MINLP superstructures for heat exchanger network synthesis allow the simultaneous optimization of utility heat loads, the number of heat exchanger units and their area requirements. The algorithms used to solve these nonlinear non-convex optimization problems solve MILP and NLP sub-problems iteratively to find optimal solutions. If these sub-problems are tightened, which means that the solution space is reduced but still includes all feasible integer solutions, the algorithms can potentially go through the solution space faster as branches can be excluded earlier. In this work, tightening measures for a commonly used MINLP stage-wise superstructure formulation are proposed and the impact of tighter variable bounds and additional inequality constraints is investigated using various case-studies taken from literature. It is shown that tighter formulations help the solver to find global optimal solutions and that the duality gap can be reduced significantly if the test cases could not be solved to global optimality.