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Estimation of the energy requirement of bread during baking by inverse heat transfer method
Abstract
Inverse heat transfer is a more efficient method for estimating unknown quantities of variable interest. The aim of present research work is to successfully predict the energy consumption to bake the bread at different baking oven temperatures during baking processes using the inverse heat transfer method. This inverse technique allows researchers to avoid the usage of intricate and expensive instrumentation. This study also compared different numerical techniques for estimating accurate sensitivity coefficients. The inverse heat transfer problem is presented as a multi-parameter estimation of heat flux and solved by the Levenberg–Marquardt algorithm. The finite element method is applied to solve the transient standard heat transfer problem while considering nonlinear two-dimensional heat transfer. The results demonstrated that the complex variable differentiation method was given the satisfactory results than the forward difference method and central difference approximation method. In order to demonstrate the accuracy of the results, statistical analysis is performed for estimated parameters. A good agreement of results is obtained with help of the inverse heat transfer problem. This developed model provides the information to enable the energy required to cook any food product in food thermal processing accurately.
... The energy requirement in the baking process is reportedly between 500 and 7300 kJ/kg. [2][3][4][5] It is normally agreed that baking, from an energy consumption point of view, is similar to drying semisolid food. 6 The inverse heat transfer problems (IHTPs) are currently generating a lot of interest in the field of science and engineering. ...
... [10][11][12][13][14][15] The inverse applications have been widely applied in the area of heat transfer, mechanics, and also in physics, chemistry, and biology. 2,7,16 However, this concept in food engineering is rapidly gaining more prominence. The estimation of thermal properties and mass transfer parameters by inverse problems in different food and thermal processes have been studied. ...
Direct heat transfer problems can be solved analytically or numerically to predict the temperature profile when the thermal properties, boundary conditions, and other relevant parameters are known. Though it is common practice to measure temperature experimentally, heat transfer parameters and boundary conditions are more challenging to measure and can instead be inferred through the use of inverse heat transfer (IHT) techniques, which can be solved through optimization. In this study, the IHT method with the conjugate gradient method is used to determine the energy consumption of bread during the cooking process in a developed baking oven with and without a reflector. A complex variable differentiation method is integrated to calculate the accurate sensitivity coefficient matrix. The results demonstrated that the estimated heat flux is very close to the exact heat flux and relative error is less than measurement errors.
... For example, research in [26] highlights the estimation of moisture diffusivity by inverting a finite element model applied to various solid foods such as salami, biscuits, and flatbread by minimizing the distance between the numerical model and experimental results through the Levenberg-Marquardt algorithm. Just as ensuring the properties of heat-treated foods is of scientific and technological interest, efficient energy consumption in the processes is also an aspect that may be optimized using similar techniques [27]. In [28], the authors highlight that using the bootstrap technique can estimate both specific heat and thermal conductivity in a freezing range, comparing the obtained results with experimental data. ...
This article proposes a framework for estimating temperature fields in food-freezing applications that significantly reduces computational load while ensuring accurate temperature monitoring, representing a promising technological tool for optimizing and controlling food engineering processes. The strategy is based on (i) a mathematical model of a convection-dominated problem coupling thermal convection and turbulence and (ii) a least-squares approach for solving the inverse data assimilation problem, regularized by projecting the governing dynamics onto a reduced-order model (ROM). The unsteady freezing process considers an idealized salmon slice in a freezer cabinet, modeled with temperature-dependent thermophysical properties. The forward problem is approximated using a third-order WENO finite volume solver, including an optimized second-order backward scheme for time discretization. We employ our data assimilation framework to reconstruct the temperature field from a limited number of sensor data and to estimate temperature distributions within frozen food. Sensor placement is optimized using a new greedy algorithm, relying on maximizing the observability of the reduced-order dynamics for a fixed set of sensors. The proposed approach allows efficient extrapolation from external sensor measurements to the internal temperature of the food, which is crucial for maintaining food quality.
... For example, one may cite the estimation of the thermal conductivity of polymeric materials [16], the attainment of the unknown functional form of a time-dependent heat transfer coefficient [17], and the estimation of the thermal conductivity and volumetric heat capacity of living tissue using a recent noninvasive measurement method [18]. In addition, there are IHTP applications in food science and engineering for assessing process parameters, for example, the energy consumption during baking [19] or temperature-dependent food properties such as moisture diffusivity [20]. ...
Thermally characterizing high-thermal conductivity materials is challenging, especially considering high temperatures. However, the modeling of heat transfer processes requires specific material information. The present study addresses an inverse approach to estimate the thermal conductivity of SAE 1020 relative to temperature during an autogenous LASER Beam Welding (LBW) experiment. The temperature profile during LBW is computed with the aid of an in-house CUDA-C algorithm. Here, the governing three-dimensional heat diffusion equation is discretized through the Finite Volume Method (FVM) and solved using the Successive Over-Relaxation (SOR) parallelized iterative solver. With temperature information, one may employ a minimization procedure to assess thermal properties or process parameters. In this work, the Quadrilateral Optimization Method (QOM) is applied to perform estimations because it allows for the simultaneous optimization of variables with no quantity restriction and renders the assessment of parameters in unsteady states valid, thereby preventing the requirement for steady-state experiments. We extended QOM’s prior applicability to account for more parameters concurrently. In Case I, the optimization of the three parameters that compose the second-degree polynomial function model of thermal conductivity is performed. In Case II, the heat distribution model’s gross heat rate (Ω) is also estimated in addition to the previous parameters. Ω [W] quantifies the power the sample receives and is related to the process’s efficiency. The method’s suitability for estimating the parameters was confirmed by investigating the reduced sensitivity coefficients, while the method’s stability was corroborated by performing the estimates with noisy data. There is a good agreement between the reference and estimated values. Hence, this study introduces a proper methodology for estimating a temperature-dependent thermal property and an LBW parameter. As the performance of the present algorithm is increased using parallel computation, a pondered solution between estimation reliability and computational cost is achieved.
Numerical codes are important in providing solutions to partial differential equations in many areas, such as the heat transfer problem. However, verification of these codes is critical. A methodology is presented in this work as an intrinsic verification method (IVM) to the solution to the partial differential equation. Derivation of the dimensionless form of scaled sensitivity coefficients is presented, and the sum of scaled sensitivity coefficients is used in the dimensionless form to provide a method for verification. Intrinsic verification methodology is demonstrated using examples of heat transfer problems in Cartesian and cylindrical coordinate. The IVM presented here is applicable to analytical as well as numerical solutions to partial differential equations.
This work presents a study of the mass transfer of osmotically dehydrated West Indian cherry, also known as acerola. The experiments were performed by immersing the fruits in a sucrose solution at 65ºBrix for 12 h at ambient temperature (27 °C), using three different fruit:solution ratios (1:4; 1:10 and 1:15 (w:w)). The kinetics of water loss, solids gain and weight reduction was determined. Effective mass diffusivity was calculated by the inverse method, using the Levenberg-Marquardt minimisation algorithm. The mathematical model used to describe the physical phenomenon of osmotic dehydration was based on Fick’s Second Law, considering the fruits as geometrically perfect spheres. The influence of the fruit:solution ratio was not significant in the range of this study. Diffusivity ranged from 1.558 × 10−10 to 1.760 × 10−10 m2 s−1.
Baking of bread involves temperature, moisture content and volume changes that are strongly coupled. Previous work usually treated these processes separately or semi-empirically, without adequate accounting for the interaction. A model is developed considering heat and moisture transport that is fully coupled with large volume change. Viscoelastic material property is used and is a function of temperature. The model is applied to a conventional baking process and solved using a finite element method. Results from modeling agree with the experimental observations in terms of temperature, moisture, volume changes and surface browning. The model shows that higher than normal baking temperature reduces the baked bread volume. Microwave baking tends to heat the bread from the inside. Therefore, microwave baking, without other special treatment, creates no firm crust and collapsed dough. Sensitivity analyses have been performed for several material properties due to inaccurate or unavailable data for the model. Larger water activity, capillary diffusivity and smaller gas intrinsic permeability tend to over-predict the volume expansion and moisture loss.
Baking is a complex process involving simultaneous heat and mass transfer besides chemical reactions and structural transformations. This study built a finely controlled cavity to measure the physical changes in baking bread and correlate the process variables and bread physical properties throughout the baking process. The temporal evolution of temperature, moisture, weight loss, volume, porosity, crumb structure, and color change was assessed. The proposed experimental system showed longitudinal symmetry, ensuring greater control of the bread changes during baking. The bread lost 11, 15, and 17% of its moisture when baked at 180, 200, and 220 °C, respectively. In the baking at 200 °C, only the crust lost moisture, while the bread center gained up to 4.2% in moisture content. The bread volumetric expansion was higher at lower temperatures, i.e., 27.6% at 180 °C, 23% at 200 °C, and 23.5% at 220 °C. The color evolution presented a sigmoid profile for all CIE L*a*b* color space parameters. A proposed model described the development of color parameters as a function of temperature with good R² and RMSE values, which allows predicting bread color throughout baking for each oven condition. The correlations obtained under controlled baking conditions accurately predict bread physical properties for each baking setup, allowing oven cavity design and optimization.
With the increase in the computational power, numerical simulations play an increasing role in designing, development, and optimization of various food processing operations. Inverse and ill-posed problems have been studied extensively in many branches of science and engineering, including mechanical, chemical, aerospace, biology, physics, and chemistry. The inverse techniques have been currently witnessing a growing trend in the food processing field for a decade. Inverse problems are usually performed when direct measurements of heat and mass transfer properties and boundary conditions, are not feasible. They are very sensitive to measurement errors; require optimization methods to tackle the inverse problem. It is necessary to consider the coupling heat and mass transfer for the solution of inverse problem since food process involves simultaneous heat and mass transfer within the food products. To date, inverse methods have been applied in few food processing operations, including drying, baking, freezing, and thermal processing. However, studies related to these areas on the estimation of unknown quantities for various fruits and vegetables, and food products are limited. This review mainly focuses on statistical concepts which include confidence interval, confidence region, model validation and minimization techniques discussed in detail. The optimal experimental design, D-optimality and Fisher Information Matrix, and sensitivity analysis are all extensively presented. The optimization techniques and their algorithms used in the area of food processing are explained. Finally, it covers the inverse estimation of unknown quantities, namely, heat and mass transfer parameters in different food processing operations.
In the present article, a novel experimental methodology is proposed for the simultaneous estimation of unknown thermal conductivity (k) and specific heat capacity (cp) of engineering materials. The experimental method involves radiative heating of the test sample and measuring its temperature response at two different locations. Proper arrangements have been made to realize one-dimensional heat conduction within the test sample. The one-dimensional heat conduction equation with convection-radiation heat loss boundary condition is considered as the forward model as it best mimics the experimental heat transfer. The inverse problem is formulated as a parameter estimation problem, and solved using Levenberg-Marquardt algorithm. First, numerical estimations were carried out with synthetic temperature data. It was found that the time-dependent heat flux boundary condition is more suitable to estimate the properties of higher thermal conductivity materials. Then, real-time temperature measurements were carried out on Stainless Steel (SS), Mild Steel (MS), Brass (BS), Aluminium (Al), and Copper (Cu) test samples. The estimated k of SS, MS, BS, Al, and Cu in (W/(m K)) is 14.03, 54.63, 103.21, 221.53 and 379.89, respectively. And, the estimated cp of SS, MS, BS, Al, and Cu in (J/(kg K)) is 500.44, 460.84, 395.01, 939.58, and 398.43 respectively. The estimated thermal properties are also validated against experimentally measured values obtained using the Modified Transient Plane Source (MTPS) sensor. It was found that the estimated k and cp in case of SS has a deviation of 7 % and 4 %, respectively.
One-dimensional model for heat and mass transfer within a cake during a one-sided baking process is presented. Thermophysical properties from literature do not simulate temperature and mean moisture content trends close enough to experimental measurements. This makes reliable estimation of these properties extremely important. Hence, thermophysical properties are approximated as effective constant parameters using an inverse procedure. To ease parameter estimation, a coupled mathematical model is derived in nondimensional form showing up the key parameters driving the baking process. Complex step differentiation method is utilized to compute sensitivities in order to improve their precision. Optimal location and minimum number of temperature sensors are picked from D-optimality criterion. Effect of deformation is neglected in this study. Sensitivity analysis shows that the parameter derived from Darcy law representing a ratio of permeability to dynamic viscosity of gas seems to perturb the temperature and mean moisture content minutely. Weighted least square method with significant weight given to temperature measurements results in better approximation than ordinary and scaled least square objective functions. Inverse procedure is unable to estimate temperature profile with expected precision at boundary where heat flux enters above 100 °C as the model fails to address dough transformation.
In many applications, it is essential to evaluate the heat flux transferred by contact between two materials. Heat flux sensors must be properly designed to reduce sources of measurement bias while maintaining a high sensitivity without disrupting the flux itself. The aim of this study is to estimate the heat flux exchanged between two materials in contact at different temperatures. When materials are rapidly brought into contact (less than one second) the heat flux exchanged presents large variations. In this study, the heat fluxes transferred are computed using an inverse heat conduction problem. Initially, a particular attention is focused on the numerical analysis of the method and its reliability. A first experimental study is conducted on a prototype where an elastomer plate is suddenly placed on a hot iron disk. Based on temperature measurements recorded in each medium, the heat flux exchanged is estimated by two ways and the results are compared with each other. These first results show the efficiency of the method and its ability to extract information from temperature measurements. Then, a second experimental study is carried out with a cereal batter spread on the iron disk. The estimation of the heat flux from cast iron temperatures is compared with that deduced from an energy balance performed on the batter. The energy balance takes into account the sensible heat, the mass losses and the heat flux exchanged by radiation and convection with ambiance.
The moisture diffusivity of food is a very important physical parameter to model baking processes. Unfortunately, specific moisture diffusivity values are not easily found in literature, especially measured or calculated during the baking processes. The main methods used to estimate the moisture diffusivity are based on the second Fick's law, but there are significant differences in the way of applying these laws. The aim of this work is to estimate the effective moisture diffusivity in baking of a small flat cake by using the inversion of a numerical model based on the coupling of heat and mass transfer. The temperature and moisture dependency of the diffusivity is evaluated. Results are compared with those obtained by using a fitting method. The inverse method allows to estimate the effective moisture diffusivity close to those reported in literature for similar bakery products (constant values). The impact of the moisture concentration appears to be very restricted and it tends to decrease with increasing the oven temperature. Subsequently, the obtained effective diffusivity was implemented on a direct Finite Element (FE) model simulating a whole cake for different baking temperatures. The direct model validation (mean moisture content and temperature) shows determination coefficients ranging from 0.921 to 0.996 confirming the robustness of the diffusivity parameters obtained by the inverse method.
Inverse heat techniques have been employed to evaluate the unknown parameters such as heat transfer coefficient, thermal properties, mass diffusivity, and to estimate boundary conditions by utilizing the temperature measurement within the solid material. These parameters with direct measurements are difficult, or sometimes even impossible. The heat flux is one of the essential parameters which would provide information about baking performance and the quality of the product. Generally, direct measurement of heat flux with sensors is difficult under unsteady state conditions. Therefore, in the present investigation, heat flux was estimated with simple temperature data during bread baking by the inverse method at a temperature of 200 °C under natural and forced convections with and without deformation. The Levenberg-Marquardt algorithm was chosen for the solution of the inverse problem. The predicted temperatures at the desired location were obtained by a direct method in COMSOL Multiphysics, and these were validated with experimental temperature data for the efficacy of the inverse problem. The convective, radiative heat flux, and heat transfer coefficient were also computed from the obtained heat flux data. The obtained results in this investigation were well correlated with findings reported in the literature.
Currently the total energy demand quantity is already larger than the total supply quantity, and the structural contradiction is serious, how to improve energy utilization efficiency of petrochemical enterprises is a problem that should be solved quickly. In order to correctly evaluate the energy utilization efficiency of refinery and petrochemical enterprises, the construction method of constructing energy utilization efficiency evaluation model of refining unit in petrochemical enterprises is established based on proposed Contourlet neural network optimized by improved grey algorithm. The energy utilization efficiency evaluation index system is confirmed considering its impacts. The Contourlet neural network is constructed through using Contourlet as excitation function to improve the evaluation precision. The nonlinear change strategy of controlling parameter aof traditional grey wolf optimization algorithm is proposed, the Cubic chaotic value is used as the perturbation operator of location of grey wolf to generate the new solution, and the individual memory function of Cuckoo algorithm is introduced to improve the location updating algorithm, and then the improved grey wolf optimization algorithm has better global optimization capability that is used to optimize the parameters of Contourlet neural network. The evaluation analysis of energy utilization efficiency for 250 devices for removing sulphur alcohol of liquefied gas is carried out. Results show that the proposed evaluation model has best evaluation efficiency and precision. In addition, the proposed evaluation model has highest evaluation precision of energy utilization efficiency of refinery and petrochemical enterprises. The evaluation results can effectively evaluate the energy utilization level of petrochemical enterprises, which can offer favorable theoretical basis of establishing the energy saving measurements for petrochemical enterprises.
Performance ratio is an important parameter of measuring the quality and efficiency of photovoltaic (PV) pumping system, which should be predicted correctly to provide guidance for regulating the measurements of reducing losses of every part, therefore this research proposed a prediction model of performance ratio of PV pumping system based on grey clustering and second curvelet neural network. The meaning of performance ratio is analyzed and the main affecting factors of the performance level for PV pumping system are also summarized. The second curvelet neural network is constructed combing the second curvelet transform and feed forward neural network, and the structure of second curvelet neural network is designed. The classification of training and testing samples are confirmed based on the improved grey clustering model, and the corresponding mathematical models are studied. The firefly algorithm is used to optimize the second curvelet neural network. The grey classifications of training samples are confirmed based on grey clustering, which are used to train the second curvelet neural network with different structure optimized by firefly with different parameters, and then the testing samples are used to carry out prediction analysis. Simulation results show that the second curvelet neural network has highest prediction precision and efficiency, which can correctly and efficiently predict the performance ratio of PV pumping system.
To reduce energy consumption, the optimization of oven operating conditions requires a thorough understanding of the influence of heat input on the baking kinetics. The objective of this study is to simulate the hydrothermal behaviour and the deformation of bread dough during the baking stage. The complete baking process is simulated in a 2-dimensional model including conductive transfer with hearth, steam injection and convective and radiative heat transfer. In this article, the simulations are realized with the use of experimental boundary conditions (infrared flux, air temperature and vapour pressure, hearth and wall temperature). The numerical evolutions of different variables are compared to measured data: temperature (product and hearth), mass loss, local moisture content, total volume and gas pressure in the dough. A sensitivity analysis is performed to understand the impact of carbon dioxide generation and heat transfer on the process. Finally, the simulated energy required during baking is evaluated, observing direct conduction with the hearth, and the impact of the transfer of convective and radiative heat.
Baking is a high energy demanding process, which requires special attention in order to know and improve its efficiency. In this work, energy consumption associated to sponge cake baking is investigated. A wide range of operative conditions (two ovens, three convection modes, three oven temperatures) were compared. Experimental oven energy consumption was estimated taking into account the heating resistances power and a usage factor. Product energy demand was estimated from both experimental and modeling approaches considering sensible and latent heat. Oven energy consumption results showed that high oven temperature and forced convection mode favours energy savings. Regarding product energy demand, forced convection produced faster and higher weight loss inducing a higher energy demand. Besides, this parameter was satisfactorily estimated by the baking model applied, with an average error between experimental and simulated values in a range of 8.0 to 10.1 %. Finally, the energy efficiency results indicated that it increased linearly with the effective oven temperature and that the greatest efficiency corresponded to the forced convection mode.
Damping factor is a key parameter in Levenberg-Marquardt algorithm for solving inverse problems, which significantly affects the efficiency and the convergence stability of Levenberg-Marquardt algorithm and needs to be updated with the iterative number. A new approach is presented for determining damping factors in Levenberg-Marquardt algorithm in this paper, which relates the damping factor directly to the dimensionless objective function in inverse problems. Then, the accuracy, the efficiency, the convergence stability as well as the robustness of Levenberg-Marquardt algorithm are investigated in detail, for identifying temperature-dependent thermal conductivities by solving an inverse heat conduction problem. The results show that the LM algorithm using the new approach for determining damping factors is accurate and robust for identifying temperature-dependent thermal conductivities. The efficiency and the convergence stability of the LM algorithm are improved by using the new approach for determining damping factors. Moreover, the new approach is easy to implement.
International home energy rating regulations are forcing to use efficient cooking equipment and processes towards energy saving and sustainability. For this reason gas ovens are replaced by the electric ones, to get the highest energy rating. Due to this fact, the study of the technologies related to the energy efficiency in cooking is increasingly developing. Indeed, big industries are working to the energy optimization of their processes since decades, while there is still a lot of room in energy optimization of single household appliances. The achievement of a higher efficiency can have a big impact on the society only if the use of modern equipment gets widespread. The combination of several energy sources (e.g. forced convection, irradiation, microwave, etc.) and their optimization is an emerging target for oven manufacturers towards optimal oven design.
In this work, an energy consumption analysis and optimization is applied to the case of bread baking. Each source of energy gets the due importance and the process conditions are compared. A basic quality standard is guaranteed by taking into account some quality markers, which are relevant based on a consumer viewpoint.
Despite numerous studies of Levenberg–Marquardt (LM) algorithms for solving inverse heat conduction problems, sensitivity coefficients are mainly evaluated by numerical differentiation methods. However, sensitivity coefficients are difficult to be precisely calculated by numerical differentiation methods, if multi-dimensions, transient states and nonlinearities are involved. To the best knowledge of the authors, there has not been a general method for accurately calculating sensitivity coefficients for a LM algorithm, except numerical differentiation methods. In this study, a modified LM algorithm is presented by introducing the complex-variable-differentiation method for sensitivity analysis, and multi-parameters of boundary heat flux are simultaneously recovered by solving transient nonlinear inverse heat conduction problems. The results show that the modified LM algorithm has the advantages of the conventional LM algorithm, that are effective, accurate and robust for simultaneous estimation of multi-parameters of boundary heat flux. Meanwhile, the efficiency and the convergence stability of the modified LM algorithm are improved, which are attributed to accurate evaluation of sensitivity coefficients, compared with the conventional LM algorithm.
The relaxation factor is a key parameter in gradient-based inversion and optimization methods, as well as in solving nonlinear equations using iterative techniques. In gradient-based inversion methods, the relaxation factor directly affects the inversion efficiency and the convergence stability. In general, the bigger the relaxation factor is, the faster the inversion process is. However, divergences may occur if the relaxation factor is too big. Therefore, there should be an optimal value of the relaxation factor at each iteration, guaranteeing a high inversion efficiency and a good convergence stability. In the present work, an optimization technique is proposed, using which the relaxation factor is adaptively updated at each iteration, rather than a constant during the whole iteration process. Based on this, a new inverse analysis method is developed for solving multi-dimensional transient nonlinear inverse heat conduction problems. One- and two-dimensional transient nonlinear inverse heat conduction problems are involved, and the instability issues occurred in the previous works are reconsidered. The results show that the new inverse analysis method in the present work has the same high accuracy, the same good robustness, and a higher inversion efficiency, compared with the previous least-squares method. Most importantly, the new method is more stable by innovatively optimizing and adaptively updating the relaxation factor at each iteration.
The objective of this study is to improve energy efficiency of an oven used for the conventional baking of French bread. Experiments performed to validate a numerical model and test different infrared emitters are presented. In order to provide a relevant experimental database, we first instrumented an industrial electrical static oven. The modifications made to the oven and instrumentation installed allows monitoring baking kinetics. The quantities measured are bread mass, temperature field, volume expansion and pressure. An energy balance is calculated to define the energy necessary to cook one “baguette”. Heat is provided by natural convection, direct conduction and mainly by infrared radiation to the dough. To improve energy efficiency, short infrared emitters are arranged on the vault instead of traditional electrical resistances made of reinforced metal alloy. These emitters allow increasing radiation heat transfer. Then, baking under short infrared emitters is carried out at lower air temperature, for the same total baking time. Energy consumption is analyzed and compared in both cases.
In response to increasing energy costs and legislative requirements energy efficient high-speed air impingement jet baking systems are now being developed. In this paper, a multi-objective optimisation framework for oven designs is presented which uses experimentally verified heat transfer correlations and high fidelity Computational Fluid Dynamics (CFD) analyses to identify optimal combinations of design features which maximise desirable characteristics such as temperature uniformity in the oven and overall energy efficiency of baking. A surrogate-assisted multi-objective optimisation framework is proposed and used to explore a range of practical oven designs, providing information on overall temperature uniformity within the oven together with ensuing energy usage and potential savings.
Changing legislation and rising energy costs are bringing the need for more efficient baking processes into much sharper focus. High-speed air impingement bread-baking ovens are complex systems used to entrain thermal air flow. In this paper, Computational Fluid Dynamics (CFD) is combined with a multi-objective optimization framework to develop a tool for the rapid generation of forced convection oven designs. A design parameterization of a three-dimensional generic oven model is carried out to enable optimization, for a wide range of oven sizes and flow conditions, to be performed subject to appropriate objective functions measuring desirable features such as temperature uniformity throughout the oven, energy efficiency and manufacturability. Optimal Latin Hypercubes for surrogate model building and model validation points are constructed using a permutation genetic algorithm and design points are evaluated using CFD. Surrogate models are built using a Moving Least Squares approach. A series of optimizations for various oven sizes and flow conditions are performed using a genetic algorithm with responses calculated from the surrogates. This approach results in a set of optimized designs, from which appropriate oven designs for a wide range of specific applications can be inferred. Results from various oven design and objective functions under investigation are presented together with ensuing energy usage and savings. Analysis suggests that 10% energy savings can be achieved for the baking process.
Energy usage in bread ovens is analysed using a generic methodology applicable to all types of mass-production tunnel ovens. The presented methodology quantifies the energy required to bake the dough, and to conduct a detailed analysis of the breakdown of losses from the oven. In addition, a computational fluid dynamics (CFD) optimisation study is undertaken, resulting in improved operating conditions for bread baking with reduced energy usage and baking time. Overall, by combining the two approaches, the analyses suggest that bake time can be reduced by up to 10% and the specific energy required to bake each loaf by approximately 2%. For UK industry, these savings equate to more than £0.5 million cost and carbon reduction of more than 5000 tonnes CO2 per year.
Most of the published methods for estimating temperature history during heating/cooling/freezing solid food require data on thermal properties of the product and any relevant heat transfer coefficients. However, there are some difficulties of obtaining thermal data for use in industrial heating/cooling/freezing of food. In this paper the development of a new procedure for estimating the temperature history is briefly reviewed, a procedure which does not require the knowledge of thermal data of the food being heated/frozen. This procedure collects a series of time/temperature data at a point in the food in the early stages of heating/freezing, analyzes these data to predict the thermal parameters which are responsible for heat conduction, and predicts the time/temperature relationship for the remainder of the heating/cooling/freezing phases.
Thermal properties are essential requirements for mathematical modeling and simulation of various thermal processing operations. Knowledge of accurate values of thermal properties will result in accurate prediction of temperature profiles needed for process selection, design and optimization. The aim of this work was to develop a simple, accurate and robust method for determination of thermal diffusivity of solid foods. An inverse numerical method was developed to estimate thermal diffusivity from transient temperature measurements at any position in a food sample using sequential parameter estimation technique. The developed algorithm was validated using computer generated temperatures in addition to actual experimental temperatures data. The estimated thermal diffusivity from the computer generated data is in excellent agreement with the true value, and the estimated thermal diffusivity from the experimental measurements is in good agreement with literature data. The method presented here was demonstrated to be simple accurate and robust.
This article presents some results on the energy demand in conventional bread baking and in the processing of frozen part baked breads, resulting from the “EU-FRESHBAKE” European project (FP6). Bread baking is one of the most energy demanding processes (around 4MJ/kg), compared with other thermal processes such as canning. However, there is a large variability of data in the literature. For partial baking, bread has to be baked twice. It may also be frozen after part baking, which will increase the total energy demand. Results obtained with equipment used by craft bakeries are presented. Conventional and frozen part baked processes are compared. The effect of occupation ratio on the overall energy demand is also assessed. It was observed that 15–20% of the total energy is used for heating up the dough and 10–20% for crust drying. Pre-heating of the oven represents another significant energy demand. The energy demand for freezing is comparable to that for baking. An Energy Efficiency Index is used to assess the ratio of energy effectively transferred to the dough during baking. Part baked frozen technology demands about 2.2 times as much energy as conventional bread making process.
Simulation of heat conduction processes in foods requires a knowledge of the temperature dependence of their thermal properties, conductivity (k) and specific heat (ce). In this work a new procedure is proposed for the inverse estimation of ce in a 1-D solid under convective and adiabatic boundary conditions, in which conductivity is constant or temperature-dependent. The inverse problem is treated as a function estimation problem so that no prior information is needed about the form of the dependence. The estimation is obtained by means of a piece-wise function, whose stretches (both slope and size) is continuously adjusted by an iterative routine that minimizes the classical functional for this kind of problems. This routine is combined with the Network simulation method as the numerical technique. The input data of this inverse problem are obtained from the solution of the direct problem, at a particular location in the food, modified by a normal distribution random error. Applications to real products are realized proving that it is possible to reach precise estimations of ce with the proposed method.
This paper presents a mathematical model for describing processes involving simultaneous heat and mass transfer with phase transition in foods undergoing volume change, i.e. shrinkage and/or expansion. We focused on processes where the phase transition occurs in a moving front, such as thawing, freezing, drying, frying and baking. The model is based on a moving boundary problem formulation with equivalent thermophysical properties. The transport problem is solved by using the finite element method and the Arbitrary Lagrangian–Eulerian method is used to describe the motion of the boundary. The formulation is assessed by simulating the bread baking process and comparing numerical results with experimental data. Simulated temperature and water content profiles are in good agreement with experimental data obtained from bread baking tests. The model well describes the stated general problem and it is expected to be useful for other food processes involving similar phenomena.
Thin-Layer drying by coating of a viscous product on a hot metallic surface is studied from the point of view of heat and mass tranfer. An experimental device has been built up in order both to provide and to estimate the heat flux density at the interface between the hot plate and the drying sample. Results are presented for alumina sludge drying. The determination of the interfacial heat flux is odtained from temperature measurements in the metallic plate and the solution of an inverse conduction problem. An analytical direct model is made using the quadrupole formalism and the system transfer function is calculated. The inverse problem is solved using Beck's sequential function specification method. Intrinsic drying kinetics are obtained by an energy balance.
A method is proposed for the simultaneous determination of the effective diffusivity and convective mass transfer coefficient in solids which can be considered as infinite cylinders. The inverse method was used to fit the analytical solution of the diffusion equation with convective boundary condition to experimental data of thin-layer drying kinetics of products with cylindrical shape. The proposed method was applied to the drying kinetic of rough rice, using experimental data available in the literature. The statistical indicators show that describing the diffusion process with convective boundary condition is more accurate than the description with boundary condition of the first kind, commonly found in the literature.
The main purpose of this paper is to introduce the five following papers on inverse problems, and to relate the field of inverse problems to measurements. Inverse techniques are an emerging suite of methods which, when fully embraced, promise to provide better experiments and improved understanding of physical processes. This paper provides an overview of the general procedure and concepts related to identification of parameters or functions by inverse techniques. A discussion of errors and their implication for an appropriate function for minimization in inverse procedures is presented, and two methods for achieving this minimization are discussed. Some sequential concepts for parameter estimation are presented, along with a discussion of residuals and confidence intervals. Experiment design and optimization are reviewed, and a discussion of residuals and their relation to model building is presented.
A model of multiphase transport in a porous medium coupled with large deformation of the porous matrix is developed and applied to the process of bread baking. Transport-governing equations are based on energy conservation and mass conservation of water, water vapor, and CO2 produced during baking. Deformation is caused by the pressure gradient from internal evaporation and CO2 generation. Temperature, moisture, and pressure changes in turn are affected by deformation. Bread is assumed to be viscoelastic, mechanical properties of which are functions of temperature. Geometric nonlinear effects are considered in the mechanics problem. Results are compared with those from baking experiments and literature data. Vapor pressure inside the matrix is likely to be lower than the equilibrium vapor pressure. Convective heat transfer is small compared to heat conduction and evaporation–condensation of water vapor promotes heat transfer to the inside. Rate of CO2 generation, mechanical properties of dough, and gravity together determine the final shape of the bread. © 2005 American Institute of Chemical Engineers AIChE J, 2005
A bakery pilot oven is modeled using computational fluid dynamics software. This approach relies on integration of an instrument into modeled geometry. The instrument is a heat flux measuring device that can be used in the industrial baking process. All three heat transfer mechanisms are considered and coupled with turbulent flow. Turbulence is taken into account via the k–ε realizable model whereas the surface-to-surface model simulates the radiation. Additionally, buoyancy forces are introduced by means of a weakly compressible formulation. The model predictions show a good qualitative agreement with the experimental measurements. A quantitative agreement was obtained to some extent. Limitations came from the difficulty to measure the temperature of the radiant surfaces of the oven. Operating conditions used are typical of bakery products and, as expected, radiation was the dominant mode of heat transfer. The integration of the instrument was useful for assessing the model. Since it is designed for industrial use, it may be a valuable tool for future challenges in the field, such as simulation of an industrial scale oven.
This paper presents a new inverse radiation analysis approach in an absorbing, emitting and non-gray participating medium. The inverted unknowns are treated as the optimization variables, and the errors to be minimized are the differences between the calculated temperatures and the experimental ones. The measured temperatures are simulated by the solution of the direct problem, in which a modified zonal method is employed to solve the radiative transfer equation, and the Edwards exponential wide-band model together with the Leckner series is adopted to obtain the radiative property of the non-gray medium. The contribution is to introduce the complex-variable-differentiation method into inverse radiation problems for calculating sensitivity coefficients. The least-square method is employed in the inverse problem. The effectiveness and efficiency of the inverse problem are demonstrated by an example. In addition, the effects of the measurement error and the thermal operating parameters on the inverse analysis are investigated.HighlightsThe paper has the following obvious new contributions: ► This paper is first to invert equivalent gray radiative property of non-gray medium. ► The distinct aspect is the use of complex-variable-differentiation method in the inverse problem. ► Effect of thermal operating parameters on equivalent gray property is estimated. ► Effects are obtained for both conventional and regenerative reheating furnaces. ► Results point to promising engineering applications in reheating furnace.
The influence the quality and shelf life of baked product has previously been reported to be effected by the time and temperature of the baking process. In this study, dough was baked at 219 °C by using different ovens (conventional, impingement or hybrid) or with doughs of different sizes (large or small) for varying times. During baking the temperature profile at the dough center was recorded. Texture, thermal properties and pasting characteristics of baked product with reference to baking conditions were investigated. Small breads baked in the hybrid oven had the highest heating rate (25.1 °C/min) while large breads baked in conventional oven had the lowest heating rate (6.0 °C/min). When the data are viewed as a function of heating rate in this study, the enthalpy of amylopectin recrystallization, rate of bread firmness and the amount of soluble amylose were all-lower at the slower heating rate. The differences observed in product firmness following storage are potentially a consequence of the extent of starch granule hydration, swelling, dispersion and extent of reassociation; all of which are affected by the heating rate during baking.
A study and modeling of the mass transfer phenomenon in the drying process of agricultural products, using Fick’s second law of diffusion adapted to infinite flat plate geometry, are described. An analytical solution methodology, the classical integral transform technique (CITT), is employed. Predicted results are compared with experimental data on the drying of sliced mushrooms in different operating conditions. A phenomenological analysis is made using graphics and tables. The data obtained by CITT and experimentally show negligible differences. The mass diffusion coefficient during hot air drying of mushrooms is estimated. Experimental drying kinetics are applied to the product at different air temperatures and air flow rates aiming to find a solution for the mass transfer equation, using an inverse problem with the Levenberg–Marquardt optimization technique in successive trials. Experimental data are also modified by a normal distribution of random errors. Statistical analyses show good agreement between experimental and predicted curves.
This study determined the diffusion coefficient during hot air drying of mushrooms (Agaricus blazei). Experimental drying kinetics were applied to sliced mushroom at different air temperatures (45, 60, 75, and 80 °C) and air velocities (1, 1.2, 1.75, 2.3, and 2.5 m/s) in order to find an analytical solution to the mass transfer equation for drying using an inverse problem with two distinct optimization techniques: Levenberg–Marquardt and Differential Evolution, in successive trials. The experimental data were also modified by a normal distribution random error. Statistical analyses showed no significant differences between reported and estimated curves. The diffusion coefficients are greater in mushrooms dried at higher temperatures, and those dried at higher drying air velocities.
Nowadays, study of processes is being performed using computer simulations. The parameters of the models used have to be determined either using experiments or using identification methods called inverse methods. The aim of this article is to present an inverse method to determine the thermal conductivity of sandwich bread by recording the temperature during the convection cooling process. Seven sandwich-bread-cooling experiments were performed, in natural or forced convection, to estimate the variation in thermal conductivity with temperature and local water content. Sensitivity studies show that to obtain reliable results, the temperature probe has to be placed in center and the data on specific heat has to be precise. The estimated thermal conductivity is given as a temperature and local water content polynomial.
A new approach for the estimation of apparent thermal diffusivity of foods at different drying temperatures was explored, analysed and discussed in this work. Temperature versus time was obtained numerically at the center of the food (banana, “nanicão” variety) using the 1D Fourier equation with drying temperatures in the range between approximately 17–65 °C and moisture content in the range between 0.01 and 3.43 (dry basis). The solution of the partial differential equation is made with a finite difference method coupled to an optimization technique of Differential Evolution used in inverse method. The mathematical model proposed considered the effects of shrinkage and convective heat transfer at surface of fruit. Parameters of two functions, the first dependent of the moisture content and the second dependent also of the temperature were obtained by inverse method modelling the apparent thermal diffusivity. Such parameters that provide the best least square fit between the experimental and predicted time-temperatures curves are presented in this work. This study demonstrated that a small change in the temperature and moisture content of banana cause an abrupt change in the apparent thermal diffusivity, which decrease with the decreasing of the moisture. Statistical analysis shows the excellent agreement between reported and estimated curves.
Finite element models are presented for the prediction of the non-linear response of fluid-structure systems exposed to transient dynamic loading. An arbitrary Lagrangian-Eulerian kinematical description of the fluid domain is adopted in which the grid points can be displaced independently of the fluid motion. This formulation leads to an easy and accurate treatment of fluid-structure interfaces and permits significant fluid sloshing and swirling to occur without producing excessive distorsions of the computational mesh. Several applications are presented to illustrate the potential of the proposed modelling procedures.
To understand the phenomena governing the heat and mass transfers during the freezing of par-baked bread, simulation of freezing in an infinite two-layer cylinder was performed. A fixed grid finite difference method was used in the solution of simultaneous heat and moisture transfer (SHMT) equations according to Lees' three-level scheme. The model accommodates the effects of temperature dependent variables such as apparent specific heat, enthalpy, thermal conductivity, and water activity, and predicts the temperature profiles and weight losses. The model was verified by freezing of par-backed breads with cylindrical shape within an experimental freezer. The RMSE between experimental results and model predictions is 0.505 °C for surface temperature, and 1.8 °C for center temperature. The precision on the loss water is 10%. These results indicate that the method could be successfully applied to SHMT operations in foods freezing.
This article deals with the numerical approximation of Navier–Stokes equations in a domain with moving boundaries. Using the arbitrary Lagrangian–Eulerian (ALE) method we transform the problem from a moving domain to a fixed reference domain through an artificial domain velocity. Then, we apply the characteristic method to solve the Navier–Stokes equation in the fixed domain. Suitable boundary conditions are used for the interfaces. Three examples are provided to show the efficiency of the present method.