Moisture buffer value for hygroscopic materials with different thickness

Moisture buffering is the capacity of hygroscopic building materials or adsorbents to mitigate the humidity fluctuations in the indoor environment through the adsorption/desorption phenomenon. NORDTEST proposed the moisture buffer value to quantitively characterize the moisture buffering capacity of porous materials. Materials with high moisture buffer values have a higher ability to passively control indoor humidity conditions and thus reduce building energy consumption and improve indoor thermal comfort. However, the ideal moisture buffer value theory is based on the assumption that the thickness of the studied materials should exceed the penetration depth of that material. This assumption significantly limits the application of moisture buffer value theory to thin-layer materials, such as textiles, desiccant coating, wallpaper, wood wall panels, etc, which represent a large proportion of moisture buffering in the built environment. In this study, we developed a new numerical model to calculate moisture buffer values for hygroscopic materials with different thicknesses. Experimental measurements were also carried out to validate the solutions. The moisture buffer value of a new kind of desiccant (metal-organic frameworks) under different thicknesses were measured by climatic chamber tests. Simulated results showed agreement with both the measurements and theory. The results indicate that the new moisture buffer value model can be used to find the optimal thickness of desiccant coating or hygroscopic materials for autonomous indoor humidity control.


Introduction
In many places of the world, humidity control is a large part of the operational energy of a building [1][2][3][4][5].A new class of novel materials, metal-organic frameworks (MOFs), excel at moisture buffering, and have shown the potential to drastically reduce the energy used on humidity control.These materials, however, are often used in thin layers, which current theory does not apply to, making it harder to design with them.
The moisture buffer value (MBV) is a parameter proposed by the NORDTEST [6,7], that characterizes the moisture buffering capacity of a material.From knowing simple material properties, the ideal MBV can be calculated, however, this value is based on the assumption that the material is semi-infinite in thickness, and therefore only reliably applies when the material exceeds the penetration depth.This means the theory is hardly applicable to cases where only a thin layer of material is used.
Thin-layer materials are used in many scenarios, from standard building materials such as wall panels to desiccant coatings for air conditioning systems [8,9].Results from a recent study investigating the influence of material parameters on moisture buffering in office buildings in Chinese cities from Harbin to Kunming shows that material thickness significantly impacts the energy used on dehumidification and in some cities humidification [10].In some cases, knowing the range of thickness, where a material is used as optimally as possible, can also prove beneficial economically, since many of these novel materials are harder to come by than standard building materials and, thus more expensive to acquire.For these reasons it would be very useful to have an easy way of determining the MBV of a material at a certain thickness, and to quickly determine what the maximum thickness of the material is, at which the material is still being used cost-efficiently.In this paper a numerical model is developed for this purpose.
A relationship between moisture buffering capacity (MBC) and thickness has previously been found both analytically and numerically [11,12].MBV is an expression of the MBC divided by the difference in relative humidity.These solutions show the relationship between MBC as being linear with thickness at low thicknesses and independent of thickness at higher thickness, as one might expect intuitively.Both of these papers also provide a way of finding the optimal thickness.However, the analytical solution assumes a linear sorption curve, and the numerical solution uses the average between adsorption and desorption curves.Both of these assumptions make them more accurate for a classic building material and less for MOFs.
This paper aims to investigate the relationship between the MBV and the thickness of a material.A MOF class material, MIL-100(Fe), will be used for experiments, and its properties for simulations.First, the MBV of the material will be measured with different sample thicknesses.Second, a numerical model will be developed and run using the material properties of MIL-100(Fe) found in other literature.The numerical model should work with any material as an easy way to find the MBV at any thickness, when material properties are already known.
Due to the limitations of the study, MIL-100(Fe) properties were gathered from existing literature rather than doing a characterization.Additionally, the MBV experiments could not be performed with samples thicker than 1 cm.

Moisture buffer value
The moisture buffer value (MBV) is a parameter that describes the ability a material has to adsorb and desorb moisture from its surroundings, and thus work as a buffer for the humidity level of, for example, a room in a building.MBV is defined as the moisture uptake and release per surface area of a material per RH change during a cycle of high and low RH.
An idealized version of the MBV called the ideal MBV or the basic MBV is a value that can be calculated from a few material properties [6,7].This calculation is, among other things based on the assumptions that the material is semi-infinite, isotropic with a linear water sorption curve and that the convective surface resistance can be neglected.
The ideal MBV is expressed as: Where MBVideal is the ideal MBV [g m −2 % −1 ], ps is the saturated water vapor pressure The MBV can also be measured directly without knowing any material properties.This value is the practical MBV.The practical MBV is measured on a sample of a material placed in an environment with a constant temperature and RH varying between high and low.It is reported as the change of mass of the sample over the duration of a cycle per open surface area per % of RH variance, with the same unit as the ideal MBV above.
The ideal MBV and practical MBV will give different values with lower material thicknesses, because the ideal MBV is based on the assumption, that the material is semi-infinite.

Numerical model
The purpose of the model is to be able to simulate the practical MBV, when some material properties are known, as an alternative to the ideal MBV.

Theory
The model divides the sample into elements, where each element has a thickness, dx.It then tracks the vapor pressure and the total moisture content (vapor and adsorbed) in each element.
To calculate the moisture content evolution over time, the model starts by calculating the convective moisture transfer between elements.The transportation of moisture happens in this case through vapor diffusion.This diffusion is driven by the vapor pressure gradient through the material and resisted by the material permeability.Using Fick's law, the vapor transferred between two elements in one time step, dt, can be expressed as: Where m is the mass of moisture in an element [kg m −2 ], pv is the vapor pressure and dpv, the difference between two elements For the transfer between the ambient air and the first element, diffusion through at the surface must be considered.Here the convective moisture transfer coefficient is introduced.
Where β is the convective moisture transfer coefficient From this, the change of moisture content is calculated.Defining positive flow as flow into the material, the change of moisture content in the n'th element, is calculated as follows.Division with element thickness to get mass per volume.
Where w is total moisture content [kg m −3 ], n is the element number.Second, after calculating the moisture content in each element, the new water vapor pressure is calculated, which will again drive diffusion in the next time step.This is done using the function shown in Fig. 1.Plugging water content into this function yields vapor pressure (consider the graph being flipped to have uptake/moisture content as x and RH/pressure as y).Due to hysteresis, this function is in two parts; one for adsorption and one for desorption, and the moisture state will be between these curves, when changing between adsorption and desorption.

Material properties
The material used in this paper is MIL-100(Fe) in powder form.The permeability, dry density and isothermal sorption curve around 23 •C for this material is needed when running simulations with the model to compare the results from the model with the experiments.The permeability of MIL-100(Fe) has been reported to be between 4.38•10 −11 and 6.37•10 −11 kg m −1 s −1 Pa −1 from 3 different sources [2,[13][14][15], with the mean being 5.27•10 −11 kg m −1 s −1 Pa −1 .The mean value was used for simulating.
The isothermal water sorption curve has been reported using DVS [1].A function has been made to fit one of these curves to be used for calculations in the numerical model, see Fig. 1.What matters most is the fit in the range of 33% to 75% RH, since this range was used for the MBV measurements.

Validation: semi-infinite case
Consider an infinite slab where the water vapor pressure throughout the slab is the same and constant.
If the water vapor pressure suddenly changed at the surface of the slab, then the total mass of water vapor that has moved through the surface of the slab at any given time can be calculated as: Where A is the surface area of the slab [m 2 ], pv,0 is the vapor pressure at the start [Pa], pv,a is the ambient vapor pressure [Pa], t is the time passed after the vapor pressure change at the surface [s].
b can be calculated as: Since the term for MIL-100(Fe) is pressure dependent, solving Eq. 5 is not feasible, so it must be assumed that it is constant, as if the vapor sorption curve had been linear between uptake and RH/vapor pressure.Fig. 2 shows that the model and analytical solutions match almost perfectly for materials with a linear sorption curve.

Validation: Lumped analysis case
Lumped analysis can be used when the Biot number is very small, in other words, that the resistance at the surface is much larger than the internal resistance.It assumes that the moisture content in the material is always uniform.The differential equation governing the vapor pressure in the material can be expressed as: Comparison with the lumped analysis theory showed that the vapor content throught the material fits well with the constant vapor content calculated using theory, as seen in Fig. 3.The Biot number for this simulation was 0.05.

Validation: MBV measurements
Samples ranging from 1 to 8 mm are prepared to compare the model to real life and to validate the model in the range of thickness that is found between the lumped and the semi-infinite case.

Preparation of samples
Each sample used for the experiment, as seen in Fig. 4, consisted of the MIL-100(Fe) powder placed in a petri-dish, leveled out to have a plane surface.The petri-dishes have a 30 mm diameter, and the thickness of each different sample is calculated from this diameter, the weight of the powder in the sample and its density.
The walls of the petri-dish are 10 mm and will affect air movement at the surface of the material, and will do so differently for each sample, because of the varying thickness.The convection is assumed to be halfway between forced and natural at 10 mm thickness, β = 8.5•10 −8 kg m −2 s −1 Pa −1 , decreasing to natural convection at 3 mm thickness, β = 2•10 −8 kg m −2 s −1 Pa −1 .

MBV measurements
The MBV measurements, each sample is placed on the scale in the climate chamber as seen on figure 5, where it is continuously weighed every 5 minutes.The temperature in the climate chamber is kept close to a constant 23 • C, and the RH cycles between 75% for 8 hours and 33% for 16 hours.The RH sensor used in the climate chamber is estimated not to work with accuracy no better than ±3% RH.
MBV values calculated from the obtained mass and RH measurements are shown with error bars representing the 95% confidence interval in Fig. 6.The simulated results are presented alongside these values for comparison.Additionally, the range in which the model gives results given the estimated RH sensor accuracy of 3% RH.The simulated results did not fall within the 95% confidence value of the results for most thicknesses.Some factors may have contributed this.The RH fluctuated with a standard deviation of 0.07% during low RH and 0.16% during high RH.This fluctuation along with hysteresis results in increased MBV values, and has a larger impact the smaller the thickness of the sample.The RH took around 1 hour to adjust from low to high RH and around 20 minutes to adjust from high to low RH.These things were not accounted for in the model, where the mean value of the high and low RH periods were simply used.Additionally, the accuracy of the humidity sensor has a very large impact, because the gradient of the desorption curve is steep around 33% RH.
The simulation was run again with 30/72% RH and 36/78% RH to get a range corresponding to the sensor accuracy.All measured values agree with this range.

Results
Fig. 7 shows the results from the simulation, where the MBV is seen to increase linearly with thickness for small thicknesses.This indicates that the material reaches equilibrium throughout the material.At high thickness, the MBV is close to constant with a small positive slope.Increasing the thickness only has a slight effect due to the fact that the moisture transfer barely reaches the back of the material.In between the high and low thickness the regimes, the relation is more complicated.At some point the MBV decreases as the thickness increases.This has been explained as being caused by the added material in the back not being reached by the diffusion of the indoor air, but rather acting in a similar way on the material at the surface as the indoor air, being that it grabs or injects moisture into the material in the front.This results in a small humidity gradient through the material and thus a slower rate of transfer [12].
The MBV to thickness ratio is higher the smaller the thickness, but remains near-constant until curve smooths out.It is therefore most economically efficient to use this material in the thickness-regime where the curve remains linear.The greatest thickness in this regime, Lopt, can be approximated with the bi-linear method [12], which involves finding the intersection between the linear relationship at low thicknesses and the constant MBV at higher thicknesses.For this only two points of the curve is needed.One at very small thickness and one at higher thickness than the hump, for example at penetration depth, see Fig. 8.  MBV curve with approximated greatest optimal thickness using bi-linear method.

Discussion
Current theory on the MBV does not support the use of thinner materials and does not take into account the relation between MBV and material thickness.It also assumes no convective surface resistance and that the slope of the sorption curve,

𝜕𝜕𝜕𝜕 𝜕𝜕𝑝𝑝 𝑣𝑣
, is constant.The ideal MBV for the simulated case is 5.645 g m −2 % −1 , and the highest simulated MBV value is 2.5 g m −2 % −1 .When the simulation is run with the same assumptions used to calculate the ideal MBV, the MBV is 5.647 g m −2 % −1 at 4 cm thickness, matching the ideal MBV.This demonstrates that the ideal MBV theory does not apply to thin layers of an MOF like MIL-100(Fe), and that models like the one proposed in this paper are needed.Some numerical models have been proposed which have been shown to work well with conventional building materials, however, these models are modeled based on the average of the adsorption and desorption data.This is a bad assumption for a material like MIL-100(Fe), which experiences a large amount of hysteresis.Running this model with this average sorption curve results in MBV values more than two twice as high.
It is important to have a model, that can yields precise results for these MOFs considering, that they are more limited than regular building materials, and have an added economical aspect, that encourages applying them to a building as efficiently as possible.From an economical point of view, the optimal thickness, Lopt, would be described as the greatest thickness, where the MBV/thickness curve is still close to linear for the most efficient MBV to thickness ratio.This thickness can be identified by simulating using the proposed model to find the MBV/thickness curve, and Lopt can be approximated quickly using the bi-linear method.
The model was validated with both measurements and theory.The difference between results and theory is negligible.The experimental results fell within the range of simulated results when considering sensor accuracy, and shows a similar trend.Other than sensor accuracy, perfect validation from the experiments is hard to get, when properties are estimated from literature rather than found through characterization, but the agreement with the numerical model is still satisfying.

Conclusion
To describe the moisture buffering capabilities of a material, MBV is used as a material property.The MBV is highly dependent on the thickness of the material.A new class of materials called MOFs, that are highly efficient in terms of moisture buffering, are emerging.These materials are less accessible than regular building materials, and it is therefore interesting to know the range of thickness, where the moisture buffering per thickness is highest, in order to use the materials as efficiently as possible.This paper has proposed a numerical model, where only few material properties are needed in order to calculate the MBV of a material at different thicknesses in order to identify this optimal range of thicknesses.
The MBV of a material increases linearly a low thicknesses until it rounds off and remains close to constant at high thicknesses, only increasing slightly.The optimal thickness range is found where the curve increases linearly, which means it should simply be lower than a certain thickness, Lopt.This thickness depends not only on material properties, but also the environment in which it is placed.
The numerical model was validated with lumped moisture transfer theory, semi-infinite slab moisture transfer theory and MBV measurements.The simulated results showed agreement for each validation.
Lopt can be found either from simulating at several thicknesses to observe the full curve, or it can be found quickly using the bi-linear method.

Figure 2 .
Figure 2. Analytic calculation of moisture uptake of a semi-infinite slab compared with the numerical calculation of moisture uptake of every sample.

Figure 3 .
Figure 3. Vapor pressure distribution over 8 hours with a Biot number of 0.05.

Figure 5 .
Figure 5.The climate chamber used for the MBV-experiment.