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V =(VS −V )−(1r2 (2cos−1((h −r )/r ))− enthalpy of the working fluid at point 3 and h4s is the enthalpy of the working fluid at the outlet of the turbine at isentropic condition. For a reversible adiabatic (isentropic) process, the entropy at the inlet of the turbine must be equal to the entropy at the outlet: s3 =s4s (21) The losses in the turbine are primarily those related to the flow of the working fluid through the turbine. Heat transfer to the surroundings also represents a loss, but this is of usually of secondary importance. The governing procedures may also cause a loss in the turbine, particularly if a throttling process is used to govern the turbine. The losses associated with irreversibilities cause the turbine efficiency to deteriorate from the ideal. The efficiency of a turbine (isentropic efficiency) is defied as: l tube 2 shell (h −r ) (2hr l shell shell If the liquid pentane level is lower than the top of the tube channel, hl can be calculated iteratively from the following equation: V =(VS −V )−(1r2 (2cos−1((h −r )/r ))− l tube 2 shell (h−r )(2hr −h2))l+ (1r2 (2cos−1((h −r )/r ))− 2 tube l tube tube (h−r )(2hr −h2))l l tube ltube l V l shell shell Sohel et al. l shell l shell −h2))l l V (16) l shell l shell l V Where, hl is the height of the pentane liquid level from the bottom of the vaporizer, rshell is the internal radius of the shell, rtube is the outer radius of the tube, Vtube is the volume (external) of the tube side channel and lv is the length of the vaporizer. The liquid level, hl is explained in Figure 6: hl = rshell + b (18) The percentage of liquid level is calculated from the following equation, l = (hl −hl,min) ×100 (19) % (hl,max −hl,min) Equations 7 - 10 and 16-17 are pure algebraic equations and are a part of a system of differential equations forming a typical higher index problem. An index greater than 1 implies that there are algebraic relations between dynamic variables. A higher index problem can be solved by reducing its index to 1 using dummy derivatives (Mattsson and Siiderlind 1993). gPROMS can solve a high index automatically. However, one must provide consistent initial conditions to start simulation. The gPROMS model was exported to Simulink via gO:Simulink interface. 2.3 The turbine model The response time of a turbine can be in the order of 10-20 s (Jurado et al. 2003) to 1-3 minutes. The available data of the plant operation is for one-hour interval so we are interested in plant performance where unit time is 60 min or an hour. Therefore, the turbines can be assumed to have static behaviour with respect to the unit time of our interest. Ou Bai et al. and Wei et al. (Bai et al. 2004; Wei et al. 2008) have used such static representation of turbine models used for dynamic modelling of binary cycles. Figure 1 presents T-s presentation of a Rankine cycle with the process 3-4s as an ideal turbine. The work done by an ideal turbine can be calculated knowing the state points 3 and 4s and calculated as: η = h3 −h4 s h3 −h4,s The work done by a real turbine is calculated as: (22) • WT =m(h3 −h4s) Where, Tamb is the ambient air temperature in K, rp is the turbine pressure ratio (inlet to utlet), ηs is the isentropic efficiency and a, b, c are constants. Values of the constants are optimized using available plant operation data. 2.4 The recuperator model The residence time of pentane in the recuperator is in the order of few minutes. Therefore, it can be assumed to have static characteristics. The heat transfer from the turbine outlet fluid to condenser outlet fluid is calculated from the following equation: Where, WT is work done by the turbine, m is the mass flow rate of the working fluid through the turbine, h3 is the • (17) (20) QRec =URecARec∆tRec m (26) 5 • WT =m(h3−h4) (23) Heat transferred to the surrounding is not incorporated in the equation 23, as it is the less significant part. In case the heat transfer becomes significant, following equation should be used to calculated actual work done: Wa =WT −QL (24) Where, Wa presents the actual work done by the turbine and QL is the heat transfer to the surroundings. The value of isentropic efficiency is calculated from a developed turbine map. It is reported by many authors i.e. (Erdem and Sevilgen 2006) that ambient temperature has a prominent effect on turbine performance. Therefore, the ambient temperature has been taken into account to develop the maps for this work: η = a +b.r +c sTp amb (25)PDF Image | Dynamic Modelling and Simulation of an Organic Rankine Cycle Unit of a Geothermal Power Plant
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