Simulation of Mass, Linear Momentum, and Energy Transport in Concrete with Varying Moisture Content during Cooling to Cryogenic Temperatures

Abstract

A set of governing equations comprising linear momentum, mass, and heat transfer is presented for thermoelastic freezing of a porous material. The theory of unsaturated freezing porous media is introduced to model deformation of concrete, a traditional building material, whose pore network is pressurized by the wet air, frozen ice, and unfrozen water. A general solution scheme is provided for the appropriate boundary conditions pertaining to the primary concrete containment in a liquefied natural gas tank, and simulated results are analyzed for fully and partially saturated non-air-entrained concrete and fully saturated air-entrained concrete. Effect of cooling rate is also demonstrated. It is found that high cooling rate results in high expansion provoked by high hydraulic pore pressure and the corresponding suppression of pore liquid freezing temperature. It is also revealed that air-entrained concrete, by allowing quick dissipation of the displaced pore water and accommodating the ensuing ice formation, shows less contraction and subsequently less crack initiating stresses than the high-porosity, non-air-entrained concrete. Similar outcomes are observed near the concrete surfaces subjected to evaporation prior to cryogenic freezing. High hydraulic pressure, induced by the delayed dissipation of excess pore water, is likely to generate at the center of surface-dried concrete walls.

Appendix

1.1 Thermodynamics of Partially Saturated Freezing Concrete

The theory presented here is an application of the model proposed by Coussy (2006) for isothermal drying induced crystallization, allowing for the thermal expansion induced by the temperature change. Let us consider a porous material whose pore space is occupied by three phases: a gas referred to by index $J = G$, a liquid referred to by index $J = L$, and a solid crystal referred to by index $J = C$. If $\hbox {d}V_0$ is the initial volume of an infinitesimal representative element from the porous solid, $\phi _0$ and $\phi$ are the initial and current porosity, respectively, and the reference porous volume of the element is given by $\phi _0 \hbox {d}V_0$ and the current porous volume is $\phi \hbox {d}V_0$. The porous solid and any of its phases are capable of exhibiting infinitesimal deformation gradients. Furthermore, $\phi _J$ if the current volume occupied by the phase J, the current overall porosity, $\phi$ can be written as

$$\begin{aligned} \phi = \phi _C + \phi _G + \phi _L \end{aligned}$$

(45)

If the current number of moles of phase J per unit initial volume is $n_J$, temperature T, the current molar entropy is $s_J$, and the current molar chemical potential $\mu _J$, the Gibbs–Duhem equality for each phase J can be written in the form

$$\begin{aligned} \phi _J \hbox {d}p_J - n_J s_J \hbox {d}T - n_J \hbox {d}\mu _J = 0 \end{aligned}$$

(46)

such that

$$\begin{aligned} \begin{array}{l} \phi _G \hbox {d}p_G = n_a s_a \hbox {d}T + n_a \hbox {d}\mu _a + n_v s_v \hbox {d}T + n_v \hbox {d}\mu _v, \\ \phi _C \hbox {d}p_{C} = n_C s_C \hbox {d}T + n_C \hbox {d}\mu _C,\hbox { and} \\ \phi _L \hbox {d}p_{L} = n_L s_L \hbox {d}T + n_L \hbox {d}\mu _L. \\ \end{array} \end{aligned}$$

(47)

When no dissipation occurs, the first and second laws of thermodynamics combine to provide a Clausius–Duhem equality (for equilibrium) for a poroelastic material as given by

$$\begin{aligned} \sigma _{ij} \hbox {d}\varepsilon _{ij} + \mu _C \hbox {d}n_C + \mu _a \hbox {d}n_a + \mu _v \hbox {d}n_v + \mu _L \hbox {d}n_L - \Sigma \hbox {d}T - \hbox {d}F = 0, \end{aligned}$$

(48)

where $\sigma _{ij}$ is the current overall stress components, $\varepsilon _{ij}$ is the current overall strain component, $\Sigma$ is the entropy of the open element per unit of its initial volume, and F is the overall Helmholtz free energy. Subscripts a and v represent air and vapor, respectively.

Let $F_{sk}$ and $\Sigma _{sk}$ be the Helmholtz free energy and entropy per unit initial volume of the skeleton. The term skeleton includes interfaces between the solid matrix and the phases present in the pore network and excludes the bulk phases in the pores. Due to the additive characteristics of free energy, $F_{sk}$ and $\Sigma _{sk}$ can be written as

$$\begin{aligned} \begin{array}{l} \Sigma _{sk} = \Sigma - n_C s_C - n_a s_a - n_v s_v - n_L s_L \hbox { and} \\ F_{sk} = F-\left( {n_C s_C - \phi _C p_{C} + n_v s_v + n_a s_a - \phi _G p_G + n_L s_L - \phi _L p_{L} } \right) . \\ \end{array} \end{aligned}$$

(49)

With the help of (48), (49) gives us the free energy balance related to the poroelastic skeleton as

$$\begin{aligned} \sigma _{ij} \hbox {d}\varepsilon _{ij} + p_{C} \hbox {d}\phi _C + p_G \hbox {d}\phi _G + p_{L} \hbox {d}\phi _L - \Sigma _{sk} \hbox {d}T - \hbox {d}F_{sk} = 0. \end{aligned}$$

(50)

Therefore, the unsaturated poroelastic state equations can be written as

$$\begin{aligned} \begin{array}{l} F_{sk} = F_{sk} \left( {\varepsilon _{ij}, \phi _J, T} \right) , \\ \sigma _{ij} = \frac{\partial F_{sk} }{\partial \varepsilon _{ij} }, \\ p_J = \frac{\partial F_{sk} }{\partial \phi _J },\hbox { and} \\ \Sigma _{sk} = - \frac{\partial F_{sk} }{\partial T}. \\ \end{array} \end{aligned}$$

(51)

Change in partial porosities $\phi _J$ results from the change in saturation, $S_J$ and skeletal deformation, $\phi _J$ resulting from the action of pressure on phases, $p_J$. Therefore, $\phi _J$ can be given in the form

$$\begin{aligned} \phi _J = \phi _0 S_J + \phi _J \end{aligned}$$

(52)

with the constraint

$$\begin{aligned} S_C + S_G + S_L = 1. \end{aligned}$$

(53)

Therefore, the current overall porosity is

$$\begin{aligned} \phi = \phi _0 + \phi _C + \phi _G + \phi _L. \end{aligned}$$

(54)

State equations (51) account for both the energy required for skeletal deformation and energy required to create new interfaces. Using (52) with (51), these two energy variables are separated as

$$\begin{aligned} F_{sk} \left( {\varepsilon _{ij}, \phi _J, T} \right) = \psi _{sk} \left( {\varepsilon _{ij}, \phi _J, S_C, S_G, T} \right) . \end{aligned}$$

(55)

Substituting (52) and (55) into (50) and applying the constraint (53), we find

$$\begin{aligned} \begin{array}{l} \sigma _{ij} \hbox {d}\varepsilon _{ij} + p_{C} \hbox {d}\phi _C + p_G \hbox {d}\phi _G + p_{L} \hbox {d}\phi _L \\ \quad +\,\phi _0 \left( {p_{C} - p_{L} } \right) \hbox {d}S_C + \phi _0 \left( {p_G - p_{L} } \right) \hbox {d}S_G - \Sigma _{sk} \hbox {d}T - \hbox {d}\psi _{sk} = 0 \\ \end{array}, \end{aligned}$$

(56)

which leads to the following equations

$$\begin{aligned}&\sigma _{ij} = \frac{\partial \psi _{sk} }{\partial \varepsilon _{ij} },\nonumber \\&p_J = \frac{\partial \psi _{sk} }{\partial \phi _J },\nonumber \\&\phi _0 \left( {p_{C} - p_{L} } \right) = \frac{\partial \psi _{sk} }{\partial S_C },\nonumber \\&\phi _0 \left( {p_G - p_{L} } \right) = \frac{\partial \psi _{sk} }{\partial S_G },\quad \hbox { and}\nonumber \\&\Sigma _{sk} = -\frac{\partial \psi _{sk} }{\partial T}. \end{aligned}$$

(57)

$F_{sk}$ can be divided into two parts: elastic free energy, W of the porous solid excluding the interfaces, and the interface energy, U per unit of initial porous volume $\phi _0 \hbox {d}V_0$. Assuming that U does not significantly vary with the skeletal deformation for constant saturation $S_{J}$, we find

$$\begin{aligned} \psi _{sk} = p_0 \phi + W\left( {\varepsilon _{ij}, \phi _J, S_C, S_G, T} \right) + \phi _0 U\left( { S_C, S_G } \right) . \end{aligned}$$

(58)

Substituting (58) into state Eq. (57), we obtain

$$\begin{aligned} \begin{array}{l} \sigma _{ij} = \frac{\partial W}{\partial \varepsilon _{ij} }, \\ p_J - p_0 = \frac{\partial W}{\partial \phi _J }, \\ \phi _0 \left( {p_{C} - p_{L} } \right) = \frac{\partial W}{\partial S_C } + \phi _0 \frac{\partial U}{\partial S_C }, \\ \phi _0 \left( {p_G - p_{L} } \right) = \frac{\partial W}{\partial S_G } + \phi _0 \frac{\partial U}{\partial S_G },\hbox { and} \\ \Sigma _{sk} = -\frac{\partial W}{\partial T}. \\ \end{array} \end{aligned}$$

(59)

Introducing the Legendre–Fenchel transform $W^{*}$ of W with respect to $\phi _J$ such that

$$\begin{aligned} W^{*} = W - \sum _{J=C,G,L} {\left( {p_J - p_0 } \right) } \phi _J, \end{aligned}$$

(60)

results in

$$\begin{aligned} \begin{array}{l} \sigma _{ij} = \frac{\partial W^{*}}{\partial \varepsilon _{ij} }, \\ \phi _J = - \frac{\partial W^{*}}{\partial \left( {p_J - p_0 } \right) },\hbox { and} \\ \Sigma _{sk} = -\frac{\partial W^{*}}{\partial T}. \\ \end{array} \end{aligned}$$

(61)

Equation (61) are the generalized state equations of unsaturated thermoporoelasticity. For a linear, isotropic thermoporoelastic material

$$\begin{aligned} \sigma _{ij}= & {} \left( {K-\frac{2 G}{3}} \right) \varepsilon _{kk} \delta _{ij} + 2 G \varepsilon _{ij} -3 \alpha _s K \left( {T - T_0 } \right) \delta _{ij}\nonumber \\&-\, b_C \left( {p_{C} - p_0 } \right) \delta _{ij} - b_L \left( {p_{L} - p_0 } \right) \delta _{ij} - b_G \left( {p_G - p_0 } \right) \delta _{ij} \end{aligned}$$

(62)

and

$$\begin{aligned} \begin{array}{l} \varphi _C = b_C \varepsilon _{kk} + \frac{p_{C} - p_0 }{N_{CC} } + \frac{p_G - p_0 }{N_{CG} } + \frac{p_{L} - p_0 }{N_{CL} } - 3 a_C \left( {T - T_0 } \right) , \\ \varphi _G = b_G \varepsilon _{kk} + \frac{p_{C} - p_0 }{N_{GC} } + \frac{p_G - p_0 }{N_{GG} } + \frac{p_{L} - p_0 }{N_{GL} } - 3 a_G \left( {T - T_0 } \right) ,\hbox { and} \\ \varphi _L = b_L \varepsilon _{kk} + \frac{p_{C} - p_0 }{N_{LC} } + \frac{p_G - p_0 }{N_{LG} } + \frac{p_{L} - p_0 }{N_{LL} } - 3 a_L \left( {T - T_0 } \right) . \\ \end{array} \end{aligned}$$

(63)

Here, $a_J$ is the linear coefficient of thermal expansion of the pore volume occupied by the phase J, and $N_{JK}$ is the generalized Biot coupling moduli, with $N_{JK} = N_{KJ}$ owing to the Maxwell’s symmetry relations, such that,

$$\begin{aligned}&\displaystyle \sum _{K=C,G,L} {\frac{1}{N_{JK} }} = \frac{b_J - \phi _0 S_J }{K_s }\hbox {,}\quad \hbox {and} \end{aligned}$$

(64)

$$\begin{aligned}&\displaystyle b = b_C + b_{G } + b_{L } = 1- \frac{K}{K_s }. \end{aligned}$$

(65)

Here a link between the macroscopic and the microscopic properties needs to be established to determine $b_J$ separately. This link can be set based on the microporomechanics considerations presented by (Dormieux et al. 2002) and applied by (Coussy 2006). At any stage of the freezing process, the gaseous phase and the ice crystals occupy the largest pores, and the remaining smaller pores are still filled up by the liquid water, and the crystal- and gas-filled pores can presumably act roughly the same as if they were enclosed in a porous body whose pore network was only constituted of liquid filled pores. Accordingly,

$$\begin{aligned} b_C + b_{G } = 1- \frac{K}{{K}'_s }, \end{aligned}$$

(66)

where ${K}'$ is the bulk modulus of the newly defined porous body such that,

$$\begin{aligned} {K}'_s = \frac{4 G_s K_s \left( {1-{\phi }'_0 } \right) }{3 {\phi }'_0 K_s + 4 G_s }. \end{aligned}$$

(67)

The matrix bulk and shear moduli, $K_s$and $G_s$, are intrinsic properties and do not depend on the properties of the pore network. The porosity ${\phi }'_0$ of this porous solid is given by

$$\begin{aligned} {\phi }'_0 = \frac{\phi _0 S_L }{1 - \phi _0 \left( {S_C + S_G } \right) }. \end{aligned}$$

(68)

Similarly, since the largest pores are occupied by the gas only, and the remaining smaller pores are filled with ice crystals and liquid water, it is therefore reasonable to assume that the wet air-filled pores are embedded in a porous matrix whose porous volume includes both the liquid filled pores and the ice filled pores. Accordingly, analogous to (66)–(68),

$$\begin{aligned} b_{G } = 1- \frac{K}{{K}''_s }, \end{aligned}$$

(69)

where

$$\begin{aligned} {K}''_s = \frac{4 G_s K_s \left( {1-{\phi }''_0 } \right) }{3 {\phi }''_0 K_s + 4 G_s } \end{aligned}$$

(70)

and

$$\begin{aligned} {\phi }''_0 = \frac{\phi _0 \left( {S_C + S_L } \right) }{1 - \phi _0 S_G }. \end{aligned}$$

(71)