*6.2. Three Selected Parameters:* κ/*n*0*, n and A*0/κ

The model parameters (κ/*n*0, *n* and *A*0/κ) required to perform numerical simulations were identified experimentally. The laboratory tests involved the concrete temperature measurements in the cubic samples curing under isothermal and semi-adiabatic conditions. For this purpose, four 150 mm cubic specimens made according to the recipe of the analyzed concrete were prepared. Immediately after molding, two cubes were placed in a water bath at a constant temperature of 24 ◦C and the concrete temperature changes under isothermal conditions were recorded. Two more cubes were inserted in specially prepared styrofoam containers with 10 cm thick insulation layer. The insulation around the specimens slows down the rate of heat loss and limits the heat exchange with the environment, thus cubes can be cured in semi-adiabatic conditions. The measurement boxes allow free samples placement in the mold and to insert the temperature sensors (Figure 12a). The experimental set-up illustrated in Figure 12b consists of a computer, 1-wire temperature sensors, semi-adiabatic containers and a water bath.

**Figure 12.** (**a**) Styrofoam container to semi-adiabatic concrete curing and (**b**) experimental set-up.

The maximum temperature in concrete cubes reached under semi-adiabatic conditions was 10.4 ◦C higher than in isothermal conditions and the rate of temperature increase was more than twice bigger (Figure 13). The tests have proved how large the influence of boundary conditions on the temperature is, even with such a small volume of concrete. The average characteristic values from two concrete cubes are summarized in Table 3.

**Figure 13.** Temperature evolution in the centre of a concrete cubes cured in isothermal and semi-adiabatic conditions.


**Table 3.** Characteristic temperature values.

6.2.1. Determination of Model Parameters Using the Martinelli's Approach

The partial differential equation of heat flow, reported by Cervera [8] and Martinelli [9], was solved using the finite difference method in the author's programs written in the MATLAB environment. For the models described in paragraph No. 2, the validation process was successfully performed based on literature data [8,9] and then used to analyze the own cases.

Experimental results, presented in above paragraph (Figure 13) were used to determine the adiabatic heat curve. Regression coefficients (*a*, *b*) for the function (11) and the heat transfer coefficients of the concrete hardening under isothermal (α*<sup>w</sup> nc*) and semi-adiabatic conditions (α*s*) were calculated by the least squares regression. Table 4 reports the input values adopted in the model.


**Table 4.** Model parameters according to Martinelli et al. [9].

The adiabatic temperature evolution represents the response of the simulations of the semi-adiabatic and isothermal temperature development (Figure 14). In both cases, the maximum temperature of self-heating of concrete hardening in adiabatic conditions was mutually convergent and the temperature value equals:

$$T\_{\text{max}} = T\_0 + \frac{\mathbb{C} \cdot \mathbb{X}\_{\text{max}} \cdot Q\_{\text{max}}}{\mathbb{c} \cdot \rho} = 23.2 + \frac{440 \cdot 0.65 \cdot 330}{0.84 \cdot 2570} = 23.3 + 43.7 = 66.9 \, (^{\circ}\text{C}). \tag{26}$$

**Figure 14.** Adiabatic hydration curve for two various conditions: (**a**) Semi-adiabatic and (**b**) isothermal.

In addition, this value was consistent with the maximum concrete temperature (67.8 ◦C) measured in the middle of the 93 cm thick bottom slab (point p5, Figure 6b), for which conditions close to adiabatic could be assumed. Very good agreement between real measured value of *T*max and numerical response was confirmed.

The results of calculations using the finite difference method (FDM) for the considered specimens allowed us to plot the development of hydration degree <sup>ξ</sup> versus time and chemical affinity *<sup>A</sup>*.*test* as a function of hydration degree according to equation 8, (Figure 15a). Founded on the nonlinear approximation of affinity *<sup>A</sup>*.(ξ) to experimental data (Figure 15b), three sought parameters of the model were obtained: <sup>κ</sup>/*n*<sup>0</sup> = 6.6 · 106 <sup>h</sup><sup>−</sup>1, *<sup>n</sup>* = 5.2 and *<sup>A</sup>*0/<sup>κ</sup> = <sup>1</sup> · <sup>10</sup><sup>−</sup>4.

**Figure 15.** (**a**) Time evolution of the hydration degree and (**b**) chemical affinity vs. hydration degree.

6.2.2. Determination of Model Parameters Using the Cervera's Approach

The next step included verification of the determined constants in reliance on temperature simulations of the concrete specimens according to the Cervera's approach and also involved a possible modification of the three parameters. For this purpose two simulations were carried out: "v1" for designated constants: <sup>κ</sup>/*n*<sup>0</sup> = 6.6 · 106 <sup>h</sup><sup>−</sup>1, *<sup>n</sup>* = 5.2, *<sup>A</sup>*0/<sup>κ</sup> = <sup>1</sup> · <sup>10</sup>−<sup>4</sup> and "v2" for coefficient *<sup>A</sup>*0/<sup>κ</sup> = 1 · 10<sup>−</sup>5. The actualization of *<sup>A</sup>*0/<sup>κ</sup> parameter improved the solution, which agreed with the measurement data (Figure 16).

**Figure 16.** Temperature of concrete specimens cured in semi-adiabatic conditions: (**a**) In the time domain and (**b**) space variation of the temperature field in a 150-mm cube at different curing times (simulations + four measured values).

The predicted adiabatic temperature evolution was consistent with Martinelli's proposal (Figure 17). The results received in this part of the work constituted a check of the models themselves as well as determined parameters and indicated the correctness of the adopted assumptions.

**Figure 17.** The concrete temperature in adiabatic conditions for two approaches.

Additionally, in order to recognize the influence of κ/*n*0, *n* and *A*0/κ parameters on the temperature distribution of concrete, three cases were analyzed. In each case, one of the mentioned parameters was changed, and two subsequent remained constant:


Figure 18 shows the results of calculations for the temperature development in a concrete cubic sample. The temperature measured experimentally with a dashed, black line was marked. On the basis of Figure 18 it was found, that the coefficient κ/*n*<sup>0</sup> influenced the reaction rate, *n* was responsible for the extreme temperature value and *A*0/κ for the time of its occurrence. If the parameter κ/*n*<sup>0</sup> increased, the reaction rate also rose. The maximum peak value of concrete temperature increased every ~1 ◦C and time occurrence of peak was reduced twice from <sup>κ</sup>/*n*<sup>0</sup> = <sup>4</sup> · 106 <sup>h</sup>−<sup>1</sup> to <sup>κ</sup>/*n*<sup>0</sup> = <sup>12</sup> · <sup>10</sup><sup>6</sup> <sup>h</sup><sup>−</sup>1. When *<sup>n</sup>* rose, the extreme concrete temperature dropped gradually from 2 to 1 ◦C for every half from *n* = 7 to *n* = 4. If we consider coefficient *A*0/κ the time occurrence of peak was shorter every ~3 h form *<sup>A</sup>*0/<sup>κ</sup> = <sup>1</sup> · <sup>10</sup>−<sup>7</sup> to *<sup>A</sup>*0/<sup>κ</sup> = <sup>1</sup> · <sup>10</sup>−<sup>2</sup> with step equals 10<sup>−</sup>1.

**Figure 18.** Three considered model parameters: (**a**) κ/*n*0; (**b**) *n* and (**c**) *A*0/κ.
