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Group Activity 2: Off-design Steam Cycle & Heat rejection Mechanical Engineering - Energy Systems Authors: •Group xx: xxxxx xxxxx (xxxxxxxx), xxxxx xxxxx (xxxxxxxx) •Group xx: xxxxx xxxxx (xxxxxxxx), Fabio Santoro (xxxxxxxx) Lecturer:Prof. Stefano Consonni Teaching Assistants:Riccardo Cremona, Hamidreza Heydari and Nima Razmjoo Academic year:2025-2026Contents Introduction2 1 Design Operation2 2 Off-Design Operations72.1 Scenario #1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 2.2 Scenario #2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 2.3 Scenario #3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 2.4 Scenario #4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 2.5 Scenario #5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 3 Design and Off-Design results comparison18 4 Heat Rejection22 GA Assessment25 References25 1 Energy Systems- Report GA2 - Groups xx and xx Introduction The scope of this activity is the design of a low-performance and low-cost subcritical superheated steam cycle plant. The working fluid (steam) is heated by the combustion of a fuel in a firing boiler. A simplified scheme of the steam plant is shown in Figure1. In the boiler, there are three heat exchangers — the evaporator, superheater, and economizer — which heat the working fluid from sub-cooled to superheated conditions. The burners inside the boiler can be tilted to modify the heat transfer properties of the evaporator (U AE V A). Between the superheater and the steam turbine, it is possible to adjust the fluid pressure through the action of a throttling valve. At the turbine outlet, the fluid pressure remains well above atmospheric pressure; therefore, the plant is not susceptible to air infiltration in the condenser, and a deaerator is not required. The feedwater pump increases the pressure of the stream — from saturated conditions at the condenser exit to the conditions at the economizer inlet. To regulate the temperature at the economizer outlet, it is possible to recycle part of the water stream within the economizer. All three heat exchangers in the boiler have counter-current configuration although in the scheme the representation don’t match the description. We also assume no pressure drop in the piping system. It is also required to evaluate the performances of the plant for five off-design case (Section2).Figure 1:Plant Scheme 1.Design Operation Given the data in table1we can write all the formulations to determine the thermodynamic properties of each point of the steam side of the plant scheme1while, all the numerical re- sults are collected in table2. The remaining properties are evaluated using an Excel add-in database table [3]. •Point 1 The pump inlet water stream is at saturated con-ditions, the temperatureT 1=T cond. The other propertiesp1,h 1,s 1,v 1andx 1are evaluated with the Excel add-in. •Point 2is This is the outlet isoentropic compression in the pumpp2is=p E V A:h 2is=h(p 2is, s 1)and we are in sub-cooled liquid conditions. The other prop- ertiesT2isandv 2isare evaluated with the Excel add-in. 2ECO EVA SH ST ~ SD from ECO to SD %rec WD 3 1 2 5 4/4h iso h valve 4 3sv 3sl 3sv 3sl 3 air+fuel ϑ: tilting angle burner to stack feed-water pump condenser b a c d 2rec Energy Systems- Report GA2 - Groups xx and xx Table 1:Design plant data SymbValueu.o.m.DescriptionTDN prop ˙m F G27 kg/s Flue gases mass flow ratec p,F G1.1 kJ/(kgK) Flue gases specific heat capacityT a2000◦ C Flue gases initial temperature ap E V A35 bar Evaporation pressure 3, 3sl, 3sv, 4p turb,in35 bar Turbine inlet pressure 4hT S H480◦ C Superheated steam temperature 4∆T sc,eva10◦ C Subcooling at evaporator inletT cond105◦ C Condensation temperature 1, (5is), (5), (5sv)T F G,stack180◦ C Flue gases stack temperature d% rec,eco0%Recycle stream in economizerη T70%Turbine efficiencyη P85%Pump efficiency•Point 2 The real compression of the pump (p2=p 2is) depends on the efficiencyηP: h2=h 1+h 2is−h 1η P The liquid is at sub-cooled conditions, the other propertiesT2,s 2andv 2are evaluated with the Excel add-in.•Point 3 This is sub-cooled liquid at that enters the steam drum (SD)p3=p 2andT E V A=T sat(p 3= pE V A): T3=T E V A−∆T sc,eva the other propertiesh3,s 3andv 3are evaluated with the Excel add-in.•Point 3sl The saturated liquid in the steam drumT3sl= TE V Aandp 3sl=p 3. The other propertiesh 3sl, s3sl,v 3slandx 3slare evaluated with the Excel add-in.•Point 3sv The saturated vapor exiting the steam drum T3sv=T E V Aandp 3sv=p 3. The other proper- tiesh3sv,s 3sv,v 3svandx 3svare evaluated with the Excel add-in. •Point 2rec If there is recycle between the outlet and the in- let of the economizer we change some thermody- namic properties at the inlet (sub-cooled liquid).p 2rec=p 2and the enthalpy is obtained by the energy balance: h2rec=h 2+h 3sl·% rec,eco1 + % rec,eco The other propertiesT2rec,s 2recandv 2recare evaluated with the Excel add-in. •Point 4 Superheated steam that exits the superheater at pressurep4=p 3at temperatureT 4=T S H. The other propertiesh4,s 4andv 4are evaluated with the Excel add-in. •Point 4h For this system there is the possibility to reduce the pressure (p4h=p turb,in) at the inlet of the turbine by throttling the steam flow in an isoen- thalpic valve (h4h=h 4). If the inlet pressure of the turbine coincides with the point 4. The other propertiesT4h,s 4handv 4hare evaluated with the Excel add-in. •Point 5is Outlet of the isoentropic expansion of the steam turbines5is=s 4hto the condensation pressure pcond=p sat(T cond). The other propertiesT 5is, v5isand eventuallyx 5isare evaluated with the Excel add-in. This point can be either super- heated steam, saturated steam or liquid-vapor conditions depending on the properties of the turbine inlet. 3 Energy Systems- Report GA2 - Groups xx and xx •Point 5 Real turbine expansionp5=p 5is: h5=h 4h−η T(h 4h−h 5is) The other propertiesT5,v 5and eventuallyx 5are evaluated with the Excel add-in. This point can be either superheated steam, saturated steam or liquid-vapor conditions depending on the prop- erties of the turbine inlet.•Point 5sv If the point 5 falls in the superheated steam re- gion, a portion of the condenser contains steam. The point 5sv is at saturated vapor conditions p5sv=p 5,T 5sv=T 1and the otherh 5svs 5sv,v 5sv andx5svare evaluated with the Excel add-in.Figure 2:T−sdiagram of Design operation steam cycle In the design operation conditions, the ideal tur- bine expansion falls in the 2-phase region while the real expansion is superheated steam. We don’t touch the throttling valve and we don’t re- cycle water across the economizer so, the points 2rec and 4h coincide respectively with 2 and 4. Here in the numerical results of the thermody- namic properties of the design steam cycle are collected in Table2. We can compute the total heat transfer power inside the boiler as: ˙ Qin,tot= ˙m F G·c p,F G·(T a−T F G,stack) With this we can estimate the mass flow rate of the steam cycle: ˙msteam=˙ Qin,toth 4−h 2We can compute the power of the turbine and the pump as: ˙ WS T= ˙m steam·(h 5−h 4h) ˙ WP= ˙m steam·(h 2−h 1) The cycle efficiency is: ηnet,cycle=˙ WS T−˙ WP˙ Qin,tot The maximum available heat power of the Flue Gases is assuming the ambient temperature be- ing25◦ C: ˙ QF G,avail= ˙m F G·c p,F G·(T a[◦ C]−25) 4`´`ˆ`˜`¨`˝`˚`ˇ`˘ `¯`˙`¸`˛`‚`‹`› `“`”`„`«`»`–`„`—`‌ ``ı `˝`ȷ `´`ȷ`ȷ `´`˝`ȷ `ˆ`ȷ`ȷ `ˆ`˝`ȷ `˜`ȷ`ȷ `˜`˝`ȷ `¨`ȷ`ȷ `¨`˝`ȷ `˝`ȷ`ȷ `ff`fi`fl `‹`fi`˛`ffi`¸`ffl`˛`fi `ff `”`´`␣`ı `´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`˝`“`" `˝`#`“`$`fi`“`#`—`˙`␣`›`%`!`fi Energy Systems- Report GA2 - Groups xx and xx Table 2:Thermodynamic Properties (steam side): Design Operation ipThsvxstatus [bar][ ◦ C][kJ/kg][kJ/(kgK)][m 3 /kg]1 1.209 105.00 440.21 1.3632 0.00105 0 SL 2is 35 105.25 443.76 1.3632 0.00105 - sub-cool 2 35 105.61 445.28 1.3672 0.00105 - sub-cool 2rec 35 105.61 445.28 1.3672 0.00105 - sub-cool 3 35 232.56 1002.35 2.6325 0.00121 - sub-cool 3sl 35 242.56 1049.78 2.7254 0.00123 0 SL 3sv 35 242.56 2802.74 6.1245 0.05706 1 SV 4 35 480.00 3406.08 7.0995 0.09633 - steam 5is 1.209 105.00 2609.44 7.0995 1.37174 0.9670 L-V 5 1.209 127.10 2728.93 7.4121 1.50794 - steam 5sv 1.209 105.00 2683.39 7.2951 1.41848 1 SV We can estimate the boiler efficiencyη boilerwith the expression and the overall efficiencyη overall: ηboiler=˙ Qtot,in˙ QF G,availη overall=˙ WS T−˙ WP˙ QF G,avail=η cycle·η boiler Now we apply the definition reduction coefficient which is a characteristic of the steam turbine that doesn’t change if we work in design or off-design conditions: mdesign red=˙m design steam·qR·T design 4hp design 4h·Adesign eq= ˙m of f−design steam·qR·T of f−design 4hp of f−design 4h·Aof f−design eq=m of f−design red We can divide everything by the constant√Rand the equivalent area in design conditions. We introduce the partialization coefficientχ≜A of f−design eqA design eq. In the end, there are two different expressions for the pseudo-coefficientbmredif we are in design or in off-design conditions but, their value must be equal: bmdesign red=˙m design steam·qT design 4hp design 4hbm of f−design red=˙m of f−design steam·qT of f−design 4hp of f−design 4h·χ Now we can compute the heat transfer power in each heat exchanger inside the boiler:˙ QS H= ˙m steam·(h 4−h 3sv)˙ QE V A= ˙m steam·(h 3sv−h 3)˙ QE C O= ˙m steam·(h 3−h 2) The Flue Gases temperature in every section of the boiler. We can verify that the temperature ind coincides with the temperature at stack: Tb=T a−˙ QE V A˙m F G·c p,F GT c=T b−˙ QS H˙m F G·c p,F GT d=T c−˙ QE C O˙m F G·c p,F G≡T F G,stack With these temperatures and the ones in the steam cycle, we can compute theLM T Din each heat exchanger and the respective coefficientU A: LM T DE V A=(T a−T 3sv)−(T b−T 3sl)ln  Ta−T 3svT b−T 3sl U A E V A=˙ QE V ALM T D E V A 5 Energy Systems- Report GA2 - Groups xx and xx LM T D S H=(T b−T 4)−(T c−T 3sv)ln  Tb−T 4T c−T 3sv U A S H=˙ QS HLM T D S H LM T DE C O=(T c−T 2)−(T d−T 2rec)ln  Tc−T 3T d−T 2rec U A E C O=˙ QE C OLM T D E C O All the results of these formulations are summarized in Table9at page18. We can plot theT–˙ Q diagram for the boiler, and even though the specific heat capacity of the steam in the superheater is not constant, it can be approximated as constant, allowing us to represent it with a straight line.Figure 3:BoilerT− ˙ Qdiagram of design case 6 Energy Systems- Report GA2 - Groups xx and xx 2.Off-Design Operations In this Section we are required to determine some parameters for five scenarios in which the system operates at different conditions respect to the design operation. For all scenarios we consider a reduction of the mass flow rate of Flue Gases equal to 50% of the design value then, we can operated different changes (one ore more at the same time) in the system: •Throttling: we can act on throttling valve between 4 and 4h to reduce the pressure at the inletof the steam turbine •Burner tilting: we can change the orientation of the burners in the combustion chamber, inthis way the heat exchanger coefficientU AE V Achanges value •Pressure sliding: we operate the steam cycle at evaporation pressure •ECO recycling: we recycle hot water from the outlet of the economizer toward the inlet tocontrol the water temperature at the steam drum inlet •Partialization: we can reduce the equivalent area of the steam turbine We need to take into account that the reduction of Gas mass flow rate causes a reduction of the boiler heat exchanger coefficientU Arespect to the design values of 10% for the evaporator and 25% for superheater and economizer. We keep the same solution methodology adopted in Section1with the data of table1but, for the solution of the off-design scenarios, we need to optimize the system considering variables and constraints. The numerical solution is found using the Microsoft Excel Solver add-in [2]. Here in Table3, we can find the final results for all the off-design cases. We introduced the partialization factorχand we need to remember the reduction of FG mass flow rate, the other parameters remain constant. For each scenario we clarify the nature of each parameter. The numerical results of the methodology are collected in Table9at page18. Table 3:Off-design table. Legend:data,variableandconstrained SymbDesignOff 1Off 2Off 3Off 4Off 5u.o.m. p E V A35353517.8417.6535 bar p turb,in3518.0317.8517.8417.6535 bar ∆T sc,eva1069.7834.364.141034.36 ◦ CT F G,stack180123.14131.71130.00149.51131.71 ◦ CT S H480347.93480480480480 ◦ C% rec,eco0%0%0%0%33.57%0% - χ-100%100%100%100%51.33% - 7 Energy Systems- Report GA2 - Groups xx and xx 2.1.Scenario #1 In this scenario we decouple the steam pressure in the boiler and the inlet pressure of the turbine by acting on the throttling valve, in this way it changes also the the fluid temperature that enters the steam drum and the FG temperature at stack. We set as variable also the superheated steam temperature because the FG heat power changes between the heat exchanges in the boiler. The optimization algorithm is represented here:Algorithm 1Optimization algorithm: Off-Design Scenario 1 ▷Set the variables:p turb,in,∆T sc,eva,T F G,stackandT S H ▷Compute all parameters with the same methodology of Design operation ▷Set constraints:□The evaporator coefficient must stay constant:U Aof f1 E V A−(1−10%)U Adesign E V A= 0 □The economizer coefficient must stay constant:U Aof f1 E C O−(1−25%)U Adesign E C O= 0 □The steam turbine pseudo-coefficient must stay constant:bmof f1 red−bmdesign red= 0 ▷Compute the errorerr=U Aof f1 S H−(1−25%)U Adesign S H ▷Update the variables till reach the error zeroFigure 4:T−sdiagram of Off-Design operation 1 Here in the figure we can see the difference between the design and off-design case. In the latter, we reach a lower superheated temperature and the very last stages of the turbine expansion deal with 2 phases fluid. The Table4contains all thermodynamic properties for the current configuration while Table9we see all the other values. In figure5we represented theT−˙ Qdiagram of the boiler. 8`´`ˆ`˜`¨`˝`˚`ˇ`˘ `¯`˙`¸`˛`‚`‹`› `“`”`„`«`»`–`„`—`‌ ``ı `˝`ȷ `´`ȷ`ȷ `´`˝`ȷ `ˆ`ȷ`ȷ `ˆ`˝`ȷ `˜`ȷ`ȷ `˜`˝`ȷ `¨`ȷ`ȷ `¨`˝`ȷ `˝`ȷ`ȷ `ff`fi`fl `‹`fi`˛`ffi`¸`ffl`˛`fi `ff `”`´`␣`ı `´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`˝`“`" `˝`#`“`´`ˆ`˜`˜`“`! `˜`“`"`¨`¨`$`˝`#`“`˝`%`fi`“`#`—`˙`␣`›`&`!`fi `’`´`(`%`fi`“`#`—`˙`␣`›`&`!`fi Energy Systems- Report GA2 - Groups xx and xx Table 4:Thermodynamic Properties: Off-Design Operation #1 ipThsvxstatus [bar][ ◦ C][kJ/kg][kJ/(kgK)][m 3 /kg]1 1.209 105.00 440.21 1.3632 0.00105 0 SL 2is 35 105.25 443.76 1.3632 0.00105 - sub-cool 2 35 105.61 445.28 1.3672 0.00105 - sub-cool 2rec 35 105.61 445.28 1.3672 0.00105 - sub-cool 3 35 172.78 732.82 2.0658 0.00112 - sub-cool 3sl 35 242.56 1049.78 2.7254 0.00123 0 SL 3sv 35 242.56 2802.74 6.1245 0.05706 1 SV 4 35 347.93 3099.82 6.6520 0.07647 - steam 4h 18.029 331.07 3099.82 6.9427 0.14898 - steam 5is 1.209 105.00 2550.13 6.9427 1.33427 0.9406 L-V 5 1.209 105.00 2632.59 7.1608 1.38637 0.9774 L-V Figure 5:BoilerT− ˙ Qdiagram of off-design case 1 9 Energy Systems- Report GA2 - Groups xx and xx 2.2.Scenario #2 In this scenario we decouple the steam pressure in the boiler and the inlet pressure of the turbine by acting on the throttling valve, in this way it changes also the the fluid temperature that enters the steam drum and the FG temperature at stack. In this case the superheated temperature matches the design conditions, we can achieve this by tilting the burners so, we remove the constraint of the evaporator heat transfer coefficient. The optimization algorithm is represented here:Algorithm 2Optimization algorithm: Off-Design Scenario 2 ▷Set the variables:p turb,in,∆T sc,evaandT F G,stack ▷Compute all parameters with the same methodology of Design operation ▷Set constraints:□The economizer coefficient must stay constant:U Aof f1 E C O−(1−25%)U Adesign E C O= 0 □The steam turbine pseudo-coefficient must stay constant:bmof f1 red−bmdesign red= 0 ▷Compute the errorerr=U Aof f2 S H−(1−25%)U Adesign S H ▷Update the variables till reach the error zeroFigure 6:T−sdiagram of Off-Design operation 2 Here in the figure we can see the difference between the design and off-design case. In the latter, even if we still exit the turbine in the superheated steam conditions, the total work extracted by the turbine is way less than before even if higher than the off-design 1. The Table6contains all thermodynamic properties for the current configuration while Table9we see all the other values. In figure9we represented theT−˙ Qdiagram of the boiler. 10`´`ˆ`˜`¨`˝`˚`ˇ`˘ `¯`˙`¸`˛`‚`‹`› `“`”`„`«`»`–`„`—`‌ ``ı `ȷ`˝`ȷ `´`ȷ`ȷ `´`˝`ȷ `ˆ`ȷ`ȷ `ˆ`˝`ȷ `˜`ȷ`ȷ `˜`˝`ȷ `¨`ȷ`ȷ `¨`˝`ȷ `˝`ȷ`ȷ `ff`fi`fl `‹`fi`˛`ffi`¸`ffl`˛`fi `ff `”`´`␣`ı `´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`˝`“`" `˝`#`“`´`ˆ`˜`˜`“`! `˜`“`"`¨`¨`$`˝`#`“ `˝`˝`“`" Energy Systems- Report GA2 - Groups xx and xx Table 5:Thermodynamic Properties: Off-Design Operation #2 ipThsvxstatus [bar][ ◦ C][kJ/kg][kJ/(kgK)][m 3 /kg]1 1.209 105.00 440.21 1.3632 0.00105 0 SL 2is 35 105.25 443.76 1.3632 0.00105 - sub-cool 2 35 105.61 445.28 1.3672 0.00105 - sub-cool 2rec 35 105.61 445.28 1.3672 0.00105 - sub-cool 3 35 208.20 890.12 2.4051 0.00117 - sub-cool 3sl 35 242.56 1049.78 2.7254 0.00123 0 SL 3sv 35 242.56 2802.74 6.1245 0.05706 1 SV 4 35 480.00 3406.08 7.0995 0.09633 - steam 4h 17.853 470.65 3406.08 7.4033 0.18922 - steam 5is 1.209 125.35 2725.37 7.4033 1.50094 - steam 5 1.209 176.31 2827.48 7.6444 1.70247 - steam 5sv 1.209 105.00 2683.39 7.2951 1.41848 1 SV Figure 7:BoilerT− ˙ Qdiagram of off-design case 2 11 Energy Systems- Report GA2 - Groups xx and xx 2.3.Scenario #3 In this scenario we operated the so called pressure sliding in which, we set the evaporation pressure as a variable and we force the equality between the evaporation and turbine inlet pressure, in other words we do not throttle the flux. Again, also in this case the superheated temperature matches the design conditions, we can achieve this by tilting the burners so, we remove the constraint of the evaporator heat transfer coefficient. The optimization algorithm is represented here:Algorithm 3Optimization algorithm: Off-Design Scenario 3 ▷Set the variables:p E V A,∆T sc,evaandT F G,stack ▷We set the equalitypin,turb=p E V A ▷Compute all parameters with the same methodology of Design operation ▷Set constraints:□The economizer coefficient must stay constant:U Aof f1 E C O−(1−25%)U Adesign E C O= 0 □The steam turbine pseudo-coefficient must stay constant:bmof f3 red−bmdesign red= 0 ▷Compute the errorerr=U Aof f3 S H−(1−25%)U Adesign S H ▷Update the variables till reach the error zeroFigure 8:T−sdiagram of Off-Design operation 3 Here in the figure we can see the difference between the design and off-design case. In the latter, even if we still exit the turbine in the superheated steam conditions, the total work extracted by the turbine is way less than before and it is slightly higher than the off-design 2. The Table6contains all thermodynamic properties for the current configuration while Table9we see all the other values. In figure9we represented theT−˙ Qdiagram of the boiler. 12`´`ˆ`˜`¨`˝`˚`ˇ`˘ `¯`˙`¸`˛`‚`‹`› `“`”`„`«`»`–`„`—`‌ ``ı `˝`ȷ `´`ȷ`ȷ `´`˝`ȷ `ˆ`ȷ`ȷ `ˆ`˝`ȷ `˜`ȷ`ȷ `˜`˝`ȷ `¨`ȷ`ȷ `¨`˝`ȷ `˝`ȷ`ȷ `ff`fi`fl `‹`fi`˛`ffi`¸`ffl`˛`fi `ff `”`´`␣`ı `´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`˝`“`" `˝`#`“`´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`#`“ `˝`˝`“`" `$`fi`“`#`—`˙`␣`›`%`!`fi `&`´`’`$`fi`“`#`—`˙`␣`›`%`!`fi Energy Systems- Report GA2 - Groups xx and xx Table 6:Thermodynamic Properties: Off-Design Operation #3 ipThsvxstatus [bar][ ◦ C][kJ/kg][kJ/(kgK)][m 3 /kg]1 1.209 105.00 440.21 1.3632 0.00105 0 SL 2is 17.842 105.12 441.97 1.3632 0.00105 - sub-cool 2 17.842 105.30 442.72 1.3652 0.00105 - sub-cool 2rec 17.842 105.30 442.72 1.3652 0.00105 - sub-cool 3 17.842 202.54 863.93 2.3545 0.00116 - sub-cool 3sl 17.842 206.6842 882.64 2.3937 0.00117 0 SL 3sv 17.842 206.6842 2795.77 6.3808 0.11131 1 SV 4 17.842 480.00 3426.56 7.4309 0.19187 - steam 5is 1.209 130.82 2736.46 7.4309 1.52282 - steam 5 1.209 182.60 2839.98 7.6721 1.72709 - steam 5sv 1.209 105.00 2683.39 7.2951 1.41848 1 SV Figure 9:BoilerT− ˙ Qdiagram of off-design case 3 13 Energy Systems- Report GA2 - Groups xx and xx 2.4.Scenario #4 This scenario matches very closely the case 3 but, in order to have a better control of the water temperature exiting the economizer, we are required to recycle a portion of stream from the outlet to the inlet of the economizer. In this case we set a fixed value for∆Tsc,evabut we make the% rec,eco variable. The other conditions are equal to scenario 3. The optimization algorithm is represented here:Algorithm 4Optimization algorithm: Off-Design Scenario 4 ▷Set the variables:p E V A,% rec,ecoandT F G,stack ▷We set the equalitypin,turb=p E V A ▷Compute all parameters with the same methodology of Design operation ▷Set constraints: □The economizer coefficient must stay constant:U Aof f1 E C O−(1−25%)U Adesign E C O= 0 □The steam turbine pseudo-coefficient must stay constant:bmof f3 red−bmdesign red= 0 ▷Compute the errorerr=U Aof f4 S H−(1−25%)U Adesign S H ▷Update the variables till reach the error zeroFigure 10:T−sdiagram of Off-Design operation 4 TheT−sdiagram is very similar to the one of scenario 3 but, only in this case we are able to distinguish the 2 and 2rec thermodynamic points. There is a slight difference of evaporation pressure and the distance between 3 and 3sl is higher because we set the same of the design case (10◦ C), but it is higher than the case 3 (≈4.1◦ C). The Table7contains all thermodynamic properties for the current configuration, which its values are very close to the previous one, while Table9we see all the other values. In figure11we represented theT−˙ Qdiagram of the boiler. 14`´`ˆ`˜`¨`˝`˚`ˇ`˘ `¯`˙`¸`˛`‚`‹`› `“`”`„`«`»`–`„`—`‌ ``ı `˝`ȷ `´`ȷ`ȷ `´`˝`ȷ `ˆ`ȷ`ȷ `ˆ`˝`ȷ `˜`ȷ`ȷ `˜`˝`ȷ `¨`ȷ`ȷ `¨`˝`ȷ `˝`ȷ`ȷ `ff`fi`fl `‹`fi`˛`ffi`¸`ffl`˛`fi `ff `”`´`␣`ı `´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`˝`“`" `˝`#`“`´`ˆ `ˆ`˛`fi`$`˜`˜`“`! `˜`“`"`¨`˝`#`“ `˝`˝`“`" `%`fi`“`#`—`˙`␣`›`$`!`fi `&`´`’`%`fi`“`#`—`˙`␣`›`$`!`fi Energy Systems- Report GA2 - Groups xx and xx Table 7:Thermodynamic Properties: Off-Design Operation #4 ipThsvxstatus [bar][ ◦ C][kJ/kg][kJ/(kgK)][m 3 /kg]1 1.209 105.00 440.21 1.3632 0.00105 0 SL 2is 17.654 105.12 441.95 1.3632 0.00105 - sub-cool 2 17.654 105.30 442.69 1.3652 0.00105 - sub-cool 2rec 17.654 131.25 552.67 1.6464 0.00107 - sub-cool 3 17.654 196.16 835.28 2.2940 0.00115 - sub-cool 3sl 17.654 206.1639 880.27 2.3888 0.00117 0 SL 3sv 17.654 206.1639 2795.50 6.3846 0.11246 1 SV 4 17.654 480.00 3426.78 7.4360 0.19394 - steam 5is 1.209 131.84 2738.53 7.4360 1.52689 - steam 5 1.209 183.50 2841.76 7.6760 1.73060 - steam 5sv 1.209 105.00 2683.39 7.2951 1.41848 1 SV Figure 11:BoilerT− ˙ Qdiagram of off-design case 4 15 Energy Systems- Report GA2 - Groups xx and xx 2.5.Scenario #5 This scenario is the only one in which we allow the reduction of the turbine equivalent area by changing the variableχ. We are very similar to the design conditions but, still we keep other two elements as variables: we tilt the burners which means set free∆Tsc,eva. The optimization algorithm is represented here:Algorithm 5Optimization algorithm: Off-Design Scenario 5 ▷Set the variables:∆T sc,eva,T F G,stackandχ ▷Compute all parameters with the same methodology of Design operation ▷Set constraints:□The economizer coefficient must stay constant:U Aof f1 E C O−(1−25%)U Adesign E C O= 0 □The steam turbine pseudo-coefficient must stay constant:bmof f3 red−bmdesign red= 0 ▷Compute the errorerr=U Aof f5 S H−(1−25%)U Adesign S H ▷Update the variables till reach the error zeroFigure 12:T−sdiagram of Off-Design operation 5 TheT−sdiagram matches very closely the design case with the only exception of the point 3 which coincides with the outlet conditions of the economizer. The Table8contains all thermodynamic properties for the current configuration, while Table9we see all the other values. In figure13we represented theT−˙ Qdiagram of the boiler. 16`´`ˆ`˜`¨`˝`˚`ˇ`˘ `¯`˙`¸`˛`‚`‹`› `“`”`„`«`»`–`„`—`‌ ``ı `˝`ȷ `´`ȷ`ȷ `´`˝`ȷ `ˆ`ȷ`ȷ `ˆ`˝`ȷ `˜`ȷ`ȷ `˜`˝`ȷ `¨`ȷ`ȷ `¨`˝`ȷ `˝`ȷ`ȷ `ff`fi`fl `‹`fi`˛`ffi`¸`ffl`˛`fi `ff `”`´`␣`ı `´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`˝`“`" `˝`#`“`´`ˆ`˜`˜`“`! `˜`“`"`¨`˝`#`“`˝`˝`“`" `$`fi`“`#`—`˙`␣`›`%`!`fi `&`´`’`$`fi`“`#`—`˙`␣`›`%`!`fi Energy Systems- Report GA2 - Groups xx and xx Table 8:Thermodynamic Properties: Off-Design Operation #5 ipThsvxstatus [bar][ ◦ C][kJ/kg][kJ/(kgK)][m 3 /kg]1 1.209 105.00 440.21 1.3632 0.00105 0 SL 2is 35 105.25 443.76 1.3632 0.00105 - sub-cool 2 35 105.61 445.28 1.3672 0.00105 - sub-cool 2rec 35 105.61 445.28 1.3672 0.00105 - sub-cool 3 35 208.20 890.12 2.4051 0.00117 - sub-cool 3sl 35 242.56 1049.78 2.7254 0.00123 0 SL 3sv 35 242.56 2802.74 6.1245 0.05706 1 SV 4 35 480.00 3406.08 7.0995 0.09633 - steam 5is 1.209 105.00 2609.44 7.0995 1.37174 0.9670 L-V 5 1.209 127.10 2728.93 7.4121 1.50794 - steam 5sv 1.209 105.00 2683.39 7.2951 1.41848 1 SV Figure 13:BoilerT− ˙ Qdiagram of off-design case 5 17 Energy Systems- Report GA2 - Groups xx and xx 3.Design and Off-Design results comparison Here in Table9, we collected all the results and compared the design case performances with the off-design ones. Table 9:Comparison results design and off-design scenariosSymbDesignOff 1Off 2Off 3Off 4Off 5u.o.m. ˙ Qin,tot54,054.00 27,871.41 27,744.10 27,769.48 27,479.81 27,744.10 kW˙m steam18.26 10.50 9.37 9.31 9.21 9.37 kg/s˙ WS T12,362.41 4,905.69 5,421.79 5,459.11 5,387.30 6,345.21 kW˙ WP92.54 53.22 47.50 23.32 22.82 47.50 kW˙ WN12,269.87 4,852.47 5,374.30 5,435.79 5,364.48 6,297.71 kWη cycle22.70 17.41 19.37 19.57 19.52 22.70%˙ QF G,avail58,657.50 29,328.75 29,328.75 29,328.75 29,328.75 29,328.75 kWη boiler92.15 95.03 94.60 94.68 93.70 94.60%η overall20.92 16.55 18.32 18.53 18.29 21.47%bm red14.32 14.32 14.32 14.32 14.32 14.32kgs √K bar ˙ QS H11,014.87 3,119.12 5,653.56 5,870.52 5,813.27 5,653.56 kW˙ QE V A32,869.01 21,733.27 17,922.15 17,978.95 18,051.27 17,922.15 kW˙ QE C O10,170.11 3,019.02 4,168.39 3,920.02 3,615.28 4,168.39 kWT a2,000.00 2,000.00 2,000.00 2,000.00 2,000.00 2,000.00◦ CT b893.30 536.48 793.12 789.30 784.43 793.12◦ CT c522.43 326.44 412.41 393.98 392.96 412.41◦ CT d180.00 123.14 131.71 130.00 149.51 131.71◦ CU A S H32.18 24.14 24.14 24.14 24.14 24.14 kW/KU A E V A29.51 26.56 17.24 16.70 16.81 17.24 kW/KU A E C O64.19 48.15 48.15 48.15 48.15 48.15 kW/KLet’s go through the first analysis of the total heat transfer power used in the steam cycle, ˙ Qin,tot, with each portion distributed among the superheater (˙ QS H), evaporator (˙ QE V A), and economizer (˙ QE C O). We need to recall that the sum of the heat powers in each heat exchanger equals the total, as losses are not considered:˙ Qin, tot=˙ QS H+˙ QE V A+˙ QE C O In Figure14, we show the absolute values of all these heat powers. We can see that the total power in the off-design cases is almost half that of the design case, mainly due to the reduction in gas mass flow rate. The diagram also shows that, in any case, most of the power is transferred in the evaporator, though the distribution among the three heat exchangers differs. In fact, in Figure15, we represent the relative distribution for each case. For the design scenario, the evaporation portion is about 60%; in off-design 1, it is the highest, at almost 80%; and in the other cases, it is around 65%. Apart from off-design 1, the relative portion of the economizer remains approximately constant. The superheater’s relative contribution is minimal in off-design 1, while in the other off-design cases it is slightly lower than in the design case. It is worth noting that the values in Table9for off-design case 2 are almost identical to those of case 5, except that the turbine in case 5 generates more power, and consequently, system 5 is more efficient. In both Figures14and15, the values for these two cases coincide. 18 Energy Systems- Report GA2 - Groups xx and xx Figure 14:Heat transfer power (Absolute values) Figure 15:Heat transfer power (Relative) In Figure16, we show the behavior of the cycle efficiencyηcycleand the overall efficiencyη overall. The two exhibit similar trends, with the overall efficiency always lower than the cycle efficiency because the boiler efficiency is not equal to unity. Off-design case 1 is the least efficient, while case 5 matches the cycle efficiency of the design case and also shows a higher overall efficiency, as the boiler is more efficient than in the design condition. The higher boiler efficiency results from the lower flue gas temperature at the stack. Even though the system configuration at condition 5 is more efficient, the power generated in absolute terms is lower.Figure 16:Cycle and overall efficiencies plot 19 Energy Systems- Report GA2 - Groups xx and xx The next aspect to analyze is the comparison between the boiler efficiencyη boilerand the stack temper- atureTd. In Figure17, we show the values of these two quantities, noting that the stack temperature is represented in inverse order. We can observe that, with this configuration, the two curves exhibit the same trend. This occurs becauseηboilerandT dare perfectly correlated, with a correlation coefficient equal to−1, as they are related by a linear function with a negative slope: ηboiler=˙ Qin,tot˙ Qmax,avail=· · ·= T a−T dT a−25[◦ C]ρ ηboiler,T d=−1 All off-design scenarios have higher boiler efficiency than the design case becauseTdis always lower. The maximum efficiency is achieved at the minimumTd, which corresponds to off-design scenario 1. As with the heat transfer power, we can also notice that the values for off-design cases 2 and 5 coincide. In Figure18, we present theηboiler–T drelationship for all cases. It is possible to further improve the boiler efficiency (with a corresponding reduction in stack temperature) by introducing a Ljungstrom preheater into the boiler system.Figure 17:Boiler efficiency - stack temperature comparison Figure 18:Boiler efficiency - stack temperature relation 20 Energy Systems- Report GA2 - Groups xx and xx In Figure19, we show the values of the heat exchanger coefficientsU Aand their reduction for each off-design case. The coefficients related to the superheater and the economizer are reduced by 25% with respect to the design value for all off-design scenarios (the coefficient associated with the superheater was the ob jective function of the optimization algorithms). Only in scenario 1 was the reduction for the evaporator limited to 10%, while in the other cases this coefficient was not constrained, resulting in a higher reduction. Cases 2 and 5 show matching values.Figure 19:Heat transfer coefficients(U A) In Table10, we report the final results of the variables used to compute the steam turbine mass reduc- tion pseudo-coefficientbmred, whose formulation—different for the design and off-design conditions—is described in Section1. Thebmredvalue is constant. In Figure20, we provide a graphical representation of these values. Table 10:Steam Turbine reduction pseudo-coefficient results tableSymbDesignOff 1Off 2Off 3Off 4Off 5u.o.m. ˙m steam18.26 10.50 9.37 9.31 9.21 9.37 kg/sT turb,in480 331.07 470.65 480 480 480◦ Cp turb,in35 18.03 17.85 17.84 17.65 35 barχ- 100 100 100 100 51.33% bm red14.32 14.32 14.32 14.32 14.32 14.32kgs √K bar Figure 20:Steam Turbine Pseudo-reduction coefficient results diagrams 21 Energy Systems- Report GA2 - Groups xx and xx 4.Heat Rejection Figure 21:Wet cooling tower The cooling tower (Figure21) operates on the principle of heat exchange between water and air, involving both sensible and latent heat transfer processes. A portion of the circulating water evaporates, removing heat from the remaining liquid and thereby reducing its temperature. The cooled water is then recirculated as a refrigerant fluid within the system. As a preliminary step, we analyzed the condenser. From the design data, we determined the thermal power released by the steam during condensation. This thermal power corresponds to the product of the steam mass flow rate and the change in enthalpy across the phase change. ˙ Qcond= ˙m steam·(h cond,in−h cond,out) = ˙m steam·(h 5−h 1)≈41784.13kW By applying an energy balance, the heat rejected on the hot side (steam) must equal the heat absorbed on the cold side (cooling water). Given the temperature rise of the cooling water (∆Tw= 10◦ C) and assuming a constant specific heat capacitycp,w= 4.186kJ/(kgK), the required water mass flow rate can be calculated. This ensures that the condenser effectively removes the heat released by the condensing steam. ˙ Qcond= ˙m w·c p,w·∆T w⇒˙m w=˙ Qcondc p,w·∆T w≈998.19kg/s In such a system, the cooling tower serves to reduce the temperature of the water, enabling its reuse as a cooling medium. The heat that must be dissipated to the air is therefore equal to the heat transferred to the water in the condenser. The design specifications of the cooling tower provide the dry-bulb temperatures of the air at both the inlet and outlet, as well as the corresponding relative humidities. From these data, all other thermodynamic properties of the air–water mixture can be derived using the psychrometric chart22 illustrated in the next page. Moreover, the air velocity is provided as part of the design data which is vair2.7 m/s. The table11contains the approximated numerical values extracted from the chart. 22 Energy Systems- Report GA2 - Groups xx and xx Figure 22:ASHRAE Psychrometric Chart No. 1 [1] 23 Energy Systems- Report GA2 - Groups xx and xx Table 11:Wet tower data. Valuesxandxextracted from chart22 Symbinletoutletu.o.m. T db14 24◦ CT wb9 24 ◦ CΦ50% 100%kg H2O/kg H2Osatx519g H2O/kg airh2772 kJ/kg v0.820.865 m 3 /kgTo determine the air mass flow rate, we recall that the heat absorbed by the air—previously identified as equal to that rejected in the condenser thus, the required air mass flow rate is obtained by dividing the total heat load˙ Qcondby the difference in specific enthalpy between the inlet and outlet air conditions: ˙ Qcond= ˙m air·(h outlet−h inlet)⇒˙m air=˙ Qcondh outlet−h inlet≈928.54kg/s Once the air mass flow rate is known, the liquid-to-gas ratio (L/G) can be easily calculated as the ratio of the water mass flow rate to the air mass flow rate. LG = ˙m w˙m air≈1.075 Knowing the inlet and outlet absolute humidity values, the mass flow rate of evaporated water can be found by multiplying the air mass flow rate by the difference in absolute humidityx: ˙mevap= ˙m air·(x outlet−x inlet)≈12.99kg/s Furthermore, since the air mass flow rate can be expressed as the product of air velocity, cross-sectional area, and inverse specific volume, the tower cross-sectional areaAtowercan be calculated using outlet conditions for both velocity and specific volume. From this area, the tower diameterdtowercan then be determined. Atower=˙m air·v airv air≈297.48m 2 dtower= 19.47m Finally, a make-up water flow rate is required to maintain the system’s overall water balance. This make-up compensates for three main losses:1.Evaporation losses, corresponding to the water that evaporates and carries away the latent heat; 2.Drift losses, due to small droplets entrained in the outgoing air stream; 3.Blowdown (purge) losses, necessary to control the accumulation of dissolved solids in the circu- lating water. In this system, the concentration of dissolved solids in the cooling tower water is allowed to reach at most three times the concentration of solids present in the make-up water. This condition defines the required blowdown rate and ensures proper water quality and stable operation of the cooling tower. ˙mmake−up= ˙m eva+ ˙m drif t+ ˙m purge⇒                ˙m drif t= 0.002·˙m eva≈1.996kg/s ˙mpurge=˙m eva3−1 −˙m drif t≈4.503kg/s ˙mmake−up≈19.499kg/s 24 Energy Systems- Report GA2 - Groups xx and xx GA Assessment GroupMember NameContribution Elisa Trevisiol 3 out of 3 09 Alessandro Luigi Valenti 3 out of 3Vijay Mani 3 out of 3 46 Fabio Santoro 3 out of 3References [1]American Society of Heating, Refrigerating and Air-Conditioning Engineers. Ashrae psychrometricchart no. 1: Si units, normal temperature, sea level (101.325 kpa), 1981. Psychrometric chart copyrighted by ASHRAE. [2]Microsoft Corporation.Microsoft Excel Solver add-in, 2024. [Computer software]. [3]Wilhelm Wagner and Norbert Kurzeja. IAPWS-IF97BO: Software for the Industrial FormulationIAPWS-IF97 (Excel Add-In / DLL). Web page, Chair of Thermodynamics, Ruhr-Universität Bochum, 2024. Includes Excel add-in IF97BO.xlam and reference spreadsheet IF97BO.xlsm; last update Nov 20, 2024. 25