The evolution equations of moments for the Brownian coagulation of nanoparticles in both continuum and free molecule regimes are analytically studied.These equations are derived using a Taylor-expansion technique.The self-preserving size distribution is investigated using a newly defined dimensionless parameter,and the asymptotic values for this parameter are theoretically determined.The dimensionless time required for an initial size distribution to achieve self-preservation is also derived in both regimes.Once the size distribution becomes self-preserving,the time evolution of the zeroth and second moments can be theoretically obtained,and it is found that the second moment varies linearly with time in the continuum regime.Equivalent equations,rather than the original ones from which they are derived,can be employed to improve the accuracy of the results and reduce the computational cost for Brownian coagulation in the continuum regime as well as the free molecule regime. The evolution equations of moments for the Brownian coagulation of nanoparticles in both continuum and free molecule regimes are analyzed. The equations are derived using a Taylor-expansion technique. The self-preserving size distribution is investigated using newly-defined dimensionless parameter, and the asymptotic values ​​for this parameter are theoretically determined. dimensionless time required for an initial size distribution to achieve self-preservation is also derived in both regimes.Once the size distribution becomes self-preserving, the time evolution of the zeroth and second moments can be theoretically obtained, and it is found that the second moment varies linearly with time in the continuum regime.Equivalent equations, rather than the original ones from which they are derived, can be employed to improve the accuracy of the results and reduce the computational cost for Brownian coagulation in the continuum regime as well as the free molecule regime.