Formalization of thermo-hydration ====================================== 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] _ : +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------+ .. _RefEquation 2.1-1 |: |:math: `\ begin {array} {c}\ mathrm {\ rho} {C} {C} _ {p}\ frac {\ mathit {dt}} {\ mathit {dt}} +\ text {div}}\ mathrm {v}}\ mathrm {q} =Q\ frac {q} = Q\ frac {d\ mathrm {\ xi}} {\ mathit {dt}} +s\\\ mathrm {q}} +s\\\ mathrm {q}}} =-\ mathrm {\ lambda}\ text {grad}\ text {grad} T\ end {array}\}` | eq 2.1-1 | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------+ where: * :math:`\mathrm{q}` is the heat flow, * :math:`s` is an internal heat source (in :math:`J\mathrm{/}s\mathrm{\cdot }{m}^{3}`), * :math:`\rho {C}_{p}` is the volume heat at constant pressure (in :math:`J\mathrm{/}{m}^{3}\mathrm{\cdot }K`), * :math:`\lambda` is thermal conductivity (:math:`W\mathrm{/}{m}^{2}\mathrm{\cdot }K`), and specifically to hydration: * :math:`\mathrm{\xi }` is the degree of hydration, by definition :math:`\xi \mathrm{\in }\mathrm{[}0;1\mathrm{]}`; * :math:`Q` is the heat of hydration (in :math:`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 :math:`\xi (t,T)`, and therefore :math:`\frac{d\xi }{\mathit{dt}}(t,T)`, is known. In *Code_Aster* we prefer parameter :math:`d\xi \mathrm{/}\mathit{dt}` over the hydration itself, and time :math:`t` is thus eliminated. The corresponding function is called **affinity** in*Code_Aster*: +------------------------------------------------------------------------------------+----------+ .. _RefEquation 2.1-2 |: |:math: `\ frac {d\ mathrm {\ xi}} {\ mathit {dt}}\ mathrm {:} =\ mathit {AFF} (\ mathrm {\ xi}, T)` | eq 2.1-2 | +------------------------------------------------------------------------------------+----------+ In *Code_Aster*, the heat of hydration :math:`Q` and the function :math:`\mathit{AFF}(\xi ,T)` must be entered by the user under the keyword THER_HYDR of DEFI_MATERIAU (see [:external:ref:`U4.43.01 `]). Their experimental determination is carried out using an adiabatic test (see [§ :ref:`2.2 `]). .. _RefNumPara__3190_825074820: 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 :math:`{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 :math:`\text{div}\mathrm{q}\mathrm{=}0`. In addition, :math:`s\mathrm{=}0` because it is considered that the only source of heat is the hydration of concrete. The integration of [] from the beginning (:math:`\xi \mathrm{=}0`) to the degree of hydration :math:`\xi` then gives the expression: +--------------------------------------------------------------------------------------------------+----------+ .. _RefEquation 2.2-1 |: |: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 :math:`\xi \mathrm{=}1` (end of the test), we obtain the heat of hydration: +----------------------------------------------------------------------------------------+----------+ .. _RefEquation 2.2-2 |: |: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 :math:`{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 :math:`d\xi \mathrm{/}\mathit{dt}`), in the form: +-------------------------------------------------------------------------------------------------------------------------------------------------+----------+ .. _RefEquation 2.2-3 |: |: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 :math:`\xi ({T}^{\mathit{ad}})` and :math:`{T}^{\mathit{ad}}(t)`. The parameters of the :math:`\mathit{AFF}(T,\xi )` affinity function can be identified from the measurement points obtained during the adiabatic test, :math:`\mathit{AFF}({T}^{\mathit{ad}},{\xi }^{\mathit{ad}})`. An expression for :math:`\mathit{AFF}(T,\xi )` was suggested by :ref:`[Bib.2] ` in the form of a temperature exponential: +-----------------------------------------------------------------------------------------------------------------------------------+----------+ .. _RefEquation 2.2-4 |: |:math: `\ frac {d\ xi} {\ 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 :math:`{E}_{a}\mathrm{/}R` is the Arrhenius constant (a fairly empirical parameter varying between :math:`4000°K` and :math:`7000°K`, and considered to be equal to :math:`4000°K` in the absence of additional information). For the adiabatic test we then have: +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------+ .. _RefEquation 2.2-5 |: |:math: `A ({T} ^ {\ mathit {ad}} (\ xi))\ mathrm {=}\ frac {1} {{T} _ {\ mathrm {\ infty}}} ^ {\ mathit {ad}} ^ {\ mathit {ad}}} ^ {\ mathit {ad}}} ^ {\ mathit {ad}}}\ mathrm {-} {\ mathit {ad}}}\ mathrm {-} {T}}\ frac {{\ mathit {dT}}} ^ {\ mathit {ad}}} ^ {\ mathit {ad}}} ^ {\ mathit {ad}}}\ mathrm {-} {\ mathit {ad}}}\ mathrm {-} {T}} ad}}} {\ mathit {dt}} ({T}} ({T} ^ {\ mathit {ad}}} (\ xi))\ mathrm {\ exp} (\ frac {{E} _ {a}} {{a}}} {{\ mathit {RT}}} {{\ mathit {ad}}})\ mathrm {exp} (\ xi)} (\ xi)} (\ xi)} (\ xi)} (\ frac {E} _ {a}}}} {{a}}} {{\ mathit {RT}}} {{\ mathit {RT}}} ^ {\ mathit {ad}}} (\ xi)}) `| eq 2.2-5 | +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------+ where the inverse functions :math:`{\xi }^{\mathrm{-}1}`, :math:`{({T}^{\mathit{ad}})}^{\mathrm{-}1}` were used to eliminate the time parameter :math:`t`. **Note**: [*] can also be written as follows:* .. math:: : 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* :math:`t` *as being the ratio of the amount of heat released up to the instant* :math:`t` *to the total amount of heat released at the end of the hydration process.* .. [1]. Note: in the context of nonlinear thermics [R5.02.02], the first of the equations is often written in the equivalent form: :math:`\frac{d\beta (T)}{\mathit{dt}}+\text{div}\mathrm{q}\mathrm{=}Q\frac{d\xi (T)}{\mathit{dt}}+s`, :math:`\beta` being the volume enthalpy. In fact, in the Code_Aster command file it is necessary to enter the enthalpy and not the volume heat, see [R5.02.02].