Politecnico Di Milano, Italy Modelling Heat Exchangers By .
Proceedingsof the 4th International Modelica Conference,Hamburg, March 7-8, 2005,Gerhard Schmitz (editor)S. Micheletti, S. Perotto, F. SchiavoPolitecnico di Milano, ItalyModelling Heat Exchangers by the Finite Element Method with GridAdaption in Modelicapp. 219-228Paper presented at the 4th International Modelica Conference, March 7-8, 2005,Hamburg University of Technology, Hamburg-Harburg, Germany,organized by The Modelica Association and the Department of Thermodynamics, Hamburg Universityof TechnologyAll papers of this conference can be downloaded Program Committee Prof. Gerhard Schmitz, Hamburg University of Technology, Germany (Program chair). Prof. Bernhard Bachmann, University of Applied Sciences Bielefeld, Germany. Dr. Francesco Casella, Politecnico di Milano, Italy. Dr. Hilding Elmqvist, Dynasim AB, Sweden. Prof. Peter Fritzson, University of Linkping, Sweden Prof. Martin Otter, DLR, Germany Dr. Michael Tiller, Ford Motor Company, USA Dr. Hubertus Tummescheit, Scynamics HB, SwedenLocal Organization: Gerhard Schmitz, Katrin Prölß, Wilson Casas, Henning Knigge, Jens Vasel,Stefan Wischhusen, TuTech Innovation GmbH
Modelling Heat Exchangers by the Finite Element Method with Grid Adaption in ModelicaModelling Heat Exchangers by the Finite Element Method withGrid Adaption in ModelicaStefano Micheletti , Simona Perotto , Francesco Schiavo†Politecnico di Milano,P.zza Leonardo da Vinci 3220133 Milano, ItalyAbstractPDEs discretization, adopting either a finite volumemethod (FVM) or a finite element method (FEM), withIn this paper we present a new Modelica model for different strategies for single-phase or two-phase fluidheat exchangers, to be used within the ThermoPower flow [5]. Furthermore, a moving-boundary evaporatorlibrary. The novelty of this work is a combined em- model has been recently added to the library.ployment of finite elements with grid adaption.The modelling of a generic single-phase 1-D heat exchanger is discussed, along with its approximation via In this paper we present a new model for single-phasethe Stabilized Galerkin/Least-Squares method. The HEs, based on the use of the finite element methodgrid adaption procedure is first introduced from a gen- with grid adaption. The objectives of this work areeral viewpoint and then within the Modelica frame- twofold: to develop a new HE model with high accuracy and reduced computational complexity and towork. Finally, some preliminary results are shown.show how complex mathematical techniques can besuccessfully used in Modelica for the modelling ofdistributed-parameters physical systems.1 IntroductionHeat exchangers (HEs) play a relevant role in manypower-production processes, so that their accuratemodelling, at least for control-oriented analysis, is akey task for any simulation suite [13].Accurate modelling of such devices is usually a complex task, the reason being that the control-relevantphenomena are associated with thermal dynamics described by Partial Differential Equations (PDEs). Onthe other hand, different complexity levels of representation may be necessary, depending on the specificsimulation experiment to be performed.Within this framework, the power-plant modelling library ThermoPower [5] exploits the Modelica language modularity features, offering to the users several interchangeable component models, with varyinglevels of detail.As for the HEs, the models currently provided are differentiated by the numerical scheme employed for theThe proposed model is an improvement of the actualFEM model [6], obtained by a grid adaption technique: the grid nodes (i.e., the points where the solution is computed) change their positions so as to adaptdynamically to the solution variations. Such modelcan significantly improve the modelling accuracy, byremoving the non-physical solution oscillations observed for the actual FEM model, whilst using fewernodes and containing the computational burden.The paper is organized as follows: in Section 2.1 werecall the modelling of a generic single-phase 1-D heatexchanger, while in Section 2.2 we discuss its approximation via the Stabilized Galerkin/Least-Squaresmethod. In the third section the grid adaption problemis introduced from a general viewpoint, while in Section 4 we address the moving mesh method on whichthe Modelica implementation, analyzed in Section 5, MOX,Dipartimento di Matematica “F. Brioschi”, is based. Some preliminary numerical results are ded in Section 6. Finally, the last section drawsmi.it† Corresponding author, Dipartimento di Elettronica e Infor- some conclusions and outlines possible future developments.mazione, francesco.schiavo@elet.polimi.itThe Modelica Association219Modelica 2005, March 7-8, 2005
S. Micheletti, S. Perotto, F. Schiavo2 The Heat Exchanger Modelmomentum and energy can be formulated as follows:In the context of object-oriented modelling, it is convenient to split the model of a generic heat exchanger(HE) into several interacting parts, belonging to threedifferent classes [5]: the model of the fluid within agiven volume, the model of the metal walls enclosingthe fluid and the model of the heat transfer between thefluid and the metal, or between the metal and the outerworld. In this paper, we focus on the modelling of thefirst class. We improve the framework proposed in [6]by introducing suitable grid adaption techniques.The model presented in this paper can represent singlephase HEs, which constitute a significative part of theindustrial applications (e.g., the primary side of a Pressurized Water Reactor nuclear power plant [3]). However, also two-phase flows could be handled as well.2.1 The Fluid ModelLet us deal with a compressible fluid within a pipeshaped volume V with a rigid boundary wall, exchanging mass and energy through the inlet and outletflanges, and thermal energy through the lateral surface.We assume that ρ w 0, t x1 w pdz C f ωw w 0 , ρg A t xdx 2 ρA3 h w h 1 p ω φe , t ρA x ρ t ρAA(1)(2)(3)where A is the pipe cross-sectional area, ρ the fluiddensity, w the mass flow-rate, p the fluid pressure, gthe acceleration of gravity, z the pipe height, C f theFanning friction factor, ω the wet perimeter, h the specific enthalpy, φe the heat flux entering the pipe acrossthe lateral surface. The fluid velocity can be definedas u w/(ρA). Notice that in (2) and (3) we have neglected the kinetic and the diffusion term, respectively.In the case of water-steam flows it is convenient tochoose the pressure and the specific enthalpy as thethermodynamic state variables, so that the expressionsof the balance equations have the same form for singlephase and two-phase flows [12]: thus all the fluid properties, such as the temperature T , the density ρ and thepartial derivatives ρ/ h and ρ/ p can be computedas functions of p and h. the longitudinal dimension x is far more relevant 2.2 The Approximation Procedurethan the other two;In view of power generation plant modelling, the mostrelevant phenomenon is described by equation (3), so the volume V is “sufficiently” regular (i.e., the that the focus for the present paper is the approximacross-sectional area is uniform and V is such that tion of this latter by FEM and grid adaption. Actuthe fluid motion along x is not interrupted);ally, the mass and momentum equations (1) and (2) describe the fast pressure and flow rate dynamics, while there are no phase-changes (that is the fluid is al- the energy one (3) describes the slower dynamics ofways either single-phase or two-phase);heat transport by the fluid velocity. These faster modesare typically not taken into account in HEs modelling the Reynolds number Re is such that turbulent [6]. In particular, note that, assuming the pressure pflow conditions are assured along all the pipe, uniform along x (with possible jumps at the HE boundwhich in turn guarantees almost uniform veloc- ary) and neglecting the inertial term w/ t in (2), theity and thermodynamic state of the fluid across integration of the mass and momentum balance equations (1) and (2) is reduced tothe radial direction.Notice that, when water or steam is assumed as theworking fluid, the last hypothesis does not hold atvery low flow rates (laminar flow regime). However,in practice, most industrial processes never operate insuch conditions.Under the hypotheses above it is possible to define allthe thermodynamic intensive variables as functions ofthe longitudinal abscissa x and time t. Within thisframework, the dynamic balance equations for mass,The Modelica Associationwin wout Apin poutZ L ρdx , t pF pH ,(4)0(5)where win , wout , pin , and pout are the mass flow-rateand pressure at the HE inlet and outlet, while pF and pH are the pressure drops due to friction and fluidhead, respectively. For further details on the approximation for equation (1) and (2) we refer to [6].220Modelica 2005, March 7-8, 2005
Modelling Heat Exchangers by the Finite Element Method with Grid Adaption in Modelicadependent. This unavoidably leads to an increase ofthe number of unknowns since the displacement of thegrid nodes is to be determined as well.As for the test functions involved in the GALS method,they are defined byψi (x,t) ϕi (x,t) Figure 1: Some typical hat functionsEquation (3) is discretized with the stabilized PetrovGalerkin method GALS (Galerkin/Least-Squares), using suitable Dirichlet weak boundary conditions at theinflow [11].We refer to [6] for further details about the applicationof the GALS method to heat exchangers.In the following we provide some details about the approximation procedure by means of piecewise linearfinite elements of equation (3), while referring to [16]for an exhaustive coverage of the finite element approximation theory.We remark that we generalize the standard GALSmethod to the case of time-dependent shape and testfunctions, since, using the grid adaption strategy, thelength of each mesh element varies in time.Let the spatial domain [0, L] be subdivided into N 1elements identified by N ( 3) nodes. The length ofthe i-th element is denoted as i (t), while the abscissaof the i-th node is indicated in the sequel with δi (t).On this partition we introduce the space of the piecewise linear functions, whose typical basis (hat) functions are shown in Fig. 1.Their analytical expressions are the following:ϕ1 (x,t) δ2 (t) x 1 (t)0ϕi (x,t)where α (0 α 1) is a stabilization coefficient. Notice that for α 0 the standard (i.e., non-stabilized)method is obtained.For the reader’s ease, we provide also the expression ofthe time derivative ϕ̇i ϕi (x,t)/ t of the basis function ϕi , namely δ̇i 1 (x δi 1 ) i 1 2i 1 ϕ̇i (x,t) δ̇i 1 (δi 1 x) i 2i 0δi x δi 1 ,otherwise .(9)Nh(x,t) hi (t)ϕi (x,t) h(t)T ϕ(x,t), h [h1 · · · hN ]T ,i 1Nρ(x,t) ρi (t)ϕi (x,t) ρ(t)T ϕ(x,t), ρ [ρ1 · · · ρN ]T ,i 1Nw(x,t) wi (t)ϕi (x,t) w(t)T ϕ(x,t), w [w1 · · · wN ]T ,i 1Ni 1(10)otherwise ,with ϕ̄(x,t) [ϕ1 (x,t), · · · , ϕN (x,t)]T .Applying the GALS finite element method to (3) leadsto the following set of N ODEs:NZ Lδi 1 (t) x δi (t) , ḣi ϕi0δi (t) x δi 1 (t) ,with i 2, · · · , N 1 and wherei 1 j (t) , for i 1 . . . N .(7) Ωinṗ0 Ni 1 ρi ϕiZNotice that, in view of the grid adaption procedure, thebasis functions defined in (6) are both space and time221Z L Ωin0N hi ϕ̇i!ψ j dx i 1!dϕi hi dx ψ j dx i 1! Ni 1 wi ϕiA Ni 1 ρi ϕiZ Lj 1ψ j dx Ni 1 wi ϕiA Ni 1 ρi ϕi0otherwise ,!i 1Z L(6)The Modelica Associationδi 1 x δi ,Let us expand the quantities h, ρ , w, φe in terms of thebasis functions ϕi as:Zδi (t) (8)φe (x,t) φi (t)ϕi (x,t) φ(t)T ϕ(x,t), φ [φ1 · · · φN ]T ,0 x 1 (t) , x δN 1 (t)δN 1 (t) x L ,ϕN (x,t) N 1 (t)0otherwise , x δi 1 (t) i 1 (t)δi (t) x i (t) 0α ϕi (x,t),2 xNN hi ϕiψ j dx Z Lω Ni 1 φi ϕi0A Ni 1 ρi ϕii 1ψ j dx ! Ni 1 wi ϕihin ψ j dx,A Ni 1 ρi ϕi!ψ j dx ψ j with j 1, · · · , N ,(11)Modelica 2005, March 7-8, 2005
S. Micheletti, S. Perotto, F. Schiavowhere hin is the fluid specific enthalpy at the inflow of FEM, can be cast into the framework of model reboundary Ωin . Such set of ODEs can be represented duction, i.e., the approximation by a finite dimensionalby the following compact matrix notation:model of a conceptually infinite dimensional one. Several parameters (e.g., the mesh spacing, the degree of11ω1M h MD h F h C h R ṗ Y φ K w , (12) the polynomial finite elements, tuning parameters reAAAAlated to the discretization procedure) govern the acwhere M, MD , F, C, R, Y , K are defined as follows:curacy of the approximation. As an effective toolto assess such approximation property, some estimaZ LZ Ltors/indicators, as the local cell residual, are typicallyM ji ϕi ψ j dx , MD ji ϕ̇i ψ j dx ,employed [1, 9, 18]. Once the error indicator has been00computed on a given mesh, the information that it conZ L N k 1 wk ϕk dϕitains can be used to generate a better mesh that givesFji ψ j dx ,N0 k 1 ρk ϕk dxmore accuracy. This is the basis of adaptive error conZtrol. Nk 1 wk ϕkC ji ϕi ψ j dx ,NMany engineering problems are characterized by solu Ωin k 1 ρk ϕktions exhibiting a complex structure, e.g., singularitiesZ LZ Lϕiψjnear corners, boundary layers or shocks. In such cases,dx,Y ψdx,Rj jijNN0 k 1 ρk ϕk0 k 1 ρk ϕkthe idea is to distribute the mesh spacings accordingZhinto local features of the solution, that is to concentrateϕi ψ j dx .K ji Ninthe elements in the regions where the solution changes Ω k 1 ρk ϕk(13) rapidly and, vice versa, to coarsen them where the solution is smoother, with the aim of obtaining a solutionThe matrices C and K, which enforce the boundary sufficiently accurate and with a reasonable computaconditions into equation (12), depend on the inflow tional load.boundary Ωin . It can be noted that, as we are consid- Typically an adaptive error control procedure consistsering the 1-D case, the inflow boundary is constituted,of a discretization method combined with an adaptiveat most, by the points x 0 and x L, depending onthe sign of w win . Thus the only test functions that algorithm. There are three main types of adaptive techare non-zero at the inflow are ψ1 and ψN and the only niques for FEM: i) the h-method: the mesh is refinedand coarsened locally according to certain error esnon-vanishing entries of the matrices C and K aretimators; ii) the p-method: the polynomial degree is ! wα1 chosen in each element according to some smoothness1 w x 0 0 ,C11 ρ12indicator; iii) the r-method: the element vertices are 0otherwise ,relocated to concentrate them in desired regions on thebasis of a monitor function. ! α wNIn the following we focus on this last philosophy1 w x L 0 ,CNN ρN2which is usually referred to as moving mesh method 0otherwise ,[7, 10, 14, 15, 17]. In this method, a mesh equation in !(14)volving the nodes speed is solved to compute the mesh α hin x 01 w x 0 0 ,points location together with the solution of the differK11 ρ12 ential equation at hand. In principle, starting from a0otherwise ,given mesh, the idea is to move the mesh nodes, while !keeping their number fixed, towards regions of rapid α hin x L1 w x L 0 ,solution variations, e.g., steep wave fronts and shocks.KNN ρN2 0otherwise .3.1Grid Adaption as a Control ProblemThe matrices C and K are consequently diagonal.An interesting point of view to tackle the grid adaption procedure is to state it as a control problem. As amatter of fact, the grid adaption is based on a feedback3 The Grid Adaption Philosophymechanism that can be represented as in Fig. 2.The discretization of complex phenomena described Within this framework, the process is represented byby systems of partial differential equations by means the N ODES obtained from GALS discretization, theThe Modelica Association222Modelica 2005, March 7-8, 2005
Modelling Heat Exchangers by the Finite Element Method with Grid Adaption in ModelicaFigure 3: The spring model for grid adaptionFigure 2: Grid adaption as a control problemsensor is represented by some estimate of the discretization error and the controller is defined by thegrid adaption strategy. The time-varying boundaryvalues for the HE (hin and win ) and the heat flux entering its lateral surface (φe ) are, from the point of viewof feedback grid adaption, process disturbances, whilethe length of the elements ( i ) can be regarded as the(vectorial) control variable .The aim of the control system is to minimize the estimated error. In this paper we adopt the equidistribution principle [2] to design the controller (i.e., themesh adaption strategy): the aim is to dynamically obtain an equidistributed error over the elements.4The Moving Mesh Method in ModelicaThe application of the GALS method to equation (3)leads to a set of N ODEs whose unknowns are thenodal values for the fluid specific enthalpy. Moreover, due to the grid adaption strategy, we have to include other N 1 unknowns, i.e., the lengths i of theelements. The coupled equations yield the so-calledDAE-system.The mesh point positions have to be calculated in sucha way thatto correct such values so that the constitutive and completeness constraints are satisfied.On the other hand, when dealing with a declarativelanguage such as Modelica, a different approach hasto be taken: the constitutive and completeness constraints have to be intrinsically fulfilled. Such resultcan be easily obtained using a physical approach forthe implementation of the adaption procedure.Let us consider Fig. 3: each element can be identifiedwith a spring of length i and specific elastic constantki , with the first and the last spring fixed to the domainboundaries x 0 and x L, respectively.Let Fi, j be the force that the i-th spring exerts on thej-th one. Usually it is assumed thatFi, j 0 j 6 i 1, i 1 ,(15)that means that each spring interacts only with the twoadjacent ones. Furthermore, the force that two adjacent springs exert on each other can be expressed asFi,i 1 ki iFi 1,i ki 1 i 1 .(16)Supposing that the spring constants ki are nonnegative, an effective choice for the unknowns i interms of the ki is: i kiL,N 1 j 1 k j i 1···N 1 .(17)This automatically guarantees the completeness constraint as1) the length of each element is strictly positive (constitutive constraint: i 0 i 1 · · · N 1 , t 0 );N 1N 1kiLN 1i 1 j 1 k j i i 1 L.(18)2) the total length of the elements is equal to L (com- Moreover, if all the spring constants are positive, thenpleteness constraint: N 1the constitutive constraint is fulfilled as well. It is imi 1 i L , t 0 ).portant to notice that such strategy is independent ofThese constraints can be easily fulfilled when dealing the particular grid adaption procedure at hand.with imperative languages (i.e., algorithm oriented). To make effective the chosen adaption procedure it isIn such a case, a specific grid adaption procedure is necessary to relate the elastic constants ki to the localfirst allowed to yield a mesh characterized by values monitor function εi , asfor the lengths i “illegal” with respect to the criteria1) and 2). Then a suitable refinement algorithm is usedThe Modelica Association223ki 1, i 1, · · · , N 1 . i εi(19)Modelica 2005, March 7-8, 2005
S. Micheletti, S. Perotto, F. SchiavoThe strategy we adopt aims at concentrating the gridpoints in the domain regions where the monitor function ε is larger. This can be justified by analyzing equations (19) and (17): the larger the monitor function,the smaller the associated spring constant and, consequently, the smaller the length of the correspondingelement.The monitor εi is usually defined as a function of a“residual”, identified in the sequel with the symbol ζi ,directly related to the approximate solution obtainedwit
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