NUMERICAL SIMULATION APPLIED TO MILK COOLING

A model is a representation of a real system that can be analysed and yield predictions under different operating conditions. The aim of this study was to model a milk cooling tank that cools milk to 4 °C to preserve its quality after milking at the farm. The model was developed and simulated using the software Ansys for finite element analysis. The results from the simulations were compared to experimental data. The model simulated milk cooling in the tank with an error lower than 2%, which is considered acceptable for numerical simulations. In other words, the model satisfactorily represents the real system. Thus, alternatives can be directly tested in the computational model to improve and optimise the milk cooling process and to better use the system without actually implementing them in the real system.


INTRODUCTION
Brazil produced 35.3 billion litres of milk in 2018, and its annual growth rate will be between 2.1 and 2.9% in the next 10 years (MINISTÉRIO DA AGRICULTURA, PECUÁRIA E ABASTECIMENTO, 2018). Efforts have been concentrated on improving the production process and on accelerating the incorporation of technologies, especially in medium and large dairy farms (EMBRAPA, 2011).
This sector has also tightened product quality requirements by adopting measures aimed at improving Brazilian milk standards. The search for quality seeks not only gains in the productivity and profitability of dairy farmers and the industry but also guaranteed food quality and safety and, mainly, consumer health (EMBRAPA, 2011). As the consumption of milk and dairy products is linked to principles of nutrition and health, ensuring the safety and quality of these products is essential for the industry.
Thus, the focus has been on dairy farmers because that is where the quality process starts. The cooling temperature of milk is directly related to the preservation of its quality. According to Normative Instruction No. 62 (2011), the cooling temperature should be 7 °C at most on the farm, and the milk should reach this temperature within 3 hours after milking. Even at this temperature, the bacterial load may still increase during storage. However, if the temperature remains lower than 4 °C, the bacterial count practically does not change in 48 hours (CLÍNICA DO LEITE, 2016). RAGSDALE (2009) stated that a computational model is a representation of a real-world problem or system and that when using a model, decision alternatives can be analysed before a specific plan is chosen for implementation in the real system. PENG et al. (2017) mention that when using a model, improvements can be proposed and alternatives can be tested and optimised to predict the best approach without performing physical experiments. Computational models are used for heat transfer analysis in a wide range of applications, as observed in Atangana (2016), Sheikholeslami andGanji (2014), andSheikholeslami et al. (2015).
With the significant increase in computational capacity, the finite element method (FEM) has been developed and widely used for its high reliability and accuracy of results (NIMDUM et al., 2015). The FEM is widely used for the thermal analysis of systems (ZHAO, 2014).
This study was performed because numerical simulation has never been applied to processing in the dairy industry in similar research.
Thus, this study aims to model a milk cooling tank and to compare the simulation data to experimental results.

MATERIALS AND METHODS
At the farm where the study was conducted, in the municipality of Ingaí -Minas Gerais (MG), Brazil, all milk from both daily milkings is stored in a bulk tank until being hauled to the dairy industry. The bulk tank has a maximum capacity of 1650 L, with a height of 96 cm and a base diameter of 152 cm. The coating material is stainless steel, with inner and outer layers and with (2-cm-thick) expanded polystyrene between the two layers. The tank cools the milk through a heat exchange system, where chlorodifluoromethane gas (also known as R22) is circulated between the bottom of the tank (heat loss), and the outside (cooling), resuming the cycle. Temperature homogenisation is provided by a rectangular single-blade impeller (10 cm height and 60 cm length). Spinning at a constant speed of 2.07 m s -1 , the stainless-steel paddle stands 10 cm from the bottom of the tank supported by a bar vertically fixed to the lid of the tank. The vertical bar was disregarded in the simulations. Figure 1 shows one of the geometries used to model milk in the bulk tank, with the milk volume and blade impeller.
The milk temperatures inside the tank were measured using the internal temperature monitoring system of the tank, consisting of a probe located 10 cm above the bottom of the tank on the sidewall.
The heat fluxes assessed when cooling specific milk volumes on different days were calculated using Equation 1. (1) where is the heat flux (J.s -1 .m -2 ); is the amount of heat (J); is the time (in seconds); and is the area of contact between milk and the bottom of the tank (m²).
The computational model developed and used to simulate the milk cooling process assumes the following working hypotheses: a transient state (as a dynamic system over time), a three-dimensional flow, and an isothermal system. As a boundary condition, a non-slip condition was considered for the boundaries with solid surfaces.
The momentum conservation equation used by the model is shown below (Equation 2): (2) where is the density; is the static pressure; is the stress tensor; and is the gravitational force.
The k-ε model was used to predict the turbulence inside the system during the simulation and is given by Equation 3: (3) where is the density; is the turbulent kinetic energy; is the component of the velocity in the corresponding direction; is the component of the deformation rate; and represents the turbulent viscosity, where ; The energy dissipation inside the cooling system is given by  where is the energy; is the temperature; is the effective thermal conductivity; and is the heat source term.
Computer simulations were performed using the commercial software Ansys version 14.2. The tank was considered thermally insulated on the side and bottom walls. To model the system, only the milk volume was considered in the simulations, that is, a cylindrical volume based on a diameter of 152 cm and a variable height, which was calculated based on the volume occupied in a given situation.
The meshes of the geometries used for each milk volume modelled in this study have tetrahedral elements and are shown below, in Figure 2. Table 1 shows the numbers of nodes and elements for the meshes of the three milk volumes tested in this study.
The physical properties considered in modelling the system are outlined in Table 2.

RESULTS AND DISCUSSION
The results from the observations of the milk volume in the tank, initial and final temperatures, total time, and heat flux at the bottom of the tank during cooling are outlined in Table 3. Figures 3, 4, and 5 show the temperature distribution in the tank at 1800 seconds (30 minutes) after starting the simulation, for the volumes of 305 L, 550 L, and 775 L, respectively. REZENDE, R. P. et al. Table 4 outlines the time required to cool the milk to 4 °C in the real system and the time required to cool the milk to 4 °C through simulation. The absolute percentage error was calculated for each observation of each simulated milk volume using Equation 7.
The overall mean absolute error calculated for the three milk volumes simulated was 1.20%. According to MOHAPATRA and RAO (2005), deviations of up to 10% between the real values and values determined by a computational model simulation are considered satisfactory and indicate that the computational model can be used to represent the real situation. Thus, the model developed in this study performs satisfactorily and can be used to represent the real milk cooling tank under specific conditions.

CONCLUSION
• A computational model was built and simulated to represent a milk cooling tank used to cool milk to 4 °C within 3 hours. The model was tested for three volumes in the tank, with different temperatures at the beginning of the process. The mean absolute percentage error between the time elapsed in the real situation and the equivalent time in the simulation for the simulated milk volumes was 1.20%.  • The findings show that the model can be used to represent the real milk cooling tank within an acceptable error without having to directly test alternatives in the real system. Any changes introduced to optimise the process, whether in the efficiency of the cooling system or in the energy efficiency of the equipment, among others, can be initially tested in the computational model, which provides a good estimate of how the real system would behave under such changes, thereby increasing the efficiency whilst reducing costs and streamlining the experimentation process.