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entering from the expander to the cold stream entering from the pump: firstly, the internal difference (∆Tint) between the entering cold stream (Tc,i) and the exiting hot stream (Th,o), and secondly, the external difference (∆Text) that allows the heat transfer fluid to be cooled down to a specific temperature thereby limiting the inlet temperature of working fluid to the boiler. The inlet con- ditions to the recuperator are known from the pump and expander equations, and with no pressure loss applied, the recuperator was described by: hc,1 = hp,o (12) Tc,o = Th,i − ∆Tsh (13) hc,n+1 = h(Pc,i, Tc,o) (14) hh,n+1 = h(Ph,i, Th,i) (15) ∆hstep = (hc,n+1 − hc,1)/n (16) hc,j = hc,1 + (j − 1)∆hstep (17) Tc,j =T(Pc,i,hc,j) (18) hh,j = hh,j+1 − (m ̇c/m ̇ h)(hc,j+1 − hc,j) (19) Th,j =T(Ph,i,hh,j) (20) Th,1 = T(Ph,i,hh,1) (21) Tc,1 = T(Pc,i,hc,1) (22) ∆Tj = Th,j − Tc,j (23) ∆Tmin = fMin(∆Tj) (24) m ̇c(hc,n+1 − hc,1) − m ̇ h(hh,n+1 − hh,1) = 0 (25) where fMin is a Matlab function that finds the min- imum value in an array of values. Subscript c is cold stream, h is hot stream, and min is minimum. Subscript p is the stream from the pump. To find the optimum superheater approach, a Matlab fminbnd optimisation al- gorithm was applied, using the Golden section search and Parabolic interpolation methods [15]. This approach is essential for the methodology because it accommodates all types of process scenarios. In subcrit- ical cases the ∆Tpp between the hot and cold sides will be at the start of evaporation. In supercritical cases and when using mixtures, the location of the pinch point cannot be as easily predicted (to the authors’ knowledge). Additionally, the approach does not distinguish between cases with or without preheater and with or without superheater, which provides the freedom to test processes and fluids without committing to a specific scenario. As the approach presented here aims at providing a generic approach not dependent on the physical design of the heat exchangers, the optimisation was simplified by assuming zero pressure losses in the cycles. 2.3. Objective weights Potentially weights may be defined for the optimisa- tion process to provide a weighted compromise solution enabling multiple objectives. Alternatively, a Pareto front may be the desired result of an optimisation using two ob- jectives. In the present work, weights were applied simply to discard inconsistent or unwanted solutions. The follow- ing weights were implemented: • The physical, health and fire hazard levels of the fluid must meet requirements of the process design. • Theexpandervapourqualitywascheckedtobeabove a specified minimum. (1) (2) (3) (4) (5) (6) then Tc,o = Th,i − ∆Tint if Tc,o > Th,o − ∆Text (10) then Tc,o = Th,o − ∆Text (11) where h is specific enthalpy, P is pressure, subscript i is inlet and o is outlet. Depending on the conditions, hc,o was updated according to the temperature Tc,o. Following this procedure, the second law of thermodynamics is not violated and recuperation will happen to the maximum possible degree. Th,i = T (Ph,i , hh,i ) Tc,i = T (Pc,i , hc,i ) Th,o = Tc,i + ∆Tint hh,o = h(Ph,o , Th,o ) ∆hmax = hh,i − hh,o hc,o = hc,i + ∆hmax Tc,o = T (Pc,o , hc,o ) (7) if Tc,o > Th,i − ∆Tint (8) n+1 n+1 HX nn HX (9) 221 HX 1 Figure 2: Sketch of heat exchangers with numbering Modelling the boiler economiser, evaporator and su- perheater was done as one heat exchanger divided into n divisions, in the presented cases n = 30. The number of 30 was found to be a reasonable compromise between accu- racy in the determination of the pinch point temperature difference and the computational time for the optimisa- tion. Figure 2 is a sketch of the boiler heat exchangers with numbering. The heat source enters at the upper left and exits at the lower right, while the working fluid enters at the bottom and leaves at the top. With j = 2, 3, ..., n+1: 4PDF Image | organic Rankine cycles for waste heat recovery in marine settings
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