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Konopka, D., Bachtiar, E., Niemz, P., and Kaliske, M. (2017). "Experimental and numerical analysis of moisture transport in walnut and cherry wood in radial and tangential material directions," BioRes. 12(4), 8920-8936.


The diffusion properties of walnut (Juglans regia L.) and cherry (Prunus avium L.) wood perpendicular to the grain are presented. The wet and dry cup tests were carried out in normal climate conditions. Simultaneously, inverse analyses by means of the finite element (FE) method were conducted based on the experimental sorption data. The results show that the numerically derived diffusion model can predict the mass changes during the experiments with sufficient accuracy with a maximum global relative error of 0.43%. However, the numerically determined diffusion coefficients do not show an agreement with the experimental data.

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Experimental and Numerical Analysis of Moisture Transport in Walnut and Cherry Wood in Radial and Tangential Material Directions

Daniel Konopka,a Erik V. Bachtiar,b Peter Niemz,b,c and Michael Kaliske a,*

The diffusion properties of walnut (Juglans regia L.) and cherry (Prunus avium L.) wood perpendicular to the grain are presented. The wet and dry cup tests were carried out in normal climate conditions. Simultaneously, inverse analyses by means of the finite element (FE) method were conducted based on the experimental sorption data. The results show that the numerically derived diffusion model can predict the mass changes during the experiments with sufficient accuracy with a maximum global relative error of 0.43%. However, the numerically determined diffusion coefficients do not show an agreement with the experimental data.

Keywords: Cherry; Walnut; Diffusion coefficient; Moisture dependency; Cup test; Inverse analysis

Contact information: a: Institute for Structural Analysis, Technische Universität Dresden, D-01062 Dresden, Germany; b: Institute for Building Materials, ETH Zurich, 8093 Zürich, Switzerland; c: Institute for Material and Wood Technology, Bern University of Applied Science, 6096 Biel, Switzerland;

* Corresponding author:


Valuable hardwood is often used for decorative artworks and frequently found in wooden cultural heritage objects. Walnut and cherry wood were investigated in this work since these species, compared e.g. to beech, oak or coniferous wood like spruce, have only been scarcely studied with respect to their moisture transport behavior. The knowledge on these wood species provided by this study can be used for any conservation purpose in the future.

As a hygroscopic material, wood adjusts its moisture to the surrounding conditions by the adsorption or desorption and diffusion of water vapor until it reaches an equilibrium moisture content (EMC) with a steady-state distribution. During this period, which requires days to months depending on the wood specimen size, swelling or shrinking will also occur as wood attaches and detaches the water molecules. In applications where wood is exposed to uncontrolled climatic conditions, extreme and repetitive swelling or shrinkage may damage the wood (Lamb 1992).

The moisture transport/diffusion properties of wood have been studied thoroughly (Klopfer 1974; Siau 1995; Sonderegger et al. 2011; Mannes et al. 2012). The diffusion properties of wood are experimentally measured in two ways, i.e., transient and steady-state. The transient/unsteady-state method is based on the water adsorption or desorption rate of wood. The diffusion coefficient (D) is estimated based on the change of mass before the wood specimen reaches an EMC. In contrast, the steady-state method calculates D based on the mass of the moisture being transported through the wood with EMC. Because transient conditions often appear at the beginning within the experimental procedure and the steady-state appears later, they are also referred to as short- and long-term moisture transport conditions, respectively (Sonderegger et al. 2011). The diffusion characteristics change during both states. Differences of approximately one order of magnitude may occur, depending on the wood species, the fiber direction, and the moisture difference (Pfriem 2006). In this study, both, unsteady- and steady-state analyses are performed based on the conventional cup method (DIN EN ISO 12572 2001).

Within an inverse analysis of the transient diffusion process, cup test specific moisture-dependent diffusion coefficients D(m) for the numerical analysis in the scope of the finite element method (FEM) were determined in this study. The inverse analysis, which was introduced by Hrcka and Babiak (1999) and Olek et al. (2005) for the bound water diffusion coefficient identification in wood, was further adapted for special diffusion experiments on certain wood species (Koc and Houska 2002; Koc et al. 2003; Olek et al. 2005; Mannes et al. 2009; Hering 2011; Olek et al. 2011; Sonderegger 2011). With the determined D(m), the experimental test setups can be modelled. The FE-analysis allows insight into the specimens’ time-dependent inner moisture distribution over the length section, i.e., the direction of moisture transport. Moreover, the species-dependent coefficients enable a direct quantitative comparison to other similar cup test diffusion experiments, e.g., different wood species, wood products, or composite materials (Sonderegger and Niemz 2009). The same analysis method was applied for scots pine (Pinus sylvestris L.), beech wood (Fagus sylvatica L.), and Norway spruce (Picea abies L.) (Olek et al. 2005; Mannes et al. 2009; Olek et al. 2011), although the quantitative comparison was restricted due to different geometries, humidity levels, and sorption directions (ad-/desorption).


Materials and Specimens

The measurements were carried out for two hardwood species, walnut (Juglans regia L.) and cherry (Prunus avium L.) wood grown in the Caucasus region. The average wood densities for walnut (ρ = 678 ± 60 kg m-3) and cherry (ρ = 550 ± 23 kg m-3) were measured at normal climatic conditions (temperature = 20 °C, relative humidity RH = 65%). In these conditions, the wood moisture contents were mwalnut = 9 ± 0.2% and mcherry = 10 ± 0.2%. The specimens used for the experimental tests were cylindrical wooden discs with a diameter of 140 mm and a thickness of 10 mm (Fig. 1) without any natural growth characteristics such as knots or reaction wood. Of the six specimens that were prepared for each sample type, three samples were tested in dry conditions and the other three in wet conditions (Table 1).

Fig. 1. Water vapor diffusion specimens in radial (R) and tangential (T) direction

Table 1. Tested Specimens

Fig. 2. Test setup: dry cup container filled with desiccants, wet cup container filled with desalinated water. All dimensions are in mm.

Testing Method

Before the tests, all specimens were conditioned under normal climate conditions (20 °C, 65% RH) for at least two months. The water vapor uptake investigation combined with the steady-state wet and dry cup diffusion tests based on DIN EN ISO 12572 (2001) were performed in the same ambient climate by using the wood specimens as the lids of the cylindrical glass containers. The containers with a height of 120 mm and an outer diameter of 140 mm were filled with either desiccant, i.e., silica gel/SiO2 (dry cup), or desalinated water (wet cup) and an air layer of 15 mm (Fig. 2). Tight rubber seals were used to seal the area between the specimens and their containers. Hence, the water vapor could only transmit (in or out of the containers) through the specimens in one material direction (Sonderegger and Niemz 2009).

All variables used in calculations are defined in Table 2.

As soon as the specimens were placed, their moisture contents slowly adjusted to their new environment (dry on one side and wet on the other side). During this period, the adsorption and desorption rates of the specimens were not equal. The water vapor sorption of the wood specimens (without containers) was observed by a regular mass measurement.

Table 2. List of Symbols

Removal, mass measurement, and re-installation were quickly performed one by one for each specimen. The mass was also used as a parameter to determine whether the specimens had reached equilibrium conditions or not. The next experimental state, i.e., the steady-state water vapor diffusion could only be continued once equilibrium was reached. The unsteady-state/transient diffusion coefficients were calculated based on the ad- and desorption data (Klopfer 1974). The water sorption coefficients of bound water (Abw) were calculated as,

 [kg m-2 s-0.5]. (1)

Therefore, the diffusion coefficients (DUS), which are assumed to be constant at the beginning of the measurements, are calculated as,

 [m2 s-1] . (2)

After 20 days, the specimens reached their EMC. In this state, the permeation of water vapor became a constant process with the three simultaneous main transport phenomena:

  1. adsorption of water vapor from higher humidity environment into the wood,
  2. diffusion of water through the wood,
  3. desorption of water on the opposite surface of the wood to lower humidity environment.

The mass of combined wood specimen and container (whole system) was measured regularly. It is assumed that every mass change of the system was due to the transmission of water in or out of the cup through the wood. The measured data are used to estimate water vapor diffusion properties. This measurement was conducted for approximately 14 days. According to DIN EN ISO 12572 (2001), the diffusion parameters that can be estimated based on the experimental data are water vapor transmission rate per surface area (g),

 [kg m-2 s-1] (3)

water vapor transmission coefficient (Wc) corrected by the air layer between the base of the specimen and the desiccant or water (see Fig. 2),

 [kg m-2 s-1 Pa-1] (4)

water vapor resistance factor (µ),

 [-] (5)

diffusion coefficient (D) based on Siau (1995),

 [m2 s-1] (6)

and water vapor diffusion-equivalent air layer thickness (sd),

 [m]. (7)

Additionally, complemented parameters required in Eqs. 3 to 7 are saturated water vapor pressure (pv)

 [kg m-2 s-1] (8)

and water vapor permeability of air with respect to the partial vapor pressure,

 [kg m-1 s-1 Pa-1]. (9)

Table 3. Sorption Isotherm Data

Inverse Analysis of the Diffusion Coefficients

Within the scope of numerical FE-analyses, the diffusion parameters are modelled as a function of moisture. A single discrete material point at a certain time relates to a certain moisture and a diffusion coefficient. In contrast, the experimentally determined steady- and unsteady-state diffusion coefficients only estimate the average values over the entire specimens. Therefore, by integrating the numerically estimated diffusion parameters over the thickness at a certain time, both experimental and numerical are comparable.

Material Model

Fick’s second law for the description of the moisture transport in solid materials [see e.g. (Crank 1975; Hanhijarvi 1997)] is used assuming a single-phase diffusion. The assumption of Fick’s law is valid in the simulation of steady-state transport processes. The transient form, however, often leads to divergence of simulation and experiment, denoted as “non-Fick’ian-behavior” (Wadso 1994). Moisture flux (qm) is proportional to the gradient of moisture content (m), the diffusion coefficient (D), and the density in absolute dry conditions (ρ0) serve as constants of proportionality

 . (10)

The time-dependent form of Fick’s law,


for transient simulations merely gives an approximation of the real transport behavior.

For the internal transport, different approaches in modelling the diffusion coefficient (D) were tested for the configuration WT dry (cf. Table 1). The steady-state values determined in the Section “Testing Method” were compared to constant, linear, and the following exponential moisture-dependent approach

 [mm2 s-1], (12)

with moisture content m [-]. As analyzed in Hering (2011), a more complex polynomial exponent is not suggested. The two optimization parameters D0 and α are considered for the exponential formulation. In the linear approach, two variables are necessary, too, whereas in the constant approach only D0 is captured (see Section “Results and Discussion”). Besides the optimized configurations WT1 to WT3, further models were tested. In configurations WT4 and WT5, the constant D of the experimental result was combined with different final equilibrium moisture contents at the inner cup surface mend, originating either from a linear distribution assumption (Sonderegger 2011), or optimization result of configuration WT1. In WT6 and WT7, a micro-structural approach for the determination of steady-state diffusion coefficients of walnut used in (Eitelberger 2011; Reichel 2015) was selected for comparison purposes.

The changing climate was caused by the inner cup conditions. Thus, only the surface emission coefficient (SE-coefficient) on the inner surface is considered as a Neumann boundary condition,


with the SE-coefficient (S), the potential between the relative humidity on the bulk materials surface RHs [-] and the air inside the cup RHair [-].

As comprehensively studied by Reichel (2015), no reliable knowledge is available on the characteristics of SE. Although, numerous experimental studies and material models are published, it is not possible to derive consistent emission properties. Thus, according to e.g. (Olek et al. 2005; Mannes et al. 2009; Eitelberger 2011; Olek et al. 2011) for the limitation of the number of optimization parameters, S is approximated as constant, i.e. m-independent. Time-dependency, like in Olek et al. (2011) due to reorganization processes of the wood ultra-structure, is neglected.

The sorption behavior is considered by an isotherm model published in Avramidis (1989),

 [-] (14)

with relative humidity RH [-] and temperature [K]. The applied regression coefficients are derived for a general and wood species independent formulation and used for both wood species in this study.

Numerical Analysis

In pre-simulations with various different FE-models, the highest accuracy with a minimum number of degrees of freedom (DOF) was obtained by modelling the cup specimens with one 20-node solid element over the whole thickness using a quadratic shape function (Bathe 2002) combined with a convective element on the inner surface, which is exposed to the climate change. The pre-study (see Fig. 3) showed that a finer discretization only increased the computational effort without any significant increase in accuracy. Although, for the analysis of the internal moisture distribution a finer discretization is recommended, as discussed in the Section “Results and Discussion”, it is not required for the correct average moisture content determination.

The samples’ average moisture content (MC) is determined by integrating the shape function of the bound water content (m) along the length section (specimen thickness).

Fig. 3. Comparison of the accuracy in MC(t)-determination with linear and quadratic shape functions and different discretizations of the FE-model

The inverse analysis optimization is carried out using an internal algorithm coupled to the in-house WoodFEM software. The set of optimization parameters (X) contains a maximum of four variables,


including the two parameters for Eq. 12, the final water content at the inner cup surface (mend), and a constant SE-coefficient (S). Although the experimental conditions for the cup climate are defined (see Section “Testing Method”), mend is regarded as an optimization variable as well.

The optimization method leads to the minimum of the objective function (Sr), which was chosen as the sum of squared residuals of the experimentally and numerically determined specimens’ average moisture contents (MC [-]) at measurement times (ti)


Due to the larger number of experimental data at the beginning, there is a higher weight on these results, which is a positive effect for the fitting of the numerical calculation. It means, that during the more pronounced non-equilibrium, which is dependent on the inner moisture distribution, more points for Eq. 16 are provided. This helps to decrease possible inaccuracies by not measuring the inner moisture distribution with the applied experimental method, described in the Section “Testing Method”. The optimization is performed via a bound constrained optimization based on a trust region approach, since it was found to be the most efficient one for this purpose. Finally, a global relative error [e.g. (Olek et al. 2011)] in relation to the experimental results can be calculated as:

 [%]. (17)

Results and Discussion

The results for the global parameters of the specimens, determined as described in Sections “Testing Method” and “Inverse Analysis of the Diffusion Coefficients” are documented in Table 4. Small differences of the experimental diffusion coefficients (Dus and Dss) were observed for cherry wood comparing the material directions T and R. The same behavior is known from other wood species, e.g.spruce and beech wood (Koc et al. 2003; Sonderegger et al. 2011). However, the diffusion coefficient (D) of WT showed a significantly lower value in comparison to D of WR. One main issue would be the high density difference between T and R specimens of walnut wood (Table 1). Moreover, D is also dependent on the moisture potential within the specimens (ΔM). Wet cups with a higher potential [difference between MC(95% RH) and MC(65% RH), see Table 3] tend to have higher D in comparison to dry cup [difference between MC(5% RH) and MC(65% RH)]. Furthermore, the experimentally determined steady-state diffusion coefficients (DSS) mostly do not show good correlation to the unsteady-state/transient ones (DUS). These findings correspond to those of other publications, e.g. (Wadso 1994; Pfriem 2006; Sonderegger 2011).

Table 4. Experimentally Determined Steady-State (DSS) and Unsteady-State (DUS), and Inversely Determined D(MCend) for the Specimens’ Global Moisture Content MCend


The results of the determination of an accurate approach for D(m), based on the reference sample WT dry (walnut, tangential diffusion, dry cup) are presented in Fig. 4 with the corresponding results of the unknown variables in Table 5. The most accurate approach was the exponential configuration WT1 (Sr=1.059E-6). The assumption of a linear moisture distribution over the samples length (thickness) m(x,t) in the final state (configurations WT4 and WT6) with higher corresponding mend [cf.Sonderegger 2011] led to a larger deviation.

While the predictions of MC(t) showed good agreement, the time-dependent moisture distribution within the specimen m(x,t) cannot be validated with the present experimental results. In comparison to other validated investigations, e.g. (Mannes et al. 2009), m(x,t) showed a qualitatively good comparison.

Fig. 4. Accuracy of different diffusion coefficient models D(m) (Table 5)

The distribution with respect to time and the shape of the steady-state condition in equilibrium remains uncertain since the profile of m(x,t) is strongly affected by the diffusion coefficient D(m). Further investigations shall visualize the effects on the internal conditions.

Table 5. Diffusion Coefficients D(m)


Fig. 5. Distribution of moisture content m(x,t) over length section x for specific experimental times (left) and as density plot (right)

Fig. 6. m(x,tend) in steady-state conditions together with its secant and its average value


Fig. 7. Diffusion coefficient D(mend,x), moisture gradient ∂m/∂x(mend,x) for the constant flux qm(mend,x) = 7.6E-14 kg mm-2s-1

Figures 5 to 7 present the results of WT dry specimen. The development of m(x,t) from initial to the final state (after about 1052 h) is shown in Fig. 5. The final non-linear equilibrium distribution ofm(x,tend) is plotted in Fig. 6 against its secant to visualize the non-linearity. The average of m(x) in the graph represents the quantity, which is compared to the experimentally determined global moisture content of the specimen (MC) in steady-state conditions (cf. Fig. 8).

Figure 7 shows the variables of Fick’s first diffusion law in Eq. 10. The exponential approach of D(m) leads to a linear distribution over the length section x

 . (18)

Together with Eq. 12, a logarithmic distribution of the steady-state m(x) is derived,

 [-] (19)

while the moisture flux qm needs to be constant over the length section.

The numerically determined D(m) of the final average moisture content MCend leads to

 [m²s-1] (20)

and is comparable to the experimentally determined DSS.

Fig. 8. Global moisture content MC(t) of the investigated cup tests (cf. Table 1)

The results in Table 4 show an appropriate comparison in the tangential dry cup experiments with a maximum deviation of 10%. All other cup tests, however, show large deviations between the experimentally and numerically determined values. These variations may be caused by several different assumptions within the analyses. In the experimentally based calculation, ΔH/ΔM in Eq. 6 is always assumed as a linear sorption isotherm distribution over the thickness. In fact, the sorption isotherm is a highly non-linear function and even more pronounced in high humidity, depending on the sorption direction (hysteresis) as well (Engelund et al. 2010). This may explain larger variations in the wet cup compared to the dry cup experiments.

During the experiment, curvature of the specimens due to moisture gradient and induced swelling was unavoidable and pronounced in wet cup tests (high moisture gradient). However, its effect on the experimental results are assumed to be negligible. The tight rubber seals (Fig. 2) ensured that the gap between the top of the cup and the wood specimens remained closed until the end of the measurement.

Moreover, a more sensitive effect such as the natural variability and the limited number of the specimens are suspected to be another reason for the incomparability between the experimental and the numerical results. As shown in Fig. 8, the MC varies. The initial MC0 of walnut wood spread between 0.085 < MC0,walnut < 0.095. The final values MCend, which were assumed as EMC, do not show a good agreement between R and T direction within the same wood species (Table 4). Furthermore, WT wet cup specimens still did not reach their EMC during the measurements (Fig. 8).

Additionally, the unsteady-state diffusion DUS, calculated by Eq. 2 for the global specimen is incomparable to any numerically obtained D. It is based on the continual change of mass of water versus the time interval of the unsteady experiment state (first state) and is supposed to be an easy estimation to describe the transient process until the equilibrium is reached. In the simulation, which describes more accurately the real transient and non-linear behavior, D(MC) describes the continuous process in dependency of the local m at a certain time, i.e. it is applicable to determine the diffusion coefficient at every moment. Thus, D(MC) is also changing simultaneously with the moisture change.

Table 6. Cup Test Specific Diffusion Coefficients D(m) and the Quality of Fit with the Sum of Squared Residuals Sr (Eq. 16) and the Global Relative Error e (Eq. 17)

The applied moisture transport models fit well for the description of the global moisture contents of the specimens’ MC. During the unsteady-state, the moisture distribution is significantly dependent on the variable D(m) and could lead to imprecise predictions of the shapes of m(x,t). Since the parameters for wet cup tests (Table 6) differ from those of the respective dry cup, another approach would become necessary for the determination of universal diffusion coefficients.

The results of the inverse analyses for the fitted curves of all MC(t) functions are visualized in Fig. 8 with their diffusion parameters for Eq. 12 in Table 6. The low values of the objective function Sr and the global relative errors e (Eqs. 16 and 17) verify that the determined D(m) led to appropriate results in the global moisture content prediction over time. For any configuration with varying material direction, wood species, and humidity load, different parameters were determined. The differences in the first two points are physically and naturally justifiable due to anisotropy and natural variations in micro-structure and density. Furthermore, walnut has a higher extractive content, which increases the inner moisture transport resistance [e.g. (Wagenführ 2007)]. The different parameters for dry and wet conditions show the deficits of the applied diffusion coefficient model with respect to a universal material model of D(m). While each test configuration will be represented with a minimal deviation, the general usage over the whole humidity spectrum is not appropriate. An additional investigation was carried out to determine one diffusion model for different conditions. The respective dry and wet cup specimens were considered together in one inverse analysis with a single exponential diffusion equation for both tests. Since the result showed a large error, it is not considered further. Additional investigations with respect to sorption directions and different humidity steps are necessary. Moreover, the influence of the density on D = D(m,ρ) becomes visible, which needs to be considered in a further step (Eitelberger 2011).

Experimental diffusion tests on walnut and cherry wood with the cup test method have been conducted with respect to the average diffusion coefficients in unsteady- and steady-state conditions. The experiments served as a basis for an inverse analysis, coupled to an FE-simulation for the numerical determination of moisture-dependent and cup test specific diffusion coefficients D(m).

The numerical diffusion model was applied to investigate the internal, time-dependent moisture profile and to assess the quality of the experimentally determined steady-state diffusion coefficients. Both the experimental DSS and the numerical D(MC) showed better correlation for the dry cup experiments, especially for T-direction. Large deviations between the wet cup results could have occurred due to the natural inhomogeneity and the limited number of specimens.

The determined diffusion parameters enable the simulation of the cup-test experiments. They are not universally appropriate for the prediction of transient conditions in the scope of FE-analyses with several humidity changes over the whole humidity spectrum. For a general moisture dependent diffusion model and for the consideration of the density-dependency, additional cup tests with further and smaller moisture differences considering a larger spectrum of relative humidity steps need to be tested (Gereke 2009) and further regression functions need to be assessed. Alternatively, multi-scale based approaches, like in (Eitelberger 2011) (c.f. configurations WT6, WT7 for walnut in Table 5) or multi-Fick’ian transport models (Konopka and Kaliske 2016) could be applied.

Further experiments in cup tests are required for a better determination of a universal single phased Fick’ian diffusion model, as presented by Olek et al. (2011). Alternatively, detailed analyses of local distribution of moisture content may also lead to reliable results (Hering 2011). In Mannes et al.(2009) and Mannes et al. (2012), the local moisture content of the specimens along the diffusion direction were investigated by neutron imaging. In Koc and Houska (2002) and Koc et al. (2003), the samples were cut and weighted separately, leading to a discrete and rough moisture profile along the longitudinal section.

However, the results enable the comparison to similar tests with different materials, e.g. other wood species, wood products, or composite materials. The determined characteristics will support further investigations of composite materials, like glued wood, to analyze and to compare the different characteristics and the influence of e.g. bond lines in relation to specific glues, on the internal moisture transport.


  1. The average diffusion coefficients in unsteady- and steady-state conditions for walnut and cherry wood in radial and tangential material directions were determined experimentally with the cup test method.
  2. The experiments served as a basis for an inverse analysis, coupled with a FE-simulation for the numerical determination of moisture-dependent and cup test specific diffusion coefficients D(m) of these wood species.
  3. The numerical diffusion model was applied to investigate the internal, time-dependent moisture profile and to assess the quality of the experimentally determined steady-state diffusion coefficients. Both the experimental DSS and the numerical D(MC) showed better correlation for the dry cup experiments, especially for T-direction. Large deviations were observed between the wet cup results.
  4. The determined diffusion parameters D(m) for walnut and cherry enable the simulation of the cup-test experiments and the comparison to similar tests with different materials, e.g. other wood species, wood products, or composite materials, like glued wood.
  5. The diffusion parameters D(m) are not universally applicable for the prediction of arbitrary transient conditions in the scope of FE-analyses.


The authors gratefully acknowledge the financial support by the German Research Foundation (DFG) (project KA 1163/25) and Swiss National Science Foundation (SNF) (project 14762) within the collaboration “Modelling and Characterization of the Structural Behaviour of Wooden Cultural Heritage under Hygro-mechanical Loading.”


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Article submitted: June 26, 2017; Peer review completed: September 4, 2017; Revised version received and accepted: September 26, 2017; Published: October 10, 2017.

DOI: 10.15376/biores.12.4.8920-8936