CH.5. BALANCE PRINCIPLES. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

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1 CH.5. BALANCE PRINCIPLES Coninuum Mechanics Course (MMC) - ETSECCPB - UPC

2 Overview Balance Principles Convecive Flux or Flux by Mass Transpor Local and Maerial Derivaive of a olume Inegral Conservaion of Mass Spaial Form Maerial Form Reynolds Transpor Theorem Reynolds Lemma General Balance Equaion Linear Momenum Balance Global Form Local Form 2

3 Overview (con d) Angular Momenum Balance Global Spaial Local Form Mechanical Energy Balance Exernal Mechanical Power Mechanical Energy Balance Exernal Thermal Power Energy Balance Thermodynamic Conceps Firs Law of Thermodynamics Inernal Energy Balance in Local and Global Forms Reversible and Irreversible Processes Second Law of Thermodynamics Clausius-Planck Inequaliy 3

4 Overview (con d) Governing Equaions Governing Equaions Consiuive Equaions The Uncoupled Thermo-mechanical Problem 4

5 5.1. Balance Principles Ch.5. Balance Principles 5

6 Balance Principles The following principles govern he way sress and deformaion vary in he neighborhood of a poin wih ime. The conservaion/balance principles: Conservaion of mass Linear momenum balance principle Angular momenum balance principle Energy balance principle or firs hermodynamic balance principle The resricion principle: Second hermodynamic law The mahemaical expressions of hese principles will be given in, Global (or inegral) form Local (or srong) form REMARK These principles are always valid, regardless of he ype of maerial and he range of displacemens or deformaions. 6

7 5.2. Convecive Flux Ch.5. Balance Principles 7

8 Convecion The erm convecion is associaed o mass ranspor, i.e., paricle movemen. Properies associaed o mass will be ranspored wih he mass when here is mass ranspor (paricles moion) convecive ranspor Convecive flux of an arbirary propery A hrough a conrol surface S : S amounofa crossing S uniof ime 8

9 Convecive Flux or Flux by Mass Transpor Consider: A An arbirary propery of a coninuum medium (of any ensor order) The descripion of he amoun of he propery per uni of mass, (specific conen of he propery A ). The volume of paricles d crossing a differenial surface ds during he inerval, is d ds dh vn ds dm d vn ds Then, The amoun of he propery per uni of mass crossing he differenial surface per uni of ime is: dm ds vn ds x, 9

10 Convecive Flux or Flux by Mass Transpor Consider: An arbirary propery A of a coninuum medium (of any ensor order) The specific conen of A (he amoun per uni of mass) x,. inflow vn 0 ouflow vn 0 Then, The convecive flux of A hrough a spaial surface, S, wih uni normal n is: S vn ds v is velociy Where: s is densiy If he surface is a closed surface, S, he ne convecive flux is: vn ds = ouflow - inflow 10 11/11/2015 MMC - ETSECCPB - UPC

11 Convecive Flux REMARK 1 The convecive flux hrough a maerial surface is always null. REMARK 2 Non-convecive flux (advecion, diffusion, conducion). Some properies can be ranspored wihou being associaed o a cerain mass of paricles. Examples of non-convecive ranspor are: hea ransfer by conducion, elecric curren flow, ec. Non-convecive ranspor of a cerain propery is characerized by he nonconvecive flux vecor (or ensor) qx, : non- convecive flux qn ds ; convecive flux vn s non-convecive flux vecor s convecive flux vecor ds 11 MMC - ETSECCPB - UPC 11/11/2015

12 Example Compue he magniude and he convecive flux which correspond o he S following properies: a) volume b) mass c) linear momenum d) kineic energy 12

13 Example - Soluion S vn s ds a) If he arbirary propery is he volume of he paricles: A The magniude propery conen per uni of mass is volume per uni of mass, i.e., he inverse of densiy: M 1 The convecive flux of he volume of he paricles hrough he surface S is: 1 S vn ds s vn s ds OLUME FLUX 13

14 Example - Soluion S vn s ds b) If he arbirary propery is he mass of he paricles: A M The magniude propery per uni of mass is mass per uni of mass, i.e., he uni value: M M 1 The convecive flux of he mass of he paricles M hrough he surface S is: 1 vn ds vn S s s ds MASS FLUX 14

15 Example - Soluion S vn s ds c) If he arbirary propery is he linear momenum of he paricles: A M v The magniude propery per uni of mass is mass imes velociy per uni of mass, i.e., velociy: M v M v The convecive flux of he linear momenum of he paricles M v hrough he surface S is: S s v vn ds MOMENTUM FLUX 15

16 Example - Soluion S vn s ds d) If he arbirary propery is he kineic energy of he paricles: 1 2 A M v 2 The magniude propery per uni of mass is kineic energy per uni of mass, i.e.: 1 M v 2 M 2 1 v The convecive flux of he kineic energy of he paricles hrough he 2 M v surface S is: 1 2 S v vn ds KINETIC ENERGY FLUX s 2 16

17 5.3. Local and Maerial Derivaive of a olume Inegral Ch.5. Balance Principles 17

18 Derivaive of a olume Inegral Consider: An arbirary propery A of a coninuum medium (of any ensor order) The descripion of he amoun of he propery per uni of volume (densiy of he propery A ), x, The oal amoun of he propery in an arbirary volume is: x, Q d The ime derivaive of his volume inegral is: Q Q Q lim 0 REMARK and are relaed hrough. Q Q 18 11/11/2015 MMC - ETSECCPB - UPC

19 Local Derivaive of a olume Inegral local derivaive 19 Consider: The volume inegral The local derivaive of Q is:,, no x x x, d lim 0 I can be compued as: Q Q x d x x x, x, d d, lim lim 0 0 x, Q d d d [,, ] d lim lim d 0 0, x Q x, x, x, Q Conrol olume, REMARK The volume is fixed in space (conrol volume). d

20 Maerial Derivaive of a olume Inegral Consider: The volume inegral x, Q d maerial derivaive 20 The maerial derivaive of Q is: no d lim 0 x, I can be proven ha: d x, x, d d ( ) ( ), d Q Q REMARK The volume is mobile in space and can move, roae and deform (maerial volume). d d x d v d v d v d maerial local convecive derivaive of derivaive of derivaive of he inegral he inegral he inegral

21 5.4. Conservaion of Mass Ch.5. Balance Principles 21

22 Principle of Mass Conservaion I is posulaed ha during a moion here are neiher mass sources nor mass sinks, so he mass of a coninuum body is a conserved quaniy (for any par of he body). The oal mass M of he sysem saisfies: M M 0 Where: x, x, M d M d 22

23 Conservaion of Mass in Spaial Form 23 Conservaion of mass requires ha he maerial ime derivaive of he mass M be zero for any region of a maerial volume, M M d M lim d 0, 0 The global or inegral spaial form of mass conservaion principle: d d d, d ( ) d x v d ( x, d ) v d0, By a localizaion process we obain he local or differenial spaial form of mass conservaion principle: for d ( x, ) (localizaion process) CONTINUITY EQUATION d( x, ) ( x, ) ( v)( x, ) ( v)( x, ) 0 x,

24 Conservaion of Mass in Maerial Form 0 0 Consider he relaions: d F 1 F v ( v) F d F d0 d F The global or inegral maerial form of mass conservaion principle can be rewrien as: d d 1 d F ( X, ) F ( X, ) v ( ) ( F ( X, ) ) d d d F 0 ( X, ) F d0 FX (, ) F X, d , 24 The local maerial form of mass conservaion principle reads : 0, F X X F X F X 1, X 0 F 0

25 5.5. Reynolds Transpor Theorem Ch.5. Balance Principles 25

26 Reynolds Lemma d v 0 Consider: An arbirary propery of a coninuum medium (of any ensor order) The spaial descripion of he amoun of he propery per uni of mass, x, The amoun of he propery in he coninuum body a ime for an arbirary maerial volume is: Using he maerial ime derivaive leads o, Thus, d d A A d d Q REYNOLDS LEMMA d d d d d Q d ( ) ( ) d d v v d d =0 (coninuiy equaion) 26 MMC - ETSECCPB - UPC 11/11/2015

27 Reynolds Transpor Theorem d x, d d v d A The amoun of he propery in he coninuum body a ime for an arbirary fixed conrol volume is: Q d Using he maerial ime derivaive leads o, d d d d v And, inroducing he Reynolds Lemma and Divergence Theorem: d d d vn ds REMARK The Divergence Theorem: v d nvds vn ds 27 d d n v ê 3 ê 1 d ê 2 d d d

28 Reynolds Transpor Theorem d d d vn ds The eq. can be rewrien as: d d d ds vn REYNOLDS TRANSPORT THEOREM Rae of change of he oal amoun of A. wihin he conrol volume a ime. Ne ouward flux of A hrough he surface ha surrounds he conrol volume. d A Rae of change of he amoun of in a maerial volume which insananeously coincides wih he conrol volume. d ê 3 ê 2 d ê 1 28

29 Reynolds Transpor Theorem d d d ds vn REYNOLDS TRANSPORT THEOREM (inegral form) d d d ds vn ( ) d d ( ) d [ ( )] d v d ( ) ( v) x ( v) d ê 3 ê 1 ê 2 d d d REYNOLDS TRANSPORT THEOREM (local form) 29

30 5.6. General Balance Equaion Ch.5. Balance Principles 30

31 General Balance Equaion Consider: A An arbirary propery of a coninuum medium (of any ensor order) The amoun of he propery per uni of mass, x, The rae of change per uni of ime of he amoun of A in he conrol volume is due o: a) Generaion of he propery per uni mas and ime ime due o a source: b) The convecive (ne incoming) flux across he surface of he volume. c) The non-convecive (ne incoming) flux across he surface of he volume: So, he global form of he general balance equaion is: 31 d ka d vn ds jands a b c k A ( x, ) ja ( x, ) non-convecive flux vecor

32 d k d vn ds j nds A General Balance Equaion A The global form is rewrien using he Divergence Theorem and he definiion of local derivaive: d vn ds v d k j d A A d (Reynolds Theorem) d d k j d A A The local spaial form of he general balance equaion is: REMARK d For only convecive ranspor ( ja 0) hen k A and he variaion of he conens of in a given paricle is only due o he inernal generaion k. A d k j A 32 11/11/2015 MMC - ETSECCPB - UPC A

33 Example A If he propery is associaed o mass AM, hen: The amoun of he propery per uni of mass is 1. The mass generaion source erm is k 0. The mass conservaion principle saes mass canno be generaed. The non-convecive flux vecor is j 0. Mass canno be ranspored in a non-convecive form. d k j 0 A A 0 0 Then, he local spaial form of he general balance equaion is: d ( ) ( v) 0 ( v) d ( v) v 0 x M M Two equivalen forms of he coninuiy equaion. 33

34 5.7. Linear Momenum Balance Ch.5. Balance Principles 34

35 Linear Momenum in Classical Mechanics Applying Newon s 2 nd Law o he discree sysem formed by n paricles, he resuling force acing on he sysem is: n n n dvi R fi miai mi Resuling force on he sysem i1 i1 i1 d dm dp n n i mivi vi i1 i1 mass conservaion principle: dm i 0 P linear momenum For a sysem in equilibrium, R 0, : dp 0 P cn CONSERATION OF THE LINEAR MOMENTUM 35

36 Linear Momenum in Coninuum Mechanics P n i1 m i v i The linear momenum of a maerial volume of a coninuum medium wih mass M is: vx, M x, vx, P d d M dm d 36

37 Linear Momenum Balance Principle The ime-variaion of he linear momenum of a maerial volume is equal o he resulan force acing on he maerial volume. dp d Where: body forces v d R R b d ds surface forces If he body is in equilibrium, he linear momenum is conserved: dp R 0 0 P cn 37

38 Global Form of he Linear Momenum Balance Principle The global form of he linear momenum balance principle: d dp R b d ds v d, P Inroducing n and using he Divergence Theorem, So, he global form is rewrien: bd ds ds n ds d d b+ d v d, 38 MMC - ETSECCPB - UPC 11/11/2015

39 Local Form of he Linear Momenum Balance Principle Applying Reynolds Lemma o he global form of he principle: d dv b d v d d, Localizing, he local spaial form of he linear momenum balance principle reads: d(,) x dvx (,) (,) x b(,) x a(,) x x, LOCAL FORM OF THE LINEAR MOMENTUM BALANCE (CAUCHY S EQUATION OF MOTION) 39

40 5.8. Angular Momenum Balance Ch.5. Balance Principles 40

41 Angular Momenum in Classical Mechanics Applying Newon s 2 nd Law o he discree sysem formed by n paricles, he resuling orque acing on he sysem is: dv n n i O i i imi i1 i1 =0 n n n dri ri mivi mivi ri mivi i1 i1 i1 M r f r d d dl L angular momenum v i d M O L For a sysem in equilibrium, MO 0, : dl 0 L cn CONSERATION OF THE ANGULAR MOMENTUM 41

42 Angular Momenum in Coninuum Mechanics The angular momenum of a maerial volume of a coninuum medium wih mass M is: rx, vx, M rx, x, vx, L d d M r dm d Where is he posiion vecor wih respec o a fixed poin. 42

43 Angular Momenum Balance Principle The ime-variaion of he angular momenum of a maerial volume wih respec o a fixed poin is equal o he resulan momen wih respec his fixed poin. dl d r v d M O Where: O M r b d r ds orque due o body forces orque due o surface forces 43

44 Global Form of he Angular Momenum Balance Principle The global form of he angular momenum balance principle: d r b d r ds r v d Inroducing n and using he Divergence Theorem, T T r ds r n ds r n ds r n ds T r I can be proven ha, d T r r m mmeˆ i i ; mi eijk jk REMARK e ijk is he Levi-Civia permuaion symbol /11/2015 MMC - ETSECCPB - UPC

45 Global Form of he Angular Momenum Balance Principle Applying Reynolds Lemma o he righ-hand erm of he global form equaion: d d d r v d rvd rvd =0 Reynold's Lemma dr dv dv vr d r v d Then, he global form is rewrien: dv r b e eˆ ijk jk i d r d 45

46 Local Form of he Angular Momenum Balance Principle Rearranging he equaion: =0 (Cauchy s Eq.) dv r b m d 0 (, ) d m x 0, Localizing mx (, ) 0 m e 0 ; i, j, k 1,2,3 ; x, i 1 e e i 2 e e i 3 e e i ijk jk T (,) x (,) x x, SYMMETRY OF THE CAUCHY S STRESS TENSOR 46 11/11/2015 MMC - ETSECCPB - UPC

47 5.9. Mechanical Energy Balance Ch.5. Balance Principles 47

48 Power Power, W, is he work performed in he sysem per uni of ime. In some cases, he power is an exac ime-differenial of a funcion (hen ermed) energy E : d W E I will be assumed ha he coninuous medium absorbs power from he exerior hrough: Mechanical Power: he work performed by he mechanical acions (body and surface forces) acing on he medium. Thermal Power: he hea enering he medium. 48

49 Exernal Mechanical Power The exernal mechanical power is he work done by he body forces and surface forces per uni of ime. In spaial form i is defined as: e P b v d v ds dr b d v dr ds v 49

50 Mechanical Energy Balance Using n and he Divergence Theorem, he racion conribuion reads, v n v : v v v l Taking ino accoun he ideniy : spaial velociy l d w gradien ensor n ds ds d d :l :d :w Divergence Theorem =0 So, : vds vd dd 50

51 Mechanical Energy Balance b dv Subsiuing and collecing erms, he exernal mechanical power in spaial form is, v ds P bv d vd : dd e dv bvd : dd vd : dd dv Reynold's Lemma d 1 d 1 vv ( v 2 ) 2 2v v d 1 d 1 P d :d d d :d d 2 2 e ( v ) ( v )

52 Mechanical Energy Balance. Theorem of he expended power. Sress power d 1 2 Pe bv d v ds v d :dd 2 exernal mechanical power enering he medium d Pe K P K kineic energy P sress power Theorem of he expended mechanical power REMARK The sress power is he mechanical power enering he sysem which is no spen in changing he kineic energy. I can be inerpreed as he work by uni of ime done by he sress in he deformaion process of he medium. A rigid solid will produce zero sress power ( d 0). 52

53 Exernal Thermal Power The exernal hermal power is incoming hea in he coninuum medium per uni of ime. The incoming hea can be due o: Non-convecive hea ransfer across he volume s surface. incoming hea qx (, ) nds uni of ime hea conducion flux vecor Inernal hea sources r( x, ) d specific inernal hea producion hea generaed by an inernal source uni of ime 53

54 Exernal Thermal Power The exernal hermal power is incoming hea in he coninuum medium per uni of ime. In spaial form i is defined as: where: Qe r d qn ds ( r q) d q x, r x, nq ds q ) d is he hea flux per uni of spaial surface area. is an inernal hea source rae per uni of mass. 54

55 Toal Power The oal power enering he coninuous medium is: d 1 v :d qn 2 Pe Q e d d r d ds 2 55

56 5.10. Energy Balance Ch.5. Balance Principles 56

57 Thermodynamic Conceps A hermodynamic sysem is a macroscopic region of he coninuous medium, always formed by he same collecion of coninuous maer (maerial volume). I can be: ISOLATED SYSTEM OPEN SYSTEM MATTER Thermodynamic space HEAT A hermodynamic sysem is characerized and defined by a se of hermodynamic variables 1, 2,... n which define he hermodynamic space The se of hermodynamic variables necessary o uniquely define a sysem is called he hermodynamic sae of a sysem. 57

58 Thermodynamic Conceps A hermodynamic process is he energeic developmen of a hermodynamic sysem which undergoes successive hermodynamic saes, changing from an iniial sae o a final sae Trajecory in he hermodynamic space. If he final sae coincides wih he iniial sae, i is a closed cycle process. A sae funcion is a scalar, vecor or ensor eniy defined univocally as a funcion of he hermodynamic variables for a given sysem. I is a propery whose value does no depend on he pah aken o reach ha specific value. 58

59 Sae Funcion Is a funcion,..., 1 n uniquely valued in erms of he hermodynamic sae or, equivalenly, in erms of he hermodynamic variables 1, 2,, n Consider a funcion 1, 2, ha is no a sae funcion, implicily defined in he hermodynamic space by he differenial form: f1 1, 2 d1 f2 1, 2 d2 The hermodynamic processes and yield: 1 2 B A f2( 1, 2) ' 1 B B 2 B' A f2( 1, 2) For o be a sae funcion, he differenial form mus an exac differenial:, i.e., mus be inegrable: 59 d The necessary and sufficien condiion for his is he equaliy of cross-derivaives: f i,..., f j1 n,..., 1 n i, j1,... n j i d

60 Firs Law of Thermodynamics POSTULATES: 1. There exiss a sae funcion E named oal energy of he sysem, such ha is maerial ime derivaive is equal o he oal power enering he sysem: d d 1 2 E : Pe Qe v d :d d r d q n ds 2 P () Qe () 2. There exiss a funcion U named he inernal energy of he sysem, such ha: 60 I is an exensive propery, so i can be defined in erms of a specific inernal energy (or inernal energy per uni of mass) u x, : U : u d The variaion of he oal energy of he sysem is: d E d K d U e REMARK de and d K are exac differenials, herefore, so is du de dk. Then, he inernal energy is a sae funcion.

61 Global Form of he Inernal Energy Balance Inroducing he expression for he oal power ino he firs posulae: K d d 1 2 E v d :d d r d qn ds 2 Comparing his o he expression in he second posulae: d E d K d U 61 The inernal energy of he sysem mus be: d d U u d :d d r d qn ds P sress power Q, e exernal hermal power GLOBAL FORM OF THE INTERNAL ENERGY BALANCE

62 Local Spaial Form of he Inernal Energy Balance Applying Reynolds Lemma o he global form of he balance equaion, and using he Divergence Theorem: d d du U u d d :d d r d q n ds d Then, he local spaial form of he linear momenum balance principle is obained hrough localizaion d( x, ) as: du U() q du d :d d r d q d, :d rq x LOCAL FORM OF THE ENERGY BALANCE (Energy equaion) 62

63 Second Law of Thermodynamics The oal energy is balanced in all hermodynamics processes following: de dk du P Q e e In an isolaed sysem (no work can ener or exi he sysem) de du dk e 0 P Q e However, i is no esablished if he energy exchange can happen in boh senses or no: du dk 0 0 There is no resricion indicaing if an imagined arbirary process is physically possible or no. du 0 dk

64 Second Law of Thermodynamics The concep of energy in he firs law does no accoun for he observaion ha naural processes have a preferred direcion of progress. For example: If a brake is applied on a spinning wheel, he speed is reduced due o he conversion of kineic energy ino hea (inernal energy). This process never occurs he oher way round. Sponaneously, hea always flows o regions of lower emperaure, never o regions of higher emperaure /11/2015 MMC - ETSECCPB - UPC

65 Reversible and Irreversible Processes A reversible process can be reversed by means of infiniesimal changes in some propery of he sysem. I is possible o reurn from he final sae o he iniial sae along he same pah. A process ha is no reversible is ermed irreversible. REERSIBLE PROCESS IRREERSIBLE PROCESS The second law of hermodynamics allows discriminaing: IMPOSSIBLE hermodynamic processes REERSIBLE POSSIBLE IRREERSIBLE 65

66 Second Law of Thermodynamics POSTULATES: 1. There exiss a sae funcion x, denoed absolue emperaure, which is always posiive. 2. There exiss a sae funcion S named enropy, such ha: I is an exensive propery, so i can be defined in erms of a specific enropy or enropy per uni of mass s : S () s( x,) d The following inequaliy holds rue: d d r S() sd d ds q n = reversible process > irreversible process Global form of he 2 nd Law of Thermodynamics 66

67 Second Law of Thermodynamics SECOND LAW OF THERMODYNAMICS IN CONTINUUM MECHANICS The rae of he oal enropy of he sysem is equal o greaer han he rae of hea per uni of emperaure e d d r S() sd d ds q n = reversible process > irreversible process Q r d ds Global form of he 2 nd Law of Thermodynamics qn rae of he oal amoun of he eniy hea, per uni of ime, (exernal hermal power) enering ino he sysem e r e d ds q n rae of he oal amoun of he eniy hea per uni of absolue emperaure, per uni of ime (exernal hea/uni of emperaure power) enering ino he sysem 67

68 Second Law of Thermodynamics Consider he decomposiion of enropy ino wo (exensive) counerpars: Enropy generaed inside he coninuous medium: Enropy generaed by ineracion wih he ouside medium: S S i i s x, d e e s x, i i e e S S S ds ds ds d 68

69 Second Law of Thermodynamics If one esablishes, e ds r e d ds q n i Then he following mus hold rue: And hus, e i e ds ds ds r d ds q n ds ds ds ds r q d ds 0 n REPHRASED SECOND LAW OF THERMODYNAMICS : i The inernally generaed enropy of he sysem, S, never decreases along ime ds e () 69

70 Local Spaial Form of he Second Law of Thermodynamics The previous eq. can be rewrien as: d i d r q s d s d d ds 0 n Applying he Reynolds Lemma and he Divergence Theorem: i ds ds r d d d d 0 q i Then, he local spaial form of he second law of hermodynamics is: ds ds r 0, q x = reversible process > irreversible process Local (spaial) form of he 2 nd Law of Thermodynamics (Clausius-Duhem inequaliy) 70

71 Local Spaial Form of he Second Law of Thermodynamics Considering ha, q 1 1 q q 2 The Clausius-Duhem inequaliy can be wrien as r 1 s q 0 i s s i ds ds r 1 1 q q 0 2 i s local CLAUSIUS-PLANCK 1 INEQUALITY 2 i s cond q 0 REMARK (Sronger posulae) Inernally generaed enropy can i be generaed locally, s local, or by i hermal conducion, s cond, and boh mus be non-negaive. HEAT FLOW INEQUALITY Because densiy and absolue emperaure are always posiive, i is deduced ha q 0, which is he mahemaical expression for he fac ha hea flows by conducion from he ho pars of he medium o he cold ones. 71

72 Alernaive Forms of he Clausius-Planck Inequaliy Subsiuing he inernal energy balance equaion given by du no u : drq ino he Clausius-Planck inequaliy, i s : s rq 0 local q r : d u yields, s : d u 0 u s : d 0 Clausius-Planck Inequaliy in erms of he specific inernal energy 72

73 Alernaive Forms of he Clausius-Planck Inequaliy The Helmholz free energy per uni of mass or specific free energy,, is defined as: : us Taking is maerial ime derivaive, : us s us s and inroducing i ino he Clausius-Planck inequaliy in erms of he specific inernal energy: u s : d 0 s : d 0 REMARK For infiniesimal deformaion, d, and he Clausius-Planck inequaliy becomes: ( s ) : 0 Clausius-Planck Inequaliy in erms of he specific free energy 73

74 5.11. Governing Equaions Ch.5. Balance Principles 74

75 Governing Equaions in Spaial Form v 0 bv T Conservaion of Mass. Coninuiy Equaion. Linear Momenum Balance. Firs Cauchy s Moion Equaion. Angular Momenum Balance. Symmery of Cauchy Sress Tensor. 1 eqn. 3 eqns. 3 eqns. u : d rq Energy Balance. Firs Law of Thermodynamics. 1 eqn. 8 PDE + 2 resricions u s :d 0 1 q 0 2 Second Law of Thermodynamics. Clausius-Planck Inequaliy. Hea flow inequaliy 2 resricions 75

76 Governing Equaions in Spaial Form The fundamenal governing equaions involve he following variables: densiy 1 variable v velociy vecor field 3 variables Cauchy s sress ensor field 9 variables u specific inernal energy 1 variable q hea flux per uni of surface vecor field 3 variables 19 scalar unknowns s absolue emperaure specific enropy 1 variable 1 variable A leas 11 equaions more (assuming hey do no involve new unknowns), are needed o solve he problem, plus a suiable se of boundary and iniial condiions. 76

77 Consiuive Equaions in Spaial Form v,, Thermo-Mechanical Consiuive Equaions. 6 eqns. s sv,, Enropy Consiuive Equaion. 1 eqn. qq v, K Thermal Consiuive Equaion. Fourier s Law of Conducion. 3 eqns. i u f, v,,,, 0 1,2,..., F i p Hea Kineic se of new hermodynamic variables: 1, 2,..., p. Sae Equaions. REMARK 1 The srain ensor is no considered an unknown as hey can be obained hrough he moion equaions, i.e., v. (1+p) eqns. (19+p) PDE + (19+p) unknowns REMARK 2 These equaions are specific o each maerial. 77

78 The Coupled Thermo-Mechanical Problem v 0 Conservaion of Mass. Coninuiy Mass Equaion. 1 eqn. 16 scalar unknowns Linear Momenum Balance. Firs Cauchy s Moion Equaion. 3 eqns. 10 equaions ( ( v), ) Mechanical consiuive equaions. 6 eqns. Energy Balance. Firs Law of Thermodynamics. 1 eqn. Second Law of Thermodynamics. Clausius-Planck Inequaliy. 2 resricions. 78 MMC - ETSECCPB - UPC

79 The Uncoupled Thermo-Mechanical Problem The mechanical and hermal problem can be uncoupled if The emperaure disribuion x, is known a priori or does no inervene in he hermo-mechanical consiuive equaions. The consiuive equaions involved do no inroduce new hermodynamic variables,. Then, he mechanical problem can be solved independenly. 79

80 The Uncoupled Thermo-Mechanical Problem v 0 Conservaion of Mass. Coninuiy Mass Equaion. 1 eqn. 10 scalar unknowns Linear Momenum Balance. Firs Cauchy s Moion Equaion. 3 eqns. Mechanical problem ( ( v), ) Mechanical consiuive equaions. 6 eqns. Energy Balance. Firs Law of Thermodynamics. Second Law of Thermodynamics. Clausius-Planck Inequaliy. 1 eqn. 2 resricions. Thermal problem 80

81 The Uncoupled Thermo-Mechanical Problem Then, he variables involved in he mechanical problem are: densiy 1 variable Mechanical variables v velociy vecor field Cauchy s sress ensor field 3 variables 6variables u specific inernal energy 1 variable Thermal variables q hea flux per uni of surface vecor field absolue emperaure 3 variables 1 variable s specific enropy 1 variable 81

82 Summary Ch.5. Balance Principles 82

83 Summary The convecive flux of A hrough a spaial surface S wih uni normal n is: S vn ds Where: s Time derivaives of a volume inegral: A is an arbirary propery x, is he descripion of he amoun of he propery per uni of mass. no local x, derivaive no maerial d x, derivaive d d inflow vn 0 ouflow vn 0 d x, d d v d 83

84 Summary (con d) Conservaion of mass: he mass of a coninuum body is a conserved quaniy. d d 0 d v v 0 Global spaial form Local spaial form (Coninuiy Equaion) Reynolds Lemma: d d d d Reynolds Transpor Theorem: v d d ds d vn ds Divergence Theorem 84

85 Summary (con d) Linear Momenum Balance: d b d ds v d dv b + x, Global spaial form Local spaial form (Cauchy s Equaion of Moion) Angular Momenum Balance: d r b d r ds r v d T x, Global spaial form Local spaial form (Symmery of he Cauchy sress ensor) 85

86 Summary (con d) Mechanical Energy Balance: d 1 2 Pe bv d ds v d d v 2 :d exernal mechanical power enering he medium K kineic energy P sress power Exernal Thermal Power: e qn q x, is he hea flux per uni of Where: Q r d ds r spaial surface area. x, is an inernal hea source rae per uni of mass. Toal Power P e Q e 86

87 Summary (con d) Firs Law of Thermodynamics. Inernal Energy Balance. P Qe d d u d :d d r d qn ds Global spaial form du, :d rq x Local spaial form (Energy Equaion) Second Law of Thermodynamics. d d r S s d d ds q n = reversible process > irreversible process Global spaial form i ds ds r 0, q x Local spaial form (Clausius-Duhem inequaliy) r 1 s q 0 CLAUSIUS-PLANK INEQUALITY 87

88 Summary (con d) Governing equaions of he hermo-mechanical problem: v 0 Conservaion of Mass. Coninuiy Mass Equaion. 1 eqn. b v Linear Momenum Balance. Firs Cauchy s Moion Equaion. 3 eqns. 8 PDE + 2 resricions T u :d rq Angular Momenum Balance. Symmery of Cauchy Sress Tensor. Energy Balance. Firs Law of Thermodynamics. 3 eqns. 1 eqn. u s :d 0 1 q 0 2 Second Law of Thermodynamics. Clausius-Planck Inequaliy. 2 resricions 19 scalar unknowns:, v,, u, q,, s. 88

89 Summary (con d) Consiuive equaions of he hermo-mechanical problem: v,, Thermo-Mechanical Consiuive Equaions. 6 eqns. (19+p) PDE + (19+p) unknowns Enropy s sv,, Consiuive Equaion. 1 eqn. qq v, K Thermal Consiuive Equaion. Fourier s Law of Conducion. 3 eqns. i u f, v,,,, 0 1,2,..., F i p Hea Kineic Sae Equaions. (1+p) eqns. se of new hermodynamic variables:. 1, 2,..., p The mechanical and hermal problem can be uncoupled if he emperaure disribuion is known a priori or does no inervene in he consiuive eqns. and if he consiuive eqns. involved do no inroduce new hermodynamic variables. 89