*2.1. Cervera's et al. Model*

The process of cement hydration is related to the change of the phase composition of the medium. Phase transitions are accompanied by thermal phenomena associated with the heat release, as a result of hydration reactions of cement components. As a reason of heat conduction, the temperature field appears in every area of the body. In the macroscopic approach, concrete can be treated as an isotropic and homogeneous material with constant conductivity (λ = *const*.)

The model proposed by Cervera et al. [8] consists of two coupled equations: Thermal (1) and chemical kinetic equation, expressed as a function of evolution of hydration degree (2):

$$
\nabla \cdot \lambda \nabla T + Q\_{\xi} \dot{\xi} = \rho c \dot{T} \,\tag{1}
$$

$$\dot{\xi} = \widetilde{A}(\xi) \exp\{-\frac{E\_a}{R \cdot T}\},\tag{2}$$

where the following parameters are responsible for heat transfer: Concrete density ρ, specific heat of concrete *c*, thermal conductivity λ, material constant *Q*<sup>ξ</sup> (heat of cement hydration in concrete), activation energy *Ea* and gas constant *R*. In this model, a normalized internal variable, i.e., a degree of hydration ξ, which evolution allows us to predict the advancement of the hardening process has been introduced. Due to the thermally activated nature of cement hydration, a strong dependence ξ on temperature *T* is observed, and the Arrhenius type law is responsible for the kinetics of these changes. The normalized chemical affinity *<sup>A</sup>*.(ξ) is expressed through the chemical affinity *<sup>A</sup>* and permeability <sup>η</sup> [8]. Hence, the rate of hydration . ξ can be finally obtained:

$$\dot{\xi} = \underbrace{\frac{\kappa}{n\_0} \Big(\frac{A\_0}{\kappa} \frac{1}{\underline{\zeta}\_{\text{max}}} + \underline{\xi}\Big) (\underline{\xi}\_{\text{max}} - \underline{\xi}) \exp\big(-\overline{n}\frac{\underline{\xi}}{\underline{\zeta}\_{\text{max}}}\Big)}\_{\widetilde{A}(\xi) - \widetilde{A}\eta} \underbrace{\exp\big(-\frac{E\_d}{R \cdot T}\right)}\_{(Arh\'emis\ \text{type law})},\tag{3}$$

where κ/*n*0, *A*0/κ and *n* are material constants. The value of the final degree of hydration ξmax depends on the water-cement ratio *w*/*c* and can be determined in an approximate way, using the formula suggested by Mills [13]:

$$
\zeta\_{\text{max}} = (1.031 \cdot w/c)/(0.194 + w/c),
\tag{4}
$$

or Waller [14]:

$$
\zeta\_{\text{max}} = 1 - 1/\exp(3.38(w/c - \delta)),
\tag{5}
$$

where without fly ash and silica fume additives δ = 0. A comparison of both approaches is presented in Figure 1. Two propositions give similar results for the ratio *w*/*c* < 0.27. However, the above formulas do not include the type and fineness of cement.

**Figure 1.** Final hydration degree ξmax versus *w*/*c* ratio.

The normalized chemical affinity is directly measurable during an adiabatic test in which heat is not exchanged with the environment, thus the Equation (1) is simplified to:

$$Q\_{\vec{\xi}}\dot{\vec{\xi}} = \rho c \dot{T},\tag{6}$$

At the end of an adiabatic test *T* = *T*max and ξ = ξmax. Marking the initial temperature of concrete by *T*0, it is possible to express a *Q*<sup>ξ</sup> constant as:

$$Q\_{\xi} = \rho c \left( T\_{\text{max}} - T\_0 \right) / \xi\_{\text{max}} \tag{7}$$

By substituting the Equation (2) and (7) into (6) we get:

$$\widetilde{A}\_{\text{test}} = \frac{\xi\_{\text{max}} \dot{T}}{(T\_{\text{max}} - T\_0) \exp(-E\_a/(\mathbb{R} \cdot T))} \,\tag{8}$$

Formula No. 8 allows us to calculate the normalized affinity by measuring the temperature rate . *T* during the adiabatic calorimetric test. Parameters κ/*n*0, *A*0/κ and *n* can be determined from the regression analysis by fitting the function *<sup>A</sup>*.(ξ) to the results of the experiment (*A*.*test*) [8].

The described model [8] is suitable for both, ordinary and high performance concrete, and it's capabilities are presented by a wide set of experimental studies (application on the viaduct bridge, located between Denmark and Sweden).
