**3. Initial-Boundary Conditions**

Assuming that the model describes the temperature development immediately after concrete mixing and placing, the initial condition can be written in the following form:

$$T(\mathbf{x}, y, z; \mathbf{t}) = T\_0(\mathbf{x}, y, z),\tag{17}$$

where *T*<sup>0</sup> is the initial temperature of concrete.

When considering boundary conditions, two mechanisms of heat transfer between a solid and the environment should be distinguished: Convection (natural and forced) and radiation (Figure 3). In natural convection, the heat flow from the concrete surface takes place as a result of the difference between the concrete and ambient temperature. In the process of forced convection, an additional factor accelerates the heat exchange, e.g., the action of wind. The second mechanism of heat transport is radiation. Longwave radiation applies to any solid substance that emits and receives radiation from the environment. Shortwave radiation is a type of energy transmission usually considered in relation to the energy emitted by the sun. It needs to be marked, that when the steel formwork is used, it can absorb large energy from the sun. As a result, the temperature of concrete can be significantly affected by the conduction heat between the steel formwork and concrete.

**Figure 3.** Heat transfer between a solid and the environment: (**a**) Natural convection; (**b**) forced convection; (**c**) longwave radiation and (**d**) shortwave radiation [12].

The equation of the energy balance is supplemented by the Newton's [18] or Stefan-Boltzmann's condition. From the physical point of view, heat exchange between the surface and the environment takes place, both, through convection and radiation. Formally, the heat transfer coefficient is a convective type factor [18]. The heat flow coming to the concrete surface must be absorbed by the surrounding air. Irrespective of the driving force of the air movement, the convective heat transfer can be expressed by Newton's law:

$$q\_0 = \alpha \left( T\_{surf} - T\_{env} \right), \tag{18}$$

where α is the convective heat transfer coefficient and *Tenv* and *Tsur f* means the ambient and concrete surface temperature. Although, many theoretical attempts have been made to establish prediction equations for the heat transfer coefficient, accurate predictions are available only for very simple geometries and controlled environmental conditions. In irregular cases, such as variable external conditions and the complex geometry of engineering structures, the forecasts were limited to empirical correlations [12]. According to the Institute of Physics of the Cracow University of Technology [19], the heat transfer coefficient on the concrete surface for natural convection <sup>α</sup>*nc* is 7 W/(m2·K). Klemczak gives the value of 6 W/(m2·K) [20] and also proposes a formula [21]:

$$
\alpha\_{\rm nc} = 2.62 \left( T\_{surf} - T\_{\rm env} \right)^{0.25}. \tag{19}
$$

For forced convection, the heat transfer coefficient on the concrete surface α*f c* depends on the wind speed *vw* and can be calculated according to the McAdams [22], Jonasson [23], Branco [24], Ruitz [25], Silveira [26] and Jayamaha [27] (Figure 4). It should be emphasized that suggestions shown in Figure 4 apply to the perfect reproduction of previous tests, in other situations they can only provide approximate solutions [12].

**Figure 4.** The convective heat transfer coefficient α*f c* versus wind speed *vw*.

On the construction site, an additional concrete surface insulation in the form of the styrofoam layer or foil is often used. For thermally fixed conditions, where the heat flux is perpendicular to the surface, a substitute heat transfer coefficient α*<sup>s</sup>* can be determined taking into account the thickness *li* and conduction coefficient of a particular layer λ*i*:

$$\alpha\_s = \left(\frac{1}{a} + \sum\_{i=1}^{i=n} \frac{l\_i}{\lambda\_i}\right)^{-1}.\tag{20}$$

Jonasson states that for young concrete the heat transfer coefficient is theoretically correct only for free surface, but the above formula can be applied as some approximation, when the insulation has a lower heat capacity than concrete and the volume of other materials in the formwork (plywood and styrofoam) it is small, i.e., the heat accumulated in the formwork can be omitted compared with the thermal energy in the concrete [18]. In case where there is insulation between the formwork girders, the heat flux is not perpendicular to the surface and equation 20 cannot be used [18].
