(1)
For the fresh cement pastes, heat capacities of 1.55 J/(g·K) and 1.73 J/(g·K) were measured for the w/c=0.3 and w/c=0.4 fresh cement pastes, respectively. Using the known heat capacity of water [22] (4.18 J/(g·K) at 23 oC) and a value of 0.75 J/(g·K) for cement powder (based on the measured values for tricalcium silicate and dicalcium silicate [5] as:
(1)
where Mfwater is the mass fraction of water in the fresh paste, and Mfcem = 1 - Mfwater is the mass fraction of cement powder. When this equation is applied for w/c=0.3 (Mfwater =(0.3/1.3)=0.231) and w/c=0.4 (Mfwater =(0.4/1.4)=0.286), estimated heat capacities of 1.54 J/(g·K) and 1.73 J/(g·K), respectively, are obtained, in excellent agreement with the measured values provided above. For a w/c=0.5 cement paste, equation (1) would predict a heat capacity of 1.89 J/(g·K), while a value of 1.92 ± 0.05 J/(g·K) at 1.5 h maturity has been obtained previously [15]. The value of about 1.7 J/(g·K) reported for a w/c=0.4 fresh cement paste in [13] is also in good agreement with the values measured experimentally and calculated according to equation (1) here.
The measured heat capacities for the hydrating cement pastes cured under sealed and saturated conditions are provided in Figure 2. For both w/c under sealed curing conditions, the heat capacity is observed to decrease rapidly at early hydration and then level off to a nearly constant value. Apparently, as the pore water becomes chemically and physically bound within the (gel) hydration products during the early stages of hydration, its heat capacity decreases significantly. For bulk water, a significant contribution to its unusually high heat capacity is the energy consumed in the breaking and bending of hydrogen bonds. It is possible that less bending and breaking will occur in “more restricted bound” water, leading to a lower heat capacity. The heat capacity of ice is about one half of the value of liquid water [24], supporting this hypothesis that as the water molecules become less mobile during cement hydration, their heat capacity would decrease.
Fig. 2 - Measured heat capacities of hydrating cement pastes as a function of degree of hydration for curing under saturated and sealed conditions. Error bars indicate a reproducibility of ± 2 %.
Previously, Hansen et al. [15] have plotted measured heat capacities vs. the log of maturity (in h), while De Schutter and Taerwe [5] have hypothesized a linear (decreasing) relationship between heat capacity and degree of hydration. Here, to adequately capture the observed initial rapid decrease in measured heat capacity, the data obtained in this study for hydration under sealed conditions were fitted to a relationship as a function of the measured degree of hydration, α, of the form:
(2)
with the obtained fitting constants A=0.26 and B=2.9 for both w/c=0.3 and w/c=0.4 cement pastes hydrated under sealed conditions. The observed rapid decrease in heat capacity during initial hydration followed by a leveling off with continued hydration as described by equation (2) would be consistent with the previously measured development of the surface area of hydrating cement paste by neutron scattering techniques [25].
For the specimens hydrated under saturated conditions, the experimentally measured heat capacities soon bifurcate from those measured under sealed conditions. For the w/c=0.3 pastes, the capillary pores depercolate after a few days hydration [26], so that it is not possible to easily maintain saturation for the 5 mm to 6 mm thick disks employed in this study. Thus, in Figure 2, the heat capacity values for these pastes reach a local maximum after 3 d of hydration (degree of hydration = 0.5) and then return towards those measured for sealed curing conditions, before increasing once more. In support of this hypothesis concerning depercolation and saturation, the measured masses of imbibed water for these “saturated” specimens follow the same trend as the presented heat capacities. This same decreasing trend in heat capacities at later ages is observed for the saturated w/c=0.4 pastes hydrated for 8 d, 15 d, and 28 d (the last three data points of their data set in Figure 2), in agreement with the expectation that this higher w/c paste would achieve depercolation of its capillary pores somewhere between 7 d and 14 d of hydration [26].
Prior to depercolation of the capillary porosity, the maintenance of saturated conditions results in a set of relatively large water-filled capillary pores remaining in the hydrating cement paste microstructure; under sealed curing conditions, these pores will be the first to empty due to self-desiccation [20, 27]. Since the heat capacity of “free” water is 4.18 J/(g·K), the imbibition and subsequent presence of this pore water would be expected to increase the heat capacity of the composite relative to that achieved under sealed curing conditions, as is observed in Figure 2. An additional effect of this extra curing water is that the degree of hydrations of the saturated specimens are increased relative to those achieved by the sealed specimens at the same hydration age [20, 27]. In Figure 2, an attempt has been made to apply the law of mixtures to the heat capacity obtained for the saturated specimens, by considering them to be a mixture of a sealed specimen and (extra) imbibed water. Knowing the chemical shrinkage (CS) of the cement used at complete hydration (0.073 g of water per 1 g of cement), the amount of imbibed water per gram of original cement paste can be conveniently estimated as αCS/(1+w/c). Then, applying the law of mixtures results in:
(3)
Equation 3 was found to provide a reasonable fit to the experimental data only when the heat capacity of the imbibed water was assumed to be 9 J/(g·K), about twice the value of 4.18 J/(g·K) for bulk water. This would suggest that the presence of the additional imbibed water and the maintenance of “saturated” conditions also increase the heat capacity of the remainder of the (original) water in the hydrated cement paste, relative to its value in an equivalent sealed system. Under sealed conditions, due to the chemical shrinkage and self-desiccation that accompanies the hydration [28], large internal stresses are generated within the pore water that might further decrease its mobility and thus, its heat capacity, as discussed above.
To extend these heat capacity measurements to concrete at various ages, the law of mixtures could again be applied [3]:
(4)
where Cppaste represents the heat capacity of the hydrating cement paste at the age (degree of hydration) of interest and can be estimated using either equation (2) or equation (3) for sealed or saturated curing conditions, respectively.
While the thermal conductivity of liquid water (k1) is well known [22], that of cement powder (k2) could not be found in the literature. An estimate, however, can be readily determined based on the measured thermal conductivities of the fresh w/c=0.3 and w/c=0.4 cement pastes and application of the Hashin-Shtrikman (H-S) bounds for the thermal conductivity of a two-phase (cement particles in water) material. For k2≥ k1, the Hashin-Shtrikman lower (kl) and upper (ku) bounds for the thermal conductivity of a two-phase composite, with volume fractions of water (x1) and cement powder (x2=1-x1), are given by [29]:
(5)
(6)
Figure 3 provides a plot of these bounds along with the measured data for the two fresh cement pastes utilizing thermal conductivity values of 0.604 W/(m·K) and 1.55 W/(m·K) for water and the cement powder, respectively. The latter value for the cement powder provides H-S bounds that encompass both of the experimental data points for the fresh cement pastes.
The measured thermal conductivities as a function of degree of hydration are provided in Figure 4. Within the reproducibility of the measurements, there is little variation in the thermal conductivity with degree of hydration and nominally a value of 1.0 W/(m·K) would provide a reasonable estimate for both w/c pastes and both curing conditions at all evaluated degrees of hydration. This value is in reasonable agreement with several of the previously measured values that were summarized in Table 1 (those ranging from 0.77 W/(m·K) to 1.16 W/(m·K), for example), and especially with the values provided recently in [12] for a w/c=0.348 cement paste and [13] for a w/c=0.4 cement paste. However, the previously measured data for crushed cement paste specimens (2.85 W/(m·K)) [8] are both significantly different from the values obtained in this (and the other previous) studies. Furthermore, in [8], a heat capacity of 0.736 J/(g·K) is reported for a w/c=0.35 cement paste hydrated at 100 % RH for 28 d, also in considerable disagreement with the values measured in the present (see Figure 2) and other previous studies.
Fig. 3 - Measured thermal conductivities of fresh cement pastes as a function of initial water volume fraction. Error bars indicate a reproducibility of ± 2 %.
Fig. 4 - Measured thermal conductivities of hydrated cement pastes as a function of measured degree of hydration. Error bars indicate a reproducibility of ± 2 %.
The H-S bounds can also be applied in extending thermal conductivity predictions to concrete [6], by considering it to be a two-phase composite consisting of aggregates in hydrated cement paste (effectively ignoring any air entrainment). Then, knowing the thermal conductivity of the specific aggregates [30] and assuming a value of 1.0 W/(m·K) for the hydrated cement paste, equations (5) and (6) can be applied to determine upper and lower bounds for the thermal conductivity of any concrete composite of known mixture proportions, typically considering cement paste as phase 1 and aggregates as phase 2, since the thermal conductivity of most aggregates is higher than the cement paste’s nominal value of 1.0 W/(m·K) and equations (5) and (6) require k2≥ k1. (Lightweight aggregates with their much lower thermal conductivity could be an exception to this; in that case, the aggregates could be considered as phase 1 and the higher thermal conductivity cement paste as phase 2.) Finally, a reasonable estimate of the thermal conductivity of the concrete of interest could be taken as the mean of these upper and lower bounds. As an example, Figure 5 shows the computed H-S bounds for a concrete containing limestone aggregates (k2 ≈ 3. W/(m·K) [7, 31]). For the typical cement paste volume fraction of 30 % to 35 %, the concrete would be expected to have a thermal conductivity of (2.1 to 2.2) W/(m·K). While the H-S bounds are fairly tight in Figure 5, for siliceous aggregates such as quartz with their higher thermal conductivity (≈ 5. to 8. W/(m·K) [6, 7, 30]), the H-S bounds will be wider and the inaccuracy of using the mean H-S value as an estimate for the concrete will increase.
Fig. 5 - Estimates based on the H-S bounds for the thermal conductivity of a concrete as a function of the volume fraction of paste, assuming that the cement paste has k1 = 1. W/(m·K) and the (limestone) aggregate has k2 = 3. W/(m·K).
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