transformation conductivity and water transport in MOF adsorbent coatings for heat Frequency response analysis for the determination of thermal International Journal of Heat and Mass Transfer

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International Journal of Heat and Mass Transfer

journalhomepage:www.elsevier.com/locate/hmt

Frequency response analysis for the determination of thermal

conductivity and water transport in MOF adsorbent coatings for heat transformation

Eric Laurenz

^a,b,∗

, Gerrit Füldner

^a,∗

, Andreas Velte

^a

, Lena Schnabel

^a

, Gerhard Schmitz

^b

aFraunhofer Institute for Solar Energy Systems ISE, Department of Heating and Cooling Technologies, Heidenhofstr. 2, 79110 Freiburg, Germany

bHamburg University of Technology, Institute of Engineering Thermodynamics, Denickestr. 17, 21073 Hamburg, Germany

a rt i c l e i nf o

Article history:

Received 15 July 2020 Revised 5 November 2020 Accepted 5 December 2020

Keywords:

Adsorption kinetics Adsorption dynamics Heat transfer Thermal conductivity Aluminum fumarate Metal organic framework Adsorbent coating Adsorption chillers

Temperature frequency response Non-isothermal

Thermal frequency response

a b s t r a c t

Inthispaperwefocusonthedifferentiationandquantificationofdifferentheatandmasstransferphe- nomenagoverningtheoverallsorptiondynamics,fortheexampleofabinder-basedaluminiumfumarate (Alfum)coatingforheattransformationapplicationswithwaterasrefrigerant. Themethodologicalem- phasisisonextendingthevolumeswingfrequencyresponse(FR)methodtoproblemswithstrongheat transferlimitation.Theheatandmasstransferparametersaremappedtothesampletemperatureand loadingstate,inordertobeabletoreproducethestronglynon-linearbehaviourexhibitedunderapplica- tionconditions.Basedonamodelwithdiscretisedheattransferandlineardrivingforce(LDF)-simplified micropore diffusion, the thermal conductivity of the samples was identified as about 0.07 W/(m K), and the LDFtimeconstant between0.1 and 3 s^–1 at40°C withaU-shaped loadingdependency and an Arrhenius-type temperaturedependency. The methodis validatedby comparing ameasured large temperaturejumpexperimenttotheresultsfromanon-linearsimulationinformedsolelybythesepa- rametersobtainedfromthenewFR-basedmethod.

© 2021TheAuthor(s).PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Adsorptionchillersareanenvironmentallyfriendlysolutionfor the valorisationofwaste orsolarheat forcoolingdemands [1,2].

The metal organic framework (MOF) aluminium fumarate is a promising candidate in applications with low differences among the driving temperature, rejection temperatureand cooling tem- perature[3],likedatacentrecoolingwherewater-cooledCPUspro- videdrivingtemperaturesofupto60°Cormore[4].

Binder-basedcoatingsareapromisingapproachtoincreasethe volume-specific cooling power (VSCP) in order to reduce costs [5] while keeping a reasonably high coefficient of performance.

Coatings allow forsubstantially better heat transfer compared to state-of-the-artloose-grainconfigurations[6].

Forefficientdesignandoptimisationofadsorberheatexchang- ers (Ad-HX), non-linear dynamic models of the sorption process arerequiredinordertoavoidcostlyandtime-consumingtrial-and- errorprototyping[7].Alongsidewiththeadsorptionequilibria,the

∗Corresponding authors.

E-mail addresses: eric.laurenz@ise.fraunhofer.de (E. Laurenz), gerrit.fueldner@ise.fraunhofer.de (G. Füldner).

adsorption enthalpy andthe specific heat capacity, these models requiredetaileddescriptions ofthephysicalheatandmasstrans- ferprocessesintheadsorbentcoating.

Frequencyresponse analysis(FRA)is a provenmethod forthe determination of heat andmass transfer processes inadsorption systems sincethe 1960s [8–14]:From aset equilibrium state,an adsorbentsampleisexposedtoan,e.g.,sinusoidal,periodicalfluc- tuation ofone variable (excitation) while the responseof one or moreothervariablesisrecorded.Thecomplexresponse,i.e.ampli- tudeandphaseshift,asafunctionofthefrequencyoftheexcita- tionsignalcontainstheentiredynamicinformationofthesample.

Themethodhasseveralprincipaladvantagescomparedtothestep responsemethodsusuallyused[7]inthefieldofheattransforma- tionapplications:

- Separationof different transport processes asthey are visible atdifferentfrequencies dependingontheir time constant and withdifferentpatternsdependingonthetransportmechanisms - Highresolutionofthethermodynamicstate as,duringtheex- periment,the systemis kept ina small regionaround a con- stantequilibriumstate

https://doi.org/10.1016/j.ijheatmasstransfer.2021.120921

0017-9310/© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Nomenclature

Abbreviations

0D Zero-dimensional,lumped Ad-HX Adsorberheatexchanger CAS Computeralgebrasystem FR Frequencyresponse FRA Frequencyresponseanalysis HT Heattransfer

LDF Lineardrivingforce LTJ Largetemperaturejump maD Macroporediffusion miD Microporediffusion

NETD Noiseequivalenttemperaturedifference RMSD Rootmeansquaredeviation

PFGNMR Pulsed-fieldgradientnuclearmagneticresonance

Variables

T Temperature(K) p Pressure(Pa)

X Loading(kg_adsorbed/kgsorbent,dry)

X_eff Effectiveloading(kg_adsorbed/kgcomposite,dry) D Diffusivity(m²/s)

h (Effective)Heattransfercoefficient(W/(m²K)) d Layerthickness,diameter(m)

P Vectorofidentifiedparameters(anyunit)

E Errorfunctional,weightedrootmeansquareddiffer- ence(-)

λ

^Thermalconductivity(W/(mK)) V Volume(m³)

A Surface area (m²), adsorption potential (J/kg),Am- plitude(anyunit)

R Universalgasconstant(J/(molK)) Rw Specificgasconstantofwater(J/(kgK))

ρ

^{Density(kg/m3)}

m Mass(kg)

τ

^{Timeconstant(s)}

f Frequency(Hz)

φ

^{Phaseshift(rad)}

ψ

^{Emptychamberpressurecorrectionfactor(-)}

t Time(s)

s Laplacevariable(1/s)

G_pV Complexpressurevolumetransferfunction(Pa/m³) GXV Complex loading volume transfer function

((kg/kg)/m³)

cp Specificheatcapacity(J/(kgK))

G_TX Complex temperature loading transfer function (K/(kg/kg))

G_TV Complex temperature volume transfer function (K/m³)

j Imaginaryunit(-) M Molarmass(kg/mol) k Kineticcoefficient(1/s)

˙

q Heatflux(W/m²) x Spacevariable(m)

˙

n Molarflux(mol/(m²s))

r Radius(m)

K_T Slopeoftheisobar((kg/kg)/K)

Kp Slopeoftheisothermperpressure((kg/kg)/Pa) c Concentration(mol/m³)

ε

P Macroporosity(m³/m³)

α

^{Modelcoefficient:}

α

=Kp

ρ

^sRwT0/

ε

P(-)

β

^{Modelcoefficient:}

β

=Ah+cpmss(W/K)

Kc Slope of the isotherm per concentration ((kg/kg)/(mol/m³))

γ

^{Modelcoefficient:}

γ

=^hsK_Tms(J/K) ^hs Differentialadsorptionenthalpy(J/kg_adsorbed) N Numberofobservations

y Vectorofallmeasurandsperfrequency(anyunit) W Weighingmatrix(anyunit)

k Vectorofindividualmeasurandsoverallfrequencies (anyunit)

σ

^{LDFcurvaturefactor(-)}

Ea Activationenergy(J/mol) Indices&superscripts

∼ Laplacetransformedofthedeviationfromthetem- poralmeanvalue

^ Amplitude,deviationfromthetemporalmean,best estimate

− Spatialmeanvalue

sat Atsaturationcondition(liquidvapourequilibrium) sp (Ad)sorbedphase,adsorbate

w Water

s (Ad)sorbent,(ad)sorption 0 Temporalmeanvalue

cal Atcalibrationconditions,blankmeasurement ch (Measurement)chamber

Btm Bottom of the composite (at the interface to the samplesupport)

mi,miD Micropore(diffusion,LDF-approximated) maD Macroporediffusion

LDF Lineardrivingforce eq Atequilibrium

srf Surfaceofthecomposite/sample cmp Composite

ct Coating

f PerFrequency exp Experimental rel Relative ad Adsorbate

g Gas

- Easilyscalableaccuracybychangingthedurationoftheexper- iment, i.e.changing thenumber ofperiods measured per fre- quency

- Usage of locally linearised models with constant coefficients without accuracy loss as, in the small state region, non- linearitiesmaytypicallybeneglected

- Measurements in the frequency domain allow for usage of model solutions directly in the Laplace domain without the needforback-transformationtothetimedomain

- Computationally inexpensive parameter identification as, for linearised models in the Laplace domain, analytical solutions canbederivedinmostcases

Theseadvantagesoutweightheconsiderableexperimental and data processing effort, especially if a quantitative differentiation betweendifferenttransportprocessesandtheirdependencyonthe thermodynamic state (temperature, loading,pressure) of the sys- temisrequired.

In the literature on FRA, the “heat effect”, i.e. the sample’s non-isothermal temperature evolution [11,15], is (if at all) han- dled as a small parasitic effect disturbing the understanding of themasstransferprocesses.It waseithermitigatedbyapplyinga flow-through system[12],where a continuousgasflow increases the heat transfer by convection, or taken into account and cor- rected using the thermal frequency response [16,17], where the

Fig. 1. Sorption kinetic setup for frequency response (FR), large temperature jump (LTJ) and large pressure jump (LPJ) experiments with absolute and differential equilibrium measurements (source: [18] ).

response samplesurface temperature is measured using a costly high-precisionfastIRdetectorcooledwithliquidhelium.Forheat transformation applications, the case is different, with the heat effect being the principal application motivation. This requires a soundunderstandingoftheheattransferprocessesasmuchasof the mass transfer processes,andthe heat effectbecomes an im- portantsourceofinformation.

Inthispaper,wepresentaFRA-baseddeterminationoftheheat andmass transferprocesses,namelythermal conductionandmi- cropore diffusion, in aluminium fumarate coatings that are rele- vantforheattransformationapplications.Thisispartofacompre- hensive approach that also allows obtaining adsorption equilibria andadsorptionenthalpies[18]alongsidewiththesorptiondynam- ics, in a singlemeasurement procedure, under exactly the same measurementconditionsandforaverynarrowregioninthether- modynamic statespace.Theloadingandtemperaturedependency ofthe obtainedtransportpropertiesis mapped.Thisallowsusto simulatealargetemperaturejump(LTJ)underrealisticapplication conditions andto validate the parametrised modelby comparing thesimulationtoexperimentalresults.

2. Experiments 2.1. Material

Aluminium sample plates (5 × 5 cm²) were partly coated (Act=18.9cm²)withaluminiumfumarate (Basolite^R A520, BASF, [19,20]) using a silicon binder, and characterised as described before [18]. The coating thickness d_ct was varied (0.14, 0.24, 0.61 mm), resulting in different coating dry masses (134, 217, 563 mg). The coating dry density was calculated as

ρ

^cmp= m_cmp/(^Actd_ct) ^{(0.51, 0.48, 0.49 g/cm³). The dryadsorbent content} ofthecoating,bymass,wasaimedat0.75andwasconfirmed,on average, for the three samples (0.72, 0.79, 0.80) by uptake com- parison[18].SampledetailsaregivenintheSupplementary Infor- mation (TableS1)andinaseparatepublicationonthethermody- namicequilibriumpropertiesofthesamples[18].Thespecificheat capacityofthecoatingswasmeasuredfordifferentwaterloadings (0–0.35kg/kgcmp)andtemperatures(30–90°C)bydifferentialscan- ning calorimetry (publication in preparation). Results were fitted

as cp= f(^X,T) ^{per dry mass of composite, withvalues between} 1.1and3.0kJ/(kgK)at0kg/kg,30°Cand0.35kg/kg,90°C,respec- tively.

Themacroporosityofthecoating

ε

P isestimatedto0.5, based on the composite drydensity andthe mass fractions anddensi- tiesof thebinder (

ρ

=1.1 g/cm³, calculatedfrom Kummeretal.

[21] where the same octagonal stainless-steel masks were used) and the apparent dry density of the aluminium fumarate crys- tals. As reported measurements of the latter are lacking, it was estimatedbased onthe structuraldataobtainedby Alvarezetal.

[19]byRietveldrefinement.Theyestimatedthevolumeofafully hydrated unit cell, consisting of four metal Al(OH) octahedrals (m=4·44.0u)andfourfumarateC₄O₄H₂linkers(m=4·114.1u), as990 ˚A³, yielding

ρ

=1.06 g/cm³. It should be notedthat this value does not account for water-induced shrinking or swelling, norforanydefectsorimpuritiesthatarelikelytobefoundinthe actualadsorbentparticles.

Theparticlesizeoftheinitialadsorbentpowderwasmeasured at18μm,onaverage(Fig.S1).

2.2. Apparatus

Frequencyresponse(FR)measurementsaredonewithacustom setupdetailedearlier(Fig.1,Table1) [18].In addition,thissetup allowslargetemperaturejump(LTJ),largepressurejump(LPJ)and smallpressurejump(SPJ)experimentswithwater[22–24].

The setupcan be classifiedasa volume swingthermalFR in- spiredbyapproachesfromotherauthors[13,25–28]butextended byathoroughcontrolofthesample’sthermalcontactandatem- peratureresponsemeasurement inordertodiscriminatebetween heatandmasstransfereffects.Toourknowledge,comparablemea- surements have only been published by LIMSI-CNRS in the late 1990s [16,17,29] butwithimmense efforts regardingtemperature measurements (liquid helium-cooled detector) to reach an NETD of1.5–15 mKat5 ms response time. Here,an off-the-shelftem- peraturesensorisemployedattheexpenseofanuptotwoorders ofmagnitude highernoise level.The noise levelis minimised by increasingthetimeconstantforlowfrequenciesofthesensorand compensatedforbylongermeasurementtimesforhigherfrequen- cies,inordertoreachsatisfyingprecision.

Table 1

Principal characteristics of the measurement quantities; details have been published in [18] . Mean values of sample temperature and chamber pressure are obtained from slow and precise sensors (Pt100, MKS), whereas fast and less precise sensors (Heitronics, STS) are used for assessing frequency response values.

Quantity Range Typical uncertainty Time constant Device

Chamber volume 849–922 ml 0.4 ml (20 °C), 1.3 ml (80 °C)

1.3 ms Schreiber Messtechnik LVDT Chamber pressure slow 0–100 mbar 0.05 mbar (5 mbar),

0.15 mbar (100 mbar)

40 ms MKS Baratron 627B

Chamber pressure fast 0–100 mbar 0.4 mbar (5 mbar, 30 °C),

5 mbar (100 mbar, 60 °C) 1 ms STS ATM

Cold-plate temperature 20–95 °C 0.1 K ~1 s 4-wired Pt100

Sample surface temperature 20–80 °C 25 mK ( τ= 1 s)

1.1–0.6 K ( τ= 5 ms, 20–100 °C) ¹

≥5 ms (adjustable) Heitronics KT15

1Noise equivalent temperature difference (NETD), depending on chosen time constant and measured temperature.

2.3. Procedure

The measurement procedure consists of three steps: (i) pre- conditioningtothedesiredstate,(ii)determinationoftheequilib- rium slopes and(iii)theactual FR measurement.Throughoutthe measurement, the temperature-controlled cabinet is kept at the measurementmeantemperatureT₀.Measurementsarecarriedout at30–60°Candintheentireloadingrange.

Pre-conditioningincludesdesorptionatp<0.01mbarand95°C overnight anddosingvapourfroma calibratedvolume tothe ac- tual loading X₀. The local slopes of the isotherm and isobar are determinedwithsmallvolumeandtemperaturevariations,respec- tively.Thelatteralsoservestoexcludelocalhysteresisinthesorp- tion equilibriumby comparing the loading reached atV₀ andT₀ coming from above andfrom below. Detailsare givenelsewhere [18].

The FR measurement consists of a set of single-frequency si- nusoidal excitationsofthemeasurement chamber volumespaced in an (almost) geometric sequence between 10^–3 and 5 Hz. This is realised by rotating the stepper motorwith constant velocity, which is convertedinto a sinusoidal linearmotion ofthe bellow throughaspeciallydesignedcam.Thesteppermotoriscontrolled with the same DAQ device (Keysight U2351A) that records the bellow position, chamber pressure and sample surface tempera- ture so that data acquisition andvolume excitation are synchro- nised.Thesingle-frequencysinusoidalsarechosenforthestraight- forwardimplementationandasthemostrobust excitationsignal.

Thesignal-to-noiseratioismaximisedbyconcentratingthewhole signal power to a single frequency [30]. The number of periods is increased with the frequency to compensate for the reducing signal-to-noiseratio.

2.4. Dataprocessing

Signalsfromsensorsforbellowposition,chamberpressureand samplesurfacetemperaturearerecordedwithaKeysightU2351A at a samplingrate of10 kHzand 16-bit resolution.During mea- surement, data is down-sampled by integer averaging to a sam- plingrateof1000timesthebellowfrequency.Thisresultsinlower noise levels for lower bellow frequencies and a considerably re- duced amount of raw data while the filter disturbance on the signal remains negligible. The complex responses V˜, p˜ and T˜ at the knownbellowfrequency f areobtainedfromthefastFourier transformation (FFT) implemented in R [31]. Spectral leakage is avoidedbycarefullychoosingbellowfrequenciesthatallowforan integer amountof(averaged)samplesper bellowperiodandcut- tingtimeseriesdatatomultiplesofcompletebellowperiods.From acomplexresponsey˜,thesignalinthetimedomaincanberecon- structedas

y

(

^t

)

^−y0=Aysin

(

²

π

^f+

φ

^y

)

⁽¹⁾

whereAy=

|

^y˜

|

^{isthesignalamplitudeand}

φ

^y=arg(^y˜)^isthephase shift.y˜can alsobe considered asa real 2-dimensional vector of Re(^y˜)^{and Im}(^y˜)^{,which is lessintuitive butmore convenientfor} furtherdataprocessingastheyareboth∈R,comparedto

|

^y˜

|

⁽≥0) andarg(^y˜)⁽∈[0, 2

π

^]).

Correctionsareappliedtothevolume perturbationandtheFR signals ofthe sample surfacetemperature and the measurement chamberpressure,asdetailedinthefollowing.

Chambervolumeandsurfacetemperaturearecorrectedforlow pass filtereffectsof thesensors by applyingan inverselow pass filtertothemeasuredresponse.Cut-off frequenciesare800Hzfor the volumeand 200Hz forthe temperaturemeasurement. Thus, theeffect,mainlyaffectingthephaseshift,remainssmallevenfor the highest measurement frequency of 5 Hz. In addition, the IR temperaturesignal is corrected forlinearity errorsby comparing thedifference of steady-statevaluesfrom smalljumps around T₀ tothosemeasuredbythePt100ofthecoldplate.

The pressure signal is typically corrected based on responses from blank measurements compared to the expected isothermal response according to the mass balance Eq. (6) with no sample (ms=0) [9,32,33]. This correction accountsfor effects fromnon- isothermal compression, adsorption at the walls, heterogeneous pressuredistributionandsensordamping.Thepressurecorrection consists of a simple multiplication of the measured pressure re- sponsep˜msrwithacorrectionfactor:

˜

pcor=

ψ

^p˜msr. (2)

The pressure correction factor

ψ

=−V^V˜₀ p₀

˜

p_cal, with the blank measurementresult p˜_cal,canbededuced fromthechamber mass balanceEq.(8).Results for

ψ

^{fromawide rangeof}temperatures, pressures and frequencies are fitted using a scaled and centred multivariate Kriging model withnoisy observations implemented inR[34]toobtainacorrectionfunctioncoveringthewholeopera- tionalrangeofthesetup.Aftercorrection,thenon-idealitiesofthe pressureresponse are below2% oftherelative pressure response

˜ p/p₀.

2.5. Uncertaintyevaluation

Uncertainty analysis was carried out based on GUM [35], in- cludingtheuncertaintyfromsamplevariance (typeA)anduncer- taintiesfromimperfectcalibrationandcorrection(typeB).

Theuncertaintyestimationforthecomplexresponses isbased onMonteCarloexperimentsonmeasurednoisefromblankexper- imentswith stoppedmotor (typeA) andall type Buncertainties that affect linearity. As uncertainties affecting only the absolute value(likeoffsets)areirrelevanttothecomplexresponse,amuch higherprecisioncanbereachedascomparedtoabsolutemeanval- ues.Complexerrorpropagationiscarriedoutasabivariatecombi- nationofrealandimaginarypartuncertainties;incontrast,acom- binationofmodulusandargumentuncertaintieshasastrongbias

forvaluescloseto0+i0andisadvisedagainst[36].Uncertainties are stronglydependentonthebellow frequency f,thenumberof periods evaluatedandtheactual measurementvalue. Typicalval- uesofrealandimaginarypartsare0.1–5μlforV˜,(0.5 × 10^–3)–

(5 × 10^–3)mbarforp˜and0.8–30mKforT˜.

Otheruncertaintiesofthesetupwerediscussedbefore[18]. 3. Modelling

Frequency response analysisis a model-basedmethod forthe determination oftransport processes.Thesorptionkinetic models derived in the following are as essential asthe actual measure- ment.Botharecombinedintheparameteridentificationprocedure describedattheendofthissection.

3.1. Principalapproach

The frameworkforFR modellingismainly basedonthetrans- ferfunctionapproachsuggestedin[37].Theprincipalassumptions are:

1 Perturbations of the loading, pressure/concentration and temperature are sufficiently small to allow linearisation of all equations with constant coefficients. These coefficients areassumedlocalforthegivenmeanstate butmaybedif- ferentfordifferentmeanstates,i.e.differentexperiments.

2 Thepressuredistributioninthechamberisuniform.

3 The temperatureofthe gasinthechamberisconstant and equaltothetemperatureofthesamplesupport.

4 Adsorptionoccursonlyonthesampleandnotonthecham- berwalls.

5 The chamber contains a pure working fluid atmosphere (single-componentadsorption).

6 Thegasmaybedescribedasidealgas,whichisaverygood approximation forwaterinthe pressure(< 100 mbar)and temperature(20–100°C)regionofinteresthere.Inprinciple, the modelscanbe extendedby differentequationsofstate that are locally linearised. This might be relevant,e.g., for high-temperatureapplicationsorworkingfluidsoperatedin ahigherpressurerangelikeammonia.

Non-ideal effectscontradicting assumptions2–4are accounted forintheexperimentsapplyingapressurecorrection(see2.4).

Themassbalanceofthemeasurementchamberis 0= dX¯

dtms+dmg

dt (3)

where X¯ denotes the averageloadingof thesample(mass ofad- sorbedworkingfluidpermassofdryadsorbent),m_sisthedryad- sorbent mass, andmg is the gasmass in the chamber.Withthe idealgaslawandconstantgastemperatureT₀,thisbecomes:

0= dX¯ dtms+

d dt

(

^pchV

)

RwT0

= dX¯

dtms+ 1 RwT0

Vdpch

dt +pch

dV dt

. (4)

For convenience, the variables are transformed to deviations fromthetemporalmeanwith

Xˆ=X−X0

ˆ

p=p−p0

ˆ c=c−c₀ Tˆ=T−T0

(5)

and Laplace transformed (X˜=L(^Xˆ)^{). Thus, the Laplace-} transformedmassbalanceis

0=sX¯ms+ 1 RwT0

V0sp˜ch +p0sV˜

. (6)

Themassbalancecanbeexplicitlysolvedforthecomplexpres- surevolumetransferfunction

GpV

(

^s

)

^{= p}_˜^˜ V =−

X¯ V˜

msRwT₀ V0 + p₀

V0

=− GXV

(

^s

)

^ms_V^RwT0

0 +p0

V0

. (7)

This allows calculating the pressure response directly from known constants and the loading volume transfer function G_XV(^s)=_V^X_˜^¯.

Fortheoverallenergybalance ofthesample, webasicallyuse thesameapproachthat wasused before[32,37], assumingan ef- fective heattransfercoefficient hbetweenthecoating, whichhas the temperature T_btm at the interface, and the support, which is constantly at the mean temperature T₀. Convective or radiative heattransferattheadsorbentsurfaceisneglected.Theenergybal- anceisthen

mscp

dT¯

dt =ms

^hs

dX¯

dt −Ah

(

^Tbtm−T0

)

⁽⁸⁾

withtheadsorbentmass-specificheatcapacitycp,theenthalpyof adsorption ^{hs and thesample surfacearea A (i.e. thereference} surfaceforh). Laplace transformation withdeviationvariables as aboveyields:

mscpsT¯=ms

^{hssX¯−AhT˜}btm. (9) For models assuming homogeneous temperature distribution, alltemperaturesare equal(T˜=T¯=T˜_btm)andEq.(9)simplifies to theformusedbyWangetal.[37],whichcanbedirectlysolvedfor T˜to formulate the temperature loading transfer function

GTX

(

^s

)

^=T˜

X¯ = ms

^hss

mscps+Ah. (10)

This is slightly different fromthe formulation used by Wang etal.[37],butitismoreconvenientforthecalculationofthemea- suredtemperatureandpressureresponsesdirectlyfromG_XV(^s)^:~

T˜=G

TX

(

^s

)

^GXV

(

^s

)

GTV(^s)

V˜ (11)

˜

p=− G_XV

(

^s

)

^ms_V^RwT0

0

+ p₀ V₀

V˜. (12)

IncaseswheretheassumptionsforEq.(10)donotapply,adi- rectformulationforG_TV(^s)^{willberequiredandthereductiontoa} singletransferfunctionG_XV(^s)^{isnolongerpossible.}

G_XV(^s)^(orGpV(^s)^andGTV(^s)^{) maybecomputeddirectlyfrom} constants (transport parameters as well as material and setup properties) and the complex variable s according to analytical model solutions from the following section. With s=j2

π

^{f, this}

allows calculating the pressure and temperature frequency re- sponses,whichcanthenbefittedtotheexperimentalresults.

3.2. Heatandmasstransfermodels

Analyticalsolutions in the formof complextransfer functions convertible to G_XV(^s) ^{are available for two} non-isothermal cases withdifferentcomplexitiesinthemasstransfermodelling[37]:

a) “0DHT_LDF”:lumpedheattransfer+lineardrivingforce(LDF), b) “0DHT_miD_3D”: lumped heat transfer + micropore(loading- driven)diffusion inradiallysymmetric spheres(notapplicable here).

Forthe modellingofmacropore diffusioninevenlayers ofad- sorbentcoatingandintra-crystalline (micropore)diffusioninnon- isotropicmaterials witha singlepreferreddiffusiondirectionlike

Fig. 2. Principles of different non-isothermal approaches to adsorption kinetic modelling with boundary conditions. FR solutions for (a) and (b) are available [37] in a similar formulation (c.f. S3), (c)–(d) are developed in this work (HT: heat transfer, miD: micro-diffusion, maD: macro-diffusion, LDF: linear driving force).

Table 2

Summary of FR model solutions developed in this work; details are given in the supplement as referenced.

Model Loading volume transfer function G XV(S) Reference

c) Lumped HT + even miD −^K_V^pp₀⁰[ _s

kmicoth (_s

kmi)+ ^msK_V^p₀^RwT0−_β^s] ⁻¹with k mi= ^D_r^mi2 and β⁼ (Ah + c pm ss )/ h sK Tm s S3.1 d) Lumped HT + even maD −^K_V^pp₀⁰[ _s

kma(1 + α)(1 −_β(1+^s_α))coth (_s

kma(1 + α))+ α(^ε_ρ^P_s^m_V0^s−_β(1+^s_α))] ⁻¹with α= K pρsR wT 0/ εP, k ma= ^D_d^ma2 and βas above

S3.2

e) Lumped HT + even maD + micro-LDF

−^K_V^pp₀⁰[

s kma

α+γ

γ γ(1 −_β(α^s_+γ))coth (

s kma

α+γ

γ )+ α(^ε_ρ^P_s^mV0^s−_β(α^s_+γ))] ⁻¹ with γ= 1 + s/ k LDFand α, β, k maas above

S3.3 f) 1D HT + micro-LDF G XV(s ) is not useful in this case. G pV(s ) and G TV(s ) are given separately in the reference. S3.4

aluminium fumarate [19,38], the diffusion equation needs to be solved onan even-plateorslabgeometry.Moreover,tomodelthe heat transfer effect of the coating thickness, a 1D heat transfer modelwithcontactresistanceisrequired.

For this work, we developed the FR solutions for differently simplifiedmodels(c.f.Fig.2):

c) “0DHT_miD”:lumpedheattransferwithmicroporediffusionon a slab geometry (homogeneous concentration in the macrop- oresandhomogeneoustemperatureinthecoating),

d)“0DHT_maD”: lumped heat transfer with macropore diffusion ona flat-plategeometry(homogeneousloadinginthecrystals andhomogeneoustemperatureinthecoating),

e) “0DHT_maD_LDF”:lumpedheattransferwithmacroporediffu- sion combined with an LDF approach formicropore diffusion (homogeneoustemperatureinthecoating)

f) “1DHT_miLDF”:heatconductioninthecoating witha thermal contactresistancetothesupportandwithanLDFapproachfor microporediffusion(homogeneousconcentrationinthemacro- pores)

A computer algebra system (CAS) was used (Wolfram Mathematica^R in our case) to efficiently solve the equation systems.The solutionsare summarisedinTable2anddetailedin SI3.

In the last two models, the micropore diffusion is approxi- mated withthe LDF approachby analogyto the isothermal case withlinearisedequilibriumandhomogeneous uptakedistribution [39]:

Dmi≈ kLDFr²

σ ( σ

+2

)

^{, (13)}

wherethe curvaturefactor

σ

= ^rA_V =1for theeven-plategeome- tryofthe1Dchannelstructureofthealuminiumfumaratecrystals (

σ

=2forcylinderand

σ

=3forspheregeometries). risthedis- tancebetweenthesurfaceandtheno-fluxboundary,i.e.theparti- cleradiushere.

Simultaneous space discretisation of both heat and macrop- orediffusioncouldprincipallybesolvedanalytically.However,the straightforward approach applied to the other cases failsto pro- duce usefulresults and isthus omitted here – thesolutions ob- tained by CAS are so fragmented and extensive that numerical evaluationruns into large rounding errors. These approaches are typically found as numerical non-linear models in the time do- main [22,23,40]. Also non-straightforward CAS-based approaches asproposedearlier[41]mightbeusefulherebutareleftforfuture studies.

Fig. 3. Example for the measured frequency response of water on the sample Ct_610 at p = 18.5 mbar, T = 40 °C and X eff= 0.09 g/g in comparison to best-fit re- sults of different models for a geometric sequence of 14 frequencies between 0.01 and 4 Hz, shown as non-normalized real and imaginary part of the complex devia- tion variables of the chamber pressure ˜ p and the surface temperature T ˜ .

3.3. Parameteridentification

Most model parameters can be determined externally (c.f.

Section2.1)orinthefirstpartofthemeasurementprocedure[18], whereas the parametersforheat andmasstransferare identified throughageneralisedweightedleast-squaresmethod.Thebestes- timate for the parameter vector P (denoted as Pˆ) is gained by minimisingtheerrorfunctionalE(^P)^{.Aconfidenceregionisgiven} around Pˆ based on the contour of E(^P)^{, following an approach} from Marsili-Libelli[42].The value ofE(^Pˆ) ^{is an averagerelative}

deviationbetween thefitted model andthe experiment. For fur- therdetails,refertoS4inthesupplement.

4. Results

Theresults fromtheFR measurements (Fig.3) show thetypi- cal bimodalshape indicatingseveraldistinct transferprocessesat differentcharacteristicfrequencies.Thesignal-to-noiseratioissuf- ficient,andforlowerfrequencies,morethansufficient.Thismight allowfurtherreductionofthemeasurementtime,e.g.byreplacing thesingle-frequencysinusoidalsignalbyaphase-shiftedmultisine withahighsignalpower,i.e.alowcrestfactor[30].

4.1. Modelcomparison

ThemeasuredFR (Fig.3) showsthetypical bimodalshapeob- servedbeforefornon-isothermaladsorptionsystems[16,32].Note thathereIm(^p˜) ^andIm(^T˜)^are proportionalto thenegative“out- of-phasecurves” duetodifferentphaseshiftdefinitions.Themax- imum temperatureamplitude is in the order ofseveral 100 mK, whichistypicalofadsorptionwithwatermeasuredinourappara- tusandallowsforagoodsignal-to-noiseratio,andisaboutoneor- derofmagnitudehigherthanthevaluesreportedbyBourdinetal.

[16].

Forthe example shown in Fig. 3,the simple macro- andmi- croporediffusionmodels0DHT_maDand0DHT_miDfail torepre- sentthe experimentaldata(weightedresidualerrorE(^Pˆ)=0.20).

Due to the mathematically similar structure, both models yield thesamebest-fitshapeforall measurements.Asthemismatchis observed throughoutall experiments,both models will be disre- gardedinthefollowing.Abetterfit(E(^Pˆ)=0.08)isachievedwith the macropore diffusion + micro-LDF model (0DHT_maD_LDF).

However, the best fit(E(^Pˆ)= 0.04) results when discretising the heattransferwiththe1DHT_miLDF model.Thesimple0DHT_LDF model returns the same fit result as 0DHT_maD_LDF, indicating thatthemacroporediffusionisnotrelevanttotheexampleshown.

Fora similar reason, the 0DHT_LDFmodel converges toward the results from the 1DHT_miLDF model when fitting other experi- ments:Forlargevaluesof

λ

^{(smallBiotnumbers),thesolutionof}

the1DHT_miLDF model simplifiesto that ofthe 0HT_LDFmodel, asthetemperaturedistributionbecomeshomogeneous.Therefore, andforitslimitedinformationcontent,the0HT_LDFisdisregarded inthefollowing.

Furtherassessmentshowsthatthemodel1DHT_miLDFismore appropriatethanthemodel0DHT_maD_LDFtoplausiblyrepresent the observed behaviour of all measurements. To this end, both models are further analysed looking at two aspects:the relative residual error E(^Pˆ) ^{(Fig. 4, Fig. S2) and the identified parame-} ters Pˆ (Fig. 5). Starting with E(^Pˆ)^{, the overall} best-fitting model is1DHT_miLDF(Fig.S2).Furthermore,withthesamplelayerthick- ness, theresidual error E(^Pˆ) ^{rises for}0DHT_maD_LDF and drops for1DHT_miLDF (Fig. 4). The difference betweenthetwo models liesinwhethertheheatconductionorthemacroporediffusionis the dominant transport mechanism perpendicular to the coating layer.Thisbecomesmorerelevantinthe caseofthickercoatings, indicating that the heat transport discretisation (1DHT_miLDF) is morerelevanttocatchtheoverallbehaviour.

Thesecond aspect,theidentified transportparameters(Fig.5), supports this hypothesis by showing plausible results only for 1DHT_miLDF. Distributed transport parameters like diffusivities andthermalconductivityshouldnot bedependentonthecoating thickness.However,forthe0DHT_maD_LDFmodel,acoatingthick- ness dependency is found forall parameters, whereas it is only plausiblefortheoverallheattransfercoefficienthtodropwithris- ingcoatingthickness;k_LDF andDma shouldbeindependentofthe

Fig. 4. Residual error, i.e. E(P ^ˆ ), for the two best-fitting models over the coating thickness.

thickness, asall samples have the same composition. D_ma is ex- pectedtorisewiththepressure,butitshowsa U-shapedloading dependency, which is not plausible. The heat transfer coefficient isexpectedtodependonthepressureandthegrain-graincontact properties,butit alsoshowsa slightlychaotic U-shape-likeload- ing dependency,whichisnot plausible.Forthe1DHT_LDFmodel, a coating-thickness-independent U-shaped loadingdependency is found forthe microdiffusivity,i.e.the transport diffusivity,which canplausiblybeexplainedbythestrongchangeofthethermody- namic factor, aswe will discusslater. Yet, thetwo heat transfer- relatedparameters,

λ

^{andh,arefluctuating}arbitrarilybyordersof magnitudeforthe thinnersamples.Only fitsto measurements of the thickest sample yielda plausible thermalconductivity inthe orderof0.1W/(mK),independentoftheloading.Inthefollowing, theevaluationwillthusfocusonthe1HT_miLDFmodel.

4.2. Identifiedtransportparameters

Based on the model with discretised heat transfer and LDF- simplifiedmicroporediffusion(1DHT_miLDF),thethermalconduc- tivity

λ

^{of the samples was identified as about 0.07 W/(m K).}

At 40°C, values for k_LDF are between 0.1 and 3 s^–1, leading to an LDF-approximatedmicroporetransportdiffusivityD_mi between 3 ×10⁻¹² and1 ×10⁻¹⁰ m²/switha characteristicallyU-shaped loading dependency (Fig. 8). The heat transfer coefficient h for the contact between coating and support was identified as ≥ 4 ×10³ W/(m²K). An upper boundary forh cannot be given, as it proved to be irrelevant to the overall transport process (non- identifiable).Detailswillbediscussedinthefollowing.

The sensitivity of the individual fit results to h is extremely low (Fig. S3). Forthe thinnersamples,anycombinationof

λ

^and

h resultinginthe sametotal heat transferresistance yields simi- larvaluesoftheerrorfunctional,whereasforthethickersample, onlyhighenoughvaluesofhyieldgoodfits.Thisexplainsthear- bitrary oscillationof

λ

^{andh forCt_240 inFig. 5: Thefitting al-}

gorithm will arbitrarilyreturn anyvaluealong thelong stretched minimum“valley” ofE(^P)^.As

λ

^,handDmi shouldbeequalforall samplesandtoincreasethepoweroftheparameteridentification procedure,thefitprocedurewasappliedtofindasinglesetofpa- rametersfittingwelltomultiplesamples(“multi-fit”).Forthis,ex-

perimentalresultsfromallthreesampleswere pooledforsimilar X,T-statesand fittedto asingleP whileall other specific model parameterswerekept attheprecisestateofeachindividualmea- surementandsample. The contourplotof E(^P) ^{inthe Dˆ}mi plane (Fig. 6) revealsthat values below4 ×10³ W/(m²K) are unlikely forh,butnoupperboundarycanbegiven.

In the following, we will therefore present the model results basedonan ideal(i.e.non-limiting)thermalcontact(i.e.h→∞).

As the values obtained for

λ

^{rise with decreasing h, this ap-}

proach yields conservative estimates for

λ

^{. In the case analysed}

in Fig. 6,this leads to a maximum underestimation of less than 0.02W/(mK)withinthestandardconfidencelevel.

Forthe ideal contactmodel, the formof the confidence limit of D_mi and

λ

^{showsthat the two variables can be} distinguished clearly(Fig.7).Themaximalextentofthestandardconfidencere- gionasdepictedwillbeusedasadefinitionforerrorbarsinthe other plots.Athree-dimensional evaluationofthestandard confi- denceregionispossibleinprinciple butomittedhereforcompu- tationalconstraintsandlimitedinsightgain.

The parameter values identified with the idealcontact model donot showanysignificant dependencyonthecoating thickness (Fig. 8). The thermal conductivity of the coating is found to be constant throughoutall measurement points, withvaluesaround 0.07 W/(m K), which seems plausible if compared to, e.g., the truethermalconductivityof0.14–0.2W/(mK)foundforsilicagels [43,44].It canbeexpectedthat thethermalconductivityislower fortheporouscoatingthanforthepureadsorbent,duetocontact resistances andincreased path lengths. The missing loading and pressure dependency that one might expect, and which was re- portedbeforefor,e.g.,granularbedsofsilicagel/water[45],could be explainedby a heattransfer limitedby binder/adsorbentcon- tactresistancesandconductioninthebinderand/orthegasphase.

The – possibly loading dependent – thermal conductivity of the adsorbent crystalwould just add a low conductive resistance in series. Assuming macropores in the order of the adsorbent par- ticles (d = 18 μm), the inverse Knudsen number under experi- mental conditions is in the order of 5–25, which leads to only minordeviationsfromthe(pressure-independent)gasphasether- mal conductivity [46]. Given that the macropore diffusion is not limitingthe overall dynamics,the thermalconductivity mightbe worth addressing by, e.g., densifying the coating. The results for micro-transportdiffusivityD_mihardlydifferfromthenon-idealised model(Fig.5). Thepronounced loadingdependencyofD_mi might be explained withthe Darkenfactor asdetailedin the following section.

4.3. Interpretationofmicroporemasstransfer

Tentatively,theadsorbatediffusivityD_ad,i.e.theself-diffusivity, maybecalculatedfromtheLDF-approximatedmicroporetransport diffusivityD_mitakingintoaccounttheDarkenfactor _d^dln_lnX^p (alsore- ferredtoasthermodynamicfactor)[47]:

Dad=Dmi

dlnX dlnp

=D_midX dp

p₀

X0. (14)

This correction compensates for most of the loading depen- dency observed for the transport diffusivity D_mi with values for D_ad ofabout1 ×10⁻¹¹m²/sat40°C(Fig.9).Pulsed-fieldgradient nuclearmagneticresonance(PFGNMR)measurementsofwaterin saturatedaluminiumfumarate at25°C yieldedself-diffusivities of D=(⁶.0±0.4) ×10⁻¹⁰ m²/sandD_⊥=(⁵.0±0.4) ×10⁻¹²m²/s forthediffusionin thedirectionofthe1D channelsandperpen- dicular tothem [38],while theoverall diffusion isdominated by D. Our estimate of D_ad directly depends on our assumptions of

Fig. 5. Identified parameters using different models (upper panels: 0DHT_maD_LDF; lower panels: 1DHT_miLDF) over the effective loading for 40 °C and different coating thicknesses (different colours and symbols); the LDF parameter k LDFis expressed as the micropore transport diffusivity D miaccording to Eq. (13) . Lines are only meant as guides to the eye.

Fig. 6. Contour plot of E(^P) over λ and h for the simultaneous fit of measure- ment results of the three samples (Ct140, Ct240, Ct610) under similar conditions ( X 0,eff≈0.16 g/g, T 0 = 40 °C, p 0 ≈20.7 mbar) to the model 1DHT_miLDF with D mi

fixed at the best estimate ( 2 . 6 ×10 ⁻¹²m ²/s). Open circle: minimum found by the fitting algorithm; red cross: minimum found by parameter variation for creating this plot; bold line: standard confidence limit (c.f. Eq. (S27)).

Fig. 7. Contour plot of E(P) for the same fit as in Fig. 6 but over D mi and λ^for model 1DHT_miLDF with ideal thermal contact ( h → ∞ ). Bold line: standard (68%) confidence limit; error bars depict the definition confidence limits used in the other plots; open circle: minimum found by the fitting algorithm; red cross: minimum found by parameter variation for creating this plot.

Fig. 8. Estimated parameters from the model 1DHT_miLDF with ideal thermal contact between support and coating for different coating thicknesses over the effective loading. “multi” (filled symbols) indicates results from simultaneous fit of all samples for similar X, T conditions with non-ideal thermal contact. Lines are only meant as guides to the eye.

Fig. 9. Inverse Darken factor calculated from measured absolute and differential equilibria [18] and adsorbate diffusivity calculated from the identified micro-transport diffusivity ( Fig. 8 ).

the validityofthe LDFapproximation,c.f. Eq.(13),andthe diffu- sion path length. However, a wrongdiffusion path length is not abletoexplain thediscrepancyofabouttwoordersofmagnitude betweenourD_ad andthePFGNMR D: Thediffusionpathlength of the micropore diffusion, i.e. the length from the particle sur- facetoitscentre,wouldneedtobeintheorderof100μm,which is impossiblegiventhemeasured adsorbentparticlesize beingin theorderof18μm(SupplementaryInformationS2).Itseemsmore likelythatderivingamicroporediffusioncoefficientfromk_LDF(LDF approximation) is misleading here. While the LDF model repro- duces the experimental FR results particularly well, k_LDF might have a differentphysical meaning, e.g. a masstransfer barrier of unknown nature.Thisremainsan openquestiontobe furtherin-

vestigated,which,however,doesnot affectthequality ofthe de- scriptionofthemacroscopicprocesses,aswewillshowinthenext section.

Irrespective of the physical interpretation, the data allows an Arrhenius plot of points with similar loading X_eff≈0.15 g/g (Fig. 10).It showsthelinearformofthe inversetemperaturede- pendencyexpected:

lnD_ad=lnD_ad,0− Ea

RT (15)

withtheactivationenergyEa=55.0±3.1kJ/mol.

Fig. 10. Arrhenius plot of the adsorbate (self) diffusion of water in aluminium fu- marate with linear fit and standard error (blue line, grey area) for points with sim- ilar loading ( X eff= 0 . 16 ±0 . 02 kg/kg cmp) of all three coating thicknesses evaluated;

the fit refers to the bottom and left axes.

4.4. ModelvalidationwithLTJresults

The identified transport andthermodynamic parameters were used foranon-linear simulationofan LTJresponse, usingan ex- isting implementation of the 1DHT_miLDF model (Fig. 2f) inthe time domain[22].Theresultsarecomparedtoearlierexperimen- tal results on the same samples [48]. The simulation is based on loading-andtemperature-dependentmicrodiffusivity,constant thermal conductivity (see Section 4.2), constant adsorption en- thalpy andthe fitted adsorption equilibrium as identified before [18].DetailsoftheparametrisationaregivenintheSupplementary InformationS6.Thesimulationissolelybasedonparametersiden- tifiedbyFRA,withoutanyfittinginthetimedomain.

The resultsshowvery goodagreement(Fig.11), validatingthe aforementioned procedures and identification results. The devia- tionbetweenmeasurementandsimulationshowan improvement comparedtopreviousworkswheresimilartransportmodelswere

fitteddirectlyinthetimedomain[22,49].Ourresultsindicatethat thedeviationobservedbeforewasratherduetotheincompletely mappedstatedependencyofthetransfercoefficientsandnotdue to the generalmodel simplificationlike the reduction of particle sizeandlayerthicknessdistributionstosinglevalues.

5. Conclusion

In this work, we developed and validated a method for local (in the thermodynamic state space) measurements of the sorp- tion dynamicsbased onFRA. Combined with linearisedheat and masstransfermodelsintheLaplacedomain,physicaltransportco- efficients can be determined. These coefficients, with their state dependency, allow precise non-linear modelling of application- relevantprocessesinthetimedomain.Asthesemodelsareexplic- itly including geometrical parameters, they may be directly used forthe design andoptimisation of Ad-HX. In particular, a model wasdevelopedthattakesthespatialtemperaturedistributioninto account.

Themeasurementprocedurewasdevelopedfurthersothatthe informationrequiredforthemodelson(a)sorptionequilibriaand (b)sorption enthalpies can be determined in thesame measure- mentsequenceunderexactlythesameconditions.Theonlyquan- tityrequiredfromexternalsourcesistheheatcapacityofthesam- pleforallthermodynamicstatesofinterest.Literatureontheheat capacityofsorptionmaterials,especiallyasafunctionoftheload- ing,is scarceanddirectmeasurements are costly.Thus, its inclu- sioninourmeasurementprocedure shouldbe aimedatinfuture research.

Samplescoatedwithacompositeofaluminiumfumarateparti- cleswithSilres^R asabinderindifferentthicknesses(140–610μm) were examined extensively with this method at30–60°C and in the entire loading range. The relevant transport mechanisms are heattransferbyconductioninthecoating layer(0.07W/mK)and mass transfer into the particles. The latter was successfully de- scribed with an LDF approach. The LDF time constant showed a pronouncedtemperatureandloadingdependency,whichcouldbe describedby interpretingthe masstransfer asa microporediffu- sionprocess. While theactual physicalnature ofthe masstrans- port process at the micro-level remains an open question, the macroscopic dynamicscan be predicted withhigh precision in a large range of operational conditions.The thermal contact resis-

Fig. 11. Response of the surface temperature (left) and the chamber pressure (right) to a large jump of the cold-plate temperature (desorption) measured for sample Ct_610 and simulated with the T, X-dependent transport and equilibrium parameters identified in this work.