The main objective of the present work is to solve numerically the flow of liquid and gases isothermal or not, in the transient regime, in industrial pipes with variable area. It also has as objective to investigate the velocity, pressure and temperature field in the presence of fluid leak in the pipe. The numerical code was implemented to solve the conservation of mass, momentum, and energy in its conservative form. This approach is very convenient to study the perturbations in the velocity and temperature fields, once a leakage is detected in some point along the point, in a certain time. The numerical solution for the one-dimensional flow in rectangular coordinates is obtained by the finite volume method. The spatial discretization is based on the upwind method and totally implicit time integration is employed. The conservation of mass and linear momentum are directly solved through an hepta-diagonal matrix algorithm, followed by the solution of the energy equation by a three-diagonal algorithm. Since the conservation equations are non- linear, an iterative procedure is necessary. To validate the methodology presented, several tests of different case tests available in the literature were solved, as well as tests with analytical solution. A comparison between the conservative and non-conservative approach is presented. Finally, some test cases with leakage are examined.