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Building and Environment 44 (2009) 2176–2184

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Building and Environment

journal homepage: www.elsevier .com/locate /bui ldenv

Coupled simulation of heat and moisture transport in air and porousmaterials for the assessment of moisture related damage

H.-J. Steeman a,*, M. Van Belleghem a, A. Janssens b, M. De Paepe a

a Department of Flow, Heat and Combustion Mechanics, Ghent UniversitydUGent, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgiumb Department of Architecture and Urbanism, Ghent UniversitydUGent, Jozef Plateaustraat 22, 9000 Gent, Belgium

a r t i c l e i n f o

Article history:Received 25 September 2008Received in revised form4 March 2009Accepted 23 March 2009

Keywords:Computational fluid dynamicsPorous materialsMoisture related damageMicroclimate vitrine

* Corresponding author. Tel.: þ32 9 264 3289; Fax.E-mail address: hendrikjan.steeman@ugent.be (H.

0360-1323/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.buildenv.2009.03.016

a b s t r a c t

This paper describes the coupling of a model for heat and moisture transport in porous materials toa commercial Computational Fluid Dynamics (CFD) package. The combination of CFD and the materialmodel makes it possible to assess the risk of moisture related damage in valuable objects for cases withlarge temperature or humidity gradients in the air. To couple both models the choice was made tointegrate the porous material model into the CFD package. This requires the heat and moisture transportequations in the air and the porous material to be written down in function of the same transportedvariables. Validation with benchmark experiments proved the good functionality of the coupled model. Asimulation study of a microclimate vitrine for paintings shows that phenomena observed in thesevitrines are well predicted by the model and that data generated by the model provides additionalinsights in the physical mechanisms behind these phenomena.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The conservation of culturally or historically valuable objectsposes a complex problem. To preserve these objects they should bemaintained in an environment as stable as possible. As many ofthese artworks are made of organic materials (e.g. wood, paper)which swell/shrink with increasing/decreasing moisture content,especially fluctuations in relative humidity should be avoided asthey lead to drying or wetting of the objects [1]. However, therequirement of a stable environment conflicts with the wish toexhibit the objects, thereby exposing them to an indoor climatewhich might not be accurately controlled under all circumstances.Hence, when valuable objects are exhibited an assessment of therisk of moisture related damage should be made.

Using numerical models for heat and moisture transfer inporous materials the temperature and relative humidity distribu-tions inside the objects can be predicted, which allows for theassessment of the risk of moisture related damage. However, toaccurately predict the distributions inside the material the correctboundary conditions at the air–material interface need to beprovided to the material model. Due to the effect of stratification,draft, etc. strong distributions can occur in the air which may resultin local microclimates in the vicinity of the objects of interest [2].

: þ32 9 264 3575.-J. Steeman).

All rights reserved.

Hence the effect of distributions in the air has to be taken intoaccount in the assessment.

Computational Fluid Dynamics (CFD) is able to accuratelypredict distributions of velocity, temperature and humidity in theair under known boundary conditions. If the interest of a study liesin simulating the local hygrothermal response of building compo-nents or objects facing the indoor air, a strong coupling betweenthe air and the porous materials can occur. Opposed to the outdoorenvironment, which is independent of the situation in the buildingcomponent, the temperature and relative humidity of the indoor airare influenced by thermal and hygric buffering in the buildingcomponents. Hence not only the effect of local air distributions hasto be taken into account in the damage assessment, but also buff-ering in the surrounding walls has to be included as this affects theair temperature and humidity. It is thus not sufficient to passinformation from the CFD model to the hygrothermal model, buta two-way coupling between both models is necessary.

Recently several models which feature a two way couplingbetween CFD and a hygrothermal material model have beendeveloped. These models can be divided in two categories: thedirectly coupled models and the indirectly coupled models. Directlycoupled models are those models which solve both the fluid domainas the porous material domain with one solver. Models which use anindirect coupling solve the fluid and the porous domain witha different solver and exchange information between both solvers.Examples of indirectly coupled models can be found in [3] and [4]. In[3] a 3D CFD model is coupled to a 1D hygrothermal material model,

Nomenclature

C Thermal capacity (J/kgK)D Diffusion coefficient of water vapour in air (m2/s)E Energy content of material (J/m3)g Water vapour diffusion flux (kg/m2 s)I Solar load (W/m2K)k Thermal conductivity (W/mK)Lvap Latent heat of vaporization (J/kg)m Iteration stepRH Relative humidity (–)T Temperature (�C)Y Mass fraction / Specific humidity (kg/kg)v Velocity (m/s)w Moisture content of material (kg/m3)

Greek symbolsa Absorbance (–)

f Open porosity (–)m Vapour resistance factor (–)r Air density (kg/m3)s Transmittance (–)z Reflectance (–)

Subscriptsair Dry airad Main adsorptionde Main desorptionliq Liquid watermat Dry porous materialvap Water vapour

Superscriptst Timem Iteration step

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–2184 2177

while in [4] a 2D CFD and a 2D hygrothermal material model arecoupled. An example of a direct coupled model can be found in [5]. Inthis model a 3D steady state CFD solver was adapted to include thegoverning equations for heat and moisture transfer in porousmaterials. The transport equations in the air and the porous materialwere separated and an algorithm was developed that predicts themoisture flux at the air-material interface and reconciles the watervapour content at both sides of this interface.

It is the aim of this paper to develop a coupled CFD–materialmodel capable of modelling the full complexity of the heat andmoisture transfer in the air, in the porous material and at theinterface. Such a model is very useful for the assessment of moistureinduced damage related to local indoor microclimates. The modelwill need to be three dimensional, able to simulate transient effectsand capable of taking the effect of hysteresis in the moisture sorp-tion/desorption process into account. To our knowledge, none of thecurrent available models complies with all these demands in boththe air and the porous material. As the new model is intended for theprediction of damage due to fluctuations in the indoor climate it issufficient to model moisture transport in the porous material asapparent water vapour transport (liquid transport is only dominantwhen the material is nearly saturated, i.e. when it is wet).

The newly developed model will be validated using an experi-ment designed for the benchmarking of Heat, Air Moisture models(HAM models), i.e. hygrothermal models for porous materials [6].Next the model will be used to simulate the hygrothermal responseof the air and wood in a microclimate vitrine for paintings [7]. Thisis a challenging simulation study as there is a very strong interac-tion between the velocity, temperature and relative humidity ofthe air inside the vitrine and the hygrothermal response of thepainting. In addition to this, this particular case is one where theassessment of damage is of vital importance.

2. Model

The new model has to be able to simulate air flows which arestrongly influenced (and sometimes even driven by) heat andmoisture transport in the porous materials. Therefore the choicewas made to directly couple the CFD code and the hygrothermalmaterial model. The strategy followed to develop the directlycoupled CFD–material model is integrating the governing equa-tions for heat and moisture transport in hygroscopic porousmaterials into the commercial CFD solver Fluent�. The advantage ofusing a commercial CFD solver is that new advances in fluid

modelling are integrated in the CFD solver with every new release.In the CFD solver transport equations for heat and moisture transferin fluids are available. By writing the governing equations for heatand moisture transport in porous materials as function of the sametransported variables as those used in the CFD solver, source terms,diffusion terms, convection terms and unsteady terms are obtainedwhich allow to convert the standard CFD heat and moisturetransport equations into transport equations for porous media. TheCFD solver is used to solve the transport equations in the entiredomain (fluid and porous zones) taking into account the changedequations in the porous zone.

2.1. Heat and moisture transfer in air

The air can be modelled as an incompressible fluid in which casethe species (moisture) and energy transport equations can besimplified to:

v

vtðrYÞ þ V:ðr v!YÞ ¼ V:ðrDVðYÞÞ ¼ �V: g! (1)

v

vtðrCTÞ þ V:ð v!rCTÞ ¼ V:

�kVðTÞ �

�Cvap � Cair

�g!T�

(2)

with

C ¼ YCvap þ ð1� YÞCair (3)

where r is the density of humid air, k the thermal conductivity, Cvap

the thermal capacity of the water vapour, g the water vapourdiffusion flux and D the diffusion coefficient of water vapour in air.The transported variables are the temperature T and the massfraction of water vapour in the air (or specific humidity), Y. Threedifferent terms can be distinguished in the transport equations(Eqs. (1), (2)): the first term in the left hand side is the storage termand the second term represents the convective transport; the righthand side represents the transport by diffusion.

2.2. Heat and moisture transfer in porous materials

2.2.1. Governing equationsThe modelled materials are capillary active and are character-

ized by a sorption curve giving the moisture content w as functionof the relative humidity. As the model is intended for use in thehygroscopic range (RH < 98%), moisture transport in the porous

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–21842178

material can be modelled as equivalent water vapour diffusion: inthe hygroscopic range water vapour diffusion is the dominantmoisture transfer process and the small contribution of liquidtransfer can be accounted for by defining a water vapour perme-ability which depends on the relative humidity and therebyincludes the liquid transfer. Heat and moisture transfer in theporous material are modelled under the assumptions that:

� No air transfer occurs� Liquid transfer is not dominant (no continuous water phase in

the pore system)� Moisture storage only depends on relative humidity� The temperature remains below the boiling point� There is no radiative transfer inside the porous material

Under these assumptions the moisture transport equation andthe heat transport equation can be written in function of Y and T inEqs. (4) and (5)

dwdt¼ �V: g!5

vwvRH

vRHvY

vYvtþ vw

vRHvRHvT

vTvt¼ V:

�r

Dm

VðYÞ�

(4)

dEdt¼ V:

�kVðTÞ �

��Cvap � Cair

�T þ Lvap

�g!�5

rmatCvTvtþ CliqT

vwliq

vtþ�CvapT þ Lvap

�vwvap

vt

¼ V:�kVðTÞ �

��Cvap � Cair

�T þ Lvap

�g!�

(5)

with

E ¼ rmatCmatT þ CliqwliqT þ�CvapT þ Lvap

�wvap (6)

C ¼ Cmat þCliqwliq

rmatþ Cvapwvap

rmat(7)

wliq ¼f� w

rvap

1rliq� 1

rvap

(8)

wvap ¼w

rliq� f

1rliq� 1

rvap

(9)

where Lvap is the latent heat of vaporization, D is the diffusion coeffi-cient of water vapour in air, m is the vapour resistance factor, thesubscript mat refers to dry material conditions, f is the open porosity ofthe material and the right hand side of Eq. (4) gives the moisturetransfer represented as equivalent water vapour diffusion. To solve Eqs.(4) and (5) the following properties of the porous material have to beknown: the sorption isotherm which states the relation between theequilibrium moisture content (w) and the relative humidity (RH), thevapour resistance factor (m) and thermal conductivity (k) as function ofthe relative humidity, the density (rmat) and heat capacity (Cmat) of thedry material and the open porosity (f). As the solution is not verysensitive to the value of the open porosity, it is sufficient to use a real-istic estimation of this value as input.

Contrary to the heat transport equation in air, Eq. (5) featuresthe latent heat of vaporization. The contribution of latent heat wascancelled out of the heat transport equation in air as there is nocondensation present. The complete heat transfer equation can berestored by multiplying Eq. (1) with Lvap and adding this to Eq. (2).In the porous material water vapour condenses in the pores due tocapillary action, hence the latent heat does not cancel in the heattransport equation.

2.2.2. Conservative implementationBecause of the non-linear nature of the transport equations (Eqs.

(4), (5)) mass and energy conservation is not guaranteed: e.g. theunsteady term vw=vRH,vRH=Y varies with Yand is not constant duringa time step. To solve this problem Janssen et al. proposed an iterativesolution procedure in which the property that has to be conserved isestimated by a truncated Taylor series [8]. This approach proved to bevery effective and is implemented here. Together with the segregatedsolution procedure of the CFD solver, this leads to the following dis-cretization of the unsteady terms in Eqs. (4) and (5):

�vw

vRHvRHvY

�tþDt;mYtþDt;mþ1 � YtþDt;m

DtþwtþDt;m �wt

Dt(10)

rmatCtþDt;mTtþDt;mþ1 � TtþDt;m

Dtþ CliqTtþDt;m

wtþDt;mþ1liq �wtþDt;m

liq

Dt

þ�

CvapTtþDt;m þ L�wtþDt;mþ1

vap �wtþDt;mvap

Dtþ EtþDt;m � Et

Dt(11)

Thanks to the truncated Taylor series the non-linear factor is movedfrom the time step level (t) to the iteration level (m) in Eqs. (10) and(11). After every iteration the conserved property (w or E) is eval-uated from the new value of the transported variable (Y or T) and asconvergence is reached the non-linear term will eventually cancel(iteration m þ 1 and iteration m will result in the same value).

Although Eqs. (4) and (5) are coupled (both transported vari-ables occur in both transport equations), the segregated solversolves each equation separately to one single transport variable anditerates between both equations. More specifically, the moisturetransport equation is solved to the mass fraction Y and the heattransport equation is solved to the temperature T. This is clearlyvisible for the discretized unsteady term of the moisture transportequation (Eq.(10)): only Y is solved for the new iteration step m þ 1.As in the iteration scheme the moisture transport equation is solvedbefore the heat transport equation, the choice was made to includethe new information on the moisture transfer in the discretizedheat transfer equation. This explains why the liquid and vapourmoisture content can already be evaluated at iteration m þ 1 in Eq.(11). The solution method can thus be considered as ‘semi-coupled’.

2.3. Hysteresis

The equations and solution strategy for heat and moisturetransfer in porous materials put forward in the previous paragraphsrequire the knowledge of the relation between the moisturecontent w and the relative humidity RH. For some materials this isa unique relation called the sorption isotherm. However for manymaterials (e.g. wood) hysteresis occurs in the sorption process,which means that the relation between moisture content andrelative humidity can no longer be expressed by a unique rela-tionship and a model for hysteresis has to be included when solvingthe transport equations. A popular explanation for the occurrenceof hysteresis during sorption in porous materials is the so called ‘inkbottle effect’. This theory states that during desorption small poressurrounding a larger pore can block the water inside the large poreand prevent evaporation from this large pore. The water can only beevacuated when the relative humidity has dropped enough toevaporate the water in the surrounding small pores: the large porehence acts as a small pore during desorption. In case of adsorptionstarting from dry conditions all pores are initially open and thewater will first condensate in the small pores due to capillaryforces. Condensation in the large pores will only occur at higherrelative humidity, as is expected based on their pore diameter. Thisdifferent behaviour during adsorption and desorption is what

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–2184 2179

causes the hysteretic effect. Besides the ink bottle effect, otherphenomena such as contact angle hysteresis and dynamic effectscontribute to the hysteretic behaviour of the porous material.

To take hysteresis into account a model based on the ‘ink bottleeffect’ was included in the CFD solver. This model was originallydeveloped by Mualem [9] and later simplified by Milly [10]. Themodel only requires the main adsorption isotherm (adsorptionfrom dry conditions) and main desorption isotherm (desorptionfrom saturated conditions) as input. By keeping track of theconditions (relative humidity and moisture content) at the lastswitch between adsorption and desorption the information presentin the main sorption isotherms can be used to reconstruct the effectof hysteresis on the moisture content in the material. A moredetailed explanation of the hysteresis model can be found in [9,10].

Hysteresis in the sorption process not only affects the relationbetween moisture content and relative humidity, also the relationbetween the vapour resistance factor (m)and relative humidity isaltered. This can easily be understood when one considers that theapparent (or phenomenological) vapour permeability depends onthe amount of pores filled with water (these pores representshortcuts for moisture transfer): for the same relative humidity thisamount is different in adsorption and desorption due to hystereticeffects. By relating the vapour resistance factor to moisture contentinstead of relative humidity when using the hysteresis model dis-cussed in the previous paragraph, the variations in vapourpermeability are linked to hysteresis effects in the moisture storageand are modelled in a physically more concise manner. Thisapproach is followed in this paper.

3. Validation study

In the validation study it is checked whether the hygric andthermal interaction between the air flow and the porous materialsimulated with the newly developed model agree with reality. Tothis end an experiment for the benchmarking of 1D transient heatand moisture transfer models for hygroscopic materials [11,12] issimulated. This particular benchmark experiment is well suited forthe validation of the new model as the temperature, humidity andvelocity of the airflow above the material are accurately controlled.This makes it possible to model the heat and moisture transport inboth the air flow and the porous material.

3.1. Experiment

The experimental set up is elaborately described in [11]. The testcase simulated in this paper is not the case discussed in [12], but isa test case developed within the frame of IEA Annex 41 [6]. Themost important characteristics of the set up and test case are brieflydiscussed here and the test section is depicted in Fig. 1. During theexperiment conditioned air is supplied through a duct which

Fig. 1. Overview of the test setup of the benchmark experiment. Th

passes over a porous specimen. This test specimen is placed in animpermeable container with adiabatic walls. A step change in thehumidity of the conditioned air is imposed and the resultingtemperature and relative humidity change inside the porousspecimen are measured. The cross section of the duct has a heightof 20.5 mm and a width of 298 mm. The porous specimen hasa height of 37.5 mm and a length of 498 mm. Temperature andrelative humidity sensors are placed inside the porous specimen ata depth of 12.5 mm and 25 mm.

The porous material used in the validation experiment isgypsum board. Three different experiments were carried out: theresponse of uncoated gypsum board (Test1), gypsum board coatedwith 0.1 mm acrylic paint (Test2) and gypsum board coated with0.1 mm latex paint (Test3) were measured in the test set-up. Thematerial properties (sorption isotherm and vapour resistancefactor) of the gypsum board, acrylic paint and latex paint weremeasured in [6] and were used as input for the numerical model.The average velocity of the airflow in the duct is 0.82 m/s whichcorresponds with a Re number of 2000. The test conditions for thethree different validation cases are given in Table 1. In all threetests the relative humidity of the air flow is high during the first24 h. Next a step change is imposed to the air resulting in a lowrelative humidity.

3.2. Model settings

To model the material properties measured in [6] as accuratelyas possible different kind of functions are used for the sorptionisotherms and vapour permeability curves of the gypsum board,acrylic paint and latex paint. For the gypsum board the followingfunctions are used:

wad ¼ RH�0:81655RH2þ0:85157RHþ0:011176

wde ¼ 13:91382�

1� lnðRHÞ0:079139

�� 11:944272

(12)

m ¼ �0:0088w3 þ 0:244w2 � 2:3558wþ 13:213 (13)

The acrylic paint is modelled as:

w ¼ 2325:6RH5 � 4778:1RH4 þ 3644:3RH3 � 1231:3RH2

þ 194:16RHþ 17:074

(14)

m ¼ �10122RH5 þ 29974RH4 � 26971RH3 þ 5947:4RH2

� 1628:5RHþ 3112:9

(15)

and the latex paint as:

e red dotted line indicates the section modelled in this paper.

0 4 8 12 16 20 24 28 32 36 40 44 4830

35

40

45

50

55

60

65

70

75a

RH

(%

)

time (h)

0 4 8 12 16 20 24 28 32 36 40 44 4830

35

40

45

50

55

60

65

70

75b

RH

(%

)

time (h)

60

65

70

75c

Table 1Test conditions for the validation cases.

Test Used material Initial conditions Airflow conditions

T (�C) RH (%) T (�C) RH (%)

1 Uncoated gypsum 23.3 30 23.8 71.922.5 29.6

2 Acrylic coated gypsum 24 34.6 23.2 72.223.2 30.8

3 Latex coated gypsum 24.1 31.4 23.4 70.923.4 31.2

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–21842180

w ¼ 671:8RH5 � 1481:9RH4 þ 1241:4RH3 � 492:92RH2

þ 121:8RHþ 21:899 (16)

m ¼ �522699RH5 þ 1407017RH4 � 1241740RH3

þ 371286RH2 � 51661RHþ 38379 (17)

Hysteresis is thus only considered for the gypsum board. Theporosity of the gypsum board is known to be 0.419. The porosity ofthe acrylic and latex paint is unknown and is estimated to have thesame value as the gypsum board. The thermal conductivity of theporous material is taken constant. This results in the followingthermal properties for the gypsum board: rmat ¼ 690 kg/m3,Cmat ¼ 840 J/kgK, k ¼ 0.198 W/mK; for the acrylic paint:rmat ¼ 2285 kg/m3, Cmat ¼ 1470 J/kgK, k ¼ 0.5 W/mK; and for thelatex paint: rmat ¼ 1950 kg/m3, Cmat ¼ 1470 J/kgK, k ¼ 0.5 W/mK.

The air flow in the duct is assumed to be laminar. This is inagreement with measurements performed by Iskra [13]. The timestep used in the transient simulation is 60 s. The effect of the timestep size is evaluated by performing a simulation with a time step of30 s. No appreciable effect was found.

The computational grid used is a 2D structured grid counting33,800 rectangular cells. The grid is dense near the air–materialinterface and gradually coarsens towards the bottom of the porousmaterial and the centre of the duct. For the two test cases where theporous material is coated with a paint layer three cells are locatedinside the thickness of the paint. The grid dependency of thesimulations is investigated by performing Richardson extrapola-tion: the original grid is refined with a factor 2 and a factor 4 in boththe X as the Y direction and the mass flow through the interface iscalculated for each grid. Using the Richardson extrapolation theexact mass flow can be calculated out of the different mass flows forthe different grids. As the difference between the exact value andthe value for the original grid was smaller than 1% it can beconcluded that the simulations are grid independent.

A second order upwind scheme is used for the discretization ofthe convective terms in the transport equations in order to reducenumerical diffusion. The SIMPLE algorithm is used for the pressure–velocity coupling. A double precision representation of realnumbers is used to reduce round-off errors.

0 4 8 12 16 20 24 28 32 36 40 44 4830

35

40

45

50

55

RH

(%

)

time (h)

Fig. 2. Comparison of the measured (,) and simulated evolution of the relativehumidity in the test specimen with (blue dashed line) and without (red solid line)hysteresis model for Test1 (a), Test2 (b) and Test3 (c) at x ¼ 0.4 m and a depth of12.5 mm.

3.3. Results

In Fig. 2, the evolution of the relative humidity at a depth of1.25 cm inside the porous material is compared for the numericalmodel and the experiments. Results are given for the simulationswith and without hysteresis model. The effect of the hysteresismodel is only visible during the desorption phase: during the initialadsorption the main adsorption isotherm is followed which is alsoused as input for the model without hysteresis. A good agreement isfound between experiments and simulations. The use of thehysteresis model also results in a better prediction of the sorption

process. By comparing Fig. 2a, b and c it can be seen that thehumidity rise in the gypsum board decreases as the coatingbecomes more vapour tight. This phenomenon is accurately pre-dicted by the numerical model. The validation cases hence showthat, taking into account the uncertainty of the experiment of 2%RH (the uncertainty only related to the RH measurement), the newmodel is capable of accurately simulating the hygric response ofa porous material to a change in the properties of the air flowingover it.

In Fig. 3, the temperature evolution at a depth of 1.25 cm insidethe porous material is compared for the numerical model and theexperiments. Contrary to the response of the relative humidity, thesimulated temperature variations are almost independent from theeffects of hysteresis. The agreement found between simulationsand experiments (uncertainty of 0.1 �C) is good, especially whenone considers the uncertainty on the imposed boundary conditions.As an example of this uncertainty the wiggles in the experimentalmeasured temperature of Fig. 3b can be mentioned.

0 4 8 12 16 20 24 28 32 36 40 44 4821

22

23

24

25a

T (°C

)

time (h)

0 4 8 12 16 20 24 28 32 36 40 44 4821

22

23

24

25b

T (°C

)

time (h)

0 4 8 12 16 20 24 28 32 36 40 44 4821

22

23

24

25c

T (°C

)

time (h)

Fig. 3. Comparison of the measured (-) and simulated evolution of the temperaturein the test specimen with (blue dashed line) and without (red solid line) hysteresismodel for Test1 (a), Test2 (b) and Test3 (c) at x ¼ 0.4 m and a depth of 12.5 mm.

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–2184 2181

The outcome of the simulations closely agrees with the resultsof the different models benchmarked in [6] for both temperatureand relative humidity. The sensitivity of the results to modelsettings such as time step and computational grid was checked inthe previous section. The sensitivity of the simulation results to thematerial properties (sorption isotherm and water vapour perme-ability) was tested in [6]. It was found that the uncertainty of themeasured material properties had an important effect on thesimulation results. Hence accurate measurements of materialproperties are required as input for the model.

Fig. 4. Detail of the microclimate vitrine: horizontal cross section.

4. Simulation study: Microclimate vitrine for paintings

4.1. Background

As an example of the possibilities offered by the newly devel-oped model, a case study is presented in which a microclimatevitrine for paintings is modelled. Such a vitrine not only consists ofa protective glass in front of the painting, protecting it against UV

radiation and contact with visitors, it also forms a vapour tightenclosure around the painting. As a result outside absolutehumidity fluctuations will no longer affect the moisture balance ofthe artwork. Yet heat transfer into the vitrine, due to outsidetemperature fluctuations or incident radiation, could still result inrelative humidity fluctuations inside the vitrine. However, if theenclosed air volume is small enough and free of leakage, a release oruptake of a small amount of moisture by the hygroscopic materialsinside the vitrine (e.g. wooden panel painting) would be sufficientto maintain the relative humidity at its original value. This designwould hence protect the painting against large fluctuations inrelative humidity and the associated shrinking and swelling.

The operating principle of the vitrine is based on the assumptionthat due to the moisture buffering capacity of the hygroscopicmaterial in the vitrine and the small enclosed air volume, therelative humidity in the material can be kept stabile irrespective ofthe temperature fluctuations. It can however be suspected that dueto temperature distributions in the enclosed air the relativehumidity will vary locally inside the vitrine. Classical HAM modelscan only predict the average behaviour of the air and porousmaterial in the vitrine providing that the heat and mass transfercoefficients are known. Using the newly developed model local 3Deffects can be simulated without prior knowledge of the transfercoefficients, which is an advantage when performing damageanalysis. In this paper the effect of incident radiation and changes inambient temperature on the vitrine’s capability of maintaining theoriginal relative humidity and avoiding local fluctuations aroundthe artwork will be simulated with the new model.

4.2. Studied case

The type of microclimate vitrine that will be studied is thedesign proposed by Sozzani [7]. Typical for this design is thatthe picture’s frame is used as the vitrine body, which reduces themanufacturing cost of the vitrine. The use of this type of microcli-mate vitrine is recommended by the Netherlands Institute forCultural Heritage (ICN) when exhibiting valuable paintings in anuncontrolled environment [14]. Fig. 4 gives a detail of the crosssection of the vitrine. Bear in mind that the supports are onlylocated at the four corners of the picture.

In the test case studied in this paper a wooden panel painting of53 cm � 77 cm � 2 cm is protected by a microclimate vitrine. The4 mm thick glass of the vitrine is located at a distance of 5 mm fromthe painting surface and the 1 cm thick plastic back plate is locatedat 2 cm from the back of the painting. The picture frame hasa thickness of 1 cm and is mounted in such a way that an 8 mm gapexists between picture and frame. At the front side of the vitrine thepicture frame covers a strip of 2 cm at the edges of the glass. The

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–21842182

type of wood used for as well picture as frame is pine. In all thedifferent scenarios that will be simulated the initial temperatureand relative humidity in the vitrine are respectively 20 �C and 50%RH. The exterior surface of the vitrine is assumed to be imperme-able for moisture transport and as boundary condition for heattransfer at the exterior surface a convective transfer coefficient of3 W/m2K is assumed with a fixed ambient temperature of 20 �C atall sides (unless mentioned otherwise in the results).

As the air velocity inside the vitrine will be low, thermal radi-ation will play an important role in the heat balance of thevitrine: longwave radiation emitted by the vitrine’s interiorsurfaces contributes to the redistribution of the interior tempera-ture in the vitrine. A surface-to-surface radiation model is used inthis paper to model the longwave radiation. In the surface-to-surface model it is assumed that the absorption of radiation in aircan be neglected and that the material surfaces are not transparentfor longwave radiation. The radiative heat flux leaving a surface iscalculated using view factors.

50

X

Y

-0.2 0 0.2

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4a

50

X

Y

-0.2 0 0.2

-0.3

-0.2

-0.1

0

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0.2

0.3

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44

45

45

46

4748

X

Y

-0.2 0 0.2

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-0.2

-0.1

0

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5152

54 53

X

Y

-0.2 0 0.2

-0.3

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0

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0.4d

Fig. 5. Relative humidity at the painting surface after a heat load of 8 h: (a) frontsurface and (b) back surface in case of changing ambient temperature; (c) front surfaceand (d) back surface in case of incident radiation.

4.3. Model settings

To model the material properties of pine, data from an IEAreport was used [15]. The moisture content and vapour resistancefactor are modelled as:

w ¼ 100�

1� lnðRHÞ0:642

�� 10:64

(18)

m ¼ 1

0:01679þ 0:2217RH8:65 (19)

other material properties used are: for pine rmat ¼ 400 kg/m3,Cmat ¼ 1880 J/kgK, k ¼ 0.11 W/mK, f ¼ 0.6 and the absorbancea ¼ 0.9 for shortwave radiation and a ¼ 0.85 for long wave radia-tion; for plastic rmat ¼ 1200 kg/m3, Cmat ¼ 1214.2 J/kgK, k ¼ 0.2 W/mK and the longwave absorbance a ¼ 0.88; for glassrmat ¼ 2600 kg/m3, Cmat ¼ 840 J/kgK, k ¼ 0.917 W/mK, shortwavetransmittance s ¼ 0.83 reflectance z ¼ 0.08 and absorbancea ¼ 0.09, the longwave absorbance is a ¼ 0.92. In case of longwaveradiation the transmittance for all considered materials is zero andthe emissivity is equal to the absorbance. Note that no desorptiondata is given. Due to the specific nature of the studied case therelative humidity variations inside the pine are small and the effectof hysteresis is assumed to be very limited.

The air flow inside the vitrine is assumed to be laminar. Theaspect ratio (height/thickness) of the cavity is 157.2 in front of thepainting and 39.3 behind the painting which means that a criticalRayleigh number exists above which the flow enters a turbulenttransition regime [16,17]. As for the studied case this critical Ray-leigh number is not reached, the laminar assumption is valid. Tocapture the evolution of the temperature and humidity in the airand painting transient simulations are performed with a time stepof 60 s. The effect of the time step is investigated by comparing thesimulation results with a control simulation with a time step of30 s. The magnitude of the time step proved to have a negligibleeffect. The grid used in the simulations is a 3D structured gridcounting 125,664 elements for the vitrine with frame (68,000without frame). The grid is fine at the interfaces between fluid andporous materials and gradually coarsens towards the centre of thedifferent volumes. The grid dependence was checked by refiningthe original grid with a factor 2. Most of the temperature differ-ences between the two grids are smaller than 0.1 �C and themaximum difference found is limited to 0.3 �C, hence the originalgrid proved to give sufficiently accurate results. Like in the

validation cases second order upwind discretization and doubleprecision numbers are used, yet now PISO is used for the pressurevelocity coupling. Unlike the validation cases the modelled controlvolume is closed for fluid flow across its boundaries. This meansthat the operating pressure can vary inside the computationaldomain (e.g. due to temperature increase in the domain). This effectis taken into account by using a variable operating pressure in thecalculation of density with the incompressible ideal gas law.

4.4. Results and discussion

The main issue investigated in this paper is how well themicroclimate vitrine fulfils its function to stabilize fluctuations inrelative humidity. To this end the response of the Sozzani micro-climate vitrine is simulated for two different heat loads: in the firstcase a step change from 20 �C to 30 �C is imposed to the ambient(outside) temperature and in the second case the ambienttemperature is kept constant at 20 �C while the vitrine is subjectedto 40 W/m2 of incident radiation on the glass. Part of this radiationreflects on the glass, part is absorbed in the glass and the remainderis transmitted to the painting in accordance with the reflectance,emissivity and transmittance of the glass as defined previously.

Fig. 5 gives the relative humidity distribution at the front andback surface of the painting after a heat load of 8 h for both thecase where the ambient temperature is raised and for the casewith incident radiation. For the case with raised ambienttemperature, the deviations from the initial relative humidity of

0 1 2 3 4 5 6 7 845

46

47

48

49

50

51

52

53

54

55

RH

(%

)

time (h)

Fig. 6. Evolution of the average relative humidity in the enclosed air of the vitrine incase of uniformly changing ambient temperature (red solid line) and in case of incidentradiation (blue dashed line).

Z

Y

-0.04-0.02 0

-0.4

-0.2

0

3

24

24

23

25

2627

28

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Y

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56

5050

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52

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54

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2

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0.4b

50

60

56

0.2

0.4c

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–2184 2183

50% are smaller than 1% RH. For the case with incident radiationrelative humidity values between 43.8% and 58% are found.Considering that during these 8 h the temperature increased from20 �C to 30 �C for the case with increased ambient temperatureand to a maximum value of 28.5 �C at the front surface for the casewith incident radiation, the stabilizing effect is excellent: in casethere would be no moisture buffering effect (thus constant vapourpressure in vitrine) a temperature increase from 20 �C, 50% RH to30 �C would cause a drop in relative humidity to 27.5% RH.However, it is obvious that in case of incident radiation on the

0 1 2 3 4 5 6 7 845

46

47

48

49

50

51

52

53

54

55

RH

(%

)

time (h)

Fig. 7. Evolution of the average relative humidity in the enclosed air of the vitrine incase of uniformly changing ambient temperature (red solid line), in case of incidentradiation (blue dashed line) and in case of a non-uniform change in ambienttemperature (green dotted line).

plastic painting glass

Fig. 8. Situation after 8 h of incident radiation at the central, vertical plane in thevitrine: (a) velocity field, (b) Temperature field and (c) relative humidity field.

vitrine, the painting experiences less favourable conditions than incase of varying ambient temperature.

When the average relative humidity in the air is plotted (Fig. 6)for the two studied cases it is found that they act differently. In caseof an increase in ambient temperature the relative humidity in thevitrine initially drops and is subsequently restored to its initialvalue due to the hygroscopic action of the porous materials.However when the temperature rise in the vitrine is caused byincident radiation, the relative humidity in the air increases withincreasing temperature. These findings seem to contradict but canbe explained when analyzing the temperature and humiditydistributions in the vitrine. In case of a rise in ambient temperaturethe air in the vitrine will first heat up at the edges of the vitrinecausing a drop in average relative humidity in the air, next theentire air volume and painting heat up causing a moisture releasefrom the painting and picture frame. As the temperature increase islimited to the ambient temperature, the temperature differences inthe vitrine will eventually vanish and the painting will achieveequilibrium with the entire air volume. In case of incident radiationon the vitrine a completely different situation arises. The radiationwill directly heat up the painting itself, which will give off moistureto the surrounding air. As this air is at elevated temperature

H.-J. Steeman et al. / Building and Environment 44 (2009) 2176–21842184

compared to the air near the vitrine surfaces it will cool down whenmixing with the rest of the air. This results in an increase of theaverage relative humidity in the air. Unlike the case with increasedambient temperature, the temperature differences in the vitrinewill continue to exist as long as radiation hits the vitrine. Thisexplains why the initial relative humidity is not restored for thiscase. Measurements performed by Baan et al. confirm these find-ings: they monitored a microclimate vitrine directly hit by radiationand also found the relative humidity in the air to increase withincreasing temperature [18]. Sozzani, however, reports increasingrelative humidity with increasing ambient temperature ina microclimate vitrine containing hygroscopic materials [7]. Thisseems to contradict with our findings. Yet in the simulations it wasassumed that uniform boundary conditions were present at all theouter surfaces of the vitrine while in the experiment the paintingswere hung against a wall. This can lead to different boundaryconditions at the front side and back side of the painting. To testthis hypothesis a new simulation is performed in which theambient temperature is raised to 30 �C except at the backside of thevitrine where the new ambient temperature is 25 �C. Fig. 7 showsthat due to different temperature boundary conditions the relativehumidity in the vitrine will rise with increasing temperature. As forthe case with incident radiation this relative humidity increase iscaused by the cooling down of warm, humid air transported fromthe vicinity of the painting to the cold back wall. Hence an expla-nation is provided for the behaviour found by Sozzani.

In the studied case featuring a step change in ambient condi-tions the vitrine almost heated up uniformly which resulted in verysmall distributions of temperature and humidity and only a weakair flow. The risk of moisture related damage is obviously very lowfor this case. For the case of a non-uniform step change in ambienttemperature the maximum relative humidity fluctuation inside thepainting was smaller than 3.5% RH. Yet in the case with directradiation larger gradients are found, so this case will be studiedmore in detail. Thanks to the new model it becomes possible to lookinto local, three dimensional effects inside the vitrine. Fig. 8, forinstance, shows the velocity, temperature and relative humiditydistribution in a vertical cross section in the centre of the vitrine,perpendicular to the painting. Warm, humid air rises in front of thepainting and subsequently falls at the backside of the painting nearthe colder back plate. As the falling air cools down, the relativehumidity increases, resulting in a locally increased relativehumidity near the bottom of the backside of the painting: values upto 58% RH are reached.

The simulations indicate that uniform heating of the vitrinedoes not lead to time dependent variations in the relative humidity.However, when non-uniform temperature distributions aregenerated inside the vitrine, spatial and temporal relative humidityvariations can occur. Such non-uniform temperature distributionsare inherently present when the vitrine is heated by radiation. Asthese relative humidity variations are responsible for the degra-dation of the painting, large temperature gradients around thevitrine and direct exposure to radiation should be avoided.

5. Conclusions

In this paper a 3D model for heat and water vapour transport inporous materials was integrated into a commercial CFD package.This new model makes it possible to take the effect of indoor airdistributions into account when simulating the hygric response ofporous objects. Validation of the new model with a benchmarkexperiment for transient heat and moisture transfer in hygroscopicmaterials proved the reliability of the new model. The validationstudy also showed that the use of a hysteresis model leads toa better agreement with the benchmark experiment.

The new model was applied to a problem where the assessmentof moisture related damage is of the highest importance: a micro-climate vitrine for paintings. The model was able to correctlypredict and explain the increase of relative humidity in the vitrinewith increasing temperature, also observed in practice. The simu-lations showed that the microclimate vitrine is an excellentprotection against fluctuations in relative humidity, yet directradiation on the vitrine should be avoided as much as possible.

Summarizing, it can be stated that an approach to combine CFDand hygrothermal material simulation was presented in this paper.This approach proved to be reliable and applicable to problemsencountered in practise.

Acknowledgements

The results presented in this paper have been obtained withinthe frame of the FWO project B/05836/02 funded by the FWO-Flanders (Research Fund Flanders) and of the SBO IWT 050154Project, funded by ‘IWT Vlaanderen’, the Institute for the Promo-tion of Innovation by Science and Technology in Flanders. Thisfinancial support is gratefully acknowledged.

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