Vartak  has shown by enumeration that the Knoecker product of two PBIB designs with $s$ and $t$ associate classes is again a PBIB with at most $s + t + st$ associated classes. In this paper the same result is established more easily with the help of association matrices. In addition to this, it is shown that the association matrices of a PBIB which is the Kronecker product of two known designs, are the Kronecker product of those of the original designs and that the "augmented matrices of the parameters of the second kind" of the resulting design are the Kronecker product of the corresponding matrices of the given designs.
P. U. Surendran. "Association Matrices and the Kronecker Product of Designs." Ann. Math. Statist. 39 (2) 676 - 680, April, 1968. https://doi.org/10.1214/aoms/1177698427