Given a sparse set of differential and algebraic equations (DAEs), it is always recommended to exploit the structure of the system's sparsity (e.g., tridiagonal blocks matrix, band rnatrix, and staircase matrix, etc.), thus to use tailored numerical solvers in order to reduce the computation time. Very frequently, though, while highly structured, a couple of elements enter the description which make it difficult for the solvers to reach a solution. They are common in process control applications, where the states added to the plant description by the integral parts of the controllers introduce unstructured elements in the otherwise very structured Jacobian of the mathematical model. Such systems are characterized by a partially structured Jacobian, which inhibits the use of the solvers tailored to fit problems with fully structured matrices. In such cases, one can either use a solver with lower performance, resulting in larger computation times, or alternatively one seeks an approximation for the unstructured points. A solution to the handling of "dirty" Jacobians is presented, which is implemented in a DAE solver package available freely on the Internet. This novel DAE solver fully exploits the overall structure of the system's sparsity, without compromising CPU computation time and precision of the results. A numerical comparison with different approaches is given by solving a DAE model representing an existing nonequilibrium distillation column.