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Chapter 4: Modeling In the case of the secondary fluid, since a constant heat capacity is assumed and since the fluid is incompressible, Eq. (48) simplifies into: V ⋅ρ⋅c̄⋅∂T i=M ̇⋅c̄⋅(T * −T *)+Q ̇ (49) i i p ∂t p i−1 i i The energy balance over the wall is written: cw⋅Mw ,i⋅dTw ,i =Q ̇ sf ,i−Q ̇ f ,i (50) Heat transfer dt When using traditional dynamic simulation tools, it is generally difficult to use correlations for the heat transfers and the pressure drops, since they slow down the calculation process and can even lead to numerical instabilities and simulation failures. To propose a heat exchanger model that is robust and fast, the following methodology is proposed: 1. A complete model of the heat exchanger is developed in steady-state, with the proper heat transfer and pressure drop correlations, as described in section 2.2 . 2. A nominal heat transfer coefficient is identified with the steady-state model for the nominal conditions. 3. The heat transfer correlations are simplified and a simple law is proposed to compute the variation of the nominal heat transfer coefficient with the most relevant working conditions. In total, four different heat transfer coefficients are calculated: one on the secondary fluid side and three (vapor, two-phase, liquid) on the refrigerant side. For single-phase heat transfer, Eq. 30 can be rewritten: N u ⋅ k k M ̇ ⋅ L m n ̇ m L ( m − 1 ) k ⋅ P r n U= L =L⋅(A⋅μ)⋅Pr=M⋅ A ⋅μm (51) The second factor (L(m−1)/A) in Eq. (51) is related to the geometry and is therefore constant for a given heat exchanger. The third factor ( k Prn/μm ) contains transport properties of the working fluid. This factor varies in a narrow range if the working fluid remains unchanged and if the operating pressure and temperature levels are kept more or less constant, which is the case in an ORC evaporator/condenser. The main term influencing the heat transfer coefficient being the mass flow rate, the following approximation can be written: U = U ⋅ M ̇ m n ( M ̇ n ) (52) 29PDF Image | Organic Rankine Cycles for Waste Heat Recovery and Solar Uses
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