Class Note for ECE 449 at UA 5
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Date Created: 02/06/15
A I we 449549 Giantinnons System mutualitth Thermodynamics 0 Until now we have ignored the thermal domain However it is fundamental for the understanding of physics 0 We mentioned that energy can neither be generated nor destroyed yet we immediately turned around and introduced elements such as sources and resistors which shouldn t eXist at all in accordance with the above statement 0 In today s lecture we shall analyze these phenomena in more depth October 13 2003 start Presentation AL I we 449549 Glontinnnns System mutualitth Table of Contents 0 Energy sources and sinks Irreversible thermodynamics Heat conduction Heat ow 0 Thermal resistors and capacitors Radiation October 13 2003 start Presentation I we 449549 ontinnons System mutualitth Energy Sources and Sinks Thermal model Phys cal sys e39quot exlemal model bound ary the lsl LL am wall hi mH ADO Clix Rilim k39Uo SEzuu Uq V1 1 V2 n2 Rm T ink mV in a 3912 mg mn z ic c Wall Outlet The olher side of the Wall battery Elf extenml model Emm man u an manmu vxlillmnnl mumum Electrical model internal model October 13 2003 Start Presentation ltIJEIgt A I we 449549 Glontinnnns System mutualitth The resistor converts free energy irreversibly into entropy The Resistive Source RSI 43 R dt This fact is represented in the bond graph by a resistive source the RSelement The causality of the thermal side is always such that the resistor is seen there as a source of entropy never as a source of temperature 0 Sources of temperature are nonphysical October 13 2003 Start Presentation ltgt A I we 449549 ontinnons System mutualitth Heat Conduction I 0 Heat conduction in a well insulated rod can be described by the onedimensional heat equation 8T 6 T at 6623 Discretization in space leads to 3 T1zn a TB 244 271121 Tl n 81 A21 mm dz mnm 2Tunttx1 I e mltn A2 a dTa d T I 2TTa 1 October13 2003 start Presentation 1A1 I we 449549 Gluntinnnns System mutualitth Heat Conduction II 0 Consequently the following electrical equivalence circuit may be considered v 1 2 vi39 WEI October13 2003 1 start Presentation A I we 449549 ontinnons System mutualitth Heat Conduction III 0 As a consequence heat conduction can be described by a series of such Tcircuits TFTL R 1 2 R T3 R 3 T171 3 Tun1 T T T C y In bond graph representation R R we is A1 th 2 same AT ism 1 TnT o October13 2003 start Presentation Asia I we 449549 Glontinnnns System mutualitth Heat Conduction IV ATE ATm A12 ua AT 39139 T 39r T 1 1 Tn DIhsllfo qu39 xo a110 I 51 31 2 52 3m spa T A u TA n TH As c c c This bond graph is exceedingly beautiful It only has one drawback It is most certainly incorrect There are no energy sinks A resistor may make sense in an electrical circuit if the heating of the circuit is not of interest but it is most certainly not meaningful when the system to be described is itself in the thermal domain October13 2003 start Presentation A I we 449549 ontinnoux System mammal Heat Conduction V 0 The problem can be corrected easily by replacing each resistor by a resistive source RS RS RS RS ATE s1 1 T2 NEE 52 52 Ta ATnZ ISM snlx TM ATn n Sm Tn T T T T T T 2 Sn 5 T TT 39139 o ALhFbo bl 1394 0ug30 n n E S1 51 52 S n l 1amp1er A5 T3 As T C C C The temperature gradient leads to additional entropy Which is reintroduced at the nearest Ojunction October13 2003 start Presentation AL I we 449549 Gluntinnonx System mnaml Heat Conduction VI 0 This provides a good approximation of the physical reality Unfortunately the resulting bond graph is asymmetrical although the heat equation itself is symmetrical A further correction removes the asymmetry October13 2003 start Presentation lt19 A we 449549 ontinnons System mutualitth Heat Flow 0 The thermal power is the heat ow dQdt It is commonly computed as the product of two adjugate thermal variables ie It is also possible to treat heat ow as the primary physical phenomenon and derive consequently from it an equation for computing the entropy October13 2003 start Presentation I we 449549 Glontinnnns System mutualitth The Computation of R and C I 0 The capacity of a long well insulated rod to conduct heat is proportional to the temperature gradient 2 EREHETQ 9 thermal resistance 11 sgecittc thermal conductance l length at the rod A crasssection at the rod VA39 11Z Z R 0 1AA Ax length atasegment October13 2003 start Presentation A I we 449549 ontinnons System mutualitth The Computation of R and C II 0 The capacity of a long well insulated rod to store heat satisfies the capacitive law 2 7 heat cagacitl c speci c heat cagac 39 m mass at the rod p material density V volume at a segment October13 2003 start Presentation I we 449549 Glontinnnns System mutualitth The Computation of R and C III 2 TC KXiTcp A AyeT RC9y cipszolAx2 The diffusion time constant RC is independent of temperature 0 The thermal resistance is proportional to the temperature 0 The thermal capacity is inverse proportional to the temperature 0 The thermal R and C elements are contrary to their electrical and mechanical counterparts not constant October13 2003 start Presentation A I we 449549 untinnous System mutualitth Is the Thermal Capacity truly capacitive We have to verify that the derived capacitive law is not in Violation of the general rule of capacitive laws q 7 1112 I q is indeed a nanEnearfunctian of e Therefore the derived law satisfies the general rule for capacitive laws October 13 2003 start Presentation AL I we 449549 Glontinnnns System mutualitth Computation of R for the Modi ed Bond Graph 0 The resistor value has been computed for the original circuit configuration We need to analyze What the effects of the symmetrization of the bond graph have on the computation of the resistor value 0 We evidently can replace the original resistor by two resistors of double size that are connected in parallel October 13 2003 start Presentation A I we 449549 Giantinnons System mutualitth Modi cation of the Bond Graph 0 The bond graph can be modi ed by means of the diamond rule This is exactly the structure in use October 13 2003 start Presentation we 449549 Glontinnnns System mutualitth Radiation I 0 A second fundamental phenomenon of thermodynamics concerns the radiation It is described by the law of StephanBoltzmann 9 aquot T4 0 The emitted heat is proportional to the radiation and to the emitting surface Q aquot A T4 0 Consequently the emitted entropy is proportional to the third power of the absolute temperature S0quotA T3 October 13 2003 start Presentation A we 449549 ontinnons System mutualitth Radiation II 0 Radiation describes a dissipative phenomenon we know this because of its static relationship between T and S 0 Consequently the resistor can be computed as follows R TS1aquotA T2 0 The radiation resistance is thus inverse proportional to the square of the absolute temperature October13 2003 start Presentation we 449549 Glontinnnns System mutualitth Radiation III R R11 Am R lag1412 T Islxb Tl RS in st W aVT y October13 2003 start Presentation 10 A we 449549 untinnons System mmml References Cellier FE 1991 Continuous System Modeling SpringerVerlag New York Chapter 8 October13 2003 Start Presentation ltIJEIgt 11