Heat transfer analysis of cheese cooling incorporating uncertainty in temperature measurement locations: Model development and validation

Successful modelling of the thermal behaviour of food products requires accurate knowledge of both the thermal properties of the product and the cooling media to which it is exposed to and a correct approach to validation of the model output against experimental temperature histories. This procedure is complicated where there is uncertainty in the location at which temperature is measured in the product. The cooling of a soft cylindrical cheese product that is partially immersed in brine and air is examined. As the cheese is very deformable, it is not possible to exactly identify the measurement location of the temperature-recording thermocouples. The Uniform and Exponential probability distributions were found to best represent the variability in thermocouple location at the centre and surface of the product, respectively. Expressions for the mean measured temperature at the centre and surface that result from the distributed nature of thermocouple location were obtained. These expressions were used to derive equivalent locations in the cheese which should be used when comparing model and experimental temperature predictions. The theoretical approach was also checked against Monte Carlo simulations and there was excellent agreement between them. It was shown that when validating model temperature histories against experimental readings that this phenomenon must be taken into account.     

Heat transfer analysis of cheese cooling incorporating uncertainty in temperature measurement locations: Application to the industrial process

A cylindrical cheese product with an irregular cross sectional geometry is cooled by partial immersion in brine. It is essential to calculate the temperature–time history of the product during this cooling phase to ensure its microbiological safety. The heat transfer analysis is complicated by both the non-standard product geometry and the non-uniform surface heat transfer coefficients that prevail. The flotation behaviour of the cheese at the air/brine interface is analysed to determine the heat transfer areas exposed to each cooling fluid. A finite element model is developed to predict temperature histories at various points in the product and to determine the slowest cooling region in the product. The cooling performance of the cheese is also studied experimentally. Comparison of the theoretical and experimental temperature histories is complicated by uncertainty in the location of the thermocouples in the product. By statistically quantifying the imprecision in thermocouple position, the expected temperatures in each region of interest can be found from the thermal model by Monte Carlo sampling. With the model, the top region of the cheese stick has been shown to determine cooling time because the existence of the non-uniform boundary condition is significant in determining the evolution of temperature.     

Temperature evolution in a partially submerged approximate cheese cylinder

An approximate cheese cylinder is cooled while partially submerged in a flowing brine solution. Microbiological considerations require the cheese to be cooled to 4 °C or less prior to packaging. Knowledge of the temperature evolution of the product is desirable from a manufacturing perspective, and we propose a model to describe the temperature time-history of a single cheese cylinder during cooling. The combination of unusual product geometry and the discontinuity in surface temperatures motivate the use of the finite element method (FEM). The FEM solution is used to describe the cooling process, and to ascertain the minimum time necessary for the cheese to attain the required packaging temperature. The FEM solution exhibits excellent agreement with the corresponding experimental findings. The discontinuity in surface temperatures has a significant influence on the cooling regime. Future process optimisation should focus on developing methods to increase the cheese surface area in contact with the brine.     

Modeling of simultaneous heat and mass transfer during convective oven ring cake baking

The convective oven ring cake baking process was investigated experimentally and numerically as a simultaneous heat and mass transfer process. The mathematical model described previously by the authors for cup cake baking was modified to simulate ring cake baking. The heat and mass transfer mechanisms were defined by Fourier’s and Fick’s second laws, respectively. The implicit alternating direction finite difference technique was used for the numerical solution of the representative model. Prior to the utilization of the developed model in predicting the temperature and moisture profiles for ring cake baking, the results of the numerical model were compared with analytical results involving only heat or mass transfer with constant thermo-physical properties. Excellent agreement was observed. The numerical temperature and moisture contents predicted by the model were compared with the experimental profiles. They agreed generally reasonably well with the experimental temperature and moisture profiles.     

粮堆内热湿耦合传递数值模拟与试验验证

通风过程中粮堆内热湿耦合传递对储粮安全和粮仓通风有着重要的影响。建立了机械通风冷却过程中粮堆内热湿耦合传递数学模型,在此模型的基础上用C语言编写了用户自定义源项(UDF),实现了谷粒吸附和解析水分时热量交换和水分迁移的CFD数值模拟。以某高大平房仓为例,建立了通风冷却过程中粮仓的三维物理模型;以瞬态条件下外界环境随时间变化的空气温湿度为入口边界条件,通过对比CFD数值模拟结果与实测数据,验证了粮堆内热湿耦合传递数学模型,得出通风冷却过程中粮堆内热湿耦合传递规律。     

火腿鼓风冷却过程的三维数值模拟

食品的冻结速率是影响冷冻食品质量的主要因素之一。近年来数值模拟在食品冷却过程中得到了广泛的应用。本文应用有限容积法对火腿鼓风冷却过程进行数值模拟,研究了火腿冷却到指定温度3℃时的时间,并将计算结果与实验结果进行比较分析。结果表明:在外界空气温度为1℃时,大约需要12h左右,火腿的中心温度可以达到指定的冷却温度,且数值模拟的结果与实验结果符合得很好。这说明数值模拟能有效地预测食品冷却过程。     

Interactions of heat and mass transfer in steam reheating of starchy foods

Steaming is a common practice in both household and industrial food processing; however, the process of heat transfer during steaming has not been fully explored. In this study, mathematical models that consider coupled energy, water and vapor transport were developed, and were solved using finite element software (COMSOL Multiphysics) for their application to the steam reheating of non-porous starch gel radish cake and steamed buns with porous matrices. Our simulation results indicated that the reheating of non-porous starch gel is a heat conduction process. During the reheating of the porous starch matrix, the infusion of condensate and evaporation–condensation mechanism contribute to steep sigmoid-shaped temperature profiles that are different from those of normal heat conduction. The validity of our mathematical model is corroborated by our experimental work of the steam reheating of samples with and without PE film wraps in a rice cooker. It shows a good agreement between the simulation and experimental results in terms of temperature and increased moisture. The implications of these models can provide a basis for future elucidation of more complicated food steaming processes.     

Temperature evolution and mass losses during immersion vacuum cooling of cooked beef joints - A finite difference model

A finite difference model was developed to describe and predict the temperature and mass loss evolution in reconstructed beef joints during immersion vacuum cooling. Fast cooling is obtained within beef pores and at beef surface when evaporation in the surrounding liquid is high. The cooling rate diminishes as the vacuum chamber pressure stabilizes and the liquid temperature reaches its lower value. The maximum deviation between measured and calculated temperatures was within 5°C for the beef (core and surface) and within 7°C for the surrounding liquid (measured at the bottom of the container). Absolute differences between predicted and experimental mass losses for the liquid and beef sample were around 2% and 1%, respectively. Mass losses are higher during the first period when evaporation is the main mode of heat transfer. Mechanical agitation in the surrounding liquid is suggested as a way to further reduce cooling times and to prevent uneven cooling.     

仓储粮堆内部自然对流和热湿传递的数学分析及验证

本文基于仓储粮堆内部自然对流、热湿耦合传递的数学模型,采用数学分析的方法对模型中各个方程中的各项的物理意义和数量级大小进行了分析,探讨了仓储粮堆内部自然对流、热量传递和水分迁移过程的相互关系。提出了判断粮堆内部自然对流强弱的瑞利数及其影响因素,分析了仓型结构、粮种及仓外大气温度对粮堆内部自然对流、热量传递和水分迁移的影响,并通过数值模拟对数学分析结果进行了验证。结果表明,数学分析方法是分析仓储粮堆内部自然对流、热量传递和水分迁移过程的一种有效途径,数学分析的结果可以为仓型设计、储粮生态系统的模拟、仓储技术管理提供借鉴。     

Modeling cooling of ready-to-eat meats by 3D finite element analysis: Validation in meat processing facilities

A three-dimensional (3D) finite element model for simulating heat transfer during cooling of irregular-shaped ready-to-eat meat products was developed and validated. The heat transfer model considered conduction as the governing equation, subject to convection, radiation and moisture evaporation boundary conditions. A 3D finite element algorithm developed in Java™ was used to solve the model. The algorithm generated solutions for meshes containing 4-node tetrahedral volume elements and 3-node triangular boundary elements. Product geometries were generated from CT-scan images of the meat products. The model was adapted to receive input parameters that can be easily provided by a meat processors including air relative humidity, air temperature, air velocity, type of casing, duration of water shower, product weight, and estimated core temperature of product prior to entering the cooling chamber. Model validation was conducted in four commercial facilities, under normal processing conditions. Temperatures predicted by the model were in agreement with observed values. Average root-mean-square error (RMSE) was 1.19 ± 0.54 °C for core temperatures, 1.73 ± 0.48 °C for temperatures 0.05 m from core to surface, and 2.01 ± 1.01 °C for surface temperatures. The developed heat transfer model was integrated with predictive microbiology models through a food safety website: numodels4safety.unl.edu. The integration can be useful for estimating the severity of cooling deviations and resulting microbiological safety caused by unexpected cooling process disruptions.     

Uncertainty and sensitivity analysis: Mathematical model of coupled heat and mass transfer for a contact baking process

Similar to other processes, the modelling of heat and mass transfer during food processing involves uncertainty in the values of input parameters (heat and mass transfer coefficients, evaporation rate parameters, thermo-physical properties, initial and boundary conditions) which leads to uncertainty in the model predictions. The aim of the current paper is to address this uncertainty challenge in the modelling of food production processes using a combination of uncertainty and sensitivity analysis, where the uncertainty analysis and global sensitivity analysis were applied to a heat and mass transfer model of a contact baking process. The Monte Carlo procedure was applied for propagating uncertainty in the input parameters to uncertainty in the model predictions. Monte Carlo simulations and the least squares method were used in the sensitivity analysis: for each model output, a linear regression model was constructed and the standardized regression coefficients (SRCs) and R² were computed. The effect of input parameters on model predictions was calculated, and the relative impact of the parameters on each of the outputs was ranked. Results of the uncertainty and sensitivity analysis can be used to prioritize future experimental efforts, as discussed for the contact baking process.     

食品真空贮藏过程中温度变化的数值模拟与有限元分析

食品真空贮藏过程是复杂的传热传质过程，该过程涉及到扩散、传热、传质等机理。在能量守衡基础上建立数学模型，通过数值求解得到不同真空压力下食品温度随时间的变化曲线，并通过有限元分析，与数值模拟的结果进行对比分析，为下一步的实验验证及食品冷冻冷藏工程的实践提供理论依据。     

Mathematical modeling of the heat transfer process and protein denaturation during the thermal treatment of Patagonian marine crabs

Southern Ocean swimming crab Ovalipes trimaculatus and the Patagonian stone crab Platyxanthus patagonicus are fishing resources with commercial value. Thermal treatment of crabs is necessary to denature muscle proteins, facilitating meat detachment from the crab shell (picking procedure). The proximal composition, protein patterns of crab muscle, thermophysical properties and heat transfer coefficients were determined. Heat transfer during thermal processing of body (i.e., cephalothorax) and claws of both crab species was simulated using a finite element computational code; the simulations were experimentally validated. Color changes in crab muscle during the heating process were measured. Thermal denaturation kinetics of myofibrillar proteins was determined using Differential Scanning Calorimetry (DSC) in small samples previously heated in water under controlled conditions. DSC thermograms of raw crab muscle showed two peaks at 49.0 ± 0.4 and 77.5 ± 0.6 °C corresponding to myosin and actin respectively. Activation energies for the denaturation of myosin (145.70 kJ/mol) and actin (156.42 kJ/mol) were calculated from Arrhenius equation. The degree of denaturation achieved by the myofibrillar proteins at the coldest point of the muscle in body and claws during the heating process was established by considering the protein denaturation kinetics determined by DSC, the activation energies and the heat penetration curves. Adequate conditions for the detachment of meat from the crab exoskeleton were established. The obtained results may help in determining the optimal heating times during the industrialization of these crustaceans.     

Simulation of a forced-air-twin-chamber for measuring heat treatment uniformity in harvested tomatoes

A heat transfer simulation model combining convection and conduction was developed to predict accurately the temperatures of air and produce (tomatoes) placed within a custom designed experimental set-up used for the measurement of the uniformity of hot air treatment. The thermal conductivity of the produce and air humidity have significant effects on the uniformity of heat treatment, particularly on the temperature gradients generated in the design. Tomato position, signifying the length of tunnel, is another factor to be considered when designing such experimental device. While air velocity is a factor affecting heat treatment uniformity, the tomato orientation did not show any significant effect. The simulation was experimentally validated and found to be accurate. This method could also be used for other fruits.     

羽绒制品热传递的有限元仿真

针对现有羽绒制品保暖性测试方法的测试时间较长、成本较高等不足,提出一种数值模拟测试方法。采用ANSYS有限元软件建立“羽绒-织物-皮肤”的三维几何模型,模拟分析不同织物、不同填充密度、不同绗缝数量条件下模型内部与外表面的温度分布特征,探究不同条件对热量沿厚度方向和宽度方向传递的影响,计算其保暖性,并进行实验验证。结果表明：羽绒填充密度增加,热量沿模型厚度方向的传递距离增大,外表面温度及热流密度减小;绗缝数量增加,热量沿模型厚度方向的传递增大,沿模型宽度方向的传递距离减小,外表面温度及热流密度逐渐增大;织物对羽绒制品保暖性的影响较小;模拟结果与实验结果的最大相对误差为4.79%,二者吻合度较好。     

Experimental study of heat transfer and air flow in a refrigerated display cabinet

The heat transfer and air flow in a vertical open refrigerated display cabinet loaded with packages of test product was studied in order to analyse unsteady state phenomena. The product and air temperatures in the curtain were measured using tiny thermocouples, air velocity using a hot wire anemometer and PIV (Particle Image Velocimetry). A comparison between the air velocities measured by these two methods shows a good agreement at the internal side of the air curtain. However, these two methods give quite different result at the external side of the air curtain which can be explained by the complex air flow in this region. The temporal variation of the air temperature inside the air curtain is due partly to the “on/off” compressor cycle and partly to the introduction of ambient air via vortices. The latter phenomena contribute to the rapid velocity fluctuations which are greatest in the mixing layer of the air curtain. The product position in the cabinet is a determining factor of its temperature.     

CFD modelling of flow and scalar exchange of spherical food products: Turbulence and boundary-layer modelling

The performance of several steady Reynolds-averaged Navier–Stokes turbulence models and boundary-layer modelling approaches is evaluated for a single sphere, by comparison with empirical data for a Reynolds number range of 10–3.2 × 10⁴. A sphere serves here as a representative model for many spherical food products. The shear stress transport (SST) k–ω turbulence model performs exceptionally well when combined with low-Reynolds number modelling (LRNM) of the boundary layer, which confirms that the turbulence model characteristics are particularly suitable to deal with this specific flow problem. Especially the k–ε turbulence models are less accurate at higher Reynolds numbers (>10²). Boundary-layer modelling with wall functions (WFs) leads to inaccurate flow-field and scalar transfer predictions, compared to LRNM. However, LRNM grids and their inherently higher computational cost are often not practically feasible, leaving WFs as the only option. It is shown that using cell sizes on the sphere surface of a few millimetres, typical for CFD studies on food products, can compromise accuracy, and grids with smaller cell sizes are actually required.     

Performance evaluation of aluminum test cell designed for determining the heat resistance of bacterial spores in foods

Thermal inactivation kinetic studies are necessary to determine heat resistances of spores in the development of new thermal processes for low-acid shelf-stable products. Most currently available sample holders used for solid and semi solid samples in the kinetic studies take long time to reach the target sample temperature, hence fail to provide isothermal condition. In this research, novel aluminum test cells were developed to facilitate easy loading and unloading samples in a hermetically sealed 1 ml cavity to evaluate the heat resistance of bacterial spores when heated at temperatures above 100 °C. Design of the test cell was governed by minimum come-up time. A finite element model based on the commercial software ‘FEMLAB’ was used to simulate transient heat transfer and finalize the test cell dimensions. Performance of the new test cell was evaluated against capillary and aluminum thermal death time tube methods in characterizing the heat resistance of Clostridium sporogenes PA 3679 spores in a phosphate buffer and mashed potato at 121 °C. D₁₂₁ values of PA 3679 spores in both the phosphate buffer and mashed potato using the new test cells were not significantly different (P>0.05) from those by the capillary tube method. The results indicated that the new test cell is appropriate for studying the inactivation kinetics of bacterial spores in microbial validation of conventional and novel thermal processes for low-acid shelf-stable foods.