Modeling the effect of temperature gradient on moisture and ionic transport in concrete

Journal Pre-proof Modeling the effect of temperature gradient on moisture and ionic transport in concrete Yu Bai, Yao Wang, Yunping Xi PII:

S0958-9465(19)31297-1

DOI:

https://doi.org/10.1016/j.cemconcomp.2019.103454

Reference:

CECO 103454

To appear in:

Cement and Concrete Composites

Received Date: 2 April 2019 Revised Date:

7 November 2019

Accepted Date: 7 November 2019

Please cite this article as: Y. Bai, Y. Wang, Y. Xi, Modeling the effect of temperature gradient on moisture and ionic transport in concrete, Cement and Concrete Composites, https://doi.org/10.1016/ j.cemconcomp.2019.103454. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

1

Modeling the effect of temperature gradient on moisture and ionic

2

transport in concrete

3

Yu Bai1, Yao Wang2*, and Yunping Xi3

4

1

5 6

100044, China; [email protected] 2

7 8

Department of Civil, Environmental and Architectural Engineering, University of Colorado Boulder, Boulder, CO 80309, USA; [email protected]

3

Department of Civil, Environmental and Architectural Engineering, University of Colorado Boulder, Boulder, CO 80309, USA; [email protected]

9 10

School of Science, Beijing University of Civil Engineering and Architecture, Beijing,

*

Correspondence: [email protected]; Tel.: +1-303-492-8991

11 12

ABSTRACT

13

Most reinforced concrete structures are under the influence of environmental conditions, such

14

as temperature fluctuations, moisture and chloride variations simultaneously. Thus, the

15

moisture transfer in concrete is driven by the moisture gradient as well as the temperature

16

gradient. Similarly, the ionic transfer in concrete is driven by the concentration gradient of

17

the ion as well as the temperature gradient. This paper developed an analytical solution that

18

can be used for the moisture or the chloride transport in concrete considering the effect of

19

temperature gradient.

20

considered, so the problem is simplified as a one-way coupled problem. The analytical

21

model was validated by using two case studies, in which realistic material parameters of

22

concrete were used, the parameters were obtained from experiments in the literature. The

23

comparisons between analytical results and experimental results showed good agreement. It

24

indicated that the present model can successfully characterize the effect of temperature

25

gradient on moisture transfer and the effect of temperature gradient on chloride transport in

26

concrete.

The effect of mass transfer on the thermal conduction was not

1

27

Keywords: concrete; moisture transfer; chloride diffusion; temperature gradient; analytical

28

model.

29

1. INTRODUCTION

30

Under the service conditions, moisture variations and chloride ingressions in concrete

31

structures usually occur with temperature fluctuations simultaneously. Thus, non-isothermal

32

moisture and ionic transfers in concrete materials are the processes that occur frequently in

33

many applications [1-4]. In the past decades, there were many researchers who have devoted

34

to studying the macroscopic models that consider the effect of temperature on moisture and

35

chloride diffusions; in which, the moisture flux or the ionic flux in the porous medium is not

36

only generated by the moisture gradient or the ionic concentration gradient but also driven by

37

the temperature gradient [5-13]. In a pioneering study, Philip et al. [10] considered

38

discrepancies between measured water vapor transfer in soil under temperature gradients and

39

the predicted values by simple theory based on water vapor pressure gradient. They indicated

40

that improved agreement could be obtained by considering the vapor flux to be resulted by

41

the temperature gradient as well as the moisture content gradient. Kumaran et al. [11]

42

conducted a series of tests to measure moisture transport through a porous insulating medium

43

under a thermal gradient, and postulated that there exits the linear relationship between the

44

moisture transport and the driving potentials including the vapor pressure and temperature

45

gradient across the sample. Lin et al. [12] developed a diffusion model for describing the

46

diffusion of chloride ions in porous concrete under the cyclic daily temperature variations.

47

The results showed that the ambient temperature variations have a strong influence on the

48

chloride diffusion behavior in concrete.

49

Despite a lot of researches have been developed for evaluating the influence of

50

temperature gradient on moisture and chloride transfer in porous materials, there is still a lack

51

of analytical solutions available that can accurately estimate the moisture transfer and

2

52

chloride diffusion in concrete samples under non-isothermal condition [10-15]. The accurate

53

estimations of moisture transfer and chloride diffusion are important for predicting the

54

service life of concrete. For example, the corrosion damage of reinforced concrete structures

55

is a major concern of infrastructure owners and operators. It has been widely accepted that

56

the corrosion of the reinforcing steel bars is primarily due to chloride ingress from the

57

concrete surface into the concrete mass [16-18]. The internal moisture is related to the almost

58

all durability issues of concrete, such as alkali-silica reaction, damages caused by carbonation,

59

as well as nuclear irradiation [19-21].

60

Therefore, there is a pressing need to develop an analytical model that can be used to

61

characterize the coupled transport processes in concrete. This paper is aimed to develop a

62

new analytical model that can estimate the moisture and ionic transfers in concrete driven by

63

the moisture gradient or ionic concentration as well as temperature gradient. The model

64

considered the characteristics of concrete materials, including water-cement ratio, specimen

65

age, cement type, temperature and relative humidity. The simulated distribution profiles of

66

relative humidity and chloride concentration were obtained by the present model. The

67

analytical results were compared with the published experimental data and a good agreement

68

was obtained.

69

2. THEORETICAL BACKGROUND

70

The effect of temperature gradient on moisture transfer and the effect of temperature

71

gradient on chloride penetration in concrete can be treated as two separate processes,

72

however, they share some similarities. The governing equations for moisture transfer and

73

chloride penetration in concrete under isothermal condition can be both based on Fick’s first

74

law with different material parameters, and thus the analytical approach for the two non-

75

isothermal transfer processes are similar. In this paper, the moisture transfer process will be

3

76

used first as an example for introducing the procedure of the analytical model development.

77

The results will then be used for chloride transfer in concrete.

78

The movement of moisture through concrete is a complex mixture of three-phase

79

phenomenon, liquid water, vapor and dry air. It is complicated to use the gradient of moisture

80

content directly [22-23]. There exists a simplified way of presenting this combined moisture

81

transport by using the pore relative humidity (H) as the basic variable, which is generated by

82

the mixture of liquid water, vapor and dry air in fine pores of concrete while not including the

83

chemically combined water. In addition, there is another reason for using H [24]. When

84

considering the changes of evaporable water content in concrete due to hydration of cement,

85

one found that, for usual water-to-cement ratios (within the range 0.3 to 0.6), the drop in H

86

due to self-desiccation caused by hydration (as in the sealed specimens) is rather small,

87

typically H > 0.97 [24]. In fact, it can be neglected even if the hydration has not yet

88

terminated.

89 90

Therefore, the moisture flux in concrete in one-dimension under the non-isothermal condition can be represented by, modifying Fick’s first law [25]: J = - [ D HH ∇ H + D HT ∇ T ]

(1)

91

that is, the water flux J through a concrete structure depends on the gradient of pore relative

92

humidity as well as the gradient of temperature. H is the pore relative humidity. T is the

93

temperature. DHH is the humidity diffusivity. DHT is the coupling parameter for the effect of

94

temperature on moisture transfer, which is also termed as the coefficient of Soret effect [8].

95 96

Combining Equation (1) with the mass conservation law for internal water in concrete, we can obtain the Fick’s second law [26], that is, dW = − div ( J ) dt

4

(2)

97

where t is time in days and W is the total water content (for unit volume of material). By

98

substituting Equation (2) into Equation (1) and decoupling the pore relative humidity H from

99

the total water content W, the mass balance of moisture flux can be expressed as following, ∂W ∂ H ⋅ = div [ DHH ∇ H + DHT ∇T ] ∂H ∂t

100 101

To calculate the temperature distribution, we can assume that the heat conduction in concrete is independent from the moisture diffusion [27], that is,

ρc 102 103

(3)

∂T ∂2T =K 2 ∂t ∂x

(4)

where c is the heat capacity, ρ is the density, x is the depth and K is the thermal conductivity. The governing partial differential Equations (3) and (4) include six material parameters: ⁄

104

the moisture capacity

, the humidity diffusivity DHH, the coupling parameter DHT, the

105

heat capacity c, the density ρ and the thermal conductivity K. In which, moisture capacity is

106

the derivative of the equilibrium adsorption isotherm. It is related to the properties of

107

concrete, including the water-cement ratio, specimen age, cement type, temperature, and

108

humidity [24, 28-29]. The details of procedures to calculate the moisture capacity are shown

109

in [28-29], which will not be described here.

110

The present analytical model is valid under the ambient temperature conditions, in which

111

the transport parameters can be assumed as constant. For example, the maximum temperature

112

cannot be higher than 100 oC where the vaporization of water occurs and the minimum

113

temperature cannot be lower than 0 oC where the formation of ice occurs. Three combined

114

parameters can be defined as:

dHH =

DHH DHT K dHT = κ= ∂W ∂H ∂W ∂H ρc

5

115

which are called the moisture diffusion coefficient, the coupling coefficient for the effect of

116

temperature on moisture transfer, and the thermal diffusivity coefficient, respectively.

117

118

119 120

Thus, the governing equations can be written as:

∂H ∂2 H ∂2T = dHH 2 + dHT 2 ∂t ∂x ∂x

(5)

∂T ∂2T =κ 2 ∂t ∂x

(6)

and

The initial conditions and the boundary conditions for a semi-infinitely long sample are usually given as:

H ( x,0) = H0 H ( 0, t ) = Hs 121

H ( ∞, t ) = H0

(7)

T ( ∞, t ) = T0

(8)

and

T ( x,0 ) = T0 T ( 0, t ) = Ts 122

where the initial and boundary conditions are all constant. T0 and Ts are initial and boundary

123

temperatures, and H0 and Hs are initial and boundary relative humidity.

124

3. ANALYTICAL MODEL

125

By the Laplace transform and the inverse Laplace transform [30], the analytical solution

126

of Equation (6) and Equation (8) can be obtained as:   x  T = T0 + (Ts − T0 ) 1 − erf    2 κ t  

127 128

where erf

=

√

is the error function.

Now substitute Equation (9) to Equation (5), it is given the equation as:

6

(9)

x2

− ∂2T Ts − T0 x 4κ t = ⋅ e κ 2 κπ t 3 ∂x2

Denote the Laplace transform of H (x, t) as ,

129

(10)

, that is,

∞

Hˆ ( x , s ) = L ( H ( x , t ) ) = ∫ H ( x , t ) ⋅ e − st dt 0

130 131

Then use Laplace transform for the Equation (5), Equation (7) and Equation (10), and it has, T −T − d2 ˆ ˆ s ⋅ H ( x, s ) − H 0 = d HH 2 H ( x, s ) + d HT ⋅ s 0 ⋅ e dx κ H H Hˆ ( 0, s ) = s Hˆ ( ∞, s ) = 0 s s

132

where the Laplace transform of Equation (10) is ℒ

=

!

#

"

∙

x

κ

s

(11)

% ∙√' √&

.

133

For Equation (11), if keep parameter s as a constant, then the problem is an ordinary

134

differential equation with respect to x. According to the theory in [31], the solution of

135

Equation (11) can be obtained as:

H H − H0 − Hˆ ( x, s ) = 0 + s ⋅e s s +

136 137

s ⋅x d HH

( d HT κ ) ⋅ (Ts − T0 ) (1 − d HH κ ) ⋅ s

⋅e

−

( d κ ) ⋅ (Ts − T0 ) ⋅ e− − HT (1 − d HH κ ) ⋅ s s

κ

s ⋅x d HH

(12)

⋅x

Finally, by the inverse Laplace transform and the convolution theorem, the analytical expression of relative humidity distributions is shown below,

  x  H ( x, t ) = H 0 + ( H s − H 0 ) 1 − erf    2 d t   HH   d ⋅ (T − T )   x   x  + HT s 0  erf  − erf    (κ − d HH )   2 d HH t   2 κ t  

7

(13)

138

The coupling processes of relative humidity diffusion and heat conduction can be

139

governed by Equation (5) and Equation (6). Under the initial condition and the boundary

140

conditions, which are Equation (7) and Equation (8), by the Laplace transform and the

141

inverse Laplace transform we can obtain the analytical model, which are Equation (13) and

142

Equation (9). Thus, we can predict the relative humidity concentration H (x, t) and the

143

temperature profile T (x, t).

144 145

Similarly, the chloride diffusion in concrete under a temperature gradient can also be described by the Fick’s law [33], J cl = −  Dcl ∇ ( C f ) + Dcl −T ∇ (T ) 

∂Ct ∂C f ⋅ = div  Dcl ∇ ( C f ) + Dcl −T ∇ (T )  ∂C f ∂t

(14)

(15)

146

that is, JCl is the chloride flux in concrete. Ct and Cf are the total chloride concentration and

147

the free chloride concentration, respectively. DCl is the chloride diffusivity and DCl-T is the

148

coupling parameter for the effect of temperature on chloride diffusion.

149 150

The initial and boundary conditions adopted in the analytical model for the chloride diffusion are given as:

C f ( x, 0 ) = C0 C f ( 0, t ) = Cs 151

C f ( ∞, t ) = C0

(16)

T ( ∞, t ) = T0

(17)

and

T ( x,0) = T0 T ( 0, t ) = Ts 152

where the constants T0 and Ts are initial and boundary temperatures, and the constants C0 and

153

Cs are initial and boundary chloride concentrations.

8

154 155

Correspondingly, by applying the similar analytical approach with moisture content transfer, the free chloride concentration Cf (x,t) can be calculated as:

  x  C f ( x, t ) = C0 + ( Cs − C0 ) 1 − erf    2 d t   cl    d ⋅ (T − T )   x   x  + cl −T s 0  erf   − erf   (κ − dcl )   2 dcl t   2 κ t   156

(18)

In which,

dcl =

Dcl ∂Ct DHT ; dHT = ; = chloride binding capacity ∂Ct ∂C f ∂Ct ∂C f ∂C f

157

4. CASE ANALYSIS

158

In order to validate the applicability of the present analytical model in predicting the

159

moisture and ionic transfer in concrete samples, the calculating results are separately

160

compared with two published experimental data obtained by Wang and Xi (2017) [29] and

161

Isteita and Xi (2017) [33]. Thus, this section includes two cases studies: Case I involves a

162

one-dimensional moisture diffusion with temperature coupling effect in a concrete cylinder

163

[29], while Case II study is for the temperature effect on one-dimensional chloride

164

penetration in concrete [33].

165

CASE I

166

The cement was Ordinary Type I Portland Cement (OPC). The coarse aggregate was

167

gravel with average size of 12.7 mm. The fine aggregate consisted of natural river sand with a

168

fineness modulus of 2.65 and maximum size of 2 mm. The mix proportions of concrete

169

specimens are presented in Table 1.

170

Table 1. Mix proportions of concrete specimen [28]

water-cement ratio sand-cement ratio gravel-cement ratio 0.6

2.4

9

2.9

171

After demolding, the specimens were cured at 20 oC and 100% RH for 28 days and then

172

kept in the ambient environment to reach the designed initial conditions (20 oC and 50%). In

173

order to ensure the transport along one dimension, the lateral surface and bottom side were

174

sealed by several layers, including silicone, foam isolation and cable ties. In addition, the top

175

side was clamped with plastic sleeve to create a container that was filled with distilled water

176

and heated by an electric heater. The internal humidity and temperature were measured by

177

three inserted SHT75 Sensirion humidity-temperature sensors, and these three sensors were

178

located at 63.5, 139.7, 215.9 and 304.8 mm from the high concentration side, as shown in

179

Figure 1. Table 2 lists the specific initial conditions and boundary conditions for two

180

experimental configurations.

181

Table 2. Initial and boundary conditions for two sets of tested concrete specimens [29] Initial conditions Boundary conditions Specimen H0 T0 (oC) Hs Ts (oC) N-40

50%

20

100%

40

N-60

50%

20

100%

60

182

Wang and Xi (2017) [29] presented the initial and boundary conditions, and calculated

183

the values of transport parameters that were required in Equations (9) and (13). The details of

184

variables are shown in Table 3, that is, CT is the cement type; t and T are the curing age and

185

temperature, respectively; ( is the thermal diffusivity of the material; dHH is the moisture

186

diffusion coefficient of the concrete sample; dHT is the coupling coefficient for temperature

187

effect on moisture transfer.

188

Table 3. Parameters used in the analytical calculating CT

I

w/c

0.6

t

65 day

T

20 oC

( 0.0018 (m2/day)

dHH 0.0004 (%∙m2∙day-1) dHT 0.0014 (%∙m2∙day-1∙oC-1)

189 190

Figure 2(a) and (b) show the experimental and simulated temperature variations as

191

function of time and depth, respectively. It is important to notice that the parameter time (t) in

192

the figures is the duration of testing, not the age of concrete sample age. When concrete 10

193

sample is exposed to the high temperature condition, there will occur the heat transfer, that is,

194

the internal temperature increases with time increasing. The two non-isothermal cases present

195

significant differences in temperature distribution, that is, the sample with higher gradient has

196

the relative higher temperature at same depth. It is consistent with the law of heat conduction

197

[32]. Thus, it has confirmed the validity of the experimental results. In addition, experimental

198

data shows a slightly higher than the model calculations. The error may be due to inaccuracy

199

of experimental measurement.

200

Figure 3 compares the experimental measurements and computed variations of relative

201

humidity with different time and different depth. The simulations of relative humidity by the

202

present model agree well with the measured data. The internal relative humidity increases as

203

time increases. The samples with different temperature gradient give the different relative

204

humidity. Moreover, the sample with a higher temperature gradient obtains a larger relative

205

humidity. These results give an evidence of temperature gradient affecting the moisture

206

transport, and the present model can predict the distribution of relative humidity under

207

temperature coupling effect in concrete.

208

CASE II

209

Apart from the moisture transport, the present model can also predict the ionic transport

210

in concrete under non-isothermal condition. This paper cited the experimental study from

211

Isteita et al. (2017) [33] to validate the prediction accuracy of the present model. The

212

concrete samples adopted the mixing ratios shown in Table 4. After demolding, all samples

213

were immersed in lime-saturated water for 28 days at 20oC. Then the samples were placed in

214

a tank containing distilled water to exclude the effect of moisture transfer on chloride

215

penetration. Moreover, the top side of concrete sample was clamped with a plastic sleeve to

216

create a reservoir. The containers were filled with 3% NaCl (sodium chloride) solution. To

11

217

evaluate the effect of temperature gradient, the test conducted two sets samples, S1 with

218

temperature gradient while S2 without temperature gradient, and each set had three samples

219

to take into account the scattering in test data (Figure 4).

220

Table 4. Concrete mix design Content

Proportions

water-cement ratio (w/c)

0.65

3

weight of water (kg/m )

232

3

weight of cement (kg/m )

356

weight of sand (kg/m3)

847

3

weight of gravel (kg/m )

1031

221 222

It should be noticed that Isteita and Xi (2017) measured the total chloride distributions at

223

different days for each specimen. However, the present model focus on the transport of

224

moveable ionic, which mainly consist of free chloride. To convert the total chloride

225

concentration to free chloride concentration, this study adopted the chloride binding capacity

226

[34]. In addition, the chloride diffusion coefficient and the coupling coefficient were

227

determined according to the mix properties of concrete specimen and experimental conditions.

228

The values of all parameters are shown in Table 5.

229

Table 5. Parameters used in the analytical calculating )* ⁄ )+

0.85

dCl

1.60E-06 (%Cl∙m2∙day-1)

dCl-T

2.00E-04 (%Cl∙m2∙day-1∙oC-1)

230

In order to illustrate the effect of temperature gradient on chloride penetration, the

231

boundaries of S1 and S2 specimens were under different situations, with and without

232

temperature gradient, respectively. Figure 5 compares the chloride profiles at different times

233

for S1 and S2 specimens. As time increases, the internal chloride concentrations for both

234

samples increase that contributes from the existing of concentration gradient at the boundary.

235

However, at the same day, the internal chloride of S1 obtains higher concentration than the

236

chloride in S2. In the other words, concrete sample with temperature gradient presents higher

237

and steeper chloride profiles than the sample without temperature gradient. When there is

12

238

temperature difference existing in the environment and concrete sample, it can trigger the

239

heat transfer from high temperature to low temperature. This physical transport can carry free

240

chloride along with heat transfer [35]. Thus, the effect of temperature gradient is crucial to

241

obtain an accurate prediction of chloride concentration in concrete. Figure 6 compares the

242

model calculations and experimental results of chloride distributions. The measurements at

243

day 3 are much lower than the analytical results, comparing to the other two cases. It is

244

because the actual internal temperature of sample is lower than the assumption of analytical

245

model at day 3, that is, a lower temperature gradient results in a lower effect on the chloride

246

penetration. However, as time increases, the boundary temperature of the sample increases

247

and closes to the assumed temperature. Thus, it is found that the comparisons between

248

experimental data and model predictions reach a good agreement at day 24 and day 48. It

249

indicates that the present model can successfully characterize the effect of temperature

250

gradient on chloride transport in concrete.

251

5. Conclusions

252

The heat and mass transfer inside porous materials such as concrete is a complicate

253

phenomenon. Although there have been several methods and models developed for

254

evaluating the temperature coupling effect on moisture transfer and chloride transport in

255

concrete, there is still a lack of analytical solution for estimating the moisture and chloride

256

concentration distributions in concrete under non-isothermal condition. On the other hand,

257

the heat transfer in concrete is driven by the temperature gradient and also influenced by the

258

mass transport processes. So, the mass and heat transfer processes are two-way fully coupled

259

processes in concrete. In order to simplify the problem, in this paper, we neglected the effect

260

of the mass transports on the heat transfer, and thus the problem became a one-way coupled

261

problem.

13

262

•

A new analytical solution was developed using Laplace transformation and the inverse

263

Laplace transformation together with the specific initial and boundary conditions

264

commonly seen in concrete structures. The solution can be used to calculate the chloride

265

concentration profiles in concrete under the effect of ionic concentration gradient as well

266

as the temperature gradient in concrete. Similarly, the solution can be also used to

267

calculate the moisture concentration profiles in concrete under the effect of moisture

268

concentration gradient as well as the temperature gradient in concrete.

269

•

In order to verify the validity of the analytical model, the analytical results were

270

compared with two sets of published experimental data. One was for the effect of

271

temperature gradient on moisture transfer in concrete, and the other was for the effect of

272

temperature gradient on chloride penetration into concrete. The parameters used in the

273

analytical model were obtained from the experimental studies. The comparisons between

274

model predictions and experimental data agreed well. It indicated that the present model

275

can be used to estimate the internal moisture and chloride distributions in concrete under

276

temperature fluctuations.

277

•

Both analytical results and experimental results showed that the temperature gradient can

278

affect the moisture and chloride transfer in concrete. Specifically, when the temperature

279

gradient is in the same direction as the moisture gradient or the chloride concentration

280

gradient, it accelerates the moisture and chloride transfers.

281

•

The transport parameters used in the analytical solution are assumed as constant. So, the

282

one-way coupled governing equations are linear. As a result, there are some application

283

limitations of the present model. For example, it is valid under ambient temperature

284

conditions. When applying the model for the extreme conditions (fire or freezing

285

temperatures), the values of transport parameters should be revised to reflect the extreme

286

conditions.

14

287

6. Acknowledge

288

The authors wish to acknowledge the partial support by the US DOE DE-NE0000659 to

289

University of Colorado at Boulder. Opinions expressed in this paper are those of the authors

290

and do not necessarily reflect those of the sponsor.

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Figure 1. Schematic illustration of the experimental set-up [29]

Figure 2. Comparisons of experimental [29] and simulated temperature variations: (a) with same depth at different time; (b) with same time at different depths.

Figure 3. Comparisons of experimental [29] and simulated relative humidity variations: (a) with same depth at different time; (b) with same time at different depths.

Figure 4. Experimental plan and concrete samples under testing [33]

3.0

S1-day3 S1-day24 S1-day48 S2-day3 S2-day24 S2-day48

Free Chloride Concentration (% by concrete weight)

2.5

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Depth (mm) Figure 5. Comparisons of measured chloride penetrations with and without temperature effect [33]

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Free Chloride Concentration (% by concrete weight)

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day 3 - Analytical results day24 - Analytical results day48 - Analytical results day 3 - Test data day24 - Test data day48 - Test data

Ts=50oC,T0=20oC Cs=3%,C0=0.03%

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Depth (mm) Figure 6. Comparisons of experimental data [34] and simulated %Cl distributions of present model

Declarations of interests: none.