Integral and Differential Laws of Energy Conservation

Transcription

1 R. Lecky 1 Integral and Dfferental Laws of Energy Conseraton 1. State of Stress n a Flowng Flud (Reew). Recall that stress s force per area. Pressure eerted by a flud on a surface s one eample of stress (n ths case, the stress s normal snce pressure acts or pushes perpendcular to a surface). At eery pont nsde a flowng flud there s stress. Ths stress can be conceptualzed as follows. Imagne a small (more precsely, nfntesmal) olume element of flud as shown n Fg. 1. The flud element s embedded n a large sea of eternal flud that eerts stress on the flud element. The stress can arse, for eample, from collsons and "rubbng" of the partcles of ths eternal flud wth those nsde the element. Now consder the front face of the flud element, whch s orented perpendcular to the 1 as. If we could measure the stress eerted by the eternal flud on the flud nsde the element across the front face, we would n general be able to dentfy three dfferent components of ths stress: the normal stress 11, whch acts perpendcular to the surface (see Fg. 1), and two shear stresses 1 and 13 that act tangental to the surface. The total force F on the front face of the flud element s, therefore F = dd3 ( ) (1) where dd3 s the total (dfferental) area of the front face. Smlar epressons can be deeloped for the other fe faces of the flud element. In total, we would need nne components of the stress to fully specfy the forces eerted on the dfferent faces: 11, 1, 13,, 3, 1, 33, 3, and 31. All nne components are llustrated n Fg. 1. We need nne stress components because for each of 3 Fg. 1 possble orentaton of a face (.e. wth the face orented perpendcular to the 1,, or 3 as), there are 3 ndependent components of the stress to consder (one normal and two shear), so we hae 3 3 stress components n total. The nne stress components completely specfy the so-called state of stress nsde the flud. Note that the stress components are functon of poston n the flud, = (1,, 3). We also recall that the,th component of the stress tensor,, s the stress eerted n drecton on a surface that s orented perpendcular to drecton. You can erfy ths last statement by referrng to Fg. 1. The stresses arse from pressure eerted by the eternal flud on that of the flud element as well as scous (frctonal) "rubbng" between the eternal and nternal flud. Vscous stresses arse because flud partcles on the two sdes of a face of the element trael at slghtly dfferent eloctes; n other words, the aerage elocty of flud mmedately outsde the element s dfferent from that mmedately nsde. The dfference n elocty causes the flud partcles to moe relate to one another and, hence, produce "rubbng" that results n scous frctonal stresses. Vscous stresses are only present when the elocty changes from pont to pont nsde a

2 R. Lecky flud. Put another way, scous stresses are drectly related to the presence of elocty gradents. For a Newtonan flud, the stress components are obtaned from the followng general epresson (whch, as epected, has elocty gradents n t): = p + ( ) + ( + ) () In equaton (), s the Kronecker delta, p s the thermodynamc pressure, s scosty of the Newtonan flud, s called the second coeffcent of scosty, and = s the elocty of the flud. and are propertes of the flud. p n equaton () s the true thermodynamc pressure (ths s what we normally thnk of as pressure n a flud at rest); howeer, n flud mechancs t s ery common to defne a so-called mechancal pressure P = p - ( ) ( ) n order to smplfy the form of equatons (by defnng the mechancal pressure s "hdden" n the equaton of momentum conseraton and does not appear eplctly). One often specalzes to ncompressble flows n whch case there s no dfference between the mechancal and thermodynamc pressures, P and p, snce = 0 for ncompressble flows. Therefore, for purposes of ths course we wll not be concerned wth these somewhat subtle dstnctons; howeer, should you at some pont need to consder stuatons n whch Newtonan fluds are greatly and rapdly compressed or epanded (large ) you should bear n mnd the dfference between p and P and be careful to note whch defnton s beng used. Equaton () can be drectly used to wrte down epresson for the dfferent normal and shear stresses, for eample and 11 = p + ( 3 = ( ) ) Note that equaton () mples that we are usng the Cartesan coordnate system snce n the last term on the rght the derates are taken wth respect to the Cartesan coordnate arables (where s 1,, or 3). We could defne a stress tensor as follows (for our present purpose we can thnk of ths defnton as ust a notatonal conenence), = (3) The "pars" of bass ectors n equaton (3),, are referred to as bass "dyads." Note that each stress component,, has a bass dyad,, assocated wth t (11 gets 11; 31 gets 31; etc.). Ths s analogous to hang each ector component, A, assocated wth a bass ector, (.e. A1

3 R. Lecky 3 gets 1). Note that we are mplctly workng n the Cartesan coordnate system snce we are usng dyads formed from the Cartesan bass ectors 1,, and 3. The usefulness n defnng the stress tensor as n equaton (3) s that, gen any surface (e.g. face of a flud element) of arbtrary orentaton as specfed by a unt normal n to that surface, the stress S (force/area) eerted on the surface by the eternal flud (the eternal flud s the flud ponted at by n) s drectly obtaned from S = n (4) In equaton (4), n s dotted wth from the left; therefore, the dot product s formed by dottng the components of n wth the left bass ector of each dyad. For eample, for the front face of the element n Fg. 1, n = 1. Then (note that the stress components are scalar quanttes and can therefore be pulled out of the dot product): S = S = (5) Equaton (5) s eactly the same as equaton (1), f equaton (1) s dded by the area dd3 n order to conert the force F nto the stress S.. The Integral Law of Energy Conseraton (Control Volume Approach)... The Concept of the Control Volume. In transport phenomena t s partcularly conenent to apply the fundamental balance laws (.e. conseratons of mass, conseraton of momentum, and conseraton of energy) to a fed control olume through whch a flud s flowng (Fg. ). A fed control olume s smply a regon of space that does not change poston or shape or moe n any way wth respect to the chosen reference frame. Physcally, the control olume could be the nteror of a chemcal reactor, a secton of a blood essel or a ppe, a lake, the atmosphere, etc We wll now apply the law of energy conseraton to a fed control olume. In what follows, we use V' to denote the regon of space correspondng to the control olume, and to denote the surface that defnes the boundary between the nteror and eteror of V'.

4 R. Lecky 4 Flud Flow 3 1 V' Fg... Deraton of the Integral Law of Energy Conseraton. Let's say we hae a fed control olume V' of arbtrary sze and shape. Then the law of energy conseraton can be wrtten: Rate of accumulaton rate of energy transfer rate of energy transfer nto of energy n V' = nto V' by flud flowng + V' by heat and work (6) across the surface of V' Energy s transferred by conecton (1st term on rght) because the flud that enters V' brngs energy wth t. Due to ts nonzero elocty, the flud possesses knetc energy. Due to ts poston n a gratatonal feld (and / or other potental felds, such as electrc or magnetc), the flud possesses potental energy. Due to the molecular bond braton, rotaton, translaton, etc. of the flud molecules, the flud possesses nternal energy. The flud that enters V' brngs the knetc, potental, and nternal energy t has wth t, and ths s the source of the conecte term n equaton (6). The second term on the rght of equaton (4) s present because transfer of heat nto and the performance of work by the eternal flud on the flud nsde also result n a transfer of energy. For eample, een n the absence of conecton, when frst term on rght equals zero, heat can stll flow nto V' by conducton or radaton. Work can be performed on the flud n V' by arous means. Some type of machnery, such as a rotatng mpeller, could eert force on the flud nsde V', resultng n a dsplacement of the flud. The product of ths force tmes the dsplacement leads to shaft work. Another eample of work performed on the flud nsde a control olume s flow work, whch represents work done by stresses as they push flud through the surface of V'. We wll say more about these types of work later. We wll denote the amount of energy per olume as e. e s energy per unt mass of flud (specfc energy), gen by e = u + (1/) + gz (7) In equaton (7), u s nternal energy per unt mass of flud, (1/) s knetc energy per unt mass of flud, and gz s gratatonal potental energy per unt mass of flud (we wll assume that the only type of potental energy present s gratatonal). Equaton (7) s obtaned by ddng the total energy E of a mass m of flud that s mong wth speed and s at a heght z by ts mass m,

5 R. Lecky 5 e = E/m = (1/m) (U + (1/)m + mgz) = u + (1/) + gz (8) where U s the total nternal energy of the mass m of flud, (1/)m s ts knetc energy, and mgz s ts potental energy due to graty. Multplcaton of e, energy per mass, by mass per olume yelds energy per olume e. An nfntesmal olume element dv' nsde the control olume V' contans an amount of energy equal to e dv' (we are ust multplyng energy per olume, e, by the olume consdered, dv'). In general and e are both functons of poston. The total amount of energy E n the entre control olume V' s then obtaned by ntegraton (.e. summaton) oer V', E = ρe dv ' (9) The derate of ths ntegral wth respect to tme s the rate at whch the amount of energy n the control olume changes, and therefore represents the rate of accumulaton of energy n V', de/dt = d dt ρedv ' (10) Equaton (10) s the desred mathematcal epresson for the left hand accumulaton term n the law of energy conseraton stated n equaton (6). Net, an epresson for the conecte term (1st term on rght of equaton (6)) wll be dered. The rate at whch energy s carred by flud flow across a dfferental area d on the surface of V' s equal to - e ( n) d, where s the flud elocty and n s a unt normal to the surface. Let's dssect ths epresson to clarfy ts meanng. - n s the proecton of onto the normal drecton (the n drecton) to the surface, and equals the speed of the flud perpendcular to the surface (Fg. 3); the mnus sgn ensures that the speed s poste when flud flows from outsde of V' nto V'. It s customary to regard flud nflow as poste snce t contrbutes to accumulaton n the control olume, whle flud outflow s regarded as negate. Flud Flow n surface length of segment = - n Fg. 3

6 R. Lecky 6 When - n s multpled by d the result, - n d, equals the rate at whch flud olume flows across d (Fg. 4); that s, - n d s the olumetrc flowrate nto V' through the area d (n unts of olume/tme). Multplcaton of the olumetrc flowrate - n d by the energy per olume of flud e ges the conecte nflow of energy nto V' through d. Thus - e n d s the rate at whch energy (nternal + knetc + potental) s brought by the flud nto V' as t flows across d. To arre at the total conecte transport of energy nto V', all of the dfferental contrbutons - e n d must be summed (.e. ntegrated) oer the entre surface Rate of conecton of energy nto V = ρe nd (11) Fg. 4 Equaton (11) s the desred result for the conecton term (1st term on rght) n the energy conseraton law stated n equaton (6). We stll need to deal wth the last term n equaton (6), the rate of energy transfer nto V' by heat and work. For now we wll smply wrte the total rate of heat transfer nto V' as dq/dt, and the total rate at whch the flud n the control olume epends energy by dong work on ts surroundngs as dw/dt. Then, combnng the heat and work terms wth equatons (10) and (11) accordng to the energy conseraton law equaton (6) results n d dt ρedv ' ρe n dq dw d (1) dt dt Equaton (1) s the ntegral (control olume) equaton of energy conseraton. The term on the left hand sde s the rate of accumulaton of energy n the control olume and the frst term on the rght s the rate at whch energy s brought nto the control olume by conecton. The nd term on the rght s the rate at whch heat s transferred nto the control olume by processes other than conecton. The 3rd term on the rght, - dw/dt, s the rate at whch the surroundngs perform work on the flud n the control olume. Accordngly, the negate of ths term (that s, dw/dt wthout the mnus sgn n front) s the rate at whch the flud n V' performs work on the surroundngs. Equaton (1) smply states that the rate at whch energy s accumulated n V' equals the rate at whch energy s brought nto V' by the flowng flud, plus the rate at whch heat

7 R. Lecky 7 s added to V' by non-conecte processes, plus the rate at whch the surroundngs perform work on the materal contaned n V'. Equaton (1) wll be used to dere a dfferental form of the energy conseraton law. efore we do that, howeer, let's consder the rate of heat transfer term dq/dt and the rate of work term dw/dt n more detal. Heat can be added to V' by conducton of heat across the surface of the control olume nto ts nteror. Heat can also be generated nsde V' by some "eternally coupled mechansm." A common eample of such a mechansm s resste heatng of the nteror of a wre carryng an electrcal current, n whch electrcal energy proded by an outsde source s conerted to heat wthn the wre due to ts resstance to current flow. Ths mechansm of heat generaton s "eternally-coupled" n the sense that t s not proded by an energy source orgnatng from nsde V'; nstead the orgnal energy used to generate the heat comes from an eternal source (.e. an electrcal power plant) that s located outsde of V'. If the mechansm were not eternally-coupled, then we would be smply conertng one type of energy nsde V' nto another type of energy nsde V', and the net change n total energy nsde V' would be zero. Transfer of heat nto V' by conducton can be calculated from the conducte heat flu qf (see also Handout ) as follows. At each pont on the surface boundng V', we calculate the component of qf perpendcular to the surface as - n qf. Ths perpendcular component of qf contrbutes to heat flu across, and s epressed n unts of energy / (area tme). Then - n qf d equals the rate of heat flow across a dfferental area d ( - n qf d has unts of energy/tme). Integraton oer the entre surface then sums the contrbutons from all area elements d, resultng n the total rate of heat flow through by conducton: Rate of heat flow nto V' by conducton = n (energy /tme) (13) q F d In addton to heat conducted across the surface, heat can also be added drectly to V' a some eternally-coupled mechansm, such as conerson of electrcal to heat energy as dscussed aboe. Such heat s typcally thought of as beng generated nsde V' at a olumetrc rate q, wth q possessng unts of energy / (olume tme). Then, the rate of heat generaton (energy/tme) nsde a olume dv' s q dv'. The total rate of all such olumetrc rates of heat generaton, summed oer the entre olume V', s Rate of heat generaton n V' by eternally-coupled mechansms = q dv ' (14) Replacement of the dq/dt term n (1) wth the sum of equatons (13) and (14) leads to d dt ρedv ' ρe n dw d n qf d q dv ' (15) dt A few remarks are now n order regardng the rate dw/dt at whch the system does work on the surroundngs. dw/dt can be decomposed nto two contrbutons. The frst contrbuton represents work due to forces between the flud n V' and the eternal enronment; these forces act at all

8 R. Lecky 8 ponts on the surface where the nternal flud (.e. flud nsde V') comes n contact and "pushes on" and moes the materal outsde V'. Ths contrbuton represents the rate of flow work. The second contrbuton comes from performance of shaft work, and represents the rate at whch mong machnery, whch s part of the surroundngs, does work on the flud nsde V. The classcal eample of shaft work s that of a rotatng mpeller that agtates the flud; snce the mpeller eerts a force on the flud that results n a dsplacement of the flud through some dstance, ths force tmes dsplacement results n work beng done. The oerall work rate term, due to flow and shaft work, comes from performance of work by forces actng between materal nsde and outsde the control olume, and can be constructed as follows. From equaton (4) we know that the stress S (force per area), eerted by materal outsde V' on materal nsde V', s gen by S = n (recall that s the stress tensor as defned n equaton (3)). Then - n s the stress -S eerted by materal n the system on that n the surroundngs (note the mnus sgn). Multplyng - S by d results n - n d whch s the force eerted by the materal n V on the surroundngs across a dfferental surface area d at the magnary boundary between the nternal and eternal enronments. Dottng ths force wth the rate of dsplacement (.e. the elocty) produces - (n ) d, the rate at whch work s performed by the system on the surroundngs across the area d. Integratng oer the entre surface s then the total rate of work performed by forces actng between materal nsde and outsde of V', Rate of work at control olume surface = σ n d (16) In ths deraton, note that we are ncludng both flow and shaft work n equaton (16). The shaft work comes from work done at flud-machnery boundares; there s no flud flow across these regons of where the machnery, whch s part of the eternal surroundngs, comes nto contact wth the materal nsde V. The flow work comes from forces actng at regons of across whch there s flud flow. Insertng equaton (16) nto (15) results n, d dt ρedv ' ρe nd n q Fd qdv ' n σ d (17) Our man nterest n equaton (17) wll be to use t n deraton of a dfferental equaton that states the prncple of energy conseraton. We wll then be able to use the dfferental equaton to calculate energy-related quanttes, such as temperature, nsde a flowng flud... Deraton of the Dfferental Law of Energy Conseraton. We frst recall two useful defntons: () Gauss's Dergence Theorem and () the substantal derate, also called the materal derate. Gauss's Dergence Theorem. The Dergence Theorem s used to conert flu ntegrals of ectors oer the control olume surface to ntegrals oer the control olume nteror:

9 R. Lecky 9 n Gd = G dv ' (18) As before, V' s a fed control olume and s a closed surface that encloses the control olume. n s a unt normal ector to, and G s an arbtrary ector whose derates are defned throughout V'. Substantal (Materal) Derate. The "materal derate" D/ s defned as D (19) t For eample, f we apply the materal derate to T, the temperature, we obtan DT T t T (0) What s the physcal meanng of equaton (0)? The frst term on the rght represents rate of change due to tme dependence of T, whle the second term represents rate of change due to mong through a gradent of T wth a elocty. Imagne that a statonary obserer s located at a pont nsde a pool of flud that s beng heated (Fg. 5a). Then, at the fed locaton at whch the obserer s located (the elocty of the obserer s zero), the obserer would see the temperature rse as the flud becomes warmer. In ths stuaton, the temperature of the flud s not at steady state;.e. t changes wth tme. At the obserer's fed poston, the rate of rse n temperature corresponds to the unsteady state term T/t n equaton (0). Fg. 5 Now, magne that an obserer s n a pool of flud n whch the temperature s at steady state (.e. not changng wth tme) so that, at each pont n the flud, T/t = 0. Howeer, magne also that a gradent n the temperature T s mantaned from one end of the pool to the other (.e. the flud s hotter at one end of the pool but cooler at the other; Fg. 5b). If the obserer

10 R. Lecky 10 remans statonary, as tme goes on the obserer would eperence the same temperature snce T/t = 0. Howeer, f the obserer begns to moe from the cool end of the pool to the warm end, the obserer would eperence a rate of rse n temperature gen by the rate of change of temperature wth poston (ths rate s the gradent T) tmes the speed wth whch the obserer s mong through ths gradent from the cooler to the warmer end. Ths product of speed tmes the temperature gradent s the T term n equaton (0), where s the obserer's elocty. Let's call ths term the moton-through-a-gradent effect. The dot product ensures that only that component of that results n moton n the drecton of T contrbutes to the obsered rate of temperature ncrease. For eample, f the obserer were not mong up the gradent T but only across t, meanng that s perpendcular to T, then no change n temperature would be eperenced snce the obserer would moe along a cure along whch the temperature s constant;.e. along an sotherm. Now magne that the obserer s nstead an element of flud mong wth the flow elocty. Therefore, s now the same elocty that appears n the Naer-Stokes equatons, the contnuty equaton, etc. Then, for the general stuaton n whch () the flud s beng heated or cooled, so that the temperature s not at steady state and () temperature gradents est from pont to pont n the flud, the total rate of change of temperature DT/ that the flud element wll eperence s the sum of the unsteady state and the moton-through-a-gradent terms, DT T t T Ths s the materal derate, as defned n (0). The materal derate represents the rate of change of a property (temperature n the aboe eample) eperenced by a flud element as t moes wth the flow elocty. Dfferental Law of Energy Conseraton. Recall the ntegral equaton of energy conseraton d dt ρedv ' ρed n q Fd qdv ' n n σ d (17) Ths equaton states: accumulaton of nternal+knetc+potental energy n a control olume V' (lhs) equals conecton of nternal+knetc+potental energy nto V' (1st term on rght), plus conducton of heat nto V' (second term on rght), plus generaton of heat nsde V' by eternallycoupled mechansms, plus work performed on materal n V' by forces actng at the boundary between V and surroundngs. Conertng all surface ntegrals nto olume ntegrals usng the Dergence Theorem (18), and mong the operator d/dt n the accumulaton term nsde the ntegral yelds (note: we can moe the d/dt nsde the accumulaton ntegral because V' s a fed control olume, so that the lmts of ntegraton oer V' are not functons of tme), ( ρe) dv ' t ρedv ' q dv ' q dv ' ( σ )dv ' (1) F

11 R. Lecky 11 In the fourth term on the rght n equaton (17), note that s a ector obtaned by takng as defned n equaton (3) and dottng t wth the elocty ector from the rght - n ths operaton, the rght bass ector of each dyad n s dotted wth the bass ectors n. The man pont s that, snce s ust a ector, the dergence theorem s used here n the same way as for the other terms on the rght een though the notaton may look more complcated. Combnng all terms under a sngle ntegraton sgn, ( ρe) ρe q F - q ( σ ) dv ' 0 () t The olume V' oer whch the ntegraton s performed s arbtrary n that we are not allowed to select a partcular V' (.e. by defnng the lmts of ntegraton) so as to satsfy the equalty to zero n (). Rather, equaton () must hold for any control olume (as long as t's a fed control olume). In such a case, the only way to satsfy the equalty to zero s by requrng the ntegrand to be zero, ( ρe) ρe q - q F ( σ ) 0 (3) t Equaton (3) s the dfferental law of energy conseraton. It could hae been obtaned ust as well by performng an energy balance on a dfferental olume element, such as a dfferental cube. Instead, we performed an energy balance on a macroscopc control olume V', and then showed that (3) must hold at each pont n space. What are the physcal meanngs of the terms n (3)? These can be traced back to the orgnal ntegral conseraton law. For eample, the frst term n equaton (3) came from the accumulaton term n the ntegral equaton (17); therefore, ths term must represent the rate of accumulaton per olume of total (nternal + knetc + potental) energy. Snce (3) s a dfferental equaton, ths term apples to a pont n space rather than to some large control olume. Smlarly, - e can be traced to be the rate at whch energy s conected to that pont, - qf the rate at whch heat s conducted to that pont, q the rate at whch energy s generated at that pont by eternally-coupled processes, and () the rate at whch surroundngs perform work on the flud at that pont. We wll now rearrange the dfferental energy balance, equaton (3). Frst, usng the product rule of dfferentaton we recognze that e/t = e/t + e/t. Second, we could show that e = e + e (e.g. see dentty 30d n handout 1). Wth these changes, (3) can be wrtten e ρ ρ e e ρ ρ e q q ( σ ) t t F = 0 (4) We recall the equaton of contnuty (the dfferental law of mass conseraton) from flud mechancs, ρ ρ 0 t (5)

12 R. Lecky 1 Accordng to (5), the sum of the nd and 3rd terms n (4) s zero; furthermore, the sum of the 1st plus 4th terms can be wrtten as e/t + e) = De/ where we used the materal derate notaton. The dfferental law of energy conseraton can then be rewrtten as De ρ q F q ( σ ) (6) Equaton (6) states that the rate of change of the total energy (nternal + knetc + potental) of a flud element mong wth the flow (left hand sde; note that a flud element mong wth the flow s a system wth zero net conecton of matter n or out) s equal to the rate of heat flow nto the element by conducton (1st term on rght), plus the rate of heat generaton wthn the flud element from eternally coupled sources (nd term on the rght), plus the rate at whch work s done on the flud element by the surroundng flud (last term on the rght). The last term can be further broken down nto useful work and work that s dsspated to nternal energy, as dscussed later. For now, we recognze that ths last term looks somewhat confusng, so to clarfy how t would be ealuated n practce let's work t out n detal. Takng the defnton of from equaton (3), we hae () = { ( ) ( ) } Carryng out the dot product between and, n whch the rght ector of each dyad s dotted nto the three components of (note we are usng the rght dyad ector because s dotted nto from the rght) results n () = { 1111 ( ) + 11 ( ) ( ) + 11 ( ) + ( ) + 33 ( ) ( ) + 33 ( ) ( ) } = { } Insertng the defnton of the gradent operator (n Cartesan coordnates) () = ( δ1 δ δ3 ) ( ) 3

13 R. Lecky 13 () = ( σ 11 1 σ1 σ133) ( σ11 σ σ33) ( σ311 σ3 σ333) 1 3 (7) Alternately, usng the summaton conenton, (7) can be wrtten more compactly as () = ( σ ) (8) (can you reerse epand (8) nto (7)?) If we knew the dependence of all the stress and elocty components ('s and 's) on the coordnate arables we could drectly ealuate (8) and arre at a formula for the rate at whch surroundngs do work on the flud at a pont. There's not much n the aboe manpulatons n terms of dffculty, but they can be confusng when seen for the frst tme and can become large n terms of paper consumed (hence the frequent use of the summaton conenton). Note that (8) s wrtten for the Cartesan coordnate system. Conseraton of energy s a physcal law; therefore equaton (6) s general. It can be specalzed to Newtonan fluds by nsertng n the Newtonan flud model for the stress (equaton ()), and to fluds obeyng Fourer's Law by substtutng that law for the conducte heat flu qf. We wll make these substtutons after rearrangng (6) nto a more commonly used form. 3. Related Energy Equatons. 3.. Mechancal Energy alance. The Mechancal Energy Equaton s concerned wth knetc and potental energes only. The nternal energy s not ncluded. Therefore, ths equaton wll not represent the law of energy conseraton, snce t s not a total energy balance. The Mechancal Energy Equaton can be dered drectly from the rate of change of momentum equaton studed n flud mechancs. The momentum equaton of nterest s D1 D D3 ρ δ1 ρ δ ρ δ3 ρg1δ1 ρgδ ρg3δ3 σ11 σ 1 σ31 σ1 σ σ3 σ13 σ 3 ( ) δ1 ( ) δ ( σ33 ) δ3 3 (9) Here we wrote the momentum balance (9) as one ector equaton, rather than as three scalar equatons (one for each coordnate drecton). y combnng the ector components, we could equalently hae also wrtten equaton (9) as D ρ D t ρg σ (30)

14 R. Lecky 14 where = , g = g11 + g + g33, and s gen n equaton (3). You could erfy that the last term n (30) epands nto the correspondng terms n (9) by performng smlar manpulatons as those used to dere (7). Takng the dot product of equaton (30) wth the elocty results n D ρ ( ρg σ) (31) We could proe that D D( ) D( / ) (3) Usng (3), equaton (31) becomes D( / ) ρ = g + ( ) (33) Now we make use of some facts: () gz, the gratatonal potental per unt mass at a pont at heght z, does not change wth tme so that (gz)/t = 0, and () f the gratatonal acceleraton g s constant, then the gratatonal force per unt mass g = - gz = -gz = - (gz). Here, Z = z s a unt ector pontng n the poste z drecton. Addng (gz)/t to the left hand sde and substtutng - (gz) for g, equaton (33) rearranges to D( / ) ( gz) ρ + { t + (gz)} = ( ) It was alrght to add (gz)/t only to the left hand sde snce ths term equals 0. Usng the ( gz) defnton of the materal derate, + (gz) = D(gz)/, and the aboe epresson t rearranges to D( / ρ gz) = ( ) (34) Equaton (34) s called the Mechancal Energy alance. The term "mechancal" emphaszes that the equaton s only concerned wth macroscopc knetc and gratatonal potental energes and does not nclude the molecular nternal energy. Howeer, t wll net be used to help dere a dfferental energy balance on nternal energy. Dfferental alance on Internal Energy. A balance on nternal energy s obtaned by subtractng the balance on mechancal (knetc + potental) energy, equaton (34), from the balance on total (nternal + knetc + potental) energy, equaton (6). We hae

15 R. Lecky 15 De ρ q F q ( σ ) (total energy balance) D( / ρ gz) = ( ) (mechancal energy balance) Recallng that e = u + / + gz, subtracton of the mechancal energy equaton from the total energy balance leads to D( u ρ / gz / gz) = qf q ( σ ) - ( ) (35) or D u ρ = q F q ( σ ) - ( ) (36) D t Equaton (36) has not referred to any partcular coordnate system as yet. Howeer, for the rest of the deraton we wll make use of Cartesan coordnates. Ths s not necessary, but may be easer to follow than stayng n the general ector notaton. We wll need the summaton and Kronecker delta conentons, so you may wsh to refresh these concepts from Handout 1. We also recall that the dergence of a ector A s A = ( δ1 δ δ3 ) (A11 + A + A33) = 1 3 A 1 1 A A 3 3 A = A (37) Second, from equaton (8) () = ( σ ) (8) Thrd, by procedures smlar to those used to dere equaton (8), we could show that ( ) = σ (38) Usng equatons (37), (8), and (38), equaton (36) rearranges to

16 R. Lecky 16 t u ρ D D = F q + q + σ ) ( σ (39) Accordng to the product rule of dfferentaton σ ) ( = σ + σ, thus t u ρ D D = F q + q + σ (40) Usng Fourer's Law, qf = - k T/, and the Newtonan flud epresson for from equaton () (we are now specalzng the nternal energy balance to Newtonan fluds that obey Fourer's Law): t u ρ D D = T k + q + [{ p + ( k k ) + ( + )} ] (41) The last term on the rght (the term nsde the [ ] parentheses) becomes p ( k k ) + ( + ) Summng oer n the two terms nolng the Kronecker delta results n (recall that the Kronecker delta equals zero unless = ): p ( k k ) + ( + ) (4) Usng epresson (37) for the dergence of a ector, we see that the p n (4) s multpled by ( ) and the n (4) s multpled by ( ). Replacng the last term n (41) wth epresson (4) rearranges equaton (41) to t u ρ D D = T k + q p ( ) ( ) + ( + ) (43) The last two terms n (43) are usually combned nto the dsspaton functon, = ( ) + ( + ) (44) or, summng oer the and ndces,

17 R. Lecky 17 = ( ) (45) To get to (45), the dsspaton functon was frst epanded by summng oer both and ndces n Equaton (44) and then smplfed by collectng terms. Usng the dsspaton functon, the nternal energy balance (43) becomes, ρ D u T = D t k + q p ( ) (46) Equaton (46) states that the rate of change of nternal energy of a flud element mong wth the flow (left hand sde; recall such an element s a closed system as no materal enters or leaes) equals the rate of heat flow nto the element by conducton (1st term on the rght), plus any eternally-coupled rate of heat generaton wthn the flud element (nd term on rght), plus the rate at whch work s performed on the element by pressure forces (3rd term on rght), plus the rate at whch mechancal energy s conerted to nternal energy wthn the element due to frctonal dsspaton (4th term on rght). The physcal meanng of the thrd term on the rght can be deduced by showng that s drectly proportonal to the olumetrc rate of epanson of a flud element (we wll omt ths proof); ths nterpretaton of then leads to the concluson that p ( ) s the rate at whch so-called "pv work" (perhaps smlar termnology s famlar from thermodynamcs) s performed on the flud element by usng pressure to compress or to epand the materal nsde t. The rgorous dentfcaton of as the rate at whch mechancal energy s conerted to nternal energy by scous dsspaton requres appeal to the second law of thermodynamcs. We wll omt ths proof here, but want to recognze that represents a ery common phenomenon. From eperence we know that rubbng two obects together can heat them. Ths heatng comes from dsspaton of mechancal energy of the obects moton to nternal energy. An analogous process occurs n a flowng flud n whch dfferent parts of the flud that are n contact moe at dfferent eloctes, leadng to such "rubbng" and resultng n a frctonal dsspaton of mechancal energy to nternal energy (why s there no dsspaton term n the total energy balance, equaton (6)?). For some materals, such as deal gases and strctly ncompressble lquds and solds, the nternal energy u s only a functon of temperature so that u = u(t). In that case, the constant olume specfc heat capacty ĉv s defned by so that ĉv = du/dt (47)

18 R. Lecky 18 du = ĉv dt (48) Usng equaton (48), the rate of change of the nternal energy of a flud element mong wth the flow, Du/, s equal to Du/ = ĉv DT/ (deal gases and ncompressble lquds) (49) where DT/ s the rate of temperature change of the flud element. Usng equaton (49), the nternal energy balance equaton (46) becomes DT ρcˆ V = D t T k + q p ( ) (deal gases & ncompressble lquds) (50) or, f the substantal derate s epanded, T ρcˆ V T = t T k + q p ( ) (51) (deal gases & ncompressble lquds) After much manpulaton, we hae arred at a useful form of the energy balance that wll allow us to calculate temperature felds, a quantty of obous engneerng mportance. In ths course we wll only be concerned wth ncompressble lquds or solds, or deal gases, so that equatons (50) and (51) apply. Howeer, t s useful to brefly consder also the case when these equatons are not good appromatons because the nternal energy depends on other arables besdes temperature. For sngle component systems (e.g. pure fluds) u becomes a functon of two arables such as T and 1/. 1/s olume of flud per unt mass of flud, and s customarly referred to as the "specfc olume" V. If u = u(t, V), then du = (u/t)v dt + (u/v)t dv. y defnton, the derate (u/t)v= ĉv; therefore du = ĉv dt + (u/v)t dv. ecause of the etra dependence of u on V, equatons (48) (51) would pck up another term correspondng to the (u/v)t dv dependence of nternal energy on specfc olume. For many common stuatons equatons (50) and (51) smplfy nto more user-frendly forms. For eample, when there s no heat generaton n the materal by eternally-coupled processes, q = 0. Furthermore, f the thermal conductty k s assumed constant, then k can be taken n front of the / derate. The remanng / (T/) = T, the Laplacan of T. Wth these modfcatons, equaton (51) would read as T ρcˆ V T = k T p ( ) (k constant; q = 0) (5) t Furthermore, f the flow s ncompressble so that = constant, the terms n equatons (50) and (51) and n the dsspaton functon equaton (45) dsappear. Thus

19 R. Lecky 19 T ρcˆ V T = k T (, k constant; q = 0) (53) t where s gen by equaton (45) but wthout the term. If, n addton, we are consderng the temperature dstrbuton n a sold materal or a flud at rest, then = 0. When = 0, also equals zero (see equaton (45)), and (53) becomes T t k = T (, k constant; = 0; q = 0) (54) ρcˆ V For strctly ncompressble solds and lquds the densty does not depend on temperature, and the constant olume and constant pressure specfc heat capactes are equal, ĉv = ĉp (one could proe ths from thermodynamc relatons and the defntons of ĉ V and ĉ P). Snce most solds and lquds typcally do not epand ery much wth temperature ths equalty s often a good appromaton, and ĉp can be used nstead of ĉv n the aboe equatons (howeer, for an deal gas, a compressble materal, ĉp = ĉv + R). If ĉp s used nstead of ĉv n equaton (54), the combnaton of parameters n front of T would be k/ĉp. Ths partcular combnaton s referred to as the thermal dffusty, = k/ĉp (55a) and (54) may be wrtten as T t = T (, k constant; = 0; q =0; ĉv = ĉp) (55b) Other smplfcatons are possble. In any problem the smplfcatons must reflect the physcal nature and statement of the problem. 4. Eamples of oundary Condtons. 4. Condtons on Temperature. A ery common boundary condton s one n whch temperature s specfed at a pont n space or an nstant n tme. For eample, T ( = 0) = To (56) T (t = 0) = To (57) where the alue of To s gen. 4. Condtons on Heat Flu. One eample of a boundary condton on heat flu deals wth heat conducton at an adabatc boundary or wall. y defnton, an adabatc wall does not transmt heat. From Fourer's Law, the conducte heat flu qf normal to the boundary s gen by (the subscrpt "" ndcates the component of heat flu normal to the boundary)

20 R. Lecky 0 dt qf = -kt = -k d z (58) f z s taken to be the coordnate perpendcular to the boundary, as n Fg. 6. Snce the boundary s adabatc, there can be no conducte heat flu perpendcular to t; therefore, qf must equal zero at the boundary,.e. at z = 0. Ths condton mples that Fg. 6 dt dz z 0 = 0 (adabatc boundary) (59) The subscrpt z = 0 on the temperature derate ndcates that the derate s to be ealuated at the boundary, at z = 0. Now magne a heat-permeable (dathermal) boundary between systems I and II, as n Fg. 7, and that steady state apples. Under steady state, there can be no accumulaton of heat at the boundary. If heat conducton s the only means by whch heat arres and leaes the boundary, then the rate at whch heat s conducted to the boundary, say from system I, must equal the rate at whch t s conducted away from the boundary nto system II. Ths means that dt dt ki k (60) dz I, z 0 dz II, z 0 dt where ki and kii are heat conducttes n systems I and II, respectely, and and dz I, z 0 dt are the temperature gradents n the two systems, ealuated at the boundary. dz II, z 0 Fg. 7

21 R. Lecky 1 A thrd eample we consder s that of a heterogeneous reacton occurrng at a boundary between a system and an adabatc wall, Fg. 8. Under steady state condtons there can be no accumulaton of heat at the boundary; therefore, the rate at whch heat of reacton s generated at the boundary must equal the rate at whch t leaes. Snce the wall s adabatc, no heat can flow nto the wall; thus, heat can only flow nto the system along the drecton of ncreasng z. If conducton s the only mechansm of heat transfer away from the surface, then n 1 r S = k H dt dz z 0 (61) where r S s the rate at whch mass of speces s produced by chemcal reactons at the boundary n unts of mass / (area tme), n s the number of speces partcpatng n the reactons, and Ĥ s the specfc enthalpy (unts: energy / mass) of speces. The left hand sde of equaton (61) s the heat generated per area per tme by chemcal reactons, and the rght hand sde s the rate at whch heat flows away from the surface by conducton. oth terms hae unts of energy / (area tme). Fg. 8 The aboe eamples hae noled conducton of heat to or from a boundary as the domnant mechansm of heat transfer. Snce conducton was noled, Fourer's Law was used to calculate the heat flu. Howeer, n all these eamples the heat flu to or from a boundary could also hae been modeled usng the heat transfer coeffcent approach, qf = h (TS - T) (6) wth TS the temperature at the boundary and T the temperature n the bulk of the phase from whch the heat occurs. In the case of radate heat transfer, epressons ntroduced n the pror handout could be used to calculate radate heat flu at a boundary.