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Chapter 4: Modeling same number of equations and unknowns. An example of initial value assignment is the steady-state hypothesis: in this case, all the time derivatives are set to zero and the simulation starts in steady-state. Even with correctly-assigned start and initial values, highly non-linear systems might still fail to converge during initialization. To improve the convergence, a strategy similar to the one proposed by (P. Li et al., 2010) is applied. In this approach, the initial system is simplified: heat transfer coefficients and pressure drops are set to constant values during a certain amount of time. When the system stabilizes after a few seconds of simulation, the heat transfer and pressure drop equations are activated one by one. The activation is performed by an interpolation function similar to the one proposed in Eq (55) between the constant value and the value computed by the model. The activation time and length are set as model parameters. This strategy makes the initialization problem faster and much easier to solve. Phase transitions Simulation failures during the integration phase often occur in the heat exchangers. Most of the time, these failures are due to a phase transition (liquid to two-phase) which involves a discontinuity in the first derivative of the density. This is illustrated in Figure 58: the discontinuity appears at x=0, and the numerical integration leads to unsteadinesses (chattering) in the calculation of the mass balance for each cell. Figure 59 is an example of such chattering. The simulation is performed for one heat exchanger with a two-phase flow under highly transient conditions. The red curve corresponds to the case with a 20-cells discretization, while the blue curve corresponds to a 100-cells discretization. Each oscillation on the flow rate prediction between t=13s and t=18s corresponds to a phase transition in one of the cell: the high density derivative is applied to a cell with a non-negligible working fluid mass. High flow rates transitions are therefore generated between cells and spread trough the heat exchanger. Figure 58: First derivative of the density as a function of the vapor quality 32PDF Image | Organic Rankine Cycles for Waste Heat Recovery and Solar Uses
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