2. Formalization of thermo-hydration#

2.1. Thermohydration equation#

As mentioned in the introduction, the hydration of concrete is a highly exothermic reaction. Its inclusion in the heat equation as a source term is therefore necessary (see [R5.02.02]). The second member that contains the internal heat sources can then be enriched in the following way [1] _

|:math: begin {array} {c}mathrm {rho} {C} {C} _ {p}frac {mathit {dt}} {mathit {dt}} +text {div}}mathrm {v}}mathrm {q} =Qfrac {q} = Qfrac {dmathrm {xi}} {mathit {dt}} +s\mathrm {q}} +s\mathrm {q}}} =-mathrm {lambda}text {grad}text {grad} Tend {array}} | eq 2.1-1 | +————————————————————————————————————————————————————————————————–+———-+

where:

\(\mathrm{q}\) is the heat flow,
\(s\) is an internal heat source (in \(J\mathrm{/}s\mathrm{\cdot }{m}^{3}\)),
\(\rho {C}_{p}\) is the volume heat at constant pressure (in \(J\mathrm{/}{m}^{3}\mathrm{\cdot }K\)),
\(\lambda\) is thermal conductivity (\(W\mathrm{/}{m}^{2}\mathrm{\cdot }K\)),

and specifically to hydration:

\(\mathrm{\xi }\) is the degree of hydration, by definition \(\xi \mathrm{\in }\mathrm{[}0;1\mathrm{]}\);
\(Q\) is the heat of hydration (in \(J\mathrm{/}{m}^{3}\)), i.e. the heat produced by the hydration of a volume unit of concrete.

The evolution of hydration depends on the composition of the concrete and on the temperature; a high temperature accelerates the hydration reaction.

The equation [] can be solved if the function \(\xi (t,T)\), and therefore \(\frac{d\xi }{\mathit{dt}}(t,T)\), is known.

In Code_Aster we prefer parameter \(d\xi \mathrm{/}\mathit{dt}\) over the hydration itself, and time \(t\) is thus eliminated. The corresponding function is called affinity in*Code_Aster*:

|:math: frac {dmathrm {xi}} {mathit {dt}}mathrm {:} =mathit {AFF} (mathrm {xi}, T) | eq 2.1-2 | +————————————————————————————+———-+

In Code_Aster, the heat of hydration \(Q\) and the function \(\mathit{AFF}(\xi ,T)\) must be entered by the user under the keyword THER_HYDR of DEFI_MATERIAU (see [U4.43.01]). Their experimental determination is carried out using an adiabatic test (see [§ 2.2]).

2.2. Exploitation of the adiabatic test for the determination of the affinity function and the heat of hydration.#

In an adiabatic test, a sample of fresh and thermally insulated concrete is immersed in a calorimeter and the evolution of temperature \({T}^{\mathit{ad}}(t)\) is measured over time until hardening.

This test can be used to determine the heat of hydration as well as the affinity function. Indeed, in adiabatic [] simplifies because \(\text{div}\mathrm{q}\mathrm{=}0\). In addition, \(s\mathrm{=}0\) because it is considered that the only source of heat is the hydration of concrete. The integration of [] from the beginning (\(\xi \mathrm{=}0\)) to the degree of hydration \(\xi\) then gives the expression:

|:math: `Q\ xi ({T} ^ {\ mathit {ad}} ^ {\ mathit {ad}} (t))\ mathrm {=}\ rho {C} _ {p} ({T} ^ {\ mathit {ad}}} ^ {\ mathit {ad}}}} ^ {\ mathit {ad}}}} (t) (t)\ mathrm {-} {T} _ {0}) `| eq 2.2-1 | +————————————————————————————————–+———-+

If in [], we make the assumption that \(\xi \mathrm{=}1\) (end of the test), we obtain the heat of hydration:

|:math: `Q\ mathrm {=}\ rho {C} _ {C} _ {p} ({T} _ {\ mathrm {\ infty}} ^ {\ mathit {ad}}}\ mathrm {-} {C} _ {0}) `| eq 2.2-2 |}mathrm {-} {-} {T} _ {0})` | eq 2.2-2 | +—————————————————————————————-+———-+

where \({T}_{\mathrm{0,}}{T}_{\mathrm{\infty }}^{\mathit{ad}}\) are the temperatures measured at the start and end of the adiabatic test.

By replacing the expression for the heat of hydration [] in [], we obtain the evolution of hydration (and therefore of its derivative \(d\xi \mathrm{/}\mathit{dt}\)), in the form:

|:math: xi ({T} ^ {mathit {ad}} ^ {mathit {ad}} (t))mathrm {=}frac {{T} ^ {mathit {ad}}} (t)mathrm {-} ^ {-} {-} {-} {-} {mathit {ad}} (t)mathrm {-} {-} {T}} {mathit {ad}} (t)mathrm {-} {-} {T} _ {0}} {{T}} _ {mathrm {infty}} ^ {mathit {ad}} (t)mathrm {-} {-} {T}} -} {T} _ {0}}} | eq 2.2-3 | +————————————————————————————————————————————————-+———-+

The adiabatic test therefore provides functions \(\xi ({T}^{\mathit{ad}})\) and \({T}^{\mathit{ad}}(t)\). The parameters of the \(\mathit{AFF}(T,\xi )\) affinity function can be identified from the measurement points obtained during the adiabatic test, \(\mathit{AFF}({T}^{\mathit{ad}},{\xi }^{\mathit{ad}})\).

An expression for \(\mathit{AFF}(T,\xi )\) was suggested by [Bib.2] in the form of a temperature exponential:

|:math: frac {dxi} {mathit {xi}} {mathit {dt}}mathrm {=}mathit {AFF} (xi, T)mathrm {=} A (xi)text {exp}} (text {exp}} (mathrm {-}}mathrm {-}}mathrm {-}frac {E} _ {a}} {mathit {RT}}}) | eq 2.2-4 | +———————————————————————————————————————————–+———-+

where \({E}_{a}\mathrm{/}R\) is the Arrhenius constant (a fairly empirical parameter varying between \(4000°K\) and \(7000°K\), and considered to be equal to \(4000°K\) in the absence of additional information). For the adiabatic test we then have:

where the inverse functions \({\xi }^{\mathrm{-}1}\), \({({T}^{\mathit{ad}})}^{\mathrm{-}1}\) were used to eliminate the time parameter \(t\).

Note: [*] can also be written as follows:*

\[\]

: label: EQ-None

xi (t)mathrm {=}frac {{T} ^ {mathit {ad}} (t)mathrm {-} {T}} {{T}} {{T} _ {mathrm {infty}}} ^ {mathrm {infty}}} ^ {mathit {ad}}} ^ {mathit {ad}}} ^ {mathit {ad}}}mathrm {-} {T}} {{T}} _ {0}} _ {T} _ {0}} _ {T} _ {0}} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ {T} _ ac {Q (T (t))} {Q ({T} _ {mathrm {infty}})})}

In fact, you can generally define the degree of hydration at each instant \(t\) as being the ratio of the amount of heat released up to the instant \(t\) to the total amount of heat released at the end of the hydration process.