An efficient numerical scheme for the thermo-hydraulic simulations of thermal grids

Renewable and smart energy systems require district heating and cooling grids able to operate with variable flow rates, to manage different supply temperatures, and to support distributed energy production involving bidirectional flows. Due to these requirements, accurate and effective thermo-hydraulic models are essential to correctly simulate the flow rates, the head drops and the temperature transients for supporting the design, management and optimisation of thermal distribution networks. In this article, an efficient numerical scheme for the simulation of thermal grids based on a thermo-hydraulic model with a quasi-dynamic approach is proposed. Global gradient algorithm of Todini together with a uniformly distributed representation of demand along the pipes is used for steady-state hydraulic simulations of the networks modelled according to the graph theory. The temperature distribution is computed by solving a first order hyperbolic PDE which accounts for heat advection in the flux term and heat dissipation in an algebraic source term. A very efficient second-order Eulerian-Lagrangian finite volume scheme is employed on a staggered mesh, which evolves in time the temperature distribution starting from the velocity distribution given by the hydraulic solver. The usage of a Eulerian-Lagrangian algorithm for the discretisation of the convective terms, allows the proposed model to be unconditionally stable for every time step size. As such, the main advantages of the proposed model are the flexibility due to the admissibility of any spatial-temporal discretisation, the second order accuracy provided by both the demand schematisation and the thermo-hydraulic solver, and the computational efficiency guaranteed by the decoupled modelling approach. The resulting algorithm is extensively validated on four tests consisting of thermal grids of various complexity that have been carefully designed in order to capture different features and behaviours of the model. The accuracy of the results in terms of velocity, pressure and temperature is tested by separately checking the symmetry, the advection term, the heat loss component and, finally, by simulating a complex grid configuration with multiple heat sources. The article aims to present a proof-of-concept concerning a breakthrough numerical scheme for the efficient thermo-hydraulic simulation of pipeline networks, which proves to be suitably implemented in the modelling of district heating and cooling networks.