# Technologies and Principles of Hot Recycling and Investigation of Preheated Reclaimed Asphalt
## 2 Theoretical Formulation of the Problem
### Model of Technologies of RAP Recycling
\(K_{i}\) shall be computed by the general permissible deviations \(N_{adm,i}\) depending on the property of RAP and the type of recycled HMA mixture:  
\[K_{i}=\frac{0.50N_{adm,i}}{a_{i}}\ \times\ 100 \tag{1}\]  
or  
\[K_{i}=\frac{0.33N_{adm,i}}{a_{i}}\ \times\ 100, \tag{2}\]  
where \(a_{i}\) is the range between the max \(x_{i,max}\) and the min \(x_{i,min}\) values of RAP \(i\)-property indicator estimated for the sample (min sample size \(n=5\)). With respect to all of the properties of base and surface-base course mixtures, Equation (1) shall apply. It shall be also applied for the softening point of binder course and surface course mixtures. With respect to all other properties, Equation (2) shall apply [42]. RAP is not only homogeneous, but it is also dosed with systemic and random errors of a certain magnitude in the AMP that increase the deviations of the RHMA component content from JMF and variation [38]. Derivation of the procedure for calculation of the maximum allowable RAP content is as follows. ASTM D 4460 gives equations for calculating standard deviation values for quantities determined from calculations involving two other values. From these equations, the following formula for calculating the standard deviation of a blend of two materials can be derived:  
\[\sigma_{m}=\sqrt{\alpha^{2}\sigma_{a}^{2}\ +\ (1\ -\ \alpha)^{2}\sigma_{b}^{2}\ +\ \Big{(}\overline{X}_{a}^{2}\ +\ \overline{X}_{b}^{2}\Big{)}\sigma_{a}^{2}}, \tag{3}\]  
where \(\sigma_{m}\)--standard deviation of the mixture; \(\sigma_{a}\)--standard deviation of component \"a\"; \(\sigma_{b}\)--standard deviation of component \"b\"; \(\alpha\)--proportion of component \"a\" in the mixture; \(\overline{X}_{a}^{2}\)-- mean value for component \"a\"; \(\overline{X}_{b}^{2}\)--mean value for component \"b\"; and, \(\sigma_{a}\)--standard deviation of the proportions. We can rewrite this for percent passing for a selected sieve for HMA mixtures consisting of a blend of new HMA materials with RAP [48]:  
\[\sigma_{PM}=\sqrt{w_{R}^{2}\sigma_{PR}^{2}+w_{N}^{2}\sigma_{PN}^{2}+\Big{(} \overline{P}_{R}^{2}+\overline{P}_{N}^{2}\Big{)}\sigma_{W}^{2}}, \tag{4}\]  
where \(\sigma_{PM}\)--standard deviation of percent passing for a selected sieve for the mixture with RAP (i.e., RHMA); \(w_{R}\)--weight fraction of RAP in the mixture; \(\sigma_{PR}\)--standard deviation of percent passing for the selected sieve for the RAP; \(w_{N}\)--weight fraction of new materials (new HMA) in the mixture \(=(1-w_{R})\); \(\sigma_{PN}\)--standard deviation of percent passing for the selected sieve for the new HMA; \(\overline{P}_{R}\)--mean value for RAP% passing for the selected sieve; \(\overline{P}_{N}\)--mean value for new HMA% passing for the selected sieve; and, \(\sigma_{w}\)--standard deviation of the weight fractions, also called \"batching variability\". Equation (3) can be solved for the maximum amount of RAP that can be added to new material without increasing the standard deviation for percent passing on the selected sieve above a selected maximum value by application of the quadratic equation:  
\[Max.RAP\%=\frac{-b+\sqrt{b^{2}-4 ac}}{2a}100\%, \tag{5}\]where \(Max.RAP\)%--maximum amount of RAP that can be added to the mix, weight%; \(a=\sigma_{PR}^{2}+\sigma_{PN}^{2}\); \(b=-2\sigma_{PN}^{2}\); \(c=\sigma_{PN}^{2}+\left(\overline{P_{PN}^{2}}+P_{N}^{2}\right)\sigma_{w}^{2}- \sigma_{PM/Max}^{2}\); and, \(\sigma_{PM/Max}^{2}\) is the maximum allowable standard deviation for percent passing for the selected sieve. Because asphalt is temperature sensitive, good preheating of RAP is helpful to eliminate the agglomerates of RAP to promote the dispersion of RAP during the hot recycling process [49]. When applying the first heat transfer method, superheating temperature of virgin mineral materials (aggregate) is calculated according to the following empirical formula applied in practice [50]:  
\[T_{v.a.}=\frac{T_{RHMA}-S\cdot T_{RAP}}{1-S}\ +\ \frac{4\cdot S\cdot W_{RAP}}{1-S}(6 37-T_{RAP}), \tag{6}\]  
where \(T_{v.a.