This paper investigates the design optimization of double pipe heat exchanger using mathematical programming. The heat exchanger area is minimized and the thermo-fluid dynamic conditions are considered for the use of the right transport correlations, together with design specifications, such as, maximum pressure drops and minimum excess area. The modular nature of this kind of heat exchanger and the allocation of the streams (inside the inner tube or in the annulus) are also contemplated. Two mixed-integer nonlinear programming (MINLP) approaches are proposed. One approach relates the binary variables to the nonlinear constraints directly. In the second, the resulting nonlinearities involving binary variables are formally linearized, without loss of rigor (e.g., no use of truncated Taylor series). The proposed methodology can get better solutions than traditional trial and error procedures. The flexibility of the model is illustrated, together with a comparison between the performances of both MINLP formulations. Additionally, computational time and local optimality issues are discussed.