Modeling and Numerical Simulation of Pipe Flow Problems in Water Supply Systems

We consider partial differential algebraic systems (PDAEs) describing water transportation networks. Similar to the approach in [6], we follow the method of lines for the discretization. However, we do not consider free surface flow models but pressure flow models covering hydraulic shocks. Moreover, we include switching models reflecting the on/off state of pumpes and valves. Aiming at a stable numerical simulation of the PDAEs we present a topology based spatial discretization that results in a differential algebraic system (DAE) of index 1. Furthermore we show that the DAE index can be higher than 1 if the spatial discretization is not adapted to the position of reservoirs and demand nodes within the network.Let the water network graph have n E edges and n N nodes. Then, the relation between the edges and the nodes of the network graph can be easily described by the incidence matricesThe well-known incidence matrix A for network graphs is given by A = A l + A r . The flow balance at each node provideswith q E l and q E r describing the flows at the left and right ends of the edges. Here, we use the convention that each edge directs from its left to its right end. The flow vector q N contains the in/out flows at the nodes. We consider water transport networks consisting of junctions (J), tanks (T) and reservoirs (S) as nodal model elements as well as pipes (P) and valves/pumps (R) as edge model elements satisfying the characteristic equationsHere, q set is the given demand at the junctions (equals zero at each node without a demand), C is the tank capacity, p set is the given pressure at the reservoirs, f R is a monotone, given function describing the resistance of valves/pumps. The switching parameter s is time-dependent. It equals 1 if the valve/pump is open/on and zero otherwise. In transport pipes one has usually only velocities up to 5m/s. Therefore, we can neglect the convective terms of the Euler equations and consider the pipe model (see [2])with the pipe diameter d, the pipe cross-section area a, the pipe wall sound velocity c, the fluid density , the Darcy friction factor λ, the gravitation acceleration g and the pipe slope angle α. Equation (1.2) describes the continuity equation and the mass flow balance through the pipe. We do not assume ∂ x q P to be constant in order to cover also hydraulic shocks. The boundary conditions are given as p(x l ) = p P l , p(x r ) = p P r , q(x l ) = q P l , q(x r ) = q P r .(1.3)Combining the equations (1.1)-(1.3) and using the splitting

show abstract

“…

…”

mentioning

confidence: 99%

A Topology Based Discretization of PDAEs Describing Water Transportation Networks

Huck

Jansen

Tischendorf

2014

Proc Appl Math and Mech

View full text Add to dashboard Cite

show abstract

“…This is an alternative expression of (1.2) without the energy balance by setting q = ρV , where we do not consider vacuum (ρ = 0), but need an additional constraints on p as it is a function of density ρ. With the Saint-Venant model (1.3), G. Steinebach et al [68] discuss the pipe flow problems in water supply systems, which consists of storage tanks, water pipes, pumps and valves, and this supply system is operated by the characteristics (or property) of the water supplier, the consumer and the pumps. Their model has some typical simulation results of the network with numerical methods such as finite volume methods (FVM) [1,29,62], a local Lax-Friedrich splitting [13,19,60] and central weighted essentially nonoscillatory (WENO) reconstruction [8,40,53].…”

Section: Introductionmentioning

confidence: 99%

“…In addition to PDE models, a new topic arises from the network system proposed by [68] due to the implementation of the boundary and coupling conditions regarded as algebraic equations, which leads to a system of differential-algebraic equations (DAE) in time. K. Rukhsana and S. Trenn also study this with a different approach in [35] which is called switched differential algebraic equations (sDAE).…”

Section: Introductionmentioning

confidence: 99%

“…As stated in Chapter 1, the paper [68] gives a mathematical framework to describe the water flow in the pipeline systems using some variant of this PDE and contains numerical simulations by the Finite Volume Method. However, this article fails to take into account the non-linear compatibility condition and there is no proof of numerical convergence.…”

Section: Introductionmentioning

confidence: 99%

“…Through the abuse of notation, we write 2 as the space of the solution even though we mean the spaces before and after the valve closure. We construct a numerical Lax-Wendroff scheme for (4.2) which has a rigorous error analysis compared to the engineering ODE model proposed by [35,68] with modelling based on the method of characteristics. This new dynamical equation set (4.2) also works for gas or petroleum pipelines industry [6,17,69], and can also refer to blood flowing through artificial valves in the human heart [23,42,46,54].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations