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Heat Transfer in Cylindrical Bodies Controlled by a Thermoelectric Converter ⁎
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IFAC PapersOnLine 52-15 (2019) 139–144
Heat Heat Transfer Transfer in in Cylindrical Cylindrical Bodies Bodies Heat Transfer in CylindricalConverter Bodies Controlled by a Thermoelectric Heat Transfer in CylindricalConverter Bodies Controlled by a Thermoelectric Controlled by a Thermoelectric Converter Controlled by∗∗ Andreas a Thermoelectric Converter ∗ Georgy Kostin Rauh ∗∗ ∗∗ Alexander Gavrikov ∗
∗ Andreas Rauh ∗∗ Alexander Gavrikov ∗ Georgy Kostin Georgy Dmitri Kostin Knyazkov ∗∗ ∗ Andreas ∗∗ Alexander ∗ Harald AschemannGavrikov Andreas ∗∗∗Rauh ∗∗ Georgy Dmitri Kostin Knyazkov Rauh Alexander Gavrikov ∗∗ Harald ∗ ∗∗ Aschemann ∗ Dmitri Knyazkov Harald Aschemann ∗ ∗∗ Georgy Dmitri Kostin Knyazkov Andreas Rauh Alexander Gavrikov Harald Aschemann ∗ Dmitri Knyazkov ∗ Harald Aschemann ∗∗ Institute for Problems in Mechanics RAS, ∗ ∗ Ishlinsky Institute for in RAS, Ishlinsky Institute for Problems Problems in Mechanics Mechanics RAS, ∗ Ishlinsky Vernadskogo 101-1, 119526 Moscow, Russia Ishlinsky Institute for Problems in Mechanics RAS, Vernadskogo 101-1, 119526 Moscow, Russia ∗ Vernadskogo 101-1, 119526 Moscow, Russia Ishlinsky Institute for Problems in Mechanics RAS, (e-mail: {kostin, gavrikov, knyazkov}@ipmnet.ru). Vernadskogo 101-1, 119526 Moscow, Russia (e-mail: {kostin, 101-1, gavrikov, knyazkov}@ipmnet.ru). (e-mail: {kostin, gavrikov, knyazkov}@ipmnet.ru). ∗∗ Vernadskogo 119526 Moscow, Russia Chair of Mechatronics, University of Rostock, ∗∗ (e-mail: gavrikov, knyazkov}@ipmnet.ru). ∗∗ Chair{kostin, of Mechatronics, University of Rostock, of Mechatronics, University of ∗∗ Chair{kostin, (e-mail: gavrikov, knyazkov}@ipmnet.ru). Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany Chair of Mechatronics, University of Rostock, Rostock, Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany ∗∗ Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany Chair of Mechatronics, University of Rostock, (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). Abstract: In this paper, the mathematical modeling of heat transfer in two aluminum cylinders Abstract: In paper, the modeling of in cylinders Abstract:by In athis this paper, the mathematical mathematical modeling of heat heat transfer transfer in two two aluminum aluminum cylinders controlled Peltier element is presented. The corresponding initial-boundary value problem Abstract: In this paper, the mathematical modeling of heat transfer in two aluminum cylinders controlled by a Peltier element is presented. The corresponding initial-boundary value problem controlled by a Peltier element is presented. The corresponding initial-boundary value problem Abstract: In this paper, the mathematical modeling of heat transfer two aluminum cylinders is derived based on the Peltier and Seebeck thermoelectric effects in the scope of the linear controlled by a Peltier element isand presented. The corresponding initial-boundary value problem is derived based on the Peltier Seebeck thermoelectric effects in the scope of the linear is derived based on the Peltier and Seebeck thermoelectric effects in the scope of the linear controlled by a transfer. Peltier element isand presented. The corresponding initial-boundary value problem theory of heat To validate the reliability of the mathematical model and identify its is derived based on the Peltier Seebeck thermoelectric effects in the scope of the linear theory of heat transfer. To validate the reliability of the mathematical model and identify its theory of heat transfer. validate reliability ofperformed the mathematical model and its is derived based on sets the To Peltier and the Seebeck thermoelectric effects in originally the scope ofidentify the linear parameters, several of experiments have been on an designed test theory of heat transfer. To validate the reliability of the mathematical model and identify its parameters, several sets of experiments have been performed on an originally designed test parameters, several sets of experiments have been performed on an originally designed test theory of heat transfer. To validate the reliability of the mathematical model and identify its rig. The method of separation of variables is used to reduce the original three-dimensional parameters, several sets of experiments have been performed on an originally designed test rig. The method of separation of variables is used to reduce the original three-dimensional rig. The method of separation of variables is used to reduce the original three-dimensional parameters, several sets of experiments have been performed on an originally designed test system with distributed parameters to a spatially one-dimensional system which is nonlinear rig. The method of separation of variables is used to reduce the original three-dimensional system with distributed parameters to aa spatially system which is system with distributed parameters to spatially one-dimensional system which is nonlinear nonlinear rig. The method of input separation of as variables is usedone-dimensional toThe reduce the original three-dimensional with respect to the voltage control function. eigenvalues and eigenforms for the system with distributed parameters to a spatially one-dimensional system which is nonlinear with respect to the input voltage as control function. The eigenvalues and eigenforms for the with respect to the input voltage as control function. The eigenvalues and eigenforms for the system with distributed parameters to a spatially one-dimensional system which is nonlinear linearized problem are found by applying a Fourier analysis. The steady-state heat flow under with respect to the input voltage as control function. The eigenvalues and eigenforms for the linearized problem are found by applying a Fourier analysis. The steady-state heat flow under linearized problem are found by applying a Fourier analysis. The steady-state heat flow under with respect to the input voltage as control function. The eigenvalues and eigenforms for the a constant input signal is obtained for the considered structure as well. The corresponding linearized problem are found by applying a Fourier analysis. The steady-state heat flow under aa constant input signal is obtained for the considered structure as well. The corresponding constant input signal is obtained for the considered structure as well. The corresponding linearized problem arediscussed. found by applying a Fourier analysis. The steady-state heat flow under numerical results are a constant input signal is obtained for the considered structure as well. The corresponding numerical discussed. numerical results are discussed. a constantresults input are signal is obtained for the considered structure as well. The corresponding numerical results are discussed. © 2019, IFAC (International Federation of Automatic Control) Hosting equations, by Elsevier Ltd. All rights reserved. numerical results are discussed. Keywords: Heat convection and conduction, partial differential temperature control, Keywords: Heat convection and conduction, partial differential equations, temperature control, Keywords: Heat convection and conduction, partial differential equations, temperature control, Fourier analysis, thermoelectric effects. Keywords: Heat convection and conduction, partial differential equations, temperature control, Fourier analysis, thermoelectric effects. Fourier analysis, thermoelectric effects. Keywords: Heat convection and conduction, partial differential equations, temperature control, Fourier analysis, thermoelectric effects. Fourier1.analysis, thermoelectric effects. conversion of electric potential energy into heat flux is fur1. INTRODUCTION INTRODUCTION conversion of potential energy into flux fur1. conversion of electric electric by potential energy into heat heat flux is is furthermore influenced the electric current flowing inside 1. INTRODUCTION INTRODUCTION conversion of electric potential energy into heat flux is furthermore influenced by the electric current flowing inside thermore influenced by the electric current flowing inside 1. INTRODUCTION conversion of electric potential energy into heat flux is furthe Peltier element. For the purpose of reliable modeling, Reliable approaches to mathematical modeling of thermothermore influenced by the electric current flowing inside the Peltierinfluenced element. For For the electric purposecurrent of reliable reliable modeling, Reliable approaches approaches to to mathematical mathematical modeling modeling of of thermothermo- the Peltier element. the purpose of modeling, Reliable thermore by the flowing inside we should take into account different phenomena: partial electric processes in bodies of elementary shape are importhe Peltier element. For the purpose of reliable modeling, Reliable approaches to mathematical modeling should into account different partial electric processes in bodies of elementary shape of arethermoimpor- we we should take into account different phenomena: partial electric processes in of shape are importhe Peltier take element. For the purpose ofphenomena: reliable modeling, Reliable approaches to mathematical modeling recuperation of heat flux back into electric energy in actant for prescribing and optimizing the controlled heating should take into account different phenomena: partial electric processes in bodies bodies of elementary elementary shape of arethermoimpor- we recuperation of heat flux back into electric energy in tant for prescribing and optimizing the controlled heating recuperation of heat flux back into electric energy in acactant for prescribing and optimizing the controlled heating we should take into account different phenomena: partial electric processes in bodies of elementary shape are imporcordance with the effect described by Seebeck (1826), inand cooling in solid structures. The foundation of control recuperation of heat flux back into electric energy in actant for prescribing and optimizing the controlled heating cordance withofthe the effect described by Seebeck (1826), inand cooling in solid solid and structures. Thethe foundation of heating control cordance with effect described by Seebeck (1826), inand cooling in structures. The foundation of control recuperation heat flux back into electric energy in actant for prescribing optimizing controlled trinsic thermal conductivity of semiconductors, heat power strategies, optimal with respect to convex functionals, was cordance with the effect described by Seebeck (1826), inand cooling in solid structures. The foundation of control trinsic thermal conductivity of semiconductors, heat power strategies, optimal with respect to convex functionals, was trinsic thermal conductivity of semiconductors, heat power strategies, optimal with respect to convex functionals, was cordance with the effect described by Seebeck (1826), inand coolingoptimal in Lions solidwith structures. The foundation of control generated according to the Joule–Lenz law, as well as established by (1971) for dynamic systems described trinsic thermal conductivity of semiconductors, heat power strategies, respect to convex functionals, was generated according to the Joule–Lenz law, as well as established by Lions Lionswith (1971) for dynamic dynamic systems described generated according tothe thesystem law, as well as established by (1971) for systems described thermal conductivity ofJoule–Lenz semiconductors, heat power strategies, optimal respect to convex functionals, was trinsic heat transfer between and the environment. by linear partial differential equations (PDEs). In simple generated according to the Joule–Lenz law, as well as established by Lions (1971) for dynamic systems described heat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple heat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple generated according to the Joule–Lenz law, as well as established by Lions (1971) for dynamic systems described These effects are often ignored or represented in a quasicases, mode-based decomposition methods (see, for examheat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple effects are often ignored or represented in a quasicases, mode-based decomposition methods (see, for for exam- These These effects are often ignored or represented in a quasicases, mode-based decomposition methods (see, examheat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple static form (Chavez et al. (2000)). Some simplified models ple, Chernousko enable one to find a distributed effects are often ignored or Some represented in amodels quasicases, mode-based(1996)) decomposition methods for exam- These static form (Chavez et al. al. (2000)). simplified ple, Chernousko (1996)) enable one one to find find(see, a distributed distributed static form (Chavez et simplified ple, (1996)) enable to a These effects areeither often ignored or Some represented in which amodels quasicases,Chernousko mode-based decomposition methods (see, forthe examare considered as aa(2000)). steady-state system is control law bringing a PDE system optimally to destatic form (Chavez et al. (2000)). Some simplified models ple, Chernousko (1996)) enable one to find a distributed are considered either as steady-state system which is control law bringing bringing a PDE PDE system optimally to the the dede- static are considered either a(2000)). steady-state system which is control law a system optimally to form (Chavez et as al.(Cernaianu Some simplified models ple, Chernousko (1996)) enable one to find astrategies distributed one-dimensional in space and Gontean (2013)) sired state in finite time. Boundary control in are considered either as a steady-state system which is control law bringing a PDE system optimally to the deone-dimensional in space (Cernaianu and Gontean (2013)) sired state in finite time. Boundary control strategies in one-dimensional in space (Cernaianu and Gontean (2013)) sired state in finite time. Boundary control strategies in are considered either as a steady-state system which is control law bringing a PDE system optimally to the deor even as one with lumped parameters (Felgner et al. linear models of heat transfer were proposed in Butkovsky one-dimensional in space (Cernaianu and Gontean (2013)) sired state in finite time. Boundary control strategies in or even lumped parameters (Felgner et linear models of heattime. transfer were proposed proposed instrategies Butkovsky or even as as one one with with lumped parameters (Felgner(2013)) et al. al. linear models of heat transfer were in Butkovsky one-dimensional in space (Cernaianu and Gontean sired state in finite Boundary control in (2014)). (1969) as well as Ahmed and Teo (1981). or even as one with lumped parameters (Felgner et al. linear models transfer proposed in Butkovsky (2014)). (1969) as well wellof asheat Ahmed and were Teo (1981). (1981). (1969) as Ahmed and Teo or even as one with lumped parameters (Felgner et al. linear models ofas heat transfer were proposed in Butkovsky (2014)). (2014)). (1969) as well as Ahmed and Teo (1981). In this paper, we study the problem of heat transfer in Thermoelectric converters, and in particular solid-state (1969) as well asconverters, Ahmed andand Teoin In this paper, study the of in Thermoelectric converters, and in(1981). particular solid-state solid-state (2014)). In this paper, we we study the problem problem of heat heat transfer transfer in Thermoelectric particular two cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements discussed in Rauh et al. In this paper, we study the problem of heat transfer in Thermoelectric converters, and discussed in particular solid-state two cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements in Rauh et al. two cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements discussed in Rauh et al. In this paper, we study the described problem of heat transfer in Thermoelectric converters, and in particular solid-state element. The original model in Sect. 2 includes (2015), are precision control devices actively cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements discussed inimplemented Rauh et al. two element. The original model described in Sect. 2 includes (2015), are precision control devices actively implemented element. The original model described in Sect.As 2 includes (2015), are precision devices actively cylindrical bodiesfactors affected by an intermediate Peltier semiconductor Peltiercontrol elements discussed inimplemented Rauh et al. two all the thermoelectric mentioned above. a result, in technical applications such as additive manufacturing, element. The original model described in Sect. 2 includes (2015), are precision control devices actively implemented all the thermoelectric thermoelectric factors mentioned above. As a result, result, in technical applications such as as additive additive manufacturing, all the factors mentioned above. a in technical applications such manufacturing, element. The original modelboundary describedconditions in Sect.As 2isincludes (2015), are precision control actively implemented aall system with specific infrared sensors, diode lasers so on. The flow of the thermoelectric factors mentioned above. As aderived result, in technical applications suchdevices asand additive manufacturing, aallPDE PDE system with specific boundary conditions is derived infrared sensors, diode lasers and so on. The flow of a PDE system with specific boundary conditions is derived infrared sensors, diode lasers and so on. The flow of the thermoelectric factors mentioned above. Asisfunction. aderived result, in technical applications such asand additive manufacturing, which is non-linear with respect to the control charged particles in such elements can transfer thermal a PDE system with specific boundary conditions infrared sensors, diode lasers so on. The flow of which is non-linear with respect to the control function. charged particles in suchlasers elements can transfer thermal non-linear with respect to the controlin charged particles in such elements can transfer thermal awhich PDE is system with specific boundary conditions isfunction. derived infraredfrom sensors, diode and so on. The flow of which Validated with the experimental data presented Sect. 3, energy a colder body to a hotter one on the basis is non-linear with respect to the control function. charged particles in such elements can transfer thermal with the data presented in Sect. 3, energy from a colder colder bodyelements to aa hotter hotter one on the the basis Validated Validated withproperties the experimental experimental data presented infunction. Sect.are 3, energy from a body to on basis which is non-linear with of respect to the control charged particles in such can one transfer thermal the dynamic the multibody structure of the effect disclosed by Peltier (1834). The direction Validated with the experimental data presented in Sect. 3, energy from a colder body to a hotter one on the basis the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction Validated with the experimental data presented in Sect. 3, energy from a colder body to a hotter one on the basis analyzed. For the identified set of parameters, we have and intensity of the thermoelectric process depend on the the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction For the identified set of parameters, we have and intensity of the thermoelectric thermoelectric process depend on the the analyzed. analyzed. For the identified set of parameters, we have and intensity of the process depend on the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction succeeded in reducing, with reasonable simplifications, the electric voltage fed to the terminals of the converter. The analyzed. For the identified set of parameters, we have and intensity of fed theto thermoelectric depend onThe the succeeded in reducing, with reasonable simplifications, the electric voltage the terminals terminalsprocess of the converter. converter. in withproblem reasonable simplifications, the electric voltage the of analyzed. Forreducing, the identified set of to parameters, we have and intensity of fed theto thermoelectric process depend onThe the succeeded original three-dimensional aa one-dimensional succeeded in reducing, withproblem reasonable simplifications, the electric voltage fed to the terminals of the the converter. The original three-dimensional to one-dimensional The study was partially supported by the Alexander von Humoriginal three-dimensional problem to a one-dimensional succeeded in reducing, with reasonable simplifications, the The study was partially supported by the Alexander von Humelectric voltage fed to the terminals of the converter. The initial-boundary value problem applying the method The study was partially supported by the Alexander von Humoriginal three-dimensional problem to a one-dimensional boldt Germany,supported Ministry of and Higher initial-boundary value problem problem applying the method method of of TheFoundation, study was partially byScience the Alexander vonEducaHuminitial-boundary value applying the of boldt Foundation, Germany, Ministry of Science and Higher Educaoriginal three-dimensional problem to a one-dimensional separation of variables. The solutions of the eigenvalue boldt Foundation, Germany, Ministry of Science and Higher Educa initial-boundary value problem applying the method as of tion within thewas framework ofsupported the Russian State Assignment (Project The study partially byScience the Alexander von Humseparation of variables. The solutions of the eigenvalue as boldt Foundation, Germany, Ministry of and Higher Educaseparation of variables. The solutions of the eigenvalue as tion within the framework of the Russian State Assignment (Project initial-boundary value problem applying the method of tion the framework of Ministry the Russian State (Project well boundary value problems discussed Sect. 4 give us separation of variables. The solutions of in the eigenvalue as Reg. within No. AAAA-A17-117021310387-0), the Assignment Russian Foundation boldt Foundation, Germany, ofand Science and Higher Education within the framework of the Russian State Assignment (Project well boundary value problems discussed in Sect. 4 give us Reg. No. and the Russian well boundary value problems discussed Sect. 4 with give us Reg. No. AAAA-A17-117021310387-0), AAAA-A17-117021310387-0), and the Assignment Russian Foundation Foundation separation of variables. The solutions of in the eigenvalue as natural distributions of temperature and heat flux or for Basic Research (grant 19-01-00173). tion within the framework of the Russian State (Project well boundary value problems discussed in Sect. 4 give us Reg.Basic No. AAAA-A17-117021310387-0), and the Russian Foundation natural distributions of and heat flux or for Research natural distributions of temperature temperature and in heat flux4 with with or for Basic Research (grant (grant 19-01-00173). 19-01-00173). well boundary value problems discussed Sect. give us Reg. No. AAAA-A17-117021310387-0), natural distributions of temperature and heat flux with or for Basic Research (grant 19-01-00173). and the Russian Foundation natural distributions of temperature and heat flux with or for Basic Research (grant 19-01-00173). 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 500 Copyright © 2019 500 Copyright © under 2019 IFAC IFAC 500 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 500 10.1016/j.ifacol.2019.11.664 Copyright © 2019 IFAC 500
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with the environment on the circumferential boundary does not change over space and time. The one-dimensional heat flux in the Peltier element is supposed to be directed along the axis of symmetry. The heat transfer with a constant voltage supplied to the element is considered either as a natural transient or stationary process in the structure. Under these assumptions, it becomes possible to reduce the original 3D eigenvalue problem to an ODE problem which is linear with constant coefficients over the considered temperature range. A series of lower eigenvalues ν is obtained in the paper for a given set of geometric and physical parameters identified experimentally. The inverse τ = ν −1 of the eigenvalues characterizes the decay time of the eigenmodes.
Fig. 1. Scheme of the thermoelectric system. without constant electric voltage supplied to the Peltier element. The resulting eigenvalues and the corresponding eigenforms can be used to construct a system of ordinary differential equations (ODEs). In future, the obtained steady state of heat transfer will help to formulate control problems with natural terminal states. 2. THERMOELECTRIC MODEL
The geometric parameters of the experimental setup have been determined (see Fig. 2): the height of the cylinders za = 100 mm, the common radius r1 = 31 mm, the height of the Peltier element 2z0 = 5 mm with the total length 2z1 = 2z0 + 2za . The identified physical parameters of the aluminum bodies include: the heat conductivity 3 λa = 254 W/m/K, the material density ρa = 2700 kg/m , the specific heat capacity ca = 896 J/kg/K. In a similar manner, we obtain for the Peltier element: the heat conductivity along the z-axis λp = 0.5 W/m/K, the average 3 volume density ρp = 3000 kg/m , the effective heat capacity cp = 500 J/kg/K, and the effective Seebeck coefficient s = 0.045 V/K.
2.1 Experimental setup In this paper, active heat exchange between two cylindrical metal bodies is considered. The cylinders are separated by a Peltier element (see Fig. 1) that is designed in the form of a thin disk with the same radius. The free ends of both cylinders are thermally insulated, and the circumferential surface of the structure is in contact with the ambient air at room temperature. The resulting stack consisting of three coaxial parts is oriented vertically. As shown in Knyazkov et al. (2017), heat exchange with the environment occurs mainly due to convective flows at the exposed boundary. It is assumed that the body is made of a homogeneous isotropic material (aluminum). The inner structure of the Peltier element suggests a strong anisotropy of heat conduction with the maximal principal direction of the conductivity tensor along the central axis. The physical parameters were identified by experiments at the setup, see Rauh et al. (2015). Both numerical and experimental estimates (Knyazkov et al. (2018)) confirm that the heat transfer coefficient can be taken as a constant for the parameters given below. The parameters obtained with experimental data are used for the current numerical simulation. An initial-boundary value problem is considered in the framework of the classical linear theory of heat conduction. Boundary conditions of the second kind are specified on the thermally insulated faces of the cylinders, whereas a condition of the third kind is taken into account for the lateral surface according to Gavrikov and Kostin (2017). The equations describing the processes inside the thermoelectric structure are derived on the basis of several simplifying assumptions. The coefficient of heat exchange 501
Fig. 2. Axial-radial domains of the structural elements. The electrical coefficients used for the control circuit of the Peltier element are the threshold voltages u+ = 1.11 V and u− = −1.29 V of the amplifier, and the overall Ohmic resistance R = 6 Ω of the electric circuit. The heat transfer coefficient is defined as α = 8.4W/m2 /K, whereas the absolute reference is given by the room temperature θ0 = 293 K. The indices a and p throughout the text stand usually for aluminum and Peltier elements, respectively. 2.2 Initial-boundary value problem The heat transfer in the aluminum cylinders is described by the first law of thermodynamics ρa ca wt (t, x) + ∇ · q(t, x) = 0. (1) Here, w is the unknown relative temperature with respect to some reference value, q = −λa ∇w denotes the heat flux vector according to Fourier’s law. These functions are defined on the space-time domain given in the cylindrical coordinate system as t > 0, x = (r, φ, z) with r < r1 and z0 < |z| < z1 . The symbol ∇ is used for the spatial gradient, the nabla with a dot ∇· is the divergence operator, whereas the subscript t denotes the time derivative.
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141
Due to a strong structural anisotropy, thermodynamic processes in the Peltier element can be considered as onedimensional by ignoring the heat flow in the directions transversal to the symmetry axis z. The first law of thermodynamics and Fourier’s law then simplify to ρp cp wt (t, x) + λp wzz (t, x) = RJ 2 (t)/Vp , (2) q = (0, 0, −sJ(θ0 + w)/A − λp wz )T in accordance with the Seebeck–Peltier law. Here, J denotes the electric current, A = πr12 is the cross section area, Vp = 2πz0 r12 is the spatial volume. The functions is given over the domain t > 0, r < r1 , 0 ≤ φ < 2π, |z| < z0 .
The right-hand side of the first equation in (2) constitutes the Joule heating caused by the current. The electric voltage U on the terminals of the Peltier element is approximately calculated with the algebraic equation U (t) ≈ RJ(t) + sδw(t), (3) where δw(t) = w(t, 0, 0, z0 ) − w(t, 0, 0, −z0 ) denotes the temperature drop between the cylinders. The electric current, in turn, depends piecewise linearly on the electromotive force E = u − sδw according to RJ = (E − u− ) · H(u− − E) + (E − u+ ) · H(E − u+ ). (4) Here, H is the Heaviside step function, whereas u represents the input voltage fed to the control circuit. The initial temperature distribution w0 is given by (5) w(0, x) = w0 (x) = w0 (0) = 0. The boundary and interface conditions have the form z = ±z1 : q(t, x) · n(x) = 0; z = ±z0 : [q(t, x) · n(x)] = 0,
[w] = 0;
Fig. 3. Experimental data of the electric circuit. resistance, the temperature difference measured by two sensors placed on the cylinders symmetrically w.r.t. the center of the Peltier element would be close to the ambient temperature. In fact, this difference, visible in Fig. 4 as a dotted line, deviates in upward direction from the air temperature due to Joule heating in the thermoelectric converter. This means that the effect of energy losses, which is quadratic with respect to the electric current J, has to be taken into account in the control-oriented modeling.
(6)
r = r1 : q(t, x) · n(x) = α(w(t, x) − w0 (t)). Here, w0 (t) is the relative temperature of the air, n(x) is the unit outer normal to the interfaces, and the square brackets denote the jump of a function on the surface. 3. ANALYSIS OF EXPERIMENTAL DATA 3.1 Sample control process Several sets of experiments were performed on the test rig to identify its parameters and to verify the proposed model. The piecewise constant voltage u supplied to the control circuit was used as a sample control law. Exemplarily, the time history of the input voltage u (blue line), the voltage U measured at the terminals of the Peltier element (green line), and the voltage drop RJ in the electric circuit (in red) are depicted in Fig. 3 for one of the experiments. The temperatures at different points on the cylinder surface yi (t) = w(t, r1 , 0, Zi ) with i = 1, ..., 7 (cf. Fig. 1) and the air temperature w0 (t) were measured during the experiment at the sampling instants t = (n − 1)∆t for n ∈ N by PT100 resistance sensors with the error ±0.1K. The sampling interval is ∆t = 1 s, and n ≤ 3 · 104 . The coordinates of the temperature measurements are: Z1 = z1 , Z2 = z1 − za /4, Z3 = z1 − za /2, Z4 = z1 − 3za /4, Z5 = z0 , Z6 = −z0 , Z7 = −z1 .
The corresponding data for the cylinder temperatures are shown in Fig. 4 by solid lines. The air temperature is represented by the dashed line. If there was no Ohmic 502
Fig. 4. Measured temperatures in the experiment. 3.2 Seebeck and Peltier effects As follows from the experimental data, the algebraic relation between thermodynamic and electric parameters in agreement with the Seebeck law (3) performs well in practice. Deviations from this law are almost entirely within the range of noise when measuring the current. The residuals are small as shown in Fig. 5, where the input signal u is again depicted in blue, whereas the effective voltage u ˆ = u± + RJ + sδw ≈ u± + U restored for a non-negligible value of J = 0 with the help of experimental data is drawn in red. Recall that s is the
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Fig. 5. Restored input voltage in accordance with the Seebeck law. Seebeck coefficient and R is the Ohmic resistance. More detailed studies have shown that it is possible to take into account some differential dependence of the current J on the temperature drop δw and the voltage u in the Peltier element. However, only a piecewise linear relationship (4) between the three functions J, δw, and u is considered in this work.
Fig. 6. Temperatures of the cylinders during a natural cooling without electrical supply.
4. NATURAL HEAT TRANSFER PROCESSES 4.1 Eigenvalue problem To identify dynamic parameters of the structure under study, natural heat transfer in the cylinders has been analyzed. The identification of the transient times in the passive regime is performed by measuring natural cooling of the system in the absence of electric currents in the circuit. It is supposed in this case that the decrease of the body temperature over time can be accurately described as the sum of a few time-dependent exponential terms. The exponential fitting between the experimental measurements y5 (t) − w0 (t) (in red), y6 (t) − w0 (t) (in green) and their corresponding theoretical least-squares approximations yˆ5 = Ae−t/τ0 + Be−t/τ1 , yˆ6 = Ae−t/τ0 − Be−t/τ1 is shown in Fig. 6. At that, the experimental data are depicted by solid lines, whereas the theoretical functions yˆ5 (in violet) and yˆ6 (in blue) are given by dashed curves. The estimated values of the decay times for the zeroth and first eigenmodes are τ0 = 4440 s and τ1 = 775 s. The root mean-square deviation of the temperature reconstruction does not exceed 0.2 K. The estimation can be refined for the first and second modes by introducing the temperature drop δy = y5 (t) − y6 (t) and its approximation δy = 2Be−t/τ1 + Ce−t/τ2 . The best least-squares fitting presented in Fig. 7 is achieved for the decay values τ1 = 775 s and τ2 = 10 s, corresponding to the first and second eigenmodes. In Fig. 7, the solid line displays the experimental results and the dashed curve is for the corresponding numerical evaluation. 503
Fig. 7. Temperature drop of the Peltier element during a natural cooling without electrical supply. As studied by Gavrikov and Kostin (2017), these three eigenvalues belong to the group of axial-symmetric modes of natural heat conductivity in cylindrical bodies. The corresponding eigenforms have been obtained based on the method of separation of variables. By taking into account the cylindrical shape of the structure, the unknown temperatures are found with the ansatz −1
w(l,m,n) (t, r, φ, z) = eν(l,m,n) t Jm (r1−1 µl r)e±imφ er1
βz
.
Here, the three-dimensional multi-index denotes the order numbers of the corresponding eigenvalues ν(l,m,n) with respect to the radial, azimuthal, and axial coordinates, respectively. Bessel functions of the first kind Jm are chosen to satisfy the homogeneous boundary conditions (6) at r = r1 for the l-th root value µl with the azimuthal value m as a nonnegative integer. The eigenvalue is expressed as ν = −τa−1 (µ2l + βn2 )
2 with τa = λ−1 a ρa Ca r1 .
The axial coefficient β is equal to
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|z| > z0 ±βn , 2 β= , where τp = λ−1 p ρp Cp r1 . ± τp−1 ν, |z| < z0
143
of the other. Again, the temperature changes moderately inside each of the cylinders with a high temperature gradient in the Peltier element.
The axial value βn , which can be exceptionally either imaginary or real, is defined such that the boundary and interface conditions at z = ±z0 and z = ±z1 are satisfied. −1 In what follows, the decay times τ(l,m,n) = −ν(l,m,n) are presented to characterize the natural behavior of the heat transfer system. The zeroth eigenmode describes cooling the stack of the three cylinders as a whole. The eigenfunction of the temperature w(0,0,0) (r, φ, z) in cylindrical coordinates does not depend on the azimuth φ; it is almost invariable over the radius r and only slightly changes along the axial coordinate z as presented in Fig. 8 for w(0,0,0) (0, 0, z). Fig. 10. Temperature distribution for the first eigenvalue. The same effect is confirmed by the function of axial heat z (0, 0, z) shown in Fig. 11. In accordance with the flux q(0,0,1) thermodynamic laws, the heat transfer is directed from the hot body to the cold one with the maximum of the flux in the central cylinder. Due to such heat pumping, the decay time is reduced to the value τ(0,0,1) = 779 s, which is also close to the estimates of the experimental fitting.
Fig. 8. Temperature distribution for the zeroth eigenvalue. The result has been obtained as the solution of the eigenvalue problem related to the initial-boundary value problem (1)–(6) subject to the condition J ≡ 0. The calculated characteristic time of the zeroth mode τ (0, 0, 0) = 4466 s for the identified system parameters is in good agreement with the exponential fitting performed for the experimental data as mentioned above. z The axial component q(0,0,0) (0, 0, z) of the heat flux vector q is depicted in Fig. 9 for the zeroth eigenmode as a function of z. The heat transfers from the central Peltier element in outward direction can be seen with a vanishing heat flow at the insulated faces of the structure.
Fig. 11. Heat flux distribution for the first eigenvalue without electrical supply. The largest decay times are summarized in Table 1. The modes with non-zero azimuthal numbers, analyzed for a similar system by Gavrikov and Kostin (2017), are not excited by the thermoelectric element due to its coaxial symmetry and have rather short transient processes. The most essential transients in the system are shaped by the first four eigenmodes with l = m = 0. The decay transients for l > 0 and n > 3 are significantly faster than the natural processes for these modes. This means that an approximate ODE system of order 4 may be rather accurate in a control-oriented modeling of the heat transfer structure under consideration. Table 1. Decay times τ(l,0,n) of the system. l=0 l=1
n=0 4412 s 0.621 s
n=1 779 s 0.618 s
n=2 10.0 s 0.592 s
n=3 9.50 s 0.517 s
n=4 0.609 s 0.495 s
Fig. 9. Heat flux distribution for the zeroth eigenvalue. 4.2 Boundary value problem The temperature eigenfunction w(0,0,1) (0, 0, z) of the first mode is depicted in Fig. 10. This distribution is responsible for synchronous heating of one metal cylinder and cooling 504
The steady state of the system for a constant non-trivial input voltage u > U+ or u < −U− can be derived from the
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initial-boundary value problem (1)–(6). For this purpose, all the partial derivatives over time in the governing relations should be set to zero. Such stationary heat flows may often be desired in controlled technical processes. As an illustrative example, the stationary distribution of the temperature over the azimuthal coordinate z at the fixed voltage u = 5 V is presented in Fig. 12. The corresponding steady-state heat flux is shown in Fig. 13. The extremal temperature difference between the first and second metal bodies characterizes the utmost capacity of the Peltier element, which is limited by the saturation properties of the controller. The heat flux is everywhere nonnegative but strongly non-symmetrical because of the Joule heat generated by the thermoelectric converter.
Fig. 12. Stationary distribution of the temperature at the constant voltage u = 5 V.
Fig. 13. Stationary distribution of the axial heat flux at the constant voltage u = 5 V. 5. CONCLUSIONS The dependence between the eigenforms of the temperature distribution on the one hand and the supplied voltage together with the internal parameters of the Peltier element on the other hand was analyzed for a system of two cylindrical bodies. The specific eigenvalues and temperature functions along the longitudinal axis of a cylindrical region were calculated. The characteristics of the modes were found by accounting for the jumps of the structural parameters on the element interfaces. The constructed set of eigenfunctions represents an orthonormal basis in the corresponding energy space which can be used to derive a countable set of ODEs describing the system behavior for the case of time-variable control. In a follow-up study, the 505
thermal contact resistance on the interface between the Peltier element and the cylinders and other phenomena are planned to be introduced in an extended model and to be identified. It is intended to validate the resulting controlled processes by using the experimental setup and to design control laws for an optimal set-point change of the thermoelectric system to desired states w.r.t. given cost functions. REFERENCES Ahmed, N.U. and Teo, K.L. (1981). Optimal Control of Distributed Parameter Systems. North Holland, New York. Butkovsky, A.G. (1969). Distributed Control Systems. Elsevier, New York. Cernaianu, M.O. and Gontean, A. (2013). Parasitic elements modelling in thermoelectric modules. IET Circuits, Devices and Systems, 7(4), 177–184. Chavez, J., Ortega, J., Salazar, J., Turo, A., and Garcia, M. (2000). Spice model of thermoelectric elements including thermal effects. In Proc. of the 17th IEEE Instrumentation and Measurement Technology Conference [Cat. No. 00CH37066]. IEEE, Baltimore. Chernousko, F. (1996). Control of elastic systems by bounded distributed forces. Applied Mathematics and Computation, 78, 103–110. Felgner, F., Exel, L., Nesarajah, M., and Frey, G. (2014). Component-oriented modeling of thermoelectric devices for energy system design. IEEE Transactions on Industrial Electronics, 61(3), 1301–1310. Gavrikov, A. and Kostin, G. (2017). Boundary control of heat transfer processes in a cylindrical body. In 2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), 192–196. IEEE, Miedzyzdroje. Knyazkov, D., Aschemann, H., Kersten, J., Kostin, G., and Rauh, A. (2018). Modeling and identification of cylindrical bodies with free convection and Peltier elements as sources for active heating. In 2018 23rd International Conference on Methods and Models in Automation and Robotics (MMAR), 424–429. IEEE, Miedzyzdroje. Knyazkov, D., Kostin, G., and Saurin, V. (2017). Influence of free convection on heat transfer in control problems for a cylindrical body. In 2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), 58–63. IEEE, Miedzyzdroje. Lions, J.L. (1971). Optimal Control of Systems Governed by Partial Differential Equations. Springer–Verlag, New York. Peltier, J.C.A. (1834). Nouvelles exp´eriences sur la caloricit´e des courants ´electrique. Annales de Chimie et de Physique, 56, 371–386. (in French). Rauh, A., Kersten, J., and Aschemann, H. (2015). Robust control for a spatially three-dimensional heat transfer process. Proc. of the 8th IFAC Symposium on Robust Control Design ROCOND 2015, Bratislava, Slovakia. IFAC-PapersOnLine, 48(14), 101–106. Seebeck, T.J. (1826). Ueber die magnetische Polarisation der Metalle und Erze durch Temperaturdifferenz. Annalen der Physik, 82(3), 253–286. (in German).
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IFAC PapersOnLine 52-15 (2019) 139–144
Heat Heat Transfer Transfer in in Cylindrical Cylindrical Bodies Bodies Heat Transfer in CylindricalConverter Bodies Controlled by a Thermoelectric Heat Transfer in CylindricalConverter Bodies Controlled by a Thermoelectric Controlled by a Thermoelectric Converter Controlled by∗∗ Andreas a Thermoelectric Converter ∗ Georgy Kostin Rauh ∗∗ ∗∗ Alexander Gavrikov ∗
∗ Andreas Rauh ∗∗ Alexander Gavrikov ∗ Georgy Kostin Georgy Dmitri Kostin Knyazkov ∗∗ ∗ Andreas ∗∗ Alexander ∗ Harald AschemannGavrikov Andreas ∗∗∗Rauh ∗∗ Georgy Dmitri Kostin Knyazkov Rauh Alexander Gavrikov ∗∗ Harald ∗ ∗∗ Aschemann ∗ Dmitri Knyazkov Harald Aschemann ∗ ∗∗ Georgy Dmitri Kostin Knyazkov Andreas Rauh Alexander Gavrikov Harald Aschemann ∗ Dmitri Knyazkov ∗ Harald Aschemann ∗∗ Institute for Problems in Mechanics RAS, ∗ ∗ Ishlinsky Institute for in RAS, Ishlinsky Institute for Problems Problems in Mechanics Mechanics RAS, ∗ Ishlinsky Vernadskogo 101-1, 119526 Moscow, Russia Ishlinsky Institute for Problems in Mechanics RAS, Vernadskogo 101-1, 119526 Moscow, Russia ∗ Vernadskogo 101-1, 119526 Moscow, Russia Ishlinsky Institute for Problems in Mechanics RAS, (e-mail: {kostin, gavrikov, knyazkov}@ipmnet.ru). Vernadskogo 101-1, 119526 Moscow, Russia (e-mail: {kostin, 101-1, gavrikov, knyazkov}@ipmnet.ru). (e-mail: {kostin, gavrikov, knyazkov}@ipmnet.ru). ∗∗ Vernadskogo 119526 Moscow, Russia Chair of Mechatronics, University of Rostock, ∗∗ (e-mail: gavrikov, knyazkov}@ipmnet.ru). ∗∗ Chair{kostin, of Mechatronics, University of Rostock, of Mechatronics, University of ∗∗ Chair{kostin, (e-mail: gavrikov, knyazkov}@ipmnet.ru). Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany Chair of Mechatronics, University of Rostock, Rostock, Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany ∗∗ Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany Chair of Mechatronics, University of Rostock, (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). (e-mail: {Andreas.Rauh, Harald.Aschemann}@uni-rostock.de). Abstract: In this paper, the mathematical modeling of heat transfer in two aluminum cylinders Abstract: In paper, the modeling of in cylinders Abstract:by In athis this paper, the mathematical mathematical modeling of heat heat transfer transfer in two two aluminum aluminum cylinders controlled Peltier element is presented. The corresponding initial-boundary value problem Abstract: In this paper, the mathematical modeling of heat transfer in two aluminum cylinders controlled by a Peltier element is presented. The corresponding initial-boundary value problem controlled by a Peltier element is presented. The corresponding initial-boundary value problem Abstract: In this paper, the mathematical modeling of heat transfer two aluminum cylinders is derived based on the Peltier and Seebeck thermoelectric effects in the scope of the linear controlled by a Peltier element isand presented. The corresponding initial-boundary value problem is derived based on the Peltier Seebeck thermoelectric effects in the scope of the linear is derived based on the Peltier and Seebeck thermoelectric effects in the scope of the linear controlled by a transfer. Peltier element isand presented. The corresponding initial-boundary value problem theory of heat To validate the reliability of the mathematical model and identify its is derived based on the Peltier Seebeck thermoelectric effects in the scope of the linear theory of heat transfer. To validate the reliability of the mathematical model and identify its theory of heat transfer. validate reliability ofperformed the mathematical model and its is derived based on sets the To Peltier and the Seebeck thermoelectric effects in originally the scope ofidentify the linear parameters, several of experiments have been on an designed test theory of heat transfer. To validate the reliability of the mathematical model and identify its parameters, several sets of experiments have been performed on an originally designed test parameters, several sets of experiments have been performed on an originally designed test theory of heat transfer. To validate the reliability of the mathematical model and identify its rig. The method of separation of variables is used to reduce the original three-dimensional parameters, several sets of experiments have been performed on an originally designed test rig. The method of separation of variables is used to reduce the original three-dimensional rig. The method of separation of variables is used to reduce the original three-dimensional parameters, several sets of experiments have been performed on an originally designed test system with distributed parameters to a spatially one-dimensional system which is nonlinear rig. The method of separation of variables is used to reduce the original three-dimensional system with distributed parameters to aa spatially system which is system with distributed parameters to spatially one-dimensional system which is nonlinear nonlinear rig. The method of input separation of as variables is usedone-dimensional toThe reduce the original three-dimensional with respect to the voltage control function. eigenvalues and eigenforms for the system with distributed parameters to a spatially one-dimensional system which is nonlinear with respect to the input voltage as control function. The eigenvalues and eigenforms for the with respect to the input voltage as control function. The eigenvalues and eigenforms for the system with distributed parameters to a spatially one-dimensional system which is nonlinear linearized problem are found by applying a Fourier analysis. The steady-state heat flow under with respect to the input voltage as control function. The eigenvalues and eigenforms for the linearized problem are found by applying a Fourier analysis. The steady-state heat flow under linearized problem are found by applying a Fourier analysis. The steady-state heat flow under with respect to the input voltage as control function. The eigenvalues and eigenforms for the a constant input signal is obtained for the considered structure as well. The corresponding linearized problem are found by applying a Fourier analysis. The steady-state heat flow under aa constant input signal is obtained for the considered structure as well. The corresponding constant input signal is obtained for the considered structure as well. The corresponding linearized problem arediscussed. found by applying a Fourier analysis. The steady-state heat flow under numerical results are a constant input signal is obtained for the considered structure as well. The corresponding numerical discussed. numerical results are discussed. a constantresults input are signal is obtained for the considered structure as well. The corresponding numerical results are discussed. © 2019, IFAC (International Federation of Automatic Control) Hosting equations, by Elsevier Ltd. All rights reserved. numerical results are discussed. Keywords: Heat convection and conduction, partial differential temperature control, Keywords: Heat convection and conduction, partial differential equations, temperature control, Keywords: Heat convection and conduction, partial differential equations, temperature control, Fourier analysis, thermoelectric effects. Keywords: Heat convection and conduction, partial differential equations, temperature control, Fourier analysis, thermoelectric effects. Fourier analysis, thermoelectric effects. Keywords: Heat convection and conduction, partial differential equations, temperature control, Fourier analysis, thermoelectric effects. Fourier1.analysis, thermoelectric effects. conversion of electric potential energy into heat flux is fur1. INTRODUCTION INTRODUCTION conversion of potential energy into flux fur1. conversion of electric electric by potential energy into heat heat flux is is furthermore influenced the electric current flowing inside 1. INTRODUCTION INTRODUCTION conversion of electric potential energy into heat flux is furthermore influenced by the electric current flowing inside thermore influenced by the electric current flowing inside 1. INTRODUCTION conversion of electric potential energy into heat flux is furthe Peltier element. For the purpose of reliable modeling, Reliable approaches to mathematical modeling of thermothermore influenced by the electric current flowing inside the Peltierinfluenced element. For For the electric purposecurrent of reliable reliable modeling, Reliable approaches approaches to to mathematical mathematical modeling modeling of of thermothermo- the Peltier element. the purpose of modeling, Reliable thermore by the flowing inside we should take into account different phenomena: partial electric processes in bodies of elementary shape are importhe Peltier element. For the purpose of reliable modeling, Reliable approaches to mathematical modeling should into account different partial electric processes in bodies of elementary shape of arethermoimpor- we we should take into account different phenomena: partial electric processes in of shape are importhe Peltier take element. For the purpose ofphenomena: reliable modeling, Reliable approaches to mathematical modeling recuperation of heat flux back into electric energy in actant for prescribing and optimizing the controlled heating should take into account different phenomena: partial electric processes in bodies bodies of elementary elementary shape of arethermoimpor- we recuperation of heat flux back into electric energy in tant for prescribing and optimizing the controlled heating recuperation of heat flux back into electric energy in acactant for prescribing and optimizing the controlled heating we should take into account different phenomena: partial electric processes in bodies of elementary shape are imporcordance with the effect described by Seebeck (1826), inand cooling in solid structures. The foundation of control recuperation of heat flux back into electric energy in actant for prescribing and optimizing the controlled heating cordance withofthe the effect described by Seebeck (1826), inand cooling in solid solid and structures. Thethe foundation of heating control cordance with effect described by Seebeck (1826), inand cooling in structures. The foundation of control recuperation heat flux back into electric energy in actant for prescribing optimizing controlled trinsic thermal conductivity of semiconductors, heat power strategies, optimal with respect to convex functionals, was cordance with the effect described by Seebeck (1826), inand cooling in solid structures. The foundation of control trinsic thermal conductivity of semiconductors, heat power strategies, optimal with respect to convex functionals, was trinsic thermal conductivity of semiconductors, heat power strategies, optimal with respect to convex functionals, was cordance with the effect described by Seebeck (1826), inand coolingoptimal in Lions solidwith structures. The foundation of control generated according to the Joule–Lenz law, as well as established by (1971) for dynamic systems described trinsic thermal conductivity of semiconductors, heat power strategies, respect to convex functionals, was generated according to the Joule–Lenz law, as well as established by Lions Lionswith (1971) for dynamic dynamic systems described generated according tothe thesystem law, as well as established by (1971) for systems described thermal conductivity ofJoule–Lenz semiconductors, heat power strategies, optimal respect to convex functionals, was trinsic heat transfer between and the environment. by linear partial differential equations (PDEs). In simple generated according to the Joule–Lenz law, as well as established by Lions (1971) for dynamic systems described heat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple heat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple generated according to the Joule–Lenz law, as well as established by Lions (1971) for dynamic systems described These effects are often ignored or represented in a quasicases, mode-based decomposition methods (see, for examheat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple effects are often ignored or represented in a quasicases, mode-based decomposition methods (see, for for exam- These These effects are often ignored or represented in a quasicases, mode-based decomposition methods (see, examheat transfer between the system and the environment. by linear partial differential equations (PDEs). In simple static form (Chavez et al. (2000)). Some simplified models ple, Chernousko enable one to find a distributed effects are often ignored or Some represented in amodels quasicases, mode-based(1996)) decomposition methods for exam- These static form (Chavez et al. al. (2000)). simplified ple, Chernousko (1996)) enable one one to find find(see, a distributed distributed static form (Chavez et simplified ple, (1996)) enable to a These effects areeither often ignored or Some represented in which amodels quasicases,Chernousko mode-based decomposition methods (see, forthe examare considered as aa(2000)). steady-state system is control law bringing a PDE system optimally to destatic form (Chavez et al. (2000)). Some simplified models ple, Chernousko (1996)) enable one to find a distributed are considered either as steady-state system which is control law bringing bringing a PDE PDE system optimally to the the dede- static are considered either a(2000)). steady-state system which is control law a system optimally to form (Chavez et as al.(Cernaianu Some simplified models ple, Chernousko (1996)) enable one to find astrategies distributed one-dimensional in space and Gontean (2013)) sired state in finite time. Boundary control in are considered either as a steady-state system which is control law bringing a PDE system optimally to the deone-dimensional in space (Cernaianu and Gontean (2013)) sired state in finite time. Boundary control strategies in one-dimensional in space (Cernaianu and Gontean (2013)) sired state in finite time. Boundary control strategies in are considered either as a steady-state system which is control law bringing a PDE system optimally to the deor even as one with lumped parameters (Felgner et al. linear models of heat transfer were proposed in Butkovsky one-dimensional in space (Cernaianu and Gontean (2013)) sired state in finite time. Boundary control strategies in or even lumped parameters (Felgner et linear models of heattime. transfer were proposed proposed instrategies Butkovsky or even as as one one with with lumped parameters (Felgner(2013)) et al. al. linear models of heat transfer were in Butkovsky one-dimensional in space (Cernaianu and Gontean sired state in finite Boundary control in (2014)). (1969) as well as Ahmed and Teo (1981). or even as one with lumped parameters (Felgner et al. linear models transfer proposed in Butkovsky (2014)). (1969) as well wellof asheat Ahmed and were Teo (1981). (1981). (1969) as Ahmed and Teo or even as one with lumped parameters (Felgner et al. linear models ofas heat transfer were proposed in Butkovsky (2014)). (2014)). (1969) as well as Ahmed and Teo (1981). In this paper, we study the problem of heat transfer in Thermoelectric converters, and in particular solid-state (1969) as well asconverters, Ahmed andand Teoin In this paper, study the of in Thermoelectric converters, and in(1981). particular solid-state solid-state (2014)). In this paper, we we study the problem problem of heat heat transfer transfer in Thermoelectric particular two cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements discussed in Rauh et al. In this paper, we study the problem of heat transfer in Thermoelectric converters, and discussed in particular solid-state two cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements in Rauh et al. two cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements discussed in Rauh et al. In this paper, we study the described problem of heat transfer in Thermoelectric converters, and in particular solid-state element. The original model in Sect. 2 includes (2015), are precision control devices actively cylindrical bodies affected by an intermediate Peltier semiconductor Peltier elements discussed inimplemented Rauh et al. two element. The original model described in Sect. 2 includes (2015), are precision control devices actively implemented element. The original model described in Sect.As 2 includes (2015), are precision devices actively cylindrical bodiesfactors affected by an intermediate Peltier semiconductor Peltiercontrol elements discussed inimplemented Rauh et al. two all the thermoelectric mentioned above. a result, in technical applications such as additive manufacturing, element. The original model described in Sect. 2 includes (2015), are precision control devices actively implemented all the thermoelectric thermoelectric factors mentioned above. As a result, result, in technical applications such as as additive additive manufacturing, all the factors mentioned above. a in technical applications such manufacturing, element. The original modelboundary describedconditions in Sect.As 2isincludes (2015), are precision control actively implemented aall system with specific infrared sensors, diode lasers so on. The flow of the thermoelectric factors mentioned above. As aderived result, in technical applications suchdevices asand additive manufacturing, aallPDE PDE system with specific boundary conditions is derived infrared sensors, diode lasers and so on. The flow of a PDE system with specific boundary conditions is derived infrared sensors, diode lasers and so on. The flow of the thermoelectric factors mentioned above. Asisfunction. aderived result, in technical applications such asand additive manufacturing, which is non-linear with respect to the control charged particles in such elements can transfer thermal a PDE system with specific boundary conditions infrared sensors, diode lasers so on. The flow of which is non-linear with respect to the control function. charged particles in suchlasers elements can transfer thermal non-linear with respect to the controlin charged particles in such elements can transfer thermal awhich PDE is system with specific boundary conditions isfunction. derived infraredfrom sensors, diode and so on. The flow of which Validated with the experimental data presented Sect. 3, energy a colder body to a hotter one on the basis is non-linear with respect to the control function. charged particles in such elements can transfer thermal with the data presented in Sect. 3, energy from a colder colder bodyelements to aa hotter hotter one on the the basis Validated Validated withproperties the experimental experimental data presented infunction. Sect.are 3, energy from a body to on basis which is non-linear with of respect to the control charged particles in such can one transfer thermal the dynamic the multibody structure of the effect disclosed by Peltier (1834). The direction Validated with the experimental data presented in Sect. 3, energy from a colder body to a hotter one on the basis the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction Validated with the experimental data presented in Sect. 3, energy from a colder body to a hotter one on the basis analyzed. For the identified set of parameters, we have and intensity of the thermoelectric process depend on the the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction For the identified set of parameters, we have and intensity of the thermoelectric thermoelectric process depend on the the analyzed. analyzed. For the identified set of parameters, we have and intensity of the process depend on the dynamic properties of the multibody structure are of the effect disclosed by Peltier (1834). The direction succeeded in reducing, with reasonable simplifications, the electric voltage fed to the terminals of the converter. The analyzed. For the identified set of parameters, we have and intensity of fed theto thermoelectric depend onThe the succeeded in reducing, with reasonable simplifications, the electric voltage the terminals terminalsprocess of the converter. converter. in withproblem reasonable simplifications, the electric voltage the of analyzed. Forreducing, the identified set of to parameters, we have and intensity of fed theto thermoelectric process depend onThe the succeeded original three-dimensional aa one-dimensional succeeded in reducing, withproblem reasonable simplifications, the electric voltage fed to the terminals of the the converter. The original three-dimensional to one-dimensional The study was partially supported by the Alexander von Humoriginal three-dimensional problem to a one-dimensional succeeded in reducing, with reasonable simplifications, the The study was partially supported by the Alexander von Humelectric voltage fed to the terminals of the converter. The initial-boundary value problem applying the method The study was partially supported by the Alexander von Humoriginal three-dimensional problem to a one-dimensional boldt Germany,supported Ministry of and Higher initial-boundary value problem problem applying the method method of of TheFoundation, study was partially byScience the Alexander vonEducaHuminitial-boundary value applying the of boldt Foundation, Germany, Ministry of Science and Higher Educaoriginal three-dimensional problem to a one-dimensional separation of variables. The solutions of the eigenvalue boldt Foundation, Germany, Ministry of Science and Higher Educa initial-boundary value problem applying the method as of tion within thewas framework ofsupported the Russian State Assignment (Project The study partially byScience the Alexander von Humseparation of variables. The solutions of the eigenvalue as boldt Foundation, Germany, Ministry of and Higher Educaseparation of variables. The solutions of the eigenvalue as tion within the framework of the Russian State Assignment (Project initial-boundary value problem applying the method of tion the framework of Ministry the Russian State (Project well boundary value problems discussed Sect. 4 give us separation of variables. The solutions of in the eigenvalue as Reg. within No. AAAA-A17-117021310387-0), the Assignment Russian Foundation boldt Foundation, Germany, ofand Science and Higher Education within the framework of the Russian State Assignment (Project well boundary value problems discussed in Sect. 4 give us Reg. No. and the Russian well boundary value problems discussed Sect. 4 with give us Reg. No. AAAA-A17-117021310387-0), AAAA-A17-117021310387-0), and the Assignment Russian Foundation Foundation separation of variables. The solutions of in the eigenvalue as natural distributions of temperature and heat flux or for Basic Research (grant 19-01-00173). tion within the framework of the Russian State (Project well boundary value problems discussed in Sect. 4 give us Reg.Basic No. AAAA-A17-117021310387-0), and the Russian Foundation natural distributions of and heat flux or for Research natural distributions of temperature temperature and in heat flux4 with with or for Basic Research (grant (grant 19-01-00173). 19-01-00173). well boundary value problems discussed Sect. give us Reg. No. AAAA-A17-117021310387-0), natural distributions of temperature and heat flux with or for Basic Research (grant 19-01-00173). and the Russian Foundation natural distributions of temperature and heat flux with or for Basic Research (grant 19-01-00173). 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
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with the environment on the circumferential boundary does not change over space and time. The one-dimensional heat flux in the Peltier element is supposed to be directed along the axis of symmetry. The heat transfer with a constant voltage supplied to the element is considered either as a natural transient or stationary process in the structure. Under these assumptions, it becomes possible to reduce the original 3D eigenvalue problem to an ODE problem which is linear with constant coefficients over the considered temperature range. A series of lower eigenvalues ν is obtained in the paper for a given set of geometric and physical parameters identified experimentally. The inverse τ = ν −1 of the eigenvalues characterizes the decay time of the eigenmodes.
Fig. 1. Scheme of the thermoelectric system. without constant electric voltage supplied to the Peltier element. The resulting eigenvalues and the corresponding eigenforms can be used to construct a system of ordinary differential equations (ODEs). In future, the obtained steady state of heat transfer will help to formulate control problems with natural terminal states. 2. THERMOELECTRIC MODEL
The geometric parameters of the experimental setup have been determined (see Fig. 2): the height of the cylinders za = 100 mm, the common radius r1 = 31 mm, the height of the Peltier element 2z0 = 5 mm with the total length 2z1 = 2z0 + 2za . The identified physical parameters of the aluminum bodies include: the heat conductivity 3 λa = 254 W/m/K, the material density ρa = 2700 kg/m , the specific heat capacity ca = 896 J/kg/K. In a similar manner, we obtain for the Peltier element: the heat conductivity along the z-axis λp = 0.5 W/m/K, the average 3 volume density ρp = 3000 kg/m , the effective heat capacity cp = 500 J/kg/K, and the effective Seebeck coefficient s = 0.045 V/K.
2.1 Experimental setup In this paper, active heat exchange between two cylindrical metal bodies is considered. The cylinders are separated by a Peltier element (see Fig. 1) that is designed in the form of a thin disk with the same radius. The free ends of both cylinders are thermally insulated, and the circumferential surface of the structure is in contact with the ambient air at room temperature. The resulting stack consisting of three coaxial parts is oriented vertically. As shown in Knyazkov et al. (2017), heat exchange with the environment occurs mainly due to convective flows at the exposed boundary. It is assumed that the body is made of a homogeneous isotropic material (aluminum). The inner structure of the Peltier element suggests a strong anisotropy of heat conduction with the maximal principal direction of the conductivity tensor along the central axis. The physical parameters were identified by experiments at the setup, see Rauh et al. (2015). Both numerical and experimental estimates (Knyazkov et al. (2018)) confirm that the heat transfer coefficient can be taken as a constant for the parameters given below. The parameters obtained with experimental data are used for the current numerical simulation. An initial-boundary value problem is considered in the framework of the classical linear theory of heat conduction. Boundary conditions of the second kind are specified on the thermally insulated faces of the cylinders, whereas a condition of the third kind is taken into account for the lateral surface according to Gavrikov and Kostin (2017). The equations describing the processes inside the thermoelectric structure are derived on the basis of several simplifying assumptions. The coefficient of heat exchange 501
Fig. 2. Axial-radial domains of the structural elements. The electrical coefficients used for the control circuit of the Peltier element are the threshold voltages u+ = 1.11 V and u− = −1.29 V of the amplifier, and the overall Ohmic resistance R = 6 Ω of the electric circuit. The heat transfer coefficient is defined as α = 8.4W/m2 /K, whereas the absolute reference is given by the room temperature θ0 = 293 K. The indices a and p throughout the text stand usually for aluminum and Peltier elements, respectively. 2.2 Initial-boundary value problem The heat transfer in the aluminum cylinders is described by the first law of thermodynamics ρa ca wt (t, x) + ∇ · q(t, x) = 0. (1) Here, w is the unknown relative temperature with respect to some reference value, q = −λa ∇w denotes the heat flux vector according to Fourier’s law. These functions are defined on the space-time domain given in the cylindrical coordinate system as t > 0, x = (r, φ, z) with r < r1 and z0 < |z| < z1 . The symbol ∇ is used for the spatial gradient, the nabla with a dot ∇· is the divergence operator, whereas the subscript t denotes the time derivative.
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Due to a strong structural anisotropy, thermodynamic processes in the Peltier element can be considered as onedimensional by ignoring the heat flow in the directions transversal to the symmetry axis z. The first law of thermodynamics and Fourier’s law then simplify to ρp cp wt (t, x) + λp wzz (t, x) = RJ 2 (t)/Vp , (2) q = (0, 0, −sJ(θ0 + w)/A − λp wz )T in accordance with the Seebeck–Peltier law. Here, J denotes the electric current, A = πr12 is the cross section area, Vp = 2πz0 r12 is the spatial volume. The functions is given over the domain t > 0, r < r1 , 0 ≤ φ < 2π, |z| < z0 .
The right-hand side of the first equation in (2) constitutes the Joule heating caused by the current. The electric voltage U on the terminals of the Peltier element is approximately calculated with the algebraic equation U (t) ≈ RJ(t) + sδw(t), (3) where δw(t) = w(t, 0, 0, z0 ) − w(t, 0, 0, −z0 ) denotes the temperature drop between the cylinders. The electric current, in turn, depends piecewise linearly on the electromotive force E = u − sδw according to RJ = (E − u− ) · H(u− − E) + (E − u+ ) · H(E − u+ ). (4) Here, H is the Heaviside step function, whereas u represents the input voltage fed to the control circuit. The initial temperature distribution w0 is given by (5) w(0, x) = w0 (x) = w0 (0) = 0. The boundary and interface conditions have the form z = ±z1 : q(t, x) · n(x) = 0; z = ±z0 : [q(t, x) · n(x)] = 0,
[w] = 0;
Fig. 3. Experimental data of the electric circuit. resistance, the temperature difference measured by two sensors placed on the cylinders symmetrically w.r.t. the center of the Peltier element would be close to the ambient temperature. In fact, this difference, visible in Fig. 4 as a dotted line, deviates in upward direction from the air temperature due to Joule heating in the thermoelectric converter. This means that the effect of energy losses, which is quadratic with respect to the electric current J, has to be taken into account in the control-oriented modeling.
(6)
r = r1 : q(t, x) · n(x) = α(w(t, x) − w0 (t)). Here, w0 (t) is the relative temperature of the air, n(x) is the unit outer normal to the interfaces, and the square brackets denote the jump of a function on the surface. 3. ANALYSIS OF EXPERIMENTAL DATA 3.1 Sample control process Several sets of experiments were performed on the test rig to identify its parameters and to verify the proposed model. The piecewise constant voltage u supplied to the control circuit was used as a sample control law. Exemplarily, the time history of the input voltage u (blue line), the voltage U measured at the terminals of the Peltier element (green line), and the voltage drop RJ in the electric circuit (in red) are depicted in Fig. 3 for one of the experiments. The temperatures at different points on the cylinder surface yi (t) = w(t, r1 , 0, Zi ) with i = 1, ..., 7 (cf. Fig. 1) and the air temperature w0 (t) were measured during the experiment at the sampling instants t = (n − 1)∆t for n ∈ N by PT100 resistance sensors with the error ±0.1K. The sampling interval is ∆t = 1 s, and n ≤ 3 · 104 . The coordinates of the temperature measurements are: Z1 = z1 , Z2 = z1 − za /4, Z3 = z1 − za /2, Z4 = z1 − 3za /4, Z5 = z0 , Z6 = −z0 , Z7 = −z1 .
The corresponding data for the cylinder temperatures are shown in Fig. 4 by solid lines. The air temperature is represented by the dashed line. If there was no Ohmic 502
Fig. 4. Measured temperatures in the experiment. 3.2 Seebeck and Peltier effects As follows from the experimental data, the algebraic relation between thermodynamic and electric parameters in agreement with the Seebeck law (3) performs well in practice. Deviations from this law are almost entirely within the range of noise when measuring the current. The residuals are small as shown in Fig. 5, where the input signal u is again depicted in blue, whereas the effective voltage u ˆ = u± + RJ + sδw ≈ u± + U restored for a non-negligible value of J = 0 with the help of experimental data is drawn in red. Recall that s is the
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Fig. 5. Restored input voltage in accordance with the Seebeck law. Seebeck coefficient and R is the Ohmic resistance. More detailed studies have shown that it is possible to take into account some differential dependence of the current J on the temperature drop δw and the voltage u in the Peltier element. However, only a piecewise linear relationship (4) between the three functions J, δw, and u is considered in this work.
Fig. 6. Temperatures of the cylinders during a natural cooling without electrical supply.
4. NATURAL HEAT TRANSFER PROCESSES 4.1 Eigenvalue problem To identify dynamic parameters of the structure under study, natural heat transfer in the cylinders has been analyzed. The identification of the transient times in the passive regime is performed by measuring natural cooling of the system in the absence of electric currents in the circuit. It is supposed in this case that the decrease of the body temperature over time can be accurately described as the sum of a few time-dependent exponential terms. The exponential fitting between the experimental measurements y5 (t) − w0 (t) (in red), y6 (t) − w0 (t) (in green) and their corresponding theoretical least-squares approximations yˆ5 = Ae−t/τ0 + Be−t/τ1 , yˆ6 = Ae−t/τ0 − Be−t/τ1 is shown in Fig. 6. At that, the experimental data are depicted by solid lines, whereas the theoretical functions yˆ5 (in violet) and yˆ6 (in blue) are given by dashed curves. The estimated values of the decay times for the zeroth and first eigenmodes are τ0 = 4440 s and τ1 = 775 s. The root mean-square deviation of the temperature reconstruction does not exceed 0.2 K. The estimation can be refined for the first and second modes by introducing the temperature drop δy = y5 (t) − y6 (t) and its approximation δy = 2Be−t/τ1 + Ce−t/τ2 . The best least-squares fitting presented in Fig. 7 is achieved for the decay values τ1 = 775 s and τ2 = 10 s, corresponding to the first and second eigenmodes. In Fig. 7, the solid line displays the experimental results and the dashed curve is for the corresponding numerical evaluation. 503
Fig. 7. Temperature drop of the Peltier element during a natural cooling without electrical supply. As studied by Gavrikov and Kostin (2017), these three eigenvalues belong to the group of axial-symmetric modes of natural heat conductivity in cylindrical bodies. The corresponding eigenforms have been obtained based on the method of separation of variables. By taking into account the cylindrical shape of the structure, the unknown temperatures are found with the ansatz −1
w(l,m,n) (t, r, φ, z) = eν(l,m,n) t Jm (r1−1 µl r)e±imφ er1
βz
.
Here, the three-dimensional multi-index denotes the order numbers of the corresponding eigenvalues ν(l,m,n) with respect to the radial, azimuthal, and axial coordinates, respectively. Bessel functions of the first kind Jm are chosen to satisfy the homogeneous boundary conditions (6) at r = r1 for the l-th root value µl with the azimuthal value m as a nonnegative integer. The eigenvalue is expressed as ν = −τa−1 (µ2l + βn2 )
2 with τa = λ−1 a ρa Ca r1 .
The axial coefficient β is equal to
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|z| > z0 ±βn , 2 β= , where τp = λ−1 p ρp Cp r1 . ± τp−1 ν, |z| < z0
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of the other. Again, the temperature changes moderately inside each of the cylinders with a high temperature gradient in the Peltier element.
The axial value βn , which can be exceptionally either imaginary or real, is defined such that the boundary and interface conditions at z = ±z0 and z = ±z1 are satisfied. −1 In what follows, the decay times τ(l,m,n) = −ν(l,m,n) are presented to characterize the natural behavior of the heat transfer system. The zeroth eigenmode describes cooling the stack of the three cylinders as a whole. The eigenfunction of the temperature w(0,0,0) (r, φ, z) in cylindrical coordinates does not depend on the azimuth φ; it is almost invariable over the radius r and only slightly changes along the axial coordinate z as presented in Fig. 8 for w(0,0,0) (0, 0, z). Fig. 10. Temperature distribution for the first eigenvalue. The same effect is confirmed by the function of axial heat z (0, 0, z) shown in Fig. 11. In accordance with the flux q(0,0,1) thermodynamic laws, the heat transfer is directed from the hot body to the cold one with the maximum of the flux in the central cylinder. Due to such heat pumping, the decay time is reduced to the value τ(0,0,1) = 779 s, which is also close to the estimates of the experimental fitting.
Fig. 8. Temperature distribution for the zeroth eigenvalue. The result has been obtained as the solution of the eigenvalue problem related to the initial-boundary value problem (1)–(6) subject to the condition J ≡ 0. The calculated characteristic time of the zeroth mode τ (0, 0, 0) = 4466 s for the identified system parameters is in good agreement with the exponential fitting performed for the experimental data as mentioned above. z The axial component q(0,0,0) (0, 0, z) of the heat flux vector q is depicted in Fig. 9 for the zeroth eigenmode as a function of z. The heat transfers from the central Peltier element in outward direction can be seen with a vanishing heat flow at the insulated faces of the structure.
Fig. 11. Heat flux distribution for the first eigenvalue without electrical supply. The largest decay times are summarized in Table 1. The modes with non-zero azimuthal numbers, analyzed for a similar system by Gavrikov and Kostin (2017), are not excited by the thermoelectric element due to its coaxial symmetry and have rather short transient processes. The most essential transients in the system are shaped by the first four eigenmodes with l = m = 0. The decay transients for l > 0 and n > 3 are significantly faster than the natural processes for these modes. This means that an approximate ODE system of order 4 may be rather accurate in a control-oriented modeling of the heat transfer structure under consideration. Table 1. Decay times τ(l,0,n) of the system. l=0 l=1
n=0 4412 s 0.621 s
n=1 779 s 0.618 s
n=2 10.0 s 0.592 s
n=3 9.50 s 0.517 s
n=4 0.609 s 0.495 s
Fig. 9. Heat flux distribution for the zeroth eigenvalue. 4.2 Boundary value problem The temperature eigenfunction w(0,0,1) (0, 0, z) of the first mode is depicted in Fig. 10. This distribution is responsible for synchronous heating of one metal cylinder and cooling 504
The steady state of the system for a constant non-trivial input voltage u > U+ or u < −U− can be derived from the
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initial-boundary value problem (1)–(6). For this purpose, all the partial derivatives over time in the governing relations should be set to zero. Such stationary heat flows may often be desired in controlled technical processes. As an illustrative example, the stationary distribution of the temperature over the azimuthal coordinate z at the fixed voltage u = 5 V is presented in Fig. 12. The corresponding steady-state heat flux is shown in Fig. 13. The extremal temperature difference between the first and second metal bodies characterizes the utmost capacity of the Peltier element, which is limited by the saturation properties of the controller. The heat flux is everywhere nonnegative but strongly non-symmetrical because of the Joule heat generated by the thermoelectric converter.
Fig. 12. Stationary distribution of the temperature at the constant voltage u = 5 V.
Fig. 13. Stationary distribution of the axial heat flux at the constant voltage u = 5 V. 5. CONCLUSIONS The dependence between the eigenforms of the temperature distribution on the one hand and the supplied voltage together with the internal parameters of the Peltier element on the other hand was analyzed for a system of two cylindrical bodies. The specific eigenvalues and temperature functions along the longitudinal axis of a cylindrical region were calculated. The characteristics of the modes were found by accounting for the jumps of the structural parameters on the element interfaces. The constructed set of eigenfunctions represents an orthonormal basis in the corresponding energy space which can be used to derive a countable set of ODEs describing the system behavior for the case of time-variable control. In a follow-up study, the 505
thermal contact resistance on the interface between the Peltier element and the cylinders and other phenomena are planned to be introduced in an extended model and to be identified. It is intended to validate the resulting controlled processes by using the experimental setup and to design control laws for an optimal set-point change of the thermoelectric system to desired states w.r.t. given cost functions. REFERENCES Ahmed, N.U. and Teo, K.L. (1981). Optimal Control of Distributed Parameter Systems. North Holland, New York. Butkovsky, A.G. (1969). Distributed Control Systems. Elsevier, New York. Cernaianu, M.O. and Gontean, A. (2013). Parasitic elements modelling in thermoelectric modules. IET Circuits, Devices and Systems, 7(4), 177–184. Chavez, J., Ortega, J., Salazar, J., Turo, A., and Garcia, M. (2000). Spice model of thermoelectric elements including thermal effects. In Proc. of the 17th IEEE Instrumentation and Measurement Technology Conference [Cat. No. 00CH37066]. IEEE, Baltimore. Chernousko, F. (1996). Control of elastic systems by bounded distributed forces. Applied Mathematics and Computation, 78, 103–110. Felgner, F., Exel, L., Nesarajah, M., and Frey, G. (2014). Component-oriented modeling of thermoelectric devices for energy system design. IEEE Transactions on Industrial Electronics, 61(3), 1301–1310. Gavrikov, A. and Kostin, G. (2017). Boundary control of heat transfer processes in a cylindrical body. In 2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), 192–196. IEEE, Miedzyzdroje. Knyazkov, D., Aschemann, H., Kersten, J., Kostin, G., and Rauh, A. (2018). Modeling and identification of cylindrical bodies with free convection and Peltier elements as sources for active heating. In 2018 23rd International Conference on Methods and Models in Automation and Robotics (MMAR), 424–429. IEEE, Miedzyzdroje. Knyazkov, D., Kostin, G., and Saurin, V. (2017). Influence of free convection on heat transfer in control problems for a cylindrical body. In 2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), 58–63. IEEE, Miedzyzdroje. Lions, J.L. (1971). Optimal Control of Systems Governed by Partial Differential Equations. Springer–Verlag, New York. Peltier, J.C.A. (1834). Nouvelles exp´eriences sur la caloricit´e des courants ´electrique. Annales de Chimie et de Physique, 56, 371–386. (in French). Rauh, A., Kersten, J., and Aschemann, H. (2015). Robust control for a spatially three-dimensional heat transfer process. Proc. of the 8th IFAC Symposium on Robust Control Design ROCOND 2015, Bratislava, Slovakia. IFAC-PapersOnLine, 48(14), 101–106. Seebeck, T.J. (1826). Ueber die magnetische Polarisation der Metalle und Erze durch Temperaturdifferenz. Annalen der Physik, 82(3), 253–286. (in German).