Dimensional Analysis.

Transcription

1 Dmensnal nalyss. Unts, r hw we chse t measure magntudes f quanttes such as sze, temperature, r mass are nt requred fr laws f nature t functn prperly. Snce equatns f mmentum, energy, and mass cnseratn are statements f natural laws, they must reman ald regardless f the unts emplyed. That means we can stll sle these equatns een f we reme all the unts by makng them dmensnless. Belw we wll lk at sme eamples f dmensnless cnseratns laws. Dmensnless Cnseratn aws. Frst, we cnsder a dmensnless mmentum balance. Fr smplcty we specalze t an ncmpressble, cnstant scsty, Newtnan flud, n the absence f bdy frces. The th cmpnent f the Naer-Stkes equatns can then be epressed as t p () Equatn () can be made dmensnless by ddng each arable by a reference, cnstant alue wth the same unts. Fr nstance, we dde all lengths by sme cnstant reference length, all pressures by a cnstant reference pressure, all elctes by a cnstant reference elcty, etc. We wll dente the reference quanttes by a subscrpt "". The reference quanttes must prde a scale fr the prblem beng cnsdered. Fr eample, say yu are cnsderng ppe flw. Then a ald reference elcty culd be the aerage elcty f the flud n the ppe, r the mamum alue f the flud elcty, r the elcty at a pnt halfway between the center and the wall f the ppe, etc. Thse wuld all wrk, as they prde nfrmatn n hw fast the flud s flwng. Hweer, the reference elcty culd nt be the elcty f yur car n the way t the supermarket, snce that has n nfrmatn n the ppe flw beng cnsdered. Hstrcally, certan cnentns hae been adpted. Fr nstance, fr ppe flw the reference elcty s usually the aerage elcty f the flud n the ppe ( = lumetrc flwrate / ppe crsssectnal area), whle fr flw arund a sphere s the unperturbed elcty far frm the sphere. Usng the reference quanttes we can frm dmensnless arables t substtute nt () = / = / p = p / p t = t ( / ) () Here s a reference length that ndcates the physcal scale f the prblem (such as ppe radus fr ppe flw), p a reference pressure quantty that sets the pressure scale (such as pressure dfference between tw ends f a ppe sectn), and / a reference tme ( / represents the tme t takes t traerse the reference dstance f mng wth the reference speed ). The resultant dmensnless arables are dented wth an astersk. T cnert back t dmensned (regular) arables, equatns () can be rearranged t = = p = p p t = t ( / ) (b)

2 Insertng epressns (b) nt the Naer-Stkes equatn () and slghtly rearrangng yelds p p t () Eery term n equatn () s dmensnless. Furthermre, tw dmensnless cmbnatns (als called dmensnless grups r dmensnless numbers) hae appeared: eynlds Number: e = / (4) Euler Number: Eu = p / ( ) (5) Usng equatns () t (5), the mmentum balance can be rewrtten e p Eu t (6) We see that by takng dfferental equatns and/r bundary cndtns fr a prblem and makng them dmensnless, as dne abe fr the Naer Stkes equatn, dmensnless grups wll be generated. n analyss can then be made t deduce the physcal nterpretatn f the dmensnless grups. Fr eample, f e s large t s clear frm (6) that the last term, representng transfer f mmentum due t scus frces, can be neglected cmpared t the ther terms. Beng famlar wth such nterpretatns can be helpful n usng the magntude f a dmensnless grup t decde whch physcal mechansms (e.g. n the case f e, cnecte s. scus transprt f mmentum) are dmnant n a prblem. In turn, ths nfrmatn can be used t smplfy mdelng. Equatn (6) s an eample f a dmensnless mmentum balance fr Newtnan fluds. We can als make ther cnseratn laws dmensnless. s anther eample, the unsteady state energy balance fr ncmpressble, cnstant k materals n the presence f cnductn (nly) can be wrtten t T = T ρc k P ˆ (7) We can then defne T = (T T)/(T T) = t = t ( / ) (8) Here, the range T - T sets the scale f the temperature dfference n the system (T and T culd cme frm bundary cndtns, fr eample). ls, we nte that the aplacan peratr has

3 unts f nerse length squared, s we can make t dmensnless by wrtng. Substtutng (8) nt (7) leads t T k = T (9) t ρcˆ P The factr (T T) used t make temperature dmensnless was present n each term and s was cancelled ut. earrangng, T t Pe k = T = T ρcˆ Pe (0) h P ρc () k h ˆP Pe h s called the Peclet number fr heat transprt, and s nterpreted as representng rat f cnecte t cnducte heat transprt. lthugh we dd nt nclude cnectn f heat n the startng equatn (7), f we had we wuld hae smlarly dered Pe h. In addtn, we can take the rat f Peclet and eynlds numbers t dere the Prandtl number Pr, whch s nterpreted as representng rat f scus transprt f mmentum t cnducte transprt f heat, h Pe ρcˆ P cˆ P Pr () e k k Equatn (0) s a smple eample f a dmensnless energy balance. We can keep gng t als lk at mass transprt f a speces n a multcmpnent system. Takng the mass balance fr speces n the absence f reactns, and assumng that the densty and dffusn ceffcent are cnstant, we preusly dered t = D ρ () Prceedng smlarly as fr the energy balance, we net defne dmensnless quanttes = ( )/( ) = = / t = t ( / ) (4) The range sets the scale f cncentratn dfferences n the system. Substtutn f (4) nt () leads t D = t (5) ρ

4 The factr ( ) used t make dmensnless was cancelled ut snce each f the three terms n (5) had ths factr. earrangng, D = t Pe = m (6) m Pe (7) D The dmensnless grup Pe m s knwn as the Peclet number fr mass transprt, and s nterpreted as representng rat f cnecte t dffuse mass transprt. The rat f Peclet and eynlds numbers s knwn as the Schmdt number Sc, anther dmensnless grup, whch s nterpreted as the rat f scus transprt f mmentum t dffuse transprt f mass, m Pe Sc (8) e D D The abe eamples llustrate hw sme f the mre famus dmensnless grups arse by rewrtng cnseratn laws nt a dmensnless frm. The Buckngham PI Therem can be used fr the same purpse. The adantage f generatng dmensnless grups frm the cnseratn laws that descrbe a prblem, nstead f the Buckngham P Therem, s that the dfferental equatns, bundary cndtns, and ther equatns that frm the mathematcal statement f a prblem are drectly dered frm ts physcal characterstcs. Therefre, these equatns wll prde nly thse dmensnless grups releant t the prblem, s lng the equatns are crrect t start wth. In cntrast, the Buckngham P Therem apprach requres that the set f parameters gernng the prblem f nterest be guessed. The dsadantage s that t may nt be easy t wrte dwn the full set f equatns needed, especally fr cmple gemetres r stuatns. In such stuatns the Buckngham PI Therem can be much easer fr derng the releant dmensnless grups. Imprtant: f tw prblems bey the same frm f dmensnless dfferental equatns and als any aulary dmensnless equatns such as bundary r ntal cndtns, they are sad t be gemetrcally smlar. If n addtn the tw prblems hae dentcal alues f the dmensnless grups (e.g. e, Pr, Sc) fund n the equatns, then the prblems are als sad t be dynamcally smlar. Therefre, tw prblems that are gemetrcally and dynamcally smlar wll pssess dentcal dmensnless prblem statements and, thus, ther dmensnless slutns fr, T, p, r ther dependent arables f nterest wll als be dentcal. The slutns d nt necessarly hae t be calculated, but can als be btaned epermentally. In such an apprach ne uses a mdel system t measure the dependent epermental arables f nterest; fr eample, the energy dsspated n a flw as a functn f flw elcty. These results are then presented as the dmensnless slutn; fr eample, dmensnless dsspatn f energy as a functn f a dmensnless flwrate. Frctn factrs fr ppe flw, n whch the dmensnless frctn factr f s pltted as a functn f the dmensnless eynlds number e, are an eample f such a crrelatn. Once these dmensnless crrelatns are establshed, they can then be used t predct the behar f ther gemetrcally smlar systems (e.g. ppe flw)

5 when perated under cndtn f dynamc smlarty (e.g. same e numbers as fr whch the crrelatn was determned). EXMPE: Dsspatn n Ppe Flw. We wll llustrate calculatn f the energy dsspatn asscated wth steady state, ncmpressble, parablc (Hagen-Pseulle) flw n ppes, usng standard and dmensnless appraches. Frm ths llustratn we wll mtate why dmensnless representatn f a prblem s useful. Cnsder the ppe flw depcted n Fgure. The ppe dameter s D = and the dstance between the entry prt and the et prt s. We want t calculate hw much mechancal energy s dsspated t nternal energy per mass f flud flwng frm prt t prt, we call ths dsspatn Wf. In ther wrds, Wf s the wrk dne t ercme retardng frctnal frces per unt mass f flud as t flws frm prt t. Fgure Wf s gen by ntegratng the preusly ntrduced dsspatn functn, representng rate f dsspatn f mechancal t nternal energy per lume, er the lume f nterest between prts and, and then ddng ths ntegral by the rate f mass flw thrugh the ppe: W (9) f 0 0 r drddz Z r drd where the dsspatn functn fr ths flw s gen by (see earler handut n dfferental energy balances) = (dz/dr) (0) The tp ntegral n (9) s the ttal rate f dsspatn f mechancal t nternal energy n the ppe lume between the tw prts, whle the denmnatr s the mass flw rate thrugh the ppe. The rat ges the desred dsspatn per mass Wf. The parablc flw s gen by Z = /(4) dp/dz (r - ) () where the pressure gradent dp/dz respnsble fr the flw s taken as cnstant. Insertng (0) and () nt (9) leads t

6 W f W f dp dz r dp r drddz dz dp dp dz ( r ) r drd 4 dz dp dz 0 ( r 0 r dr ) r dr () Equatn () s smewhat ncnenent t use snce dp/dz s nt always knwn. On the ther hand, scsty and flwrate are mre readly accessble than dp/dz. ecallng that the aerage elcty fr parablc ppe flw s gen by = rdrd lumetrc flwrate z 0 0 = - /(8) (dp/dz) () area f flw () can be rewrtten by slng () fr dp/dz and substtutng the result nt () W f 8 (4) We can further smplfy (4) by wrtng Wf per unt length f ppe. We wll call ths Ŵf, and t wll epress energy dsspated per mass f flud as t traels alng a unt length f ppe (rather than all the way frm prt t ), Ŵf = Wf / = D (5) where we als replaced wth D/ t cnnect back t Fgure. Frmula (5) was able t be calculated because we had an epressn fr the elcty prfle n the ppe. Hweer, n mre cmple stuatns, say fr turbulent flw, we wuld nt be able t d that as easly and may need t resrt t epermental measurements f Ŵf. Eamnatn f the parameters n equatn (5) suggests that such eperments shuld be perfrmed as a functn f the scsty, densty, and flwrate f the flud, as well as the dameter f ppe; n ther wrds, Ŵf = g(,,, D) (6) where the unknwn functn g wuld be determned epermentally.

7 Nw let s re-epress Ŵf n a dmensnless frmat. Snce Ŵf has unts f energy per mass per length, ne way t make t dmensnless s t dde t by /D snce ths quantty has the same unts as Ŵf. / prdes a scale f the knetc energy f the flw per mass, whle D prdes a sze scale f the flw. The new dmensnless quantty wll be dented by f. Fr parablc flw, f fllws frm (5) f = Ŵf /( /D) = D D (7) f = = D e (8) where the eynlds number e = D/. Nte that, because f s dmensnless, t can nly depend n dmensnless cmbnatns f parameters, n ths case e. Frmula (8) apples t parablc flws thrugh ppes. Fr mre cmple flws, such as turbulent flws, we may need eperments t determne an epressn fr f. Imprtantly, (8) suggests that nly the sngle parameter e wuld need t be ared, f = h(e) (9) where h s the functn t be determned epermentally. Ths s a great smplfcatn er (6); ndeed, by epressng the prblem n a dmensnless frm t was clarfed that nly a sngle parameter, e, needs t be ared t establsh the crrelatn wth f. In cntrast, equatn (6) suggested that fur parameters wuld need t be ared ndependently. The smplfcatn realzed n (9) thus greatly reduces the number f eperments that wuld be needed t characterze hw energy dsspatn depends n characterstcs f the flw. In practce, measurements f f wuld be perfrmed n a mdel ppe flw as a functn f e. These data wuld then be used t generate a plt f f s e. Once the plt s establshed, t can then be used t determne f fr gemetrcally smlar systems (meanng: ther ppe flws) under cndtns f dynamc smlarty (meanng: lk up f frm the plt fr the e alue calculated fr the gemetrcally smlar flws f nterest). s stated earler, when tw prblems are gemetrcally and dynamcally smlar, they wll hae the same dmensnless slutn; n ther wrds the same alue f f wll apply. The actual dsspatn n the gemetrcally smlar flw can be calculated frm the lked up f alue by re-arrangng (7) Ŵf = ( /D) f (0) where and D refer t alues fr the gemetrcally smlar flw f nterest. In summary, benefts f dmensnless representatn nclude:

8 (). Dmensnless representatn allws lkng up slutns fr gemetrcally and dynamcally smlar systems based n dmensnless slutns btaned (ften epermentally) n a mdel system. (). Dmensnless representatn reduces the ttal number f parameters descrbng a prblem. Fr the abe eample we went frm fur (equatn 6) t just ne (equatn 9) parameter. Ths s f tremendus help especally f the slutn needs t be btaned epermentally, as many fewer parameters need t be ared. (). Dmensnless representatn helps dentfy hw crrelatns shuld be presented, e.g. as plts f f s e. In the fllwng handuts we wll encunter ther eamples n whch a dmensnless dependent arable, such as f n the abe eample, s epressed n terms f dmensnless ndependent arables, such as e. lthugh we wll nt repeat a detaled dscussn f each such case, the mtatn fr these dmensnless crrelatns (such as equatn (8) fr f) remans the same, namely t facltate calculatns fr gemetrcally and dynamcally smlar prblems.