*2.2. Martinelli's et al. Model*

The model presented in the work of Martinelli et al. [9,15] is also a thermo-chemical model, however, it differs from the previous original proposition to simulate the temperature evolution of concrete in adiabatic and non-adiabatic conditions.

The cement hydration is the reason of concrete hardening, so the concrete maturity can be expressed by the degree of cement hydration, which in general, is defined as the ratio between the amount of hydrated cement at time *t* to the total amount of cement contained in the mixture. However, this is not evident due to the fact that there is no experiment that shows directly the degree of hydration. Byfors [16] presents five different definitions of the degree of hydration, adapted to specific research possibilities. According to one of them and to Martinelli's work [9], the degree of hydration ξ(*t*) is expressed by the formula:

$$
\xi(t) = Q(t) / Q\_{\text{max}} \tag{9}
$$

where *Q*(*t*) is the heat of hydration produced at time *t*, and *Q*max means the total amount of heat. The generated heat depends on the temperature, which, in turn, is determined by the size of the sample and the boundary conditions. The analytical relationship between the degree of hydration ξ(*t*) and the corresponding temperature increase, in the ideal adiabatic conditions Δ*Ta*(*t*) is expressed by the following relationship:

$$
\Delta T\_d(t) = \frac{\mathcal{C}}{\rho \mathcal{C}} Q\_d(t) = \frac{\mathcal{C}}{\rho \mathcal{C}} \xi(t) Q\_{\text{max}} \tag{10}
$$

where *C* is the cement content per 1 m3 volume. Based on the results of adiabatic experiments on concrete samples, the evolution of the hydration heat can be approximated using equation [9,17]:

$$Q\_a(t) = Q\_{\text{max}} \underline{\chi}\_{\text{max}} \mathbf{e}^{-\left(\frac{\mathbf{f}}{\mathbf{f}}\right)^b},\tag{11}$$

where *a* and *b* control the shape of the function.

In real structures, setting and hardening of concrete in non-adiabatic conditions take place, which generates non-stationary temperature field. In general, heat transfer through a solid can be described by the Fourier equation 1. The rate of heat source of cement hydration in concrete (*Q*<sup>ξ</sup> . ξ) expressed in equation 1 is described by the formula [9]:

$$q\_{\mathcal{L}}(\mathbf{x},t) = \mathbb{C}\frac{dQ\_{\mathcal{L}}(t)}{dt}.\tag{12}$$

The current value of the concrete temperature has a significant effect on the rate of heat source *qc*, while the temperature depends on the produced heat. Therefore, the feedback effect between *qc* and *T* occurs. The temperature influence on the rate of chemical reactions is given by the relation:

$$k(T) = A\_k \exp\left(-\frac{E\_d}{R \cdot T}\right). \tag{13}$$

Arrhenius constant *Ak* and apparent activation energy *Ea* can be determined by measuring the reaction rate *k*(*T*) versus curing temperature. American standard ASTM C1074 [2] provides procedures for the calculation of mentioned values. The Arrhenius equation 13 is useful to represent the relationship between the actual heat source *qc*(*T*) and the corresponding one *qa*(*Ta*), measured in adiabatic conditions at the same stage of reaction (Figure 2a). If concrete hardening in non-adiabatic conditions has reached the degree of hydration ξ(*t*) at time *t*, it is possible to define an equivalent time *teq*, in which the same degree of hydration will be achieved under adiabatic conditions (ξ*a*(*teq*) = ξ(*t*)) [9]. Thus, it will be assumed that *Qa*(*teq*) = ξ(*t*) · *Q*max. An equivalent time *teq* may be computed from Equation (11):

$$t\_{eq} = a / \left( -\ln \left( \frac{\xi'(t)}{\xi\_{\text{max}}} \right) \right)^{\frac{1}{b}}.\tag{14}$$

**Figure 2.** Adiabatic and semi-adiabatic conditions: (**a**) Hydration heat and (**b**) temperature development [9].

Although the heat released in non-adiabatic conditions *Qc*(*T*(*t*)) at time *t* is equal to the heat generated in the adiabatic process *Qa*(*Ta*(*teq*)) at time *teq*, the temperature values *T*(*t*) and *Ta*(*teq*) are not equal (Figure 2b). Considering the above dependence, the ratio of heat source of cement hydration in concrete, in both indicated conditions, can be expressed using Equation (13):

$$\frac{q\_c(T(t))}{q\_a(T\_a(t\_{cq}))} = \frac{\mathbf{e}^{-\frac{E\_q}{RT(t)}}}{\mathbf{e}^{-\frac{E\_q}{RT\_a(t\_{cq})}}} = \mathbf{e}^{-\frac{E\_q}{R}\frac{T\_a(t\_{cq}) - T(t)}{T\_a(t\_{cq}) - T(t)}}.\tag{15}$$

Thus, describing the heat source *qc*(*T*(*t*)) using the formula 15, the partial differential equation representing the thermal energy balance in one-dimensional space is equal to:

$$\rho \lambda \cdot \frac{\partial^2 T}{\partial \mathbf{x}^2} + q\_d(T\_d(t\_{\text{eq}})) \cdot \mathbf{e}^{-\frac{\mathbb{E}}{\mathcal{R}} \frac{T\_d(t\_{\text{eq}}) - T(t)}{T\_d(t\_{\text{eq}}) \cdot T(t)}} = \rho \mathbf{c} \frac{\partial T}{\partial t} \,. \tag{16}$$

Equation (16) with the defined initial-boundary conditions allows us to determine the temperature distribution of the concrete in non-adiabatic conditions.

In analyzed paper [9], the validation of the mathematical model was carried out using 150 mm cubic samples and prismatic specimens. The experimental and numerical results showed a high compatibility and the possibility to estimate the compressive strength of concrete.
