# Water Demand (or Specific Surface) of Aggregate as a Dominating Factor for SCC Composition Design
## 2 Methods
### The Method for Determining the Relative Thickness of Aggregate Coating with Cement Paste  
The previous own research [13; 14] showed that the self-compaction limit of concrete is strongly correlated with the dimensionless \(t_{rel}\) value, which is the relative thickness of the aggregate grains coating with cement paste, defined as  
\[t_{rel}=t_{m}/d_{m} \tag{10}\]  
where:  
\(t_{m}\)--average distance of aggregate grains in the concrete mix;  
\(d_{m}\)--average grain size of aggregate in the aggregate composition;  
\(t_{m}\) is determined from the formula given by [PERSON] et al. [9].  
\[t_{m}=V_{exc}/S_{at}=\left(1-\frac{\varphi_{a}}{\varphi_{solids}}\right)\frac {10}{\varphi_{a}S_{so}} \tag{11}\]  
In this formula:  
\(V_{exc}\)--volume of the excess paste in the mix (i.e., the volume of paste overfilling the intergranular spaces in the mix) (mm\({}^{3}\));  
\(S_{SO}\)--specific surface area of the aggregate (mm\({}^{2}\)/mm\({}^{3}\)) calculated from Formula (7);  
\(\varphi_{solids}\)--volume concentration of the solid phase (i.e., aggregate (\(a\)) + cement (\(c\)) + additions (\(ad\))).  
In the previous research program [3; 13] only the ONT model was tested, with very high correlation (R\({}^{2}\approx 0.95\)). The current research program was devoted to validate these results on the new set of data and compare them to the remaining three models described above. Next, in the case of negative validation (R\({}^{2}\) < 0.9 was assumed), it was considered a trial to find a new model.  
Formula (1) shows that \(w_{a}\) is directly proportional to \(S_{at}\) through the thickness of the water film, which is a consistency-dependent variable. In the previous studies [13; 14] the fluidity of the paste used for testing was kept constant (in accordance with the recommendations of the methodology described in Ref. [3]). It allowed for the purposes of the current research program to treat \(t_{c}\) as the constant used for converting units from \(w_{a}\) to \(S_{at}\). Therefore, in Formula (11), one can substitute the remaining values (\(RSS\) or \(w_{a}\)) instead of \(S_{at}\) simply by introducing the appropriate unit correction constant in each of these cases. Therefore, \(t_{rel}\) can be written generally as:  
\[t_{rel}=\frac{\chi}{d_{m}}\frac{\Sigma_{p}}{\Sigma_{a}} \tag{12}\]  
where \(\chi\) is the constant including aspect ratio of the aggregate particles, model constants, unit conversion, and consistency effects. For the ONT model, \(\chi\) is equal to \(\chi_{O}\) = 10 \(k\)/\(f\), which gives 1.66 for ideal spheres, 1.54 for rounded aggregate, and 1.33 for crushed aggregate. \(\Sigma_{a}\) is the influence of the aggregate given by the formula  
\[\Sigma_{a}=\varphi_{a}S_{at} \tag{13}\]  
and \(\Sigma_{p}\) is the influence of the cement paste with constant fluidity/consistency given by the formula  
\[\Sigma_{p}=\left(1-\frac{\varphi_{a}}{\varphi_{solids}}\right)\!=\left(\frac {\varphi_{c}+\varphi_{ad}}{1-\varphi_{w}}\right)\!=\left(\frac{\varphi_{c+ad} }{\varphi_{a}+\varphi_{c+ad}}\right)=\left(1-\frac{\varphi_{a}}{\varphi_{a}+ \varphi_{c+ad}}\right) \tag{14}\]where the indices denote the individual basic concrete ingredients (\(c\)--cement, \(ad\)--additive, \(w\)--water, \(a\)--aggregate, _solids_--the sum of all solid ingredients). As a result, in specific cases in consideration, one can obtain:  
\[t_{\textit{rtIO}}=\frac{\chi_{O}}{d_{m}}\frac{\Sigma_{p}}{\sum_{i=1}^{n}\frac{ \Phi_{\textit{id}}}{d_{\textit{rtI}}}} \tag{15}\]  
\[t_{\textit{rtI}B}=\frac{\chi_{B}}{d_{m}\varphi_{a}}\frac{\Sigma_{p}}{\sum_{j=1 }^{n}y_{j}d_{j}^{-\frac{2}{3}}} \tag{16}\]  
\[t_{\textit{rtI}S}=\frac{\chi_{S}}{d_{m}\varphi_{a}}\frac{\Sigma_{p}}{\sum_{j=1 }^{n}y_{j}\left(log\frac{d_{j}}{2}+\log d_{j}\right)^{-3}} \tag{17}\]  
\[t_{\textit{rtI}R}=\frac{\chi_{R}}{d_{m}\varphi_{a}}\frac{\Sigma_{p}}{\sum_{j=1 }^{n}\frac{y_{j}}{d_{\textit{rtI}}}} \tag{18}\]  
The following part of the article presents experimental verification of all these four models.