## 6 Mortar Air Void Content Calculation Equation  
Based on the preparation work, the method to calculate the designed mortar air void content with various NMPS and different mixture types was introduced. The calculation method consists of several equations from Equations (7)-(13), and the calculation results are shown in Figure 8.  
Equation (7) is the calculation for the mixture air void content:  
\[VV=\left(1-\frac{\gamma_{f}}{\gamma_{t}}\right)\times 100 \tag{7}\]  
where \(VV\) is the designed mixture air void content (%); \(\gamma_{f}\) is the measured sample density (g/cm\({}^{3}\)); and \(\gamma_{t}\) is the theoretical maximum density (g/cm\({}^{3}\)) of the mixture sample.  
The effect density of the mortar is determined by Equation (5), which is similar to the effect density of the mixture aggregate. Equations (8)-(12) determining the effect density for the mortar, which is vital for calculating the mortar voids.  
The bulk density, \(\gamma_{sb-r}\) (g/cm\({}^{3}\)), of the mortar aggregate with \(r\) NMPS is calculated as per Equation (8):  
\[\gamma_{sb-r}=\frac{100}{\frac{P_{i-r}}{\gamma_{1-r}}+\frac{P_{2-r}}{\gamma_{ 2-r}}+\cdots+\frac{P_{n-r}}{\gamma_{n-r}}} \tag{8}\]  
where \(P_{i-r}\) is the mass content of aggregate particle \(i\) (%); \(\gamma_{i-r}\) is the bulk density of mortar particle \(i\) (g/cm\({}^{3}\)), \(1\leq i\leq n\).  
The apparent density, \(\gamma_{sa-r}\) (g/cm\({}^{3}\)), of the mortar aggregate with \(r\) NMPS is calculated as per Equation (9):  
\[\gamma_{sa-r}=\frac{100}{\frac{P_{1-r}}{\gamma_{1}^{2}}+\frac{P_{2-r}}{\gamma _{2}^{2}}+\cdots+\frac{P_{n-r}}{\gamma_{n}^{2}}} \tag{9}\]  
where \(\gamma_{i}^{\prime}\) is the bulk density of mortar particle \(i\) (g/cm\({}^{3}\)), \(1\leq i\leq n\).  
Figure 8: Mortar air void content calculation results.  
The effective density, \(\gamma_{se-r}\) (g/cm\({}^{3}\)), of the mortar aggregate with \(r\) NMPS is calculated as per Equation (10):  
\[\gamma_{se-r}=C\times\gamma_{sa-r}+(1-C)\times\gamma_{sb-r} \tag{10}\]  
where \(C\) is the aggregate absorption index (%).  
The aggregate absorption index, \(C\), is calculated by Equation (11), which is a widely used empirical equation in China and also recommended in the standard for the use of modified asphalt mixtures. Equation (11) is as follows:  
\[C=0.033W_{x-r}^{2}-0.296W_{x-r}+0.9339 \tag{11}\]  
where \(W_{x-r}\) is the water absorption percent (%).  
The water absorption percent, \(W_{x-r}\), is calculated as per Equation (12):  
\[W_{x-r}=\left(\frac{1}{\gamma_{sb-r}}-\frac{1}{\gamma_{sa-r}}\right)\times 100 \tag{12}\]  
Finally, the air void content of a given mortar NMPS can by calculated by Equation (13), which clarifies each phase's volume, including the air voids in the mortar, mortar aggregate and asphalt. In this equation, \(v_{p-r}\) is the void percent of the mortar (%); \(P_{a}\) is the mortar aggregate passing rate (%), and \(P_{a}\) = 100% in this equation; \(\gamma_{a}\) is the density of the asphalt (g/cm\({}^{3}\)); and \(VV_{m-r}\) is the air void content of the mortar with NMPS \(r\) (%). The final step is the calculation of the mortar voids, as shown in Equation (13); the top part of the fraction are the air voids in the mortar, and the bottom part of the fraction is the accumulated volume of voids, mortar aggregate and mortar asphalt. This equation is the calculation for a certain mortar NMPS and asphalt mixture type.  
The air void content, \(VV_{m-r}\), of a mortar with an NMPS is calculated as per Equation (13):  
\[VV_{m-r}=\frac{VV\times v_{p-r}}{\left(VV\times v_{p-r}+\frac{p_{i-r}}{\gamma_{ sa-r}}+\frac{p_{i-r}}{\gamma_{a}}\right)}\times 100 \tag{13}\]