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Modeling and control of Rankine based waste heat recovery systems for heavy duty trucks
June 7-10, 2015. Whistler, British Columbia, Canadaof Chemical Processes 9th 9th International International Symposium Symposium on on Advanced Advanced Control Control of Chemical Processes 9th International Symposium on Advanced Control of Chemical Processes June 7-10, 2015. Whistler, British Columbia, Canada Available online at www.sciencedirect.com June 7-10, 2015. Whistler, British Columbia, Canada June 7-10, 2015. Whistler, British Columbia, Canada
ScienceDirect IFAC-PapersOnLine 48-8 (2015) 568–573 Modeling and control of Rankine based waste heat recovery systems for heavy duty Modeling and control of Rankine basedtrucks waste heat recovery systems for heavy duty Modeling waste Modeling and and control control of of Rankine Rankine based basedtrucks waste heat heat recovery recovery systems systems for for heavy heavy duty duty trucks 1,2,3 , P. Dufour2*, V. Grelettrucks 1,2,3
2*
2* 2 1 3 1,2,3 2*, ,, P. Dufour V. , T.1,2,3 Reiche ,V. Lemort M. Nadri V. Grelet Grelet 1,2,3 2*, 2 1 2 1 , P. P. Dufour Dufour , 333 V. Grelet 2, T. Reiche 1,V. Lemort M. Nadri M. 2, T. Reiche1,V. Lemort3 1 , T. Reiche ,V. Lemort M. Nadri Nadri Volvo Trucks Global Trucks Technology Advanced Technology and Research, F-69800, Saint Priest, France 111 2 Lyon 1, Villeurbanne, Volvo Trucks Global Trucks Technology Advanced Technology Research, F-69800, Saint Priest, France University of Lyon, F-69622, Lyon, France – University France – CNRS, UMR 5007, LAGEP and Research, F-69800, Saint Priest, France 1Volvo Trucks Global Trucks Technology Advanced Technology and 2 2 3 Volvo Trucks Global Trucks Technology Advanced Technology andB49 Research, F-69800, Saint Priest, France 2 University of Lyon, F-69622, Lyon, France – University Lyon 1, Villeurbanne, France – CNRS, UMR 5007, University of Liège, Campus du Sart Tilman Bât. B4000 Liège, Belgium 2 University of Lyon,3 F-69622, Lyon, France – University Lyon 1, Villeurbanne, France – CNRS, UMR 5007, LAGEP University of*Corresponding Lyon,33 F-69622, Lyon, France – University Lyon 1, Bât. Villeurbanne, France –Belgium CNRS, UMR 5007, LAGEP LAGEP Universityauthor. of Liège, du Sart Tilman B49 B4000 Liège, TelCampus +33472431878. E-mail address: [email protected] of Liège, Campus du Sart Tilman Bât. B49 B4000 Liège, Belgium 3 University Universityauthor. of Liège, Campus du Sart Tilman Bât. B49 B4000 Liège, Belgium *Corresponding Tel +33472431878. E-mail address: [email protected] *Corresponding *Corresponding author. author. Tel Tel +33472431878. +33472431878. E-mail E-mail address: address: [email protected] [email protected]
Abstract: This paper presents a control oriented model development for waste heat recovery Rankine Abstract: This paper presents control oriented model for waste recovery Rankine based control systems in heavyaa duty trucks. Waste heat development recovery systems, suchheat as Rankine cycle, are Abstract: This paper control oriented model development for heat recovery Rankine Abstract: This papertopresents presents a the control oriented model development for waste waste heat recovery Rankine based control systems in heavy duty trucks. Waste heat recovery systems, such as Rankine cycle, are promising solutions improve fuel efficiency of heavy duty engines. Due to the highly transient based control systems in heavy duty trucks. Waste heat recovery systems, such as Rankine cycle, are based control systems in heavy the duty trucks. Waste heat systems, such as the Rankine are promising solutions to improve efficiency of heavy duty engines. to highly transient operating conditions, improving thefuel control strategy of recovery those systems is Due an important stepcycle, to their promising solutions to improve the fuel efficiency of heavy duty engines. Due to the highly transient promising solutions to improve the fuel efficiency of heavy duty engines. Due to the highly transient operating conditions, improving the control strategy those systems is an important to their integration into a vehicle. The system considered here isof recovering heat from both EGR andstep exhaust in a operating conditions, improving the control strategy of those systems is important step to operating conditions, improving the considered control strategy ofrecovering those systems is an an important step to their their integration into a vehicle. The system here is heat from both EGR and exhaust in a serial arrangement and use a mixture of water and ethanol as working fluid. The paper focuses on integration into aa vehicle. The system considered here is recovering heat from both EGR and exhaust in integration into vehicle. The system considered here isstate recovering heat from both EGR and exhaust in aaa serial arrangement and use a mixture of water and ethanol as working fluid. The paper focuses on comparison of a classical PID controller which is the of the art in the automotive industry and serial and aa mixture of water ethanol as working fluid. The paper focuses on serial arrangement arrangement and use use mixture water and and ethanol asthe working fluid. The paper focuses on aaa comparison of aa classical PID controller which is the state of art in the automotive industry and nonlinear model based controller in a of simulation environment. The nonlinear model based controller comparison of classical PID controller which is the state of the art in the automotive industry and comparison of a classical PID the controller which is the state of theThe artof inthe thesystem. automotive industry and aa nonlinear model based controller in aa one simulation environment. nonlinear model based controller shows better performance than PID and ensures safe operation nonlinear model based controller in simulation environment. The nonlinear model based controller nonlinear model based controller in a one simulation environment. The of nonlinear model based controller shows better performance than the PID and ensures safe operation the system. shows performance than the PID and ensures safe operation the Keywords: Waste Heat Recovery, Control, PID. © 2015,better IFAC (International Federation Automatic Control) byof Elsevier Ltd. All rights reserved. shows better performance than the Modeling, PIDofone one and ensures safe Hosting operation of the system. system. Keywords: PID. Keywords: Waste Waste Heat Heat Recovery, Recovery, Modeling, Modeling, Control, Control, PID. Keywords: Waste Heat Recovery, Modeling, Control, PID.
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION The idea of recovering heat and utilize it as another 1. waste INTRODUCTION The of recovering waste heat it as another form idea of energy is not new. Most ofand the utilize actual power plants The idea of recovering waste heat and utilize it as another The idea of recovering waste heat and utilize it as another form of energy is not new. Most of the actual power use the principle of new. co or even tri-generation. Inplants the form of energy is not Most of the actual power plants form the of energy is not new. Most of the actualofpower plants use principle of co or even tri-generation. In the automotive industry, the most applied form waste heat use the principle of co or even tri-generation. In the use the system principle ofthe comost or even tri-generation. Inwaste the automotive industry, applied form of waste heat recovery is the turbocharger which converts the automotive industry, the most applied form of waste heat automotive industry, the most applied form of waste heat recovery the turbocharger which converts the waste heat into system aeraulicis energy by compressing the fresh air via an recovery system is the which converts the waste recovery system isenergy the turbocharger turbocharger which converts the via waste heat into aeraulic by compressing the fresh air an exhaust driven turbine. Driven by future emissions heat into energy by compressing the fresh air via an heat into aeraulic aeraulic energy byinDriven compressing the fresh emissions air viaheat an exhaust driven turbine. by future legislations and increase fuel prices, engine gas exhaust driven turbine. Driven by future emissions exhaust driven turbine. Driven by future emissions legislations and increase in fuel prices, engine recovering has recently attracted a lot of interest. Ingas the heat past legislations and increase in fuel prices, engine gas heat legislations and increase in studies fuela lot prices, enginethe gas heat recovering recently attracted of interest. In the past few years, ahas high number of have shown interest recovering has recently attracted a lot of interest. In the past recovering has recently attracted a lot of interest. In the past few years, a high number of studies have shown the interest of energy recovery Rankine based systems for heavy duty few years, a high number of studies have shown the interest few years, a high number of studies have shown the interest of energy recovery Rankine based systems for heavy duty trucks engine compounding [Sprouse et al. (2013), Espinosa of energy recovery Rankine based systems for duty of energy recovery Rankine based for heavy heavy duty trucks engine compounding et (2013), Espinosa (2011)]. Recent studies have[Sprouse broughtsystems a al. significant potential trucks engine compounding [Sprouse et al. (2013), Espinosa trucks engine compounding [Sprouse et al. (2013), Espinosa (2011)]. have brought aa significant potential for such Recent a systemstudies in a Heavy Duty (HD) vehicle which can (2011)]. Recent studies have brought potential (2011)]. Recent studies haveconsumption brought a significant significant potential for such a system in a Heavy Duty (HD) vehicle which can lead to a decrease in fuel of about 5% and for such aa system in Duty (HD) vehicle which can for such system ininaa Heavy Heavy Duty (HD) vehicle which can lead to a decrease fuel consumption of about 5% and reduce engine emissions. Yet many challenges have to be lead to a decrease in fuel consumption of about 5% and lead to a decrease in fuel consumption of about 5% and reduce engine emissions. Yet many challenges have to be faced before the vehicle integration. The first challenge deals reduce engine emissions. Yet many challenges have be reducethe engine emissions. manyand challenges have to to be faced before the vehicle The first challenge deals with correct choiceintegration. ofYetfluids system architecture faced before the vehicle integration. The first challenge deals faced before the vehicle integration. The first challenge deals with the correct choice of fluids and system architecture [Magotheet correct al. (2007), Grelet et al. and (2014)] and architecture shows that with choice of system with correct choice of fluids fluids architecture [Mago et al. (2007), et al. (2014)] and showswork. that systemthesimulation is a Grelet critical part ofand the system development [Mago et al. (2007), Grelet et al. (2014)] and that [Mago etof al. (2007), Grelet etpart al. (2014)] and shows shows that system simulation is a critical of the development work. The use water-alcohol mixture can bring some advantages system simulation is a critical part of the development work. system simulation is a critical part of the development work. The of water-alcohol can bring some advantages in theuse power recuperationmixture and overcome both disadvantages The use of water-alcohol mixture can some advantages The usepower offluids: water-alcohol mixture can bring bring some advantages in the recuperation and overcome both disadvantages of these high freezing temperature of water and in the power recuperation and overcome both disadvantages in the power recuperation and overcome both disadvantages of these fluids: high freezing temperature of water and flammability of alcohol [Latz et al. (2012)]. In those blends, of these fluids: high freezing temperature of water and of theseEthanol fluids: high freezing temperature ofthose water and flammability of alcohol [Latz et al. (2012)]. In blends, Water is quite promising and is compliant with flammability of alcohol [Latz et al. (2012)]. In those blends, flammability of alcohol [Latz et al. (2012)]. In those blends, Water Ethanol is quite promising and is compliant with vehicle Ethanol integration where both pure fluids are not.with In Water is quite promising and is Water Ethanol quite promising and is compliant compliant with vehicle integration where both fluids not. In comparison withis stationary plant,pure where theare system is vehicle integration where both pure fluids are not. In vehicle integration where both pure fluids are not. In comparison with stationary plant, where the system is designed to run at stationary its nominal plant, point, where the vehicle integration comparison with the system is comparison with stationary plant, where the integration system is designed to run at its nominal point, the vehicle has to face a second challenge such as the limited cooling designed to run at its nominal point, the vehicle integration designed to run at its nominal point, the vehicle integration has to face challenge such as the limited cooling capacity andaa second the highly transient behaviour of the heat has to second challenge such as limited cooling has to face face adeal second challenge suchbehaviour as the the limited cooling capacity and the highly transient of the heat sources. To with that, an effective control strategy is capacity and the highly transient behaviour of the heat capacity anddeal thetowith highly transient behaviour ofand theensure heat sources. To that, an effective control strategy is really important maximize power recuperation sources. To that, effective control strategy is sources. To deal dealtowith with that, an an effective control and strategy is really important maximize power recuperation ensure really important to maximize power recuperation and ensure really important to maximize power recuperation and ensure
a safe operation of the system. Although many papers aaconcerning safe operation the system. many papers Rankineof components andAlthough system optimization for safe operation of the system. Although many papers aconcerning safe application operation of the system. Although many Rankine components and system optimization mobile can be found in the literature [Seherpapers et for al. concerning Rankine components and system optimization for concerning Rankineetcan components and system for mobile application found in the literature [Seher al. (2012), Mavridou al.be (2010)], only few ofoptimization them deal et with mobile application can be found in the literature [Seher et al. mobile application can be found in the literature [Seher et al. (2012), Mavridou et al. (2010)], only few of them deal with control development and operating strategy improvement (2012), Mavridou et al. (2010)], only few of them deal with (2012), Mavridou et al.and (2010)], only few of them deal with control development operating strategy improvement [Peralez et al. (2013), Willems et al. (2012)]. One key control development and operating strategy improvement control development operating strategy improvement [Peralez et al. (2013), et al. (2012)]. One variable to control is and theWillems working fluid temperature at key the [Peralez et al. (2013), Willems et al. (2012)]. One key [Peralez et al. (2013), Willems et al. (2012)]. Oneathas key variable to control is the working fluid temperature the evaporator outlet / expander inlet since this temperature a variable to control is the fluid temperature at the variable to outlet control is performance the working working fluid temperature athas the evaporator / expander inlet since this temperature aa big impact on system [Quoilin et al. (2011)]. In evaporator outlet // expander inlet since this temperature has evaporator outlet expander inlet since this as temperature has a big impact on system performance [Quoilin et al. (2011)]. In abig system perspective, it has to be as close possible to the impact on system performance [Quoilin et al. (2011)]. In big impact on system performance [Quoilin et al. (2011)]. In avapor system perspective, it has to be as close as possible to the saturation line to increase the system efficiency. An aa system perspective, it has to as close as to system perspective, it temperature has to be be the aswill close as possible possible to the the vapor saturation line increase system efficiency. An effective control of thisto allow to have longer vapor saturation line to increase the system efficiency. An vapor saturation line to increase the system efficiency. An effective control of this temperature will allow to have longer recovery control period by increasing the time where the expander is effective of this temperature will allow to have longer effective control of thiscriterion temperature will allowthe to by have longer recovery period by increasing the time where expander is fed with vapour. This can be achieved reducing recovery period by increasing the time where the expander is recovery period by increasing the time where the expander is fed with vapour. This criterion can be achieved by reducing the standard deviation to the set point (SP). This paper is fed with vapour. This criterion can be achieved by reducing fed with vapour. This criterion can be achieved by reducing the standard deviation to the set point (SP). This paper is organized as follows. Section 2 presents the principle and the the standard deviation to point (SP). This paper is the standard deviation to the the32set set point the (SP). Thismodelling paper is organized as follows. Section presents principle and the studied system. Section approaches the organized as follows. Section 2 presents the principle and the organized as follows. Section 2 presents the principle and the studied system. Section 3 the methodology and the resulting partial differential equation studied system. Section 3 approaches approaches the modelling modelling studied system. Section approaches modelling methodology and the resulting partial equation (PDE) system. Section 4 3shows thedifferential twotheimplemented methodology and the resulting partial differential equation methodology and the resulting partial differential equation (PDE) system. Section 4 shows the two implemented controllers when section 5 compares their performances. (PDE) system. Section 4 shows the two implemented (PDE) system. Section5 compares 4 shows their the performances. two implemented controllers when section controllers section their PRINCIPLES AND STUDIED SYSTEM controllers2.when when section 5 5 compares compares their performances. performances. 2. PRINCIPLES AND STUDIED SYSTEM 2. STUDIED SYSTEM All the variables used in the AND following are explicitly defined 2. PRINCIPLES PRINCIPLES AND STUDIED SYSTEM All the variables used in the following are explicitly defined in tables 2, 3 and 4 of the appendix. All the used in the All the variables variables used in appendix. the following following are are explicitly explicitly defined defined in tables 2, 3 and 4 of the in 3 of 2.1. 2, Rankine in tables tables 2, 3 and and 4 4Process of the the appendix. appendix. 2.1. Rankine Process 2.1. Rankine cycle is Process a widely used power generation cycle to 2.1. Rankine Rankine Process Rankine cycle is aa widely used power generation cycle to turn heat into mechanical or electrical power. First the Rankine cycle is widely used power generation cycle to Rankine cycle is apumped widely from used power cyclethe to turn heat into or electrical First working fluid ismechanical a tank generation atpower. the condensing turn heat into mechanical or electrical power. First the turn heat into mechanical or electrical power. First the working fluid is pumped from a tank at the condensing pressure to the is evaporator atfrom the evaporating pressure. Then working fluid pumped aa tank condensing working fluid is pumpedfluid from tank at at the the condensing pressure to the evaporator at the pressure. Then the pressurized working is evaporating pre-heated, vaporized and pressure to the evaporator at the evaporating pressure. Then pressure to the evaporator at the evaporating pressure. Then the pressurized working fluid is pre-heated, vaporized and superheated in one or several heat exchangers (HEX), also the pressurized working fluid is pre-heated, vaporized and the pressurized working fluid heat is pre-heated, vaporized and superheated in one or several exchangers (HEX), also known as boilers. These HEX are linked to the heat source. superheated in one or several heat exchangers (HEX), also superheated in one or several heat exchangers (HEX), also known as boilers.vapor These HEX are linked to the heat source. The superheated expands from evaporating pressure to known as These HEX linked to source. known as boilers. boilers.vapor Theseexpands HEX are are linked to the the heat heat source. The superheated from evaporating pressure to The The superheated superheated vapor vapor expands expands from from evaporating evaporating pressure pressure to to
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condensing pressure in an expansion device converting the pressure and enthalpy drop into mechanical work. Finally the expanded vapour condenses through a condenser releasing heat into the heat sink (e.g. ambient air) and returns to the working fluid reservoir. In this process the changes of states in both the pump and the expander are irreversible and increase the fluid entropy to a certain extent.
outlet of the boiler, in order not to disturb the emissions control strategy and the internal combustion process. 3.1. Model Assumptions Several assumptions have been done to simplify the problem in a great extent. They are usually admitted when coming to heat exchanger modelling [Feru et al. (2013), Vaja (2009)]: The transfer fluid is always considered in single phase i.e. no condensation in the EGR/exhaust gases is taken into account. The conductive heat fluxes are neglected since the predominant phenomenon is the convection. The pressure drops on each fluids side (transfer and working fluids) are not considered. Both boilers are represented by a straight pipe in pipe counterflow heat exchanger, similarly to Vaja (2009), divided into n lumped sub-volumes in the longitudinal direction. Fluid properties are evaluated at the outlet of each sub-volume i.e. linear profile is considered between inlet and outlet of each node. Pressure dynamics is neglected since its time scale is very small considered to the HEX time scale. Working fluid mass flow rate is supposed constant along the heat exchanger.
2.2. Studied System The Waste Heat Recovery System (WHRS) is compounded on a turbocharged 6 cylinder 11L 320kW HD engine using exhaust gas recirculation (EGR) and a selective catalyst reduction system (SCR) to reduce the NOx emissions. The studied Rankine cycle is recovering heat from both EGR and Exhaust applying a serial configuration of two heat exchangers. Figure 1 shows a schematic of the studied system. The mass flow rate through the two boilers is controlled by the pump speed. The expansion machine is a turbine, which has a higher power density than volumetric expanders [Seher et al. (2012), Lemort et al. (2013)]. The working fluid is then condensed through an indirect condenser fed by coolant. Moreover the cycle is equipped with two bypass valves one located in the exhaust stream to control the amount of energy introduced in the system and a second in front of the expansion device to prevent liquid to enter in the turbine and avoid blade erosion caused by liquid droplets into a high speed rotor.
DPF
3.2. Governing Equations Since the mass is assumed constant the continuity equation is neglected and the model is only based on the energy conservation for the working fluid (1), the gas (2) and the energy balance at the internal (3) and external wall (4). 𝜕𝜕𝑚𝑚̇𝑤𝑤𝑤𝑤 ℎ𝑤𝑤𝑤𝑤 𝜕𝜕ℎ𝑤𝑤𝑤𝑤 (1) − Q̇ =ρ V .
SCR
conv wf int 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ∂T − Q̇ conv g int − Q̇ conv g ext = ρ𝑔𝑔 V𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 ) 𝑔𝑔 . 𝜕𝜕𝜕𝜕 ∂t ∂Twall int Q̇ conv g int + Q̇ conv wf int = ρwall Vwall int cp wall . ∂t ∂Twall ext Q̇ conv g ext + Q̇ conv amb ext = ρwall Vwall ext cp wall , ∂t ̇ with Q conv j k = 𝛼𝛼𝑗𝑗 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑗𝑗 𝑘𝑘 (𝑇𝑇𝑗𝑗 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑘𝑘 ),
𝜕𝜕ṁ𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 )T𝑔𝑔
WHR fluid Charge Air
(2) (3) (4)
(5) where j = g, wf, amb and k = int, ext. Furthermore, to complete the system we need boundary and initial conditions. Time-dependent boundary conditions are used in z=0 and z=L:
Coolant Exhaust gas
Fig 1 Studied system The chosen working fluid is a predefined mixture of Ethanol and Water which gives the best compromise concerning performance and reduced flammability. 3.
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𝑚𝑚̇wf (𝑡𝑡, 0) = 𝑚𝑚̇wf 0 (𝑡𝑡), 𝑚𝑚̇g (𝑡𝑡, 𝐿𝐿) = 𝑚𝑚̇g L (𝑡𝑡),
ℎ𝑤𝑤𝑤𝑤 (𝑡𝑡, 0) = ℎ𝑤𝑤𝑤𝑤 0 (𝑡𝑡), 𝑇𝑇𝑔𝑔 (𝑡𝑡, 𝐿𝐿) = 𝑇𝑇𝑔𝑔 𝐿𝐿 (𝑡𝑡).
𝑇𝑇𝑔𝑔 (0, 𝑧𝑧) = 𝑇𝑇𝑔𝑔 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧), ℎ𝑤𝑤𝑤𝑤 (0, 𝑧𝑧) = ℎ𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧).
𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 (0, 𝑧𝑧) = 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧), 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 (0, 𝑧𝑧) = 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧),
The initial conditions for the gas and wall temperatures and working fluid enthalpy are given by:
HEAT EXCHANGER CONTROL ORIENTED MODELING
The manipulated variable (MV) is the working fluid mass flow 𝑚𝑚̇wf 0 (𝑡𝑡), whereas the controlled variable (CV) is the working fluid enthalpy ℎ𝑤𝑤𝑤𝑤 (𝑡𝑡, 𝐿𝐿). However the enthalpy is impossible to measure directly we have to use pressure and temperature measurement to compute it.
The system dynamic is controlled by the HEX behaviour (i.e. evaporators and condenser) and models of these components are developed to dynamically predict temperature and enthalpy of transfer and working fluid at the outlet of each boiler. This is critical when coming to control design to ensure a safe operation and a proper operation of the EGR function. Safe operation means that the fluid is completely vaporized when entering the turbine in order not to destroy it. For the EGR, the aim is to have low gas temperature at the
3.3. Heat Transfer To model the convection from the transfer fluid to the pipe walls and from the internal pipe to the working fluid a heat transfer coefficient (α) is needed. The convection from a 570
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boundary to a moving fluid is usually represented by the dimensionless Nusselt number (Nu) which is the ratio of convective to conductive heat transfer. 𝛼𝛼 ∗ 𝑙𝑙 (6) 𝑁𝑁𝑁𝑁 = ,
In single phase (liq and vap) the coefficient a. b. and c. are evaluated thanks to third order polynomial function of pressure similar to: 3 2 𝑎𝑎𝑑𝑑 = 𝑎𝑎𝑑𝑑3 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑎𝑎𝑑𝑑2 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑎𝑎𝑑𝑑1 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑎𝑎𝑑𝑑0 . (14) In two-phase region ad 2φ and 𝑏𝑏d 2φ are approximated with: 1 (15a) a = ,
𝜆𝜆
where l here represents a characteristic length and is in this case the hydraulic diameter. Numerous correlations to approach this number can be found in the literature and are usually derived from experiments, see for example Thome (2010). Those correlations depend on the flow regime, the number of phases and the geometry studied. In single phase, the following correlation is implemented: 𝑁𝑁𝑁𝑁 = 𝐴𝐴 𝑅𝑅𝑅𝑅 𝑛𝑛 𝑃𝑃𝑃𝑃 𝑚𝑚 , (7) where A is a constant Re and Pr are dimensionless number (respectively Reynolds and Prandtl number). By developing these two numbers (7) becomes: 4𝑚𝑚̇ 𝑛𝑛 𝑐𝑐𝑝𝑝 𝜇𝜇 𝑚𝑚 (8) 𝑁𝑁𝑁𝑁 = 𝐴𝐴 ( ) ( ) . 𝜋𝜋𝑑𝑑ℎ 𝜇𝜇
𝑏𝑏𝑑𝑑 2𝜑𝜑
3.5. Discretization To implement such a model, we now have to discretize the continuous heat exchanger model with respect to space based on finite differences method. The HEX is split into n longitudinal cell where a backward Euler scheme in space is applied. The system of equations defining the response of the ith cell for transfer fluid, pipe and working fluid can be represented under the following formalism. 𝑥𝑥̇ = 𝑓𝑓(𝑥𝑥, 𝑢𝑢), (16) where 𝑢𝑢 = [𝑚𝑚̇𝑤𝑤𝑤𝑤 0 𝑃𝑃𝑤𝑤𝑤𝑤 0 ℎ𝑤𝑤𝑤𝑤 0 𝑚𝑚̇𝑔𝑔 𝐿𝐿 𝑇𝑇𝑔𝑔 𝐿𝐿 ]. (17) The vector u contains the only MV (𝑚𝑚̇𝑤𝑤𝑤𝑤 0 ) and four inputs disturbances: inlet gas mass flow and temperature (𝑚𝑚̇𝑔𝑔 𝐿𝐿 ) and (𝑇𝑇𝑔𝑔 𝐿𝐿 ) and inlet working fluid pressure and enthalpy (𝑃𝑃𝑤𝑤𝑤𝑤 0 ) and (ℎ𝑤𝑤𝑤𝑤 0 ). And xi = [hwf i Twall int i Tg i Twall ext i ], (18a) 𝑓𝑓𝑖𝑖 (𝑥𝑥, 𝑢𝑢) = (18b)
𝜆𝜆
𝑤𝑤𝑤𝑤 𝑙𝑙𝑙𝑙𝑙𝑙
𝑞𝑞0,01 [
𝛼𝛼𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣𝑣𝑣 𝛼𝛼𝑤𝑤𝑤𝑤 𝑙𝑙𝑙𝑙𝑙𝑙
(1 + 8(1 − 𝑞𝑞)0,7
𝜌𝜌𝑤𝑤𝑤𝑤 𝑙𝑙𝑙𝑙𝑙𝑙 0,67
𝜌𝜌𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣𝑣𝑣
)] }
.
(𝑚𝑚̇𝑤𝑤𝑤𝑤 (ℎ𝑤𝑤𝑤𝑤 𝑖𝑖−1 −ℎ𝑤𝑤𝑤𝑤 𝑖𝑖 )−𝛼𝛼𝑤𝑤𝑤𝑤𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑤𝑤𝑤𝑤 (𝑇𝑇𝑤𝑤𝑤𝑤 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 ))
This correlation corresponds to a tube arrangement and is practical since it creates continuity between single and two phase heat transfer coefficients and does not need transport properties such as viscosity or heat conductivity.
𝜌𝜌𝑤𝑤𝑤𝑤 𝑖𝑖 𝑉𝑉𝑤𝑤𝑤𝑤
𝛼𝛼𝑤𝑤𝑤𝑤 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑤𝑤𝑤𝑤 (𝑇𝑇𝑤𝑤𝑤𝑤 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 )+𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑖𝑖𝑖𝑖𝑖𝑖 (𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 ) (
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑤𝑤𝑤𝑤
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
𝑤𝑤𝑤𝑤
𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙
𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤
𝑖𝑖𝑖𝑖 ℎ𝑤𝑤𝑤𝑤 ≥ ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣
𝜌𝜌𝑤𝑤𝑤𝑤 =
𝑎𝑎𝑑𝑑 2𝜑𝜑 ℎ𝑤𝑤𝑤𝑤 + 𝑏𝑏𝑑𝑑 2𝜑𝜑 {𝑎𝑎𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ℎ𝑤𝑤𝑤𝑤 ² + 𝑏𝑏𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ℎ𝑤𝑤𝑤𝑤 + 𝑐𝑐𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝑤𝑤𝑤𝑤
𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 },
. ]
CONTROL DEVELOPMENT
4.1. PI Controller The main problem when considering PI or PID control structure is that identification for heat exchangers is very sensitive to the inputs considered [Peralez et al. (2013), Horst et al. (2013)]. This is well described when considering
𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙
𝑖𝑖𝑖𝑖 ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 < ℎ𝑤𝑤𝑤𝑤 < ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 . 𝑖𝑖𝑖𝑖 ℎ𝑤𝑤𝑤𝑤 ≥ ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣
)
In this paper we focus on the temperature control with only regards to power generation. Our goal here is to improve the temperature stability around a given set point. For that we can manipulate the working fluid mass flow to regulate the working fluid enthalpy (or temperature) at the outlet of the exhaust boiler. Due to time issue, at the moment where this paper is written, no experimental tests of the control system are possible. The validation is based on the representative simulation model. The comparison is done on the tracking error either on a step load change or on a complete driving cycle. On each case, we start from the same equilibrium.
ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 − ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝜌𝜌𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑐𝑐𝑐𝑐𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑉𝑉 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒
4.
The saturation temperature (Tsat) is approximated with the Wagner equation with adapted coefficient for liquid and vapor saturation [Kleiber and Joh (2010)] and allows to make the transition between each phase. Density model: 𝑎𝑎 ℎ ² + 𝑏𝑏 ℎ + 𝑐𝑐 𝑖𝑖𝑖𝑖 ℎ ≤ ℎ (13) 1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑤𝑤𝑤𝑤
𝜌𝜌𝑔𝑔 𝑉𝑉𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 )
[
where a. b. and c. are first order polynomial expressions function of pressure and q is the fluid quality defined as the quantity of vapor present in the two phase flow. ℎ𝑤𝑤𝑤𝑤 − ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 (12) 𝑞𝑞 = .
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑤𝑤𝑤𝑤
−𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑒𝑒𝑒𝑒𝑒𝑒(𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖 )
𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑎𝑎𝑎𝑎𝑎𝑎 (𝑇𝑇𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖 )+𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑒𝑒𝑒𝑒𝑒𝑒(𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖 )
Similarly to Kleiber and Joh (2010), the working fluid properties are approximated using mathematical description. This allows not to rely on thermochemical database such as Lemmon et al. (2011) and creates continuity in derivative terms during the single / two phase transition. The fluid properties are only function of pressure and enthalpy. Temperature model: 𝑎𝑎 ℎ ² + 𝑏𝑏 ℎ + 𝑐𝑐 𝑖𝑖𝑖𝑖 ℎ ≤ ℎ (11) 𝑇𝑇 + 𝑞𝑞(𝑇𝑇 − 𝑇𝑇 ) 𝑖𝑖𝑖𝑖 ℎ <ℎ <ℎ 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑤𝑤𝑤𝑤
𝜌𝜌𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑐𝑐𝑐𝑐𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑉𝑉 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖
𝑚𝑚̇𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 )(𝑇𝑇𝑔𝑔 𝑖𝑖−1 −𝑇𝑇𝑔𝑔 𝑖𝑖 )−𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑖𝑖𝑖𝑖𝑖𝑖 (𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 )
3.4. Working Fluid Properties
𝑇𝑇𝑤𝑤𝑤𝑤 = { 𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ℎ𝑤𝑤𝑤𝑤 ² + 𝑏𝑏𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ℎ𝑤𝑤𝑤𝑤 + 𝑐𝑐𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
(15b)
All coefficients are evaluated by fitting routines in Matlab using REFPROP as thermodynamic properties database [Lemmon et al. (2011)].
Assuming the viscosity (µ) the specific heat (cp), the heat conductivity (λ) and the characteristic length (l) constant in single phase region for the working fluid and the gas we get: 𝛼𝛼𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑟𝑟𝑟𝑟𝑟𝑟𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑛𝑛𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 , (9a) 𝛼𝛼𝑣𝑣𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑟𝑟𝑟𝑟𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤 𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑛𝑛𝑣𝑣𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤 , (9b) 𝛼𝛼𝑔𝑔 = 𝛼𝛼𝑟𝑟𝑟𝑟𝑟𝑟𝑔𝑔 𝑚𝑚̇𝑔𝑔 𝑛𝑛𝑔𝑔 , (9c) Where the constant αref . and the exponent n. have to be identified in liquid and vapor region for the working fluid and in single phase for the gas. In the two phase region, a more complex correlation (10) is used to enhance the single phase heat transfer coefficient [Kleiber and Joh (2010)]. 0,37 −2,2 𝜌𝜌 (10) 𝛼𝛼2𝜑𝜑 𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 {(1 − 𝑞𝑞)0,01 [(1 − 𝑞𝑞) + 1,2𝑞𝑞0,4 ] + 𝜌𝜌𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣𝑣𝑣 −2 −0,5
ad 2φ1 𝑃𝑃𝑤𝑤𝑤𝑤 + ad 2φ 0 1 = . 𝑏𝑏𝑑𝑑 2𝜑𝜑1 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑏𝑏𝑑𝑑 2𝜑𝜑 0
d 2φ
}
571
IFAC ADCHEM 2015 June 7-10, 2015. Whistler, BC, Canada
V. Grelet et al. / IFAC-PapersOnLine 48-8 (2015) 568–573
several operating points. Most of the results on PID tuning are reducing the process to a first order plus time delay (FOPTD) transfer function [Skogestad (2003), Madhauranthakam et al. (2009)]. The dynamic relation between the variations of WF mass flow and the temperature is: Δ𝑇𝑇𝑤𝑤𝑤𝑤 𝐿𝐿 (𝑠𝑠) 𝐺𝐺 (19) = 𝑒𝑒 −𝐷𝐷𝐷𝐷 . Δ𝑚𝑚̇𝑤𝑤𝑤𝑤 0 (𝑠𝑠)
manipulated variable is here preferred to take into account the thermal inertia of the system [Rasmussen and Alleyne (2010)]. Comparison on a step load change for several PID tuning method is shown in table 1. On two criteria: the integrated absolute error (IAE) and the total variation (TV) which are defined as: ∞ (21) IAE = ∫ |𝐶𝐶𝐶𝐶(𝑡𝑡) − 𝑆𝑆𝑆𝑆(𝑡𝑡)|𝑑𝑑𝑑𝑑.
1 + 𝜏𝜏𝜏𝜏
0
𝑇𝑇𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 +𝑇𝑇𝑒𝑒𝑒𝑒ℎ 𝑖𝑖𝑖𝑖 2
TWF L
20
.
0 FOPTD Reference Model R² = 93.8929 % 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s)
TWF L
20
(22)
FOPTD Identication on high load operating point
Table 1 PID tuning method comparison
0 FOPTD Reference Model R² = 99.9591 % 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s)
2
Fig 2 FOPTD model identification validation 0 x 10
Controlled Variable Error vs Time
0
Error to SP(K)
-20 0
G (K/kg/s)
0
𝑑𝑑𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑑𝑑. 𝑑𝑑𝑑𝑑
The method proposed by Madhuranthakam et al. (2008) is minimizing the IAE with the highest TV, i.e. the control effort is more important but still acceptable. The IMC and Tyreus-Luyben tuning methods seem to be conservative and too slow since they present the highest IAE. Figure 4 shows the tracking error versus time for the five tuning methods. Tuning Method IAE TV Ziegler Nichols in Closed Loop 84.6325 0.0065 Tyreus-Luyben 743.1427 0.0053 [Skogestad (2003)] IMC 940.3763 0.0064 SIMC 208.5099 0.0047 [Skogestad (2003)] Madhuranthakam et al. 48.6754 0.0204 [Madhuranthakam et al. (2008)]
FOPTD Identication on low load operating point
-20 0
∞
𝑇𝑇𝑇𝑇 = ∫
For the identification a pseudo random binary sequence (PRBS) is the same for all operating points. As it can be seen in figure 2, this model structure is confirmed by comparing results of the reference model and an identified FOPTD around an operating point since on the worst point the agreement is around 94%. Figure 3 shows the FOPTD parameters for several inputs disturbance. For convenience they are represented versus the total heat flow rate entering: ∗ 𝑄𝑄̇𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇 ) (𝑚𝑚̇𝑒𝑒𝑒𝑒𝑒𝑒 (𝑇𝑇𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤 0) + 𝑚𝑚̇𝑒𝑒𝑒𝑒ℎ (𝑇𝑇𝑒𝑒𝑒𝑒ℎ 𝑖𝑖𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠 )). (20) where 𝑇𝑇 ∗ =
571
4
G
-2
-2 -4 -6 -8
-4
-10 0
500
100
200
300 Time (s)
(s)
400
ZN CL TL IMC SIMC Mad 500 600
Fig 4 Tracking error comparison 0
D (s)
40
D
The PID tuning of Madhuranthakam et al. (2008) is now used.
180
4.2. Nonlinear model inversion
20 0 20
40
60
80 100 120 Heat Flow Rate (kW)
140
160
Despite of their very good agreement, the FOPTD could not be inversed to be used as controller since they do not take account of the disturbances and therefore every correction action would only be done through a feedback on the manipulated variable. In order to improve the performance of the classical “PID” controller we use the inverse of the system model (16-18) as feedforward to compute the first part of the MV corresponding to the desired set point (Fig. 5). Details of nonlinear inversion are approached and detailed in [Peralez et al. (2013)]. To simplify the model inversion, the heat transfer coefficient for the WF is assumed constant in liquid and vapor phase. 𝑛𝑛 (23) (𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚 )
Fig 3 FOPTD model parameter FOPTD parameters are evaluated on twenty different engine operating points, which correspond to the input disturbances, representing the overall engine map. As it can be seen below, the FOPTD parameters change a lot, according to the nonlinearity of the model developed in section 3. To improve the system control several PID tuning methods are compared to reduce both the tracking error and the control effort [Skogestad (2003), Madhuranthakam et al. (2008)]. Moreover, to obtain better results on a realistic driving cycle, gains are scheduled as a function of the working fluid mass flow rate entering in the system. The scheduling of each gain is realized by interpolating Kp, Ti and Td according to the working fluid mass flow feedback signal. A normal scheduling would be to select the appropriate gain in function of the operating point but schedule it as a function of the
𝛼𝛼𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑤𝑤𝑤𝑤 𝑟𝑟𝑟𝑟𝑟𝑟
2
,
with maximum and minimum mass flow set as the mass flow corresponding to maximum and minimum WF pump speed: N (24) ṁ =ρ Cc . 𝑤𝑤𝑤𝑤 0
572
wf 0
60
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
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Controlled Variable vs Time
Set Point U PID
External Conditions
Rankine System
SP CV InvHEXModel CV PID
220 200 180 160 0
500
1000
1500
2000
Time (s) Manipulated Variable vs Time
disturbances 0.04
outputs
mWF(kg/s)
Inverse HEX Ufeedforward Model
TWFoutExhBoiler (°C)
As the WF mass flow is considered homogenous in the entire heat exchanger it can be computed from the system (16).
MV InvHEXModel MV PID
0.03 0.02
.
Ufeedback
2500
0.01 0 0
Fig 5 New control structure proposed 5.
500
1000
1500
Time (s)
2000
2500
Fig 7 Controllers performance comparison
COMPARISON OF CONTROLLERS PERFORMANCES
30
TWFoutExhBoiler Error(°C)
20
This two proposed control strategies are evaluated on a validated Rankine model (at the time of writing the experimental set up was not available). Gases data (exhaust and EGR) are coming from an 11L long haul truck on a highway driving cycle (Fig. 6). For the other perturbations, namely called hwf 0 and Pwf 0 in (19), the EGR boiler inlet WF enthalpy is kept constant whereas the inlet WF pressure depends on the expansion turbine inlet conditions. Sensors and actuators dynamics are represented by first order models where time constant are fitted thanks to manufacturer data. As previously said, the controlled variable in the model is the WF enthalpy but as this one is not physically measurable and for convenience we track the WF temperature at the second boiler outlet. In order to well assess the performance, the set point changes during the simulation. Temperature (°C)
6.
Mass Flow (kg/s)
0.4 0.3
1500 2000 Time (s) Gases Mass Flow vs Time
500
1000
2500
3000
2500
3000
Egr Exhaust
0.2 0.1 0 0
1500 Time (s)
2000
1000
1500
Time (s)
2000
CONCLUSION
This paper reports control strategy development for waste heat recovery Rankine based system used on heavy duty engines. Comparison of both controllers shows a better accuracy for the inverse HEX model when comparing the tracking error. The objective to stabilize as close as possible the temperature around a given set point is achieved by using this strategy. We conclude that the nonlinear controller leads to the best performances, at the cost of developing an accurate first principle nonlinear model with all known parameters, which underline the problem of identification. Yet it still needs to be validated on test bench level to see the robustness with respect to modelling errors and parameters mismatch. However the model is compliant with classical control unit used in the automotive industry since the computational needs is low. Future work will focus on experimental validation and optimal control strategy.
Egr Exhaust 1000
500
InvHEXModel PID 2500
Fig 8 Tracking error comparison
300
500
-10
-30 0
400
200 0
0
-20
Gases Temperature vs Time
500
10
Fig 6 Input disturbances Figure 7 and 8 show comparison of the two implemented control strategies: the PID based on Madhuranthakam et al. (2008) and the nonlinear control presented in 4.2. Except at the beginning, where the PID is not handling the initial SP variation and the large change in disturbances appearing around 30s, it keeps the error within +/-10K, which could be acceptable when considering a volumetric expander, which are less sensitive to liquid droplets. Here a kinetic expansion machine is used and the temperature has to be as close as possible to the set point, which is better achieved by the nonlinear controller: the error remains in +/- 3K. The control efforts seem similar but are slightly different (PID presents a delay compared to nonlinear controller).
ACKNOWLEDGMENT This PhD thesis is collaboration between UCBL1, ULg and Volvo Trucks which is gratefully acknowledged for the funding. The French ministry of higher education and research for the financial support of the CIFRE PhD thesis 2012/549 is also acknowledged.
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REFERENCE
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Strategy of Waste Heat Recovery Organic rankine Cycles”. Applied Energy, vol 88, 2183–2190. Rasmussen, B.P. and Alleyne, A.G. (2010). Gain Scheduled Control of an Air Conditioning System Using the Youla Parameterization. IEEE Transactions on Control Systems Technology, 18(5), 1216–1225. Seher, D. Lengenfelder, T. Gerhardt, J. Eisenmenger, N. Hackner, M. Krinn, I. (2012). ”Waste Heat recovery for Commercial Vehicles with a Rankine Process”. 21st Aachen Colloquium Automobile and Engine Technology. Skogestad, S. (2003). “Simple analityc rules for model reduction and PID controller tuning”. Journal of Process and Control, vol 13, 291-309. Sprouse, C. Depcik, C. (2013). “Review of Organic Rankine Cycles for Internal Combustion Engine Exhaust Waste Heat Recovery”. Applied Thermal Engineering, vol 51, 711-722. Thome, J.R. (2010). ”Wolverine Tube Inc Engineering Data Book III”. Web based book available at www.wlv.com/heat-transfer-databook/ Vaja, I. (2009). “Definition of an object oriented library for the dynamic simulation of advanced energy systems: Methodologies, Tools and Applications to Combined ICE&ORC Power Plants”. PhD Thesis University of Parma
Espinosa, N. (2011). “Contribution to the Study of Waste Heat Recovery Systems on Commercial Truck Diesel Engines”. PhD Thesis University of Liege and National Polytechnic Institute of Lorraine. Feru, E. Kupper, F. Rojer, C. Seykens, X. Scappin, F. Willems, F. Smits, J. De Jager, B. Steinbuch, M. (2013). ”Experimental Validation of a Dynamic Waste Heat Recovery System Model for Control Purposes”. SAE Technical Paper, 2013-01-1647. Grelet, V. Reiche, T. Guillaume, L. Lemort, V. (2014). “Optimal WHR Rankine based for Heavy Duty Application”. FISITA Conference, NL, F2014-CET-151 Horst, T. Rottengruber, H. Seifert, M. Ringler, J.(2013) ”Dynamic Heat Exchanger Model for Performance Prediction and Control System Design of Automotive Waste Heat recovery Systems”. Applied Energy, Vol 105, 293-303. Kleiber, M. Joh, R. (2010). VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen. VDI Heat Atlas Second Edition. Part D Thermophysical properties. Springer Berlin Germany. Latz, G. Andersson, S. Munch, K. (2012). ”Comparison of Working Fluids in Both Subcritical and Supercritical Rankine Cycles for Waste Heat Recovert Systems in Heavy Duty Vehicles”. SAE Technical Papers, 2012-011200. Lemmon, E.W. Huber, M.L. McLinden, M.O. (2011). REFPROP Nist Standard Reference Database 23 (Version 9.0), Thermophysical Properties Division, National Institute of Standards and Technology, Boulder, CO. Lemort, V. Guillaume, L. Legros, A. Declay, S. Quoilin, S. (2013). “A Comparison of Piston, Screw and Scroll Expanders for Small Scale Rankine Cycle Systems”. Proceedings of the 3rd International Conference on Microgeneration and Related Technologies. Madhauranthakam, C.R. Elkamel, A. Budman, H. (2008). “Optimal tuning of PID controllers for FOTPD, SOPTD and SOPTD with lead processes”. Chemical Engineering and Processing, vol 47, 251-264. Mago, P. Chamra, L. Somayajii. (2007). “Performance Analysis of Different Working Fluids for Use in Organic Rankine Cycles”. Journal of Power and Energy, vol 221, 255-263. Mavridou, S. Mavropoulos, G. Bouris, D. Hountalas, D. Bergeles, G. (2010). “Comparative Design Study of a Diesel Exhaust Gas Heat Exchanger for Truck Applications with Conventional and State of the Art Heat Transfer Enhancements”. Applied Thermal Engineering, vol 30, 935-947 Peralez, J. Tona, P. Lepreux, O. Sciaretta, A. Voise, L. Dufour, P. Nadri, M. (2013). “Improving the Control Performance of an Organic Rankine Cycle System for Waste Heat Recovery from a Heavy Duty Diesel Engine using a Model Based Approach”. IEEE Conference on Decision and Control (CDC), pp 6830-6836. Quoilin, S. Aumann, R. Grill, A. Schuster, A. Lemort, V. (2011). ”Dynamic Modeling and Optimal Control
APPENDIX A Cc cp D G h 𝑚𝑚̇ N P Q̇ s t T V 𝑊𝑊̇ z
Area Cubic capacity Specific heat Transfer function delay Transfer function Gain Enthalpy Mass flow Speed Pressure Heat flow rate Entropy time Temperature Volume Power Space length
m² m3 J/kg/K s K/kg/s J/kg kg/s rpm Pa W J/kg/K s K / °C m3 W m
Table 2 Latin letters α µ λ ρ τ
Heat Transfer Coefficient Viscosity Heat Conductivity Density Transfer function time constant Table 3 Greek letters
574
W/m²/K Pa.s W/K kg/ m3 s
ScienceDirect IFAC-PapersOnLine 48-8 (2015) 568–573 Modeling and control of Rankine based waste heat recovery systems for heavy duty Modeling and control of Rankine basedtrucks waste heat recovery systems for heavy duty Modeling waste Modeling and and control control of of Rankine Rankine based basedtrucks waste heat heat recovery recovery systems systems for for heavy heavy duty duty trucks 1,2,3 , P. Dufour2*, V. Grelettrucks 1,2,3
2*
2* 2 1 3 1,2,3 2*, ,, P. Dufour V. , T.1,2,3 Reiche ,V. Lemort M. Nadri V. Grelet Grelet 1,2,3 2*, 2 1 2 1 , P. P. Dufour Dufour , 333 V. Grelet 2, T. Reiche 1,V. Lemort M. Nadri M. 2, T. Reiche1,V. Lemort3 1 , T. Reiche ,V. Lemort M. Nadri Nadri Volvo Trucks Global Trucks Technology Advanced Technology and Research, F-69800, Saint Priest, France 111 2 Lyon 1, Villeurbanne, Volvo Trucks Global Trucks Technology Advanced Technology Research, F-69800, Saint Priest, France University of Lyon, F-69622, Lyon, France – University France – CNRS, UMR 5007, LAGEP and Research, F-69800, Saint Priest, France 1Volvo Trucks Global Trucks Technology Advanced Technology and 2 2 3 Volvo Trucks Global Trucks Technology Advanced Technology andB49 Research, F-69800, Saint Priest, France 2 University of Lyon, F-69622, Lyon, France – University Lyon 1, Villeurbanne, France – CNRS, UMR 5007, University of Liège, Campus du Sart Tilman Bât. B4000 Liège, Belgium 2 University of Lyon,3 F-69622, Lyon, France – University Lyon 1, Villeurbanne, France – CNRS, UMR 5007, LAGEP University of*Corresponding Lyon,33 F-69622, Lyon, France – University Lyon 1, Bât. Villeurbanne, France –Belgium CNRS, UMR 5007, LAGEP LAGEP Universityauthor. of Liège, du Sart Tilman B49 B4000 Liège, TelCampus +33472431878. E-mail address: [email protected] of Liège, Campus du Sart Tilman Bât. B49 B4000 Liège, Belgium 3 University Universityauthor. of Liège, Campus du Sart Tilman Bât. B49 B4000 Liège, Belgium *Corresponding Tel +33472431878. E-mail address: [email protected] *Corresponding *Corresponding author. author. Tel Tel +33472431878. +33472431878. E-mail E-mail address: address: [email protected] [email protected]
Abstract: This paper presents a control oriented model development for waste heat recovery Rankine Abstract: This paper presents control oriented model for waste recovery Rankine based control systems in heavyaa duty trucks. Waste heat development recovery systems, suchheat as Rankine cycle, are Abstract: This paper control oriented model development for heat recovery Rankine Abstract: This papertopresents presents a the control oriented model development for waste waste heat recovery Rankine based control systems in heavy duty trucks. Waste heat recovery systems, such as Rankine cycle, are promising solutions improve fuel efficiency of heavy duty engines. Due to the highly transient based control systems in heavy duty trucks. Waste heat recovery systems, such as Rankine cycle, are based control systems in heavy the duty trucks. Waste heat systems, such as the Rankine are promising solutions to improve efficiency of heavy duty engines. to highly transient operating conditions, improving thefuel control strategy of recovery those systems is Due an important stepcycle, to their promising solutions to improve the fuel efficiency of heavy duty engines. Due to the highly transient promising solutions to improve the fuel efficiency of heavy duty engines. Due to the highly transient operating conditions, improving the control strategy those systems is an important to their integration into a vehicle. The system considered here isof recovering heat from both EGR andstep exhaust in a operating conditions, improving the control strategy of those systems is important step to operating conditions, improving the considered control strategy ofrecovering those systems is an an important step to their their integration into a vehicle. The system here is heat from both EGR and exhaust in a serial arrangement and use a mixture of water and ethanol as working fluid. The paper focuses on integration into aa vehicle. The system considered here is recovering heat from both EGR and exhaust in integration into vehicle. The system considered here isstate recovering heat from both EGR and exhaust in aaa serial arrangement and use a mixture of water and ethanol as working fluid. The paper focuses on comparison of a classical PID controller which is the of the art in the automotive industry and serial and aa mixture of water ethanol as working fluid. The paper focuses on serial arrangement arrangement and use use mixture water and and ethanol asthe working fluid. The paper focuses on aaa comparison of aa classical PID controller which is the state of art in the automotive industry and nonlinear model based controller in a of simulation environment. The nonlinear model based controller comparison of classical PID controller which is the state of the art in the automotive industry and comparison of a classical PID the controller which is the state of theThe artof inthe thesystem. automotive industry and aa nonlinear model based controller in aa one simulation environment. nonlinear model based controller shows better performance than PID and ensures safe operation nonlinear model based controller in simulation environment. The nonlinear model based controller nonlinear model based controller in a one simulation environment. The of nonlinear model based controller shows better performance than the PID and ensures safe operation the system. shows performance than the PID and ensures safe operation the Keywords: Waste Heat Recovery, Control, PID. © 2015,better IFAC (International Federation Automatic Control) byof Elsevier Ltd. All rights reserved. shows better performance than the Modeling, PIDofone one and ensures safe Hosting operation of the system. system. Keywords: PID. Keywords: Waste Waste Heat Heat Recovery, Recovery, Modeling, Modeling, Control, Control, PID. Keywords: Waste Heat Recovery, Modeling, Control, PID.
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION The idea of recovering heat and utilize it as another 1. waste INTRODUCTION The of recovering waste heat it as another form idea of energy is not new. Most ofand the utilize actual power plants The idea of recovering waste heat and utilize it as another The idea of recovering waste heat and utilize it as another form of energy is not new. Most of the actual power use the principle of new. co or even tri-generation. Inplants the form of energy is not Most of the actual power plants form the of energy is not new. Most of the actualofpower plants use principle of co or even tri-generation. In the automotive industry, the most applied form waste heat use the principle of co or even tri-generation. In the use the system principle ofthe comost or even tri-generation. Inwaste the automotive industry, applied form of waste heat recovery is the turbocharger which converts the automotive industry, the most applied form of waste heat automotive industry, the most applied form of waste heat recovery the turbocharger which converts the waste heat into system aeraulicis energy by compressing the fresh air via an recovery system is the which converts the waste recovery system isenergy the turbocharger turbocharger which converts the via waste heat into aeraulic by compressing the fresh air an exhaust driven turbine. Driven by future emissions heat into energy by compressing the fresh air via an heat into aeraulic aeraulic energy byinDriven compressing the fresh emissions air viaheat an exhaust driven turbine. by future legislations and increase fuel prices, engine gas exhaust driven turbine. Driven by future emissions exhaust driven turbine. Driven by future emissions legislations and increase in fuel prices, engine recovering has recently attracted a lot of interest. Ingas the heat past legislations and increase in fuel prices, engine gas heat legislations and increase in studies fuela lot prices, enginethe gas heat recovering recently attracted of interest. In the past few years, ahas high number of have shown interest recovering has recently attracted a lot of interest. In the past recovering has recently attracted a lot of interest. In the past few years, a high number of studies have shown the interest of energy recovery Rankine based systems for heavy duty few years, a high number of studies have shown the interest few years, a high number of studies have shown the interest of energy recovery Rankine based systems for heavy duty trucks engine compounding [Sprouse et al. (2013), Espinosa of energy recovery Rankine based systems for duty of energy recovery Rankine based for heavy heavy duty trucks engine compounding et (2013), Espinosa (2011)]. Recent studies have[Sprouse broughtsystems a al. significant potential trucks engine compounding [Sprouse et al. (2013), Espinosa trucks engine compounding [Sprouse et al. (2013), Espinosa (2011)]. have brought aa significant potential for such Recent a systemstudies in a Heavy Duty (HD) vehicle which can (2011)]. Recent studies have brought potential (2011)]. Recent studies haveconsumption brought a significant significant potential for such a system in a Heavy Duty (HD) vehicle which can lead to a decrease in fuel of about 5% and for such aa system in Duty (HD) vehicle which can for such system ininaa Heavy Heavy Duty (HD) vehicle which can lead to a decrease fuel consumption of about 5% and reduce engine emissions. Yet many challenges have to be lead to a decrease in fuel consumption of about 5% and lead to a decrease in fuel consumption of about 5% and reduce engine emissions. Yet many challenges have to be faced before the vehicle integration. The first challenge deals reduce engine emissions. Yet many challenges have be reducethe engine emissions. manyand challenges have to to be faced before the vehicle The first challenge deals with correct choiceintegration. ofYetfluids system architecture faced before the vehicle integration. The first challenge deals faced before the vehicle integration. The first challenge deals with the correct choice of fluids and system architecture [Magotheet correct al. (2007), Grelet et al. and (2014)] and architecture shows that with choice of system with correct choice of fluids fluids architecture [Mago et al. (2007), et al. (2014)] and showswork. that systemthesimulation is a Grelet critical part ofand the system development [Mago et al. (2007), Grelet et al. (2014)] and that [Mago etof al. (2007), Grelet etpart al. (2014)] and shows shows that system simulation is a critical of the development work. The use water-alcohol mixture can bring some advantages system simulation is a critical part of the development work. system simulation is a critical part of the development work. The of water-alcohol can bring some advantages in theuse power recuperationmixture and overcome both disadvantages The use of water-alcohol mixture can some advantages The usepower offluids: water-alcohol mixture can bring bring some advantages in the recuperation and overcome both disadvantages of these high freezing temperature of water and in the power recuperation and overcome both disadvantages in the power recuperation and overcome both disadvantages of these fluids: high freezing temperature of water and flammability of alcohol [Latz et al. (2012)]. In those blends, of these fluids: high freezing temperature of water and of theseEthanol fluids: high freezing temperature ofthose water and flammability of alcohol [Latz et al. (2012)]. In blends, Water is quite promising and is compliant with flammability of alcohol [Latz et al. (2012)]. In those blends, flammability of alcohol [Latz et al. (2012)]. In those blends, Water Ethanol is quite promising and is compliant with vehicle Ethanol integration where both pure fluids are not.with In Water is quite promising and is Water Ethanol quite promising and is compliant compliant with vehicle integration where both fluids not. In comparison withis stationary plant,pure where theare system is vehicle integration where both pure fluids are not. In vehicle integration where both pure fluids are not. In comparison with stationary plant, where the system is designed to run at stationary its nominal plant, point, where the vehicle integration comparison with the system is comparison with stationary plant, where the integration system is designed to run at its nominal point, the vehicle has to face a second challenge such as the limited cooling designed to run at its nominal point, the vehicle integration designed to run at its nominal point, the vehicle integration has to face challenge such as the limited cooling capacity andaa second the highly transient behaviour of the heat has to second challenge such as limited cooling has to face face adeal second challenge suchbehaviour as the the limited cooling capacity and the highly transient of the heat sources. To with that, an effective control strategy is capacity and the highly transient behaviour of the heat capacity anddeal thetowith highly transient behaviour ofand theensure heat sources. To that, an effective control strategy is really important maximize power recuperation sources. To that, effective control strategy is sources. To deal dealtowith with that, an an effective control and strategy is really important maximize power recuperation ensure really important to maximize power recuperation and ensure really important to maximize power recuperation and ensure
a safe operation of the system. Although many papers aaconcerning safe operation the system. many papers Rankineof components andAlthough system optimization for safe operation of the system. Although many papers aconcerning safe application operation of the system. Although many Rankine components and system optimization mobile can be found in the literature [Seherpapers et for al. concerning Rankine components and system optimization for concerning Rankineetcan components and system for mobile application found in the literature [Seher al. (2012), Mavridou al.be (2010)], only few ofoptimization them deal et with mobile application can be found in the literature [Seher et al. mobile application can be found in the literature [Seher et al. (2012), Mavridou et al. (2010)], only few of them deal with control development and operating strategy improvement (2012), Mavridou et al. (2010)], only few of them deal with (2012), Mavridou et al.and (2010)], only few of them deal with control development operating strategy improvement [Peralez et al. (2013), Willems et al. (2012)]. One key control development and operating strategy improvement control development operating strategy improvement [Peralez et al. (2013), et al. (2012)]. One variable to control is and theWillems working fluid temperature at key the [Peralez et al. (2013), Willems et al. (2012)]. One key [Peralez et al. (2013), Willems et al. (2012)]. Oneathas key variable to control is the working fluid temperature the evaporator outlet / expander inlet since this temperature a variable to control is the fluid temperature at the variable to outlet control is performance the working working fluid temperature athas the evaporator / expander inlet since this temperature aa big impact on system [Quoilin et al. (2011)]. In evaporator outlet // expander inlet since this temperature has evaporator outlet expander inlet since this as temperature has a big impact on system performance [Quoilin et al. (2011)]. In abig system perspective, it has to be as close possible to the impact on system performance [Quoilin et al. (2011)]. In big impact on system performance [Quoilin et al. (2011)]. In avapor system perspective, it has to be as close as possible to the saturation line to increase the system efficiency. An aa system perspective, it has to as close as to system perspective, it temperature has to be be the aswill close as possible possible to the the vapor saturation line increase system efficiency. An effective control of thisto allow to have longer vapor saturation line to increase the system efficiency. An vapor saturation line to increase the system efficiency. An effective control of this temperature will allow to have longer recovery control period by increasing the time where the expander is effective of this temperature will allow to have longer effective control of thiscriterion temperature will allowthe to by have longer recovery period by increasing the time where expander is fed with vapour. This can be achieved reducing recovery period by increasing the time where the expander is recovery period by increasing the time where the expander is fed with vapour. This criterion can be achieved by reducing the standard deviation to the set point (SP). This paper is fed with vapour. This criterion can be achieved by reducing fed with vapour. This criterion can be achieved by reducing the standard deviation to the set point (SP). This paper is organized as follows. Section 2 presents the principle and the the standard deviation to point (SP). This paper is the standard deviation to the the32set set point the (SP). Thismodelling paper is organized as follows. Section presents principle and the studied system. Section approaches the organized as follows. Section 2 presents the principle and the organized as follows. Section 2 presents the principle and the studied system. Section 3 the methodology and the resulting partial differential equation studied system. Section 3 approaches approaches the modelling modelling studied system. Section approaches modelling methodology and the resulting partial equation (PDE) system. Section 4 3shows thedifferential twotheimplemented methodology and the resulting partial differential equation methodology and the resulting partial differential equation (PDE) system. Section 4 shows the two implemented controllers when section 5 compares their performances. (PDE) system. Section 4 shows the two implemented (PDE) system. Section5 compares 4 shows their the performances. two implemented controllers when section controllers section their PRINCIPLES AND STUDIED SYSTEM controllers2.when when section 5 5 compares compares their performances. performances. 2. PRINCIPLES AND STUDIED SYSTEM 2. STUDIED SYSTEM All the variables used in the AND following are explicitly defined 2. PRINCIPLES PRINCIPLES AND STUDIED SYSTEM All the variables used in the following are explicitly defined in tables 2, 3 and 4 of the appendix. All the used in the All the variables variables used in appendix. the following following are are explicitly explicitly defined defined in tables 2, 3 and 4 of the in 3 of 2.1. 2, Rankine in tables tables 2, 3 and and 4 4Process of the the appendix. appendix. 2.1. Rankine Process 2.1. Rankine cycle is Process a widely used power generation cycle to 2.1. Rankine Rankine Process Rankine cycle is aa widely used power generation cycle to turn heat into mechanical or electrical power. First the Rankine cycle is widely used power generation cycle to Rankine cycle is apumped widely from used power cyclethe to turn heat into or electrical First working fluid ismechanical a tank generation atpower. the condensing turn heat into mechanical or electrical power. First the turn heat into mechanical or electrical power. First the working fluid is pumped from a tank at the condensing pressure to the is evaporator atfrom the evaporating pressure. Then working fluid pumped aa tank condensing working fluid is pumpedfluid from tank at at the the condensing pressure to the evaporator at the pressure. Then the pressurized working is evaporating pre-heated, vaporized and pressure to the evaporator at the evaporating pressure. Then pressure to the evaporator at the evaporating pressure. Then the pressurized working fluid is pre-heated, vaporized and superheated in one or several heat exchangers (HEX), also the pressurized working fluid is pre-heated, vaporized and the pressurized working fluid heat is pre-heated, vaporized and superheated in one or several exchangers (HEX), also known as boilers. These HEX are linked to the heat source. superheated in one or several heat exchangers (HEX), also superheated in one or several heat exchangers (HEX), also known as boilers.vapor These HEX are linked to the heat source. The superheated expands from evaporating pressure to known as These HEX linked to source. known as boilers. boilers.vapor Theseexpands HEX are are linked to the the heat heat source. The superheated from evaporating pressure to The The superheated superheated vapor vapor expands expands from from evaporating evaporating pressure pressure to to
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condensing pressure in an expansion device converting the pressure and enthalpy drop into mechanical work. Finally the expanded vapour condenses through a condenser releasing heat into the heat sink (e.g. ambient air) and returns to the working fluid reservoir. In this process the changes of states in both the pump and the expander are irreversible and increase the fluid entropy to a certain extent.
outlet of the boiler, in order not to disturb the emissions control strategy and the internal combustion process. 3.1. Model Assumptions Several assumptions have been done to simplify the problem in a great extent. They are usually admitted when coming to heat exchanger modelling [Feru et al. (2013), Vaja (2009)]: The transfer fluid is always considered in single phase i.e. no condensation in the EGR/exhaust gases is taken into account. The conductive heat fluxes are neglected since the predominant phenomenon is the convection. The pressure drops on each fluids side (transfer and working fluids) are not considered. Both boilers are represented by a straight pipe in pipe counterflow heat exchanger, similarly to Vaja (2009), divided into n lumped sub-volumes in the longitudinal direction. Fluid properties are evaluated at the outlet of each sub-volume i.e. linear profile is considered between inlet and outlet of each node. Pressure dynamics is neglected since its time scale is very small considered to the HEX time scale. Working fluid mass flow rate is supposed constant along the heat exchanger.
2.2. Studied System The Waste Heat Recovery System (WHRS) is compounded on a turbocharged 6 cylinder 11L 320kW HD engine using exhaust gas recirculation (EGR) and a selective catalyst reduction system (SCR) to reduce the NOx emissions. The studied Rankine cycle is recovering heat from both EGR and Exhaust applying a serial configuration of two heat exchangers. Figure 1 shows a schematic of the studied system. The mass flow rate through the two boilers is controlled by the pump speed. The expansion machine is a turbine, which has a higher power density than volumetric expanders [Seher et al. (2012), Lemort et al. (2013)]. The working fluid is then condensed through an indirect condenser fed by coolant. Moreover the cycle is equipped with two bypass valves one located in the exhaust stream to control the amount of energy introduced in the system and a second in front of the expansion device to prevent liquid to enter in the turbine and avoid blade erosion caused by liquid droplets into a high speed rotor.
DPF
3.2. Governing Equations Since the mass is assumed constant the continuity equation is neglected and the model is only based on the energy conservation for the working fluid (1), the gas (2) and the energy balance at the internal (3) and external wall (4). 𝜕𝜕𝑚𝑚̇𝑤𝑤𝑤𝑤 ℎ𝑤𝑤𝑤𝑤 𝜕𝜕ℎ𝑤𝑤𝑤𝑤 (1) − Q̇ =ρ V .
SCR
conv wf int 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ∂T − Q̇ conv g int − Q̇ conv g ext = ρ𝑔𝑔 V𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 ) 𝑔𝑔 . 𝜕𝜕𝜕𝜕 ∂t ∂Twall int Q̇ conv g int + Q̇ conv wf int = ρwall Vwall int cp wall . ∂t ∂Twall ext Q̇ conv g ext + Q̇ conv amb ext = ρwall Vwall ext cp wall , ∂t ̇ with Q conv j k = 𝛼𝛼𝑗𝑗 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑗𝑗 𝑘𝑘 (𝑇𝑇𝑗𝑗 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑘𝑘 ),
𝜕𝜕ṁ𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 )T𝑔𝑔
WHR fluid Charge Air
(2) (3) (4)
(5) where j = g, wf, amb and k = int, ext. Furthermore, to complete the system we need boundary and initial conditions. Time-dependent boundary conditions are used in z=0 and z=L:
Coolant Exhaust gas
Fig 1 Studied system The chosen working fluid is a predefined mixture of Ethanol and Water which gives the best compromise concerning performance and reduced flammability. 3.
569
𝑚𝑚̇wf (𝑡𝑡, 0) = 𝑚𝑚̇wf 0 (𝑡𝑡), 𝑚𝑚̇g (𝑡𝑡, 𝐿𝐿) = 𝑚𝑚̇g L (𝑡𝑡),
ℎ𝑤𝑤𝑤𝑤 (𝑡𝑡, 0) = ℎ𝑤𝑤𝑤𝑤 0 (𝑡𝑡), 𝑇𝑇𝑔𝑔 (𝑡𝑡, 𝐿𝐿) = 𝑇𝑇𝑔𝑔 𝐿𝐿 (𝑡𝑡).
𝑇𝑇𝑔𝑔 (0, 𝑧𝑧) = 𝑇𝑇𝑔𝑔 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧), ℎ𝑤𝑤𝑤𝑤 (0, 𝑧𝑧) = ℎ𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧).
𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 (0, 𝑧𝑧) = 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧), 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 (0, 𝑧𝑧) = 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 (𝑧𝑧),
The initial conditions for the gas and wall temperatures and working fluid enthalpy are given by:
HEAT EXCHANGER CONTROL ORIENTED MODELING
The manipulated variable (MV) is the working fluid mass flow 𝑚𝑚̇wf 0 (𝑡𝑡), whereas the controlled variable (CV) is the working fluid enthalpy ℎ𝑤𝑤𝑤𝑤 (𝑡𝑡, 𝐿𝐿). However the enthalpy is impossible to measure directly we have to use pressure and temperature measurement to compute it.
The system dynamic is controlled by the HEX behaviour (i.e. evaporators and condenser) and models of these components are developed to dynamically predict temperature and enthalpy of transfer and working fluid at the outlet of each boiler. This is critical when coming to control design to ensure a safe operation and a proper operation of the EGR function. Safe operation means that the fluid is completely vaporized when entering the turbine in order not to destroy it. For the EGR, the aim is to have low gas temperature at the
3.3. Heat Transfer To model the convection from the transfer fluid to the pipe walls and from the internal pipe to the working fluid a heat transfer coefficient (α) is needed. The convection from a 570
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V. Grelet et al. / IFAC-PapersOnLine 48-8 (2015) 568–573
boundary to a moving fluid is usually represented by the dimensionless Nusselt number (Nu) which is the ratio of convective to conductive heat transfer. 𝛼𝛼 ∗ 𝑙𝑙 (6) 𝑁𝑁𝑁𝑁 = ,
In single phase (liq and vap) the coefficient a. b. and c. are evaluated thanks to third order polynomial function of pressure similar to: 3 2 𝑎𝑎𝑑𝑑 = 𝑎𝑎𝑑𝑑3 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑎𝑎𝑑𝑑2 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑎𝑎𝑑𝑑1 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑎𝑎𝑑𝑑0 . (14) In two-phase region ad 2φ and 𝑏𝑏d 2φ are approximated with: 1 (15a) a = ,
𝜆𝜆
where l here represents a characteristic length and is in this case the hydraulic diameter. Numerous correlations to approach this number can be found in the literature and are usually derived from experiments, see for example Thome (2010). Those correlations depend on the flow regime, the number of phases and the geometry studied. In single phase, the following correlation is implemented: 𝑁𝑁𝑁𝑁 = 𝐴𝐴 𝑅𝑅𝑅𝑅 𝑛𝑛 𝑃𝑃𝑃𝑃 𝑚𝑚 , (7) where A is a constant Re and Pr are dimensionless number (respectively Reynolds and Prandtl number). By developing these two numbers (7) becomes: 4𝑚𝑚̇ 𝑛𝑛 𝑐𝑐𝑝𝑝 𝜇𝜇 𝑚𝑚 (8) 𝑁𝑁𝑁𝑁 = 𝐴𝐴 ( ) ( ) . 𝜋𝜋𝑑𝑑ℎ 𝜇𝜇
𝑏𝑏𝑑𝑑 2𝜑𝜑
3.5. Discretization To implement such a model, we now have to discretize the continuous heat exchanger model with respect to space based on finite differences method. The HEX is split into n longitudinal cell where a backward Euler scheme in space is applied. The system of equations defining the response of the ith cell for transfer fluid, pipe and working fluid can be represented under the following formalism. 𝑥𝑥̇ = 𝑓𝑓(𝑥𝑥, 𝑢𝑢), (16) where 𝑢𝑢 = [𝑚𝑚̇𝑤𝑤𝑤𝑤 0 𝑃𝑃𝑤𝑤𝑤𝑤 0 ℎ𝑤𝑤𝑤𝑤 0 𝑚𝑚̇𝑔𝑔 𝐿𝐿 𝑇𝑇𝑔𝑔 𝐿𝐿 ]. (17) The vector u contains the only MV (𝑚𝑚̇𝑤𝑤𝑤𝑤 0 ) and four inputs disturbances: inlet gas mass flow and temperature (𝑚𝑚̇𝑔𝑔 𝐿𝐿 ) and (𝑇𝑇𝑔𝑔 𝐿𝐿 ) and inlet working fluid pressure and enthalpy (𝑃𝑃𝑤𝑤𝑤𝑤 0 ) and (ℎ𝑤𝑤𝑤𝑤 0 ). And xi = [hwf i Twall int i Tg i Twall ext i ], (18a) 𝑓𝑓𝑖𝑖 (𝑥𝑥, 𝑢𝑢) = (18b)
𝜆𝜆
𝑤𝑤𝑤𝑤 𝑙𝑙𝑙𝑙𝑙𝑙
𝑞𝑞0,01 [
𝛼𝛼𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣𝑣𝑣 𝛼𝛼𝑤𝑤𝑤𝑤 𝑙𝑙𝑙𝑙𝑙𝑙
(1 + 8(1 − 𝑞𝑞)0,7
𝜌𝜌𝑤𝑤𝑤𝑤 𝑙𝑙𝑙𝑙𝑙𝑙 0,67
𝜌𝜌𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣𝑣𝑣
)] }
.
(𝑚𝑚̇𝑤𝑤𝑤𝑤 (ℎ𝑤𝑤𝑤𝑤 𝑖𝑖−1 −ℎ𝑤𝑤𝑤𝑤 𝑖𝑖 )−𝛼𝛼𝑤𝑤𝑤𝑤𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑤𝑤𝑤𝑤 (𝑇𝑇𝑤𝑤𝑤𝑤 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 ))
This correlation corresponds to a tube arrangement and is practical since it creates continuity between single and two phase heat transfer coefficients and does not need transport properties such as viscosity or heat conductivity.
𝜌𝜌𝑤𝑤𝑤𝑤 𝑖𝑖 𝑉𝑉𝑤𝑤𝑤𝑤
𝛼𝛼𝑤𝑤𝑤𝑤 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑤𝑤𝑤𝑤 (𝑇𝑇𝑤𝑤𝑤𝑤 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 )+𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑖𝑖𝑖𝑖𝑖𝑖 (𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 ) (
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑤𝑤𝑤𝑤
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
𝑤𝑤𝑤𝑤
𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙
𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤
𝑖𝑖𝑖𝑖 ℎ𝑤𝑤𝑤𝑤 ≥ ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣
𝜌𝜌𝑤𝑤𝑤𝑤 =
𝑎𝑎𝑑𝑑 2𝜑𝜑 ℎ𝑤𝑤𝑤𝑤 + 𝑏𝑏𝑑𝑑 2𝜑𝜑 {𝑎𝑎𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ℎ𝑤𝑤𝑤𝑤 ² + 𝑏𝑏𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ℎ𝑤𝑤𝑤𝑤 + 𝑐𝑐𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝑤𝑤𝑤𝑤
𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 },
. ]
CONTROL DEVELOPMENT
4.1. PI Controller The main problem when considering PI or PID control structure is that identification for heat exchangers is very sensitive to the inputs considered [Peralez et al. (2013), Horst et al. (2013)]. This is well described when considering
𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙
𝑖𝑖𝑖𝑖 ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 < ℎ𝑤𝑤𝑤𝑤 < ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 . 𝑖𝑖𝑖𝑖 ℎ𝑤𝑤𝑤𝑤 ≥ ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣
)
In this paper we focus on the temperature control with only regards to power generation. Our goal here is to improve the temperature stability around a given set point. For that we can manipulate the working fluid mass flow to regulate the working fluid enthalpy (or temperature) at the outlet of the exhaust boiler. Due to time issue, at the moment where this paper is written, no experimental tests of the control system are possible. The validation is based on the representative simulation model. The comparison is done on the tracking error either on a step load change or on a complete driving cycle. On each case, we start from the same equilibrium.
ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 − ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝜌𝜌𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑐𝑐𝑐𝑐𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑉𝑉 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒
4.
The saturation temperature (Tsat) is approximated with the Wagner equation with adapted coefficient for liquid and vapor saturation [Kleiber and Joh (2010)] and allows to make the transition between each phase. Density model: 𝑎𝑎 ℎ ² + 𝑏𝑏 ℎ + 𝑐𝑐 𝑖𝑖𝑖𝑖 ℎ ≤ ℎ (13) 1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑤𝑤𝑤𝑤
𝜌𝜌𝑔𝑔 𝑉𝑉𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 )
[
where a. b. and c. are first order polynomial expressions function of pressure and q is the fluid quality defined as the quantity of vapor present in the two phase flow. ℎ𝑤𝑤𝑤𝑤 − ℎ𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 (12) 𝑞𝑞 = .
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑤𝑤𝑤𝑤
−𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑒𝑒𝑒𝑒𝑒𝑒(𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖 )
𝛼𝛼𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑎𝑎𝑎𝑎𝑎𝑎 (𝑇𝑇𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖 )+𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑒𝑒𝑒𝑒𝑒𝑒(𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖 )
Similarly to Kleiber and Joh (2010), the working fluid properties are approximated using mathematical description. This allows not to rely on thermochemical database such as Lemmon et al. (2011) and creates continuity in derivative terms during the single / two phase transition. The fluid properties are only function of pressure and enthalpy. Temperature model: 𝑎𝑎 ℎ ² + 𝑏𝑏 ℎ + 𝑐𝑐 𝑖𝑖𝑖𝑖 ℎ ≤ ℎ (11) 𝑇𝑇 + 𝑞𝑞(𝑇𝑇 − 𝑇𝑇 ) 𝑖𝑖𝑖𝑖 ℎ <ℎ <ℎ 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑤𝑤𝑤𝑤
𝜌𝜌𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑐𝑐𝑐𝑐𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑉𝑉 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖
𝑚𝑚̇𝑔𝑔 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇𝑔𝑔 )(𝑇𝑇𝑔𝑔 𝑖𝑖−1 −𝑇𝑇𝑔𝑔 𝑖𝑖 )−𝛼𝛼𝑔𝑔 𝐴𝐴𝑒𝑒𝑒𝑒𝑒𝑒ℎ 𝑔𝑔 𝑖𝑖𝑖𝑖𝑖𝑖 (𝑇𝑇𝑔𝑔 𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖 )
3.4. Working Fluid Properties
𝑇𝑇𝑤𝑤𝑤𝑤 = { 𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ℎ𝑤𝑤𝑤𝑤 ² + 𝑏𝑏𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ℎ𝑤𝑤𝑤𝑤 + 𝑐𝑐𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇
(15b)
All coefficients are evaluated by fitting routines in Matlab using REFPROP as thermodynamic properties database [Lemmon et al. (2011)].
Assuming the viscosity (µ) the specific heat (cp), the heat conductivity (λ) and the characteristic length (l) constant in single phase region for the working fluid and the gas we get: 𝛼𝛼𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑟𝑟𝑟𝑟𝑟𝑟𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑛𝑛𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 , (9a) 𝛼𝛼𝑣𝑣𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑟𝑟𝑟𝑟𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤 𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑛𝑛𝑣𝑣𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤 , (9b) 𝛼𝛼𝑔𝑔 = 𝛼𝛼𝑟𝑟𝑟𝑟𝑟𝑟𝑔𝑔 𝑚𝑚̇𝑔𝑔 𝑛𝑛𝑔𝑔 , (9c) Where the constant αref . and the exponent n. have to be identified in liquid and vapor region for the working fluid and in single phase for the gas. In the two phase region, a more complex correlation (10) is used to enhance the single phase heat transfer coefficient [Kleiber and Joh (2010)]. 0,37 −2,2 𝜌𝜌 (10) 𝛼𝛼2𝜑𝜑 𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑙𝑙𝑙𝑙𝑙𝑙 𝑤𝑤𝑤𝑤 {(1 − 𝑞𝑞)0,01 [(1 − 𝑞𝑞) + 1,2𝑞𝑞0,4 ] + 𝜌𝜌𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣𝑣𝑣 −2 −0,5
ad 2φ1 𝑃𝑃𝑤𝑤𝑤𝑤 + ad 2φ 0 1 = . 𝑏𝑏𝑑𝑑 2𝜑𝜑1 𝑃𝑃𝑤𝑤𝑤𝑤 + 𝑏𝑏𝑑𝑑 2𝜑𝜑 0
d 2φ
}
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several operating points. Most of the results on PID tuning are reducing the process to a first order plus time delay (FOPTD) transfer function [Skogestad (2003), Madhauranthakam et al. (2009)]. The dynamic relation between the variations of WF mass flow and the temperature is: Δ𝑇𝑇𝑤𝑤𝑤𝑤 𝐿𝐿 (𝑠𝑠) 𝐺𝐺 (19) = 𝑒𝑒 −𝐷𝐷𝐷𝐷 . Δ𝑚𝑚̇𝑤𝑤𝑤𝑤 0 (𝑠𝑠)
manipulated variable is here preferred to take into account the thermal inertia of the system [Rasmussen and Alleyne (2010)]. Comparison on a step load change for several PID tuning method is shown in table 1. On two criteria: the integrated absolute error (IAE) and the total variation (TV) which are defined as: ∞ (21) IAE = ∫ |𝐶𝐶𝐶𝐶(𝑡𝑡) − 𝑆𝑆𝑆𝑆(𝑡𝑡)|𝑑𝑑𝑑𝑑.
1 + 𝜏𝜏𝜏𝜏
0
𝑇𝑇𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 +𝑇𝑇𝑒𝑒𝑒𝑒ℎ 𝑖𝑖𝑖𝑖 2
TWF L
20
.
0 FOPTD Reference Model R² = 93.8929 % 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s)
TWF L
20
(22)
FOPTD Identication on high load operating point
Table 1 PID tuning method comparison
0 FOPTD Reference Model R² = 99.9591 % 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (s)
2
Fig 2 FOPTD model identification validation 0 x 10
Controlled Variable Error vs Time
0
Error to SP(K)
-20 0
G (K/kg/s)
0
𝑑𝑑𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑑𝑑. 𝑑𝑑𝑑𝑑
The method proposed by Madhuranthakam et al. (2008) is minimizing the IAE with the highest TV, i.e. the control effort is more important but still acceptable. The IMC and Tyreus-Luyben tuning methods seem to be conservative and too slow since they present the highest IAE. Figure 4 shows the tracking error versus time for the five tuning methods. Tuning Method IAE TV Ziegler Nichols in Closed Loop 84.6325 0.0065 Tyreus-Luyben 743.1427 0.0053 [Skogestad (2003)] IMC 940.3763 0.0064 SIMC 208.5099 0.0047 [Skogestad (2003)] Madhuranthakam et al. 48.6754 0.0204 [Madhuranthakam et al. (2008)]
FOPTD Identication on low load operating point
-20 0
∞
𝑇𝑇𝑇𝑇 = ∫
For the identification a pseudo random binary sequence (PRBS) is the same for all operating points. As it can be seen in figure 2, this model structure is confirmed by comparing results of the reference model and an identified FOPTD around an operating point since on the worst point the agreement is around 94%. Figure 3 shows the FOPTD parameters for several inputs disturbance. For convenience they are represented versus the total heat flow rate entering: ∗ 𝑄𝑄̇𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑐𝑐𝑝𝑝 𝑔𝑔 (𝑇𝑇 ) (𝑚𝑚̇𝑒𝑒𝑒𝑒𝑒𝑒 (𝑇𝑇𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤 0) + 𝑚𝑚̇𝑒𝑒𝑒𝑒ℎ (𝑇𝑇𝑒𝑒𝑒𝑒ℎ 𝑖𝑖𝑖𝑖 − 𝑇𝑇𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠 )). (20) where 𝑇𝑇 ∗ =
571
4
G
-2
-2 -4 -6 -8
-4
-10 0
500
100
200
300 Time (s)
(s)
400
ZN CL TL IMC SIMC Mad 500 600
Fig 4 Tracking error comparison 0
D (s)
40
D
The PID tuning of Madhuranthakam et al. (2008) is now used.
180
4.2. Nonlinear model inversion
20 0 20
40
60
80 100 120 Heat Flow Rate (kW)
140
160
Despite of their very good agreement, the FOPTD could not be inversed to be used as controller since they do not take account of the disturbances and therefore every correction action would only be done through a feedback on the manipulated variable. In order to improve the performance of the classical “PID” controller we use the inverse of the system model (16-18) as feedforward to compute the first part of the MV corresponding to the desired set point (Fig. 5). Details of nonlinear inversion are approached and detailed in [Peralez et al. (2013)]. To simplify the model inversion, the heat transfer coefficient for the WF is assumed constant in liquid and vapor phase. 𝑛𝑛 (23) (𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑚𝑚̇𝑤𝑤𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚 )
Fig 3 FOPTD model parameter FOPTD parameters are evaluated on twenty different engine operating points, which correspond to the input disturbances, representing the overall engine map. As it can be seen below, the FOPTD parameters change a lot, according to the nonlinearity of the model developed in section 3. To improve the system control several PID tuning methods are compared to reduce both the tracking error and the control effort [Skogestad (2003), Madhuranthakam et al. (2008)]. Moreover, to obtain better results on a realistic driving cycle, gains are scheduled as a function of the working fluid mass flow rate entering in the system. The scheduling of each gain is realized by interpolating Kp, Ti and Td according to the working fluid mass flow feedback signal. A normal scheduling would be to select the appropriate gain in function of the operating point but schedule it as a function of the
𝛼𝛼𝑤𝑤𝑤𝑤 = 𝛼𝛼𝑤𝑤𝑤𝑤 𝑟𝑟𝑟𝑟𝑟𝑟
2
,
with maximum and minimum mass flow set as the mass flow corresponding to maximum and minimum WF pump speed: N (24) ṁ =ρ Cc . 𝑤𝑤𝑤𝑤 0
572
wf 0
60
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
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Controlled Variable vs Time
Set Point U PID
External Conditions
Rankine System
SP CV InvHEXModel CV PID
220 200 180 160 0
500
1000
1500
2000
Time (s) Manipulated Variable vs Time
disturbances 0.04
outputs
mWF(kg/s)
Inverse HEX Ufeedforward Model
TWFoutExhBoiler (°C)
As the WF mass flow is considered homogenous in the entire heat exchanger it can be computed from the system (16).
MV InvHEXModel MV PID
0.03 0.02
.
Ufeedback
2500
0.01 0 0
Fig 5 New control structure proposed 5.
500
1000
1500
Time (s)
2000
2500
Fig 7 Controllers performance comparison
COMPARISON OF CONTROLLERS PERFORMANCES
30
TWFoutExhBoiler Error(°C)
20
This two proposed control strategies are evaluated on a validated Rankine model (at the time of writing the experimental set up was not available). Gases data (exhaust and EGR) are coming from an 11L long haul truck on a highway driving cycle (Fig. 6). For the other perturbations, namely called hwf 0 and Pwf 0 in (19), the EGR boiler inlet WF enthalpy is kept constant whereas the inlet WF pressure depends on the expansion turbine inlet conditions. Sensors and actuators dynamics are represented by first order models where time constant are fitted thanks to manufacturer data. As previously said, the controlled variable in the model is the WF enthalpy but as this one is not physically measurable and for convenience we track the WF temperature at the second boiler outlet. In order to well assess the performance, the set point changes during the simulation. Temperature (°C)
6.
Mass Flow (kg/s)
0.4 0.3
1500 2000 Time (s) Gases Mass Flow vs Time
500
1000
2500
3000
2500
3000
Egr Exhaust
0.2 0.1 0 0
1500 Time (s)
2000
1000
1500
Time (s)
2000
CONCLUSION
This paper reports control strategy development for waste heat recovery Rankine based system used on heavy duty engines. Comparison of both controllers shows a better accuracy for the inverse HEX model when comparing the tracking error. The objective to stabilize as close as possible the temperature around a given set point is achieved by using this strategy. We conclude that the nonlinear controller leads to the best performances, at the cost of developing an accurate first principle nonlinear model with all known parameters, which underline the problem of identification. Yet it still needs to be validated on test bench level to see the robustness with respect to modelling errors and parameters mismatch. However the model is compliant with classical control unit used in the automotive industry since the computational needs is low. Future work will focus on experimental validation and optimal control strategy.
Egr Exhaust 1000
500
InvHEXModel PID 2500
Fig 8 Tracking error comparison
300
500
-10
-30 0
400
200 0
0
-20
Gases Temperature vs Time
500
10
Fig 6 Input disturbances Figure 7 and 8 show comparison of the two implemented control strategies: the PID based on Madhuranthakam et al. (2008) and the nonlinear control presented in 4.2. Except at the beginning, where the PID is not handling the initial SP variation and the large change in disturbances appearing around 30s, it keeps the error within +/-10K, which could be acceptable when considering a volumetric expander, which are less sensitive to liquid droplets. Here a kinetic expansion machine is used and the temperature has to be as close as possible to the set point, which is better achieved by the nonlinear controller: the error remains in +/- 3K. The control efforts seem similar but are slightly different (PID presents a delay compared to nonlinear controller).
ACKNOWLEDGMENT This PhD thesis is collaboration between UCBL1, ULg and Volvo Trucks which is gratefully acknowledged for the funding. The French ministry of higher education and research for the financial support of the CIFRE PhD thesis 2012/549 is also acknowledged.
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REFERENCE
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Espinosa, N. (2011). “Contribution to the Study of Waste Heat Recovery Systems on Commercial Truck Diesel Engines”. PhD Thesis University of Liege and National Polytechnic Institute of Lorraine. Feru, E. Kupper, F. Rojer, C. Seykens, X. Scappin, F. Willems, F. Smits, J. De Jager, B. Steinbuch, M. (2013). ”Experimental Validation of a Dynamic Waste Heat Recovery System Model for Control Purposes”. SAE Technical Paper, 2013-01-1647. Grelet, V. Reiche, T. Guillaume, L. Lemort, V. (2014). “Optimal WHR Rankine based for Heavy Duty Application”. FISITA Conference, NL, F2014-CET-151 Horst, T. Rottengruber, H. Seifert, M. Ringler, J.(2013) ”Dynamic Heat Exchanger Model for Performance Prediction and Control System Design of Automotive Waste Heat recovery Systems”. Applied Energy, Vol 105, 293-303. Kleiber, M. Joh, R. (2010). VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen. VDI Heat Atlas Second Edition. Part D Thermophysical properties. Springer Berlin Germany. Latz, G. Andersson, S. Munch, K. (2012). ”Comparison of Working Fluids in Both Subcritical and Supercritical Rankine Cycles for Waste Heat Recovert Systems in Heavy Duty Vehicles”. SAE Technical Papers, 2012-011200. Lemmon, E.W. Huber, M.L. McLinden, M.O. (2011). REFPROP Nist Standard Reference Database 23 (Version 9.0), Thermophysical Properties Division, National Institute of Standards and Technology, Boulder, CO. Lemort, V. Guillaume, L. Legros, A. Declay, S. Quoilin, S. (2013). “A Comparison of Piston, Screw and Scroll Expanders for Small Scale Rankine Cycle Systems”. Proceedings of the 3rd International Conference on Microgeneration and Related Technologies. Madhauranthakam, C.R. Elkamel, A. Budman, H. (2008). “Optimal tuning of PID controllers for FOTPD, SOPTD and SOPTD with lead processes”. Chemical Engineering and Processing, vol 47, 251-264. Mago, P. Chamra, L. Somayajii. (2007). “Performance Analysis of Different Working Fluids for Use in Organic Rankine Cycles”. Journal of Power and Energy, vol 221, 255-263. Mavridou, S. Mavropoulos, G. Bouris, D. Hountalas, D. Bergeles, G. (2010). “Comparative Design Study of a Diesel Exhaust Gas Heat Exchanger for Truck Applications with Conventional and State of the Art Heat Transfer Enhancements”. Applied Thermal Engineering, vol 30, 935-947 Peralez, J. Tona, P. Lepreux, O. Sciaretta, A. Voise, L. Dufour, P. Nadri, M. (2013). “Improving the Control Performance of an Organic Rankine Cycle System for Waste Heat Recovery from a Heavy Duty Diesel Engine using a Model Based Approach”. IEEE Conference on Decision and Control (CDC), pp 6830-6836. Quoilin, S. Aumann, R. Grill, A. Schuster, A. Lemort, V. (2011). ”Dynamic Modeling and Optimal Control
APPENDIX A Cc cp D G h 𝑚𝑚̇ N P Q̇ s t T V 𝑊𝑊̇ z
Area Cubic capacity Specific heat Transfer function delay Transfer function Gain Enthalpy Mass flow Speed Pressure Heat flow rate Entropy time Temperature Volume Power Space length
m² m3 J/kg/K s K/kg/s J/kg kg/s rpm Pa W J/kg/K s K / °C m3 W m
Table 2 Latin letters α µ λ ρ τ
Heat Transfer Coefficient Viscosity Heat Conductivity Density Transfer function time constant Table 3 Greek letters
574
W/m²/K Pa.s W/K kg/ m3 s