Measurement and optimization of estimation method for moisture diffusivity of porous building materials with different salt content

Abstract

Sintered bricks, common porous building materials in architectural heritage, necessitates an accurate understanding of moisture diffusivity for effective moisture transfer analysis. Using the “ruler method” proposed by Evangelides et al., based on pure water capillary absorption experiments, we determined the moisture diffusivity (D_l) of Chinese sintered blue brick containing sodium chloride (NaCl) crystals. Before the experiments, the samples were immersed in 0%, 1%, 5%, and 10% NaCl solutions and dried to contain salt crystals. Subsequently, based on the D_l values measured by the ruler method, a heat and moisture coupled transfer model was developed to discuss the applicability of the moisture diffusivity obtained from the ruler method by reproducing the pure water capillary absorption process of sintered bricks with different salt contents. Experimental results showed that the capillary absorption coefficient (A_cap) and capillary moisture content (w_cap) of sintered bricks both decreased with increasing NaCl content, with maximum reductions of 18% and 8.6%, respectively. The D_l increased with elevated salt content in lower moisture content range but decreased in higher moisture content range. The simulation results indicated that the choice of boundary moisture content played an important role in the determination of moisture diffusivity in porous building materials and using saturated moisture content (θ_sat) over capillary moisture content (θ_cap) can give more suitable moisture diffusivity and the moisture content above θ_cap cannot be ignored in the ruler method. This research not only extends the ruler method’s applicability for sintered bricks but also provides support data for moisture simulations and conservation initiatives related to sintered brick architectural heritage.

Keywords

Sintered brick salt content moisture diffusivity ruler method heat and moisture simulation

Introduction

Sintered brick stands out as a prevalent porous building material worldwide (D Ayala and Aktas, 2016; Jin et al., 2017; Takatori et al., 2021). In China, the architectural heritage, such as the Great Wall, Nanjing Ming City Wall and a large number of ancient dwellings, are all made of the sintered brick which is fired with clay but has a blue color due to cooled in an anoxic environment. The exposure of these structures to fluctuating environmental conditions triggers salt migration through capillary water within the bricks. This, in turn, leads to diverse forms of deterioration resulting from salt crystallization, including spalling, hollowing, salting out, and even structural safety issues (Desarnaud et al., 2013; Shahidzadeh and Desarnaud, 2012).

The weathering of sintered bricks by salt is a consequence of the intricate interplay between temperature, moisture, and salt. In the natural state, salt exists in mixtures and the mixture composition determines the crystallization behavior. The damage mechanism caused by different salts are strongly different. Yue et al. (2022) conducted salt crystallization tests on gray bricks using 1%, 3%, and 5% NaCl, CaCl₂, and MgSO₄ solutions, and found that NaCl and MgSO₄ caused significant damage to the surface of the gray bricks while CaCl₂ did not cause significant damage on the surface due to the differences in crystallization pressures of different salts. Godts et al. (2023) analyzed the critical crystallization and dissolution ranges of different mixture composition. The results show that the critical crystallization RH of mixture salts are lower than the single salts, and the crystallization RH of NaCl in a sulfate-rich mixture and calcium-rich mixture decreased to about 68% and 57% respectively. Salt migration is intricately tied to the distribution of temperature and moisture within the material, with the transfer of temperature and moisture being influenced by both salt migration and phase change. To comprehend the mechanism of salt weathering and mitigate salt-induced damage, it becomes imperative to elucidate the distribution of temperature and humidity within the material. Numerical simulation serves as a common tool for calculating heat and moisture (HAM) migration in porous materials (Li et al., 2020; Nicolai, 2017; Zhao et al., 2022a). Achieving accurate simulation results requires an in-depth understanding of the moisture transfer properties of the materials involved, including moisture capacity, water characteristic curve, moisture diffusivity, vapor permeability coefficient et al. (Colinart et al., 2020; D’Altri et al., 2021; Steiger et al., 2008). The salt crystals can naturally exist in building materials during manufacturing process (bricks or mortars), or from the external environment including rainfall, rising damp, atmospheric pollution et al., the presence of which can significantly affect their moisture transfer properties (Todorović and Janssen, 2018). However, the impact of salt crystals on the moisture transfer of porous materials has not been extensively explored.

Most studies have predominantly focused on the effect of salt solution on the capillary absorption properties, where the samples were usually placed in salt solution. For instance, Britoand Diaz Gonçalves (2013) conducted capillary absorption experiments on stones using pure water and varying concentrations of salt solutions (NaCl or NaNO₃). The results indicated the sorptivity of stones decreased linearly as salt solution concentration increased because of the difference in surface tension and viscosity. Koronthalyova and Bagel (2015) analyzed the impact of low NaCl salt contamination on the capillary absorption process of ceramic bricks, observing practically identical capillary absorption processes for pure water and 4% and 6% NaCl solutions. These studies generally reflect the infiltration process of salt-contaminated groundwater or rainwater into salt-free brick walls. However, an important but often overlooked reality is that brick walls usually already contain salt. When relatively pure rainwater hits such walls or water vapor from the outside air intrudes into the wall, adsorbed, and condenses, it can be assumed that the pure water is diffusing through the salt containing brick rather than salt solution infiltrating the brick. In such cases, the diffusion coefficient of the salt solution may not accurately represent the process of water diffusion. Therefore, it is necessary to explore the effect of salt crystals within materials on moisture transfer.

Recently, Xie et al. (2023) conducted the capillary absorption experiment on saline mortar. The results show that as the salt content increased (from 0% to 0.6%), the capillary absorption coefficient (A_cap) first increased and then decreased, and the capillary moisture content (w_cap) gradually decreased. This is because the interaction between the promoting effect of dissolved salt on water transport and the adverse effect of salt crystals on the microscopic pore structure. Espinosa-Marzal and Scherer (2013) compared the capillary absorption process of limestone with NaCl, NaSO₄, MgSO₄ crystals respectively using water-insoluble decane, and they found that the presence of salt crystals decreased the capillary absorption coefficient due to the pore clogging resulting from the interplay between pore size distribution and salt properties. Generally, the moisture transfer properties of porous materials vary with the moisture content (Abdou Ibro et al., 2021). However, the A_cap only represents the average value of capillary water absorption rate and may not adequately characterize moisture transfer properties as moisture content changes. The moisture content distribution inside the porous material is usually nonuniform during the moisture transfer process. Thus, it also cannot be directly applied to numerical simulations of moisture transfer processes, which may lead to a large error. The moisture transfer parameters governing the transfer of liquid water in porous materials are generally categorized into permeability and diffusivity based on the driving potential for moisture transfer. In unsaturated conditions, the driving potential for moisture permeability is capillary pressure, a challenging parameter to directly measure. Consequently, moisture diffusivity is typically measured, and the permeability coefficient is then calculated by integrating it with the water retention curve (Janetti and Wagner, 2017). Therefore, it is crucial to determine the moisture diffusivity of porous materials influenced by salt crystals. Moreover, there are many gaps in knowledge regarding the moisture diffusivity of salt-containing sintered bricks, particularly the Chinese sintered blue brick. Chinese sintered blue bricks are the main building material of many cultural heritages including the Great Wall. It is blue in color due to cooling in an oxygen deficient environment.

Moisture diffusivity is commonly obtained indirectly through the resolution of the one-dimensional diffusion equation using the Boltzmann transform method, which relies on capillary absorption experiments (Evangelides et al., 2018). During capillary absorption experiments, the transient moisture distribution inside the material is obtained by monitoring, and then the one-dimensional diffusion equation can be solved by defining Boltzmann variable (two independent variables are unified into one variable). Finally, the moisture diffusivity can be obtained. Therefore, obtaining transient moisture distribution in the material is necessary for obtaining the moisture diffusivity. Destructive methods, such as the slice-dry-weight method (Záleská et al., 2017), exist but are unsuitable for continuous measurement. In response, non-destructive methods have been developed, including X-ray or γ-ray attenuation (Fukui and Takada, 2024; Yang et al., 2019), nuclear magnetic resonance (Deckers and Janssen, 2024; Zhao et al., 2021), time domain reflectometry (Pavlík et al., 2006), thermal imaging (Marynowicz and Kucharczyk, 2021), among others. Koronthalyova et al. (2021) monitored the 1D water uptake process of calcium silicate board using the X-ray method. They examined the impact of different moisture diffusivity functions on simulation results for the capillary absorption process using the 1D simulation tool WUFI. Their results demonstrated that simplified moisture diffusivity functions could accurately evaluate capillary absorption. Kočí et al. (2019) determined the moisture diffusivity of autoclaved aerated concrete at various room temperatures, describing moisture profiles with spline curves generated from cubic polynomial functions. They concluded that the material’s moisture diffusivity decreased by 20% when cooled to 5°C. Meanwhile, Zhao et al. (2022b) introduced a novel neutron radiography imaging method to visualize the moisture diffusivity of sandstone. Their analysis identified the anisotropy and heterogeneity of the matrix and nonuniform geometric boundaries as key factors influencing anomalous diffusivity phenomena. Nevertheless, these methods face challenges in standardization for continuous measurement and are not widely adopted due to the expensive facilities required and the complexity of data processing.

In response to these challenges, various simplified mathematical methods have been proposed for determining moisture diffusivity. These include the multi-step method (Bianchi, 2018), ruler method (Evangelides et al., 2010), Kießl-Künzel method (Kunzel, 1995), among others. Ren et al. (2019) conducted a comparative study of these three simplifying mathematical methods for the non-destructive determination of moisture diffusivity in calcium silicate, ceramic brick, and lime mortar. Their findings indicated that the ruler method yielded results closest to those obtained through the well-established X-ray attenuation method, with a more straightforward process compared to the multi-step and Kießl-Künzel methods. However, the use of the ruler method in different porous building materials and the factors affecting the accuracy of the moisture diffusivity obtained are still worthy of further investigation.

The aim of this study was to elucidate the effect of sodium chloride (NaCl) content on the moisture diffusivity of sintered blue bricks using the ruler method proposed by Evangelides et al. (2010) and to propose a fitting equation for the moisture diffusivity, in order to allow for a simple and fast simulation of the moisture transfer process in salt-containing blue bricks using the heat-moisture coupling model. The bricks were firstly immersed in 0%, 1%, 5%, and 10% NaCl solutions and completely dried until the mass of bricks didn’t change to contain salt crystals. Then the capillary absorption experiments with pure water were conducted on the bricks containing different NaCl content to determine moisture diffusivity (D_l) using the ruler method. The analysis, based on capillary pore distribution measured by mercury intrusion porosimetry (MIP), delves into the effect of salt on the capillary absorption coefficient (A_cap), capillary moisture content (w_cap), and moisture diffusivity (D_l). Subsequently, a coupled heat and moisture transfer model was developed to simulate the pure water capillary absorption process using the experimentally determined D_l, further exploring the applicability of the ruler method. Besides, the factors influencing the ruler method’s measurement results was analyzed. The outcomes not only furnish fundamental data for moisture simulation and enhance our understanding of the influence of NaCl on moisture transfer in sintered bricks, but also improve the knowledge for the ruler method.

Materials and methods

Materials

In this study, the sintered blue bricks produced in Jiangsu Province of China were used, as shown in Figure 1(a). The bulk density, porosity, and pore size distribution of three samples from different sintered bricks were tested by mercury intrusion porosimetry (MIP) method (MicroActive-AutoPore V 9600, America). The bulk density of three samples was 1.80, 1.82, and 1.81 g/cm³, and the corresponding porosity were 31.25%, 32.21% and 32.19%, respectively. The pore diameter distributions obtained from MIP method are illustrated in Figure 1(b). The major pore size of the samples ranged from 500–5000 nm, and the average pore diameter were 396, 409, and 428 nm, respectively. The MIP results showed a very high degree of consistency among the three random samples.

Experimental test

Salt intrusion method

Five brick samples of the size of 50 × 50 × 50 mm³ were chosen to eliminate errors caused by individual differences. Figure 2 shows the capillary absorption experimental process of sintered blue bricks with different sodium chloride contents. Before the experiment, five brick samples were soaked in distilled water, and the soaking solution was changed every 24 h until its electrical conductivity decreased to 10 μs/cm, which was considered to be desalted completely (Ottosen and Rörig-Dalgaard, 2009). The process of brick desalting usually lasts 3–5 days. Next, (1) the samples were dried in a 105°C drying oven until constant weight for 2 days, and the dry mass (M_dry, kg) of samples was recorded.

To determine the moisture diffusivity of sintered blue bricks with different sodium chloride content, four series of experiments with different salt content were carried out using the same five brick samples. In this experiment, pure water and three mass fractions (salt mass/pure water mass) of sodium chloride (NaCl) solution were prepared: 0% (no contamination), 1 %, 5%, and 10%. Firstly, (2) the capillary absorption experiment of salt free bricks was conducted. After that, (3) the samples were dried and then (4) immersed into NaCl solution for 48 h. Then, (5) the samples were dried again, and the dry mass with salt (M_{dry, salt}, kg) of samples were recorded. After (6) the pure water capillary absorption experiment under each salt content was completed, (7) the samples were soaked in distilled water for desalting. The average salt content of the brick samples (S, %) can be calculated by the change of mass before and after soaking in NaCl solution, as expressed in equation (1).

S = \frac{M_{dry, salt} - M_{dry}}{M_{dry}} \times 100 %

(1)

Capillary absorption experiment

The capillary absorption experiment is one of the most commonly used physical property tests of materials, which was conducted based on ISO 15148-2002 standard (2002), as shown in Figure 3(a).

In Figure 3(b), the slope of the first stage is defined as the capillary absorption coefficient (A_cap, kg/(m² s^−1/2)), and the moisture content at the cross point of the fitting lines for the first and the second stages is defined as the capillary moisture content (w_cap, kg/m³; Feng and Janssen, 2018; Indekeu et al., 2022), as shown in equations (2) and (3).

A_{cap} = \frac{Δ M}{A \cdot \sqrt{t}}

(2)

where ΔM is the moisture mass absorbed by samples, kg; A is the water absorption area equal to the bottom area of the sample, m²; t is the time, s.

w_{cap} = \frac{Δ M}{A \cdot H}

(3)

where H is the height of water front, m.

In this study, the volume of solution (pure water) absorbed by the samples is defined as the capillary volumetric moisture content (θ_cap, m³/m³), which can be calculated by equation (4):

θ_{cap} = \frac{w_{cap}}{ρ_{w}}

(4)

where ρ_w is the density of water, kg/m³.

The experiment was carried out at a temperature of 15 (±2)°C and a relative humidity of 50 (±2) %, which was determined according to the annual average climatic conditions in Nanjing, China. Figure 4(a) shows the capillary absorption experimental setup, and main devices include a shallow tank, an electric balance with an accuracy of 0.01 g, a ruler with an accuracy of 1 mm for measurement of water front height, a CT-3030L conductivity meter with a range of 0–1999 μs/cm and an automatic electric drying oven. Prior to the experiment, the five surfaces of the samples, except the water absorption bottom surface, were sealed by plastic films to prevent the influence of water evaporation. To inhibit the liquid water from rising along the gap between the sample and the plastic films, the plastic films wrapped on four side surfaces were set aside a 10 mm gap from the bottom. Moreover, a certain number of exhaust holes were made on the top surface plastic film of the samples using a fine needle to maintain atmospheric air pressure (only vapor transmission through plastic film), which were extremely small.

During the standard measurement, the dry samples were placed in a shallow tank for one-dimensional (1D) isothermal free water uptake. The water level in the tank was kept constant at 5 mm above the bottom surface of samples. When the samples touched water, the timer was started. In the first stage, the samples were removed from water every 3–5 min, and the free water attached to the bottom was wiped with a damp cloth. The wet mass (M_wet, kg) of samples was measured regularly with their corresponding times (t, s). Additionally, the height of the water front (x_t, m) at different times was measured simultaneously to obtain Boltzmann variable λ_t as detailed in Section 2.2.3. In the second stage, the wet mass of samples was recorded at intervals of 30–60 min and the experiment stopped when at least five sets of data were recorded.

The water mass absorbed by the samples in a period of t can be calculated by equation (5).

Δ M = M_{wet} - M_{dry}

(5)

Then, the capillary absorption process of sintered brick samples was described by plotting the unit area water absorption quality against the square root of time t.

Ruler method to determine moisture diffusivity

The ruler method is a simplified version to determine the moisture diffusivity based on capillary absorption test, which does not require complicated facilities to determine moisture distribution in porous materials although irregular moisture fronts may occur that affect measurement accuracy (Evangelides et al., 2010). During the first stage of the capillary absorption process, the height of the water front (x_t, m), that is the position at which surface color changes sharply due to wetting, was also measured visually using a ruler as shown in Figure 3(a). The one-dimensional moisture diffusion in porous materials under isothermal conditions can be described as:

\frac{\partial θ}{\partial t} = \frac{\partial}{\partial x} (D_{l} \frac{\partial θ}{\partial x})

(6)

where θ is the moisture content, m³/m³; D_l is the moisture diffusivity, m²/s; x is the position, m.

By defining the Boltzmann variable λ (m/s^0.5) as:

λ = \frac{x}{\sqrt{t}}

(7)

The moisture diffusivity with different moisture content can be express as:

D_{l} = - \frac{1}{2} \cdot \frac{\int_{θ_{0}}^{θ} λ d θ}{{(\frac{d θ}{d λ})}_{θ}}

(8)

where θ₀ is the initial moisture content, m³/m³.

Figure 4(a) shows the θ-λ profile in a typical case (such as using X-ray method, Yang et al., 2019). Carmeliet et al. (2004) presented that the θ-λ profile consists of three zones: a boundary layer with moisture content above the capillary volumetric moisture content θ_cap (Zone 1), a transport zone with a moisture content distribution around θ_cap (Zone 2) and a steep water front (Zone 3). Among them, the moisture transport in Zone 2 is mainly driven by capillary pressure. When the moisture content exceeds θ_cap (Zone 1), the samples will continue to absorb water slowly until saturated moisture content θ_sat because the pores of materials are initially occupied by air partly, and then gradually filled through water vapor diffusion. Actually, the moisture transport process in Zone 1 and 3 is slow compared with the rapid capillary absorption process, which was usually ignored and the θ_sat is usually replaced by the θ_cap in most studies (Indekeu et al., 2022; Roels and Carmeliet, 2006). According to reference (Evangelides et al., 2010), the θ-λ profile can be approximated with an empirical function as follows:

λ (θ) = \frac{1}{b} [\tan (- \frac{θ + θ_{cap}}{a}) - c]

(9)

where a, b, and c are all fitting parameters, which are obtained by combining the following three boundary conditions, as shown in equations (10)–(12).

θ = θ_{cap} : λ = 0

(10)

θ = 0 : λ = λ_{f}

(11)

A_{cap} = \int_{0}^{θ_{cap}} λ d θ

(12)

where λ_f is the average value of the Boltzmann variable λ at different t, m/s^0.5. During the first stage of the capillary absorption process, the height of the water front (x_t, m) at different times is measured regularly. For each measurement, a Boltzmann variable λ_t can be obtained by equation (7). Then the average value λ_f can be calculated using equation (13).

λ_{f} = \frac{λ_{t 1} + λ_{t 2} + . . . + λ_{tn}}{n}

(13)

Figure 4(b) shows the θ-λ profile in the ruler method. According to the boundary conditions, the abscissa of the intersection of the curve and the X axis is the average value of the Boltzmann variable λ which represents the rising speed of moisture front of materials, and the ordinate of the intersection of the curve and the Y axis is the capillary volumetric moisture content (θ_cap). The area of the θ-λ curve surrounded by X and Y axes represents the capillary absorption coefficient (A_cap), as shown in equation (12). Finally, the moisture diffusivity is calculated by substituting the determined θ-λ profile into equation (8).

Error analysis

Five samples were used for each set of experiment in our study. It is necessary to quantitatively analyze the difference between samples owing to the inhomogeneity of sintered blue bricks. The relative standard deviation (RSD) of sintered blue bricks was determined by equations (14) and (15) (Feng et al., 2015).

sd = \sqrt{\frac{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}}{n - 1}}

(14)

RSD = \frac{sd}{\bar{x}} \times 100 %

(15)

where sd is the standard deviation; x_i is the measured value; n is the number of samples, $\bar{x}$ is the average value.

Numerical simulation

The coupled heat and moisture transfer model can simulate the moisture migration inside materials and has been widely used to evaluate the thermal and moisture behavior of porous materials. To examine the applicability of the moisture diffusivity obtained from the ruler method, a heat and moisture transfer model was established to simulate the capillary absorption process of sintered blue bricks. The coupled heat and moisture transfer model proposed by Matsumoto and Iwamae (1988; Li et al., 2014) was adopted in this paper, which have been continuously developed and widely applied in the past decades (Ishikawa et al., 2021; Ma et al., 2024). In the model, the heat and moisture transfer are driven by a temperature gradient and water chemical potential gradient, respectively.

Heat and moisture transport equations

In the model, the water chemical potential (μ) was used to express the moisture state, as shown in equation (16). It is a function of temperature and relative humidity. The temperature has only a slight effect on it, while changes in relative humidity play a decisive role.

\begin{matrix} μ = R_{v} Tln (h) \end{matrix}

(16)

where μ is the water chemical potential of the material, J/kg; R_v is the gas constant of water vapor; T is the temperature, K; h is the relative humidity, -.

The heat and moisture balance equations are expressed in equations (17) and (18), respectively.

c ρ \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} [(λ + r λ_{Tg}') \frac{\partial T}{\partial x} + r λ_{μ g}' (\frac{\partial μ}{\partial x} - n_{x} g)]

(17)

ρ_{w} \frac{\partial θ}{\partial μ} \frac{\partial μ}{\partial t} = \frac{\partial}{\partial x} [λ_{T}' \frac{\partial T}{\partial x} + λ_{μ}' (\frac{\partial μ}{\partial x} - n_{x} g)]

(18)

where c is the heat capacity of the material, J/(kg·K); ρ is the density of the material, kg/m³; λ_k is the thermal conductivity of the material, W/(m·K); $λ_{Tg}'$ is the water vapor conductivity under the gradient of temperature, kg/(m·s·K); $λ_{μ g}'$ is the water vapor conductivity under the gradient of water chemical potential, kg/(m·s·(J/kg)); r is the phase change heat of the vaporization of water, J/kg; ρ_w is the density of the water, kg/m³; θ is the volumetric moisture content of the material, m³/m³; $λ_{T}'$ is the water conductivity under the gradient of temperature, kg/(m·s·K), $λ_{T}' = λ_{Tg}' + λ_{Tl}'$ ; $λ_{μ}'$ is the water conductivity under the gradient of water chemical potential, kg/(m·s·(J/kg)), $λ_{μ}' = λ_{μ g}' + λ_{μ l}'$ ; g is the gravity acceleration, 9.8 m/s²; n_x is the direction of gravity vector (vertical direction is 1 and horizontal direction is 0).

Among them, the vapor conductivity under the temperature gradient, $λ_{Tg}'$ (kg/(m s K)) and under the water chemical potential gradient, $λ_{μ g}'$ (kg/(m s·(J/kg)) are given in equations (19) and (20), respectively.

λ_{Tg}' = λ_{m}' {(\frac{\partial P_{v}}{\partial T})}_{μ}

(19)

λ_{μ g}' = λ_{m}' {(\frac{\partial P_{v}}{\partial μ})}_{T}

(20)

where $λ_{m}'$ is the vapor permeability, kg/(m·s·Pa); P_v is vapor pressure, Pa.

The liquid water conductivity under the temperature gradient, $λ_{Tl}'$ (kg/(m s K)) and under the moisture chemical potential gradient, $λ_{μ l}'$ (kg/(m s·(J/kg)) are obtained from moisture diffusivity, given in equations (21) and (22), respectively.

λ_{Tl}' = ρ_{w} D_{l} {(\frac{\partial θ_{l}}{\partial T})}_{μ}

(21)

λ_{μ l}' = ρ_{w} D_{l} {(\frac{\partial θ_{l}}{\partial μ})}_{T}

(22)

where D_l is the moisture diffusivity, m²/s.

Model setting

A two-dimensional hygrothermal model was developed based on equations (16) and (17) using Fortran language programming, as shown in Figure 5(a). The model was divided into three areas: brick (gray part), water (blue part), and air (white part). The size of brick was 5 cm × 5 cm and it was divided into 10 (horizontal) × 25 (vertical) meshes uniformly. Hence the size of each grid of brick was 5 mm × 2 mm. The boundary and initial conditions of the model are set based on capillary absorption experiments. Specifically, the bottom of brick directly contacted with water, and the side surface and the upper surface was adiabatic and moisture insulation because they were wrapped with plastic film in our experiment, which does not exchange heat and moisture with outside air. The initial temperature of brick was 15°C (that was the experimental temperature), and the chemical potential of water was −100,000 J/kg (dry state). The finite difference method was used to discretize the coupled heat and moisture transfer equations. The time increment for calculation was 0.0001 s.

Material property parameters

The basic properties of the sintered blue brick were listed in Table 1. The moisture diffusivity of sintered bricks with different salt content was determined by the ruler method described in Section 2.2, and the water retention curve of salt free bricks was obtained according to the MIP test, as shown in Figure 5(b). When simulating the capillary water absorption process, the maximum moisture content of sintered bricks with different salt content was set according to the values of capillary moisture content obtained from the experiment.

Table 1.

Main properties of sintered blue brick in the model.

Parameter	Density ρ kg/m³	Porosity %	Heat capacity c J/kg·K	Thermal conductivity λ_k W/(m·K)	Vapor permeability λ_m kg/(m s Pa)
Value	1,800	32	840	0.22	3.98 × 10⁻¹²

Results of experiment and analysis

Experiment results

Salt content results

The average salt content of five samples immersed into different NaCl solution was determined and shown in Table 2. In our experiment, a very thin layer of NaCl crystal was observed on the surfaces of brick samples (efflorescence) with higher NaCl content (0.72%S and 1.37%S, S given by equation (1)) after a drying at 105°C. To examine the salt crystals distribution inside bricks, another brick sample was immersed into 10% NaCl solution for 48 h. Then, the sample was cut into thin slices after drying at 105°C for 48 h. After that, a microscope (KEYENCE VHX-2000E) was used to observe salt crystal distribution of the brick slice. Two areas of the slice (Area 1 and Area 2) were randomly selected, as shown in Figure 6. It can be seen that the salt crystals distribution inside bricks was relatively uniform inside bricks. After that, three slices were soaked in distilled water for desalting, and then dried in a 105°C drying oven. The salt content of each slice can be calculated by equation (1), which were 1.80%, 1.30%, and 1.66% respectively. The salt content of Slice 1 and 3 was slightly higher than Slice 2 due to the efflorescence on the surface, while the salt content difference between 3 slices was sufficiently small and negligible. This further illustrated the salt crystal distribution inside bricks was relatively uniform.

Table 2.

Average salt content of five samples immersed into different NaCl solution.

Salt solution concentration	Salt content S (%)
0%NaCl	0
1%NaCl	0.16 ± 0.02
5%NaCl	0.72 ± 0.02
10%NaCl	1.37 ± 0.05

Capillary absorption process

Figure 7 shows the capillary absorption process of samples. Among them, Figure 7(a) show the experimental results of five salt free samples, and Figure 7(b)–(f) compares the capillary absorption process of five samples under different salt content. In Figure 7(a), the capillary absorption process of five salt free samples was slightly different. To decrease the errors caused by individual differences as far as possible, the same five bricks were used for four series of experiments with different salt content instead of using different samples, and there were five parallel samples per salt content. Figure 7(f) presents the capillary absorption process of Brick 5 sample with different salt content. It can be seen that the capillary absorption process was divided into two stages: fast absorption and slow absorption. The first phase with a rapid water absorption lasted approximately 1–1.5 h and then moved on to the second phase with a slow water absorption. The first phase duration of the salt free sample was shorter than those of saline sample and it gradually extended as salt content increased. Besides, the slope of the first stage and the corresponding moisture content at the cross point of the fitting lines for the first and the second stages both decreased gradually as salt content increased. With respect to other samples, the effect of salt content on the capillary absorption process was basically consistent with Brick 5, as shown in Figure 7(b)–(e).

Capillary absorption coefficient and capillary moisture content

Figure 8 illustrates the capillary absorption coefficient (A_cap), capillary moisture content (w_cap), and capillary volumetric moisture content (θ_cap) of five bricks with different salt contents, and the specific values were presented in Tables 3 and 4.

Table 3.

Measured capillary absorption coefficient and capillary moisture content of samples.

Samples	0%S		0.16%S		0.72%S		1.37%S
	A_cap kg/(m² s^1/2)	w_cap kg/m³	A_cap kg/(m² s^1/2)	w_cap kg/m³	A_cap kg/(m² s^1/2)	w_cap kg/m³	A_cap kg/(m² s^1/2)	w_cap kg/m³
Brick 1	0.209	269.60	0.184	265.28	0.202	246.88	0.162	250.32
Brick 2	0.246	282.56	0.226	280.72	0.204	270.00	0.218	260.40
Brick 3	0.260	256.96	0.225	253.20	0.223	248.00	0.195	229.28
Brick 4	0.229	287.60	0.192	284.88	0.198	276.40	0.191	267.12
Brick 5	0.207	281.12	0.184	276.40	0.180	260.08	0.179	252.56
Average	0.230	275.57	0.202	272.10	0.201	260.27	0.189	251.94
RSD	10.0%	4.7%	10.7%	4.7%	7.6%	5.0%	10.9%	5.7%

Table 4.

Capillary volumetric moisture content θ_cap (m³/m³).

Samples	0%S	0.16%S	0.72%S	1.37%S
Brick 1	0.270	0.265	0.261	0.250
Brick 2	0.283	0.281	0.270	0.260
Brick 3	0.257	0.253	0.248	0.229
Brick 4	0.288	0.285	0.276	0.267
Brick 5	0.281	0.276	0.260	0.253
Average	0.276	0.272	0.263	0.252
RSD	4.7%	4.7%	5.0%	5.7%

Among them, the A_cap of five salt free brick samples was 0.209, 0.246, 0.260, 0.229, and 0.207 kg/(m² s^−1/2), and the average value was 0.230 kg/(m² s^1/2). As shown in Figure 8(a), the A_cap of saline bricks was smaller than that of salt free bricks for all five sample. As presented in Table 3, the average A_cap of five brick samples with four salt content was 0.230, 0.202, 0.201, and 0.189 kg/(m² s^1/2), respectively, which showed a decreasing trend with the increase of salt content. The average A_cap of brick samples with 1.37% NaCl content was 18% smaller than that of salt free bricks. The relative standard deviations (RSDs) of samples with the same salt content were approximately 10%.

Figure 8(b) indicates the w_cap of five samples decreased with the increasing salt content of the bricks. As shown in Table 3, the average w_cap of five brick samples with four salt content was 275.6, 272.1, 260.3, 251.9 kg/m³, respectively. Specifically, the average w_cap of brick samples with 1.37% NaCl content was 8.6% smaller than that of salt free bricks. The RSDs of samples with different salt content were approximately 5%.

The capillary volumetric moisture content (θ_cap) of five brick samples with different salt content was also calculated, as shown in Figure 8(c). The θ_cap of five samples decreased significantly with the increase of salt content of bricks. As shown in Table 4, the average θ_cap of five brick samples with four salt content was 0.276, 0.272, 0.260, and 0.252 m³/m³, respectively.

Moisture diffusivity

In Figure 9, the θ-λ profiles and the moisture diffusivity (D_l) of sintered blue bricks with different salt content obtained from the ruler method described in Section 2.1 are presented, where the average results of five samples are used. A closer look at Figure 9(a) reflects that the moisture content of bricks slowly decreased first, and then sudden decline appeared as the λ increased for all brick samples with four salt content, indicating the sharp waterfront in our experiment. The detailed results of fitting parameters in equation (9) are shown in Table 5. According to equation (8), the D_l under different moisture content was calculated, as shown in Figure 9(b). The D_l of saline bricks was obviously higher than that of salt free bricks, and it significantly increased with the increase of salt content in almost all moisture content range.

Table 5.

Fitting parameters obtained from the ruler method.

S %	A_cap cm³/(cm² min^0.5)	θ_cap cm³/cm³	λ_f cm/min^0.5	a	b	c
0	0.178	0.276	0.683	0.354	108.192	−74.882
0.16	0.156	0.272	0.663	0.358	28.648	−19.945
0.72	0.151	0.263	0.674	0.348	23.954	−17.088
1.37	0.137	0.252	0.665	0.339	16.519	−11.905

Simulation results

Figure 10 shows the capillary absorption process of sintered bricks with different salt content obtained from the experiment and simulation. For the simulation, the average moisture content at different time of the brick was calculated by the average value of each point in the grid. For the experiment, the average results of five samples with different salt content were chosen. The simulation results showed that the capillary absorption coefficient (A_cap) and capillary moisture content (w_cap) both decreased as salt content increased, which was consistent with the experimental results. However, we can see that the A_cap obtained from simulation was much larger than the experimental value. The A_cap of bricks with four salt content was 0.755, 0.387, 0.378, and 0.324 kg/(m² s^−1/2) respectively. Among them, that the simulated A_cap of salt free bricks was larger than experimental results may be due to the large moisture diffusivity (D_l) in high moisture content range, as presented in Figure 9(b). This indicates that the moisture diffusivity obtained from the ruler method in our experiment may be larger than the actual value, especially in high moisture content range.

Discussion

Our experiments employed the “ruler method” to determine the moisture diffusivity of Chinese sintered blue bricks with varying salt contents. The objective was to investigate alterations in pure water absorption and moisture diffusivity of bricks when subjected to different initial contents of NaCl crystals. Several crucial findings were obtained.

Firstly, the results indicate a significant disparity in the capillary absorption coefficient between Chinese blue bricks and other porous building materials. The capillary absorption coefficient is about 1.35 times higher than that of European red bricks, which are also clay-sintered, as illustrated in Table 6. This probably because the Chinese blue bricks have larger porosity (around 30%) and higher hygroscopic capacity compared with the European red bricks (Li et al., 2021). Secondly, the inclusion of NaCl in bule bricks resulted in a decrease in the capillary absorption coefficient, while the moisture diffusivity (D_l) increases gradually in lower moisture content range but decreases in the higher moisture content range as the salt content increases. This trend became more pronounced with higher NaCl content. Furthermore, it was observed that the moisture diffusivity obtained through the “ruler method” may overestimate the values in the high moisture content range when reproducing the capillary water absorption process using heat-humidity coupled transfer simulation. In the subsequent sections, we will discuss the following two aspects: Firstly, we will delve into the reasons behind the decrease in the capillary absorption coefficient (A_cap) and capillary volumetric moisture content (θ_cap) with increasing NaCl content. Secondly, we will explore the primary factors influencing the accuracy of moisture diffusivity obtained from the “ruler method.”

Table 6.

The A_cap value of various materials obtained from different studies.

Material	A_cap (kg/(m²·s^1/2))	References
Chinese sintered blue bricks	0.230±0.030	This study
Red brick	0.17	Koronthalyova and Bagel (2015)
Sandstone	13.71	Al-Naddaf (2018)
Limestone	2.09	Al-Naddaf (2018)
Foamed cement	0.59391 ± 0.004	Wang et al. (2021)
Aerated concrete	3.9868 ± 0.143	Wang et al. (2021)
Coral sand concrete	1.77879 ± 0.030	Wang et al. (2021)

Effect of salt on moisture transport of sintered bricks

According to the experimental results, the capillary absorption coefficient (A_cap), capillary moisture content (w_cap), and capillary volumetric moisture content (θ_cap) of saline bricks were generally smaller than those of salt free bricks, illustrating that the salt in sintered bricks hinders the capillary migration of water to a certain extent. The pressure difference between the liquid surface and the bottom of the capillary tube leads to the moisture migration in the capillary pores of materials. The capillary absorption process of sintered bricks is similar to the laminar pipe flow in fluid mechanics. Due to the small Reynolds number (small radius and slow flow velocity), the capillary absorption coefficient (A_cap) can be approximately quantified by the Poiseuille equation, as shown in equation (23; Andreas, 2007; Jia et al., 2022).

A_{cap} = φ ρ_{w} \sqrt{\frac{r γ \cos α}{2 η}}

(23)

where φ is the porosity of materials, –; ρ_w is the fluid density, kg/m³; r is the capillary radius, m; γ is the surface tension of the water phase, N/m; α is the contact angle (the angle between the action direction of surface tension γ and the vertical plane), °; η is the viscosity coefficient of liquid, (N·s)/m².

According to the equation (18), the A_cap has a positive correlation with porosity and pore diameter and has a negative correlation with viscosity coefficient. In our experiment, salt will remain in the pores and reduce the porosity of bricks because the bricks have been immersed in salt solution. Furthermore, in the capillary absorption experiment, some salt in bricks dissolves and migrates with water during capillary absorption process, thus the viscosity coefficient of salt solution increases with the salt content. Therefore, the A_cap of sintered bricks will decrease as the salt content increases.

The capillary volumetric moisture content (θ_cap) is the total volume of water absorbed during the first stage of capillary absorption process. Moisture entered the pores of samples under capillary pressure during the capillary absorption test. For saline sintered bricks, some salt crystals dissolved and migrated with water in the pores, while other salt that failed to be dissolved remained in the pores because the first stage of capillary water absorption is a rapid process (about 1–1.5 h). Table 7 shows the salinity of bricks with different salt content, which was calculated by equation (24; Koniorczyk, 2010).

ϕ_{salt} = \frac{V_{salt}}{ϕ_{brick} \cdot V_{brick}} = \frac{M_{salt}}{ρ_{brick} \cdot ϕ_{brick} \cdot V_{brick}}

(24)

where Φ_salt is the salinity, which is the proportion of salt crystals occupying the pores, cm³/cm³; V_salt is the volume occupied by salt crystals, cm³; Φ_brick is porosity of bricks, cm³/cm³; V_brick is volume of bricks, cm³; M_salt is the mass of sodium chloride crystals in bricks, g; ρ_salt is the density of sodium chloride crystals, g/cm³.

Table 7.

Initial proportion of salt crystals occupying pores of bricks with different salt content.

S %	M_salt g	ρ_salt g/cm³	V_salt cm³	V_brick cm³	Φ_brick cm³/cm³	Φ_salt cm³/cm³	θ_cap cm³/cm³
0	0.00	2.165	0.000	125	0.32	0.000	0.276
0.16	0.40	2.165	0.185	125	0.32	0.005	0.272
0.72	1.87	2.165	0.864	125	0.32	0.022	0.263
1.37	3.58	2.165	1.654	125	0.32	0.041	0.252

The decrease of θ_cap compared to salt free bricks (0% NaCl) in Table 7 is slightly smaller to the salinity of bricks, which demonstrates that some salt crystals were undissolved during the first stage of capillary absorption process and remained in the pores. The salt crystals in bricks reduce the volume of open pores, thus resulting in the inability of water to enter the pores. Therefore, the θ_cap may have decreased with the increase of salt content. This phenomenon also partly explains why the water absorption of saline bricks decreased compared with that of salt free bricks during the same time interval in the first stage as shown in Figure 7.

To further investigate the impact of salt content on microscopic pore characteristics of sintered bricks, the mercury intrusion porosimetry (MIP) experiment of sintered bricks with different salt crystal content was conducted. Figure 11 shows the pore size distribution of sintered bricks with different salt content obtained from MIP. As the salt content increased, the number of small pores with a size below 500 nm decreased because these relatively small pores in bricks were filled with salt. It is these small pores that may play an important role in the capillary absorption process, resulting in a slower rate of capillary water absorption coefficient with increasing salt content. The pore size of the peak of the curve represents the main pore size in bricks. As shown in Figure 11, the number of main pore size of saline bricks decreased significantly and the number of large pores over 10⁵ nm increased compared to that of salt free bricks. Moreover, the number of pores with the size of 3000–10,000 nm increased slightly with the increase of salt content. Further statistics show that the average pore diameters of sintered bricks with 0%, 0.16%, 0.72%, and 1.37% NaCl content are 428.49, 427.80, 410.76, and 427.16 nm respectively, and the porosity of bricks are 32.19%, 31.74%, 31.70%, and 30.86% respectively. It can be seen that the porosity of brick decreases with the increase of salt content, thus lead to the decrease of A_cap.

Ruler method for determining moisture diffusivity

The capillary absorption process simulated by using the moisture diffusivity obtained from the ruler method is different from our experiment results, and the capillary absorption coefficient is larger than the measured value, as shown in Figure 10. This may be related to the value of boundary moisture content in θ-λ profile.

A popular parametric description of the moisture diffusivity was a linear approximation, as shown in equation (25).

D_{l} (θ) = D_{l} (θ_{2}) \exp (F \frac{θ - θ_{2}}{θ_{2} - θ_{1}})

(25)

where, θ₁ to θ₂ is the moisture content range where the logD_l-θ the curve is linear. F is the slope of the diffusivity in a logarithmic plot. For sintered bricks we obtain: θ₁ = 0.05 m³/m³, θ₂ = 0.22 m³/m³, F = 4 (approximately). According to the moisture diffusivity obtained from the ruler method as presented in Figure 9(b), the D_l (θ₂) values of sintered bricks with different salt content were 1.1 × 10⁻⁷, 2.9 × 10⁻⁷, 4.1 × 10⁻⁷, and 6.1 × 10⁻⁷ m²/s respectively.

Figiure 12(a) shows the calculated moisture diffusivity of sintered blue bricks without salt content using the linear fitting equation (equation (25)), while Figure 12(b) shows the calculated moisture diffusivity of sintered bricks with different salt contents. It is evident that the linearized new D_l is larger in the low moisture content range and smaller in the high moisture content range compared to the moisture diffusivity obtained by the scale method and, moreover, D_l gradually increases with the increase in salt content over the entire moisture content range. The linearized new D_l was also used to simulate the capillary water absorption process of bricks with different salt content, the simulation results of which are shown in Figure 13. Compared to the original simulation results (Figure 10), the new simulated A_cap decreased significantly, which were 0.174, 0.223, 0.239, and 0.257 kg/(m² s^−1/2) for four different salt contents, respectively, meaning the simulated results were improved. This may be because the linear approximation method declined the moisture diffusivity in the higher moisture content range. However the simulation results still deviated from the experimental results. Moreover, the new simulated A_cap showed an increasing trend with the salt content increase, which was contrary to our experimental results. Thus, the linear approximation method needs to be more improved.

To further optimize our simulation results, the capillary volumetric moisture content (θ_cap) was replaced by saturated volumetric moisture content (θ_sat) in calculating moisture diffusivity using ruler method. The θ_sat of salt free bricks is the porosity, which is 0.32 m³/m³. And the θ_sat of saline bricks was determined according to the θ_cap measured in our experiment. Figure 14 shows the comparison of θ-λ profile and moisture diffusivity of salt free bricks using θ_cap and θ_sat. In Figure 14(a), the slope of θ-λ profile using θ_sat increases significantly compared with that of the θ-λ profile using θ_cap. It is obvious that with increasing boundary moisture content (λ = 0) in the θ-λ curve from θ_cap and θ_sat, the D_l increased in the lower moisture content range and decreases in the higher moisture content range as shown in Figure 14(b). The θ-λ profiles and moisture diffusivity of sintered bricks with different salt content using θ_sat are calculated, as shown in Figure 15. The D_l increases gradually in lower moisture content range but decreases in the higher moisture content range as the salt content increases. The fitting parameters in equation (9) are shown in Table 8.

Table 8.

Fitting parameters obtained from ruler method using saturated volumetric moisture content.

S %	A_cap cm³/(cm² min^0.5)	θ_sat cm³/cm³	λ_f cm/min^0.5	A	b	c
0	0.178	0.32	0.683	0.431	15.566	−11.549
0.16	0.156	0.316	0.663	0.448	8.073	−6.204
0.72	0.151	0.311	0.674	0.452	6.332	−5.090
1.37	0.137	0.309	0.665	0.484	3.868	−3.314

Figure 16 presents the renewal simulation results of the capillary absorption process of bricks with different salt content using the moisture diffusivity obtained by using saturated volumetric moisture content (θ_sat). In the renewal simulation, the maximum volumetric moisture content with four salt content was determined according to the experimental results (as the moisture content of the last data point of the capillary water absorption experiment), which were 0.300, 0.289, 0.285, and 0.275 m³/m³, respectively. As shown in Figure 16, the simulated A_cap (the slope of the first stage) of samples with lower salt content (0% and 1%) were slightly smaller, while at higher salt content (5% and 10%) in good agreement with the experimental results. Overall, comparing with the previous simulation results (Figure 9), the simulation results using the new moisture diffusivity were closer to the experimental values with a maximum error of 5.8% and a minimum error of 2.5%, as presented in Table 9. It indicates that the value of the boundary moisture content has a large influence on the moisture diffusivity and the Zone 1 in Figure 4(a) where the moisture content is above θ_cap cannot be ignored when solving moisture diffusivity in the ruler method.

Table 9.

Capillary absorption coefficient A_cap (kg/(m² s^−1/2)) obtained from new simulation and experiment results.

	0%S	0.16%S	0.72%S	1.37%S
Experiment	0.230	0.202	0.201	0.189
Simulation	0.218	0.197	0.196	0.178
Error	5.2%	2.5%	2.5%	5.8%

In the coupled heat and moisture model, the initial salt crystals in the sample were assumed to remain as crystals during the whole absorption process. This approximation may not be so bad at least during the first phase because it is a short period from 1 to 1.5 h. Although the moisture diffusivity calculated using saturated moisture content effectively improves the prediction accuracy of the water transfer process, the coupled heat and moisture model does not consider the salt precipitation/dissolution and salt migration during capillary water absorption and has some limitations in modelling the capillary water absorption process in saline bricks. In future studies, a more refined coupled heat, moisture, and salt transfer model should be developed to better reproduce the effect of salt in the material on the water transfer process.

Conclusion

The moisture diffusivity of salt-containing porous building materials under the effect of salt is crucial to the advancement of heat-moisture-salt transfer simulations. This study endeavors to determine the moisture diffusivity (D_l) of sintered blue bricks using the ruler method during pure water capillary absorption experiments. Comparative analyses involving capillary absorption coefficient (A_cap), capillary moisture content (w_cap), and moisture diffusivity (D_l) were conducted on sintered blue bricks containing different NaCl crystals content to elucidate the influence of salt on moisture transport in these bricks. Furthermore, the applicability of the ruler method was investigated and optimized through heat and moisture transfer simulation, and the factors influencing the ruler method’s results were discussed. The conclusions drawn from the study are summarized as follows:

(1) The presence of NaCl in sintered bricks reduced the capillary absorption coefficient (A_cap), and it decreased with the increase of NaCl content overall. The average A_cap of bricks with 1.37% NaCl content was 18% lower than that of salt free bricks.

(2) As the NaCl content of sintered blue bricks increased, the capillary moisture content (w_cap) generally decreased. The average w_cap of brick samples with 1.37% NaCl content was 8.6% smaller than that of salt free bricks.

(3) The moisture diffusivity (D_l) increases gradually in lower moisture content range but decreases in the higher moisture content range as the NaCl content increases.

(4) The “ruler method” can give the moisture diffusivity, but the value of boundary moisture content has a great impact on moisture diffusivity obtained from this method. By increasing boundary moisture content (when λ = 0) in the θ-λ curve, the D_l increases in the lower moisture content range and decreases in the higher moisture content range. The use of saturated moisture content (θ_sat) in the “ruler method” seems to give more accurate moisture diffusivity than that of capillary moisture content (θ_wap), but further research is still required.

In conclusion, the experimental results in this study provide a valuable database for coupled heat and moisture simulations of sintered brick heritage. Additionally, the study enhances our understanding of the effect of NaCl on moisture transfer in sintered bricks. These results are expected to provide some insights into the conservation of masonry heritage in research and practice. The applicability of the “ruler method” for determining the moisture diffusivity of sintered bricks with varying salt content is further investigated and improved by optimized simulation. However, further research is required, such as repeating the experiment using alternative methods like X-ray attenuation and improve the coupled heat, moisture and salt model. Furthermore, this study focused only on sodium chloride salt, and future research will explore the effect of other salts and salt mixture on the moisture diffusivity of sintered bricks.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the National Natural Science Foundation of China (Grant No. 52108006 and 52278013) and the China National Key R&D Program during the 13th Five-Year Plan Period (Grant No. 2019YFC1520901).

ORCID iDs

Daisuke Ogura

Yonghui Li

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