Example 4: Distribution of compressed air

• Basic system:
  • simple pneumatic system with pressure source, tee branch, pipe and consumers
    • 
• Model in Modelica:
  • uses PneuBib library
  • PneuBib itself based on MSL Fluid library
  • MSL Fluid library basically describes quasistatic processes
  • run in Dymola →
    • Warning: Failed to solve nonlinear system using Newton solver...
      Error message from dymosim
      At time T = 5.130275e-04 ... the corrector could not converge because there were model evaluation failures and the stepsize cannot be reduced further.
      Integration will be terminated.
• Mathematical problem:
  • model consists of 170 equations, many are coupled and highly non-linear
  • proper initial values are unknown and guessed by solver
    • → Newton solver does not converge
  • well-known problem of MSL Fluid library
• Reality check:
  • system starts with ambient pressure and zero mass flow everywhere
  • switching on pumps etc.
    • → global pressure difference increases
    • → mass flows increase slowly due to mass inertia
    • → local pressure differences build up
• Extended Model:
  • introduce inertial pressure difference Δr
    • 
    • inertance L is given by geometry of a component
    • can be derived from Euler equation
  • split pressure into quasi-static part and inertial part
    • 
  • at junctions use instead of p
    • approximation of transient behaviour
    • leads to decoupling of component equations
  • consequences
    • independent small non-linear parts instead of a huge coupled system
    • total coupling by relaxation using simple linear ODE
    • new parameter L (usually having default value)
  • free DLR ThermoFluidStream Library
    • provides components for vessels, pipes and machines based on these ideas
    • use it to model pneumatic system → runs
  • result
    • 
    • approximate relaxation for t < 1
    • quasistatic behaviour for t > 1