True bond graph formulation for one-dimensional incompressible pipe flows: modeling and analytical benchmarks

In this paper, a bond graph methodology described in a previous contribution by Balino (Simul Model Pract Theory 17:35–49, 3) is used to model incompressible one-dimensional pipe flows with rigid walls. As the volumetric flow is independent of position, the balance equation corresponding to the inertial port can be simplified and readily integrated. The volumetric flow and a nodal vector of entropy are defined as bond graph state variables. The state equations and the coupling between the inertial and entropy ports are modeled with true bond graph elements. The methodology was implemented using constant piecewise shape functions and linear piecewise weight functions for the entropy port. Different problems are simulated: advective heating due to constant wall heat flux and due to constant wall temperature, advective heating due to viscous dissipation and combined advection–diffusion. The numerical results show an excellent agreement with the corresponding analytical solutions, both for transient and steady-state flow. A representation with single-bond elements is obtained by lumping the temperature matrix. Simulations were run with the lumping approximation, showing no significant deterioration of the numerical solutions.