Characterization of Concrete by Calibrating Thermo-Hydraulic Multiphase Flow Models

Abstract

Evaporation tests in concrete columns have been analysed by numerical models to characterize the thermo-hydraulic properties and the processes in concrete. Two evaporation tests were performed: a column heated by a lamp and a column kept in room conditions. The conceptual model considers unsaturated liquid flow and transport of vapour and energy. We also calculated models that take into account the dissolved salts to study its effect on vapour pressure and evaporation. A retention curve has been obtained from relative humidity and gravimetric water content measured after dismantling the tests. The models have been calibrated by adjusting the model’s results to the measured data of water loss, relative humidity and temperature inside the concrete. The parameters obtained with the calibration are the permeability, thermal conductivity, boundary conditions and a tortuosity factor for vapour diffusion. Results show that the vapour diffusion is the dominant water transport process above an evaporation front, and liquid advection is dominant below it.

Appendix: Constitutive Laws

1.1 Definition Constraints

1.1.1 Mass fractions

$$\begin{aligned}&\omega ^\mathrm{w}_\mathrm{l}+\omega ^\mathrm{a}_\mathrm{l}+\omega ^\mathrm{h}_\mathrm{l}=1 \end{aligned}$$

(8)

$$\begin{aligned}&\omega ^\mathrm{w}_\mathrm{g}+\omega ^\mathrm{a}_\mathrm{g}=1 \end{aligned}$$

(9)

1.1.2 Volumetric content

$$\begin{aligned} \theta _\mathrm{l}=\dfrac{V_\mathrm{l}}{V_\mathrm{t}}=\dfrac{1}{\rho _\mathrm{l}}w\rho _\mathrm{s}( 1-\phi ) \end{aligned}$$

(10)

1.1.3 Saturations

$$\begin{aligned}&S_\mathrm{l}+S_\mathrm{g}=1 \end{aligned}$$

(11)

$$\begin{aligned}&S_\mathrm{l}=\dfrac{V_\mathrm{l}}{V_\mathrm{p}}=\dfrac{\theta _\mathrm{l}}{\phi } \end{aligned}$$

(12)

1.1.4 Partial pressures

$$\begin{aligned} P^\mathrm{a}_\mathrm{g}+P^\mathrm{w}_\mathrm{g}=P_\mathrm{g} \end{aligned}$$

(13)

1.2 Equilibrium Constraints

1.2.1 Vapour–liquid water (psychrometric law)

$$\begin{aligned} P^\mathrm{w}_\mathrm{g}= & {} a_\mathrm{w}P^\mathrm{w}_\mathrm{g,sat}\exp \left( \frac{-(P_\mathrm{g}-P_\mathrm{l})M^\mathrm{w}}{R\rho _\mathrm{l}(273.15+T)}\right) \end{aligned}$$

(14)

$$\begin{aligned} P^\mathrm{w}_\mathrm{g,sat}= & {} 136075\exp \left( \frac{-5239.7}{273.15+T}\right) \end{aligned}$$

(15)

$$\begin{aligned} a_\mathrm{w}= & {} 1-\left( \left( \omega ^\mathrm{h}_\mathrm{l} \frac{1000}{M^\mathrm{h}-3}\right) 1.9775\times 10^{-5}T+0.035\right) \omega ^\mathrm{h}_\mathrm{l} \frac{1000}{M^\mathrm{h}-3} \end{aligned}$$

(16)

where $M^\mathrm{h}$ = 49.37 g mol$^{-1}$

$$\begin{aligned} \hbox {HR}= & {} \frac{P^\mathrm{w}_\mathrm{g}}{P^\mathrm{w}_\mathrm{g,sat}} \end{aligned}$$

(17)

$$\begin{aligned} \omega ^{w}_\mathrm{g}= & {} \frac{P^\mathrm{w}_\mathrm{g}M^\mathrm{w}}{R\rho _\mathrm{g}(273.15+T)} \end{aligned}$$

(18)

1.2.2 Dissolved-gaseous air (Henry’s law)

$$\begin{aligned} \omega ^\mathrm{a}_\mathrm{l}=\frac{P^\mathrm{a}_\mathrm{g} M^\mathrm{a}}{\textit{HM}^\mathrm{w}} \end{aligned}$$

(19)

1.3 Phase and Interphase Properties

1.3.1 Retention curve (van Genuchten law)

$$\begin{aligned} S_\mathrm{e}=\left( 1+ \left( \frac{P_\mathrm{g}-P_\mathrm{l}}{P_{0}\frac{\sigma _{T}}{\sigma _{0}}}\right) ^{\frac{1}{1-m}}\right) ^{-m} \end{aligned}$$

(20)

where $P_{0}=7.7$ Mpa, $m=0.34$, $\sigma _{T}$ is the surface tension at temperature T and $\sigma _{0}=0.072$ N m$^{-1}$ at 20 $^{\circ }$C.

1.3.2 Relative permeability

$$\begin{aligned} k_\mathrm{rl}=\textit{AS}^\mathrm{n}_\mathrm{el} \end{aligned}$$

(21)

where $A=0.01$ and $n=7$

$$\begin{aligned} k_\mathrm{rg}=\textit{AS}^\mathrm{n}_\mathrm{eg} \end{aligned}$$

(22)

where $A=1$ and $n=3$.

1.3.3 Properties of liquid

$$\begin{aligned} \rho _\mathrm{l} = \rho _{l0} \exp \left( \beta \left( P_\mathrm{l}-P_\mathrm{l0}\right) + \epsilon T+ \gamma \omega ^\mathrm{h}_\mathrm{l} \right) \end{aligned}$$

(23)

where $\rho _{l0}=1002.6$ kg m$^{-1}$, $P_{l0}=0.1\,\hbox {MPa}$, $\beta =4.5\times 10^{-4}$ MPa$^{-1}$, $\gamma =0.6923$ and $\epsilon =3.4\times 10^{-4}\,^{\circ }$C$^{-1}$

$$\begin{aligned} \mu _\mathrm{l}= & {} 2.1\times 10^{-12}\left( \frac{1808.5}{273.15+T}\right) \end{aligned}$$

(24)

$$\begin{aligned} \sigma _{T}= & {} \left( 1-0.625 \left( \frac{273.15+T}{647.3} \right) \right) \left( 0.2358 \left( \frac{273.15+T}{647.3} \right) ^{1.256} \right) + 0.04055 \omega ^\mathrm{h}_\mathrm{l} \qquad \end{aligned}$$

(25)

$$\begin{aligned} E_\mathrm{l}= & {} E^\mathrm{w}_\mathrm{l} \omega ^\mathrm{w}_\mathrm{l}+ E^\mathrm{a}_\mathrm{l} \omega ^\mathrm{a}_\mathrm{l} \end{aligned}$$

(26)

where $E^\mathrm{w}_\mathrm{l}=4184$ T J kg$^{-1}$ and $E^\mathrm{a}_\mathrm{l}=1000T$ J kg$^{-1}$.

1.3.4 Properties of gas

$$\begin{aligned} \rho _\mathrm{g}=\frac{P^\mathrm{w}_\mathrm{g}M^\mathrm{w}}{R(273.15+T)}+\frac{P^\mathrm{a}_{g}M^\mathrm{w}}{R(273.15+T)} \end{aligned}$$

(27)

where $M^\mathrm{a}_\mathrm{g}=0.02895$ kg mol$^{-1}$ and H $=$ 10,000 MPa

$$\begin{aligned} \mu _\mathrm{g}= & {} \frac{1.48 \times 10^{-12} \sqrt{273.15+T}}{1+ \dfrac{119.4}{T+273.15} } \dfrac{1}{1+ \dfrac{0.14-1.2 \times 10^{15}k_\mathrm{i}}{P_\mathrm{g}}} \end{aligned}$$

(28)

$$\begin{aligned} E_\mathrm{g}= & {} E^\mathrm{w}_\mathrm{g} \omega ^\mathrm{w}_\mathrm{g}+ E^\mathrm{a}_\mathrm{g} \omega ^\mathrm{a}_\mathrm{g} \end{aligned}$$

(29)

where $E^\mathrm{w}_\mathrm{g}=2.5 \times 10^{6}+1900T$ J kg$^{-1}$ and $E^\mathrm{a}_\mathrm{g} =1000T$ J kg$^{-1}$.

1.3.5 Properties of solid

The density of the solid phase ($\rho _\mathrm{s}$) is 2360 kg m$^{-1}$ and $E_\mathrm{s} =780T$ J kg$^{-1}$.

1.4 Fluxes

1.4.1 Darcy’s flux

$$\begin{aligned} q_{\alpha }=-\frac{k_\mathrm{i} k_\mathrm{r\alpha }}{\mu _{\alpha }}(\bigtriangledown P_{\alpha } - \rho _{\alpha }\mathrm g) \end{aligned}$$

(30)

where $k_\mathrm{i}=4.2 \times 10^{-18}$ m$^{2}$.

1.4.2 Diffusive flux (Fick’s law)

$$\begin{aligned} i^\mathrm{i}_{\alpha , \mathrm{dif}}=-(\tau \phi \rho _\alpha S_{\alpha } D^\mathrm{i}_\mathrm{m}) \bigtriangledown \omega ^\mathrm{i}_{\alpha } \end{aligned}$$

(31)

where $\tau _\mathrm{non{\text {-}}heated}=0.08$ and $\tau _\mathrm{heated}=0.3$

$$\begin{aligned} D^\mathrm{vapor}_\mathrm{m}=D \left( \frac{(273.15+T)^{n}}{P_\mathrm{g}}\right) \end{aligned}$$

(32)

where $D=5.9 \times 10^{-6}$ m s$^{-1}$ K$^{-n}$ Pa and $n=2.3$

$$\begin{aligned} D^\mathrm{solute}_\mathrm{m}=D \exp \left( \frac{-24530}{R(273.15+T)}\right) \end{aligned}$$

(33)

where $D=1.1 \times 10^{-4}$ m$^{2}$ s$^{-1}$.

1.4.3 Dispersive flux (Fick’s law)

$$\begin{aligned} i^\mathrm{i}_{\alpha , \mathrm{dis}}= & {} -\left( \rho _{\alpha } D^{\prime }_{\alpha }\right) \bigtriangledown \omega ^\mathrm{i}_{\alpha } \end{aligned}$$

(34)

$$\begin{aligned} D^{\prime }_{\alpha }= & {} d_\mathrm{l}q_{\alpha } \end{aligned}$$

(35)

where $d_\mathrm{l}=0.015$ m.

1.4.4 Mass flux

$$\begin{aligned} j^\mathrm{i}_{\alpha }=q_{\alpha }\rho _{\alpha }\omega ^\mathrm{i}_{\alpha } + i^\mathrm{i}_{\alpha ,\mathrm{dis}} + i^\mathrm{i}_{\alpha ,\mathrm{dif}} \end{aligned}$$

(36)

1.4.5 Conductive flux of heat (Fourier’s law)

$$\begin{aligned} i_\mathrm{c}= & {} - \lambda \bigtriangledown T \end{aligned}$$

(37)

$$\begin{aligned} \lambda= & {} \lambda _\mathrm{sat}^{S_\mathrm{l}} \lambda _\mathrm{dry}^{1-S_\mathrm{l}} \end{aligned}$$

(38)

where $\lambda _\mathrm{sat}=1.14$ W m K$^{-1}$ and $\lambda _\mathrm{dry}=0.66$ W m K$^{-1}$.

1.4.6 Advective flux of heat

$$\begin{aligned} j_\mathrm{E\alpha }=q_{\alpha }E_{\alpha } \end{aligned}$$

(39)