*2.1. Test Materials*

Concrete was made of cement, mineral powder, fine aggregate, and coarse aggregate. The binder material comprised P.O42.5 ordinary Portland cement and mineral powder with densities of 3060 and 2890 kg/m<sup>3</sup> respectively, in equal mass fraction. The chemical composition of cement is shown in Table 1. The fine aggregate was well-graded medium sand with a measured apparent density of 2623 kg/m<sup>3</sup> (Table 2). The coarse aggregate had a 5–20 mm continuous particle gradation, a maximum particle size of 20 mm (Table 3), and a measured apparent density of 2710 kg/m<sup>3</sup> . The fresh concrete was cast into 120 molds in dimensions of 100 mm × 100 mm × 100 mm. After curing at 20 ± 2 ◦C and 95% RH for 28 days, each specimen was cut horizontally into three pieces in a dimension of 100 mm × 100 mm × 30 mm. To avoid separation from vibration, only the middle piece was selected for testing. In total, 60 absorption and 60 desorption specimens were prepared. Table 4 provides the mix proportion of concrete in accordance with previous studies [19–21].

**Table 1.** Chemical compositions of cement in mass.


**Table 2.** Particle size distribution of fine aggregate.


**Table 3.** Particle size distribution of coarse aggregate.



The tests were performed as per ISO12571-2013 [22]. The specimens for the absorp-

261.0 562.0 514.0 960.0

**Fine Aggregate (sand) kg/m3**

**Coarse Aggregate kg/m3**

**Table 4.** Concrete mix ratio for the absorption and desorption specimens. tion test were first oven-dried at 100 °C for 24 h and then placed in a dryer to cool down

**Table 4.** Concrete mix ratio for the absorption and desorption specimens.

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**Binder Material kg/m3**

### *2.2. Test Method* with a towel and weighed. Then, the specimens were returned to the water for 24 h and weighed. This was repeated until the change in the specimen mass was less than 0.5% to

**Water kg/m3**

*2.2. Test Method* 

The tests were performed as per ISO12571-2013 [22]. The specimens for the absorption test were first oven-dried at 100 ◦C for 24 h and then placed in a dryer to cool down to 20 ◦C (room temperature) and weighed. Subsequently, all specimens were oven-dried for another 12 h, cooled down to 20 ◦C in the dryer and weighed. These steps were repeated until the change in mass was less than 0.5% to obtain a dry state. For the desorption test, all dry specimens were placed in water for seven days to be saturated, surface-dried with a towel and weighed. Then, the specimens were returned to the water for 24 h and weighed. This was repeated until the change in the specimen mass was less than 0.5% to obtain a saturated state. obtain a saturated state. Afterward, the specimens were moved to glass containers in a constant-temperature/humidity chamber with a temperature accuracy of ±1 °C and 50 ± 2% RH as shown in Figure 1. During the test, saturated salt solutions of LiCl, MgCl2, KBr and KNO3 were added at the bottom of glass containers to secure 10%, 35%, 80% and 95% RH respectively. It should be noted that for the specimens at 50% RH, there was no saturated salt solution. The testing temperatures were 20 °C, 35 °C and 50 °C. Each specimen was weighed at designated time interval until the change in mass was less than 0.1%. It should be noted that since the mass change was decreasing over time,

Afterward, the specimens were moved to glass containers in a constant-temperature/ humidity chamber with a temperature accuracy of ±1 ◦C and 50 ± 2% RH as shown in Figure 1. During the test, saturated salt solutions of LiCl, MgCl2, KBr and KNO<sup>3</sup> were added at the bottom of glass containers to secure 10%, 35%, 80% and 95% RH respectively. It should be noted that for the specimens at 50% RH, there was no saturated salt solution. The testing temperatures were 20 ◦C, 35 ◦C and 50 ◦C. the time interval of weight measurement in the early stage was shorter than in the later stage. In total, there were 12 weight measurements for the absorption test and 13 weight measurements for the desorption test. The specimens were designated as X-% and P-% for the absorption and desorption tests, respectively. The percentage represented the RH conditions. For example, specimens X-10% is the absorption specimens for 10% RH.

**Figure 1.** Isothermal adsorption devices (**a**) glass container, (**b**) constant-temperature/humidity chamber. **Figure 1.** Isothermal adsorption devices (**a**) glass container, (**b**) constant-temperature/humidity chamber.

*2.3. Determination of Saturation*  The moisture content of a specimen can be defined as follows [8]: Each specimen was weighed at designated time interval until the change in mass was less than 0.1%. It should be noted that since the mass change was decreasing over time, the time interval of weight measurement in the early stage was shorter than in the later stage. In total, there were 12 weight measurements for the absorption test and 13 weight measurements for the desorption test.

The specimens were designated as X-% and P-% for the absorption and desorption tests, respectively. The percentage represented the RH conditions. For example, specimens X-10% is the absorption specimens for 10% RH.

## *2.3. Determination of Saturation*

The moisture content of a specimen can be defined as follows [8]:

$$S = \frac{V\_S}{V\_\phi} \tag{1}$$

where *S* is the moisture saturation, *V<sup>S</sup>* is the volume occupied by water in the pores of the specimen, and *V<sup>φ</sup>* is the volume of pores in the specimen.

In order to determine adsorption and desorption isotherms, the moisture saturation *S* can be expressed based on the water content [23]:

$$S = \frac{m\_{\rm W} - m\_{\rm d}}{m\_{\rm WS} - m\_{\rm d}} \tag{2}$$

where *m<sup>w</sup>* is the mass of specimen underwater (kg), *m<sup>d</sup>* is the mass of the specimen in a dry state (kg) and *mws* is the mass of the specimen under saturation state (kg).

### **3. Results and Discussion**

### *3.1. Absorption–Desorption Curves*

Absorption mainly refers to physical absorption, which is the phenomenon that moisture enters concrete through capillary pressure, while desorption is the phenomenon that moisture in concrete is transferred from liquid to gas and released from concrete.

Absorption–desorption curves representing the changes in the moisture contents of the specimens over time at 20, 35, and 50 ◦C are shown in Figure 2.

Figure 2a shows that, at the same temperature, the absorption curves tended to be consistent between the different RH conditions, as the specimen saturation increased significantly during the initial stage of the absorption process. As absorption progressed, the saturation between the interior and exterior of concrete decreased due to decreased moisture absorption capacity and eventually stabilized. The moisture absorption behaviors of the specimens were similar at different temperatures. For a given temperature, a higher RH resulted in faster moisture absorption. When the RH was lower than 50%, the moistureabsorption process of the specimen was gradual, which reveals that temperature and RH had little effect on the time required to reach the equilibrium. The absorption rate and capacity increased sharply when the RH was greater than 50%. At 80% RH, the moisture absorption capacity and equilibrium time both increased as temperature decreased. Under low-RH conditions (i.e., less than 50% RH), the concrete exhibited less moisture absorption capacity with saturation less than 0.1, and temperature's effect on the moisture absorption rate was marginal. When RH was greater than 50%, temperature's effect on the moisture absorption became significant.

During the desorption test, a lower RH came with a greater saturation between the interior and exterior of concrete, and a longer time to reach equilibrium, as shown in Figure 2b. When the RH was lower than 80%, the increment of temperature increased desorption capacity and the time to reach equilibrium. When RH was higher than 80%, the effect of temperature on the moisture transport rate decreased. Regardless of temperature, the saturation at equilibrium was approximately 0.9.

**Figure 2.** Absorption and desorption over time: (**a**) 20 °C, (**b**) 35 °C, (**c**) 50 °C. **Figure 2.** Absorption and desorption over time: (**a**) 20 ◦C, (**b**) 35 ◦C, (**c**) 50 ◦C.

### *3.2. RH–Saturation Equilibrium 3.2. RH–Saturation Equilibrium*

The RH–saturation curves during absorption and desorption for different RH conditions are shown in Figure 3. The curves were substantially similar for different temperatures. As RH increased, the saturation gradually increased. When RH was below 50%, the absorption process was gradual. The absorption capacity significantly increased when RH was greater than 50%. Irrespective of temperature, 50% RH was an inflection point in the moisture absorption process. Based on the Kelvin equation, the maximum pore size that could be saturated was approximately 4 nm at 50% RH, and the boundary between the gel pores and capillary pores was generally 10 nm (i.e., gel pores < 10 nm < capillary pores). Therefore, when RH was below 50%, only part of the gel pores was saturated, while for RH above 50%, the capillary pores saturated gradually, thereby rapidly increasing the saturation [24]. Further, the saturation decreased with decreasing RH. As temperature decreased, the saturation considerably decreased. At 80% RH, the equilibrium saturation decreased substantially, which indicates that an RH greater than 80% could greatly affect the equilibrium of the moisture desorption process in concrete. The RH–saturation curves during absorption and desorption for different RH conditions are shown in Figure 3. The curves were substantially similar for different temperatures. As RH increased, the saturation gradually increased. When RH was below 50%, the absorption process was gradual. The absorption capacity significantly increased when RH was greater than 50%. Irrespective of temperature, 50% RH was an inflection point in the moisture absorption process. Based on the Kelvin equation, the maximum pore size that could be saturated was approximately 4 nm at 50% RH, and the boundary between the gel pores and capillary pores was generally 10 nm (i.e., gel pores < 10 nm < capillary pores). Therefore, when RH was below 50%, only part of the gel pores was saturated, while for RH above 50%, the capillary pores saturated gradually, thereby rapidly increasing the saturation [24]. Further, the saturation decreased with decreasing RH. As temperature decreased, the saturation considerably decreased. At 80% RH, the equilibrium saturation decreased substantially, which indicates that an RH greater than 80% could greatly affect the equilibrium of the moisture desorption process in concrete.

**Figure 3.** Absorption and desorption equilibrium curves: (**a**) 20 °C, (**b**) 35 °C, (**c**) 50 °C. **Figure 3.** Absorption and desorption equilibrium curves: (**a**) 20 ◦C, (**b**) 35 ◦C, (**c**) 50 ◦C.

In Figure 3, since the desorption curve is above the absorption curve, it indicates a higher saturation for a given RH. The desorption process displays a significantly hysteretic nature compared to the absorption process, which implies that the gas–liquid equilibrium in pores is nonhomogeneous. The hysteresis represents the "ink bottle effect," which refers to the effect that a small pore (bottleneck) exerts stress upon the water in a connected large pore (bottle) [25]. Larger concrete capillary pores require a higher RH to reach equilibrium saturation than small pores. For initially saturated pores, the liquid water in large pores transports through the liquid water in small pores without forming a gas–liquid interface. Therefore, the water in large pores can be discharged only when the ambient RH is below the saturated RH of small pores. For a given RH, the saturation equilibrium of the moisture absorption process is lower than that of the moisture desorption process In Figure 3, since the desorption curve is above the absorption curve, it indicates a higher saturation for a given RH. The desorption process displays a significantly hysteretic nature compared to the absorption process, which implies that the gas–liquid equilibrium in pores is nonhomogeneous. The hysteresis represents the "ink bottle effect," which refers to the effect that a small pore (bottleneck) exerts stress upon the water in a connected large pore (bottle) [25]. Larger concrete capillary pores require a higher RH to reach equilibrium saturation than small pores. For initially saturated pores, the liquid water in large pores transports through the liquid water in small pores without forming a gas–liquid interface. Therefore, the water in large pores can be discharged only when the ambient RH is below the saturated RH of small pores. For a given RH, the saturation equilibrium of the moisture absorption process is lower than that of the moisture desorption process [26].

[26]. For a given RH, the capillary condensation phenomenon is more likely to occur during the absorption process, and a higher saturation can be attained at a lower temperature. During desorption, at a higher temperature, the saturation is lower because the pore water evaporates more easily [27]. According to a study by Zeng [8], the relationship between For a given RH, the capillary condensation phenomenon is more likely to occur during the absorption process, and a higher saturation can be attained at a lower temperature. During desorption, at a higher temperature, the saturation is lower because the pore water evaporates more easily [27]. According to a study by Zeng [8], the relationship between the capillary pressure and RH is:

$$p\_{\mathbb{C}} = \frac{\rho\_{\mathbb{I}}RT}{M} \ln h \tag{3}$$

<sup>l</sup> <sup>c</sup> ln *RT p h <sup>M</sup>* ρ= (3) where *M* is the molar mass of water (kg/mol), *R* is the gas constant (given as 8.314 where *M* is the molar mass of water (kg/mol), *R* is the gas constant (given as 8.314 J/mol/K), *p*<sup>c</sup> is the capillary pressure (Pa), *h* is the RH, *T* is the absolute temperature (K) and *ρ*<sup>l</sup> is the density of liquid water (kg/m<sup>3</sup> ).

J/mol/K), <sup>c</sup> *p* is the capillary pressure (Pa), *h* is the RH, *T* is the absolute temperature (K) and ρ<sup>l</sup> is the density of liquid water (kg/m3). Using Equation (3), the isothermal equilibrium curve of RH and concrete saturation can be converted into a moisture characteristic curve, as shown in Figure 4. When the concrete was at the same saturation through absorption and desorption, the capillary pressure of the desorbed concrete was greater than that of the absorbed concrete, reflecting

the capillary pressure and RH is:

the hysteretic nature of the desorption process. When the temperature was lower during both the absorption and desorption processes, the capillary pressure was greater, as well as the equilibrium saturation corresponding to the same capillary pressure. the hysteretic nature of the desorption process. When the temperature was lower during both the absorption and desorption processes, the capillary pressure was greater, as well as the equilibrium saturation corresponding to the same capillary pressure.

Using Equation (3), the isothermal equilibrium curve of RH and concrete saturation can be converted into a moisture characteristic curve, as shown in Figure 4. When the concrete was at the same saturation through absorption and desorption, the capillary pressure of the desorbed concrete was greater than that of the absorbed concrete, reflecting

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## *3.3. RH and Saturation Equilibrium Model*

*3.3. RH and Saturation Equilibrium Model*  In [28], a model representing the absorption of water by the hygroscopic material and In [28], a model representing the absorption of water by the hygroscopic material and RH is shown as follows:

$$\mathcal{W} = \left[ \frac{-\ln(1 - h)}{A(T + B)} \right]^{\frac{1}{C}} \tag{4}$$

*<sup>C</sup> <sup>h</sup> <sup>W</sup>* where *W* is the equilibrium moisture content and *A*, *B* and *C* are coefficients.

<sup>1</sup> ln(1 ) [ ] ( ) *AT B* − − <sup>=</sup> <sup>+</sup> (4) By replacing the equilibrium moisture content with the saturation, Equation (4) can be applied to find the inverse function. The RH and saturation relationship is:

$$h = 1 - \exp(-A(T+B)S^C) \tag{5}$$

be applied to find the inverse function. The RH and saturation relationship is: where *S* is the saturation.

RH is shown as follows:

=1 exp( ( ) ) *<sup>C</sup> h AT B S* −−+ (5) where *S* is the saturation. The model of the relationship between RH and equilibrium saturation can be obtained by substituting the data from the experimental results as seen in Figure 3 into Equation (5). Nonlinear regression was used to yield Equations (6) and (7) for the absorption and desorption processes, respectively.

$$h\_{\rm abs} = 1 - \exp\left(-0.03(T - 180.7) \text{S}^{0.837}\right) \qquad \mathbb{R}^2 = 0.91\tag{6}$$

$$h\_{\rm ds} = 1 - \exp(-0.05(T - 258.76)\mathbf{S}^{5.04}) \qquad \mathbb{R}^2 = 0.91 \tag{7}$$

law. For saturated concrete, the permeability coefficient is constant and a function of sat-

### and desorption processes, respectively. **4. Moisture Transport Model**

as *h TS R* =− − − = 1 exp( 0.03( 180.7) ) 0.91 (6) *4.1. Moisture Movement in Unsaturated Concrete*

uration for unsaturated concrete.

5.04 2 ds *h T SR* =− − − = 1 exp( 0.05( 258.76) ) 0.91 (7) The transport of liquid water in unsaturated concrete can be expressed by Darcy's law. For saturated concrete, the permeability coefficient is constant and a function of saturation for unsaturated concrete.

**4. Moisture Transport Model**  According to the Darcy–Buckingham equation, the relationship between transport rate of liquid water and capillary pressure is expressed as [11]:

$$v\_{\rm l} = -\frac{k\_{\rm l}}{\mu} \text{grad}(p\_{\rm c}) \tag{8}$$

0.837 2

where *v*<sup>1</sup> is the rate (m/s); *k*<sup>1</sup> is the effective permeability for moisture transport (m<sup>2</sup> ), which is related to the moisture content in concrete and micro-geometric parameters of the pore occupied by moisture; *µ* is the dynamic viscosity coefficient of water (Pa·s).

If the conversion of liquid water to water vapor in the pores of concrete is neglected, the moisture transport in the concrete conforms to the law of mass conservation. The moisture transport equation in concrete can be written as below [29,30]:

$$\frac{\partial\theta}{\partial t} = \text{div}\left[\frac{k\_l}{\mu}\text{grad}(p\_\circ)\right] \tag{9}$$

Converting the left side of Equation (9) to pore moisture content and the right side into the pore moisture saturation gradient, it becomes:

$$\frac{\partial \theta}{\partial t} = \text{div} \left[ \frac{k\_1}{\mu} \frac{\partial p\_\mathbf{c}}{\partial S} \text{grad}(S) \right] \tag{10}$$

where *θ* is the moisture content of the concrete.

The effective permeability to moisture transport (*k*1) can be expressed as

$$k\_{\rm l} = k k\_{\rm rl} \tag{11}$$

Therefore, Equation (10) can be rewritten as follows:

$$\frac{\partial \theta}{\partial t} = \text{div} \left[ \frac{k k\_{\text{rl}}}{\mu} \frac{\partial p\_{\text{c}}}{\partial S} \text{grad}(S) \right] \tag{12}$$

where *k* is the intrinsic permeability of concrete (m<sup>2</sup> ), and *k*rl is the relative permeability, which is a parameter associated with the saturation.

Equation (12) can be expressed as:

$$\frac{\partial \theta}{\partial t} = \text{div}[D(\mathcal{S})\text{grad}(\mathcal{S})] \tag{13}$$

$$D(S) = \frac{k}{\mu} k\_{\rm rl} \frac{\partial p\_{\rm c}}{\partial S} \tag{14}$$

where *D(S)* is the transport coefficient of liquid water (m2/s).

The relationship between the dynamic viscosity coefficient of water (*µ*), and temperature can be expressed as follows: [31]

$$\mu = \frac{0.001775}{\left[1 + 0.0837(T - 273.15) + 0.000221(T - 273.15)^2\right]}\tag{15}$$

The intrinsic permeability (*k)* of concrete can be determined based on the porosity, specific surface area and tortuosity of concrete using the Kozeny–Carman model [13]:

$$k = \frac{\phi^3}{F\_\text{s} \tau\_\text{c}^2 \Omega^2 \left(1 - \phi\right)^2} \tag{16}$$

where *F*<sup>s</sup> is the factor representing the influence of pore shape (2 for a circular and tubular pore), *τ*<sup>c</sup> is the tortuosity and Ω is the specific surface.

### *4.2. Relationship between the Moisture Transport Parameters*

By substituting Equations (6) and (7), which represent the equilibrium relationship between RH and saturation, into Equation (3), the relationship between the capillary pressure and saturation during the moisture absorption and desorption processes can be expressed as Equations (17) and (18), respectively.

$$p\_{\rm as} = \frac{\rho\_1 RT}{M} \ln(1 - \exp(-0.03(T - 180.7) \text{S}^{0.837})) \tag{17}$$

$$p\_{\rm ds} = \frac{\rho\_{\rm l}RT}{M} \ln(1 - \exp(-0.05(T - 258.76)S^{5.04})) \tag{18}$$

The relationship between the relative permeability (*k*rl(*S*)), and relative saturation can be established based on previous studies [14,15,32]:

$$p\_{\mathbb{C}} = A\_{\mathbb{A}} (\mathbb{S}^{-1/m} - 1)^{1-m} \tag{19}$$

$$k\_{\rm rl}(S) = \sqrt{S} \left[ 1 - \left( 1 - S^{1/m} \right)^{m} \right]^{2} \tag{20}$$

where *m* and *A*<sup>a</sup> are undetermined coefficients.

The value of *m* can be determined through regression of the capillary pressure versus saturation curve. The absolute value of capillary pressure *m* for absorption and desorption processes at different temperatures can be derived based on Equation (19) through nonlinear fitting.

Using the data in Table 5 to fit the relationship between *m* and *T* for the absorption and desorption processes gives Equations (21) and (22), respectively.

$$m\_{\rm as} = 1/(2.89 \ln(T) - 14.89) \tag{21}$$

$$m\_{\rm ds} = 1/(21.06 - 3.16 \ln(T))\tag{22}$$

**Table 5.** *m* values corresponding to the absorption and desorption process.


Equations (21) and (22) are then substituted into Equation (20) to express the relationship between the relative permeability and saturation during the moisture absorption process.

$$k\_{\rm rl-as}(S) = \sqrt{S} [1 - (1 - S^{2.89 \ln(T) - 14.89})^{1/[2.89 \ln(T) - 14.89]}]^2 \tag{23}$$

Thus, the relationship between the relative permeability and saturation during the moisture desorption process is:

$$k\_{\rm rl-ds}(S) = \sqrt{S} [1 - (1 - S^{21.06 - 3.16 \ln(T)})^{1/[21.06 - 3.16 \ln(T)]}]^2 \tag{24}$$

### *4.3. Moisture Transport Model*

The relationships between capillary pressure and saturation during the process of moisture absorption and desorption are shown in Equations (17) and (18). These equations are then substituted into Equation (13) to develop the moisture transport model for absorption and desorption processes.

$$\frac{\partial \phi S}{\partial t} = \text{div}[D(S)\text{grad}(S)]\tag{25}$$

where *φ* is the porosity of concrete.

Transport coefficient *D*(*S*) of Equation (25) in absorption and desorption processes are then represented as Equations (26) and (27), respectively.

$$D\_{\rm as}(S) = \frac{0.02511k k\_{\rm l-as} \rho\_l RT(T - 180.7)S^{-0.163} \exp(-0.03(T - 180.7)S^{0.837})}{\mu M (1 - \exp(-0.03(T - 180.7)S^{0.837}))} \tag{26}$$

$$D\_{\rm ds}(S) = \frac{0.252 k k\_{\rm l-ds} \rho\_l RT (T - 258.76) S^{4.04} \exp(-0.05 (T - 258.76) S^{5.04})}{\mu M (1 - \exp(-0.05 (T - 258.76) S^{5.04}))} \tag{27}$$

### *4.4. Heat Balance Equation*

The heat balance equation correlated to the saturation can be established based on Fourier's law [11,33]:

$$c\rho \frac{\partial T}{\partial t} + \nabla \cdot [-\lambda\_S \nabla T] = Q \tag{28}$$

where *c* is the specific heat of material (J/kg·K), *λ<sup>S</sup>* is the thermal conductivity at different saturations (W/m·K), *<sup>ρ</sup>* is the material density (kg/m<sup>3</sup> ) and *Q* is the heat flux (J/m<sup>2</sup> ·s). The negative sign indicates that the direction of heat flow is opposite to the direction of the temperature gradient.

The transient plane source (TPS) technique was used to determine the thermal conductivity for the concrete in the same mix proportion as in the present study and at the same saturation values of 0, 0.3, 0.5, 0.7, 0.9 and 1.0 as in the moisture absorption and desorption tests described in Section 2.1. The relationship between the thermal conductivity and saturation is:

$$
\lambda\_S = (1 + 0.3432S^2)\lambda\_\text{a} \tag{29}
$$

where *λ*<sup>a</sup> is the thermal conductivity under dry state (W/m·K).

### **5. Numerical Simulation**

### *5.1. Model Parameters and Mesh Generation*

The concrete was considered as a uniform and isotropic continuous medium; thus the concrete skeleton was assumed as an impermeable material that did not react with gas or liquid phases. COMSOL®® software (COMSOL Inc., Stockholm, Sweden) was used to establish a three-dimensional numerical model based on Equations (25)–(28), and one-dimensional transmission was adopted. The physical parameters of heat and moisture transport are shown in Tables 6 and 7. The size of the simulated specimen was the same as that of the experimental specimen used in the test, namely 30 mm × 100 mm × 100 mm. A total of 9520 tetrahedral elements were used in COMSOL®® software to develop the finite element model with DOFs of 28,186 and convergence criteria of 10−<sup>5</sup> , as shown in Figure 5.





**Parameter Dry Thermal Conductivity,**

**(m2)** 

**Parameter Intrinsic Permeability,** *<sup>k</sup>*

**Parameter Dry Thermal Conductivity,**

**(W/m·K)** 

**(W/m·K)** 

**(m2)** 

**Table 6.** Heat transfer physical parameters.

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**Table 6.** Heat transfer physical parameters.

Value 1.578 900 2297 0.1796

**Table 7.** Moisture transport physical parameters.

Value 1.578 900 2297 0.1796

**Table 7.** Moisture transport physical parameters.

**Water Density,** 

**Water Density,** 

**(kg/m3)** 

Value 1.413 × 10−21 1000 8.3144 0.018

**(kg/m3)** 

λa

λa

Value 1.413 × 10−21 1000 8.3144 0.018

**Specific Heat,** *c*  **(J/kg·K)** 

**Specific Heat,** *c*  **(J/kg·K)** 

**Gas Constant** *R*

**Gas Constant** *R*

**Concrete Density, (kg/m3)** 

**(J/mol·K) Molar Mass of Water,** *<sup>M</sup>* **(kg/mol)** 

**Concrete Density, (kg/m3)** 

**(J/mol·K) Molar Mass of Water,** *<sup>M</sup>* **(kg/mol)** 

**Porosity,**  *ϕ*

**Porosity,**  *ϕ*

**Figure 5.** Mesh elements division of concrete. **Figure 5.** Mesh elements division of concrete. librium saturation. The initial temperature of inside concrete was 283.15 K, and the simu-

### *5.2. Moisture Transport Model Validation 5.2. Moisture Transport Model Validation* lation time was the hygroscopic equilibrium time for the three temperatures, which was approximately 350 h. The simulated results were compared with the experimental results,

0.2

The experimental results shown in Figure 2 were simulated for 80% and 95% RH at 20, 35 and 50 °C to validate the moisture transport model. The initial internal saturation of the concrete was set as 0.08, and the boundary saturation was the corresponding equilibrium saturation. The initial temperature of inside concrete was 283.15 K, and the simulation time was the hygroscopic equilibrium time for the three temperatures, which was approximately 350 h. The simulated results were compared with the experimental results, as shown in Figure 6. The experimental results shown in Figure 2 were simulated for 80% and 95% RH at 20, 35 and 50 ◦C to validate the moisture transport model. The initial internal saturation of the concrete was set as 0.08, and the boundary saturation was the corresponding equilibrium saturation. The initial temperature of inside concrete was 283.15 K, and the simulation time was the hygroscopic equilibrium time for the three temperatures, which was approximately 350 h. The simulated results were compared with the experimental results, as shown in Figure 6. as shown in Figure 6. The simulated and experimental results show a similar trend, and the values also agreed well. The simulated results were slightly lower than the experimental results. Under the simulated conditions, the moisture absorption rate decreased, and the moisture absorption capacity was slightly lower than that in the experiment. This could be because the moisture transport model only considers the transport of liquid water, not the moisture vapor transport that occurred under low-saturation conditions.

of the concrete was set as 0.08, and the boundary saturation was the corresponding equi-

0.3 0.3 **Figure 6.** Test and simulation values of moisture absorption process: (**a**) is 80% RH, (**b**) is 95% RH.

0.2

 (**a**) (**b**) 0 0.1 0 40 80 120 160 200 240 280 320 360 Time(h) 0 0.1 0 40 80 120 160 200 240 280 320 360 Time(h) The simulated and experimental results show a similar trend, and the values also agreed well. The simulated results were slightly lower than the experimental results. Under the simulated conditions, the moisture absorption rate decreased, and the moisture absorption capacity was slightly lower than that in the experiment. This could be because the moisture transport model only considers the transport of liquid water, not the moisture vapor transport that occurred under low-saturation conditions.

The experimental results provided in Figure 2 were applied to validate the moisture desorption model. The simulation was performed under the conditions of 10% and 35% RH at 20, 35 and 50 ◦C. The initial temperature inside the concrete was 283.15 K. The concrete interior was initially saturated and the boundary saturation was the corresponding equilibrium saturation. The simulation time was 250 h.

Figure 7 provides a comparison of the simulated and experimental results, which shows that the simulated values were in good agreement with the experimental values under 35% RH and were slightly higher than the experimental values at 10% RH. The overall results demonstrated that the simulated model of moisture desorption was valid. This also indicates that the simulated results remained accurate when the concrete was at a higher saturation level. When the saturation decreased, the simulated moisture desorption

rate was slightly lower than the experimental rate, implying that the water vapor transport in the concrete exerted limited influence under the low-saturation condition. In the marine environment, it is difficult for concrete to reach a low saturation state due to high RH. Therefore, the simulated model of moisture transport was accurate for the applications in the high RH environment. a higher saturation level. When the saturation decreased, the simulated moisture desorption rate was slightly lower than the experimental rate, implying that the water vapor transport in the concrete exerted limited influence under the low-saturation condition. In the marine environment, it is difficult for concrete to reach a low saturation state due to high RH. Therefore, the simulated model of moisture transport was accurate for the applications in the high RH environment.

equilibrium saturation. The simulation time was 250 h.

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**Figure 6.** Test and simulation values of moisture absorption process: (**a**) is 80% RH, (**b**) is 95% RH.

The experimental results provided in Figure 2 were applied to validate the moisture desorption model. The simulation was performed under the conditions of 10% and 35% RH at 20, 35 and 50 °C. The initial temperature inside the concrete was 283.15 K. The concrete interior was initially saturated and the boundary saturation was the corresponding

Figure 7 provides a comparison of the simulated and experimental results, which shows that the simulated values were in good agreement with the experimental values under 35% RH and were slightly higher than the experimental values at 10% RH. The overall results demonstrated that the simulated model of moisture desorption was valid. This also indicates that the simulated results remained accurate when the concrete was at

**Figure 7.** Test and simulation values of moisture desorption process: (**a**) is 10% RH, (**b**) is 35% RH. **Figure 7.** Test and simulation values of moisture desorption process: (**a**) is 10% RH, (**b**) is 35% RH.

### *5.3. Absorption*

The upper, lower and right ends of the two-dimensional geometric model of concrete were defined as heat and moisture insulation. Therefore, moisture absorption was from left to inside. The effect of temperature on the moisture absorption process at 95% RH was evaluated under the temperatures of 293.15, 308.15 and 323.15 K and the boundary saturation values of 0.80, 0.72, and 0.39, respectively. The simulation had an initial temperature of 273.15 K, saturation of 0.2 and time of 500 days. The upper, lower and right ends of the two-dimensional geometric model of concrete were defined as heat and moisture insulation. Therefore, moisture absorption was from left to inside. The effect of temperature on the moisture absorption process at 95% RH was evaluated under the temperatures of 293.15, 308.15 and 323.15 K and the boundary saturation values of 0.80, 0.72, and 0.39, respectively. The simulation had an initial temperature of 273.15 K, saturation of 0.2 and time of 500 days.

The simulated results are shown in Figure 8. It can be seen that the saturation was the lowest at 323.15 K (i.e., the highest temperature) but the highest at 293.15 K (i.e., the lowest temperature). For a given RH, an increased temperature decreased boundary saturation, moisture transport rate, moisture transport capacity, moisture transport distance and moisture absorption. The saturation gradient is the main driving potential of moisture transfer. A higher temperature reduced moisture absorption rate. Meanwhile, with temperature increasing, the equilibrium saturation of the concrete boundary further decreased. Such a combined effect decreased moisture absorption of concrete. The simulated results are shown in Figure 8. It can be seen that the saturation was the lowest at 323.15 K (i.e., the highest temperature) but the highest at 293.15 K (i.e., the lowest temperature). For a given RH, an increased temperature decreased boundary saturation, moisture transport rate, moisture transport capacity, moisture transport distance and moisture absorption. The saturation gradient is the main driving potential of moisture transfer. A higher temperature reduced moisture absorption rate. Meanwhile, with temperature increasing, the equilibrium saturation of the concrete boundary further decreased. Such a combined effect decreased moisture absorption of concrete. *Crystals* **2021**, *11*, x FOR PEER REVIEW 13 of 17

*5.3. Absorption* 

**Figure 8.** Simulated distribution at different temperatures. **Figure 8.** Simulated distribution at different temperatures.

The combined influences of the temperature and saturation in the moisture transport process were investigated at three conditions: high temperature and RH (boundary temperature *T* = 323.15 K, boundary saturation *S* = 0.9), medium temperature and RH (bound-The combined influences of the temperature and saturation in the moisture transport process were investigated at three conditions: high temperature and RH (boundary temperature *T* = 323.15 K, boundary saturation *S* = 0.9), medium temperature and RH (boundary

ary temperature *T* = 308.15 K, boundary saturation *S* = 0.75) and low temperature and RH

sults are provided in Figure 9, which shows the same trend among the three conditions. The transport rate is directly correlated with temperature and RH. It also shows that the saturation gradient mostly affected moisture transport rate. Although the saturation was large at high temperature, the moisture transport rate was notably high. Therefore, the effect of RH on the moisture transport rate in the concrete was greater than that of temperature. The transport process of moisture in concrete at the temperature of 293.15 K and saturation 0.6 for simulation at 500d is shown in Figure 10. It can be seen that the moisture is gradually transferred from the left end of the concrete to the interior, and the saturation

is reduced with depth increasing, which is coherent with the trend in Figure 9.

**Figure 9.** Saturation distribution during the absorption simulation.

temperature *T* = 308.15 K, boundary saturation *S* = 0.75) and low temperature and RH (boundary temperature *T* = 293.15 K, boundary saturation *S* = 0.6). The simulation had an initial temperature of 273.15 K, saturation of 0.2 and time of 500 days. The simulated results are provided in Figure 9, which shows the same trend among the three conditions. The transport rate is directly correlated with temperature and RH. It also shows that the saturation gradient mostly affected moisture transport rate. Although the saturation was large at high temperature, the moisture transport rate was notably high. Therefore, the effect of RH on the moisture transport rate in the concrete was greater than that of temperature. The transport process of moisture in concrete at the temperature of 293.15 K and saturation 0.6 for simulation at 500d is shown in Figure 10. It can be seen that the moisture is gradually transferred from the left end of the concrete to the interior, and the saturation is reduced with depth increasing, which is coherent with the trend in Figure 9. ary temperature *T* = 308.15 K, boundary saturation *S* = 0.75) and low temperature and RH (boundary temperature *T* = 293.15 K, boundary saturation *S* = 0.6). The simulation had an initial temperature of 273.15 K, saturation of 0.2 and time of 500 days. The simulated results are provided in Figure 9, which shows the same trend among the three conditions. The transport rate is directly correlated with temperature and RH. It also shows that the saturation gradient mostly affected moisture transport rate. Although the saturation was large at high temperature, the moisture transport rate was notably high. Therefore, the effect of RH on the moisture transport rate in the concrete was greater than that of temperature. The transport process of moisture in concrete at the temperature of 293.15 K and saturation 0.6 for simulation at 500d is shown in Figure 10. It can be seen that the moisture is gradually transferred from the left end of the concrete to the interior, and the saturation is reduced with depth increasing, which is coherent with the trend in Figure 9.

The combined influences of the temperature and saturation in the moisture transport process were investigated at three conditions: high temperature and RH (boundary temperature *T* = 323.15 K, boundary saturation *S* = 0.9), medium temperature and RH (bound-

**Figure 9. Figure 9.** Saturation distribution during the absorption simulation. Saturation distribution during the absorption simulation.

*Crystals* **2021**, *11*, x FOR PEER REVIEW 13 of 17

**Figure 8.** Simulated distribution at different temperatures.

**Figure 10.** Cloud atlas of moisture in absorption process at temperature of 293.15 K and saturation of 0.6. **Figure 10.** Cloud atlas of moisture in absorption process at temperature of 293.15 K and saturation of 0.6.

### *5.4. Desorption 5.4. Desorption*

The upper and lower ends of the two-dimensional geometric model of concrete were treated as heat and moisture insulation. The direction of moisture desorption was from inside to both sides. The effect of temperature on the moisture desorption process at 10% The upper and lower ends of the two-dimensional geometric model of concrete were treated as heat and moisture insulation. The direction of moisture desorption was from inside to both sides. The effect of temperature on the moisture desorption process at 10%

RH was analyzed at temperatures of 293.15, 308.15 and 323.15 K and boundary saturation

Figure 11 presents the simulated results. It illustrates that, for a given RH, higher temperatures lowered the boundary saturation. When the internal saturation of concrete was higher, the difference in saturation was greater between the outside and inside of concrete, resulting in a significant loss of moisture in the concrete. The boundary saturation was lower at higher temperatures, which increased the moisture desorption rate as

Additionally, the combined effects of temperature and saturation on the moisture transport process during desorption were investigated for three conditions: low temperature and high RH (boundary temperature *T* = 293.15 K, internal saturation *S* = 0.9), medium temperature and RH (boundary temperature *T* = 308.15 K, internal saturation *S* =

**Figure 11.** Saturation distribution during the desorption simulation.

well as the moisture desorption capacity.

RH was analyzed at temperatures of 293.15, 308.15 and 323.15 K and boundary saturation values of 0.70, 0.46 and 0.36, respectively. The simulation time was 200 days with an initial temperature of 273.15 K and saturation of 0.9. values of 0.70, 0.46 and 0.36, respectively. The simulation time was 200 days with an initial temperature of 273.15 K and saturation of 0.9. Figure 11 presents the simulated results. It illustrates that, for a given RH, higher

The upper and lower ends of the two-dimensional geometric model of concrete were treated as heat and moisture insulation. The direction of moisture desorption was from inside to both sides. The effect of temperature on the moisture desorption process at 10% RH was analyzed at temperatures of 293.15, 308.15 and 323.15 K and boundary saturation

*Crystals* **2021**, *11*, x FOR PEER REVIEW 14 of 17

**Figure 10.** Cloud atlas of moisture in absorption process at temperature of 293.15 K and saturation of 0.6.

*5.4. Desorption* 

Figure 11 presents the simulated results. It illustrates that, for a given RH, higher temperatures lowered the boundary saturation. When the internal saturation of concrete was higher, the difference in saturation was greater between the outside and inside of concrete, resulting in a significant loss of moisture in the concrete. The boundary saturation was lower at higher temperatures, which increased the moisture desorption rate as well as the moisture desorption capacity. temperatures lowered the boundary saturation. When the internal saturation of concrete was higher, the difference in saturation was greater between the outside and inside of concrete, resulting in a significant loss of moisture in the concrete. The boundary saturation was lower at higher temperatures, which increased the moisture desorption rate as well as the moisture desorption capacity.

**Figure 11.** Saturation distribution during the desorption simulation. **Figure 11.** Saturation distribution during the desorption simulation.

Additionally, the combined effects of temperature and saturation on the moisture transport process during desorption were investigated for three conditions: low temperature and high RH (boundary temperature *T* = 293.15 K, internal saturation *S* = 0.9), medium temperature and RH (boundary temperature *T* = 308.15 K, internal saturation *S* = Additionally, the combined effects of temperature and saturation on the moisture transport process during desorption were investigated for three conditions: low temperature and high RH (boundary temperature *T* = 293.15 K, internal saturation *S* = 0.9), medium temperature and RH (boundary temperature *T* = 308.15 K, internal saturation *S* = 0.75) and high temperature and low RH (boundary temperature *T* = 323.15 K, internal saturation *S* = 0.6). The simulation time was 500 days with an initial temperature of 273.15 K and saturation of 0.2.

As shown in Figure 12, the three curves displayed a similar trend. The saturation values for the three conditions at the internal depth of 50 mm were 0.552, 0.481 and 0.417, respectively. Even at low temperature, the greatest moisture transport rate was observed at the highest RH. The temperature had a greater effect on the moisture desorption process than saturation. The change of saturation of moisture in the desorption process at a temperature of 308.15 K and saturation of 0.75 is shown for the simulation at 500d in Figure 13. This indicates that the moisture was uniformly desorbed to both sides. The rate of saturation reduction was high close to both sides, while it was slow in the middle. This is due to the fact that the boundary temperature is higher at the concrete's surface, resulting in faster moisture transport.

crete's surface, resulting in faster moisture transport.

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K and saturation of 0.2.

K and saturation of 0.2.

**Figure 12.** Internal average saturation distribution during the desorption process. **Figure 12.** Internal average saturation distribution during the desorption process. **Figure 12.** Internal average saturation distribution during the desorption process.

0.75) and high temperature and low RH (boundary temperature *T* = 323.15 K, internal saturation *S* = 0.6). The simulation time was 500 days with an initial temperature of 273.15

0.75) and high temperature and low RH (boundary temperature *T* = 323.15 K, internal saturation *S* = 0.6). The simulation time was 500 days with an initial temperature of 273.15

As shown in Figure 12, the three curves displayed a similar trend. The saturation values for the three conditions at the internal depth of 50 mm were 0.552, 0.481 and 0.417, respectively. Even at low temperature, the greatest moisture transport rate was observed at the highest RH. The temperature had a greater effect on the moisture desorption process than saturation. The change of saturation of moisture in the desorption process at a temperature of 308.15 K and saturation of 0.75 is shown for the simulation at 500d in Figure 13. This indicates that the moisture was uniformly desorbed to both sides. The rate of saturation reduction was high close to both sides, while it was slow in the middle. This is due to the fact that the boundary temperature is higher at the con-

As shown in Figure 12, the three curves displayed a similar trend. The saturation values for the three conditions at the internal depth of 50 mm were 0.552, 0.481 and 0.417, respectively. Even at low temperature, the greatest moisture transport rate was observed at the highest RH. The temperature had a greater effect on the moisture desorption process than saturation. The change of saturation of moisture in the desorption process at a temperature of 308.15 K and saturation of 0.75 is shown for the simulation at 500d in Figure 13. This indicates that the moisture was uniformly desorbed to both sides. The rate of saturation reduction was high close to both sides, while it was slow in the middle. This is due to the fact that the boundary temperature is higher at the con-

**Figure 13.** Cloud atlas of moisture in the desorption process at a temperature of 308.15 K and saturation of 0.75. **Figure 13.** Cloud atlas of moisture in the desorption process at a temperature of 308.15 K and saturation of 0.75.

**Figure 13.** Cloud atlas of moisture in the desorption process at a temperature of 308.15 K and saturation of 0.75.

### **6. Conclusions 6. Conclusions**

**6. Conclusions**  This study sought to elucidate the mechanisms of moisture transport in concrete. A This study sought to elucidate the mechanisms of moisture transport in concrete. A numerical model was developed. Based on the experimental and simulated results, the following conclusions can be drawn. This study sought to elucidate the mechanisms of moisture transport in concrete. A numerical model was developed. Based on the experimental and simulated results, the following conclusions can be drawn.

numerical model was developed. Based on the experimental and simulated results, the following conclusions can be drawn. (1) Under isothermal conditions, a higher RH increased the moisture absorption rate, and the moisture absorption capacity was greater at lower temperatures. When RH exceeded 50%, the moisture absorption rate increased significantly. Similarly, a lower RH resulted in a faster desorption, and higher temperatures increased the moisture desorption capacity.

(2) A model of the relationship between RH and equilibrium saturation in absorption and desorption processes was proposed. A formula in the function of capillary pressure and moisture saturation of concrete was established, and a model of moisture transport in the concrete was developed. The experimental results verified the accuracy of the moisture transport model.

(3) Based on the simulated results, an increased temperature reduced the rate of moisture absorption. The saturation had more effects on the absorption rate than temperature. Although an increment of temperature increased the moisture desorption rate, the saturation had more effects on the moisture desorption than temperature.

(4) Under low saturation, the transport of water vapor affected the moisture desorption rate in concrete. Hence, the transport of water vapor should be further considered in the moisture transport model.

**Author Contributions:** Conceptualization, Q.Z. and H.C.; investigation, Z.K.; data curation, Q.Z., Z.K. and K.L.; methodology, Q.Z.; visualization, Y.L.; formal analysis, Z.K.; software, K.L.; writing original draft, Z.K.; writing—review and editing, Y.L., Q.Z., Z.K., H.C. and K.L.; project administration, Q.Z.; funding acquisition, Q.Z.; validation, Q.Z., Y.L. and H.C.; resources, H.C. All authors have read and agreed to the published version of the manuscript

**Funding:** This work was financially supported by the National Natural Science Foundation of China [grant number 51509084]; the Foundation of Key Laboratory of Performance Evolution and Control for Engineering Structures of Ministry of Education, Tongji University [grant number 2018KF-2]; the Cultivation Plan for Youth Backbone Teachers of Institution of Higher Education by Henan Province [grant number 2019GGJS086]; the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology [grant number 2017RCJH03]; the Cultivation Plan for Youth Backbone Teachers by Henan University of Technology; and the Innovative Funds Plan of Henan University of Technology (Grant No. 2020ZKCJ05).

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

