A computational model for calculating temperature distributions in microwave food applications

Available online at www.sciencedirect.com

Technologies 9 (2008) 374–384www.elsevier.com/locate/ifset

Innovative Food Science and Emerging

A computational model for calculating temperature distributions inmicrowave food applications

Kai Knoerzer a,b,⁎, Marc Regier b, Helmar Schubert b

a Innovative Foods Centre, Food Science Australia, Private Bag 16, Werribee, VIC, Australiab Universität Karlsruhe (TH), Institute of Engineering in Life Sciences, Dept. I: Food Process Engineering, Karlsruhe, Germany

Received 21 August 2007; accepted 21 October 2007

Abstract

By heating volumetrically, microwave processes have several advantages over conventional heating processes. The main advantage is theincrease of process rates and thus improving the quality of microwave heated, dried, pasteurised or sterilised products. However, to ensure productsafety and thus satisfy regulatory bodies, temperature distributions have to be as uniform as possible.

A new simulation approach has been developed, based on a user-friendly interface coupling two commercial software packages, to model time-dependent temperature profiles of arbitrarily shaped microwave treated products in three dimensions. Simulations have shown uneven temperaturedistributions when products were exposed to uncontrolled microwave applications.

Using this model, hot and cold spots in the products may be simulated to test appropriate microwave treatment control strategies. A furtherdevelopment of this approach, being one-of-a-kind to date, describes a feedback-control loop in the simulation which helps optimising microwaveprocesses by ensuring minimal time/temperature treatments and uniform temperature distributions. Thus microwave power pulse programs can bedeveloped and tested in a model before being implemented in a real microwave system. Validation of the model was performed using non-invasivemagnetic resonance imaging (MRI). Verification results showed that simulated data agrees well with the measured data in discrete locations(heating curves) as well as the temperature data throughout the samples.© 2007 Elsevier Ltd. All rights reserved.

Keywords: Microwaves; Microwave heating; Simulation; Modelling; Feedback control; Minimal treatment

Industrial relevance: The presented simulation approach calculates 3D temperature distribution as a function of time and thus allows for thedetermination of hot and cold spots in the products. With this, appropriate microwave treatment control strategies can be tested.

A further development of this approach, being one-of-a-kind to date, describes a feedback-control loop in the simulation which allows foroptimising microwave processes by ensuring minimal time/temperature treatments and uniform temperature distributions. Thus microwave powerpulse programs can be developed and tested in a model before being implemented in a real microwave system.

With this approach the main advantage of microwave applications, the increase of process rates due to the volumetric heating can be utilisedand at the same time the quality of the treated product can be optimised and product safety can be ensured by improving temperature uniformity.Furthermore, regulatory bodies can be satisfied.

1. Introduction

The product and process development for microwaveprocessing started off as a trial-and-error procedure. Nowadays,

⁎ Corresponding author. Innovative Foods Centre, Food Science Australia,Private Bag 16, Werribee, VIC, Australia.

E-mail address: [email protected] (K. Knoerzer).

1466-8564/$ - see front matter © 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ifset.2007.10.007

in order to design thermal process operations, the knowledgeabout the temperature distribution during treatment inside aprocessing unit is required. This is necessary both for processdesign, scale-up and specification and for governmentalregulators to approve the process. Modelling can be a helpfultool in meeting these requirements.

Calculation of realistic temperature and electromagnetic fielddistributions within microwave applicators and treated product

mailto:[email protected]

http://dx.doi.org/10.1016/j.ifset.2007.10.007

375K. Knoerzer et al. / Innovative Food Science and Emerging Technologies 9 (2008) 374–384

is complicated because of interactions of electromagnetism withheat and mass transfer and thus the necessity of coupling thegoverning partial differential equations, which have to be solvedin a parallel manner. Due to the lack of powerful computersduring the initial introduction of microwave technology it wasnearly impossible to model the coupled problem especiallywhen products were present.

Latest advances in computer technology allow for over-coming this limitation. Numerical software packages areavailable for solving the separated problems and great progresshas been made in coupling the partial differential equationproblem, which best describes the real process.

1.1. Review of microwave thermal modelling

The governing equations of electromagnetism, Maxwell'sequations, and the equations describing the interaction ofmicrowaves with different materials, the constitutive relations,have been described in detail in a number of publications (Regier& Schubert, 2001; Regier & Schubert, 2005) and will be shownin Section 2.2. Regier and Schubert (2001) described theexample of the wave solutions for the simple one-dimensionalplane wave. The resulted exponentially damped wave within amaterial with dielectric losses has often been used for the three-dimensional case to estimate the power distribution withinproducts. The development of a more sophisticated approach forcalculating realistic solutions based on a coupled EM-thermalproblem will be presented in Section 2.2 of this manuscript.

The governing equations for coupling heat and mass transferto electromagnetism start from the continuity equation, thethermal energy equation and Fick's law. Neglecting masstransfer, a general equation for heat transfer can be described by:

qCPATAt

�jd kdjTð Þ ¼ �jqR þ Qem ð1Þ

where ρ is the density, CP the specific heat capacity, λ thethermal conductivity of the materials used in the scenario, qRthe radiative power flux density and Qem the heat sourcegenerated by the dissipated microwaves. (For a complete list ofsymbols, the reader is referred to Notation of this article.)

The left side of this equation is well known from thetraditional heat conduction equation, the terms on the right sidewere added for heat transfer by radiation and for the heat sourceby the dielectric losses, respectively (Metaxas, 1996).

Assuming the product consists of a moist solid material, onlythe radiation term has to be taken into account at surfaces togaseous materials yielding additional boundary conditions andEq. (1) can be simplified to:

qCPATAt

�jd kdjTð Þ ¼ Qem: ð2Þ

Heat transfer and electromagnetic equations are coupledexplicitly by the values of the temperature and the electro-magnetic heat generation and implicitly by the temperaturedependency of the thermo-physical material properties. At theproduct surface, boundary conditions need to take the externalheat transfer into account.

Although there is a fast development of numerical calcula-tion power, a complete calculation without some simplificationsis hardly feasible to date.

In the case of pure electromagnetism, commercial numericalsoftware packages are available. A comparison of their potentialfor microwave heating has been addressed by (Yakovlev, 2000;Yakovlev, 2001a; Yakovlev, 2001b;Komarov&Yakovlev, 2001).Some customised software codes are also described in literature.Most of them originated from the telecommunication area butwere developed further for microwave heating applications.

The discretisation of the partial differential equations or theircorresponding integral equations togetherwith the suitable bound-ary conditions on a calculation grid is common to all numericaltechniques.

In practice the method of finite differences (FDM) and thefinite element method (FEM) are most common, however alsothe method of moments (MOM), the transmission line matrixmethod (TLM), the boundary element method (BEM) andmethods using optical ray tracing codes have been applied.

For short burst of high microwave power densities, the heattransfer component of the equation can be neglected, since itis much slower than the microwave heat generation. Thetemperature rise in a defined volume is then directly propor-tional to the dissipated microwave power, which can be inferredfrom the effective electric field value and the dielectric lossfactor. Some results using this approximation can be found forexample in Fu andMetaxas (1994), Liu, Turner, and Bialkowski(1994), Dibben and Metaxas (1994), Zhao and Turner (1997),and Sundberg, Kildal, and Ohlsson (1998).

To date, a growing number of papers also describe the coupledelectromagnetic field and thermal model. Examples including theheat conduction can be found inSekkak, Pichon, andRazek (1995),Ma et al. (1995), Torres and Jecko (1997), Zhao and Turner (2000),Kopyt and Celuch-Marcysiak (2002, 2003, 2004), Rabello, Silva,Saldanha, Vollaire, and Nicholas (2005), Sun, Zhu, Feng, and Xu(2007), and Zhu, Kuznetsov, and Sandeep (2007).

In addition to the heat conduction, heat transport by radiationmay be addressed by a ray tracing algorithm (Haala &Wiesbeck,2000). Since the temperatures (and temperature differences) formost food applications are more moderate in comparison toceramics sintering, where the latter software code originates from,the radiation term becomes more negligible than heat transport byconvection or evaporation.

However, most of the above mentioned publications are basedon self-developed software codes and lack the comfort of user-friendliness, full support by software vendors and extensivemanuals. Thus they are predominantly useful for the developers intheir academic research rather than being suitable for industrialuse (which does not mean that they are not used in industry). Inaddition to that most of them occupy further simplifications, likeusing water as food material, constant thermo-physical propertiesor calculating the temperatures only in one or two dimensions.

The review shows that successful application of model calcu-lations are often limited to special cases or to very simplified ones.

An example of a successful microwave heating simulation,incorporating thermal conductivity and free convection (Knoer-zer, 2006), will be shown in detail in this paper. A procedure

Fig. 1. Sketch of microwave device for introducing microwaves into MRI tomograph (a) and depiction of area of interest in the simulation process (b).

376 K. Knoerzer et al. / Innovative Food Science and Emerging Technologies 9 (2008) 374–384

allowing for simulation of arbitrarily shaped foods and thecontrol of temperature distributions by feedback-controlledsimulations will be presented.

In order to prove the validity of the model and satisfyregulatory bodies, measuring temperatures in microwaveheating applications is indispensable. However, this is a verychallenging task; the electromagnetic fields and temperaturedistributions are not easily measurable without changing themby the measurement procedure itself. A relatively old biblio-graphy of different temperature indication methods in micro-wave ovens can be found in Ringle and Donaldson (1975). Moreup-to-date bibliographies can be found in Ohlsson andBengtsson (2001) and Knoerzer, Regier, and Schubert (2005a).

In this work, a new simulation approach has been developed,based on a user-friendly interface coupling two commercialsoftware packages, to model time-dependent temperature profilesnot only of microwave treated products with simple shapes butalso of arbitrarily shaped products in three dimensions.

With this model, hot and cold spots in the products can besimulated to test appropriate microwave treatment control strat-egies. A further development of this model, describes a feedback-control loop in the simulation which helps in optimising micro-wave processes by ensuringminimal time/temperature treatmentsand uniform temperature distributions. Validation of the modelhas been performed using non-invasive magnetic resonanceimaging (MRI) allowing for measuring temperature distributionsduring the running process in three dimensions.

2. Materials and methods

2.1. The microwave system

A microwave device has been developed together withGIGATHERM AG (Grub, Switzerland), to fit into a Avance200 SWB tomograph (maximum sample diameter of 64 mm,magnetic flux density of 4.7 T; Bruker Biospin MRI GmbH,Ettlingen, Germany). The design avoids temperature equalisationbetween microwave heating and measuring temperature distribu-

tion by introducing microwave power directly into the magnet,where the sample to be heated is located (sketch see Fig. 1(a)).The system consists of a microwave generator with a workingfrequency of 2450MHz, a rectangular waveguide transmitting themicrowaves to the circular waveguide which then directs theminto themagnet of the tomograph. For a detailed description of thesystem the authors refer to Knoerzer (2006).

This device together with the MRI tomograph allowed formeasuring three-dimensional temperature distributions duringmicrowave heating in real time.

2.2. The model

2.2.1. GeometryThe geometry of the model microwave cavity is depicted in

Fig. 1(b). The model has been designed in full 3D in order to bea good approximation of the microwave chamber (h=300 mm,d=84 mm).

Samples used were predominantly simple shapes (cylinders,spheres) of a model food with known temperature dependentthermo-physical properties (Knoerzer, Regier, Erle, & Schubert,2004). However, a method was developed to allow the intro-duction of models of real food structures based on a 3D spin-density measurement technique (Knoerzer, 2006) as shown inFig. 2 and described in the section “Computational methods”.

2.2.2. Governing equationsAs mentioned in the Introduction, microwaves, being electro-

magnetic waves, are described by Maxwell's Eqs. (3)–(6):

jdYD ¼ q ð3Þ

jdYB ¼ 0 ð4Þ

j�YE ¼ �AYBAt

ð5Þ

j�YH ¼ Yj þ AYDAt

: ð6Þ

Fig. 2. The inner structure of the chicken wing as determined by MRI (middle: MRI raw data; corners: matrix subdivided into bones (a), fat (b), and meat (c)).


Maxwell's equations describe the coupled theory of theformer separately described electric and magnetic phenomena.Eqs. (3) and (4) describe the source (ρ) of an electric field Ewithout a magnetic monopole as a source for the magnetic fieldH. Eqs. (5) and (6) show the coupling between the electric andmagnetic fields.

The interaction of electromagnetism with matter is expressedby the material equations or constitutive relations Eqs. (7)–(9),where the permittivity or dielectric constant ε (the interaction ofnon-conducting matter with an electric field), the conductivity σand the permeability μ (the interaction with a magnetic field)appear tomodel their behaviour. The zero-indexed values describethe behaviour in vacuum, so that ε and μ are relative values.

YD ¼ e0 erdYE ð7Þ

YB ¼ A0 ArdYH ð8Þ

Yj ¼ rdYE : ð9ÞIn general, all these materials can be complex tensors (with

directional-dependent behaviour). In the case of food sub-

stances, some simplifications are possible: since food behavesnon-magnetically, the relative permeability can be set to μ=1and the permittivity tensor can be reduced to a complex constantwith real (e′) and imaginary part (e″), which may include theconductivity σ (Tang, 2005), as shown in Eq. (10).

eWr ¼ r2d kd f d e0

ð10Þ

PV ¼ PV

¼ 2d kd f d e0d eWrd E2: ð11Þ

Eq. (11) describes the volumetrically dissipated power PV asfunction of microwave frequency f, dielectric properties (ε0,ε″)and the electric field E.

Knowing dissipated power PV as heat source Qem, Eq. (2)can be modified to:

qCpATAt

�jd kdjTð Þ ¼ PV: ð12Þ

Electromagnetism and heat transfer are thus coupled.

Fig. 3. Exemplary flowchart of the simulation (with QuickWave-3D™- and COMSOL Multiphysics™-solution): Microwave heating of a chicken wing. (a) Interface(programmed in MATLAB 7.4™). (b) QuickWave-3D™ model. (c) COMSOL Multiphysics™ structure. (d) Solution of coupled EM-thermal solver. (Forinterpretation of the coloured scales in this figure, the reader is referred to the online version of this article.)


2.2.3. Process, initial and boundary conditions

2.2.3.1. Process conditions. In the uncontrolled microwaveheating simulation of model food cylinders, the microwavepower was set to constant values of P=19 W and P=23.1 Wbetween t=200 s and t=650 s and P=0 at all other times. In the

feedback-controlled simulation, the microwave power was set to37 Was long as the temperature in the hottest spot was below thepredefined maximum temperature 343 K and set to P=0 whenthis temperature was exceeded. A stop condition was triggeredwhen the temperature in the coldest spot is equal or greater thanthe predefined target temperature (333 K) for a certain time.

Fig. 4. Visual comparison between the simulated (left) and the measured (right)heating of a model food cylinder (at a discrete time). (For interpretation of thecoloured scales in this figure, the reader is referred to the online version of this article.)

Fig. 5. Visual comparison between the simulated (left) and the measured (right)heating of the center cross-section (h=16 mm) of the microwave heated modelfood cylinder (at a discrete time). (For interpretation of the coloured scales in thisfigure, the reader is referred to the online version of this article.)


2.2.3.2. Initial conditions. At the start of the process, bothsample and surrounding air are in thermal equilibrium at T0=298 K. In the feedback-controlled simulation the initial tem-perature of both sample and surrounding air was set to T0=295 Kfor both the pure microwave heating process and the combinedmicrowave hot air heating process.

2.2.3.3. Boundary conditions. The boundary conditions forthe electromagnetic interaction with the surface of the sampleare described by reflection and transmission.

The boundary of the cavity is assumed to be a perfectconductor and thus:

E � n ¼ 0 and Hd n ¼ 0;

where n is the normal to the surface:ð13Þ

The boundary conditions at an interface between twodielectric materials are:

E2 � E1ð Þ � n ¼ 0 ð14Þe2E2 � e1E1ð Þd n ¼ 0 ð15Þ

H2 � H1ð Þ � n ¼ 0 ð16Þ

A2H2 � A1H1ð Þd n ¼ 0: ð17ÞA heat transfer based on free convection on the boundary of

the heated object (∂V) was modelled for thermal interactions.

�kATAn

þ hd T � Tað Þ ¼ 0 on AV : ð18Þ

The heat transfer coefficient was calculated taking thermalconductivity of the surrounding medium and shape of thesample into consideration (VDI Wärmeatlas, 2002).

A good approximation for the heat transfer coefficient wasfound to be:

h ¼ kairPr

ð19Þ

with the thermal conductivity of the surrounding air (λair) and anaverage radius of the sample with H/D (height to diameter ratio)between 0.7 and 1.3.

2.2.4. Material propertiesThe thermo-physical properties, loss factor ε″, permittivity

ε′, thermal conductivity λ, density ρ, heat capacity CP andtheir variation with water content and temperature were deter-mined experimentally for a model food, developed for micro-wave heating research, and fitted with third order polynomials(Knoerzer et al., 2004). These equations were implemented inthe subdomain settings of the software packages used.

In the case of the chicken wing heating, thermo-physicalproperties of fat, meat and bone material have been taken fromliterature.

2.2.5. Computational methodsThe partial differential equations describing the electromag-

netic part of the model were solved with the finite differencetime domain method (FDTD) using QuickWave-3D™ (QWEDSp. z o.o., Warsaw, Poland), a commercial software package.The finite element method was chosen for the heat transport partand solved using COMSOL Multiphysics™ (COMSOL AB,Stockholm, Sweden) a second commercial software package.The interface between both software packages was programmedas a graphical user interface (GUI) in MATLAB 7.4™(Mathworks, Natick, MA, USA). The cross-software commu-nication was enabled by transforming the different outputformats into a universal format (flowchart see Fig. 3).

The presented approach describes a one-way simulation, i.e.after calculating the dissipated power PV by QuickWave-3D™with constant dielectric properties at a given initial temperature,the heat transfer is calculated with temperature and/or locationdependent properties. In the heat transfer balance calculations,PV (from QuickWave-3D) is used as a source term, interpolatedto describe the power density as a function f (x,y,z). FollowingEq. (11), PV is directly proportional to the loss factor (PV∝ε″)and thus a modified form can be used in order to take thedependency of the loss factor and thus the dissipated power intoconsideration.

PV T ; x; y; zð Þ ¼ PV;QuickWave�3D x; y; zð Þd eW T ; x; y; zð ÞeWinitial

ð20Þ

where PV,QuickWave-3D is the solution of the EM simulation, ε″initialthe initial loss factor of the sample, ε″(T,x,z,y) the recalculatedloss factor taking different temperatures at different locations inthe sample into account.

Fig. 8. Comparison of heating curves in a hot and a cold spot in a microwaveheated cylindrical model food sample (microwave power P=23.1 W).

Fig. 6. Visual comparison between the simulated (left) and the measured (right)heating of a cross-section near the top (h=26 mm) of the microwave heatedmodel food cylinder (at a discrete time). (For interpretation of the colouredscales in this figure, the reader is referred to the online version of this article.)


The temperature dependence of the permittivity ε′ is neg-lected in this approach. However, in food materials the changein permittivity due to changes in temperature is far less pro-nounced than the changes in loss factor.

A MATLAB 7.4™ code was used to transform the structuredata of real foods as determined by MRI into the format read-able by the simulation software packages and allocating thecorresponding material properties. In other words, differentmaterials yield different intensities in MRI spin-densitymeasurements. Thus it is possible by multi-threshold settingto subdivide a full 3D MRI image (3D matrix) into differentmatrices corresponding to the materials and allocate thematching thermo-physical properties at the specific locationsto these materials.

A further code was developed in MATLAB 7.4™ to allowfor a feedback-controlled simulation. This code was able toobserve the hottest and coldest spots in the simulated heatingscenario and switching power off after a predefined maximumtemperature in the hot spot is reached (i.e. heat source is set tozero) and turning it on (i.e. heat source=PV(T,x,y,z)) after fallingbelow that value according to the description in the “Processconditions” subsection. The simulation was programmed to stopafter a predefined temperature in the coldest spot is reached andheld for a certain time.

In QuickWave-3D™ the computational domain was sub-divided into cubic elements with 1 mm edge length (corre-

Fig. 7. Comparison of heating curves in a hot and a cold spot in a microwaveheated cylindrical model food sample (microwave power P=19 W).

sponding to 1.5 million cells); in COMSOL Multiphysics™ thenumber of elements depended on the size of the sample andvaried between 9.000 and 20.000 tetrahedral elements.

Aworkstationwith a dual-core processor (CPU speed 2.6GHz)and RAM of 2 GB running a 32-bit Windows XP professional OSwas used, which allowed the simulation to be completed withinless than 2 h.

2.2.6. Model validationValidation was performed by visually comparing simulated

and measured 3D temperature distributions results. Averagetemperature profiles at measured points (hot and cold spots)were compared with profiles provided by the model at the samelocations. A MATLAB routine was developed to plot averagedmeasured temperatures within axial cross-sections at specificpoints in time versus simulated ones. The routine was used toperform a statistical analysis in order to calculate R2.

The sophisticated method of temperature mapping usingmagnetic resonance imaging was used to measure temperaturesin three dimensions throughout the heated samples. Due to thecomplexity of this approach and the space limitations, the reader is

Fig. 9. Confidence plot of simulated temperatures versus measured temperaturesof identical locations of the axial cross-section where the hottest spot is located.(For interpretation of the coloured points, the reader is referred to the onlineversion of this article.)

Fig. 10. 3D temperature distribution in a chicken wing after microwavetreatment of 600 s (power was switched on after 200 s) as calculated using thenew simulation procedure. (For interpretation of the coloured scales in thisfigure, the reader is referred to the online version of this article.)


referred to Knoerzer et al. (2005a), Knoerzer, Regier, and Schubert(2005b), and Knoerzer (2006) for a detailed description of thismeasurement technique and the underlying physical principles.

3. Results and discussion

3.1. Comparison of simulated and measured heating profile ofa model food cylinder in real time

Simulated and measured temperature profiles throughout themicrowave treated samples are shown in three dimensions. Inthis section simulation results will be compared to measure-ments in order to validate the new simulation approach.

3.1.1. Visual comparison of 3D temperature distributionsFig. 4 shows both simulated and measured 3D temperature

distributions in a microwave heated model food cylinder at adiscrete time of 250 s and a microwave power of 19 W.

Figs. 5 and 6 provide a close-up view of two cross-sectionsof the cylinder one in the middle (h=16 mm) of the cylinder, thesecond one near the top (h=26 mm).

Fig. 11. (a) Pure microwave heating (P=37 W; external temperature Text=29

A good qualitative agreement could be found between simu-lation and measurement in all cases. Both simulation and measure-ment show almost identical temperature distribution profiles.

3.1.2. Comparison of heating curves in discrete pointsFigs. 7 and 8 show heating curves in a hot and a cold spot of the

heatedmodel food cylinder at microwave powers of 19 and 23.1W.Good quantitative agreements were found in all cases.

3.1.3. Comparison of entire cross-sectionsFig. 9 shows a confidence plot of simulated versus measured

temperatures at different times of the heating process at identicallocations of a cross-section of the microwave treated sample. Astatistical analysis showed an R2 greater than 0.95 which meansthey are not significantly different.

3.2. Simulated heating of a chicken wing

The simulated heating of the chickenwing described in Section2.2 and shown in Fig. 2 was performed at a microwave power of25 W for 600 s. Fig. 10 shows the temperature distribution in thechicken wing after heating. It is obvious that uncontrolled processconditions lead to the development of unacceptably uneven tem-perature distributions caused by a number of interacting factors asdescribed in detail in Ringle and Donaldson (1975), Sinell (1986),Risman (1992), Buffler (1993), Ohlsson and Bengtsson (2001),Ryynänen, Risman, and Ohlsson (2004), Wäppling-Raaholt andOhlsson (2005), and Knoerzer (2006).

3.3. Controlled microwave heating of a model food cylinder

In microwave heating simulation, temperature distributionsare clearly defined at any point in time. This fact allows afeedback-control loop to be simulated on the basis of arisingtemperatures to ensure uniform heating. In practice, such afeedback-control loop is difficult to accomplish in a real lifeindustrial microwave process due to the necessity of measuring3D temperature distributions in-line non-invasively.

Fig. 11 shows the heating curves for the hottest and thecoldest spot in a microwave heated cylindrical model food

5 K). (b) Combined microwave hot air heating (P=37 W; Text =333 K).

Fig. 12. Output of the on/off-control tool; microwave power pulse program(example of combined microwave hot air heating as described above).


sample. The microwave power was set to P=37 W as long asthe temperature in the hot spot was below the maximumtemperature and to P=0 as soon as this temperature wasexceeded. The maximum temperature to be reached in the hotspot was set to 343 K and the minimum temperature to bereached and hold for 30 s was set to 333 K.

Fig. 11 (a) shows the example of a pure microwave heating,i.e. the external temperature was set to the initial temperature ofthe sample (T=295 K), Fig. 11(b) shows a combinedmicrowave hot air process, i.e. the external temperature wasset to the target temperature (T=333 K).

In the case of the pure microwave heating, the maximumtemperature was reached and regulated after about 200 s. How-ever, the temperature in the coldest spot did not reach the targettemperature even after more than 20min. The reason for this is theheat loss from the heated sample to the cooler surrounding air.

When hot air is introduced in the combined process, thetemperature in the hot spot also reaches maximum temperatureafter about 200 s. In this case, the temperature in the coldest spotreached the target temperature after about 10 min. Thecombined process allows a secure minimal time/temperaturetreatment. The output of the feedback-controlled simulation isshown in Fig. 12, a microwave power pulse program that can beincorporated in the control unit of a microwave oven and thusallows for repeating the simulated scenario in reality bycontrolling (no feedback) the microwave power.

3.4. Discussion: implications for process design

The experimental data shows that it is possible to model thetemperature changes that occur as a result of a microwavetreatment. No previous publication shows a similar approach forrunning a feedback-controlled simulation based on a combinationof software packages for electromagnetism and heat transfercalculations. Furthermore, simulations of the microwave treat-ment of real heterogeneous foods with structural data provided byMRI spin-density measurements have not been published before.

4. Conclusions

Microwaves heat food volumetrically and more rapidly thanconventional heating. However, several factors associated with

volumetric heating may lead to the development of uneventemperature distributions. A model has been developed toaccurately predict temperature profiles during a heating processin a lab scale microwave system. Based on the fact that twocommercial software packages have been coupled in thisapproach, a major benefit of this approach is the user-friendlyoperation, full support by the software vendors and extensiveuser manuals. The one-way coupling, enhanced by implemen-tation of temperature dependent loss factor and thus dissipatedpower is computationally convenient and very time efficientcompared to a two-way coupling.

The model simulates both electromagnetic waves and heattransfer and thus the temperature evolution in arbitrarily shapedfoods. Good agreement between the model and actualtemperature measured in a series of experimental runs usingthe sophisticated magnetic resonance imaging 3D temperaturemapping technique was obtained. Significant temperatureheterogeneities occurred as long as the treatment was uncon-trolled while more uniformity was achieved when a control toolwas implemented in the simulation. This kind of feedback-controlled simulation is one-of-a-kind to date and allows forpredicting optimal power programs, ensuring minimal time/temperature treatments in microwave food processing. Modelsof this type could be used both to design and optimise processesand to demonstrate the safety of the process for industry andregulatory bodies.

Notation

Latin lettersB→ magnetic flux density [T=V·s/A·m]CP specific heat capacity [J/kg·K]D→ electric flux density [A·s/m2]E→, E electric field [V/m]f frequency [s−1]h, H height [mm]h heat transfer coefficient [W/m2 K]H→

, H magnetic field [A/m]j→ electric current density [A/m2]n normal to the surfaceP power [W]PV power density [W/m3]PV,QuickWave-3D power density calculated by QuickWave-3D™

[W/m3]Qem electromagnetic heat source [J/s m3]qR radiative power flux density [W/m3 s]r radius [m]R2 coefficient of determination [–]T absolute temperature [K]t time [s]V volume [m3]x,y,z coordinates

Greek lettersε0 dielectric constant of vacuum (=8.854 ·10−12 A s/Vm)εr relative permittivity [–]ε complex dielectric constant [–]


εr″ real part of complex permittivity [–]λ thermal conductivity [W/m·K]μ0 permeability of vacuum (=1/ε0c0

2=1.257·10−6 V s/Am)μr relative magnetic permeability [–]μ magnetic permeability [–]π Ludolph's number (≈3.14159)ρ charge density [A·s/m3]ρ density [kg/m3]σ conductivity [A/V m]

Operators∂ partial differential▿ Nabla operator∑ sumΔ difference▿· divergence▿× rotation

Abbreviations3D three-dimensionalBEM boundary element methodEM electromagneticFDM finite difference methodFDTD finite difference time domainFEM finite element methodGUI graphical user interfaceMOM method of momentsMRI magnetic resonance imagingTLM transmission line matrix

Acknowledgement

The authors would like to thank the German ResearchFoundation (DFG) for financial support in the NMR researchgroup FOR338.

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A computational model for calculating temperature distributions in microwave food applications

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