Evaluation of the influence of the degree of saturation, measuring time and use of a conductive paste on the determination of thermal conductivity of normal and lightweight concrete using the hot-wire method

2.1. Materials and Mix Proportions

Two different concrete mixes were designed to evaluate the influence of the three parameters mentioned before by means of a hot-wire thermal probe. In order to ensure that the measured data belonged to clearly distinct populations, a normal-weight concrete mix (NW) and a lightweight concrete mix (LW) were used. Both mixes were batched and blended using the same basic components: a CEM I 42.5R (EN 197-1), a siliceous gravel (4/12 mm) and a siliceous sand (0/4 mm). LW concrete was achieved by means of an air-entraining agent. Air entrainment reduces the thermal conductivity (22. Tatro, S. (2006) Chapter 22: Thermal properties, in significance of tests and properties of concrete and concrete-making materials, ed. J. Lamond and J. Pielert. West Conshohocken, PA: ASTM International, 226-237. Retrieved from https://doi.org/10.1520/STP37740S.
, 77. Campbell-Allen, D.; Thorne, C.P. (1963) The thermal conductivity of concrete. Mag. Concr. Res. 15 [43], 39-48. https://doi.org/10.1680/macr.1963.15.43.39.
, 88. Marshall, A.L. (1972) The thermal properties of concrete. Build. Sci. 7 [3], 167-174. https://doi.org/10.1016/0007-3628(72)90022-9.
). No other admixtures were used. Table 2 shows the nominal compositions and the real compositions after considering the air inclusion.

2.2. Test Methods

Two batches were prepared, one of NW and another of LW. Air content was determined in the fresh state (EN 12350-7). From each batch, twelve 150 mm side cubic specimens (EN 12390-1) were fabricated according to (EN 12390-2) and stored under water at 20±2°C for at least 90 days, in order to neglect the influence of continuous hydration in the thermal properties. For control purposes, compressive strength was determined (EN 12390-3) on two specimens for each batch after 90 days’ curing.

In order to measure thermal conductivity by means of a needle probe, a series of pilot-pins were inserted into ten of the specimens from each batch in the fresh state, just immediately after the troweling of the specimens (Figure 1). Pins were removed after concrete hardened. The pilot-pins were 10 cm long and 2.4 mm in diameter, equal to the measuring needle, to allow for the introduction of the probe in the hardened state. A total of 25 measuring points were molded for each set of ten concrete cubic specimens, five specimens with two measuring points and five others with three points.

Thermal conductivity was measured using the hot-wire needle method by means of a Decagon KD2 Pro Thermal Properties Analyzer (2323. Decagon Devices (2016) KD2 Pro thermal properties analyzer. Operator’s manual 68. Retrieved from http://manuals.decagon.com/Manuals/13351_KD2 Pro_Web.pdf.
) equipped with a TR-1 probe (2.4 mm x 10 cm long, range 0.1 to 4.0 W/m·K, accuracy ±10% from 0.2 to 4.0 W/m·K, ±0.02 from 0.1 to 0.2 W/m·K). The probe is heated at a constant rate for a predetermined time, while the temperature in the needle is monitored. Heating stops for the same period of time, and the cycle is applied several times depending on the instrument’s settings. Thermal conductivity can be computed from the acquired data (3838. Abramowitz, M.; Stegun, I.A. (1972) Handbook of mathematical functions, 10th ed. National Bureau of Standards. Washington, D.C. (1972) Retrieved from https://www.pdmi.ras.ru/~lowdimma/BSD/abramowitz_and_stegun.pdf.
). The longer the measuring time, the larger the data collected and the accuracy of the prediction. The apparatus provides an indication of the goodness of fit of the model to the data (3939. Kluitenberg, G.J.; Bristow, K.L.; Das, B.S. (1995) Error analysis of heat pulse method for measuring soil heat capacity, diffusivity, and conductivity. Soil Sci. Soc. Am. J. 59 [3], 719-726. https://doi.org/10.2136/sssaj1995.03615995005900030013x.
) by means of the dimensionless value err. If the err value is quite large (> 0.0100), this can be an indication of issues usually arising from contact resistance between the needle and the material, and rejection of the value must be considered. However, this is only a qualitative indicator that must be carefully interpreted by the operator of the equipment, and rejection of data must be done only after enough indications of misreading are collected.

Thermal conductivity k was first measured in three different saturation conditions for each set of ten specimens and concrete type, NW and LW, without the utilization of any thermal paste to prevent contact resistance. Firstly, the specimens were measured in a fully saturated state immediately after removing from the water tank at room temperature. After completion of all the measurements on a specimen, it was left to dry at room temperature (20±2°C, 50±5% RH) up to constant mass (Dm < 0,1%) and k was measured again at each point. Finally, the specimens were oven-dried at 70±5°C up to constant mass again, with thermal conductivity being determined once more. For each measuring point and saturation degree, k was determined for three different measuring times allowed by the device’s setting menu: 2 min, 5 min and 10 min.

Eventually, thermal conductivity was measured again, but this time using two different thermal pastes, a ceramic and a silver compound, whose purpose was to reduce contact resistance between the measuring probe and the material, improving the accuracy and precision of the measurement. The pastes were tested versus three different saturation conditions and three measuring times, meaning the specimens were again submerged in the water tank until constant mass, then dried at room temperature and finally oven-dried. On this occasion, the measurements could not be carried out with a single type of paste in all the measuring holes for each set of ten specimens, since once the hole is filled with thermal grease, the substance cannot be removed and all subsequent measurements on that point are considered to be influenced with that type of thermal paste. Furthermore, since the tolerance between the hole and the measuring probe is so tight, in some cases the needle did not fit again once the paste was introduced. Measuring points were then randomized for each type of paste.

In this regard, a total of 774 measurements of the thermal conductivity for different combinations of type of concrete, saturation degree and measuring time and thermal paste were obtained. Once all the measurements were performed, data was analyzed using the statistical analysis software IBM® SPSS® Statistics. In addition to the descriptive statistics for each set of measurements, a linear mixed-effect model (40-4240. Demidenko, E. (1987) Mixed models. John Wiley & Sons, Inc., Hoboken, NJ, USA (1987). https://doi.org/10.1002/9781118651537.
41. Lindsey, J.K. (1993) Models for repeated measurements. Oxford Statistical Science Series, Oxford University Press, New York (1993).
42. McCulloch; C.E.; Searle, S.R. (2001) Generalized, Linear; Mixed Models, 45. (2001).
) was used to fit the collected data in order to consider both fixed and random effects. Both types of concretes were analyzed separately, since it can be safely assumed that both populations are statistically independent because of the differences in thermal conductivity between both mixes. The saturation degree, use of thermal grease and measuring time are considered as fixed effects, while the specimen where measurements take place is assumed to be a random effect. Table 3 shows the dimensions of the mixed linear model for each concrete and effect discussed later, including the interactions between the effects.

3.1. Compressive Strength

The result of compressive strength after 90 days of curing is shown in Table 2 solely for the purpose of identifying the two concrete types. NW concrete reached more than 59 MPa of strength. As was expected, the inclusion of air in the LW concrete reduces significantly the strength of concrete at the approximate ratio of a 4% drop in strength for each 1% additional content of air, which acts accordingly with the rule-of-thumb usually employed for the effect of air-entraining on strength (66. Neville, A.M. (2012) Properties of concrete. 5th ed., Pearson Education Limited, Harlow, UK (2012).
).

3.2. Statistical Tests for Normality

Mixed linear models assume that the response variables follow a Gaussian distribution. This section investigates the normality of the obtained data.

Estimates for the skewness and kurtosis were calculated for each sample and combination of effects, as well as the p-values obtained in the Saphiro-Wilk (S-W) test for normality (4343. Saphiro, S.S.; Wilk, M.B. (1965) An analysis of variance test for normality (complete samples). Biometrika. 52 [3-4], 591-611. https://doi.org/10.1093/biomet/52.3-4.591.
). From the 27 data sets for NW resulting from the combination of effects, 9 of them failed both tests simultaneously, and 1 failed the skewness and kurtosis test, although with a S-W test p-value somewhat close to the limit (p = 0.051). For LW, only 3 out of 27 combinations failed both tests simultaneously.

These results might instigate certain doubts regarding the data’s normality. However, a more in-depth examination of the data rapidly revealed that the loss of normality is strongly influenced by the presence of possible outliers. First of all, skewness and kurtosis did not show any clear trend that could prove an indication of another distribution, since both parameters displayed quite random occurrences of positive and negative values, somewhat homogeneously distributed around zero.

Q-Q plots (4444. Wilk, M.B.; Gnanadesikan, R. (1968) Probability plotting methods for the analysis of data. Biometrika. 55 [1], 1-17. https://doi.org/10.1093/biomet/55.1.1.
) were drawn for all the 12 data sets that did not pass the normality tests. Figure 2 shows the Q-Q plot for the 25 thermal conductivity values measured in the NW specimens, when stored at room temperature, without thermal grease and for a measuring time of 2 min. This case is included due to its representative nature. According to the testing results, data distribution is rather removed from normality. The estimated skewness and kurtosis were 5.55 and 10.16 times their respective standard errors, and the p-value of the S-W test is 0.000. Tests for outliers identify only one possible outlier datum out of 25 points. Nonetheless, visual observation of Figure 2 quickly reveals that this single outlier is the reason of the deviation of the data points from the straight line drawn in the Q-Q plot and that indicates normality. If this value identified as the outlier is removed, the estimated skewness of the remaining 24 data points is merely 1.21 times its standard error, the kurtosis is 0.44 times, and the p-value of the S-W test becomes 0.812.

This conclusion was repeated for the rest of cases where doubts regarding normality existed, their analyses not shown herein due to space limitations. Inclusion or rejection of outliers in the subsequent analysis, always a compromised question, is discussed in the next section. In any case the authors estimate that, even before screening possible outliers out of the picture, it can be assumed the hypothesis that the measurement of the thermal conductivity in the two types of concrete follows quite reasonably a normal distribution, which makes possible the subsequent use of mixed linear models, since deviation from normality is mainly due to the presence of outliers, while the rest of the data closely follows a Gaussian distribution.

3.3. Outliers

According to the test proposed by Tukey (4545. Tukey, J.W. (1977) Exploratory data analysis. Addison-Wesley Pub. Co., Reading, Mass. (1977).
), an observation is considered as outlier if its value is 1.5 times out the interquartile range of its data set, and as far-out value if it is 3 times out. For NW concrete, 18 measurements (4.9%) out of 369 ones were initially labelled as outliers and 7 as far-out points (1.9%). For LW concrete, 8 points (2.0%) were suspected to be outliers and 3 (0.7%) far-out measurements out of 405 points.

Since this number of outliers was considered high, actions were taken in order to investigate the possible existence of side effects affecting the compromised measurements. The first action was to repeat the measurements in order to confirm the readings whenever possible. Since some of the holes were already impregnated with a type of thermal grease, it was not possible to restore the conditions for some of the no-paste cases where outliers were detected. In any event, the second set of readings confirmed the first measurements.

The decision to keep or discard the data was then taken with the help of the err function given by the Decagon KD2 Pro Thermal Properties Analyzer described in Section 2.2. It was decided that if, for a single measurement detected out of the Tukey’s fences, the err function was above 0.01, the value would be discarded. In this regard, for NW concrete 3 measurements labelled as outliers and 6 suspected far-out points were rejected, while for LW concrete 2 outliers and 1 far-out point were not considered for further analysis. Table 4 shows the number of valid measurements finally considered for each combination of effects.

It is noteworthy that the number of total measurements outside Tukey’s boundaries is larger for NW concrete than for LW concrete, even after removing probable erroneous measurements (16 versus 8). Although there is not statistical evidence to support this assumption, since the investigation submitted in this paper did not contemplate the repetition of batches for each type of concrete which could confirm this observation, a possible explanation is that, for NW concrete, the recorded thermal conductivity is closer to the upper limit of the measuring range of the probe (4.0 W/m·K), which could have a bearing on the appropriacy of the fit.

3.4. Mixed Linear Model

Table 4 shows the main descriptive statistics for the full experimental program: means, standard deviations and number of valid cases for each combination of factors and type of concrete. Table 5 shows the results of the F-tests for the main three fixed effects analyzed in this paper for both NW and LW: saturation degree, use of conductive paste and measuring time, as well as their 2 and 3-ways interactions, when considering 762 thermal conductivity measures after discarding 12 outliers.

According to the model, all three main fixed effects are significant at the 0.05 level for both types of concrete (significance p < 0.05); therefore, they are potentially significant predictors of the dependent variable thermal conductivity. When considered individually, ignoring all other variables, degree of saturation (NW: F = 525.25; LW: F = 481.73) and measuring time (NW: F = 424.13; LW: F = 287.42) have a much greater effect than the conductive paste (NW: F = 8.93; LW: F = 61.17) since their F-values are much higher than the F-value of the thermal grease effect.

Figure 3 shows the graphs of the overall means for each main effect. Visual analysis confirms the assumption that both concretes belong to different populations: LW clearly shows lower values of thermal conductivity than NW under the same conditions, as expected.

As reported previously with steady-state methods (77. Campbell-Allen, D.; Thorne, C.P. (1963) The thermal conductivity of concrete. Mag. Concr. Res. 15 [43], 39-48. https://doi.org/10.1680/macr.1963.15.43.39.
, 88. Marshall, A.L. (1972) The thermal properties of concrete. Build. Sci. 7 [3], 167-174. https://doi.org/10.1016/0007-3628(72)90022-9.
), the amount of water within the concrete increases the value of the measured thermal conductivity for NW (mean k values of 2.683, 2.989 and 3.588 W/m·K for oven-dry, room and saturated conditions, respectively). For LW, there is no significant change between oven-dry conditions (1.404 W/m·K) and room conditions (1.326 W/m·K), although thermal conductivity also increases when the specimens are saturated (1.800 W/m·K), but the rate of change is smaller than the NW. A possible explanation is that, since the entraining of air in LW produces a large number of air pores without connectivity, most of the accessible water is already removed at room temperature, but even the most energetic drying does not remove an appreciable amount of trapped water within the non-connected pores.

Regarding the measuring time, an increase in time significantly increases the value of the thermal conductivity measurements for both concrete types (mean k values of 2.610, 3.201 and 3.462 W/m·K for 2 min, 5 min and 10 min and NW, respectively; and 1.248, 1.549 and 1.728 W/m·K for 2 min, 5 min and 10 min and LW, respectively), as reported in the state-of-the-art (1616. dos Santos, W.N. (2008) Advances on the hot wire technique. J. Eur. Ceram. Soc. 28 [1], 15-20. https://doi.org/10.1016/j.jeurceramsoc.2007.04.012.
). This phenomenon is explained by the contact resistance effect, which can be significant for reduced measuring times. Indeed, for both concretes change rate of thermal conductivity measurements is greater between 2 and 5 minutes (23% and 24% for NW and LW concrete, respectively) than between 5 and 10 minutes (8% for NW concrete and 12% for LW concrete).

The use of conductive pastes is recommended by the manufacturer of the equipment in order to reduce the influence of this effect. However, as it has already been mentioned, the influence of this factor considered individually is significantly lower than the degree of saturation or the measuring time. More detailed observation of the lower graph in Figure 3 shows that the rate of change of the measurements when using no paste, with a ceramic conductive paste and with a silver paste is small (mean k values of 3.110, 3.112 and 2.991 W/m·K for NW, respectively, and 1.579, 1.425 and 1.425 W/m·K for LW), not taking into account its interaction with the other factors. Table 6 shows the estimates of the mixed model for the fixed main effect thermal paste and the results of the t-tests when inter-comparing the three levels of the effect. While the overall F-value for the use of a thermal paste was statistically significant, the t-tests between levels are statistically non-significant (p > 0.05) with these confirming the visual observation for both types of concrete which states that the use of a thermal paste in overall terms does not entail a significant change to the k value .

Regarding double interactions, for NW concrete, the double interaction saturation degree*thermal grease (F = 35.375) and grease*measuring time (F = 4.124) are significant at the 0.05 level, while saturation*time (F = 1.600) is not statistically relevant, meaning no interaction. For LW concrete, the three double interactions (saturation*grease F = 59.471; saturation*time F = 7.077; grease*time F = 7.770) are potentially significant.

For both concretes, the interaction factor between the saturation degree and the use of a thermal grease has more bearing in the model (F-value is larger). Figure 4 shows the graph of this final interaction between the saturation degree and the use of a thermal paste (upper) for the two concretes studied. The interaction is complex, as it is difficult to comprehend the effect of each principle on the combination of both of them. For NW concrete, the no-paste case shows coherent results regarding its saturation state: when increasing the water content, thermal conductivity increases as expected. The use of conductive pastes, contrary to what might be expected, increases the variability of the measurements when considering their interaction with the saturation degree (the 95% confidence interval of the means is wider). Although the use of pastes does not seem to have an effect on the NW concrete when saturated (the green line is approximately flat), the use of conductive pastes results in thermal conductivity values on dry specimens similar to those values of concrete at room equilibrium (RH = 50%).

With regard to LW concrete, the variability of the measurements using a thermal paste also increases when considering its interaction with the degree of saturation, as in the case of NW concrete, albeit to a lesser extent. Similarly, the use of paste has no effect when the concrete is saturated (green flat line). However, when the decision is made not to use conductive pastes, the conductivity results of LW concrete are not as would be expected, since although the test specimens with higher water content do show a higher conductivity, the oven-dried specimens show a significantly higher conductivity than the specimens in equilibrium with the laboratory, with values close to those of the saturated test specimens, which is not consistent. These apparently false readings for LW concrete are corrected with the use of conductive pastes, in which the conductivity increases as the degree of saturation increases. Furthermore, as observed when analyzing the fixed factors individually, the conductivity values for oven-dry concrete and concrete in equilibrium with the laboratory are similar. Both conductive pastes, ceramic and silver, behave similarly without significant differences between them.

Figure 4 also shows the interaction graph between the saturation degree and the measuring time (middle) for the two concretes studied. As it was observed when considering the effect of the measuring time individually, an increase in the measurement time produces an increase in the measured value of thermal conductivity for all saturation states in both concretes. Increasing the measuring time does not significantly affect results’ variability (error bars are approximately the same width). The graph confirms the absence of interaction for the NW concrete detected in the F-test (the three lines are quite parallel). In the case of LW concrete, there is no interaction from 2 minutes to 5 minutes between the measurement time and the degree of saturation, nor does it appear that there is interaction between the measuring time in saturated specimens and in specimens at room equilibrium. The interaction detected by the model is due to the results obtained with 10 minutes of measurement on oven-dried specimens that, contrary to what should be expected, present thermal conductivity values higher than those of the specimens in equilibrium with the laboratory, which should have a higher water content. This result is probably due to the interaction between the conductive pastes and the degree of saturation previously analyzed, which, as seen before, has a greater influence than the interaction between time and degree of saturation and may be influencing the means and confusing the graphs. This remark is supported by observation of the results for each combination of factors in Table 4. The ratio between the mean thermal conductivity of the oven-dry/no-paste/10-minutes readings and the room-equilibrium/no-paste/10-minutes case is 1.38, a strong deviation from the expected relationship, while the ratio between the oven-dry/ceramic-paste/10-minutes and the room-equilibrium/ceramic-paste/10-minutes measurements is 0.89, the foregoing being consistent with what should be expected.

Figure 4 also displays the interaction graph between the measuring time and the use of a thermal conductive paste (lower) for the two concretes studied. Again, the use of conductive pastes increases the variability of the measurements when considering their interaction with the measuring time for both types of concrete. The interaction is complex, but for both concretes, there seems to be no double effect between the use of conductive pastes and 5 and 10 minutes measuring times (green and red lines are parallel). The interaction in the conductivity results is caused by the 2 minutes measuring time. In addition, the results at 2 minutes are clearly inferior to those of 5 and 10 minutes, which is consistent with the greater measurement error expected for short times.

Regarding the 3-ways interaction, it is on the verge of significance for NW concrete, while for LW concrete it could be of relevance. Due to the complexity of physical interpretation of the 3-ways interaction and the relatively low weight in the model (F-values of 2.005 and 8.499), it is considered that this interaction is of no practical interest.

Finally, Table 7 shows the estimates of the covariance parameters between the thermal conductivity measurements and the specimens. Since the variance for both types of concrete is close to zero, it can be assumed that there was no linear relationship between the test specimens and the measurements of thermal conductivity.