Non-Isothermal Kinetics of Water Adsorption in

Compact Adsorbent Layers on a Metal Support

G. Fuldner∗1, L. Schnabel11Fraunhofer Institute Solar Energy Systems, Freiburg∗Corresponding author: Heidenhofstr.2, 79110 Freiburg, gerrit.fueldner@ise.fraunhofer.de

Abstract: Water adsorption in highlyporous materials can be used in heat trans-formation processes for the efficient use ofenergy in heat and cold production [8].To optimize the power density of compactthin layer adsorbent beds, a one-dimensionalmodel of the coupled heat and mass transferlimiting the adsorption kinetics has been setup in COMSOL Multiphysics. Results whichare presented in this paper have been vali-dated and calibrated with measurements.

Keywords: water adsorption kinetics, heattransfer, Knudsen diffusion

1 Introduction

Thermally driven chillers and heat pumpsuse heat from a high temperature heatsource (solar or waste heat, gas burner) toefficiently supply heat at a medium temper-ature level (heat pump) or cold at a low tem-perature level (chiller). One technology forsuch a thermal heat transformation is theuse of water adsorption in highly porous ad-sorbents like zeolite [8]. Adsorbers used insuch a machine are made up of a heat ex-changer fluid loop in contact with an adsor-bent bed. The volume specific power den-sity [W/Lmachine−volume] of such machinesis limited -besides the maximum water up-take of the used adsorbent- by the coupledmass transfer (water vapour) into the ad-sorbent and heat transfer (heat of adsorp-tion) to the fluid loop, see Figure 1 fora schematic drawing. To compare differ-ent new adsorber concepts, small samples(50x50mm2, see Figure 2) representing apart of such an adsorber are characterizedexperimentally at Fraunhofer ISE. Measure-ments are performed in a constant volumeset-up, where the decrease in system pres-sure can be directly converted into water up-take of the adsorbent material. The exper-

imental setup is explained in more detail insection 2.

Figure 1: Coupled heat and mass transfer in acompact adsorber heat exchanger

A one dimensional model of the coupledmass and heat transfer has been set upwithin COMSOL Multiphysics and has beenvalidated with experimental data in order toidentify the limiting factors for the adsorp-tion kinetics and to optimize such sampleswith respect to layer thickness, coupling toheat exchanger or other parameters.

2 Experimental setup

The sample characterised experimentallyconsists of a more or less homogenous layerof zeolite (UOP DDZ 70) with a thickness of0.72mm which is glued onto an aluminumsupport with a thickness of 3 mm. Betweenthis sample and a coldplate kept at constanttemperature by a thermostat, a Captec heatflux sensor is attached with heat conductingpaste. The experimental setup is shown inFigure 2.

Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover

Figure 2: Sample (50x50mm2) with aluminumcarrier on coldplate with temperature and heat

flux sensor.

The sample is desorbed at 95� against vac-uum pump pressure (1 Pa). After coolingdown the sample to 40�, a valve is openedbetween a big dosing chamber containingwater steam at 40� at a pressure of about1700 Pa and the sample chamber; the sam-ple starts to adsorb, releasing the heat ofadsorption, until adsorption equilibrium isreached.

3 Model description andimplementation

Due to symmetry reasons the mathematicalmodel is one-dimensional. It includes dif-fusion of water vapour into the adsorbentlayer, the adsorption process and heat trans-fer by conduction within the adsorbent andmetal layer. Additionally, the following as-sumptions are made:

• Adsorption equilibrium is reached ateach place and time instantaneously,depending on local pressure and tem-perature.

• Water vapour is described as an idealgas (see Eq. (9)).

• Heat transfer by convection inside theopen pore volume is ignored.

• The fluid loop is included only by aconstant temperature boundary.

To comply with the measuring procedure,the time-dependent form of the PDE’s isused since the process is not stationary.

A comprehensive overview on adsorptiontechnology and the mathematical descrip-tion of adsorption kinetics can be found in

[7]. A detailed model description of themodel used in this work is given in [11].Other related material used in the derivationof this model are the publications [1],[2], [3],[5] and [9].

3.1 Mass balance

The equation governing the mass transfer ofwater vapour inside the adsorbent layer canbe derived from a simple mass balance andreads

∂cAd∂t

=∂

∂zD∂cAd∂z

− 1M

ρdryAd

ψ

dXdt

(1)

It includes the process of adsorption as asink term. cAd is the water vapour concen-tration in [mol/m3], D is the diffusion co-efficient [m2/s] , ρdryAd the density of the dryadsorbent in [kgAd/m3] corresponding to thelayer volume, M the molar mass of water in[kg/mol], ψ the adsorbent porosity and Xthe water loading in [kgH2O/kgAd].

The equation in the pure vapour area isa standard Fickian diffusion equation with adiffusion coefficient Dv which is about twoorders of magnitude higher than the diffu-sion coefficient D within the adsorbent. Forcomparison the boundary and initial condi-tions are set equal to those given by the ex-perimental set-up:

• At t = 0, the adsorbent is at a very lowpressure (vacuum pump ending pres-sure 1Pa); a given amount of watervapour is then dosed into the vapourchamber (initial pressure at boundary4, see Figure 3), leading to a jumpin pressure there. To avoid numeri-cal problems caused by too steep gra-dients in pressure and temperature, asmoothed Heaviside function is used asinitial condition as well as for the diffu-sion coefficient to model the concentra-tion front reaching the adsorbent sur-face.

• The boundary condition at boundary3 (see Figure 3) is given by ”concen-tration”, the concentrations cAd andcv being equal at each timestep. Onboundary 4, the boundary condition isgiven by a normal flux which is propor-tional to the change in concentrationon the vapour side.

The exact boundary conditions are given inTable 2.

Figure 3: Geometry and boundaries in COMSOL

3.2 Energy balance

The energy balance within the adsorbent in-cludes the heat of adsorption released in theadsorption process as a heat source,

ρdryAd cp,Ad∂TAd∂t

= λAd∂2TAd∂z2

+ ρdryAd haddXdt(2)

where cp,Ad is the specific, loading depen-dent adsorbent heat capacity [J/(kg K)], λAdthe loading dependent heat conductivity ofthe adsorbent [W/(m K)] and had the spe-cific enthalpy of adsorption [J/kgH2O] whichis calculated according to Eq.7. Within themetal layer, heat transfer takes place by con-duction with the library values for standardaluminum. To avoid numerical problems be-cause of steep temperature gradients in thebeginning, the variable transformation

TAd,log = lnTAd (3)

is used and the differential equation is solvedfor TAd,log. The initial and boundary condi-tions are given by the experimental set-up:

• The initial temperature is for all re-gions the same and corresponds to thecoldplate temperature.

• At boundary 3 ”thermal insulation” isdefined; heat losses by radiation andconvection are ignored.

• At boundary 2, the boundary condi-tion is ”heat flux” with a heat transfercoefficient hAd,m and the two temper-atures TAd and Tm.

• At boundary 1 - due to heat flux mea-surements - a second heat transfer re-sistance exists, therefore ”heat flux”

is defined by the heat transfer coeffi-cient hHF and the temperatures Tmand TCP , which is the constant tem-perature of the cooling fluid loop.

The variable transformation also has to becarried out for the boundary and initial con-dition, so e.g. the thermal conductivity getsan extra factor of exp(TAd,log), with the timescaling coefficient δts = exp(TAd,log) = TAd.The exact formulations are given in Table 2.

3.3 Adsorption equilibrium

Loading equilibrium has been measured forthe modeled material (UOP DDZ 70) in athermogravimetric balance. From this data,a characteristic curve

W (A) =a1 + a2A+ a3A

2 + a4A3

1 + a5A+ a6A2 + a7A3(4)

has been fitted using a generalized form ofDubinin’s theory of volume filling [4, 10],where W = X/ρH2O [cm3/kgAd] is the filledvolume per adsorbent mass (the adsorbatedensity is taken equal to the density of liq-uid water) and

A =RTAdM

ln(psatp

)(5)

=RTAdM

(ln

(psatcAdR

)− TAd,log

)(6)

is called the adsorption potential [J/gH2O],psat being the saturation pressure of waterat temperature TAd. The coefficients ai aregiven in Table 1.

a1 221.16073 a5 -0.002006a2 -0.790530 a6 -2.224e-6a3 0.0014848 a7 1.6494e-8a4 -5.129e-7

Table 1: Coefficients of characteristic curveW (A)

The enthalpy of adsorption can be calculatedusing the adsorption potential:

had = hev +A (7)

where hev is the evaporation enthalpy of wa-ter at temperature TAd. The loading changewith time is given by the total differential

dXdt

=(∂X

∂p

)T

∂p

∂t+

(∂X

∂T

)p

∂T

∂t(8)

Since the characteristic curve serves as a pa-rameterization of the equilibrium data, thepartial derivations of X in the expressionabove can be calculated from it. The rela-tionship between p and cAd comes from the

ideal gas law,

p = cAdRT (9)

with R [J/(mol K)] the general gas constant.The properties of water (ρH2O, psat, hev) cor-respond to the standard data IAPWS-IF97[6].

3.4 Meshing and solversettings

The model has been solved in transient modefor 1200 s with timesteps stored every 0.2 s,so adsorption equilibrium could be reachedeven in the slowest case. The relative toler-ance has been set to 1e − 4. A customizedmesh has been created, being denser in sub-domains 2 and 3 (adsorbent and vapourlayer) and coarser in subdomain 1. Witha number of 352 elements (1118 degrees offreedom), independency of the solution fromthe mesh was sufficient to be far below theerror bar of the measurement.

Mass balance Energy balance

Vapour ∂cv

∂t = ∂∂zDv

∂cv

∂z –

Initialc(0) = c0− (c0− cAd0) ∗HS T (t = 0) = TCP = const.

with HS = flc2hs(7.5e− 4− x, 1e− 5)and Dv = 0.01− (0.01 ∗ flc2hs(3− t, 3))

Boundary 4 : n(Dv∇cv) = − Vtot

ψAr∂cv

∂t and 3 : cv = cAd

Adsorbent∂cAd

∂t = ∂∂zD

∂cAd

∂z − 1M

ρdryAd

ψdXdt δtsρ

dryAd cp,Ad

∂TAd,log

∂t =

λAdTAd∂2TAd,log

∂z2 + ρdryAd haddXdt

Initial cAd(t = 0) = cAd0 = 1Pa TAd,log(t = 0) = lnTCP

Boundary 3 : cAd = cv 2 : n(λAdTAd∇TAd,log) =hAd,m

TAd,logTAd

Metal – ρmcp,m∂Tm

∂t = λm∂2Tm

∂z2

Initial – Tm(t = 0) = TCP

Boundary – 2 : n(λm∇Tm) = hAd,m(TAd − Tm)1 : n(λm∇Tm) = hHF (TCP − Tm)

Table 2: Governing equations, initial and boundary conditions; n is the boundary normal pointing outof the domain

4 Results

Figure 4 shows simulated and experimen-tal pressure within the vapour room. Asdescribed in section 2, the loading X canbe calculated from this pressure drop whenthe volume and the dry adsorbent mass areknown exactly.

Figure 4: Simulated and experimental watervapour pressure in vapour room

At the same time, when adsorption takesplace, the temperature TAd within the adsor-bent will rise due to the release of adsorptionenthalpy hAd. Only when the initial temper-ature is reached again, thermodynamic equi-librium is reached. Figure 5 shows the simu-lated adsorbent layer surface temperature incomparison to measured results (blue dots).

Figure 5: Adsorbent surface temperature

The simulated surface temperature reaches amuch higher maximum, and also at an ear-lier point in time (Figure 6, grey line) thanthe measured temperature. This is due to aheat flux resistance between the surface andthe temperature sensor. Modeling this re-sistance leads to much better agreement be-tween measurement results and simulation(Figure 6, black lines).

Figure 6: Influence of heat flux resistance oftemperature sensor

With the same parameters, also the compar-ison of simulation results with the heat fluxmeasurement (blue dots) between the metalsupport and the coldplate shows good agree-ment (Figure 7).

Figure 7: Simulated and experimental heat fluxthrough boundary 1.

The simulated heat flux is slightly higher,since no losses (e.g. by radiation and con-vection) have been taken into account.

5 Conclusion

A 1-d model of water adsorption kinetics ina compact layer of porous zeolite on a metalcarrier has been implemented in COMSOLMultiphysics and validated by measurementresults. The initial conditions and geome-try have been adjusted to real experimentalconditions, thus avoiding unphysically steepgradients in vapour concentration and ad-sorbent temperature and providing for bet-ter numerical stability. A transformation tothe logarithm of the adsorbent layer tem-perature is used to minimize numerical er-rors which turn up using the non-logarithmicform. Simulation results show that the ad-sorbent surface temperature reached dur-ing the adsorption process is actually higher

than the measured temperature due to thedynamic behaviour of the sensor. Therefore,a different way of measuring the surface tem-perature is proposed (e.g. infrared sensor) tocheck the simulation results.

The model can now be used for param-eter variations to find out possible mass orheat transfer limitations and allowing for op-timization of layer thickness or other param-eters to obtain higher power densities of ad-sorbers in heat transformation applications.Results of this optimization process will bepresented in future publications.

References

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Nomenclature

A adsorption potential [J/gH2O]Ar sample surface area [m2]D pore diff. coeff. [m2/s]Dv self-diff. coeff. [m2/s]M molar mass [kg/mol]R general gas constant [J/(mol K)]TAd adsorbent temperature [K]TAd,log logarithmic temperatureTCP coldplate temperature 313.15 KTm metal temperature [K]Vtot vapour chamber volume [m3]W filled pore volume [cm3/kgAd]X water loading [kgH2O/kgAd]ai coeff. char. curvecAd vapour conc. Ad [kg/mol]cv vapour conc. [kg/mol]cp,Ad ads. spec. heat capacity [J/(kg K)]had spec. ads. enthalpy [J/kg]hev spec. evap. enthalpy [J/kg]hAd,m heat transf. coeff. Ad/m [W/(m2 K)]hHF heat transf. coeff. m/CP [W/(m2 K)]p pressure [Pa]psat water saturation pressure [Pa]δts time scaling coefficient [-]λAd ads. heat conductance [W/(m K)]ρdryAd dens. of dry adsorbent [kg/m3]ρH2O dens. of water [kgH2O/cm3]ψ porosity of adsorbent [-]

Acknowledgements

Funding by PhD scholarship of DBU(Deutsche Bundesstiftung Umwelt) to bothauthors is gratefully acknowledged.