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Chapter 4: Modeling Figure 59: Exhaust flow rate chattering The case with N=100 also generates oscillations, but since the density transition is applied to a much smaller cell, the oscillations are relatively smaller and occur at a higher frequency. It should be noted that these oscillations only occur for "stressed systems", i.e. systems under highly transient conditions and with a high ratio between heat exchanger internal volume and mass flow rate. The underlying reason is derived mathematically in Appendix F. In the scope of this work, special attention is paid to the model robustness, even in extreme conditions. Several methods are therefore developed and tested to avoid these oscillations: 1. Constant node flow rate: the flow rate variations are reported on the last node, while a constant flow rate is assumed for all the other nodes. 2. Truncated density derivative: dρ/dx (Figure 58) is truncated to limit the instabilities generated during a phase transition. The parametric value of the maximum value of dρ/dt must be provided by the user. This is internally converted into maximum values for ∂ρ/∂h and ∂ρ/∂p using arbitrary typical values for the derivatives of the state variables. 3. Filtered density derivative: the phase transition is smoothed by applying a first order filter on the time derivative of the density. The time constant of the filter must be provided by the user. 4. High number of cells: even though the occurrence of phase changes is higher with highly discretized models, their impact is lower, as shown in Figure 59. A thorough description of these 4 methods is provided in appendix F, with the comparison of their accuracy, and simulation speed. A significant increase of the simulation robustness is stated with these different methods. This increase is however achieved at the cost of a small decrease of the simulation accuracy. 33PDF Image | Organic Rankine Cycles for Waste Heat Recovery and Solar Uses
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