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Global monitoring of fluidized-bed processes by means of microwave cavityresonances

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Measurement: Journal of the International Measurement Confederation (ISSN: 0263-2241)

Citation for the published paper:Nohlert, J. ; Cerullo, L. ; Winges, J. (2014) "Global monitoring of fluidized-bed processes bymeans of microwave cavity resonances". Measurement: Journal of the InternationalMeasurement Confederation, vol. 55 pp. 520-535.

http://dx.doi.org/10.1016/j.measurement.2014.05.034

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Measurement 55 (2014) 520–535

Contents lists available at ScienceDirect

Measurement

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

Global monitoring of fluidized-bed processes by meansof microwave cavity resonances

http://dx.doi.org/10.1016/j.measurement.2014.05.0340263-2241/� 2014 The Authors. Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

⇑ Corresponding author. Tel.: +46 317721710.E-mail address: [email protected] (J. Nohlert).

Johan Nohlert a,⇑, Livia Cerullo a, Johan Winges a, Thomas Rylander a, Tomas McKelvey a,Anders Holmgren b, Lubomir Gradinarsky b, Staffan Folestad b, Mats Viberg a,Anders Rasmuson c

a Department of Signals and Systems, Chalmers University of Technology, SE-41296 Göteborg, Swedenb AstraZeneca R&D, Pharmaceutical Development, SE-431 83 Mölndal, Swedenc Department of Chemical and Biological Engineering, Chalmers University of Technology, SE-41296 Göteborg, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 October 2013Accepted 23 May 2014Available online 19 June 2014

Keywords:Pharmaceutical fluidized-bed processMicrowave measurement systemPermittivity estimationElectromagnetic homogenizationFinite element methodSensitivity analysis

We present an electromagnetic measurement system for monitoring of the effective per-mittivity in closed metal vessels, which are commonly used in the process industry. Themeasurement system exploits the process vessel as a microwave cavity resonator andthe relative change in its complex resonance frequencies is related to the complex effectivepermittivity inside the vessel. Also, thermal expansion of the process vessel is taken intoaccount and we compensate for its influence on the resonance frequencies by means ofa priori information derived from a set of temperature measurements. The sensitivities,that relate the process state to the measured resonance frequencies, are computed bymeans of a detailed finite element model. The usefulness of the proposed measurementsystem is successfully demonstrated for a pharmaceutical fluidized-bed process, wherethe water and solid contents inside the process vessel is of interest.� 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY

license (http://creativecommons.org/licenses/by/3.0/).

1. Introduction scopic methods such as near infra-red (NIR) and Raman

In the pharmaceutical industry, monitoring and controlof manufacturing processes are important in order toachieve high-quality products to a low production-cost. Acommon step in the manufacturing process for many phar-maceutical products is coating and drying of particles influidized-bed processes. Measurement techniques thatare capable of monitoring the material distribution andmoisture content in this type of process are importanttools for control, optimization and improved understand-ing of the process.

Currently available methods for monitoring of moisturecontent in pharmaceutical fluidized-beds include spectro-

spectroscopy [1,2]. Although these methods yield impor-tant information on moisture content as well as physicaland chemical properties of the material, they mainly pro-vide local information for the region in the very proximityof the measurement probe, which may be insufficient ormisleading for the estimation of the global process state.Recently, Buschmüller et al. [3] developed a stray fieldmicrowave resonator technique for in-line moisture mea-surements during fluidized-bed drying. However, theirapproach is based on local fields in the vicinity of theresonator probe and, therefore, it does not yield globalinformation on the process state.

At low frequencies, electric capacitance tomography(ECT) can be applied to regions with high particle density[4]. Although ECT yields important information on the par-ticle distribution on a global scale, it suffers from limitedspatial resolution and it is not suited for the estimation

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Fig. 1. Schematic picture of the process vessel.

J. Nohlert et al. / Measurement 55 (2014) 520–535 521

of dielectric losses, which leads to difficulties in distin-guishing between material density and moisture content.

In this article, we present a microwave-based measure-ment technique that exploits electromagnetic eigenmodesthat are supported by the process vessel bounded by itsmetal walls. The process vessel is considered as a cavityresonator and the permittivity distribution inside the ves-sel is estimated based on measurements of the complexresonance frequencies of the cavity. This technique resem-bles the well-established cavity perturbation measure-ment technique, which is regularly used for accuratemeasurements of dielectric and magnetic properties ofmaterials at microwave frequencies [5,6]. In contrast totraditional cavity-perturbation measurements, our tech-nique exploits several different eigenmodes, which yieldsthree-dimensional spatial resolution and measurementinformation at multiple frequencies.

The process vessel is equipped with a set of probes thatare connected to a network analyzer. Given the scatteringparameters measured with the network analyzer, we esti-mate the complex resonance frequencies of the processvessel, where the real part corresponds to the frequencyof oscillation and the imaginary part corresponds to thedamping. We exploit a linearization of the electromagneticresponse of the process vessel and it yields a relationbetween (i) a perturbation of the resonance frequenciesand (ii) a perturbation of the complex permittivity insidethe process vessel. We parameterize the permittivity withspace dependent basis functions and determine the coeffi-cients by means of solving an estimation problem based onan over-determined system of linear equations, where theright-hand side contains the measured deviations in theresonance frequencies as compared to the empty vessel.

2. Measurement system

2.1. Process equipment

Fig. 1 shows a schematic picture of a fluidized-bed pro-cess vessel of Wurster type, which is the type of process weconsider in this article. This process is used for coating anddrying of pharmaceutical particles, such as pellets or gran-ules, and the particular vessel we consider is a custom builtlab-scale device for batches consisting of up to 1 kg of par-ticles. The particles are dielectric objects of size that nor-mally range from 250 lm to 1 mm and in this article, weconsider particles that mainly consist of microcrystallinecellulose (MCC).

The bottom plate of the process vessel, also referred to asthe distributor plate, is perforated with a large number ofsmall holes. An airflow is applied through the distributorplate, which fluidizes the particles and creates a fluidizedbed. Above the distributor plate resides a metal tube, whichis referred to as the Wurster tube. At the center of the dis-tributor plate below the Wurster tube, a pneumatic spraynozzle injects a rapidly moving mixture of air and liquid thataccelerates the particles in its immediate vicinity in anupward motion. The particles move through the Wurstertube and create a fountain-like flow pattern above the Wur-ster tube. As the particles exit the fountain region, they fallback into the fluidized bed outside the Wurster tube.

The liquid injected through the spray nozzle is a solu-tion that is sprayed onto the surface of the particles. Thesolvent evaporates mainly inside and immediately abovethe Wurster tube, which results in a deposited layer ofthe solute on the surface of the particles [7]. A commonsolvent is water and there is a large variety of solutes suchas active pharmaceutical ingredients (API), polymers usedfor film coating and various filler materials. Thus, this typeof process allows for the manufacturing of multi-layeredparticles. Here, the different layers provide designed func-tions such as protection from humidity during storage,delay of API release in the digestive system and the actualdelivery of the API.

The water and its distribution inside the vessel for theduration of the process influence the process state and,consequently, the quality of the final product. For example,API that are sensitive to long-term exposure to moisturemay be coated with a film that protects the API from ambi-ent humidity. In such cases, it is important to establish thatthe particles are sufficiently dry before this film is applied.In addition, a too fast drying rate may have negativeimpact on the coating quality. A more extreme situationis the undesirable event of excessive amounts of waterinside the vessel, which can cause extensive particle

522 J. Nohlert et al. / Measurement 55 (2014) 520–535

agglomeration and eventually process break-down. If sucha condition can be detected, it is feasible to change the pro-cess parameters in order to counter-act agglomeration. Thewater inside the vessel can occur in a number of differentforms: (i) water vapor; (ii) sorbed water inside the parti-cles; (iii) free water on the surface of the particles; (iv)water droplets in the air; (v) condensed water dropletson the vessel walls; and (vi) water in agglomerates ordense gatherings of particles.

2.2. Measurement principle

From an electromagnetic point of view, the process ves-sel that we consider is a rather good approximation of amicrowave cavity resonator. The geometry of the vessel,which is used in the electromagnetic model described laterin the article, is shown in Fig. 2. We exploit two magnetic-field probes to excite eigenmodes supported by the micro-wave resonator and the corresponding eigenfrequenciesare measured with the same set of probes. A network ana-lyzer is connected to the two probes and the 2� 2 scatter-ing matrix is measured as a function of frequency. Themeasured scattering parameters are used to extract thecomplex resonance frequencies of a selection of the lowesteigenmodes supported by the process vessel.

It should be emphasized that although these probes arerather small in comparison to the microwave resonator,i.e., the process vessel, the electromagnetic field for basi-cally all eigenmodes extends throughout most of the pro-cess vessel’s volume. The eigenfrequencies are influenced

Fig. 2. The process vessel including its two rectangular windows (W1 andW2) and the cylindrical tube used for loading of particles (H1). This figurealso illustrates four temperature probes (T1–T4) and two microwaveprobes (M1 and M2) although these are not part of the electromagneticmodel.

by the geometry of the process vessel and, more impor-tantly for this application, the material and its distributioninside the process vessel. Consequently, importantinformation on the distribution of particles and theirmoisture content can be derived from the measuredeigenfrequencies.

2.2.1. Homogenization of a mixture of air and particlesThe mixture of particles and air influences the electro-

magnetic fields inside the process vessel. The typicallength-scales associated with the lowest eigenmodes arecomparable to the size of the cavity and, as a consequence,the particles are extremely small in relation to the spatialvariation of the electromagnetic field. Thus, it is beneficialto represent the mixture of particles and air in terms of aneffective permittivity, where the possible presence of othersmall dielectric objects, such as liquid droplets, contributesto the effective permittivity in the same manner as the par-ticles. Given the material parameters and geometry of theparticles together with their spatial distribution, the effec-tive permittivity can be calculated by means of electro-magnetic mixing formulas [8]. Here, we focus on thewell-known Maxwell–Garnett mixing formula

�eff � �b�eff þ 2�b

¼ 13�b

XNn¼1

#nanVn

; ð1Þ

where �eff is the effective permittivity and �b is the permit-tivity of the background medium for a mixture with N dif-ferent types of inclusions indexed by n. In this article, weassume that �b is real, i.e., we have a lossless backgroundmedium. For the inclusions of type n, we have the polariz-ability an, the inclusion volume Vn and the volume fraction#n. For homogeneous spherical inclusions with permittiv-ity �i, we have the polarizability

a ¼ 3�bV�i � �b�i þ 2�b

; ð2Þ

where V is the volume of the particle. Layered sphericalparticles with a core and an outer shell with different per-mittivities can also be incorporated into Eq. (1) by usingthe associated polarizability for this type of inclusions [8].

A dispersive inclusion material yields a dispersivebehavior of the effective permittivity, although the disper-sion relation of the mixture may be different from that ofthe inclusions. For example, air with inclusions that aredescribed as a Debye medium yields an effective permittiv-ity that also behaves like a Debye medium, but with differ-ent values of the Debye parameters [9]. Furthermore, airand inclusion particles with non-zero conductivity yieldsDebye-characteristics of the effective permittivity, due tothe Maxwell–Wagner effect [10].

2.2.2. Density independent moisture measurementsA feasible way to extract information about moisture

content of the particles from the estimated complex effec-tive permittivity �eff ¼ �0eff � j�00eff in the process vessel, isthe approach of density-independent measurement tech-niques [10]. The main idea of this approach is that if thematerial density in a given volume changes, both �0eff and�00eff will be affected in a similar way, since they both

J. Nohlert et al. / Measuremen

depend on the total amount of polarizable material in thevolume.

In a situation where a certain material is dispersed in abackground medium with permittivity �b, the ratio

g ¼ �00eff

�0eff � �b; ð3Þ

is an example of a density-independent function that hasshown to be more or less independent on the density formany material mixtures [11,12]. This is not surprising, andit can be shown using the Maxwell–Garnett mixing formulaapplied to lossy spherical inclusions dispersed in a losslessbackground medium, that g is independent on the volumefraction # in the low-density limit. Therefore, it is expectedthat measurements of �00eff and �

0eff taken at different material

densities would fall on a straight line that intersects the ori-gin if �00eff is plotted versus �

0eff � �b, given that the particles

have constant permittivity. The slope g of this line usuallydepends on the moisture content of the inclusion materialin a characteristic manner. The slope g may therefore beused as a density-independent metric of the moisture con-tent, if it is known how g depends on the moisture contentfor the material considered. Such relations are usuallyobtained from calibration procedures which involve mea-surements on material samples with known moisture con-tent. It should be emphasized that g may depend on manyother factors as well, such as the physical state of the water(e.g. whether it is free or bound in the material) and the tem-perature. In the case of free water, its distribution in thematerial may also be of importance.

In practical measurement situations, the estimatedeffective permittivity �eff may be biased by, e.g., tempera-ture variations of the metal parts of the cavity, or by changesin the permittivity of the background medium (which isusually air) caused by, e.g., a varying humidity. In such cases,there may be non-zero residual components of �00eff and�0eff � �b at zero material density due to biased estimates of�eff , which makes a direct use of the ratio g unreliable. Wetherefore make an attempt to extend the classical density-independent measurement approach to situations wherean uncontrolled bias of the estimated �eff is present. Ourapproach is based on the covariance between �00eff and�0eff � �b, which is computed for a fairly large number of mea-surement samples. The material (in our case the particles inthe process) is re-distributed between different samplepoints which yields a variation in the material density andhence a variation in �00eff and �

0eff , which is used to calculate

the covariance. As a motivation to this approach, we con-sider a situation where lossy spherical inclusions with per-mittivity �i ¼ �0i � j�00i are dispersed in a backgroundmedium with permittivity �b with volume fraction #. If thedispersion is dilute (#� 1) and if the loss factor of the inclu-sion material is low (�00i =�

0i 6 0:1), the Maxwell–Garnett mix-

ing formula yields the following expressions for the real andimaginary parts of the effective permittivity

�0eff=�b � 1 ’ 3�0i=�b � 1�0i=�b þ 2

# ð4Þ

�00eff=�b ’ 9�00i =�b

ð�0i=�b þ 2Þ2 #: ð5Þ

If we now assume that #; �0i and �00i vary randomly around

their respective mean values, this will obviously yieldvariations in �eff . Hence, by studying the covariancebetween the real and imaginary part of �eff , one mayextract important information about the parameters #; �0iand �00i and their variabilities. In particular, we notice fromEqs. (4) and (5) that both the real and imaginary part of�eff depend linearly on the volume fraction #, whichexplains the density-independent nature of the ratio gand the reason why �00eff and �

0eff � �b obtained from differ-

ent volume fractions tend to fall on a straight line. Devi-ations from this line can be explained by means ofvariations in �0i and �

00i , since they influence the real and

imaginary part of �eff in different ways, as seen fromEqs. (4) and (5).

2.3. Experimental setup

The experimental setup is shown in Fig. 3a and itincludes the process vessel, the process machine and thetwo-port network analyzer used to perform the microwavemeasurements.

t 55 (2014) 520–535 523

2.3.1. Reduction of undesired lossesThe original process vessel has a couple of apertures, such

as windows that are used for visual inspection of the process,and such apertures damp the microwave cavity by means ofradiation losses. We are interested in measuring the dampingassociated with the moisture content of the particles and,thus, we wish to reduce other contributions to the totaldamping of the cavity. Therefore, we shield the process vesselby covering the apertures W1, W2 and H1 in Fig. 2 by alumi-num foil, and use a perforated plate to shield the microwavecavity from the air outtake in the upper part of the processvessel. In addition, the shielding mitigates possible problemswith external electromagnetic sources that may disturb themeasurements. With these arrangements we obtainunloaded Q-values of the cavity which are in the order of103—104 for the lowest eigenmodes.

2.3.2. Choice of probes and their designPharmaceutical mixtures processed in steel containers

often become electrically charged due to the triboelectriceffect [13,14]. Consequently, the statically charged parti-cles yield a quasi-static electric field as they move insidethe vessel. In order to protect the network analyzer frominduced voltages of large magnitude and electric dis-charges that may occur in the process vessel, we exploitmagnetic field probes (also referred to as H-probes) thatare mainly sensitive to the time variations of the magneticflux that passes through the loop of the H-probe.

Fig. 3b shows the process vessel equipped with two H-probes located in the upper part of the cavity for measure-ment of the azimuthal and vertical components of themagnetic field. At these locations, we can efficiently coupleto the magnetic field of the lowest eigenmodes that we usefor sensing. In addition, the size of each loop is chosen suchthat we get sufficiently good coupling to all of these modes.For industrial usage, it may not be feasible to let the probeextend into the process vessel. A possible remedy is to

(a) Experimental setup(a) Experimental setup (b) H-probes(b) H-probes

Fig. 3. Photographs of (a) the process vessel in the experimental setup and (b) the two H-probes.


create a shallow dent in the cavity wall and to cover thisdent by a dielectric ‘‘radome’’.

2.3.3. Measurement settingsThe network analyzer (Agilent E8361A) is controlled

from a PC via GPIB interface, in order to perform repeatedscans of a frequency band that includes the resonancefrequencies of the sensing modes (0.7–1.7 GHz). This fre-quency band takes approximately 2 s to scan (for excitationon one of the two ports), using totally 6400 frequency pointsand the IF bandwidth set to 5 kHz, which means that it takes1–10 ms to scan the frequency band occupied by a singleresonance. The time between two consecutive frequencysweeps is set to 10 s, where a substantial part of this timeis required for data transfer over the GPIB interface. In thepost-processing of the measurement data, we use a statisti-cal approach which benefits from a large number of mea-surement samples. Therefore, a measurement setup thatallows for faster sampling of the process would be beneficialfor this measurement technique.

3. Electromagnetic model

For the analysis of the resonance frequencies and thecorresponding eigenmodes, we use an idealized model ofthe process vessel: (i) all the metal parts of the process ves-sel are assumed to be perfect electric conductors (PEC); (ii)all apertures are completely sealed such that the interior ofthe process vessel is completely enclosed by a perfect elec-tric conductor; and (iii) the probes are excluded from themodel. Further, we assume that the material inside theprocess vessel can be described by relatively simple disper-sion relations for the frequency band of interest and, as astarting point, we consider a frequency-independent per-mittivity that may vary with respect to space. Also, weassume that the magnetization for the media involved isnegligible and use the permeability l ¼ l0.

Thus, we consider the eigenvalue problem

r� E ¼ �jxB in X; ð6Þ

r � 1l0

B ¼ ðjx�þ rÞE in X; ð7Þ

n̂� E ¼ 0 on C: ð8Þ

We seek the eigenmodes ðEm;BmÞ and the correspondingeigenfrequencies xm that satisfy the eigenvalue problem(6)–(8), where the index m labels the resonancessupported by the process vessel.

3.1. Weak formulation

The eigenvalue problem (6)–(8) is expressed in theweak form: Find ðE;B;xÞ 2 H0ðcurl; XÞ � Hðdiv; XÞ � Csuch that

cFLðE;vÞ ¼ �jxm1ðB;vÞ; ð9ÞcALðB;wÞ ¼ jxl0�0m�r ðE;wÞ þ l0mrðE;wÞ; ð10Þ

for all ðv ;wÞ 2 Hðdiv; XÞ � H0ðcurl; XÞ. Here, we expressthe real (absolute) permittivity � ¼ �0�r in terms of the cor-responding relative permittivity. Further, we have thebilinear forms

cFLðu;vÞ ¼Z

Xðr � uÞ � v dX; ð11Þ

cALðu;vÞ ¼Z

Xu � ðr � vÞdX; ð12Þ

m1ðu;vÞ ¼Z

Xu � v dX; ð13Þ

m�r ðu;vÞ ¼Z

X�ru � v dX; ð14Þ

mrðu;vÞ ¼Z

Xru � v dX: ð15Þ

Finally, we have the spaces


Hðcurl; XÞ ¼ fu : u 2 L2ðXÞ andr� u 2 L2ðXÞg; ð16ÞHðdiv; XÞ ¼ fu : u 2 L2ðXÞ andr � u 2 L2ðXÞg; ð17Þ

and H0ðcurl; XÞ is the subspace of Hðcurl; XÞ that satisfiesthe boundary condition (8).

3.2. Finite element approximation

We expand the electric field in terms of curl-conform-ing elements and the magnetic flux density in diver-gence-conforming elements, see Ref. [15] for furtherdetails. This yields the discrete eigenvalue problem

0 �CCT �Z0Mr

� �c0be

� �¼ jx

c0

M1 00 M�r

� �c0be

� �; ð18Þ

where we have scaled the magnetic flux density with thespeed of light and Ampère’s law with the impedance ofvacuum. This form is useful since it yields an eigenvalueproblem that is linear in terms of the eigenvalue jx=c0.Reference [16] contains further details on the treatmentof eigenvalue problems by means of the finite elementmethod.

Elimination of the magnetic flux density from Eq. (18)yields the corresponding eigenvalue problem

CTM�11 Cþ jxl0Mr �x2l0�0M�r� �

e ¼ 0; ð19Þ

formulated in terms of only the electric field. This eigen-value problem is nonlinear since it is quadratic in termsof the eigenfrequency x.

For problems with conducting media, it is computation-ally advantageous to solve Eq. (18). Such situations mayoccur in pharmaceutical processes that feature solutionswith ions. For dry processes, the non-conducting back-ground medium with possibly conducting inclusions, dis-plays a more complicated dispersion for the effectivepermittivity. Given the somewhat limited frequency-bandused for the measurements in this article, we choose toapproximate the effective permittivity by a frequency-independent function �ðrÞ ¼ �0ðrÞ � j�00ðrÞ. This yields theeigenvalue problem

CTM�11 C�x2l0 M�0 � jM�00ð Þ� �

e ¼ 0; ð20Þ

which is not particularly attractive for direct computationbut useful for the sensitivity analysis that follows below.

3.3. Losses due to finite conductivity in the cavity walls

It is possible to approximately determine the losses dueto cavity walls of finite conductivity based on the electro-magnetic field solution computed for a PEC boundary. Thistype of approach yields the quality factor associated with aparticular eigenmode and, consequently, its damping [17].The losses are given by

Pwall ¼ Re12

ZC

ZCjn̂�Bml0j2dC

� �; ð21Þ

where ZC ¼ ð1þ jÞ=ðrdmÞ is the surface impedance for thewall of finite conductivity. Here, dm ¼ ð2=ðxml0rÞÞ

1=2 isthe penetration depth for the eigenfrequency xm and the

magnetic flux density Bm of the corresponding eigenmodeis computed based on the approximation that the wall con-ductivity is infinite. Finally, the quality factor is given byQwall;m ¼ xmWm=Pwall;m where Wm is the time-averageenergy stored in the vessel by the eigenmode ðEm;BmÞ, i.e.,

Wm ¼14

ZX�jEmj2dXþ

14

ZX

1l0jBmj2dX: ð22Þ

4. Estimation procedure

Given the measured scattering parameters, we estimatethe complex resonance frequencies by means of a sub-space-based multivariable system identification algorithm[18]. This measurement procedure is repeated in a regularfashion throughout the entire duration of the process time,and in the following, we refer to the estimated resonancefrequencies as the measured resonance frequencies. First,the vessel is empty and the measured resonance frequen-cies are denoted �xrefm , where the overline denotes a mea-sured quantity. Next, the process vessel is loaded withparticles and the subsequent measured resonance frequen-cies �xm are used to form the deviations in the resonancefrequencies D �xm ¼ �xm � �xrefm . As the process evolves, weexploit the relative deviation D �xm= �xrefm to estimate thestate of the process and, later in this section, we presentformulas that relate a perturbation in resonance frequencyto the corresponding perturbations in (i) the materialparameters and (ii) the shape of the vessel due to thermalexpansion.

The estimation of the material and shape parameters isbased on minimization of the misfit between the measuredand computed relative deviations in resonance frequen-cies. Thus, we wish to minimize the following expression

Xm

Dxmxrefm

� D�xm

�xrefm

��

2

; ð23Þ

where Dxm ¼ xm �xrefm is the deviation in the computedresonance frequencies xm as compared to the correspond-ing (computed) result xrefm for the empty reference case.The computed resonance frequencies xm are functions ofthe material and shape parameters, and this dependencymay be expressed according to

xm ¼ xmjLP þX

n

@xm@pnjLPdpn; ð24Þ

for small deviations around a certain linearization point,which is denoted by the subscript LP. The empty cavitymay be used as linearization point, but if the permittivityinside the vessel deviates substantially from vacuum, it isbeneficial to use a linearization point with higher permit-tivity in certain regions of the vessel. The perturbationsdpn in the parameters with respect to the linearizationpoint, are stored in a vector x and we compute an estimatex̂ of x by means of the regularized approach

x̂ ¼ arg minxjjAx� bjj2 þ cjjLðx� ~xÞjj2: ð25Þ

Here, the elements of the matrix A and the right-hand-sidevector b are given by


Amn ¼1

xrefm@xm@pnjLP ð26Þ

bm ¼D �xm�xrefm

�xmjLP �xrefm

xrefm: ð27Þ

The first term in Eq. (25) resembles a system of linearequations which is over-determined, i.e., the number ofrows in A exceeds the number of columns. Hence, theresidual jjAx� bjj can be used as a metric of how wellthe column space of A is able to represent the data vectorb. The second term in Eq. (25) is a regularization term,which penalizes solutions x̂ away from a given vector ~xusing a weighted Euclidean norm. The matrix L in the reg-ularization term is diagonal with all diagonal elementsequal to unity, except for the elements that multiply theshape parameters, which are substantially larger. The bal-ance between the two terms in Eq. (25), i.e., a solutionwhich makes Ax̂ � b and a solution x̂ � ~x, is controlledby the regularization parameter c. From a Bayesian pointof view we can interpret Eq. (25) as the posterior meanestimate of x under the assumption that the measurementerrors in b are Gaussian distributed, have zero mean andcovariance matrix 12I and the parameter vector x has aGaussian prior distribution with mean value ~x and covari-ance matrix ð12=cÞðLTLÞ�1 [19].

For normal process conditions, the losses in the processvessel are small and, therefore, we compute the sensitivitymatrix A for linearization points that corresponds to thevessel without losses. Consequently, the permittivity isrepresented by � ¼ �0LP þ d�0 � jd�00, where �0LP is the permit-tivity at the linearization point. Here, the perturbationswith respect to the linearization point are denoted d�0

and d�00, which are estimated by means of Eq. (25).Thus, we use the eigenvalue problem (20) with

�0ðrÞP �0 and �00ðrÞ ¼ 0 to compute the eigenfrequenciesand eigenmodes at the linearization point, which reducesto a special case of Eq. (18). The computed eigenfrequen-cies and eigenmodes are also used to compute the sensitiv-ities as described later in this section. The down-bed regionwith the fluidized particles yields an effective permittivitythat deviates substantially from vacuum. Therefore, we usea linearization point

�0LPðrÞ ¼�0bed in the bed region;�0 outside the bed region;

�ð28Þ

where �0bed P �0. We solve the eigenvalue problem for arange of bed heights hbed and bed permittivities �0bed, wherethe solutions from the eigenvalue problem together withthe corresponding sensitivities are stored in a database. Itis noticed that the resonance frequencies and their sensi-tivities do not change significantly as hbed decreases, if�0bed is increased to an appropriate extent. This is reason-able since it merely corresponds to a redistribution of thebed material to a smaller region. During the evolution ofthe process, we continuously seek the linearization point�0LP that yields the best fit with the corresponding devia-tions in the measured resonance frequencies, where thebed height is fixed at a representative value hbed ¼ 5 cm.

Hence, this estimation procedure can be used for a pro-cess where the particles grow significantly in size, by

means of changing the linearization point �0LP. To summa-rize, Eq. (25) yields reliable estimates of the perturbationsd�0eff and d�

00eff with respect to the linearization point. This

approach is sufficient for most normal process states.

4.1. Sensitivity analysis

The sensitivity analysis is based on the eigensolutionðEm;Bm;xmÞ computed for a linearization point with�0ðrÞP �0 and no losses, i.e., �00ðrÞ ¼ 0 and rðrÞ ¼ 0. Giventhis linearization point, we formulate the sensitivities thatrelate a small perturbation dxm in the resonance frequencyxm to perturbations in (i) the material parameters d�0 andd�00 and (ii) the normal displacement dn of the cavity’sboundary. Fig. 4a shows a subdivision of the cavity’s vol-ume that is used for the parameterization of the materialand Fig. 4b shows the cavity’s wall with a set of basis func-tions used for the representation of the normal displace-ment, where both parameterizations are axisymmetric.

4.1.1. Material sensitivitiesBased on the eigenvalue problem (20), the sensitivity of

the resonance frequency xm with respect to changes in thematerial parameters p ¼ �0 and �00 is expressed as

dxmxm

¼ �md�0 ðEm;EmÞ � jmd�00 ðEm;EmÞ

2m�0 ðEm;EmÞ; ð29Þ

where we have

mdpðu;vÞ ¼Z

Xdpu � v dX; ð30Þ

for a perturbation dp with respect to the material parameter p[17,20]. From Eq. (29), we notice that d�0 affects only the realpart of the eigenfrequencies, and that d�00 only affects theimaginary part. The discrete version of this linearization is

dxmxm

¼ � eTm Md�0 � jMd�00½ �em

2eTmM�0em: ð31Þ

4.1.2. Representation of material perturbationsThe perturbation dp of the material parameter p is rep-

resented in terms of a set of N space dependent basis func-tions unðrÞ, which are multiplied with the coefficients dpn,according to

dpðrÞ ¼XNn¼1

dpnunðrÞ: ð32Þ

Here, we subdivide the vessel’s volume X into regions Xnsuch that Xn do not overlap and

SNn¼1Xn ¼ X. We exploit

piecewise constant basis functions unðrÞ that are equal toone if r 2 Xn and zero elsewhere. In particular, we associ-ate the basis functions with three regions: (i) the bedregion of height hbed with a relatively high density of par-ticles; (ii) a cylinder shaped volume that coincides withthe core of the fountain; and (iii) the rest of the processvessel. These regions are formed by combining the basicsubregions shown in Fig. 4a.

The residual jjAx̂� bjj in the solution to the estimationproblem (25) is used as a metric of how well these basisfunctions are able to represent the actual permittivity

Fig. 4. Two-dimensional projection of the process vessel showing (a) the basic subregions that may be combined in different ways to obtain the materialbasis functions un and (b) basis functions used to represent shape deformations. All dimensions are in millimeters.


distribution in the experiment. For instance, the height andradius of the fountain region is chosen such that the resid-ual is minimized. Based on measurement results presentedlater in this article, we find that the fountain region indi-cated by gray color in Fig. 4a yields the best agreementwith the measured resonance frequencies.

4.1.3. Shape sensitivitiesFor a normal displacement dn of the vessel’s boundary

C, the resulting perturbation in the resonance frequencyis given by [17,20]

dxmxm

¼ � ddnðBm;BmÞ=l0 � ddnð�0Em;EmÞ

2Wm; ð33Þ

where Wm is the time-average energy in the cavity givenby Eq. (22), and

ddnðu;vÞ ¼Z

Cu � v�dndS: ð34Þ

A positive deformation dn > 0 here corresponds to that thevolume of the cavity increases. We notice that shape defor-mations only affect the real part of the eigenfrequencies.

The corresponding discrete formulation of Eq. (33) is

dxmxm

¼ �1l0

bHDdnb� eHD�0dne1l0

bHM1bþ eHM�0e; ð35Þ

where the superscript H indicates conjugate transpose.

4.1.4. Representation of shape perturbationsWe use an axisymmetric representation of the normal

displacement dn by means of a quadratic hierarchical basisfor three different parts of the vessel’s boundary: (i) theupper lid; (ii) the cylindrical surface; and (iii) the conicalsurface. Fig. 4b shows the basis functions in relation tothe geometry of the vessel. On each of the three parts ofthe boundary, we exploit the (local) basis functions

w1ðuÞ ¼ 1� u ð36Þw2ðuÞ ¼ u ð37Þw3ðuÞ ¼ 4uð1� uÞ ð38Þ

where the part of the boundary is parameterized by thelocal coordinate u such that the interval 0 6 u 6 1 corre-sponds to the entire part of the boundary.

Table 1Computed and measured values of the resonance frequencies and unloadedQ-values of the lowest eigenmodes in the process vessel. The variabilities(here represented by one standard deviation) in the measured quantitiesarise from disassembling and reassembling the complete vessel.

m Computed Measured

f0 ðMHzÞ Q f0 ðMHzÞ Q

1 759 5100 761 ± 0.07 5400 ± 1002 894 5800 894 ± 0.05 5100 ± 4003 928 5900 930 ± 0.02 4400 ± 4004 1008 6000 1009 ± 0.03 3200 ± 8005 1051 6100 1051 ± 0.06 4700 ± 2006 1149 6300 1150 ± 0.03 3400 ± 7007 1207 5600 1206 ± 0.08 3500 ± 3008 1495 9800 1498 ± 0.05 10,300 ± 400


In order to constrain the number of degrees of freedomassociated with the cavity’s shape to a number that is prac-tical, and to suppress non-physical shape deformations, weform two new basis functions by linearly combiningdn1—dn8 according to

dnlid ¼ 0:97dn6 þ 0:24dn8 ð39Þdncone=cyl ¼ 0:69dn2 þ 0:16dn3 þ 0:71dn4: ð40Þ

Thus, dnlid describes a smooth deformation of the top lidand dncone=cyl describes a joint deformation of the cylindri-cal and conical parts of the vessel. The coefficient valuesin Eqs. (39) and (40) are chosen such that the residualjjAx̂� bjj is minimized, given only these two independentdegrees of freedom for the shape.

4.2. Temperature measurements

Before the vessel is loaded with particles, it is normallypre-heated. We exploit this part of the process to gaininformation that is used to compute the a priori informa-tion ~x for later parts of the process. During the pre-heatingof the vessel, the vessel is empty and we perform simulta-neous and synchronized measurements on (i) the reso-nance frequencies and (ii) the temperature registered bythe set of temperature probes shown in Fig. 2. Given thedeviations ydT in the temperatures (relative to the initialtemperatures) and the shape perturbations x̂dn estimatedfrom the resonance frequencies for the empty cavity, weuse the relatively simple model x̂dn ¼ BydT , where thematrix B is determined by means of linear regression fromthe data acquired during the pre-heating interval. Thismodel is used to obtain a priori estimates ~xdn ¼ BydT fromthe measured temperatures during later parts of the pro-cess, after the particles have been loaded. In the following,we exploit the a priori information

~x ¼~xd�eff~xdn

� �¼

0BydT

� �; ð41Þ

in the estimation problem (25), where ~xd�eff corresponds toall the material parameters that are incorporated in theestimation procedure.

Tests demonstrate that about 95% of the perturbationsin the resonance frequencies due to thermal expansion ofthe vessel during heating and cooling can be accountedfor by this model, when the vessel is empty. This approachis less successful once the vessel is loaded with particles,since the presence of particles yields a different tempera-ture distribution in the vessel. However, it is presumedthat it is feasible to improve the a priori information,should the metal temperature be measured at more placeson the vessel’s wall and top lid.

5. Results

In the measurement results presented in the following,we perform repeated microwave measurements on theprocess vessel as the process is running. Each processexperiment typically takes a few hours in total. The exter-nal process parameters such as the flow-rate, temperatureand humidity of the fluidizing air, the pressure and flow in

the pneumatic spray-nozzle and the flow-rate of the spray-ing liquid, are controlled from the user interface of the pro-cess machine. These external process parameters togetherwith measured temperatures at the locations indicated inFig. 2, are logged by the process machine.

5.1. Cavity without particles

Among the eigenmodes hosted by the process vessel wehave identified eight modes that are feasible to use forsensing. These modes are listed in Table 1 together withtheir resonance frequencies and unloaded Q-values. Allthese modes have similarities with the TE and TM modeshosted by a circular cylindrical cavity. In particular, themodes m ¼ 1;2;5 and 7 resemble the modesTE111;TE112;TE113, and TE114 of a circular cylindrical cavity,and the modes m ¼ 3;4 and 6 resemble the modesTM010;TM011 and TM012 respectively. The mode m ¼ 7(TE114) has a strong electric field in the lower part of thevessel, which makes it well-suited for probing the permit-tivity in the down-bed region. The mode m ¼ 8 resemblesTE011 which has an electric field that is zero on all cavitywalls, which implies that its resonance frequency is insen-sitive to dielectric material adhering to the cavity walls.This mode also has a notably high Q-value.

The loaded Q-values in the experiment can be calcu-lated from the real and imaginary part of the measuredcomplex resonance frequency x according to

Q l ¼ReðxÞ

2 � ImðxÞ ; ð42Þ

for the weakly damped cavity modes [17]. Since the cavityis loaded by two probes with different coupling coeffi-cients, the following expression is used to calculate theunloaded Q-value

Q u ¼ Q l2

jS11j þ jS22j

; ð43Þ

where the reflection coefficients S11 and S22 are evaluatedat each resonance frequency [21].

We calculate the Q-value from the electromagneticmodel using the method described in Section 3.3, and herewe assume that the cavity walls have a conductivityrc ¼ 106 S=m, which should be representative for thestainless steel that the vessel is made of.


As can be seen in Table 1, the measured and computedresonance frequencies agree well. There is also a generallygood agreement between measured and computed Q-val-ues although, for some modes, the measured Q-value is sig-nificantly lower than the computed. The latter may be aresult of insufficient electric contact in the joint betweenthe conical and cylindrical part of the cavity, or at the alu-minum foil that covers the windows. As the process vesselis disassembled and reassembled, a certain variability inthe measured resonance frequencies and Q-values is intro-duced, which is presented in Table 1. This variability is sig-nificant in relation to the perturbations in resonancefrequencies caused by changes in the state of the process.Therefore, the particles are inserted into the process vesselthrough a small hole (indicated by H1 in Fig. 2) after thenecessary reference measurements on the empty cavityhave been collected, in order to eliminate this variability.Investigations show that the variability introduced frominserting the particles in this manner is negligible.

5.1.1. Effects from air humidity and sprayIn a Wurster-process, several different components that

influence the resonance frequencies are present simulta-neously. The most important of these components are (i)particles, (ii) the temperature of the vessel (iii) liquid drop-lets from the spray nozzle and (iv) water vapor in the flu-idizing air. The temperature of the vessel affects theresonance frequencies mainly via perturbations of the cav-ity’s shape, whereas the other components are assumed toinfluence the resonance frequencies via the permittivity. Inorder to assess how each of these components contributeto the microwave response, they have been studied sepa-rately. Some of these components, such as the sprayingliquid and the temperature, are inherently interconnected,i.e., the spraying rate cannot be changed without alsoaffecting the temperature.

The humidity of the fluidizing air can be controlledindependently using the process machine and humiditiesranging from 0 to 20 ðg H20Þ=ðkg airÞ, corresponding todew-points in the range from �30 �C to 25 �C, can beachieved. Our measurements show that the humidityinfluences only the real part of the permittivity of the airand hence the resonance frequencies, to a degree that isclearly detectable.

Water sprayed from the pneumatic spray nozzle (alsoreferred to as the atomizer) yields similar changes in thepermittivity inside the cavity as the humidity does. Only�0eff is influenced by the spray, and if the same amount ofwater is added through the atomizer as via the fluidizingair, the change in �0eff is very similar. The contributions tothe permittivity of the air from humidity in the fluidizingair and from water spray are additive, as long as the spray-ing rate is sufficiently low to allow for complete evapora-tion of the liquid droplets.

5.2. Measurements with particles

In the following, we present measurement results forprocesses involving particles. The particles used are madeof MCC (Cellets

�brand) and have diameters in a distribu-

tion from 500 to 700 lm. A batch consisting of 200 g of

particles is used in all of the following experiments. Allprocess settings are identical in the following experiments,except for the type of spraying solution and the sprayingrate. The fluidization flow is set to 30 Nm3=h (where Ndenotes normal temperature and pressure) with an inlettemperature of 70 �C. The atomizer pressure is 2:0 barand the associated atomizer flow is 2:0 Nm3=h.

The spraying liquid used in the experiments is watercontaining either (i) Mannitol or (ii) NaCl. The permittivityof these spraying liquids is measured using the dielectricprobe kit 85070E from Agilent Technologies, and theresults are shown in Fig. 5.

In the beginning of each experiment, the vessel is pre-heated and the particles are loaded, before the experi-ment-specific part of the process (e.g. coating) is initiated.In the results presented in the following, we omit the pre-heating interval and focus on the experiment-specific parts.

The average error between the model and measurementis computed as the root-mean-square of the differencebetween measurements (b) and model fit (Ax̂), averagedwith respect to the number of sensing modes M. With par-ticles present in the vessel, this average error is in the orderof 10�5 during normal process conditions (no extensiveagglomeration, etc.) which should be compared with theaverage magnitude of the measurement response(jjbjj=M), which is in the order of 10�3.

According to the sensitivity analysis in Section 4.1, per-turbations in the real part of the resonance frequencies arerelated to both d�0eff and dn. Similarly, a perturbation in theimaginary part of the resonance frequency is related tod�00eff . As described in Section 4.2, dn is to a large extent deter-mined based on a priori information that exploits tempera-ture measurements. An error in the a priori information mayyield a corresponding error in d�0eff , which varies on a time-scale that is significantly slower than the fast timescalesassociated with the fluid-dynamics of the process. Basedon these two time-scales, we make the assumption�eff ðtÞ ¼ �eff ;slowðtÞ þ �eff ;fastðtÞ, where �eff ;fastðtÞ representsfast random fluctuations and �eff ;slowðtÞ represents slowchanges in the permittivity caused either by errors in the apriori information or by changes in the actual process state(such as particles growing in size or accumulation of mois-ture in the particles). We model �eff ;slowðtÞ by a low-orderpolynomial with respect to time t, and this polynomial isdetermined by means of linear regression based on the esti-mated �eff ðtÞ. Furthermore, we compute the covariancematrix from the real and imaginary part of �eff ;fastðtÞaccording to C ¼ covðz1; z2Þ where z1 ¼ ½�0eff ;fastðtnÞ=�0 � 1�

T

and z2 ¼ ½�00eff ;fastðtnÞ=�0�T are vectors containing samples of

�eff ;fast taken at discrete time instants tn. If the eigenvectorsof C are scaled by the square root of its eigenvalues (i.e.the standard deviations), they define the major and minorsemi-axes of an ellipse that represent the co-variation ofthe data. Such ellipses are included in the scatter-plots inthe presentation of the following measurement results.Variations in �eff ;fastðtÞ along the eigenvector associated withthe largest eigenvalue of C, are mainly attributed to fluctua-tions in the particle volume fraction #. Therefore, we takethe slope of this eigenvector (i.e. the major semi-axis ofthe ellipse) as a density-independent ratio which is equiva-lent to the ratio g in Section 2.2.2.

0.5 1 1.5 20

10

20

30

40

50

60

70

80

f [GHz]

/0

and

/0

[-]

(a) Mannitol

0.5 1 1.5 20

10

20

30

40

50

60

70

80

f [GHz]

/0

and

/0

[-]

(b) NaCl

Fig. 5. Relative permittivity of distilled water at room temperature that contains (a) Mannitol (13.5%) and Kollicoat IR�

(1.5%), and (b) NaCl (0.5%). Solidcurves represent �0=�0 and dashed curves �00=�0.


The relationship between this slope and underlyingphysical parameters such as the moisture content of theparticles and the concentration of salt (NaCl) in the spray-ing liquid, have been studied by means of Monte-Carlosimulations. These simulations are based on measure-ments of the complex permittivity of MCC [11] and empir-ical models for the permittivity of water with and withoutsalt [10], together with the mixing formulas described inSection 2.2.1. Thus, we generate random mixtures of MCCparticles and air by using statistical normal-distributionsto represent the volume fraction and water content ofthe particles. The water is assumed to be either absorbeduniformly in the MCC-particles or distributed in a uniformwater-layer on the surface of the particles. The case with apronounced water-layer may be representative also for sit-uations where the surface of the particles have a substan-tially higher moisture content than the core. Based onthese simulations, we observe the following trends:

1. An increase in the moisture content of the MCC parti-cles (absorbed water) increases the slope, if the initialmoisture content is less than around 7%. For highermoisture contents, the slope instead decreases withincreasing moisture content.

2. Pure water distributed on the surface of the particlesdecreases the slope.

3. Water containing salt distributed on the surface of theparticles increases the slope if the salt concentrationexceeds 1%.

4. The slope is not affected by changes in the volume frac-tion, if the volume fraction is sufficiently low (around5% or lower), whereas at higher volume fractions, theslope increases with increasing volume fraction.

5.2.1. Fast coating processHere, we present results from a process where MCC par-

ticles are coated with a mixture of Mannitol (a sugar) and apolymer labeled Kollicoat IR

�(ethylene glycol vinyl acetate

grafted polymer), where the Mannitol and the Kollicoat IR

are dissolved in de-ionized water. The dry content of thesolution (the fraction that ideally stays on the surface ofthe particles) is 15 mass% (13.5% Mannitol and 1.5% Kolli-coat IR). The complex relative permittivity for this solutionis shown in Fig. 5a, and here we notice that the response issimilar to that of pure water. In this experiment, a rela-tively high spraying rate is used to obtain fast build-up ofthe coating layer. First, the spraying rate is set to 11.3 g/min and kept constant during approximately 15 min. Next,the spraying rate is increased even further to 14.4 g/min,which eventually leads to extensive agglomeration andparticles adhering to the walls of the vessel.

Fig. 6 shows the estimated permittivity in three differ-ent regions: (i) the down-bed region; (ii) the fountainregion above the Wurster tube; and (iii) the rest of the vol-ume above the down-bed region, as function of processtime in hours. At t = 0.9 h, the particles are fluidized inthe process with the fountain off, and the temperature isin equilibrium. The following changes in process settingsare marked with dashed vertical lines: at t = 1.04 h thefountain is switched on; in the interval 1.2 h < t < 1.26 hthe spraying rate is successively increased to 11.3 g/min;at t = 1.51 h the spraying rate is increased to 14.4 g/min.Some interesting features can be observed in these figures:

1. The estimated permittivities fluctuate substantially,which is reasonably caused by the continuous re-distribution of the particles in the process which occurson a short time scale (

1 1.2 1.4 1.6

0.1

0.2

0.3

0.4

0.5

t [h]

eff/

0-

1an

deff

/0

[-]

(a) Bed

1 1.2 1.4 1.6

0

0.01

0.02

0.03

0.04

0.05

t [h]

eff/

0-1

and

e ff/

0[-]

(b) Fountain

1 1.2 1.4 1.6

−2

0

2

4

6

8

x 10−4

t [h]

eff/

0-

1an

deff

/0

[-]

(c) Volume

Fig. 6. Estimated permittivity as a function of process time in the fast coating process. Black lines represent �0eff=�0 � 1 and gray lines �00eff=�0. The dashedblack curve in (a) shows the permittivity in the bed region at the current linearization point.


compensation for temperature variations, but it couldalso be due to a lower particle density in the fountainas the spraying starts. The estimated permittivity inthe remaining volume increases as the spraying starts,reasonably due to increased air humidity inside thevessel.

4. During the interval 1.5 h < t < 1.75 h, extensive agglom-eration occurs in the process and the particles start tobuild up dense layers on parts of the cavity walls, as aresult of the high spraying rate (this could be observedvisually after the experiment was finished). This pro-cess is clearly reflected in the estimated permittivities,which are changing dramatically during the agglomera-tion interval.

In situations similar to the agglomeration intervaldescribed in item 4 above, i.e. at very moist process condi-tions that yields material build-up on the vessel walls, werepeatedly observe a larger discrepancy between themodel and measurements, i.e. an increasing residualjjAx̂� bjj. Thus, a model that allows only for a constantpermittivity in the bed, fountain and remaining volumeregions, is unable to accurately represent the material dis-tributions formed during extensive agglomeration. Thisindicates the possibility of detecting anomalous processconditions based on the residual. Studies show howeverthat it is possible to maintain a low residual also duringextensive agglomeration if the cavity is divided into sub-regions that better represent the actual material distribu-tion. The sub-regions in the direct vicinity of the cavitywalls shown in Fig. 4a may be well-suited for this purpose.

Fig. 7 shows scatter plots of �00eff=�0 versus �0eff=�0 � 1 in

the three regions before and after spraying starts. The ellip-ses in these scatter plots represent the co-variation of thedata, where the length of each semi-axis is three standarddeviations. All ellipses are highly elongated and the varia-tions along the major semi-axis is assumed to be caused byvariations in material density, according to the discussionon density-independent functions in Section 2.2.2. Theslopes in the scatter plots are shown in Table 2. Here wenotice that (i) the slope is higher in the bed region, as com-pared to the fountain and volume regions and (ii) the slopedecreases in the fountain and in the volume after the

spraying starts, but remain unchanged in the bed. Accord-ing to the results from the Monte-Carlo simulationsdescribed previously, the decreasing slope in the fountainand volume regions could indicate that the moisture con-tent at the surface of these particles increases during thespraying interval. The higher slope in the bed is in agree-ment with the Monte-Carlo simulations, since the volumefraction in the bed is substantially higher than in the otherregions.

5.2.2. Slow coating processIn this experiment, MCC particles are coated with Man-

nitol and Kollicoat IR�

with the mixing ratio described inSection 5.2.1, under relatively dry process conditions, i.e.,a relatively low spraying rate of 6:6 g=min is used.

Fig. 8 shows the estimated permittivity in the bed, foun-tain and remaining volume regions, as function of processtime in hours. The time interval 0 h < t < 0.9 h, which hasbeen omitted, includes pre-heating of the vessel and load-ing of particles, so at t = 0.9 h, the particle fountain isswitched on and the temperature is close to equilibrium.The process conditions are changed at the time instantsindicated by the dashed vertical lines: at t = 0.89 h spray-ing (coating) starts with a spraying rate of 4.6 g/min; att = 0.96 h the spraying rate is increased to the final value6.6 g/min; at t = 1.13 h the temperature is considered sta-ble and at t = 1.78 h the spraying ends. In these figures,some interesting features can be observed:

1. The estimated permittivity in the bed increases slowlyduring the coating interval 0.90 h < t < 1.78 h, whereasthe permittivities in the fountain and the remainingvolume regions remain fairly constant. This may beexplained by particles growing in size as a result ofthe coating, which results in more dielectric materialthat accumulates mainly in the bed region. The growthin particle size was verified by means of sieving andweighing the particles after the coating process wasfinished, and the mass-increase was found to beapproximately 25%. A bed that grows in size in theexperiment is interpreted by our model as an increasein the effective permittivity in the bed-region, whichhas a fixed size. This effect can however not be clearly

0.27 0.3 0.33 0.36 0.39 0.420.01

0.03

0.05

0.07

0.09

eff / 0 - 1 [-]

eff/

0[-]

0.012 0.014 0.016 0.018 0.02 0.022−2−101234 x 10

−3

eff / 0 - 1 [-]

eff/

0[-]

2.1 2.4 2.7 3 3.3 3.6x 10−4

1

3

5

7

9x 10−5

eff / 0 - 1 [-]

eff/

0[-]

0.22 0.28 0.34 0.4 0.46−0.02

0.01

0.04

0.07

0.1

eff / 0 -1 [-]

eff/

0[-]

0.004 0.008 0.012 0.016 0.02

−3

−1

1

3

5x 10−3

eff / 0 -1 [-]

eff/

0[-]

6.6 6.9 7.2 7.5 7.8 8.1x 10−4

01234567

x 10−5

eff / 0 - 1 [-]

eff/

0[-]

Fig. 7. Scatter plots of �00eff=�0 versus �0eff=�0 � 1 in the fast coating process. Values inside parentheses are the slopes of the major semi-axis of the ellipses.

The black lines starting with squares represent the slowly varying component �eff ;slow of the estimated permittivity.

Table 2Slopes of the major semi-axis in the scatter plots of �00eff versus �

0eff � 1.

Bed Fountain Volume

Spraying off 0.314 0.245 0.247Spraying 11.3 g/min 0.313 0.228 0.211


observed in the fast coating experiment described inSection 5.2.1, which is somewhat surprising given thatwe use the same coating solution in both processes.

2. The permittivity in the remaining volume region (Fig. 8)increases as the spraying starts, it increases further asthe spraying rate increases and drops as the sprayingis stopped. This may be attributed to changes in the per-mittivity of the air, due to increased humidity caused bythe evaporating water in the spray.

1 1.2 1.4 1.6 1.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t [h]

eff/

0-

1an

de ff

/0

[-]

(a) Bed

1 1.2

0

0.005

0.01

0.015

0.02

t

eff/

0-

1an

de ff

/0

[-]

(b) F

Fig. 8. Estimated permittivity as a function of process time in the slow coatindashed black curve in (a) shows the permittivity in the bed region at the curren

Fig. 9 shows scatter plots of �00eff=�0 versus �0eff=�0 � 1 for

the tree regions, during the coating interval. Here, wenotice that the slope is higher in the bed as compared tothe other two regions, which is reasonable given the highervolume fraction in the bed. We also notice how the slowlyvarying component �eff ;slow shown by black lines in Fig. 9,varies substantially in the bed region, reasonably due tothe growth in particle size.

5.2.3. Particles sprayed with salt waterIn this experiment, MCC particles are sprayed with salt

water (0.5 mass% NaCl dissolved in distilled water at roomtemperature). The complex relative permittivity for thissolution is shown in Fig. 5b and here we notice a significantincrease in �00 at low frequencies due to the conductivity

1.4 1.6 1.8

[h]

ountain

1 1.2 1.4 1.6 1.8

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10−4

t [h]

eff/

0-

1an

deff

/0

[-]

(c) Volume

g process. Black curves represent �0eff=�0 � 1 and gray curves �00eff=�0. Thet linearization point.

0.32 0.39 0.46 0.53 0.6−0.01

0.02

0.05

0.08

0.11

0.14

eff/ 0 1 [-]

eff/

0[-]

0.006 0.01 0.014 0.018 0.022−4

−2

0

2

4

x 10−3

eff / 0 1 [-]

eff/

0[-]

3.5 3.8 4.1 4.4 4.7 5x 10 −4

12345678

x 10−5

eff / 0 1 [-]

eff/

0[-]

Fig. 9. Scatter plots of �00eff=�0 versus �0eff=�0 � 1 in the slow coating process. Values inside parentheses are the slopes of the major semi-axis of the ellipses.

The black lines starting with squares represent �eff ;slow, i.e. the slowly varying component of the estimated permittivity.

1.9 2 2.1 2.2 2.3 2.4

0.1

0.2

0.3

0.4

0.5

t [h]

eff/

01

and

eff/

0[-]

(a) Bed

1.9 2 2.1 2.2 2.3 2.4

0

0.01

0.02

0.03

0.04

t [h]

eff/

01

and

e ff/

0[-]

(b) Fountain

1.9 2 2.1 2.2 2.3 2.40

2

4

6

8

10

x 10−4

t [h]

eff/

01

and

e ff/

0[-]

(c) Volume

Fig. 10. Estimated permittivity as a function of process time in the salt water experiment. Black curves represent �0eff=�0 � 1 and gray curves �00eff=�0. Thedashed black curve in (a) shows the permittivity in the bed region at the current linearization point.

0.27 0.31 0.35 0.39 0.43 0.47

0.01

0.03

0.05

0.07

0.09

0.11

eff/ 0 1 [-]

eff/

0[-]

0.012 0.015 0.018 0.021 0.024

−2−1

01234

x 10−3

eff / 0 1 [-]

eff/

0[-]

2.4 2.6 2.8 3 3.2 3.4x 10−4

1

2

3

4

5

6

x 10−5

eff / 0 1 [-]

eff/

0[-]

0.23 0.29 0.35 0.41 0.47 0.530

0.03

0.06

0.09

0.12

0.15

eff/ 0 1 [-]

eff/

0[-]

0.003 0.007 0.011 0.015 0.019 0.023−4

−2

0

2

4

6x 10−3

eff/ 0 1 [-]

eff/

0[-]

8.1 8.5 8.9 9.3 9.7x 10−4

2

4

6

8

10

x 10−5

eff/ 0 1 [-]

eff/

0[-]

Fig. 11. Scatter plots of �00eff=�0 versus �0eff=�0 � 1 in the salt water experiment. Values inside parentheses are the slopes of the major semi-axis of the ellipses.

The black lines starting with squares represent the slowly varying component �eff ;slow of the estimated permittivity.


introduced by the salt. Two different spraying rates arestudied, namely 11:6 g=min and 13:3 g=min.

Fig. 10 shows the estimated permittivity in the bed,fountain and remaining volume regions, as function of

process time in hours. At the first time instant shown inFig. 10, the particles are fluidized, the fountain is on andthe temperature is stable. The following changes in processsettings are marked with dashed vertical lines: at t = 1.84 h

Table 3Slopes of the major semi-axis in the scatter plots of �00eff versus �

0eff � 1 for

the salt water experiment.

Bed Fountain Volume

Spraying off 0.327 0.248 0.235Spraying: 11.6 g/min 0.374 0.299 0.307


the spraying starts (11.6 g/min); at t = 2 h the temperatureis considered stable; at t = 2.24 h the spraying rate isincreased to 13.3 g/min which yields very moist processconditions and presumed initiation of extensive agglomer-ation. The increase in spraying rate to 13.3 g/min is fol-lowed by clear changes in the estimated permittivities,possibly due to free water (with salt) on the surface ofthe particles in the fountain region and/or agglomerationwith particles adhering to the walls of the vessel.

Fig. 11 shows scatter plots of �00eff=�0 versus �0eff=�0 � 1 in

two intervals: (i) no spray and (ii) spraying at 11.6 g/min(which does not yield agglomeration). The slopes of themajor axis for the three regions is also presented in Table 3.Here, it is interesting to notice how the slopes in all regionsclearly increase after the spraying is switched on. This isexpected, given the high dielectric losses of the salt waterin the frequency range of the cavity resonances, and it isalso in agreement with the Monte-Carlo simulations pre-sented previously.

6. Conclusions

We have presented a microwave measurement tech-nique for industrial processes contained in closed metalvessels, which exploits a number of electromagnetic reso-nances of the metal vessel to estimate the process state.In particular, we estimate the effective permittivity insidea vessel that hosts a pharmaceutical fluidized-bed process,which is used for particle coating. This process involvesspraying a solution onto the particles, where water actsas the solvent for different types of coating substances.For this particular application, it is important to monitorthe moisture content of the particles and the particle dis-tribution. Electromagnetic waves interact strongly withwater in a non-invasive and non-destructive manner,which makes this type of measurement technology veryattractive. In addition, our measurement system is sensi-tive to changes in permittivity throughout the process ves-sel and, thus, it can be considered to perform a globalmeasurement of the process state, which is challengingfor more conventional techniques.

We exploit two magnetic field probes and measure thescattering parameters as function of frequency using a net-work analyzer. The complex resonance frequencies areestimated from the scattering parameters and these arerelated to a reference measurement of the empty vesselin order to achieve relative changes in the resonance fre-quencies. The perturbations in the resonance frequenciesare related to (i) the perturbations in the complex effectivepermittivity inside the vessel and (ii) the thermal expan-sion of the vessel caused by variations in the temperature.We exploit an estimation algorithm with regularization to

estimate the effective permittivity and the shape of thevessel, where the unknown effective permittivity andshape is parameterized with the aid of spatial basis func-tions. The real and imaginary part of the effective permit-tivity are assumed to be frequency-independent in thefrequency band of the used resonance frequencies. Wemodel the vessel by means of the finite element methodand the computed eigenmodes and eigenfrequencies areused to calculate sensitivities with respect to changes inthe material parameters and the shape of the vessel. Theestimation incorporates a priori information that is col-lected from probes that measure the temperature of thevessel’s metal walls and the air that flows through thevessel. This approach accounts for substantial parts of thetemperature variations but it is believed that it can besignificantly improved.

The measurement technique performs well for the esti-mation of the effective permittivity that is parameterizedfor a small number of sub-regions. The re-distribution ofmaterial occurring when the particle fountain is switchedon, is clearly reflected in the estimated permittivity in allthese sub-regions. The effect of particles that grow in sizeduring coating processes can be observed in the estimatedpermittivities, and we have also shown that material build-up on the cavity walls can be clearly detected. We haverelated the real part of the effective permittivity to itsimaginary part by means of scatter plots and covariancematrices, where measurements taken at different timeinstants fall approximately on a straight line. This line isdefined by the eigenvector associated with the largesteigenvalue of the covariance matrix, and the slope of theline is affected by the liquid sprayed onto the particles inthe process. In particular, the slope of this line clearlyincreases if the particles are sprayed with a liquid thatfeatures substantial dielectric losses.

Acknowledgements

Livia Cerullo was financially supported by AstraZeneca.In addition, Johan Winges was financially supported by theSwedish Research Council (dnr 2010-4627) in the project‘‘Model-based Reconstruction and Classification Based onNear-Field Microwave Measurements’’. The computationswere performed on resources at Chalmers Centre for Com-putational Science and Engineering (C3SE) provided by theSwedish National Infrastructure for Computing (SNIC).

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Global monitoring of fluidized-bed processes by means of microwave cavity resonances1 Introduction2 Measurement system2.1 Process equipment2.2 Measurement principle2.2.1 Homogenization of a mixture of air and particles2.2.2 Density independent moisture measurements

2.3 Experimental setup2.3.1 Reduction of undesired losses2.3.2 Choice of probes and their design2.3.3 Measurement settings

3 Electromagnetic model3.1 Weak formulation3.2 Finite element approximation3.3 Losses due to finite conductivity in the cavity walls

4 Estimation procedure4.1 Sensitivity analysis4.1.1 Material sensitivities4.1.2 Representation of material perturbations4.1.3 Shape sensitivities4.1.4 Representation of shape perturbations

4.2 Temperature measurements

5 Results5.1 Cavity without particles5.1.1 Effects from air humidity and spray

5.2 Measurements with particles5.2.1 Fast coating process5.2.2 Slow coating process5.2.3 Particles sprayed with salt water

6 ConclusionsAcknowledgementsReferences

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