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F Numerical issues in dynamic heat exchanger models. Figure 104: Truncature of the density derivative The maximum value of the density derivative (with respect to time) is a model input. It is converted into maximum values for ∂ρi/∂h and ∂ρi/∂p with the following relation: d ρmax ∂ρmax dhref ∂ρmax dpref dt =∂h ⋅dt +∂p ⋅dt (112) where the reference enthalpy and pressure time derivatives are set to typical values, namely: dhref dpref 5 dt =−4000 [J/kgs] ; dt =10 [Pa/s] (113) The maximum partial derivatives are then calculated by assuming that the time derivative of the second parameter is null (i.e. dp/dt is set to 0 to calculate ∂ρ/∂h and dh/dt is set to 0 to calculate ∂ρ/∂h . Filtered density derivative. In this strategy , a first order filter is applied to the fast variations of the density with respect to time. This filter acts as "mass damper" and avoids transmitting abrupt variations to the node flow rates. The filtered mass accumulation in each cell is written: 2 Vi⋅[∂ρi⋅dhi+∂ρi⋅dp]−dMi dMi = ∂h dt ∂p dt dt (114) dt2 Ti where Ti is the filter time constant, set as model input. This strategy displaces the mass variations in time but does not generate mass defaults. However the energy balance is affected because the cell density is not exactly the one corresponding to the actual node flow rates. High number of cells. As shown in Figure 103, increasing the number of cells reduces the refrigerant mass on which the non-continuous density derivative is applied, which reduces the node mass flow rate fluctuations. This method is the most accurate, but also requires a large number non-linear equations and increases computational times. To avoid simulation failures on the stressed system, a number of cells close to 100 was necessary. XXIPDF Image | Organic Rankine Cycles for Waste Heat Recovery and Solar Uses
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