at SciVerse ScienceDirect

Energy 55 (2013) 278e294

Contents lists available

Energy

journal homepage: www.elsevier .com/locate/energy

A theoretical study on the use of microwaves in reducing energyconsumption for an endothermic reaction: Role of metal coatedbounding surface

Madhuchhanda Bhattacharya a, Tanmay Basak b,*

aC2-5-4C, Delhi Avenue, Indian Institute of Technology Madras Campus, Chennai 600036, IndiabDepartment of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e i n f o

Article history:Received 4 September 2012Received in revised form27 February 2013Accepted 3 March 2013Available online 25 April 2013

Keywords:ReactionMicrowaveMetallic supportMathematical modelingEnergy savings

* Corresponding author.E-mail address: tanmay@iitm.ac.in (T. Basak).

0360-5442/$ e see front matter � 2013 Published byhttp://dx.doi.org/10.1016/j.energy.2013.03.016

a b s t r a c t

This work presents a theoretical analysis on savings of energy during an endothermic reaction undermicrowave heating compared to conventional heating and shows the use of metal coated boundingsurface to enhance the energy savings in otherwise low saving zones. Main thrust of this work is thequantification of energy savings for various probable microwave heating scenarios that may arise eitherdue to varying reactor dimension (2L) over thin, intermediate and thick regimes or due to varyingdielectric properties of the reactor. The analysis considers detailed transport equations in conjunctionwith Helmholtz equation for microwave propagation within a semiinfinite batch reactor. Simulationsshow that use of microwave can significantly save energy (as high as 60%) depending on reactorconfiguration. Simulations also show efficient use of metal coated bounding surface to enhance energysavings for reactors with 2L/leff ¼ 0.5n�0.25, where n ¼ 1, 2, 3. and leff is wavelength of microwavewithin the reactor. The enhancement is found to be 2 and 1.5 times at 2L/leff ¼ 0.25 and 0.75, respec-tively. Various regions of efficient use of metal coated bounding surface for different microwave heatingscenarios have been identified in a series of master curves.

� 2013 Published by Elsevier Ltd.

1. Introduction

Energy consumption has been one of the important issues overpast few decades triggering a series of research on finding fasterand lower energy consuming technologies. Among several otherpossibilities like ultrasonic [1e16], electro-osmotic [17e20] andpulsed corona processing [21e26], microwave processing of ma-terials came out as an alternate viable process due to its fasterprocessing and quick start up capabilities. Microwaves within fre-quency range of 300 MHz to 300 GHz interact with dipoles and cantransfer energy volumetrically at molecular level in dielectric ma-terials. Volumetric supply of energy accelerates the progress of aprocess due to lower energy transfer time compared to cases whereenergy is supplied from surfaces. Over the years, potentials of mi-crowave processing in terms of process intensification and lowerenergy consumption have been proven in many different fields[27e39] with a major potential in the field of material synthesis,

Elsevier Ltd.

where chemical reactions are reported to undergo dramatic ac-celeration under microwave radiations [40e60].

Since reaction rates are strong functions of local temperature, adetailed knowledge about temperature profile inside the reactingdomain is necessary to ascertain and quantify enhancement of re-action under microwave radiation. The situation becomes moredemanding for endothermic reaction, where occurrence of reactiondepends on local availability of heat and hence on heating pattern.It may be noted that microwave induced heating pattern may behighly complex with multiple local hot-spots within the domaindepending on dimension and dielectric properties of the medium[61e65]. However, due to the difficulties associated with detailedmeasurement of electric field within the domain, microwaveinduced heating pattern has often been approximated by ad hocmodels in literature. Use of such ad hoc models may hide some ofthe physics underlying the observed enhancement leading tospeculation about athermal effect of microwave radiation onchemical reaction [66,67]. Correspondingly, a few theoreticalstudies have been initiated to relate the enhanced reaction rate dueto microwave heating effects [68e73]. But, none of the earlier re-searchers considered detailed description of microwave propaga-tion via Helmholtz equation and thus could not quantify/analyze

Nomenclature

CA and cAdimensional (mol/m3) and dimensionless reactantconcentration

CA0 initial reactant concentration (mol/m3)cA dimensionless average reactant concentration(rCp)eff effective heat capacity (J/m3K)Deff effective reactant diffusivity within packed bed (m2/s)Dpeff effective penetration depth (m)E activation energy (kJ/kmol)Em electric field (V/m)f frequency (Hz)fp leff=2pDpeff

fw leff/l0I0 intensity of incident radiation (W/m2)keff effective thermal conductivity of packed bed (W/m K)L half-width of packed column, mNp penetration number (2L=Dpeff )NR heat-reaction number (2LDHRfRðCA0; T0Þ=I0)Nw wave number (2L/leff)QMWand qMW dimensional (W/m3) and dimensionless absorbed

power distributionqMW dimensionless average absorbed powert time, s

T temperature, KT0 initial temperature, KZ and z dimensional (m) and dimensionless spatial coordinate

Greek symbolsaeff effective thermal conductivity (keff=ðrCpÞeff )b conduction number (keffT0/2LI0)DHR heat of reaction (J/mol)gR dimensionless activation energy (E/RT0)ε0 free space permittivity (Farad/m)k0eff effective dielectric constant of packed bedk00eff effective dielectric loss of packed bedleff effective wavelength of microwave within packed

column, ml0 wavelength within free space, ms dimensionless timesD diffusion number (faeff=Deff )sfMW

dimensionless reaction time under microwave heatingsfConv dimensionless reaction time under conventional

heatingq dimensionless temperature (keff(T�T0)/2LI0)f porosityFR bulk Thiele modulus (4L2fRðCA0; T0Þ=DeffCA0)

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 279

the entire spectrum of microwave heating possibilities and theirinfluence on overall progress of reaction.

Main objective of the current study is to fundamentallyinvestigate the enhancement of reaction under various probablemicrowave heating scenarios (that can be encountered due tovarying reactor dimension or dielectric properties of the reactingmedium) and to show how the enhancement can be furtherimproved by inserting a metallic coated surface at the other sideof microwave incidence for certain specific cases. The reactionconsidered is a first order endothermic gas phase reactionoccurring in a semiinfinite batch reactor in presence of susceptors.It may be noted that most of the common gases do not absorbmicrowave energy to significant extent and thus require suscep-tors to continue the reaction. Susceptors absorb microwave en-ergy and transfer them to the gaseous reacting medium. Reactionscan also be carried out without the help of susceptors in cases ofmicrowave absorbing reacting medium e.g. dielectric liquid re-actants. However, microwave propagation depends on reactantconcentration within such system and change during the courseof reaction depending on the specific material under consider-ation. Correspondingly, simulations of such microwave absorbingreacting system remain restricted to a specific material or a groupof materials representing only a few specific microwave heatingscenarios. This directed the current work to consider microwavenon-absorbing gaseous reaction, which unleashes the limitationsof material specific simulations and allows to perform a materialinvariant analysis considering all probable microwave propaga-tion scenarios. Nevertheless, the results presented here can beextended for qualitative estimates of microwave absorbing liquidphase reactions.

The analysis is carried out via simulating progress of reactionfrom detailed mass and energy balance equations in conjunctionwith Helmholtz equation for microwave propagation. Simulationsare carried out for microwave propagation with or without metalcoated bounding surface in order to investigate the changes inreactor dynamics and probable enhancement of energy savings.Simulations are also carried out for reactions under equivalentconventional heating to quantify enhancement of reaction rate and

corresponding reduction of energy consumption under microwaveradiation in various probable scenarios of varying reactor di-mensions, presence of metallic coated bounding surface, changingsusceptor properties, varying reaction rate etc.

2. Mathematical modeling

The reacting system under consideration is a semiinfinitebatch reactor as shown in Fig. 1, where the entrapped gas un-dergoes a first order endothermic conversion given byA(g) /

kR B(g). This work considers two possible configurations ofheating the reactor via microwave as shown in Fig. 1(a) and (b).The right wall of the reactor is coated with a reflective (metallic)material in subplot (a), while the reactor does not contain anyreflective coating in subplot (b). The reflective materialcompletely reflects back the microwave and hence alters theresulting heating pattern in the former compared to the later. Inboth the cases, the reactor is exposed to microwave of intensity I0from the left end and packed with highly microwave absorbingmaterials or susceptors (e.g. SiC). Note that susceptors arerequired since gaseous species do not absorb microwave ingeneral and thus cannot be heated under microwave radiation bythemselves. The two configurations with and without reflectivecoating will be referred as “type 1” and “type 2” reactors,respectively. To compare the efficiencies of the two microwaveheating strategies, their equivalent conventional heating withsame heating rate are also considered as shown in Fig. 1(c). Itmay be noted that intensities of conventional heating corre-sponding to type 1 and type 2 configurations may differ fromeach other due to changes in microwave propagation.

It is assumed that the reaction occurs only in the gas phase andthe packing materials do not absorb any species. It is also assumedthat material properties remain constant throughout the reaction,packings are homogenous and reactor walls are impermeable andthermally insulated. With further assumption of local thermalequilibrium and negligible convection in gas phase, evolution ofreactant concentration (cA) and temperature (T) from the uniforminitial state of CA,0 and T0, respectively can be simulated from

(a) (b) (c)

Fig. 1. Schematic representation of (a) type 1 and (b) type 2 reactor configurations (with and without metallic coating, respectively) under microwave heating with subplot (c)showing equivalent reactor configuration for conventional heating.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294280

following mass and energy balance equations and the associatedboundary conditions:

fvCAvt

¼ Deffv2CAvZ2

� fRðCA; TÞ (1a)

�rCp�eff

vTvt

� keffv2TvZ2

þ fDHRRðCA; TÞ ¼�QMW Microwave0 Conventional

(1b)

@Z ¼ �L :vCAvZ

¼ 0;

8>>>>><>>>>>:

vTvZ

¼ 0 Microwave

�keffvTvZ

¼ZL�L

QMWdZ Conventional;

(1c)

@Z ¼ L :vCAvZ

¼ 0;vTvZ

¼ 0: (1d)

These equations have to be solved in conjunction with Helmholtzequation for electric field (Em) along with radiation boundaryconditions for scattering of waves at reactor boundaries given inAppendix A (see Eqs. (A.1)e(A.6) for evaluation of absorbed powerdistribution within type 1 and type 2 reactors (QMW) given by

QMW ¼ 2I0l0leffDpeff

�EmE0

��E�mE0

�; (2)

where E�m is the complex conjugate of Em, l0 and leff are wavelengthofmicrowavewith free space and reactiondomain, respectively,Dpeff

is the penetration depth of microwave within reaction domain andE0 is the incident field (see Appendix A). Other variables used in Eqs.(1a)e(1d) are 2L as the reactor width, f as porosity of packing,Deff ¼ DAf=sp, ðrCpÞeff ¼ fðrCpÞg þ ð1� fÞðrCpÞp and keff ¼ fkg þð1� fÞkp as effective diffusivity of reactant, effective heat capacityand effective thermal conductivity of the packed reaction domain,

respectively, where subscripts g and p represent gas phase andpackingmaterial, respectivelyandDAand sp aremoleculardiffusivityof species A and tortuosity of packing, respectively. In addition, DHR

is the heat of reaction and R(CA,T) is the first order reaction ratesatisfying Arrhenius temperature dependency given by

RðCA; TÞ ¼ kRCAexp�� ERT

�; (3)

where kR is the first order rate constant, E is the activation energyand R is the universal gas constant.

In terms of the following dimensionless variables

z ¼ Z þ L2L

; s ¼ aeff t4L2

; q ¼ keff ðT � T0Þ2LI0

;

cA ¼ CACA0

; qMW ¼ 2LQMW

I0

~Em ¼ EmE0

; rðcA; qÞ ¼ RðCA; TÞR�CA0

; T0�;gR ¼ E

RT0; (4)

where aeff ¼ keff=ðrCpÞeff is the effective thermal diffusivity of thepacked reaction domain, dimensionless form of governing massbalance, energy balance and Helmholtz equation and their associ-ated boundary conditions (Eqs. (1a)e(1d) and Eqs. (A.1), (A.4) and(A.5a), (A.5b)) can be rewritten as

sDvcAvs

¼ v2cAvz2

� FRrðcA; qÞ; (5a)

vq

vs� v2q

vz2þ NRrðcA; qÞ ¼

�qMW Microwave0 Conventional ; (5b)

d2~Emdz2

þ 4p2N2w

1þ ifp

2~Em ¼ 0; (5c)

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 281

8>>>>< vcA ¼ 0;�vq ¼0 Microwave

;

@z ¼ 0 : >>>>:vz vz

(qMW Conventional

12pNwfw

d~Emdz

þ i~Em ¼ 2i exp½�ipNwfw�;(5d)

@z ¼ 1 :

8>>>>>>><>>>>>>>:

vq

vz¼ vcA

vz¼ 0;

~Em ¼

8>><>>:0 Type 1

� i2pNwfw

d~Emdz

Type 2;

(5e)

supplemented with the following expressions for dimensionlessreaction rate (r(cA,q)) and dimensionless absorbed power (qMW)

rðcA; qÞ ¼ cAexp�gR

q

qþ b

�; qMW ¼ 4pNwfp

fw~Em~Em

*; (6)

where dimensionless heat flux for conventional heating given byqMW ¼ R 1

z¼0 qMWðzÞdz is the dimensionless average absorbed po-wer under microwave radiation. In this formulation, dimensionlessparameters Nw (wave number), fp and fw characterize wave prop-agation within the reactor and are defined as

Nw ¼ 2Lleff

; fp ¼ leff2pDpeff

; fw ¼ leffl0

; (7)

where Nw captures the effect of reactor thickness on wave propa-gation and dimensionless parameters fw and fp capture the effect ofdielectric properties on microwave induced heating. [It may benoted that the combination 2pNwfp ¼ 2L=Dpeff is the measure ofreactor thickness compared to the penetration depth of microwaveand is called penetration number or Np in literature [61,62,64]]. Theother four dimensionless numbers given by FR (bulk Thielemodulus), sD (diffusion number or Lewis number), NR (heat-reac-tion number) and b (conduction number) quantify relative mag-nitudes of various reaction and transport time scales within thereactor and are defined as

FR ¼ tDtR; b ¼ tH

ta; sD ¼ ftD

ta;NR ¼ tH

tDHR

; (8)

where tD ¼ 4L2=Deff , ta ¼ 4L2=aeff , tR ¼ CA0=fRðCA0

; T0Þ,tDHR

¼ ððrCpÞeffT0Þ=ðfDHRRðCA0; T0ÞÞ and tH ¼ ð2LðrCpÞeffT0Þ=I0

represent mass diffusion time, thermal diffusion time, reactiontime, cooling time due to heat absorption by reaction and heatingtime, respectively. It may be noted that FR and b capture mass andheat transport resistances within the reactor, respectively, NR

measures relative supply of heat compared that required by reac-tion and sD quantifies relative speed of mass and thermal diffusionwithin the reactor.

Fig. 2. Schematic representation of superficial reaction zone.

3. Solution technique

The governing equations (Eq. (5)) are spatially discretized viaweak formulation of Galerkin finite element method using 50quadratic elements and time integrated using Crank-Nicholsonscheme with a time step of Ds ¼ 10�4. The resulting nonlinearalgebraic equations are solved at each time step by NewtoneRaphson method following the same procedure as laid out in anearlier article [74]. It is worthwhile to mention here that simulationof endothermic reaction requires to stop the reaction locally andtemporarily wherever and whenever local temperature reduces

below a critical level (say Tcrit or qcrit in dimensionless scale). This isto avoid unrealistic temperature drop during the simulation due toabsorption of heat by reaction especially in no/low heating zoneswithin the reactor. Reaction is resumed back once local tempera-ture increases to Tcrit. This causes the reaction rate to undergo a stepjump given by

rðcA; qÞ ¼

8><>: cA exp�gR

q

qþ b

�for q � qcrit

0 otherwise

; (9)

which causes the tracing of its dynamics to be numerically chal-lenging due to introduction of Dirac delta function in the Jacobianmatrix [75]. However, following the suggestion of an earlier workon tracing the phase change of pure materials [75], the numericaldifficulties are efficiently circumvented by smoothing the stepjump with a linear function within a superficial reaction zone ofwidth 2DT around Tcrit as shown in Fig. 2. The modified rateexpression with inclusion of superficial reaction zone can be writ-ten in terms of dimensionless variables as

brðcA;qÞ ¼8>>>>><>>>>>:

0 for q� qcrit � ε

rðcA;qcrit þ εÞ�q� qcrit þ ε

2ε

�for qcrit � ε� q� qcrit þ ε;

rðcA;qÞ for q> qcrit þ ε

(10)

where ε¼ keffDT/2LI0 is half width of the superficial reaction zone indimensionless temperature scale. Convergence of numerical sim-ulations with respect to the width of superficial reaction zone areshown in Fig. 3, which shows variations of overall reactant deple-tion rate ð1� cAÞ as 3is varied from 10�3 to 10�6 (dotted, dash andsolid lines and circles). Simulations in Fig. 3 are carried out for lowintensity conventional heating corresponding to type 1 and type 2configurations at Nw ¼ 0.06, fw ¼ 0.1, fp ¼ 0.1 (qMW ¼ 0:002 and0.141, respectively). It may be noted that unrealistic temperaturedrop during the simulation become more probable for surfaceheating (at regions far away from heat source) than volumetricheating of microwave especially for low intensity of qMW. As aresult, simulations require much frequent use of superficial reac-tion zone (Eq. (10)) in cases of low intensity of conventional heat-ing. Correspondingly, convergence of numerical simulations in thetwo severe cases presented in Fig. 3 guarantees convergence for allother cases. Other parameters in Fig. 3 correspond to FR ¼ 10,sD ¼ 1, NR ¼ 0.1, b ¼ 0.1 and gR ¼ 10. Fig. 3 shows that numericalsimulations become invariant of the width of superficial reaction

Fig. 3. Convergence of numerical simulation with respect to the width of superficial reaction zone ( 3) shown in terms of 1� cAvs s at ε ¼ 10�3 (dotted lines), 10�4 (dashed lines),10�5 (solid lines) and 10�6 (circles) for two severe cases of slow surface heating corresponding to type 1 and type 2 configurations at Nw ¼ 0.06, fp ¼ 0.1 and fw ¼ 0.1 with qMW ¼0:002 and 0.141, respectively with other parameters as sD ¼ 1, NR ¼ 0.1, b ¼ 0.1, FR ¼ 10 and gR ¼ 10.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294282

zone at ε ¼ 10�5 even for these severe situations. Hence, we willuse ε ¼ 10�5 for all the simulations. It may be noted that we haveassumed Tcrit ¼ T0 in Fig. 3 and will use the same in all forthcomingillustrations.

Starting from cA ¼ 1 and q ¼ 0 at s ¼ 0, all simulations arestopped when conversion reaches 90% across the entire reactorcorresponding to max(cA) ¼ 0.1. Time required for this conversionwill be referred as reaction time and denoted by sfMW

and sfConv formicrowave and conventional heating, respectively. Simulations arecarried out for different reactor dimensions over the entire range ofNw to replicate various possible microwave heating scenarios andtheir alteration due to metallic coating. For all the cases, reactordynamics are simulated and compared for two limiting cases ofweak and strong mass transfer limitations within the reactor viaFR ¼ 0.1 and 10, respectively with fixed relative heat supply as wellas fixed heat transfer limitations at NR ¼ 0.1 and b ¼ 0.1, respec-tively. It may be noted that NR�1 ensures enough heat supply tothe reactor compared to the heat requirement for reaction. On theother hand, higher thermal diffusion time compared to heatingtime at b�1 allows to simulate distinct dynamics of microwave andconventional heating [74]. It is important to note that heating ofreactor becomes invariant of the supply pattern for b[1, wheresupplied heat spreads uniformly across the reactor instantaneously.Other parameters are selected to be sD ¼ 1 (to avoid preferentialdiffusion of either heat of mass) and gR ¼ 10.

4. Model validation

In absence of any appropriate experimental or numerical data inliterature, we validate the present simulations by comparing themwith the analytical solutions which can be obtained in the limit ofgR / 0. In this limit of gR / 0, the nonlinear mass and energybalance equations (Eqs. (5a) and (5b)) reduce to the following linearones:

sDvcAvs

¼ v2cAvz2

� FRcA; (11a)

vq

vs� v2q

vz2þ NRcA ¼

�qMW Microwave0 Conventional ; (11b)

along with the boundary conditions given by Eqs. (5d) and (5e) andthe initial conditions given by cA ¼ 1 and q ¼ 0 at s ¼ 0. Using FiniteFourier Transformations (FFT), closed form solutions of the aboveequations are obtained as

cAðsÞ ¼ exp�� FR

sDs�; (12a)

qðz; sÞ ¼ sqMW � NRsDFR

1� exp � FR

sDs

� � ��

þ PNn¼1

1� exp½�lns�ln

wnðzÞ( hqMW;wni Microwaveffiffiffi

2p

qMW Conventional;

(12b)

where ln ¼ n2p2 is the nth eigenvalue, wn(z) is the nth eigenfunc-tion given by

wnðzÞ ¼�1 n ¼ 0ffiffiffi2

pcos npz n � 1 ; (12c)

and hqMW;wni is the inner product defined as

hqMW;wni ¼Z1

z¼0

qMWðzÞwnðzÞdz: (12d)

The expressions for hqMW;wni and qMW are given in Appendix B(Eqs. (B.4) and (B.5)).

To compare numerical results, reactor dynamics in the limit ofgR / 0 are simulated from Eq. (5) by using gR ¼ 0.001 with otherparameters as FR ¼ 10, NR ¼ 0.1, sD ¼ 1.0 and b ¼ 0.1 (same as inFig. 3). Resulting concentration and temperature profiles arecompared with those obtained from the analytical expressions(Eqs. (12a) and (12b), respectively) for three different reactor di-mensions of Nw ¼ 0.25, 0.5 and 0.75 with fp ¼ 0.1 and fw ¼ 0.1 asshown in Fig. 4(a)–(b). These three reactor dimensions correspondto various probable stiffness of microwave heating patterns (asreflected in the corresponding temperature profiles) and alsocorrespond to a wide range of conventional heating intensities(qMW ¼ 0.949, 0.114 and 0.6 for type 1 configuration andqMW ¼ 0.059, 0.504 and 0.158, for type 2 configuration at Nw¼ 0.25,0.5 and 0.75, respectively). Fig. 4 shows the comparisons for bothmicrowave (MW) and conventional (Conv.) heatings, where bulletsrepresent numerical simulations and continuous lines representanalytical solutions with darker and lighter shades indicating type1 and type 2 configurations, respectively. The spatial profiles inFig. 4 are shown at s ¼ 0.15, which is half the time required forcompletion of 90% conversion [Note: analytical solution predictsrequired reaction time as s ¼ 0.2303, while simulations predictrequired reaction time to be s¼ 0.2307]. It may be noted that in thelimit of gR / 0, reactant concentration and its consumption be-comes invariant of heating mode and hence invariant of spatiallocation within the reactor (as reflected in Eq. (12a)). Accordingly,numerical simulations predict uniform concentration for both mi-crowave and conventional heating irrespective of reactor dimen-sion as shown in Fig. 4(a). In contrast to concentration, temperature

(a)

(b)

Fig. 4. Comparison between simulated results (bullets) and analytical solutions (continuous lines) for type 1 and type 2 configurations (represented by darker and lighter shades,respectively) in case of three representative reactor dimensions of Nw ¼ 0.25, 0.5 and 0.75 with fp ¼ 0.1, fw ¼ 0.1, NR ¼ 0.1, sD ¼ 1, FR ¼ 10, b ¼ 0.1 and gR ¼ 0.001. Subplot (a) showscomparison of concentration profiles at s ¼ 0.15 and subplot (b) shows comparison of temperature profiles for microwave heating (MW) and corresponding conventional heating(Conv.) at s ¼ 0.15.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 283

profiles depend on heating patterns and thus vary with heatingmode, reactor configuration and reactor dimension as shown inFig. 4(b). Nevertheless, numerical simulations are in excellentagreement with analytical solutions in all the cases showing theaccuracy of the solutions for different probable heating scenarios.

5. Microwave power absorption and influence of metalliccoating

Influence of metallic coating on microwave power absorptionare measured by two factors: (i) changes in the spatial distributionof absorbed power (qMW) and (ii) variations in the intensity of qMWor surface heat flux. Spatial distribution of absorbed power de-termines the homogeneity of microwave heating, which inconjunction with intensity of conventional heating ðqMWÞ play acritical role in determining the efficiency of microwave heating.Influence of metallic coating on the above two factors are shown inFig. 5, where the main plot shows the variations of qMW and therepresentative insets show the variations of absorbed power dis-tributions from type 1 (solid lines) to type 2 (dotted lines) config-urations for various probable microwave heating scenariosspanning the range of reactor dimension given by 0.05 � Nw � 10.The results in Fig. 5 are simulated using fp ¼ 0.1 and fw ¼ 0.1.

The insets of Fig. 5 show that entire spectrum of power ab-sorption patterns can be categorized in three distinct classes ofuniform (Nw ¼ 0.06), (ii) oscillatory (Nw ¼ 0.5 and 0.75) and (iii)exponentially attenuated (Np ¼ 3) profiles. These three patternsappear in chronological order as reactor dimension increases frommuch below the wavelength of microwave (Nw � 0.1) to muchgreater than penetration depth of microwave (Np � 3) and dividethe entire range of reactor dimensions in thin (Nw � 0.1), interme-diate (0.1 < Nw and Np < 3) and (iii) thick (Np � 3) regimes,respectively [62]. Thin regime refers to the entire range of reactordimensions below Nw ¼ 0.1, where microwave power absorptionoccurs uniformly irrespective of metallic coating as seen in the insetfor Nw ¼ 0.06. However, metallic coating significantly reduces mi-crowave power absorption in thin regime as can be seen from themagnitudes of qMW for Nw � 0.1 in Fig. 5. It may be noted that

microwave power absorption in type 1 configuration becomesinsignificantly small as Nw decreases beyond 0.05.

Thick regime refers to the reactors with dimensions muchgreater than penetration depth of microwave (Np � 3), where mi-crowave losses most its energy before reaching the other end of thereactor. Correspondingly, power absorption in thick regime occursonly near the exposed surface and remain unaltered by the pres-ence of metallic coating as seen in Fig. 5 for Np � 3. It may be seenfrom the inset for Np ¼ 3 that absorbed power attenuates expo-nentially from the exposed surface in thick regime. Power absorp-tion regime further shrinks with increasing reactor dimension andmicrowave heating approaches surface heating in the limit ofNp / N.

Intermediate regime refers to the reactors in between thin andthick regimes (0.1 < Nw < 3/2pfp), where reactor dimensions areneither much lower than wavelength of microwave nor muchgreater than penetration depth of microwave. In this regime, mi-crowave power absorption is strongly dependent on the nature ofscattering from the reactor walls. Scattering of microwaves formvarious traveling waves within the reactor and interferences be-tween them determine the resultant power absorption character-istics depending on phase-space of individual wave [62]. Sincemetallic coating alters scattering by completely reflecting micro-waves from the right wall, resulting traveling waves in type 1configuration also differ significantly from those in type 2 config-uration. As a result, intermediate regime exhibits most prominentinfluence of metallic coating as seen in Fig. 5.

Most important characteristic of intermediate regime is theoscillatory variations of qMW with reactor dimension, which resultfrom constructive and destructive interferences between travelingwaves. Constructive interference enhances power absorption,while destructive interference suppresses power absorption. Inintermediate regime, constructive interferences occur at periodicintervals as reactor dimension increases from 0.1 to 3/2pfp andevery two consecutive constructive interferences are accompaniedby a destructive interference in between [62]. Hence, qMW un-dergoes periodic enhancement and suppression over the range ofintermediate regime, which is reflected in Fig. 5 as a series of

Fig. 5. Influence of metallic coating on qMW over the entire range of thin to thick regimes (0.05 � Nw � 10) as well as influence of metallic coating on the power absorption profiles[qMW(z)] at representative reactor dimensions (Nw or Np), where solid and dotted lines represent reactors with (type 1) and without (type 2) metallic coating, respectively andparameters fw and fp are taken to be 0.1.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294284

maxima and minima of qMW for both type 1 and type 2 reactors.Enhancement of qMW at locations of its maxima are also calledresonances. Fig. 5 shows that locations of resonances dependstrongly on reactor configuration, where resonances for type 1configuration occur at the locations of minima for type 2 configu-ration and vice versa. Resonances occur at Nw ¼ 0.5n, n ¼ 1,2. fortype 2 configuration, where type 1 configuration leads tominima ofqMW. Similarly, resonances for type 1 configuration occur atNw ¼ 0.5n�0.25, n ¼ 1,2., where qMW exhibits minima for type 2configuration (see Fig. 5). Amplitudes of resonances are also muchhigher in type 1 configuration than type 2.

Similar to qMW, power absorption patterns of intermediateregime are also strongly influenced by metallic coating, where type1 and type 2 configurations lead to completely opposite absorbedpower distributions as illustrated in Fig. 5 for Nw ¼ 0.25, 0.5 and0.75. Typical feature of intermediate regime is the presence of localmaxima in absorbed power distributions due to occurrence ofspatial resonances, where larger reactor favors more spatial reso-nances as can be seen from the insets of Fig. 5 for Nw ¼ 0.25, 0.5 and0.75. In all the cases, locations of spatial maxima in type 1 config-uration correspond to locations of local minima of absorbed powerin type 2 configuration and vice versa.

6. Alteration of reactor dynamics under microwave heatingand influence of metallic coating

Alteration of reactor dynamics under thin, intermediate and thickregimes of microwave heating for type 1 and type 2 configurationsare illustrated in Figs. 6e10 for the representative reactor di-mensions corresponding to Nw ¼ 0.06 (Fig. 6), 0.25 (Fig. 7), 0.5(Fig. 8), 0.75 (Fig. 9) and Np ¼ 3 (Fig. 10). In each case, influences ofmicrowave heating are illustrated for two limiting cases of weakand strong mass transfer limitations at FR ¼ 0.1 and 10 in subplots

(a) and (b), respectively. It may be noted that reactor dynamicsunder microwave heating are shown at the top panel in eachsubplot of Figs. 6e10 and are denoted by “MW”. Reactor dynamicsunder conventional heating corresponding to type 1 and type 2configurations of various Nw’s and Np are shown at the bottompanel in each subplot of Figs. 6e10 and are denoted by “Conv”. In allthe cases, parameters fp and fw are considered to be 0.1 and type 1and type 2 configurations are represented by solid and dashedlines, respectively. Here, reactor dynamics are compared in terms ofspatial non-uniformities of reaction pattern (NRr(cA,q vs z) andreactant concentration (cA vs z) within the reactor as well as overallreactant depletion rate (cAvs s). In each figure, spatial non-uniformities within the reactor are shown at two intermediatetime levels given by sf/3 (darker shade) and 2sf/3 (lighter shade).Power absorption profiles along with magnitude of qMW for type 1and type 2 reactor configurations of Figs. 6e10 are also shown attop of each figure for ease of illustrations.

Alteration of reaction dynamics under microwave heating ispurely driven by individual power absorption patterns of type 1and type 2 configurations in thin, intermediate and thick regimes.Conventional or surface heating always results in localized reactionfrom the hot left surface, where intensity of reaction is purelydetermined by the surface heat flux or qMW. On the other hand,reaction occurs volumetrically under microwave heating but withvarying spatial profiles depending on the regime of operation andreactor configuration. Reaction occurs uniformly under thin regimeof microwave heating (Nw � 0.1) as reflected in almost uniformspatial profiles of NRr(cA,q) at Nw ¼ 0.06 for both type 1 and type 2configurations. In contrast, reaction occurs only near the exposedleft surface (similar to conventional heating) in thick regime ofmicrowave heating as seen in the spatial profiles of NRr(cA,q) forNp ¼ 3 in Fig. 10. In between these two regimes, reaction patternsunder microwave heating vary with reactor dimension as well as

Fig. 7. Spatial dynamics (NRr(cA,q) vs z and cA vs z shown at sf/3 and 2sf/3 in darker and lighter shades, respectively) and overall progress (cAvs s) of type 1 (solid lines) and type 2(dotted lines) reactors under microwave heating (MW) along with those under their equivalent convectional heating (Conv.) corresponding to Nw ¼ 0.25, fw ¼ 0.1 and fp ¼ 0.1 withsD ¼ 1.0, NR ¼ 0.1, b ¼ 0.1, gR ¼ 10 and FR ¼ 0.1 and 10 in subplots (a) and (b), respectively.

Fig. 6. Spatial dynamics (NRr(cA,q) vs z and cA vs z shown at sf/3 and 2sf/3 in darker and lighter shades, respectively) and overall progress (cAvs s) of type 1 (solid lines) and type 2(dotted lines) reactors under microwave heating (MW) along with those under their equivalent convectional heating (Conv.) in thin regime corresponding to Nw ¼ 0.06, fw ¼ 0.1 andfp ¼ 0.1 with sD ¼ 1.0, NR ¼ 0.1, b ¼ 0.1, gR ¼ 10 and FR ¼ 0.1 and 10 in subplots (a) and (b), respectively.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 285

Fig. 8. Spatial dynamics [NRr(cA,q) vs z and cA vs z shown at sf/3 and 2sf/3 in darker and lighter shades, respectively] and overall progress [cAvs s] of type 1 (solid lines) and type 2(dotted lines) reactors under microwave heating (MW) along with those under their equivalent convectional heating (Conv.) in intermediate regime corresponding to Nw ¼ 0.5,fw ¼ 0.1 and fp ¼ 0.1 with sD ¼ 1.0, NR ¼ 0.1, b ¼ 0.1, gR ¼ 10 and FR ¼ 0.1 and 10 in subplots (a) and (b), respectively.

Fig. 9. Spatial dynamics (NRr(cA,q) vs z and cA vs z shown at sf/3 and 2sf/3 in darker and lighter shades, respectively) and overall progress (cAvs s) of type 1 (solid lines) and type 2(dotted lines) reactors under microwave heating (MW) along with those under their equivalent convectional heating (Conv.) in intermediate regime corresponding to Nw ¼ 0.75,fw ¼ 0.1 and fp ¼ 0.1 with sD ¼ 1.0, NR ¼ 0.1, b ¼ 0.1, gR ¼ 10 and FR ¼ 0.1 and 10 in subplots (a) and (b), respectively.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294286

Fig. 10. Spatial dynamics (NRr(cA,q) vs z and cA vs z shown at sf/3 and 2sf/3 in darker and lighter shades, respectively) and overall progress (cAvs s) of type 1 (solid lines) and type 2(dotted lines) reactors under microwave heating (MW) along with those under their equivalent convectional heating (Conv.) in thick regime corresponding to Np ¼ 3, fw ¼ 0.1 andfp ¼ 0.1 with sD ¼ 1.0, NR ¼ 0.1, b ¼ 0.1, gR ¼ 10 and FR ¼ 0.1 and 10 in subplots (a) and (b), respectively.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 287

configuration but remain bound within uniform reaction of thinregime and surface reaction of thick regime. Reactions mostly occurvolumetrically under intermediate regime of microwave heatingexcept near the boundaries of thin and thick regimes. It is inter-esting to note that similar to thick regime, reaction under micro-wave heating also becomes highly one sided within a small regionnear the boundary of thin regime aroundNwz 0.25, where reactionis driven from the reactor boundary as seen in Fig. 7 for Nw ¼ 0.25.

Special characteristics of reaction under intermediate regime ofmicrowave heating is presence of multiple reaction fronts at thelocations of spatial resonances especially for larger reactors(Nw � 0.5). Reactor with dimensions Nw < 0.5 leads to a singlereaction front irrespective of reactor configuration as seen in Fig. 7for Nw ¼ 0.25. However, location of reaction front changes from leftto right surface as reactor configuration changes from type 1 to type2 at Nw ¼ 0.25 (see Fig. 7). Location of reaction front under mi-crowave heating also changes from the center to both the surfacesas reactor configuration is changed from type 1 to type 2 atNw ¼ 0.5. This interestingly introduces two reaction fronts in type 2configuration at Nw ¼ 0.5 in contrast to the single reaction front intype 1 configuration (see Fig. 8). Reactors with Nw � 0.75 lead tomultiple reaction fronts irrespective of reactor configurations, butreaction fronts of type 1 configuration always appear in the loca-tions of minimum reaction zones of type 2 configuration as illus-trated in Fig. 9 for Nw ¼ 0.75.

Common feature of all the reaction patterns is homogeneity ofreaction under microwave heating compared to their conventionalcounterparts except for very thick reactors with Np / N. Corre-spondingly, both type 1 and type 2 reactors exhibit homogenizedconcentration profiles under microwave heating in presence ofmass transfer limitations at FR ¼ 10 as illustrated in subplot (b) of

Figs. 6e10. Mass transfer limitations exist when diffusion of reac-tant is slower than its consumption rate or equivalently FR a 1. Insuch cases, reactant builds up in no/low reaction zones of thereactor according to the non-uniformity of reaction pattern. Volu-metric power absorption significantly enhances the uniformity ofreaction from conventional to microwave heating in thin and in-termediate regimes (except around Nw z 0.25). As a result, reactorswith Np < 3 and Nw « 0.25 show prominent homogenization ofconcentration gradients under microwave heating compared toconventional heating as shown in Figs. 6(b), 8(b) and 9(b) forNw ¼ 0.06, 0.5 and 0.75, respectively. On the other hand, highly onesided reaction from the surface significantly reduces homogeniza-tion ability of microwave heating around Nw z 0.25 as well as inthick regime as illustrated in Figs. (7b) and (10b) for Nw ¼ 0.25 andNp ¼ 3, respectively.

Homogenization ability of microwave heating also depends onthe degree of localization in its conventional counterpart and thuson the intensity of qMW. Reactor dynamics remain homogenized(uniform) even under conventional heating if intensity of qMW re-mains sufficiently low such that resulting spatial gradients can beeasily homogenized by diffusion. For such low intensity of qMW,reactor dynamics become invariant of heating pattern as is the casefor type 1 configuration at Nw ¼ 0.06 ðqMW ¼ 0:0016Þ, where mi-crowave and conventional heating lead to almost same reactordynamics at FR ¼ 0.1 as well as at FR ¼ 10 (see the profiles in solidlines). Influence of heating pattern increases with increasing qMWand becomes prominent at sufficiently high intensity of qMW,where spatial gradients build up in presence of mass transfer lim-itations (FR a 1) according to the non-uniformity of individualheating pattern. Correspondingly, entire intermediate regime ofmicrowave heating with significantly high qMW as well as with

(a)

(b)

Fig. 11. Variations of hMW with Nw over the range of thin to thick regimes(0.05 � Nw � 10) for (a) type 1 and (b) type 2 configurations corresponding to fw ¼ 0.1,fp ¼ 0.1, sD ¼ 1.0, NR ¼ 0.1, b ¼ 0.1, gR ¼ 10 and varying Thiele modulus as FR ¼ 0.1, 1 and10. Corresponding qMW vs Nw diagrams are also shown in dotted lines (correspondingscale is shown at right axis).

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294288

volumetric power absorption (except for the region aroundNw z 0.25) leads to much homogenized reactor dynamicscompared to conventional heating at FR a 1 irrespective of reactorconfiguration as seen in Figs. (8b) and (9b) for Nw ¼ 0.5 and 0.75.Microwave heating also exhibits homogenization in thin regime fortype 2 configuration, where qMW is not insignificantly small (seeprofiles of Fig. 6(b) in dashed lines). Highly one sided power ab-sorption near the region around Nw ¼ 0.25 obviously decreases thehomogenization ability of microwave heating in spite of extremelyhigh qMW for type 1 configuration and microwave heating alsoleads to significant concentration gradients within the reactor asseen in Fig. 7(b) for Nw ¼ 0.25. Similarly, microwave and conven-tional heating lead to almost similar spatial gradients at Np ¼ 3 asshown in Fig. 10.

Superior homogenization ability of microwave heating overconventional heating becomes irrelevant in the limit of FR � 1,where faster diffusion than consumption of reactant homogenizesthe concentration gradients irrespective of heating patterns.However, magnitude of FR required to reach this limit depends onthe non-uniformity of heating/reaction pattern and hence on qMWfor conventional heating. Faster diffusion of reactant at FR ¼ 0.1 isobserved to completely homogenize the concentration gradientsfor all the cases of microwave heating in volumetric power ab-sorption regimes (i.e. for Nw¼ 0.06, 0.5 and 0.75 (Figs. 6(a), 8(a) and9(a), respectively). Concentration gradients within conventionalreactors are also completely homogenized at FR ¼ 0.1 for low tomoderately high qMW corresponding to (i) both type 1 and type 2configurations at Nw ¼ 0.06 [Fig. 6(a)], (ii) type 2 configuration atNw ¼ 0.25 and 0.75 (Figs. 7(a) and 9(a), respectively) and (iii) type 1configuration at Nw ¼ 0.5 (Fig. 8(a)). However, concentration gra-dients within conventional reactors are not completely homoge-nized at FR ¼ 0.1 for cases of significantly high qMW correspondingto (i) type 1 configuration at Nw ¼ 0.25 and 0.75 and (ii) type 2configuration at Nw ¼ 0.5. In such cases, conventional reactorsexhibit slight concentration gradients at FR ¼ 0.1, which disappearwith further decrease of Thielemodulus. For the same reason, slightconcentration gradients are also observed at FR ¼ 0.1 for bothmicrowave and conventional heating at Np ¼ 3 (see Fig. 10(a)).

7. Savings of energy under microwave heating: influence ofmetallic coating

Since same heating rate is maintained, variations of energyconsumption in microwave heating result from the differences intotal processing time compared to conventional heating. For anendothermic reaction, changes in processing time are mainlycaused by the diffusion time required by the reactant to reach theheating zone/zones from the colder regions of the reactor prior tothe occurrence of reaction. It is obvious that volumetric supply ofheat reduces diffusion time in microwave heating compared to thehighly one sided conventional heating, where reactants from coldright end of the reactor have to diffuse the entire reactor width toavail the heat and react. As a result, microwave heating reduces theprocessing time as seen in all the cAvs s diagrams of Figs. 6e10especially for FR ¼ 10. Associated reduction of energy consump-tion can be quantified as

hMW ¼ 1� sfMW

sfConv

!� 100; (13)

where hMW determines percentage savings of energy in microwaveheating compared to conventional heating. Extent of energy sav-ings obviously depends on the homogenization ability of micro-wave heating and thus changes with reactor dimension (Nw) as wellas reactor configuration as reflected in the cAvs s diagrams of

Figs. 6e10. These changes are quantified in Fig. 11(a)e(b), whichshows the variations of hMW with Nw (solid lines) over the entirerange of thin to thick regimes at various Thiele modulus of FR ¼ 0.1,1 and 10 for type 1 [subplot (a)] and type 2 [subplot (b)] configu-rations. Corresponding qMW vs Nw diagrams are also shown bydotted lines in Fig. 11(a) and (b). As expected, savings of energyunder microwave heating increases with increasing mass transferlimitations (Thiele modulus) and becomes prominent for FR � 1,where microwave heating can potentially reduce energy con-sumption as reflected in the magnitude of hMW. Fig. 11 shows therole of reactor dimension (Nw) to maximize the utilization of mi-crowave energy for a given reactor configuration. Fig. 11 also showsthe role of metallic coating to further enhance the energy savings atspecific reactor dimension. It may be noted that savings of energycan be significantly enhanced at Nw ¼ 0.5n�0.25 by selecting re-actors with metallic coating as in type 1 configuration. On the otherhand, metallic coating should be avoided at Nw ¼ 0.5n as well as inthin regime (Nw � 0.1). Obviously, savings of energy remain unin-fluenced by metallic coating in thick regime.

It follows from the discussion on Figs. 6e10 that two factorsgiven by (i) variations of the homogeneity of power absorptionpattern and (ii) changes in the magnitude of qMW influence thevariations of hMW as Nw varies from thin to thick regimes. It may benoted from Fig. 5 that changes in the homogeneity of power ab-sorption with Nw are much lower compared to the changes of qMWover the entire volumetric power absorption regime of Np < 3.Correspondingly, hMW mostly follows the variations of qMW forNp < 3 in both Fig. 11(a) and (b) except for a very small region nearNw ¼ 0.5 in case of type 1 configuration as shown by shading inFig. 11(a). It may be noted that as Nw increases within the shadedregion of Fig. 11(a), hMW increases while qMW undergoes a reduc-tion. This anomaly interestingly leads to an additional peak in hMWat Nw ¼ 0.52 for type 1 configuration in presence of mass transferlimitations as may be observed from Fig. 11(a) for FR ¼ 1 and 10.

Anomalous variations of qMW within the shaded region ofFig.11(a) are due to the changes in homogeneity of absorbed power,

Fig. 12. Influence of fp on variation patterns of qMW and hMW with Nw for type 1 (solidlines) and type 2 (dotted lines) configurations at fw ¼ 0.1 and FR ¼ 10.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 289

which undergoes a significant transformation from Nw ¼ 0.1 to 0.5irrespective of the reactor configuration as shown in the insets ofFig. 5. It may be noted that homogeneity of power absorption re-duces drastically from Nw ¼ 0.1 to Nw ¼ 0.25 and then reverts backas Nw increases to 0.5. Increasing homogeneity of absorbed powercauses hMW to increase from Nw z 0.4 to Nw z 0.5 (more specif-ically from Nw ¼ 0.38 to 0.52 at FR ¼ 10 and from Nw ¼ 0.45 to 0.52at FR ¼ 1) in spite of decreasing qMW for type 1 configuration. Itmay be noted that reduction of qMW are not significant within theshaded region. On the other hand, changes of qMW are exceptionallyhigh within the range of 0:15(Nw(0:4 for type 1 configurationand hence hMW follows the same regardless of the changes of ho-mogeneity of power absorption. This causes hMW to increase fromNw ¼ 0.1 to 0.25 for type 1 configuration in spite of decreasinghomogeneity of absorbed power. Similarly, hMW decreases fromNw ¼ 0.25 to Nw z 0.4 in Fig. 11(a) although homogeneity of powerabsorption increases over that range. However, changes in thehomogeneity of power absorption suppress the variations of hMWatNw ¼ 0.25 compared to the corresponding resonating peak of qMWas may be observed from Fig. 11(a). Also, since mass transfer

Fig. 13. Variations of power absorption profiles of type 1 and type 2 configurations from fp ¼0.5 and 0.75.

limitations bring out the effect of varying homogeneity of absorbedpower distribution, the region of anomalous variations of hMW alsoshrinks with decreasing FR and disappear at FR ¼ 0.1. In contrast totype 1 configuration, influence of the variations of qMW corroboratewith influence of changing homogeneity of power absorption overthe entire range of 0.1 � Nw � 0.5 for type 2 configuration and hMWshows one-to-one correspondence with qMW in Fig. 11(b) for theentire range of Np < 3 irrespective of Thiele modulus.

8. Effect of fp on influence of metallic coating

Influence of fp on qMW over the entire range of thin to thick re-gimes (0.05 � Nw � 10) and associated changes in hMW at FR ¼ 10for type 1 and type 2 configurations are shown in Fig. 12 byincreasing fp from 0.1 to its upper bound of 1 [Note that fp is boundwithin 0e1 according to Eq. (A.3)]. In Fig. 12, solid and dotted linesrepresent type 1 and type 2 configurations, respectively. Here, masstransfer limited situations of FR ¼ 10 is selected to prominentlybring out the effect of fp on hMW. All the simulations in Fig. 12 areperformed at fw ¼ 0.1 with all other parameters to be same as inprevious figures. Fig. 12 shows that resonances of power absorptionare strongly affected by fp, where resonating peaks of qMW gradu-ally disappear as fp increases beyond 0.5 (or 1.5/p more precisely[62]). It may be noted that most of the resonating peaks of qMW,except the first few, disappear even at fp ¼ 0.3 for both the con-figurations. With suppression of resonances, intermediate regimealso shrinks and causes its volumetric power absorption to trans-form to the highly one sided distribution of thick regime as fp in-creases from 0.1 to >1.5/p as illustrated in Fig. 13. Fig. 13 shows thetransformation of power absorption profiles in type 1 and type 2configurations as fp increases from 0.1 (solid line) to 0.5 (dashedline) and 1 (dotted line) at Nw ¼ 0.06, 0.25, 0.5 and 0.75 for fw ¼ 0.1.As power absorption transforms towards the highly one sideddistribution of thick regime, influence of metallic coating alsogradually decreases with increasing fp as shown in Fig. 13. It may benoted that power absorption profiles of Nw ¼ 0.5 and 0.75 becomeinvariant of the reactor configuration beyond fp ¼ 0.5. Corre-spondingly, region of influence of reactor configuration on qMWshrinks considerably from Nw � 5 at fp ¼ 0.1 to Nw � 1.5, 0.9 and 0.5at fp ¼ 0.3, 0.5 and 1, respectively (see Fig. 12). As a result, hMW alsoshows a decreasing region of influence of reactor configurationfrom Nw � 4 to Nw � 1.5, 0.6 and 0.5 as fp increases from fp ¼ 0.1 tofp ¼ 0.3, 0.5 and 1 respectively.

Transformation to highly one sided power absorption profilesreduces the homogeneity of microwave heating as fp increases from0.1 to 1 for Nw � 0.5, Correspondingly, Fig. 12 shows a significantreduction of hMW from fp ¼ 0.1 to 1 for both type 1 and type 2configurations over the entire range of Nw � 0.5. Over this range of

0.1 (solid lines) to 0.5 (dashed lines) and 1 (dotted lines) at fw ¼ 0.1 and Nw ¼ 0.06, 0.25,

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294290

Nw � 0.5, influence of reducing homogeneity of microwave heatingon hMW either dominates or follows the influence of varying qMWfrom fp ¼ 0.1 to 1. In contrast, homogeneity of microwave heatingdoes not alter significantly with fp in thin regime (as shown inFig. 13 for Nw ¼ 0.06) and hMW follows the variations of qMW forNw � 0.1 as seen in Fig. 12. In between 0.1 � Nw � 0.5, changes inboth homogeneity of power absorption and intensity of qMWcombinedly influence hMW as reflected in its complex variationpatterns from fp ¼ 0.1 to 1 within that range (see Fig. 12).

Fig. 15. Influence of fw on variation patterns of qMW and hMW with Nw for type 1 (solidlines) and type 2 (dotted lines) configurations at fp ¼ 0.1 and FR ¼ 10.

9. Effect of fw on influence of metallic coating

Parameter fw ¼ leff/l0 alters relative magnitude of microwavewavelength within the sample compared to that in free space,which in turn alters scattering of microwaves from reactor walls viaEqs. (5d) and (5e). Associated changes in power absorption patternsfor type 1 and type 2 configurations are shown in Fig. 14 forrepresentative reactor dimensions of Nw ¼ 0.06, 0.25, 0.5 and 0.75,while the corresponding changes in qMW are shown in Fig. 15 overthe entire range of 0.05 � Nw � 10. In both the figures (Figs. 14 and15), fw is varied from 0.1 (solid line) to 0.5 (dashed line) and 1(dotted line) while fp is kept constant at 0.1. It may be noted thatsimilar to fp, fw is also bound between 0 and 1 (0 � fw � 1) aswavelength of microwave is maximum in vacuum or free space.Fig. 15 also illustrates the corresponding changes of hMW at FR ¼ 10(similar to Fig. 12), where other parameters used in the simulationsare same as in previous figures (i.e. sD ¼ 1, NR ¼ 0.1, b ¼ 0.1 andgR ¼ 10).

It may be noted from Fig. 14 that unlike fp, influence of fw onpower absorption pattern does not follow a specific trend and de-pends strongly on reactor dimension as well as reactor configura-tion. Power absorption pattern remains almost unchanged fromfw ¼ 0.1 to 1 in case of type 1 configuration, while fw significantlyalters power absorption patterns for type 2 configuration for all thecases except for Nw ¼ 0.06 corresponding to thin regime (seeFig. 14). On the other hand, both the configurations interestinglyshow similar variations of qMW with suppression of resonatingpeaks from fw ¼ 0.1 to 1 as well as higher magnitudes of qMW atfw ¼ 0.5 and 1 compared to those at fw ¼ 0.1 over the entire inter-mediate regime except near the boundary of thin regime(0.1 < Nw < 0.5) (see Fig. 15). Accordingly, hMW also shows highermagnitudes at fw ¼ 0.5 and 1 compared to those at fw ¼ 0.1 overmost of the intermediate regime along with suppression of oscil-lations as fw increases from 0.1 to 1 in Fig. 15 for both type 1 andtype 2 configurations. These transformations of hMW interestinglyalter the range of reactor dimension for selection of either type 1 ortype 2 configuration in order to enhance hMW. At fw ¼ 0.5 and 1,type 1 configuration can effectively enhance qMW or hMWover most

Fig. 14. Variations of power absorption profiles of type 1 and type 2 configurations from fw ¼0.5 and 0.75.

of the intermediate regime. This is in contrast to the case of fw ¼ 0.1,where type 1 configuration preferentially enhances qMW and hencehMWonly around the resonating peaks at Nw¼ 0.5e0.25n as may beobserved from Fig. 15. The region of preferential enhancement ofhMW by type 1 configuration expands to Nw � 0.6 and Nw � 0.45 atfw ¼ 0.5 and 1, respectively. On the other hand, type 1 configurationno longer remains effective to enhance hMW near Nw ¼ 0.25 atfw ¼ 0.5 and 1 as may be observed from Fig. 15. Instead, type 2configuration becomes efficient choice to enhance hMW till Nw� 0.6and Nw � 0.45 at fw ¼ 0.5 and 1 respectively in spite of lower qMWthan type 1 configuration around Nw ¼ 0.25. This is due to thetransformation of power absorption patternwith fw as illustrated inFig. 14, which shows that highly one sided power absorption ofNw ¼ 0.25 at fw ¼ 0.1 transforms to volumetric power absorption atfw ¼ 0.5 and 1 for type 2 configuration. On the other hand, powerabsorption for type 1 configuration remains to be one sided for allthe fw’s. Correspondingly, influence of homogenized heating undertype 2 configuration at fw¼ 0.5 and 1 dominate and hMW remains tobe higher in type 2 configuration around Nw ¼ 0.25 for fw ¼ 0.5 and1 as observed in Fig. 15.

0.1 (solid lines) to 0.5 (dashed lines) and 1 (dotted lines) at fp ¼ 0.1 and Nw ¼ 0.06, 0.25,

Table 1Literature data regarding relative savings of energy under microwave heatingcompared to conventional heating at similar operating conditions (reactor size, re-action temperature etc.

Reference Reaction Energysavings

Comment

[57] Heterogenous Suzuki 98% Preliminary calculationsFriedeleCrafts acetylation 90%

[55] Biodiesel production 4%e73% (Depending onoperating condition)

[48] SNAr reaction at 140 C 66% MARS MW unitAlkylation reaction at 150 C 59%Heck reaction at 140 C 54%

[44] DielseAlder reaction 11% & 71% MARS unit at 140 C for120 min & at 180 C for8 min, respectively

[43] FFA removal by esterification 37%[42] Hantzsch dihydropyridine

synthesis65% Multimode MW unit

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 291

10. Summary and conclusive remarks

This work presents a theoretical basis for probable energy sav-ings under microwave heating for conducting an endothermic gasphase reaction and shows the role of metallic coating in enhancingthe savings of energy in places where savings are not significantotherwise. The reaction is assumed to occur in presence of sus-ceptors within a semiinfinite batch reactor and total power con-sumption under microwave heating are compared with thoseunder equivalent conventional heating. The analysis presented inthis work are based on single mode microwave radiations, whichcan be easily achieved within any custom made waveguide. Theentire assembly of reactor with packed susceptors can be placed inthe test section of the waveguide, which can be further customizedby applying a metal coated film on the opposite side of wavepropagation to investigate the reactor dynamics with and withoutthemetal coated film. It may be noted that single modemicrowavesnot only reduce complexity of the mathematical model withoutdestroying the essential physics but also cause much higher powerabsorption than multi mode microwaves. The analysis has beenperformed by simulating reactor progress from detailed mass andenergy balance equations in conjunction with Helmholtz equationof wave propagation by using weak formulation of Galerkin finiteelement method and Crank-Nicholson time integration scheme. Inall the simulations, same heating rates are maintained in micro-wave and conventional heatings to make them comparable andequivalent to each other.

This work considers probable uniform, oscillatory and exponen-tially attenuated microwave heating patterns of thin (Nw � 0.1),intermediate [0.1 < Nw < 3/(2pfp)] and thick (2pNwfp or Np � 3)regimes and simulates them by varying reactor dimension (2L/leff)over the range 0.05 � Nw � 10 for both the cases of with andwithout metallic coating. The analysis shows that microwaveheating homogenizes reactor dynamics and can save as much as60% energy consumption based on regime of operation (reactordimension, dielectric properties of the susceptors and transportlimitations within the reactor). This analysis shows efficient use ofmetallic coating in enhancing savings of energy consumption undermicrowave heating for reactor dimensions around Nw ¼ 0.5n�0.25,n ¼ 1,2., where otherwise microwave heating leads to very slowreaction and less energy savings. For example, savings of energydoubles from 20% to 40% in presence of metallic coating atNw ¼ 0.25 for significant mass transfer limitations within thereactor corresponding to FR ¼ 10. Similarly, metallic coating en-hances saving of energy from 40% to 60% and 50% at Nw ¼ 0.75 and1.25, respectively. On the other hand, metallic coatings are shownto be avoided at resonating reactor dimensions without metalliccoating corresponding to Nw¼ 0.5n, n¼ 1,2. as well as for reactorsin thin regime.

This work quantifies probable savings of energy under micro-wave heating (hMW) as reactor dimension varies from thin to thickregimes and identifies the ranges of reactorswheremetallic coatingcan efficiently reduce energy consumption. These results can beused as basis for appropriate selection of reactor configuration(with or without metallic coating) for a given reactor dimension aswell as to select reactor dimension in order to maximize utilizationof microwave energy. This work also quantifies changes in energysavings due to changing dielectric properties of the reacting me-dium and shows how tuning of susceptor material in conjunctionwith appropriate selection of reactor configuration (with orwithout metallic coating) can play an important role in enhancingenergy utilization via microwave heating.

Reduction of energy consumption is one of the important factorsin determining sustainability of microwave heating compared toconventional heating to conduct chemical reaction. In that context,

it becomes extremely essential to analyze relative energy con-sumption under microwave heating compared to that required forconventional heating to achieve desired yield. However, majority ofthe previous work only concentrated on the chemistry aspect of theprocess and reported the extent of enhancement of reaction for avariety of reacting systems under different reaction conditions. Tilldate, only a very few studies exist, which report comparativemeasurement of energy consumption under microwave heatingand equivalent conventional heating [42e44,48,55,57]. The obser-vations of theseworks are summarized in Table 1. It is interesting tonote that current simulations also predict similar range of energysavings as measured/estimated in earlier work (especially thosereported in the last five entries of Table 1 [42e44,48,55]) eventhough the reacting systems differ completely from the presentsetup. It is important to note that Table 1 summarizes only thoseresults corresponding to similar operating parameters for micro-wave and conventional heating (e.g. reaction volume, reactiontemp etc.). As noted in the previous works [42e44,48,55,57], sav-ings of energy can vary widely depending on operating parametersand even with the specific microwave unit used for the reactions.Also, reaction volume, reacting medium, mode of operation(continuous or batch) can significantly affect the energy savingsamong many other factors. Thus, it has been recommended byprevious researchers to do a case-to-case basis energy consumptionanalysis to study the viability of the process. In that context, thepresent simulations may act as basic guidelines to design appro-priate reactor setup with or without metallic coating in order tomaximize energy utilization.

Acknowledgment

Authors would like to thank anonymous reviewers for criticalcomments and suggestions which improved the quality of themanuscript.

Appendix A. Electromagnetic equations for evaluation ofQMW

Propagation of microwave within a semiinfinite domain withdielectric packings shown in Fig. 1 can be characterized by thefollowing Helmholtz equation [65] for both type 1 and type 2configurations:

d2EmdZ2

þ 2pleff

þ i1

Dpeff

!2

Em ¼ 0 (A.1)

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294292

and associated power absorption can be expressed as

QMWðZÞ ¼ cε0l0leffDpeff

EmðZÞEmðZÞ*: (A.2)

In the above equations, Em is microwave induced electric field, c isvelocity of light, ε0 is free space permittivity, l0 ¼ c/f is wavelengthof microwave in free space and leff and Dpeff are effective wave-length and penetration depth of microwave within the packeddomain of Fig. 1, respectively given by

leff ¼ 2cf

h2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k02eff þ k002effq

þ k0effi�1=2

(A.3a)

Dpeff ¼ cpf

"2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik02eff þ k002eff

q� k0eff

r !#�1=2

; (A.3b)

where k0eff and k00eff are effective dielectric constant and effectivedielectric loss of the packed domain, respectively and f is frequencyof incident radiation. Electromagnetic wave undergoes scattering atthe interface of free space and power absorbing medium in absenceof metallic surface/coating, which can be mathematically repre-sented by radiation boundary condition for Em [76]. Radiationboundary condition at the left wall is given

qMW ¼16pNwfpfw

�c2 cosh 2Npð1� zÞ þ c4 sinh 2Npð1� zÞ

þc1 cos 4pNwð1� zÞ þ c3 sin 4pNwð1� zÞ�

�c23 � c21

�cos 4pNw þ �c22 þ c24

�cosh 2Np � 2c1c3 sin 4pNw þ 2c2c4 sinh 2Np

(B.2)

@Z ¼ �L :dEmdZ

þ i2pl0

Em ¼ i4pl0

E0 exp�� i

2pl0

L�; (A.4)

for both type 1 and type 2 configurations, where E0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2I0=cε0

pis

the incident electric field. Radiation boundary condition for theright wall of type 2 reactor with no metallic coating is given by

@Z ¼ L :dEmdZ

� i2pl0

Em ¼ 0 for type 2: (A.5a)

On the other hand, metallic coating completely reflects electro-magnetic wave in type 1 reactor and associated boundary conditioncan be written as

hqMW; wni ¼

4ffiffiffi2

pfw

266664 sinh 2Np

1 þ n2p2

4N2p

�

8>>>><>>>>:2pNwfpcos 4pNw n ¼ 4Nw

fpsin 4pNw

1 � n2p2

16p2N2w

ns4Nw

377775c1 cos 4pNw þ c2 cosh 2Np þ c3 sin 4pNw þ c4 sinh 2Np

; (B.4a)

@Z ¼ L : Em ¼ 0 for type 1: (A.5b)

The description of electromagnetic wave propagation throughpacked dielectrics under consideration is complete with appro-priate correlations k0eff and k00eff . Most widely accepted correlationswere first developed by Fricke [77], which considers the effect ofporosity (f) and dispersity of packings in addition to the dielectricproperties of packing material (k0p and k00p)

k0eff þ ik00eff ¼k0p þ ik00p

h1þ afþ a

k0p þ ik00p

ð1� fÞ

i1� fþ

k0p þ ik00p

ðaþ fÞ

; (A.6)

where a ¼ 2 for spherical and 1 for cylindrical packings.

Appendix B. Analytical expressions for absorbed power andtheir inner products

Solving Helmholtz equation (Eq. (5c)) and associated radiationboundary conditions (Eqs. (5d) and (5e)), analytical expressions forabsorbed power (qMW) can be written as

qMW ¼ 16pNwfpfw�cosh 2Npð1� zÞ � cos 4pNwð1� zÞ

c1 cos 4pNw þ c2 cosh 2Np þ c3 sin 4pNw þ c4 sinh 2Np

(B.1)

for type 1 reactor with metallic coating and

for type 2 reactor configuration without metallic coating, wherecoefficients c1ec4 are given by

c1 ¼ 1þ f 2p � f 2w; (B.3a)

c2 ¼ 1þ f 2p þ f 2w; (B.3b)

c3 ¼ �2fpfw; (B.3c)

c4 ¼ 2fw: (B.3d)

Using the above expressions for absorbed power, the expressionof hqMW;wni for n s 0 can be written as

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294 293

for type 1 reactor with metallic coating and

hqMW;wni ¼

4ffiffiffi2

pfw

2666666666666664

c2 sinh 2Np þ c4�cosh 2Np � ð�1Þn�

1þ n2p2

4N2p

þ

8>>>>><>>>>>:

2pNwfpðc1 cos 4pNw þ c3 sin 4pNwÞ n ¼ 4Nw

fp�c1 sin 4pNw þ c3

�ð�1Þn � cos 4pNw �

1� n2p2

16p2N2w

ns4Nw

3777777777777775�c23 � c21

�cos 4pNw þ �c22 þ c24

�cosh 2Np � 2c1c3sin 4pNw þ 2c2c4 sinh 2Np

(B.4b)

for type 2 reactor configurationwithout metallic coating. Similarly,the expressions for hqMW;w0ihhqMW;1i ¼ qMW can be written as

qMW ¼4fwhsinh 2Np � fpsin 4pNw

ic1 cos 4pNw þ c2 cosh 2Np þ c3 sin 4pNw þ c4 sinh 2Np

(B.5a)

and

qMW ¼4fwhc2 sinh 2Np þ c4

�cosh 2Np � 1

�þ fpðc1 sin 4pNw þ c3ð1� cos 4pNwÞÞi

�c23 � c21

�cos 4pNw þ �c22 þ c24

�cosh 2Np � 2c1c3 sin 4pNw þ 2c2c4 sinh 2Np

(B.5b)

for type 1 and type 2 reactor configurations, respectively.

References

[1] Deora NS, Misra NN, Deswal A, Mishra HN, Cullen PJ, Tiwari BK. Ultrasound forimproved crystallisation in food processing. Food Engineering Reviews2013;5(1):36e44.

[2] Rooze J, Rebrov EV, Schouten JC, Keurentjes JTF. Dissolved gas and ultrasoniccavitation e a review. Ultrasonics Sonochemistry 2013;20(1):1e11.

[3] Eren Z. Ultrasound as a basic and auxiliary process for dye remediation: areview. Journal of Environmental Management 2012;104:127e41.

[4] Veljkovic VB, Avramovic JM, Stamenkovic OS. Biodiesel production byultrasound-assisted transesterification: state of the art and the per-spectives. Renewable and Sustainable Energy Reviews 2012;16(2):1193e209.

[5] Awad TS, Moharram HA, Shaltout OE, Asker D, Youssef MM. Applications ofultrasound in analysis, processing and quality control of food: a review. FoodResearch International 2012;48(2):410e27.

[6] Shirsath SR, Sonawane SH, Gogate PR. Intensification of extraction of naturalproducts using ultrasonic irradiations e A review of current status. ChemicalEngineering and Processing 2012;53:10e23.

[7] Zhang WJ, Yao Y, He BX, Wang RS. The energy-saving characteristic of silicagel regeneration with high-intensity ultrasound. Applied Energy 2011;88(6):2146e56.

[8] Bang JH, Suslick KS. Applications of ultrasound to the synthesis of nano-structured materials. Advanced Materials 2010;22(10):1039e59.

[9] Leonelli C, Mason TJ. Microwave and ultrasonic processing: now a realisticoption for industry. Chemical Engineering and Processing 2010;49(9):885e900.

[10] Yao Y. Using power ultrasound for the regeneration of dehumidizers in desic-cant air-conditioning systems: a review of prospective studies and unexploredissues. Renewable and Sustainable Energy Reviews 2010;14(7):1860e73.

[11] Cella R, Stefani HA. Ultrasound in heterocycles chemistry. Tetrahedron2009;65(13):2619e41.

[12] Yao Y, Zhang W, Li S. Feasibility study on power ultrasound for regenerationof silica gel-A potential desiccant used in air-conditioning system. AppliedEnergy 2009;86(11):2394e400.

[13] Rokhina EV, Lens P, Virkutyte J. Low-frequency ultrasound in biotechnology:state of the art. Trends in Biotechnology 2009;27(5):298e306.

[14] Leveque JM, Cravotto G. Microwaves, power ultrasound, and ionic liquids. Anew synergy in green organic synthesis. Chimia 2006;60(6):313e20.

[15] Lifka J, Ondruschka B, Hofmann J. The use of ultrasound for the degradation oforganic compounds in water: aquasonolysis e a review. Chemie IngenieurTechnik 2002;74(4):403e13.

[16] Ince NH, Tezcanli G, Belen RK, Apikyan IG. Ultrasound as a catalyzer ofaqueous reaction systems: the state of the art and environmental applications.Applied Catalysis B Environmental 2001;29(3):167e76.

[17] Citeau M, Olivier J, Mahmoud A, Vaxelaire J, Larue O, Vorobiev E. Pressurisedelectro-osmotic dewatering of activated and anaerobically digested sludges:electrical variables analysis. Water Research 2012;46(14):4405e16.

[18] Qi RH, Tian CQ, Shao SQ, Tang MS, Lu L. Experimental investigation on per-formance improvement of electro-osmotic regeneration for solid desiccant.Applied Energy 2011;88(8):2816e23.

[19] Chien SC, Ou CY. A novel technique of harmonic waves applied electro-osmotic chemical treatment for soil improvement. Applied Clay Science2011;52(3):235e44.

[20] Ho MY, Chen GH. Enhanced electro-osmotic dewatering of fine particle sus-pension using a rotating anode. Industrial and Engineering ChemistryResearch 2001;40(8):1859e63.

M. Bhattacharya, T. Basak / Energy 55 (2013) 278e294294

[21] Locke BR, Thagard SM. Analysis and review of chemical reactions and trans-port processes in pulsed electrical discharge plasma formed directly in liquidwater. Plasma Chemistry and Plasma Processing 2012;32(5):875e917.

[22] Yamamoto T, Tanioka G, Okubo M, Kuroki T. Water vapor desorption andadsorbent regeneration for air conditioning unit using pulsed corona plasma.Journal of Electrostatics 2007;65(4):221e7.

[23] Nair SA, Yan K, Pemen AJM, Winands GJJ, van Gompel FM, van Leuken HEM,et al. A high-temperature pulsed corona plasma system for fuel gas cleaning.Journal of Electrostatics 2004;61(2):117e27.

[24] Kim HH. Nonthermal plasma processing for air-pollution control: a historicalreview, current issues, and future prospects. Plasma Processes and Polymers2004;1(2):91e110.

[25] Ravasio S, Cavallotti C. Analysis of reactivity and energy efficiency of methaneconversion through non thermal plasmas. Chemical Engineering Science2012;84:580e90.

[26] Malik MA, Malik SA. Pulsed corona discharges and their applications in toxicVOCs abatement. Chinese Journal of Chemical Engineering 1999;7(4):351e62.

[27] Chandrasekaran S, Ramanathan S, Basak T. Microwave material processing-Areview. AIChE Journal 2012;58(2):330e63.

[28] Zhang X, Liu Z. Recent advances in microwave initiated synthesis of nano-carbon materials. Nanoscale 2012;4(3):707e14.

[29] Damour C, Hamdi M, Josset C, Auvity B, Boillereaux L. Energy analysis andoptimization of a food defrosting system. Energy 2012;37(1):562e70.

[30] Motevali A, Minaei S, Khoshtaghaza MH, Amirnejat H. Comparison of energyconsumption and specific energy requirements of different methods fordrying mushroom slices. Energy 2011;36(11):6433e41.

[31] Motevali A, Minaei S, Khoshtagaza MH. Evaluation of energy consumption indifferent drying methods. Energy Conversion and Management 2011;52(2):1192e9.

[32] Kharissova OV, Kharisov BI, Valdes JJR. Review: the use of microwave irradi-ation in the processing of glasses and their composites. Industrial and Engi-neering Chemistry Research 2010;49(4):1457e66.

[33] Mutyala S, Fairbridge C, Pare JRJ, Belanger JMR, Ng S, Hawkins R. Microwaveapplications to oil sands and petroleum: a review. Fuel Processing Technology2010;91(2):127e35.

[34] Sharma GP, Prasad S. Specific energy consumption in microwave drying ofgarlic cloves. Energy 2006;31(12):1921e6.

[35] Zhang M, Tang J, Mujumdar AS, Wang S. Trends in microwave-related dryingof fruits and vegetables. Trends in Food Science and Technology 2006;17(10):524e34.

[36] Kingman SW. Recent developments in microwave processing of minerals.International Material Review 2006;51(1):1e12.

[37] Jones DA, Lelyveld TP, Mavrofidis SD, Kingman SW, Miles NJ. Microwaveheating applications in environmental engineering e a review. ResourcesConservation and Recycling 2002;34(2):75e90.

[38] Sumnu G. A review on microwave baking of foods. International Journal ofFood Science and Technology 2001;36(2):117e27.

[39] Haque KE. Microwave energy for mineral treatment processes - a brief review.International Journal of Mineral Processing 1999;57(1):1e24.

[40] Chen KS, Lin YC, Hsu KH, Wang HK. Improving biodiesel yields from wastecooking oil by using sodium methoxide and a microwave heating system.Energy 2012;38(1):151e6.

[41] Wang MJ, Huang YF, Chiueh PT, Kuan WH, Lo SL. Microwave-induced torre-faction of rice husk and sugarcane residues. Energy 2012;37(1):177e84.

[42] Devine WG, Leadbeater NE. Probing the energy efficiency of microwaveheating and continuous-flow conventional heating as tools for organicchemistry. Arkivoc 2011;Part 5:127e43.

[43] Kim D, Choi J, Kim GJ, Seol SK, Ha YC, Vijayan M, et al. Microwave-acceleratedenergy-efficient esterification of free fatty acid with a heterogeneous catalyst.Bioresource Technology 2011;102(3):3639e41.

[44] Godwin DR, Lawton SJ, Moseley JD, Welham MJ, Weston NP. Energy efficiencyof conventionally-heated pilot plant reactors compared with microwave re-actors. Energy and Fuels 2010;24:5446e53.

[45] Lam SS, Russell AD, Chase HA. Microwave pyrolysis, a novel process forrecycling waste automotive engine oil. Energy 2010;35(7):2985e91.

[46] Strauss CR, Rooney DW. Accounting for clean, fast and high yielding reactionsunder microwave conditions. Green Chemistry 2010;12(8):1340e4.

[47] Durka T, Van Gerven T, Stankiewicz A. Microwaves in heterogeneous gas-phase catalysis: experimental and numerical approaches. Chemical Engi-neering and Technology 2009;32(9):1301e12.

[48] Moseley JD, Woodman EK. Energy efficiency of microwave- and convention-ally heated reactors compared at meso scale for organic reactions. Energy andFuels 2009;23:5438e47.

[49] Polshettiwar V, Nadagouda MN, Varma RS. Microwave-assisted chemistry: arapid and sustainable route to synthesis of organics and nanomaterials.Australian Journal of Chemistry 2009;62(1):16e26.

[50] Cherbanski R, Molga E. Intensification of desorption processes by use of mi-crowaves e an overview of possible applications and industrial perspectives.Chemical Engineering and Processing 2009;48(1):48e58.

[51] Kappe CO. Microwave dielectric heating in synthetic organic chemistry.Chemical Society Reviews 2008;37(6):1127e39.

[52] Bowman MD, Holcomb JL, Kormos CM, Leadbeater NE, Williams VA. Ap-proaches for scale-up of microwave-promoted reactions. Organic ProcessResearch and Development 2008;12(1):41e57.

[53] Wei ZS, Du ZY, Lin ZH, He HM, Qiu RL. Removal of NO(x) by microwave reactorwith ammonium bicarbonate and Ga-A zeolites at low temperature. Energy2007;32(8):1455e9.

[54] Cecilia R, Kunz U, Turek T. Possibilities of process intensification using mi-crowaves applied to catalytic microreactors. Chemical Engineering and Pro-cessing 2007;46(9):870e81.

[55] Barnard TM, Leadbeater NE, Boucher MB, Stencel LM, Wilhite BA. Continuous-flow preparation of biodiesel using microwave heating. Energy and Fuels2007;21(3):1777e81.

[56] Tokuyama H, Nakamura M. Acceleration of reaction by microwave irradiation.Journal of Synthetic Organ Chemistry Japan 2005;63(5):523e38.

[57] Gronnow MJ, White RJ, Clark JH, Macquarrie DJ. Energy efficiency in chemicalreactions: a comparative study of different reaction techniques. OrganicProcess Research and Development 2005;9(4):516e8.

[58] Nuchter M, Ondruschka B, Bonrath W, Bonrath W, Gum A. Microwave assistedsynthesise a critical technology overview. Green Chemistry 2004;6(3):128e41.

[59] Nuchter M, Muller U, Ondruschka B, Tied A, Lautenschlager W. Microwave-assisted chemical reactions. Chemical Engineering and Technology2003;26(12):1207e16.

[60] Akyel C, Bilgen E. Microwave and radiofrequency curing of polymers e energy-requirements, cost and market penetration. Energy 1989;14(12):839e51.

[61] Bhattacharya M, Basak T. Detailed material-invariant analysis on spatial res-onances of power absorption for microwave-assisted material processingwith distributed sources. Industrial and Engineering Chemistry Research2007;46:750e60.

[62] Bhattacharya M, Basak T. New closed form analysis of resonances in micro-wave power for material processing. AIChE Journal 2006;52:3707e21.

[63] Basak T, Kumaran SS. A generalized analysis on material invariant charac-teristics for microwave heating of slabs. Chemical Engineering Science2005;60(20):5480e98.

[64] Ayappa KG, Davis HT, Crapiste G, Davis EA, Gordon J. Microwave heating: anevaluation of power formulations. Chemical Engineering Science 1991;46(4):1005e16.

[65] Balanis CA. Advanced engineering electromagnetics. New York: Wiley; 1989.[66] Stuerga DAC, Gaillard P. Microwave athermal effects in chemistry: a myth’s

autopsy .2. Orienting effects and thermodynamic consequences of electricfield. Journal of Microwave Power and Electromagnetic Energy 1996;31(2):101e13.

[67] Herrero MA, Kremsner JM, Kappe CO. Nonthermal microwave effects revis-ited: on the importance of internal temperature monitoring and agitation inmicrowave chemistry. Journal of Organic Chemistry 2008;73(1):36e47.

[68] Gerasev AP. Simulation of catalytic processes in a fixed bed with the use ofmicrowave radiation for performing an endothermic reaction. Kinetics andCatalysis 2011;52(6):907e13.

[69] Yeo YK, Kim HY, Kim IW, Moon II , Chung Y, Levdansky VV. Analysis of cat-alytic reaction systems under microwaves to save energy. Korean Journal ofChemical Engineering 2003;20(1):1e7.

[70] Perry WL, Datye AK, Prinja AK, Brown LF. Microwave heating of endothermiccatalytic reactions: Reforming of methanol. AIChE Journal 2002;48(4):820e31.

[71] Shin GI, Yeo YK, Jo BY, Kim HY, Levdansky VV. Analysis on effects of micro-wave heating in porous catalysts. Journal of Chemical Engineering of Japan2001;34(12):1567e74.

[72] Pipus G, Plazl I, Koloini T. Esterification of benzoic acid in microwave tubularflow reactor. Chemical Engineering Journal 2000;76(3):239e45.

[73] Plazl I, Pipus G, Koloini T. Microwave heating of the continuous flow catalyticreactor in a nonuniform electric field. AIChE Journal 1997;43(3):754e60.

[74] Bhattacharya M, Basak T, Senagala R. A comprehensive theoretical analysis forthe effect of microwave heating on the progress of a first order endothermicreaction. Chemical Engineering Science 2011;66:5832e51.

[75] Bhattacharya M, Basak T, Ayappa KG. A fixed grid finite element basedenthalpy formulation for generalized phase change problems: role of super-ficial mushy region. International Journal of Heat and Mass Transfer 2002;45:4881e98.

[76] Ayappa KG, Davis HT, Davis EA, Gordon J. Analysis of microwave-heating ofmaterials with temperature-dependent properties. AIChE Journal 1991;37(3):313e22.

[77] Fricke H. The complex conductivity of a suspension of stratified particles ofspherical or cylindrical form. Journal of Physical Chemistry 1955;59:168e70.