#
Simplified Method of Estimating the A_{300} Micropore Content in Air-Entrained Concrete

^{*}

## Abstract

**:**

_{300}. The A

_{300}parameter requires complex calculations with the estimation of the air-void size in a 3D space. The procedure is based only on one-dimensional chord lengths. The air-void distribution is used only to determine the content of micropores and has no other practical application. Based on the results of the analysis, it was found that there is a simpler way to estimate the A

_{300}parameter without the tedious calculations described in the EN 480-11 Standard. The presented approach is based on the existing linear correlation between the A

_{300}parameter and the number of chords in 28 length classes. The developed function includes only a few coefficients (eight classes) because only chord lengths of 10–350 µm are statistically significant. This fact is important not only for the simplification of calculations but may also have consequences for the methodology of testing parameters characterizing the structure of air-entrained concrete using the 2D method. The presented function allows the estimation of A

_{300}with a standard error not exceeding 0.02%, so it is useful for practical use.

## 1. Introduction

_{300}micropore content is about 1%. On the other hand, an A

_{300}condition above 2.0% does not guarantee the spacing of the air bubbles L < 0.20 mm. Many documents assume that concrete frost-resistance criteria concern both the requirement for spacing factor L < 0.20 mm and the content of A

_{300}> 1.5% [7,8] or A

_{300}> 2.0%, according to Dag Vollset et al. [9]. However, the micropore content is regarded as an auxiliary criterion of secondary importance by the ACI recommendation and the ASTM C 457 Standard. In these documents, the A

_{300}criterion is not considered at all.

_{300}could be verified, e.g., by using computed tomography (CT). At present, the resolution of the devices is too small in relation to the dimensions of the small pores, and such an estimation is unreliable [19].

_{300}, and α were determined by the Rosival linear traverse method, counting the length of the chords according to the EN-480-11 Standard. The aim of the analysis was to determine the relationship between the number of chords in individual classes and the content of A

_{300}micropores. A statistical analysis using the linear multiple regression method made it possible to determine the correlation relationships and the significance of the influence of the selected pore ranges (classes) with the A

_{300}parameter. A positive verification of the obtained regression functions would allow the proposal of a simpler, faster A

_{300}calculation method without using either tedious standard calculations or estimating the air-void distributions.

## 2. Materials and Methods

_{300}, were read directly from the distribution table as the volume attributed to all the air voids with diameters ranging from 0 to 300 μm.

_{300}, specific surface area, α, and a number of chords, N, (chord length range of 0–4000 μm).

_{300}, α, and L. The general characteristics of the set are presented in the form of a matrix diagram (Figure 2) and in Table 1.

## 3. Results

_{300}micropore content is determined. The calculated value of A

_{300}is only a certain estimate of the actual micropore content of the space.

_{0}, a

_{1}, …, a

_{28}—regression coefficients;

_{300}micropore content);

_{tot}—the length of the measuring line (total for 2 samples).

_{tot}, (Table 1) ranged from 2438 mm to 2643 mm (mean 2522 mm), which significantly affected the number of chords recorded. Therefore, it was necessary to introduce a conversion factor in the equation, bringing the line length to 2500 mm.

_{300}). The analysis was carried out in three stages, eliminating variables (chord classes) with a non-significant impact on A

_{300}and grouping the others into classes with larger dimension ranges. This led to the development of a linear function with only eight regression coefficients. A number of analyses were carried out to verify the goodness of fit of the function with the measurement data, verifying the possible autocorrelation of the independent variables and the normality of the distribution of the residuals as key conditions for the quality of the relationship obtained. All the statistical calculations were performed using Statistica software.

**Stage 1**: Analysis of the dataset including all 28 chord classes

^{2}= 0.9999, Adj.R

^{2}= 0.9999;

- where:

^{2}—coefficient of determination;

^{2}—corrected goodness of fit;

**Stage 2: Analysis of 18 chord classes**

^{2}= 0.9999, Adj. R

^{2}= 0.9999;

_{300}. Small pores with chords of 0–10 μm have a small volume and number, which has little effect on the volume gain of the A

_{300}micropores. The pores with larger dimensions, corresponding to chord lengths greater than 350 μm, do not have a significant impact on the A

_{300}value. According to the stereology assumptions, larger pores can also generate chords in smaller classes, but the number of such pores is too small to affect the A

_{300}. The results obtained provide the basis for reducing the number of necessary chord classes from 28 to 19, and thus the number of corresponding coefficients in the regression (Equation (1)).

**Stage 3: Analysis of eight chord classes**

^{2}= 0.99982, Adj. R

^{2}= 0.99981;

_{300}. The more large pores, the greater the number of chords associated with these pores in the smaller classes. This fact must be taken into account to not overestimate the value of the A

_{300}.

^{2}≈ 1.0). The estimated model meets the assumptions of the least squares method. The analysis of the residuals of the analysed model confirms its validity. The value of the Durbin–Watson test statistic (DW = 1.912) allows us to conclude that there is no autocorrelation of the residuals in the resulting model.

## 4. Conclusions

_{300}micropore content to be estimated with good accuracy as an alternative to the rather complicated method contained in the EN-480-11 Standard. It was assumed that the A

_{300}value could be estimated without the need to determine the distribution of pores with diameters up to 4000 μm. It should be emphasised that, apart from the A

_{300}value, the pore size distribution is of no practical significance.

_{300}) are random variables, the relationships are linear and without autocorrelation, and the residuals have a normal distribution. The tests confirmed the validity of these assumptions.

- −
- chords with lengths of 0–10 μm and greater than 355 μm have no statistically significant effect on the A
_{300}value; - −
- only eight classes with a chord size range of 15–350 μm are statistically significant;
- −
- the final linear multiple regression function with eight independent variables allows us to precisely predict the A
_{300}micropore content; - −
- the standard error of the calculation is approximately 0.02%, which means that the error of the A
_{300}estimate does not exceed 0.1% of the volume of concrete and is therefore sufficient for practical purposes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Matrix diagram showing the relationships between factors A, A

_{300}, α, and N (the column variables are used as X coordinates and the row variables represent the Y coordinates).

P % | T_{tot}mm | N | A % | A_{300}% | α mm ^{2}/mm^{3} | L mm | |
---|---|---|---|---|---|---|---|

mean | 28.07 | 2522.56 | 1016 | 5.66 | 2.73 | 29.37 | 0.17 |

median | 28.50 | 2537.74 | 937 | 5.59 | 2.55 | 27.19 | 0.17 |

Sd | 1.94 | 47.74 | 313 | 1.65 | 0.92 | 7.49 | 0.04 |

Min | 21.00 | 2438.62 | 251 | 0.95 | 0.61 | 15.80 | 0.08 |

Max | 35.50 | 2643.62 | 2221 | 11.83 | 6.04 | 55.80 | 0.33 |

25%_cases | 27.00 | 2479.42 | 819 | 4.74 | 2.14 | 24.33 | 0.14 |

75%_cases | 29.15 | 2551.08 | 1194 | 6.42 | 3.09 | 34.20 | 0.19 |

_{tot}—total length of measuring lines for two samples, L—spacing factor.

Concrete Number | Number of Chords in Class No/Sizes | Content of Micropores A_{300}, % | ||||
---|---|---|---|---|---|---|

1 | 2 | … | 27 | 28 | ||

0–10 | 15–20 | … | 3005–3500 | 3505–4000 | ||

1 | 14 | 94 | … | 0 | 2 | 2.39 |

2 | 18 | 63 | … | 4 | 2 | 0.99 |

3 | 6 | 76 | … | 1 | 1 | 3.63 |

… | … | … | … | … | … | … |

261 | 2 | 66 | … | 0 | 0 | 2.91 |

262 | 6 | 58 | … | 0 | 0 | 2.18 |

**Table 3.**Coefficients of the regression function with the assessment of their significance considering 18 chord classes.

#Class | BETA | St. Err. | B | St. Err. | t(243) | Statistical Sign. |
---|---|---|---|---|---|---|

Absolute | −0.0001 | 0.0000 | −2.7 | 0.0068 | ||

15–20 | 0.0215 | 0.0000 | 0.0006 | 0.0000 | 1044.0 | 0.0000 |

25–30 | 0.0502 | 0.0000 | 0.0009 | 0.0000 | 1536.0 | 0.0000 |

35–40 | 0.0704 | 0.0000 | 0.0013 | 0.0000 | 2048.9 | 0.0000 |

45–50 | 0.0700 | 0.0000 | 0.0017 | 0.0000 | 2484.3 | 0.0000 |

55–60 | 0.0788 | 0.0000 | 0.0021 | 0.0000 | 2915.9 | 0.0000 |

65–80 | 0.1379 | 0.0000 | 0.0027 | 0.0000 | 4498.1 | 0.0000 |

85–100 | 0.1343 | 0.0000 | 0.0035 | 0.0000 | 4544.0 | 0.0000 |

105–120 | 0.1230 | 0.0000 | 0.0043 | 0.0000 | 5012.8 | 0.0000 |

125–140 | 0.1164 | 0.0000 | 0.0051 | 0.0000 | 4969.8 | 0.0000 |

145–160 | 0.1151 | 0.0000 | 0.0058 | 0.0000 | 5254.5 | 0.0000 |

165–180 | 0.1051 | 0.0000 | 0.0067 | 0.0000 | 4585.7 | 0.0000 |

185–200 | 0.0992 | 0.0000 | 0.0075 | 0.0000 | 4803.6 | 0.0000 |

205–220 | 0.0952 | 0.0000 | 0.0083 | 0.0000 | 4477.1 | 0.0000 |

225–240 | 0.0919 | 0.0000 | 0.0091 | 0.0000 | 4924.1 | 0.0000 |

245–260 | 0.0851 | 0.0000 | 0.0099 | 0.0000 | 4331.9 | 0.0000 |

265–280 | 0.0753 | 0.0000 | 0.0107 | 0.0000 | 3968.6 | 0.0000 |

285–300 | 0.0731 | 0.0000 | 0.0113 | 0.0000 | 4185.6 | 0.0000 |

305–350 | −0.2430 | 0.0000 | −0.0219 | 0.0000 | −12,019.7 | 0.0000 |

**Table 4.**Coefficients of the regression function with the assessment of their significance considering only 8 chord classes.

No. | #Class | BETA | St. Err. | B | St. Err. | t(253) | Stat. Sign. |
---|---|---|---|---|---|---|---|

Absolute t. | −0.0027 | 0.0029 | −1.0 | 0.3422 | |||

1 | 15–30 | 0.0606 | 0.0019 | 0.0007 | 0.0000 | 31.4 | 0.0000 |

2 | 35–60 | 0.2102 | 0.0032 | 0.0017 | 0.0000 | 65.7 | 0.0000 |

3 | 65–100 | 0.2680 | 0.0031 | 0.0031 | 0.0000 | 85.8 | 0.0000 |

4 | 105–120 | 0.1237 | 0.0023 | 0.0043 | 0.0001 | 53.7 | 0.0000 |

5 | 125–180 | 0.3188 | 0.0028 | 0.0058 | 0.0001 | 113.7 | 0.0000 |

6 | 185–240 | 0.2688 | 0.0028 | 0.0083 | 0.0001 | 94.3 | 0.0000 |

7 | 245–300 | 0.2156 | 0.0024 | 0.0106 | 0.0001 | 90.1 | 0.0000 |

8 | 305–350 | −0.2413 | 0.0018 | −0.0218 | 0.0002 | −130.8 | 0.0000 |

Class No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

Class size | 15–30 | 35–60 | 65–100 | 105–120 | 125–180 | 185–240 | 245–300 | 305–350 |

Regression coefficient | 0.0007 | 0.0017 | 0.0031 | 0.00043 | 0.0058 | 0.0083 | 0.0106 | −0.0218 |

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**MDPI and ACS Style**

Wawrzeńczyk, J.; Kowalczyk, H. Simplified Method of Estimating the A_{300} Micropore Content in Air-Entrained Concrete. *Materials* **2023**, *16*, 2752.
https://doi.org/10.3390/ma16072752

**AMA Style**

Wawrzeńczyk J, Kowalczyk H. Simplified Method of Estimating the A_{300} Micropore Content in Air-Entrained Concrete. *Materials*. 2023; 16(7):2752.
https://doi.org/10.3390/ma16072752

**Chicago/Turabian Style**

Wawrzeńczyk, Jerzy, and Henryk Kowalczyk. 2023. "Simplified Method of Estimating the A_{300} Micropore Content in Air-Entrained Concrete" *Materials* 16, no. 7: 2752.
https://doi.org/10.3390/ma16072752