a sub-micron milling process limited by agglomeration phenomena
Chem. Eng. Sci.
Developing breakage models relating morphological data to the milling behaviour of flame synthesised titania particles
Chem. Eng. Sci.
A robust parallel algorithm of the particle swarm optimization method for large dimensional engineering problems
Appl. Math. Model.
Cited by (1)
Comparison of the Dynamic and Thermal Behavior of Different Ideal Flow Crystallizers
2023, ChemEngineering