Determining water mass flow control strategies for a turbocharged SI engine using a two-stage calculation method

Determining w ater mass flo w con trol strategies for a turb o c harged SI engine using a t w o-stage calculation metho d P eter H¨ olz a, ∗ , Thomas B¨ ohlk e b , Thomas Kr¨ amer a a Porsche AG, Porsche Motorsp ort, Porschestr. 911, 71287 Weissach, Germany b Chair for Continuum Me chanics, Institute of Engineering Me chanics, Karlsruhe Institute of T e chnology (KIT), Kaiserstr. 10, 76131 Karlsruhe, Germany Abstract Reduction of heat and friction losses is a prov en approach to increase the engine effi- ciency . Therefore, and due to a stabilized, robust combustion, a sp ecific adjustment of comp onen t temperatures is desirable in highly transien t conditions. In this paper, a turbo charged SI engine is inv estigated n umerically concerning the po- ten tial regulation of temp eratures, including heat fluxes, only by con trolling the water mass flow rate. Using tw o indep enden t models, a simplified lump ed capacity mo del and a detailed three-dimensional CFD-CHT sim ulation, an efficien t, tw o-stage calculation metho d is suggested for an optimized determination of con trol strategies and their pa- rameters. This complemen ts existing published w orks whic h usually control more than one parameter, but use one mo del. Differen t con trol strategies, like feed forward or feed- bac k controllers, are proposed and compared. In addition, a more holistic approac h is presen ted p erforming a Monte Carlo simulation which ev aluates temp eratures, as well as h ydraulic pumping losses. Using exp edien t control strategies and parameters, it could b e shown that the engine temp eratures can b e effectively regulated within a wide range. The tw o different mo dels sho w partly similar results, and the efficient, tw o-stage optimization method has prov en its w orth. Ho wev er, there are some significan t differences betw een simplified and detailed mo delling which are w orth men tioning. Keywor ds: Thermal managemen t, Mon te C arlo simulation, Conjugate heat transfer, Engine heat transfer, Lump ed element mo del, Model reduction. Copyrigh t, including manuscript, tables, illustrations or other material submitted as part of the manuscript, is assigned to the authors. Con ten ts 1 In tro duction 2 1.1 Motiv ation and state of the art . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Outline of the pap er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ∗ Corresponding author. T el.: +49 711 911 88680 Email addr ess: peter.hoelz@porsche.de (Peter H¨ olz) Pr eprint submitted to arXiv Novemb er 2, 2021 2 Optimized thermal management due to v arious feed forw ard and con- trol strategies 4 2.1 Metho d used in this pap er . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Simplified thermal mo del - lump ed capacit y model . . . . . . . . . . . . . 5 2.3 Detailed thermal mo del - CFD-CHT metho d . . . . . . . . . . . . . . . . 7 2.4 F eed forward strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Con trol strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 F eed forward and con trol strategy . . . . . . . . . . . . . . . . . . . . . . . 11 3 Results and discussion 11 3.1 F easible temp erature range and consistency . . . . . . . . . . . . . . . . . 11 3.2 Mon te Carlo sim ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Dynamic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 T emperature shifting and resulting heat sa ving . . . . . . . . . . . . . . . 17 3.5 Three-dimensional results . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Conclusions 21 5 Nomenclature 22 6 References 25 1. In tro duction 1.1. Motivation and state of the art In order to increase the engine efficiency by reducing unnecessary heat and friction losses and decrease its emissions b y effective exhaust gas treatments under a transient driv e, lots of inno v ative co oling concepts can b e found in literature. A reduction of heat losses b y application of temp erature swings insulation is presented in [ 1 ]: sp ecial materials in the combustion cham b er with low thermal conductivities and heat capacities decrease heat transfer by reducing the temp erature difference b etw een gas and solid during com- bustion. In [ 2 ], a concept is presen ted which uses the phase transition of water to control engine w all temperatures. In addition, the combustion control can be effectiv ely stabi- lized under dynamic engine conditions with v arying load p oints. This can be explained b y the fact that the inlet air temp erature, and hence the in-cylinder heat transfer, highly influences the ignition and combustion pro cess. As an example, [ 3 ] in vestigates closed- lo op com bustion control mec hanisms in a HCCI engine by using thermal management for the inlet air temp erature. A review about engine cooling tec hnologies can be found in [ 4 ]. Different asp ects, partially counterv ailing effects, are discussed. In addition to the already men tioned p oints, engine protection mo des, e.g., improv ed kno c k protection, as w ell as higher v olumetric efficiencies for lo w temperature set points, are discussed. In this con text, with regard to unnecessary heat and pumping losses, [ 5 ] in vestigates n umerically optimal cooling structures within an in ternal com bustion engine for giv en maxim um com- p onen t temp eratures. Within the framework of an own developed 1D lump ed capacity mo del, [ 6 ] also studies tw o different co olant structures for the engine blo ck and head. [ 7 ] in vestigates experimentally the effect of active engine thermal managemen t on a bi-fuel engine. A p ositiv e effect on heat release rates, fuel consumption and emissions can b e 2 pro ved. A sp ecific application of thermal managemen t systems, including waste heat re- co very , for h ybrid electric v ehicles with a con tinuously v ariable transmission is reported in [ 8 ]. In this case, a lump ed capacit y mo del is used to increase the transmission efficiency . Classical thermal management strategies for com bustion engines try to con trol the water inlet temp erature b y installing a thermostat, which can direct a part of the w ater mass flo w around the radiator. In [ 9 ] this concept is supplemented with a pump throttle in order to additionally control the heat transfer co efficien ts. Analogously , [ 10 ] inv estigates a v ariable sp eed electric pump b y using a reduced order m ultiple no de lumped parameter resistor-capacitor thermal model. In [ 11 ], a nonlinear controller for suc h applications, ad- ditionally adjusting the radiator fan sp eed, is prop osed for cases with unmeasurable heat inputs from the combustion pro cess. A global asymptotic regulation of the engine tem- p erature can b e shown. Similarly , a robust controller design with an estimate of future disturbances is presented in [ 12 ]. Using a zero-dimensional mo del, [ 13 ] presents a mathe- matically well formulated con troller design pro cedure for the co olan t flow rate, including a discretization of the engine map and its piecewise linearization. A one-dimensional, transien t numerical mo del is also developed in [ 14 ], including some strategies for feed forw ard and feedbac k controllers. F or these kinds of co oling system, [ 15 ] rep orts a low er fuel consumption of 1.1 p ercen t under NEDC (New Europ ean Driving Cycle) cyc le op eration conditions as well as a re- duction of emissions in the range of 5 p ercen t for h ydro carb on and 6 p ercen t for carb on mono xide. In addition, a simplified, holistic vehicle mo del is prop osed, whic h is mainly based on exp erimental measuremen ts. Similarly , [ 16 ] developes a transien t vehicle mo del with div erse submo dels and corresp onding interactions. Again, the application of an electrical water pump results in a fuel saving of ab out 0.75 to 1.1 p ercen t. Addition- ally , an increase of the turbine outlet temp erature can b e observed, and hence, a faster catalyst heat-up. A 1D/3D sim ulation metho d for the vehicle in tegrated thermal man- agemen t is presented in [ 17 ] and [ 18 ]. An effective coupling b et ween 1D submo dels of single subsytems with the 3D underho od structure offers suitable b oundary conditions and parameters for the simulation. [ 19 ] defines a ”p erfect co oling system” for vehicles concerning fuel consumption and simulates, with the help of an own developed mo del, p oten tial sa vings for different co oling systems. Concerning effective con trol strategies, there exist many differen t approaches and the field of control technology is large. The well-kno wn rules after Kessler are based on direct mo difications of the comp ensated system in the frequency space: the so called symmet- rical optim um is prop osed in [ 20 ], whereas [ 21 ] eliminates the largest time constant in the system. Another, completely empirical, approach is presented by [ 22 ]: the research results are based on many test cases with v arying parameters. Using a frequency domain mo del of the plant, a new metho d for the auto-calibration of PI and PID controllers is presen ted in [ 23 ]. Some adv antages with regard to the ab o ve mentioned Ziegler-Nic hols approac h are rep orted. Concerning some disadv antages of current design techniques for mo del predictive con trol, [ 24 ] proposes a more robust metho d to deal explicitly with plan t model uncertain ties. Using dimensional analysis, [ 25 ] developes a quite general form ulation of synthesis of PID con trol lo ops with delay . The fo cus is additionally set on disturbance rejections. A more sp ecific, robust tuning metho d for first order systems 3 with dela y is presented in [ 26 ], supplemen ted b y a consideration of stabilit y . 1.2. Outline of the p ap er Concerning control and feedbac k con trol systems for engine temp eratures, one of the goals of the present pap er is the systematic inv estigation of their controllabilit y by ad- justing the water mass flow rate within the water jac ket, e.g., at a constant water inlet temp erature. Based on that, the dynamic b eha vior of component temp eratures and the in ternal heat fluxes are studied under transien t engine applications. More sp ecifically , t ypical racing scenarios with their characteristic alternation b etw een braking and accel- eration is inv estigated. Therefore, contributing to the completeness of already published works, a tw o-stage cal- culation metho d is prop osed and explained in detail. In a first step, a simplified lump ed capacit y model is dev elop ed. With the help of the stationary solution and linearization, a first estimation of suitable strategies and parameters for the feed forward and feed- bac k controllers are gained. Afterwards, a Monte Carlo simulation is p erformed to get optimized control parameters: in a more holistic approach, their influences on engine temp eratures, hydraulic flow losses and heat fluxes are inv estigated. In a second step, the most promising constellations are used for v alidation and for a more detailed in- v estigation of the aforementioned questions. Therefore, a three-dimensional CFD-CHT engine simulation is used, which has already b een v alidated in [ 27 ]. In this reference, using a dynamic engine dynamometer, temp erature measuremen ts w ere conducted with an engine which w as equipp ed with 70 thermo couples. The v alidation contained highly transien t engine conditions, as it is the case in this pap er, as w ell as well-defined station- ary b oundary conditions. The latter was used for the v alidation of equation ( 1 ). Hence, b ecause of the tw o completely indep enden t mo dels, the tw o-stage metho d can b e seen as a kind of v alidation for the simplified, lump ed mo del. The research question can b e formulated as follows: With regard to engine temp eratures near the com bustion cham b er, which temp erature range, including the temperature devi- ation, its dynamic b ehavior and the resulting heat flows, is feasible only by adjusting the w ater mass flo w rate at a constan t water inlet temperature? On this basis, what are the differences b etw een the fast lump ed capacity mo del, which is suitable for optimization and sensitivit y calculations, and a more compute-in tensive, three-dimensional CFD-CHT sim ulation? 2. Optimized thermal management due to v arious feed forward and control strategies 2.1. Metho d use d in this p ap er V arious feed forw ard and con trol strategies for the engine water mass flo w rate are inv esti- gated numerically and compared with a mec hanical w ater pump with a fixed transmission ratio. Fig. 1 shows the prop osed metho d. A simplified lump ed capacity mo del of the engine serv es as a first parameter and strategy estimation with subsequent, preliminary in vestigations, including a Monte Carlo simulation. Afterwards, the most promising parameter sets and strategies are inv estigated with detailed CFD-CHT engine sim ula- tions. Sp ecial fo cus is laid on heat saving p oten tials and the dynamic b ehavior of diverse comp onen t temperatures under a transient drive. 4 Figure 1: Prop osed t wo-stage calculation method for an optimized thermal managemen t. Therefore, simplified and the detailed, three-dimensional sim ulations w ere carried out for one representativ e race lap and analysed regarding v arious criteria such as heat losses, solid temperature fluctuations or the necessary h ydraulic pow er. 2.2. Simplifie d thermal mo del - lump e d c ap acity mo del The used heat transfer mo dels are extracted from [ 28 ] and [ 27 ]. Consequently , the heat transfer coefficient for the w ater jac k et can be written as: α w ( x , t ) ≈ α ref ( x )  ˙ m w ( t ) ˙ m w | ref  m . (1) This approach separates the time and location dependency by means of a turbulence Reynolds exponent m = 0 . 7. The subscripts describ e represen tative reference states. The lo cation dep endency is omitted b ecause of the lumped-capacity model and, therefore, serv es as a calibration parameter with a certain physical meaning: c ho osing ˙ m w | ref as a represen tative mean w ater mass flo w rate, a range of α ref ∼ 10 4 W/(m 2 K) is used. In this case, the HTC is referred to the water temp erature as the reference temp erature. F ollowing approach is used for the combustion cham b er HTC: h α c i ( t ) = Z R ≥ 0 Ap α | n ( t ) ( A ) d A. (2) During transient simulations, the conditional probabilit y density function for the heat transfer co efficien t p α | n ( t ) has to b e used. It strongly dep ends on the engine sp eed n ( t ) 5 and the engine load. How ever, due to the fact that a race engine is inv estigated, which sp ends most of the time in a complete full load or coasting condition, part load conditions are neglected. F or the inv estigated race track its p ercen tage was less than 10 p ercen t. Because of the cyclic fluctuations in the pressure curves, the HTC is interpretated as a random v ariable w ith its realisation A . F or the reference temp erature, in this case the gas temp erature T gas within the combustion cham b er, an analogous pro cedure was done. Ho wev er, due to the nonlinear nature of heat transfer, a statistically mo dified reference temp erature T mod = h α c T gas i / h α c i has to b e used. In the following, it is called ACT (Av erage Cylinder T emp erature). In this paper, the determination of p α | n ( t ) is based on the W osc hni mo del [ 29 ] and high pressure indication measurements. Details can b e found in [ 28 ] and [ 27 ]. Coasting conditions are treated in a similar manner. During simulation, following ordinary differential equation for the engine temp erature T Cyl is solv ed n umerically: C v ρ ∆ x ˙ T Cyl ( t ) = χ h α c i ( t ) ( T mod ( t ) − T Cyl ( t )) + α w ( t ) ( T w ( t ) − T Cyl ( t )) , (3) T w ( t ) = T w,i + ˙ Q ( t ) / ( ˙ m w ( t ) C p,w ) . (4) T w,i is the inlet water temperature, and serv es as a mo del input parameter. ˙ Q is the heat flux which is transferred from the solid to the water. The time-dep endent v ariables α c and α w are the effective heat transfer co efficients according to equation ( 2 ) and ( 1 ). The parameter χ , which should b e smaller than one, describ es the ratio b et ween the effective surfaces of the com bustion c hamber and the water jack et. It serv es as an additional calibration parameter. In this case, a v alue of 0.3 w as chosen. ∆ x is the characteristic w all thickness, and is necessary for dimensional reasons. It can b e seen as the volume of the representativ e engine part, which is described by a lump ed capacit y , divided through the effective surface area of the w ater wetted side. Therefore, it is not completely inde- p enden t, e.g., it has to b e scaled according to the existing geometry . In this pap er, a v alue of ∆ x = 15 mm is used. C v and ρ are the heat capacity and the density of the solid, resp ectively . T ypical v alues for aluminium cylinder heads are in the range of 900 J/(kgK) and 2.7 kg/m 3 . There are lots of system parameters which influence heat flow: Prandtl and Reynolds n umber, oscillating frequency and amplitude, as w ell as the entry length of pip es. Ex- p erimen tal results with comparable parameters prev ailing in engine water c hannels can b e found in [ 30 ] or [ 31 ]. Cho osing P r ≈ 1, R e ≈ 8 10 3 − 5 10 4 and the oscillating fre- quency in the range of 1 − 5 Hz, the Nusselt num b er increase, resp ectiv ely decrease, can b e ab out 15 p ercen t. In order to ensure the mo del v alidit y and the technical feasibility , the w ater pump is mo delled as a PT1-element with limited water mass flow rates. Local Reynolds num b ers should b e noticeably larger than 2300, the transition num b er b et w een laminar and turbulen t pip e flow. In addition, if the w ater mass flow rate is very low, the amplitudes of the water temp erature will b e quite large: see equation ( 4 ). Normally , the water pressure is limited and, therefore, to a void lo cal b oiling and corresponding ca vitation damage, a minimum flo w rate must b e guaranteed. On top of that, the design of a suitable vehicle radiator is muc h complicated for large oscillations, and fluctuating w ater inlet temp eratures are undesirable for real engines. The transfer function for the w ater pump in Laplace space is 6 G p = 1 τ p s + 1 , (5) with a characteristic time constant τ p , which was set to 0 . 2 s, resulting in a base fre- quency of 5 Hz: the allow able range for the water mass flo w rate during the simulation is qualitativ ely sho wn in fig. 2 . Figure 2: Allo wable range for the water mass flow rate during the simulation in the time ( ˙ m w | min and ˙ m w | max ) and frequency (5 Hz) domain. 2.3. Detaile d thermal mo del - CFD-CHT metho d The detailed finite volume model of the engine consists of all relev ant engine components lik e the crank case, cylinder heads, v alves, pistons, as w ell as the cylinder liners. Details ab out the mo del can b e read in [ 27 ]. The transient, thermal boundary conditions in the com bustion cham b ers are giv en according to equation ( 2 ) with some spatial extensions. Details can b e found in [ 28 ]. In addition, one can find the used meshing strategy , including lo cal mesh refinements, and a mesh study for the cylinder head. The thermal b oundary conditions for the engine water jack et are given by the SST k - ω turbulence mo del by Men ter [ 32 ]. The adaption of the water reference temp erature is also mo delled according to equation ( 4 ). The operating p oint of the w ater pump is also limited according to fig. 2 . Therefore, the transfer function ( 5 ) is used. 2.4. F e e d forwar d str ate gy The blo ck diagram for the feed forward strategy is shown in fig. 3 . The time-dependent disturbance z ( t ) on the engine are transient heat fluxes caused by alternating firing and coasting conditions. The control v ariable is the cylinder head temp erature T Cyl of 7 a represen tative measuring p osition close to the combustion cham b er. The correcting v ariable for all scenarios is the water mass flow rate ˙ m w . T Cyl,t is the command v ariable. ˙ m n,w is the nominal water mass flo w rate. Figure 3: Blo c k diagram for the feed forw ard strategy . Neglecting changes in the water temp erature according to equation ( 4 ), and setting ˙ T Cyl = 0 and T Cyl = T Cyl,t in equation ( 3 ), it follows with equation ( 1 ) for the nominal w ater mass flo w rate ˙ m n,w ( t ) =  χ h α c i ( t ) ( T mod ( t ) − T Cyl,t ) − ( α ref / ˙ m w | m ref ) ( T w,i − T Cyl,t )  1 /m . (6) Due to fig. 2 , there are some restrictions regarding the feasible mass flo w. 2.5. Contr ol str ate gy The blo ck diagram for the closed control lo op is shown in fig. 4 . It is assumed that there are no measurement errors or dela y elemen ts in the feedback path. Figure 4: Blo c k diagram for the closed con trol loop. The controller is executed as PID control device with following transfer function in Laplace space: G c = k P + k I s + k D s, (7) with k P , k I and k D as the con trol parameter for the v arious comp onen ts of the controller. Equation ( 7 ) can b e rewritten in terms of tw o time constan ts T R,1 and T R,2 : G c = k I (1 + T R,1 s )(1 + T R,2 s ) s , (8) with k P = k I ( T R,1 + T R,2 ) and k D = k I T R,1 T R,2 . When c ho osing suitable control pa- rameters, it must b e noted that equation ( 3 ) is non-linear in the water mass flow rate 8 ˙ m w . Therefore, in order to mo del the engine as a PT1-element with a c haracteristic time constan t τ e and a static amplification k s,e , a linearization at a representativ e op erating p oin t is necessary . Neglecting changes in the water reference temp erature according to equation ( 4 ), it follows for the engine temp erature difference ∆ T Cyl C v ρ ∆ x α w,0 | {z } = τ e ∆ ˙ T Cyl + ∆ T Cyl = T w,0 − T Cyl,0 α w,0 ∆ α w , (9) = T w,0 − T Cyl,0 α w,0 0 . 7 ˙ m − 0 . 3 w,0 α ref ˙ m w | ref | {z } = k s,e ∆ ˙ m w , (10) with ∆ T Cyl = T Cyl − T Cyl,0 . The subscript 0 corresp onds to the representativ e op erating p oin t. The last step is the result of equation ( 1 ). As a first appro ximation for the control parameters, the rule according to Kessler [ 21 ] w as used. 9 (a) (b) (c) Figure 5: First appro ximations for the control parameters of the PID control device as functions of the water mass flow rate in the represen tativ e op erating p oin t ˙ m w,0 and the resulting system time constant of the closed control lo op τ c = 1 / ( k I k s,e ). Comp ensating b oth the time constants of the water pump τ p and the engine τ e , the transfer function for the op en control lo op G o and the closed lo op G cl in Laplace space is G o = k I k s,e s , (11) G cl = G o 1 + G o = 1 1 + 1 k I k s,e s . (12) As an example, fig. 5 shows the three control parameters of the PID control device for 10 a representativ e engine op erating p oint. The given v alues should b e regarded as first, p ossible approximations, b ecause of the strong non-linearity and the ev er c hanging engine states. According to equation ( 9 ) and ( 10 ), representativ e temp eratures T w,0 = 373 K and T Cyl,0 = 419 K are used for the time constant τ e and the amplification k s,e . The con trol parameters are giv en as functions of the w ater mass flow rate in the represen tative op erating p oin t ˙ m w,0 and the resulting system time constant of the closed control lo op τ c = 1 / ( k I k s,e ). 2.6. F e e d forwar d and c ontr ol str ate gy Of course, a combination of the aforementioned strategies is p ossible. Fig. 6 shows the resulting block diagram. Figure 6: Blo c k diagram for the feed forw ard strategy including a closed control lo op. 3. Results and discussion 3.1. F e asible temp er atur e r ange and c onsistency Firstly , the feasible temp erature range and the temp oral b eha viour was in v estigated by means of the lump ed capacity mo del. Therefore, the minimum w ater mass flow rate w as set to 0.25 kg/s, whereas the maximum rate was 4.5 kg/s. As can b e seen in fig. 7 a), a temperature range about 80 K can b e realized b y using a water pump with a v ariable w ater mass flow rate. The limitations result from the allow able parameter space in fig. 2 . 11 a) b) c) Figure 7: Lump ed capacity mo del: feasible temp erature range by using a PID controller with k P = − 1 . 4 kg/(Ks), k I = − 0 . 05 kg/(Ks 2 ) and k D = − 1 . 0 kg/K. a) Mean temp era- ture. b) Standard deviation of the temperature. c) Exemplary temp erature curves. The ratio of the (reference) mec hanical water pump w as set to reach a maxim um temp erature of 407 K. Ho wev er, as can b e seen in fig. 7 b), the standard deviation of the temp erature rises strongly with higher target temp eratures. Allowing a maxim um v alue of 6 K, whic h can also b e reached b y a conv entional mechanical water pump, a range of 60 K can b e realized. The reasons are the same as for fig 7 a): T argeting a comp onen t temp erature of 380 K, the resulting water mass flow rate was constan t at 4.5 kg/s. One gets exactly the opp osite result for a target temp erature of 470 K, e.g., a nearly constant mass flow rate of 0.25 kg/s. How ev er, the temp orary , lo cal temp erature minima are deep er b ecause of larger thermal differences b etw een firing and coasting engine conditions: fig. 7 c) shows some exemplary temp erature curv es for v arious target temp eratures. 12 3.2. Monte Carlo simulation Fig. 8 to 11 show some results of a Monte Carlo simulation for a constant target tem- p erature of 407 K. In order to inv estigate the sensitivity of the controller parameters according to fig. 4 , the corresp onding v alues of k P , k I and k D w ere v aried systematically . The results in fig. 5 serve as starting v alues. Figure 8: Lump ed capacity model: Monte Carlo sim ulation for a target temp erature of 407 K in the k P - k I space. The resulting cylinder head temp erature is shown. 13 Figure 9: Lump ed capacity model: Monte Carlo sim ulation for a target temp erature of 407 K in the k P - k D space. The resulting cylinder head temp erature is shown. Regarding regular temp erature curv es under a transient driv e, high k P and small k I absolute v alues are exp edient, see fig. 8 . Ho wev er, k P v alues higher than 0.6 do not pro vide a significan t adv antage if the k I v alue is small enough. Because op en lo ops without in tegrating parts do not lead to stationary accurate closed loops, k I lev els close to zero are not recommended: the limit theorem for Laplace-T ransformations lim t →∞ y ( t ) = lim s → 0 sY ( s ) , (13) for a function y and its Laplace transform Y , requires following condition for stationary accuracy: G cl (0) = G o (0) 1 + G o (0) = 1 . (14) This can b e only fulfilled with lim s → 0 G o ( s ) → ∞ . Therefore, k I v alues different to zero are necessary . V alues ab out -0.05 and -0.1 are recommended. In the k P - k D space, a similar b eha vior can b e observed: high k P and k D absolute v alues are exp edient. With v alues higher than 1.2 and 0.3, no significant adv antages can b e observed, see fig. 9 . Regarding hydraulic p o wer losses, a similar result can b e observed in fig. 10 and 11 . 14 Concerning a holistic consideration, e.g., taking into account the maxim um h ydraulic p o w er, its v ariance and mean v alue, a similar range in the in the k P - k I - k D space is recommended. Figure 10: Lump ed capacity mo del: Monte Carlo sim ulation for a target temp erature of 407 K in the k P - k I space. The resulting h ydraulic p o wer is shown. 15 Figure 11: Lump ed capacity mo del: Monte Carlo sim ulation for a target temp erature of 407 K in the k P - k D space. The resulting h ydraulic p o wer is shown. 3.3. Dynamic effe cts On the comparability of the following strategies, the b oundary condition for all cases is the same: a fixed maximum engine temp erature under full load state at the e nd of straigh t track. As can b e seen in fig. 12 , the pure feed forward strategy decreases signif- ican tly the temp erature standard deviation. Esp ecially in coasting situations or in the b eginning of acceleration phases, the temp erature curve is smo other. While accelerating out of corners, a further adv antage can b e gained if a controller is used simultaneously . Using a higher integrating and no differentiating part for the PID con troller, a certain temp erature o vershoot can b e realized, which results, with regard to a mechanical wa- ter pump, in a slightly higher heat saving. According to the lump ed capacity mo del, a heat saving p otential ab out one kilow att is p ossible. This corresp onds to a share of appro ximately 2.5 percent of the total heat loss. 16 Figure 12: Lump ed capacit y mo del: comparison of four different feed forward and con- trol strategies concerning heat saving p oten tials and temp erature consistency under a transien t driv e. 3.4. T emp er atur e shifting and r esulting he at saving Next, different target temp eratures are used and compared to a mechanical water pump. As can b e seen in fig. 13 , according to the lumped capacit y model, a heat saving potential of five kilow atts can b e ac hieved by shifting the temp erature up to 470 K. This corre- sp onds to a share of appro ximately 10 p ercent of the total heat loss. In terestingly , a very smo oth temp erature curve can b e realized if one uses anticyclical target temp eratures, e.g., a higher target temperature under coasting conditions than under fired situations. 17 Figure 13: Lump ed capacity mo del: comparison of v arious control strategies and target temp eratures concerning heat saving potentials and temp erature consistency under a transien t driv e. 3.5. Thr e e-dimensional r esults In the following, according to fig. 1 , transient finite volume simulations are presen ted compared with the lump ed capacity mo del. a) b) Figure 14: Detailed 3D simulation: feasible temp erature range by using a PID controller with k P = − 1 . 4 kg/(Ks), k I = − 0 . 05 kg/(Ks 2 ) and k D = − 1 . 0 kg/K. a) Mean tem- p erature. b) Mean differences in the temp erature standard deviation with regard to a mec hanical pump. The ratio of the (reference) mechanical water pump was set in order to reac h a maximum temp erature of 407 K. 18 Compared to fig. 7 , mean temp erature results, together with their standard deviations, for the detailed FVM simulation are shown in fig. 14 . In the case of the detailed sim ulation, one can notice a larger temp erature range for the feasible mean temp erature. One p ossible reason can b e the additional heat flux due to heat conduction. It seems that, with higher mean temp eratures, its relative weigh ting falls in relation to the heat flux due to water co oling. This can b e explained with the higher ov erall temp erature lev el. As a consequence, the water cooling becomes more important, e.g., the temp erature difference b etw een solid and water has to b e larger. With regard to fig. 7 b) with fig. 14 b), one can notice a smaller sensitivity with regard to the temp erature v ariance. One argumen t can also b e the additional heat flux due to heat conduction, and, therefore, the lo wer sensitivit y tow ards water mass flo w rates. In addition, the temp erature gradients in the detailed simulation cause a kind of filtering function. In the case of the lump ed mo del, there exist no spatial gradien ts, and, therefore, ev ery change in the boundary conditions directly effects the solid temperature. An ov erview of the used measurement p ositions is given in fig. 15 : all p ositions are lo cated within the cylinder head and one millimeter b elow the surfaces. The num b ers indicate different cylinders. In this case, the command v ariable was related to the measurement p osition ”Indy 1”. The second setting which is called 380 K (*) corresp onds to the an ticyclical target temp erature strategy from fig. 13 . F or the detailed simulation, it can be observed that a larger feasible temperature range can b e realized. T emp erature v alues up to 510 K are possible. Compared to the mec hanical w ater pump, low er v alues for the standard deviation of the engine temp eratures are reachable. On the other hand, the increase with higher target temp eratures is not that high. A key element app ears to b e that the other five measuremen t p oin ts near the combustion cham b er b eha ve in a manner similar to that of the command v ariable ”Indy 1”. Figure 15: Overview of the measurement p ositions in the detailed FVM sim ulation. Compared with fig. 13 , transient heat saving potentials for the detailed sim ulations are sho wn in fig. 16 . Under dynamic op erating conditions, heat savings up to 8 kW are p ossible for a target temp erature of 510 K. This corresp onds to a share of approximately 15 p ercen t of the total heat loss. Because of the large standard deviation at such high temp erature levels, this p oten tial drops b elow 4 kW under coasting conditions. 19 a) b) Figure 16: Detailed 3D simulation: a) Heat saving p oten tial with regard to a mechanical pump. The PID controller parameters are the same as in fig. 14 . a) T ransien t heat sa ving p otential. b) Mean heat sa ving p otential with corresponding spatial distribution. The ratio of the (reference) mec hanical water pump was set in order to reac h a maxim um temp erature of 407 K. As can b e seen in fig. 16 b), most of the heat is sa ved in the exhaust channels and the cylinder liners. Surprisingly , the cylinder head deck area has a small prop ortion. The higher heat saving p oten tial in the case of the detailed simulation, as can b e seen in fig. 13 and fig. 16 , is a result of higher, feasible mean temp eratures. Compared to fig. 12 , the detailed FVM sim ulations sho w a higher transient heat sa ving p oten tial in acceleration phases. As can b e seen in fig. 17 , the temp erature swing strategy delivers v alues up to 2 kW. Regarding the integrating part of the controller, it is important to note, ho wev er, that the oscillation b eha vior is more pronounced. 20 Figure 17: Detailed 3D sim ulation: comparison of four different feed forw ard and con trol strategies concerning heat saving p oten tials under a transient driv e. In terestingly , with respect to the lumped capacit y mo del, the difference b et ween the pure feed forward strategy and the com bination of pre-control and feedback control is higher. In general, the lumped capacit y model underestimates the temperature swings: the low er temp eratures during coasting are ov erestimated in the lump ed mo del. One p ossible explanation can b e the missing heat flux due to heat conduction, which is only contained in the three-dimensional sim ulation. As a result of this lo wer temperature difference b et w een the mechanical pump and the control strategy , the heat saving p oten tial is higher for the detailed simulation. In addition, the real, inhomogeneous distribution of the heat flux vector around the combustion cham b er cannot b e realized with a lump ed mo del approach. 4. Conclusions Concerning the initial research questions, the results are as follows. Within the limits of 0.25 and 4.5 kg/s for the water mass flow rate, a feasible range for the mean tem- p eratures of engine comp onen ts near the com bustion cham b er is 130 K. Compared with a conv entional mec hanical water pump, the standard deviation can b e reduced up to 6 K, which can reduce p ossible damage mechanisms through lo cal plastic deformations, e.g., low cycle fatigue or thermo-mechanical fatigue, resp ectiv ely . F or the highest mean temp erature, the standard deviation w as 6 K higher. A very imp ortant finding was the fact that, b y using one command v ariable, the other measuremen t p oin ts near the com- bustion c ham b er behav e in a similar manner. Concerning heat fluxes in full load states, it could b e shown that a heat saving p oten tial ab out 8 kW is p ossible. This corresp onds 21 to a share of approximately 15 p ercen t of the total heat loss. In re-acceleration phases, v alues ab out 2 kW are realistic. Most of the heat losses can b e reduced in the exhaust c hannels and the cylinder liners: 75 p ercent of the complete heat flow is reduced in this area. Of the remaining 25 p ercen t, half of the heat is reduced at the cylinder head dec k area. V alves and their rings contribute equally to the rest. Concerning the tw o differen t modelling ideas, the qualitativ e progressions of relev ant ph ysical quantities are v ery similar. Therefore, the lump ed capacity model can b e used for optimization and sensitivity calculations: the Mon te Carlo simulation has sho wn v aluable correlations for the control device parameters, which could b e effectively used in subsequent three-dimensional s im ulations. Compared to these detailed simulations, the feasible range for the mean temp eratures is underestimated b y 50 K. Analogously , the profit concerning standard deviations is sligh tly underestimated b y 1 K. How ever, for high temp erature lev els, it is o v erestimated b y 6 K. With regard to heat sa ving potentials, the simplified mo del gives to o lo w v alues for b oth the dynamic and the quasi-stationary case: In the first one, the difference is 3 kW, in the second one, it is 1 kW. The direct influence on the combustion, either p ositiv ely affected by reduced heat losses, and stabilized temp erature conditions, or negatively affected by higher inlet air temp er- atures, resulting in different kno cking tendencies and volumetric efficiencies, has to b e in vestigated exp erimen tally . In addition, the mo del v alidity and its scop e of application, sho wn in fig. 2 , should b e extended b y the laminar transition regime, as w ell as b y higher frequencies. 5. Nomenclature Sym b ol Description Unit A Realisation of random v ariable α W/(m 2 K) C v Sp ecific heat at constan t v olume J/(kgK) C p,w Sp ecific heat at constan t pressure for water J/(kgK) f F requency Hz G p T ransfer function for the water pump dimensionless G c T ransfer function for the controller kg/(Ks) G o T ransfer function for the op en control lo op kg/(Ks) G cl T ransfer function for the closed con trol loop kg/(Ks) k P P-comp onen t of the controller kg/(Ks) k I I-comp onen t of the controller kg/(Ks 2 ) 22 Sym b ol Description Unit k D D-comp onen t of the controller kg/K k s,e Static amplification engine (Ks)/kg m Reynolds exponent dimensionless ˙ m w W ater mass flo w rate kg/s ˙ m n,w Nominal v alue of the w ater mass flo w rate kg/s ˙ m w | max Maxim um w ater mass flo w rate kg/s ˙ m w | min Minim um w ater mass flo w rate kg/s ˙ m w | ref Reference w ater mass flow rate kg/s n Engine speed rounds per minute [rpm] p α | n Conditional probabilit y density function on α with regard to n dimensionless P r Prandtl n um b er dimensionless ˙ Q Heat flux W Re Reynolds n umber dimensionless s Laplace co ordinate 1/s t Ph ysical time s T Cyl Cylinder head temp erature K T Cyl,t Command v ariable of the cylinder head temp erature K T gas Reference (gas) temp erature K T mod Statistically modified reference (gas) temp erature K T w,i Inlet w ater temperature K T w W ater reference temperature K T R,1 Time constan t of control device s 23 Sym b ol Description Unit T R,2 Time constan t of control device s Y Laplace transform of y v arious y Arbitrary function v arious z Disturbances of all kinds on the engine v arious x P osition v ector m ∆ x Characteristic w all thic kness m Greek symbols Sym b ol Description Unit α Heat transfer coefficient W/(m 2 K) α ref Reference heat transfer co efficien t W/(m 2 K) α w Effectiv e heat transfer co efficien t for the w ater c han- nel W/(m 2 K) α c Effectiv e heat transfer co efficien t for the combustion c hamber W/(m 2 K) ρ Mass densit y kg/m 3 τ p Time constan t of water pump s τ e Time constan t of engine s τ c Time constan t of closed lo op s χ Ratio b et w een effective surfaces of the combustion c hamber and the water jack et dimensionless Mathematical Notation Sym b ol Description ∂ ( · ) ∂ t P artial time deriv ativ e 24 Sym b ol Description ∂ ( · ) ∂ x Spatial gradien t h·i Exp ectation v alue regarding time Abbreviations Abbreviation Description C F D C omputational F luid D ynamics C H T C onjugate H eat T ransfer F V M F inite V olume M ethod H T C H eat T ransfer C oefficient AC T A v erage C ylinder T emperature 6. 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