TUTORIAL No. 1 FLUID FLOW THEORY

Transcription

1 TUTOIAL N. FLUID FLOW THEOY In de t cmlete ths tutal yu shuld aleady have cmleted level have a gd basc knwledge lud mechancs equvalent t the Engneeng Cuncl at eamnatn 03. When yu have cmleted ths tutal, yu shuld be able t d the llwng. Elan the meanng vscsty. Dene the unts vscsty. Descbe the basc ncles vscmetes. Descbe nn-newtnan lw Elan and slve blems nvlvng lamna lw thugh es and between aallel suaces. Elan and slve blems nvlvng dag ce n shees. Elan and slve blems nvlvng tubulent lw. Elan and slve blems nvlvng ctn cecent. Thughut thee ae wked eamles, assgnments and tycal eam questns. Yu shuld cmlete each assgnment n de s that yu gess m ne level knwledge t anthe. Let us stat by eamnng the meanng vscsty and hw t s measued.

2 . BASIC THEOY. VISCOSITY Mlecules luds eet ces attactn n each the. In lquds ths s stng enugh t kee the mass tgethe but nt stng enugh t kee t gd. In gases these ces ae vey weak and cannt hld the mass tgethe. When a lud lws ve a suace, the laye net t the suace may becme attached t t (t wets the suace). The layes lud abve the suace ae mvng s thee must be sheang takng lace between the layes the lud. Fg.. Let us suse that the lud s lwng ve a lat suace n lamnated layes m let t ght as shwn n gue.. y s the dstance abve the sld suace (n sl suace) L s an abtay dstance m a nt usteam. dy s the thckness each laye. dl s the length the laye. d s the dstance mved by each laye elatve t the ne belw n a cesndng tme dt. u s the velcty any laye. du s the ncease n velcty between tw adjacent layes. Each laye mves a dstance d n tme dt elatve t the laye belw t. The at d/dt must be the change n velcty between layes s du d/dt. When any mateal s demed sdeways by a (shea) ce actng n the same dectn, a shea stess τ s duced between the layes and a cesndng shea stan γ s duced. Shea stan s dened as llws. γ sdeways dematn heght the laye beng demed d dy The ate shea stan s dened as llws. γ& shea stan tme taken γ dt d dt dy du dy It s und that luds such as wate, l and a, behave n such a manne that the shea stess between layes s dectly tnal t the ate shea stan. τ cnstant γ& Fluds that bey ths law ae called NEWTONIAN FLUIDS.

3 It s the cnstant n ths mula that we knw as the dynamc vscsty the lud. DYNAMIC VISCOSITY shea stess ate shea τ γ & τ dy du FOCE BALANCE AND VELOCITY DISTIBUTION A shea stess τ ests between each laye and ths nceases by dτ ve each laye. The essue deence between the dwnsteam end and the usteam end s d. The essue change s needed t vecme the shea stess. The ttal ce n a laye must be ze s balancng ces n ne laye (assumed m wde) we get the llwng. d dy dτ dl 0 dτ dy d dl It s nmally assumed that the essue deces unmly wth dstance dwnsteam s the essue d gadent s assumed cnstant. The mnus sgn ndcates that the essue alls wth dstance. dl Integatng between the n sl suace (y 0) and any heght y we get d dl d dl du d dτ dy dy dy d u...(.) dy Integatng twce t slve u we get the llwng. d du y A dl dy y d dl u Ay B A and B ae cnstants ntegatn that shuld be slved based n the knwn cndtns (bunday cndtns). F the lat suace cnsdeed n gue. ne bunday cndtn s that u 0 when y 0 (the n sl suace). Substtutn eveals the llwng B hence B 0 At sme heght abve the suace, the velcty wll each the mansteam velcty u. Ths gves us the secnd bunday cndtn u u when y. 3

4 Substtutng we nd the llwng. d u A dl d u A hence dl y d d u u dl dl u y d dl u Plttng u aganst y gves gue.. BOUNDAY LAYE. y The velcty gws m ze at the suace t a mamum at heght. In they, the value s nnty but n actce t s taken as the heght needed t btan 99% the mansteam velcty. Ths laye s called the bunday laye and s the bunday laye thckness. It s a vey mtant cncet and s dscussed me ully n late wk. The nvese gadent the bunday laye s du/dy and ths s the ate shea stan γ. Fg..

5 .. UNITS VISCOSITY.. DYNAMIC VISCOSITY The unts dynamc vscsty ae N s/m. It s nmal n the ntenatnal system (SI) t gve a name t a cmund unt. The ld metc unt was a dyne.s/cm and ths was called a POISE ate Pseulle. The SI unt s elated t the Pse as llws. 0 Pse Ns/m whch s nt an accetable multle. Snce, hweve, Cent Pse (cp) s 0.00 N s/m then the cp s the acceted SI unt. cp 0.00 N s/m. The symbl η s als cmmnly used dynamc vscsty. Thee ae the ways eessng vscsty and ths s cveed net... KINEMATIC VISCOSITY ν Ths s dened as : ν dynamc vscsty /densty ν /ρ The basc unts ae m/s. The ld metc unt was the cm/s and ths was called the STOKE ate the Btsh scentst. The SI unt s elated t the Stke as llws. Stke (St) m/s and s nt an accetable SI multle. The cent Stke (cst),hweve, s m/s and ths s an accetable multle...3 OTHE UNITS cst m/s mm/s Othe unts vscsty have cme abut because the way vscsty s measued. F eamle EDWOOD SECONDS cmes m the name the edwd vscmete. Othe unts ae Engle Degees, SAE numbes and s n. Cnvesn chats and mulae ae avalable t cnvet them nt useable engneeng SI unts... VISCOMETES The measuement vscsty s a lage and cmlcated subject. The ncles ely n the esstance t lw the esstance t mtn thugh a lud. Many these ae cveed n Btsh Standads 88. The llwng s a be desctn sme tyes. 5

6 U TUBE VISCOMETE The lud s dawn u nt a esev and allwed t un thugh a callay tube t anthe esev n the the lmb the U tube. The tme taken the level t all between the maks s cnveted nt cst by multlyng the tme by the vscmete cnstant. ν ct The cnstant c shuld be accuately btaned by calbatng the vscmete aganst a maste vscmete m a standads labaty. Fg..3 EDWOOD VISCOMETE Ths wks n the ncle allwng the lud t un thugh an ce vey accuate sze n an agate blck. 50 ml lud ae allwed t all m the level ndcat nt a measung lask. The tme taken s the vscsty n edwd secnds. Thee ae tw szes gvng edwd N. N. secnds. These unts ae cnveted nt engneeng unts wth tables. Fg.. 6

7 FALLING SPHEE VISCOMETE Ths vscmete s cveed n BS88 and s based n measung the tme a small shee t all n a vscus lud m ne level t anthe. The buyant weght the shee s balanced by the lud esstance and the shee alls wth a cnstant velcty. The they s based n Stkes Law and s nly vald vey slw velctes. The they s cveed late n the sectn n lamna lw whee t s shwn that the temnal velcty (u) the shee s elated t the dynamc vscsty () and the densty the lud and shee (ρ and ρ s ) by the mula Fg..5 F gd (ρ s -ρ )/8u F s a cectn act called the Faen cectn act, whch takes nt accunt a eductn n the velcty due t the eect the lud beng cnstaned t lw between the wall the tube and the shee. OTATIONAL TYPES Thee ae many tyes vscmetes, whch use the ncle that t eques a tque t tate scllate a dsc cyde n a lud. The tque s elated t the vscsty. Mden nstuments cnsst a small electc mt, whch sns a dsc cyde n the lud. The tsn the cnnectng shat s measued and cessed nt a dgtal eadut the vscsty n engneeng unts. Yu shuld nw nd ut me detals abut vscmetes by eadng BS88, sutable tetbks lteatue m l cmanes. ASSIGNMENT N.. Descbe the ncle eatn the llwng tyes vscmetes. a. edwd Vscmetes. b. Btsh Standad 88 glass U tube vscmete. c. Btsh Standad 88 Falg Shee Vscmete. d. Any m tatnal Vscmete Nte that ths cves the E.C. eam questn 6a m the 987 ae. 7

8 . LAMINA FLOW THEOY The llwng wk nly ales t Newtnan luds.. LAMINA FLOW A steam e s an magnay e wth n lw nmal t t, nly ag t. When the lw s lamna, the steames ae aallel and lw between tw aallel suaces we may cnsde the lw as made u aallel lamna layes. In a e these lamna layes ae cydcal and may be called steam tubes. In lamna lw, n mng ccus between adjacent layes and t ccus at lw aveage velctes.. TUBULENT FLOW The sheang cess causes enegy lss and heatng the lud. Ths nceases wth mean velcty. When a cetan ctcal velcty s eceeded, the steames beak u and mng the lud ccus. The dagam llustates eynlds clued bbn eement. Clued dye s njected nt a hzntal lw. When the lw s lamna the dye asses ag wthut mng wth the wate. When the seed the lw s nceased tubulence sets n and the dye mes wth the suundng wate. One elanatn ths tanstn s that t s necessay t change the essue lss nt the ms enegy such as angula knetc enegy as ndcated by small eddes n the lw. Fg...3 LAMINA AND TUBULENT BOUNDAY LAYES In chate t was elaned that a bunday laye s the laye n whch the velcty gws m ze at the wall (n sl suace) t 99% the mamum and the thckness the laye s dented. When the lw wthn the bunday laye becmes tubulent, the shae the bunday layes waves and when dagams ae dawn tubulent bunday layes, the mean shae s usually shwn. Cmang a lamna and tubulent bunday laye eveals that the tubulent laye s thnne than the lamna laye. Fg.. 8

9 . CITICAL VELOCITY - EYNOLDS NUMBE When a lud lws n a e at a vlumetc lw ate Q m3/s the aveage velcty s dened Q u m A s the css sectnal aea. A ρu md u md The eynlds numbe s dened as e ν I yu check the unts e yu wll see that thee ae nne and that t s a dmensnless numbe. Yu wll lean me abut such numbes n a late sectn. eynlds dscveed that t was ssble t edct the velcty lw ate at whch the tanstn m lamna t tubulent lw ccued any Newtnan lud n any e. He als dscveed that the ctcal velcty at whch t changed back agan was deent. He und that when the lw was gadually nceased, the change m lamna t tubulent always ccued at a eynlds numbe 500 and when the lw was gadually educed t changed back agan at a eynlds numbe 000. Nmally, 000 s taken as the ctcal value. WOKED EXAMPLE. Ol densty 860 kg/m 3 has a knematc vscsty 0 cst. Calculate the ctcal velcty when t lws n a e 50 mm be damete. SOLUTION u e m u md ν eν D m/s 9

10 .5 DEIVATION OF POISEUILLE'S EQUATION LAMINA FLOW Pseulle dd the gnal devatn shwn belw whch elates essue lss n a e t the velcty and vscsty LAMINA FLOW. Hs equatn s the bass measuement vscsty hence hs name has been used the unt vscsty. Cnsde a e wth lamna lw n t. Cnsde a steam tube length L at adus and thckness d. y s the dstance m the e wall. y Fg..3 dy d du dy du d The shea stess n the utsde the steam tube s τ. The ce (F s ) actng m ght t let s due t the shea stess and s und by multlyng τ by the suace aea. Fs τ π L du F a Newtnan lud, τ dy du d. Substtutng τ we get the llwng. du F s - π L d The essue deence between the let end and the ght end the sectn s. The ce due t ths (F ) s ccula aea adus. F π du Equatng ces we have - π L π d du d L In de t btan the velcty the steame at any adus we must ntegate between the lmts u 0 when and u u when. u du - d L 0 u L u L ( ) ( ) 0

11 Ths s the equatn a Paabla s the equatn s ltted t shw the bunday laye, t s seen t etend m ze at the edge t a mamum at the mddle. Fg.. F mamum velcty ut 0 and we get u L The aveage heght a aabla s hal the mamum value s the aveage velcty s u m 8 L Oten we wsh t calculate the essue d n tems damete D. Substtute D/ and eaange. 3 Lum D The vlume lw ate s aveage velcty css sectnal aea. π Q 8 L Ths s ten changed t gve the essue d as a ctn head. The ctn head a length L s und m h /ρg 3 Lum h ρgd Ths s Pseulle's equatn that ales nly t lamna lw. π 8 L πd 8 L

12 WOKED EXAMPLE. A callay tube s 30 mm g and mm be. The head equed t duce a lw ate 8 mm3/s s 30 mm. The lud densty s 800 kg/m3. Calculate the dynamc and knematc vscsty the l. SOLUTION eaangng Pseulle's equatn we get h ρgd 3Lu m πd π A mm Q 8 u m 0.8 mm/s A N s/m.cp ν m / s 30.cSt ρ 800 WOKED EXAMPLE N..3 Ol lws n a e 00 mm be wth a eynlds numbe 50. The dynamc vscsty s 0.08 Ns/m. The densty s 900 kg/m3. Detemne the essue d e mete length, the aveage velcty and the adus at whch t ccus. SOLUTION eρu m D/. Hence u m e / ρd u m ( )/(900 0.) 0.05 m/s 3L u m /D /0..88 Pascals. u { /L}( - ) whch s made equal t the aveage velcty 0.05 m/s 0.05 (.88/ 0.08)( ) m 35.3 mm.

13 .6. FLOW BETWEEN FLAT PLATES Cnsde a small element lud mvng at velcty u wth a length d and heght dy at dstance y abve a lat suace. The shea stess actng n the element nceases by dτ n the y dectn and the essue deceases by d n the d d u dectn. It was shwn eale that d dy It s assumed that d/d des nt vay wth y s t may be egaded as a ed value n the llwng wk. Fg..5 d du Integatng nce - y A d dy y d Integatng agan - u Ay B...(.6A) d A and B ae cnstants ntegatn. The slutn the equatn nw deends un the bunday cndtns that wll yeld A and B. WOKED EXAMPLE N.. Deve the equatn kng velcty u and heght y at a gven nt n the dectn when the lw s lamna between tw statnay lat aallel lates dstance h aat. G n t deve the vlume lw ate and mean velcty. SOLUTION When a lud tuches a suace, t stcks t t and mves wth t. The velcty at the lat lates s the same as the lates and n ths case s ze. The bunday cndtns ae hence u 0 when y 0 Substtutng nt equatn.6a yelds that B 0 u0 when yh Substtutng nt equatn.6a yelds that A (d/d)h/ Puttng ths nt equatn.6a yelds u (d/d)(/){y - hy} (The student shuld d the algeba ths). The esult s a aablc dstbutn smla that gven by Pseulle's equatn eale nly ths tme t s between tw lat aallel suaces. 3

14 FLOW ATE T nd the lw ate we cnsde lw thugh a small ectangula slt wdth B and heght dy at heght y. Fg..6 The lw thugh the slt s dq u Bdy (d/d)(/){y - hy} Bdy Integatng between y 0 and y h t nd Q yelds The mean velcty s hence Q -B(d/d)(h3/) u m Q/Aea Q/Bh u m -(d/d)(h/) (The student shuld d the algeba).7 CONCENTIC CYLINDES Ths culd be a shat tatng n a bush lled wth l a tatnal vscmete. Cnsde a shat tatng n a cyde wth the ga between lled wth a Newtnan lqud. Thee s n veall lw ate s equatn.a des nt aly. Fg.7 Due t the stckness the lud, the lqud stcks t bth suaces and has a velcty u ω at the nne laye and ze at the ute laye.

15 I the ga s small, t may be assumed that the change n the velcty acss the ga changes m u t ze ealy wth adus. τ du/dy But snce the change s ea du/dy u/( - ) ω /( - ) τ ω /( - ) Shea ce n cyde F shea stess suace aea π hω F π hτ Tque F 3 π hω T F In the case a tatnal vscmete we eaange s that T ( ) 3 π hω In ealty, t s unlkely that the velcty vaes ealy wth adus and the bttm the cyde wuld have an aect n the tque..8 FALLING SPHEES Ths they may be aled t atcle seaatn n tanks and t a alg shee vscmete. When a shee alls, t ntally acceleates unde the actn gavty. The esstance t mtn s due t the sheang the lqud assng aund t. At sme nt, the esstance balances the ce gavty and the shee alls at a cnstant velcty. Ths s the temnal velcty. F a bdy mmesed n a lqud, the buyant weght s W and ths s equal t the vscus esstance when the temnal velcty s eached. W vlume densty deence gavty 3 πd g( ρ s ρ ) W 6 ρ s densty the shee mateal ρ densty lud d shee damete The vscus esstance s much hade t deve m st ncles and ths wll nt be attemted hee. In geneal, we use the cncet DAG and dene the DAG COEFFICIENT as C D esstance ce Dynamc essue jected Aea 5

16 ρu The dynamc essue a lw steam s πd The jected aea a shee s 8 C D ρu πd eseach shws the llwng elatnsh between C D and e a shee. Fg..8 F e <0. the lw s called Stkes lw and Stkes shwed that 3πdu hence C D /ρ ud / e F 0. < e < 500 the lw s called Allen lw and C D 8.5 e -0.6 F 500 < e < 0 5 C D s cnstant C D 0. An emcal mula that cves the ange 0. < e < 0 5 s as llws. C D 6 e e 0. F a alg shee vscmete, Stkes lw ales. Equatng the dag ce and the buyant weght we get 3πdu (πd 3 /6)(ρ s - ρ ) g gd (ρ s - ρ )/8u a alg shee vcmete The temnal velcty Stkes lw s u d g(ρ s - ρ )8 Ths mula assumes a lud nnte wdth but n a alg shee vscmete, the lqud s squeezed between the shee and the tube walls and addtnal vscus esstance s duced. The Faen cectn act F s used t cect the esult. 6

17 .9 THUST BEAINGS Cnsde a und lat dsc adus tatng at angula velcty ω ad/s n t a lat suace and seaated m t by an l lm thckness t. Fg..9 Assume the velcty gadent s ea n whch case du/dy u/t ω/t at any adus. du ω The shea stess n the ng sτ dy t ω The shea ce s df π d t 3 ω The tque s dt df π d t The ttal tque s und by ntegatng wth esect t. T 0 π 3 ω ω d π t t In tems damete D ths s T πωd 3t Thee ae many vaatns n ths theme that yu shuld be eaed t handle. 7

18 .0 MOE ON FLOW THOUGH PIPES Cnsde an elementay thn cydcal laye that makes an element lw wthn a e. The length s, the nsde adus s and the adal thckness s d. The essue deence between the ends s and the shea stess n the suace nceases by dτ m the nne t the ute suace. The velcty at any nt s u and the dynamc vscsty s. Fg..0 The essue ce actng n the dectn lw s {π(d) -π } The shea ce sng s {(ττ)(π)(d) - τπ} Equatng, smlyng and gnng the duct tw small quanttes we have the llwng esult. τ dτ du τ Newtnan luds. d dy I y s measued m the nsde du d u d d du d u d d du d Usng atal deentatn t deentate d hence d Integatng we get du d d u d du d A du A d...(a) whee A s a cnstant ntegatn. du the e then -y and dy - d sτ d du d d d yelds the esult du d u d d 8

19 ntegatn. whee B s anthe cnstant )...( Integatng agan we get B B A u Equatns (A) and (B) may be used t deve Pseulle's equatn t may be used t slve lw thugh an annula assage..0. PIPE At the mddle 0 s m equatn (A) t llws that A0 At the wall, u0 and. Puttng ths nt equatn B yelds { } agan. s equatn and ths spseulle' 0 whee A 0 u B B A.0. ANNULUS Fg.. { } { } { } A A B A C B A B A u 0 0 subtact D m C...(D) 0 )...( 0. and at 0 The bunday cndtns ae u 0 9

20 { } { } { } { } { } { } { } { } u - ut nt equatn B Ths s 0 be btaned m C. may be substtuted back nt equatn D. The same esult wll Ths A u u B B F gven values the velcty dstbutn s smla t ths. Fg.. 0

21 ASSIGNMENT. Ol lws n a e 80 mm be damete wth a mean velcty 0. m/s. The densty s 890 kg/m3 and the vscsty s Ns/m. Shw that the lw s lamna and hence deduce the essue lss e mete length. (50 Pa e mete).. Ol lws n a e 00 mm be damete wth a eynlds Numbe 500. The densty s 800 kg/m3. Calculate the velcty a steame at a adus 0 mm. The vscsty 0.08 Ns/m. (0.36 m/s) 3. A lqud dynamc vscsty Ns/m lws thugh a callay damete 3.0 mm unde a essue gadent 800 N/m3. Evaluate the vlumetc lw ate, the mean velcty, the cente e velcty and the adal stn at whch the velcty s equal t the mean velcty. (u av 0.0 m/s, u ma 0.0 m/s.06 mm). Smla t Q6 998 a. Elan the tem Stkes lw and temnal velcty. b. Shw that a shecal atcle wth Stkes lw has a temnal velcty gven by u d g(ρ s - ρ )/8 G n t shw that C D / e c. F shecal atcles, a useul emcal mula elatng the dag cecent and the eynld s numbe s 6 C D 0. e Gven ρ 000 kg/m 3, cp and ρ s 630 kg/m 3 detemne the mamum sze shecal atcles that wll be lted uwads by a vetcal steam wate mvng at m/s. d. I the wate velcty s educed t 0.5 m/s, shw that atcles wth a damete less than 5.95 mm wll all dwnwads. e

22 5. Smla t Q5 998 A smle lud cug cnssts tw aallel und dscs adus seaated by a a ga h. One dsc s cnnected t the nut shat and tates at ω ad/s. The the dsc s cnnected t the utut shat and tates at ω ad/s. The dscs ae seaated by l dynamc vscsty and t may be assumed that the velcty gadent s ea at all ad. ( ω ω ) πd Shw that the Tque at the nut shat s gven by T 3h The nut shat tates at 900 ev/mn and tansmts 500W we. Calculate the utut seed, tque and we. (77 ev/mn, 5.3 Nm and W) Shw by alcatn ma/mn they that the utut seed s hal the nut seed when mamum utut we s btaned. 6. Shw that ully develed lamna lw a lud vscsty between hzntal aallel lates a dstance h aat, the mean velcty u m s elated t the essue gadent d/d by u m - (h/)(d/d) Fg.. shws a langed e jnt ntenal damete d cntanng vscus lud vscsty at gauge essue. The lange has an ute damete d and s meectly tghtened s that thee s a naw ga thckness h. Obtan an eessn the leakage ate the lud thugh the lange. Fg..3 Nte that ths s a adal lw blem and B n the ntes becmes π and d/d becmes -d/d. An ntegatn between nne and ute ad wll be equed t gve lw ate Q n tems essue d. The answe s Q (πh3/)/{(d /d )}

23 3. TUBULENT FLOW 3. FICTION COEFFICIENT The ctn cecent s a cnvenent dea that can be used t calculate the essue d n a e. It s dened as llws. Wall Shea Stess C Dynamc Pessue 3.. DYNAMIC PESSUE Cnsde a lud lwng wth mean velcty u m. I the knetc enegy the lud s cnveted nt lw lud enegy, the essue wuld ncease. The essue se due t ths cnvesn s called the dynamc essue. KE ½ mu m Flw Enegy Q Q s the vlume lw ate and ρ m/q Equatng ½ mu m Q mu /Q ½ ρ u m 3.. WALL SHEA STESS τ The wall shea stess s the shea stess n the laye lud net t the wall the e. Fg.3. du The shea stess n the laye net t the wall s τ dy The shea ce esstng lw s F τ πld The esultng essue d duces a ce Equatng ces gves τ D L s F πd wall 3

24 3..3 FICTION COEFFICIENT LAMINA FLOW C Wall Shea Stess Dynamc Pessue D Lρu 3 Lu m Fm Pseulle s equatn Hence D 3.. DACY FOMULA m D 3 Lu 6 C L u ρ m D ρu md 6 e Ths mula s manly used calculatng the essue lss n a e due t tubulent lw but t can be used lamna lw als. Tubulent lw n es ccus when the eynlds Numbe eceeds 500 but ths s nt a clea nt s 3000 s used t be sue. In de t calculate the ctnal lsses we use the cncet ctn cecent symbl C. Ths was dened as llws. C Wall Shea Stess Dynamc Pessue D Lρu eaangng equatn t make the subject C Lρu m D Ths s ten eessed as a ctn head h C Lu m h ρg gd Ths s the Dacy mula. In the case lamna lw, Dacy's and Pseulle's equatns must gve the same esult s equatng them gves C Lu m 3 Lu m gd ρgd C 6 ρu D m 6 Ths s the same esult as bee lamna lw. e Tubulent lw may be saely assumed n es when the eynlds Numbe eceeds In de t calculate the ctnal lsses we use the cncet ctn cecent symbl C. Nte that n lde tetbks C was wtten as but nw the symbl eesents C FLUID ESISTANCE Flud esstance s an altenatve aach t slvng blems nvlvng lsses. The abve equatns may be eessed n tems lw ate Q by substtutng u Q/A C Lu C LQ h gd gda m Substtutng A πd / we get the llwng. m 3C LQ h gπ D Q s the lud esstance estctn. 5 3C L gπ D 5

25 I we want essue lss nstead head lss the equatns ae as llws. 3ρC LQ π D ρgh Q s the lud esstance estctn. 5 3ρC L 5 π D It shuld be nted that cntans the ctn cecent and ths s a vaable wth velcty and suace ughness s shuld be used wth cae. 3. MOODY DIAGAM AND ELATIVE SUFACE OUGHNESS In geneal the ctn head s sme unctn u m such that h φu m n. Clealy lamna lw, n but tubulent lw n s between and and ts ecse value deends un the ughness the e suace. Suace ughness mtes tubulence and the eect s shwn n the llwng wk. elatve suace ughness s dened as ε k/d whee k s the mean suace ughness and D the be damete. An Amecan Engnee called Mdy cnducted ehaustve eements and came u wth the Mdy Chat. The chat s a lt C vetcally aganst e hzntally vaus values ε. In de t use ths chat yu must knw tw the thee c-dnates n de t ck ut the nt n the chat and hence ck ut the unknwn thd c-dnate. F smth es, (the bttm cuve n the dagam), vaus mulae have been deved such as thse by Blasus and Lee. BLASIUS C e 0.5 LEE C e The Mdy dagam shws that the ctn cecent educes wth eynlds numbe but at a cetan nt, t becmes cnstant. When ths nt s eached, the lw s sad t be ully develed tubulent lw. Ths nt ccus at lwe eynlds numbes ugh es. A mula that gves an amate answe any suace ughness s that gven by Haaland. 6.9 ε 3.6 lg0 C e

26 Fg. 3. CHAT 6

27 WOKED EXAMPLE 3. Detemne the ctn cecent a e 00 mm be wth a mean suace ughness 0.06 mm when a lud lws thugh t wth a eynlds numbe SOLUTION The mean suace ughness ε k/d 0.06/ Lcate the e ε k/d Tace the e untl t meets the vetcal e at e ead the value C hzntally n the let. Answe C Check usng the mula m Haaland. C C C C C 3.6 lg 3.6 lg 3.6 lg ε e WOKED EXAMPLE 3. Ol lws n a e 80 mm be wth a mean velcty m/s. The mean suace ughness s 0.0 mm and the length s 60 m. The dynamc vscsty s N s/m and the densty s 900 kg/m 3. Detemne the essue lss. SOLUTION e ρud/ ( )/ ε k/d 0.0/ Fm the chat C h C Lu/dg ( )/( ).7 m ρgh kpa. 7

28 ASSIGNMENT 3. A e s 5 km g and 80 mm be damete. The mean suace ughness s 0.03 mm. It caes l densty 85 kg/m3 at a ate 0 kg/s. The dynamc vscsty s 0.05 N s/m. Detemne the ctn cecent usng the Mdy Chat and calculate the ctn head. (Ans m.). Wate lws n a e at 0.05 m3/s. The e s 50 mm be damete. The essue d s 3 0 Pa e mete length. The densty s 000 kg/m3 and the dynamc vscsty s 0.00 N s/m. Detemne. the wall shea stess (67.75 Pa). the dynamc essue (980 Pa).. the ctn cecent ( ) v. the mean suace ughness ( mm) 3. Elan bely what s meant by ully develed lamna lw. The velcty u at any adus n ully develed lamna lw thugh a staght hzntal e ntenal adus s gven by u (/)( - )d/d d/d s the essue gadent n the dectn lw and s the dynamc vscsty. Shw that the essue d ve a length L s gven by the llwng mula. 3Lu m /D The wall skn ctn cecent s dened as C τ /( ρum ). Shw that C 6/e whee e ρumd/ and ρ s the densty, um s the mean velcty and τ s the wall shea stess.. Ol wth vscsty 0- Ns/m and densty 850 kg/m3 s umed ag a staght hzntal e wth a lw ate 5 dm3/s. The statc essue deence between tw tang nts 0 m aat s 80 N/m. Assumng lamna lw detemne the llwng.. The e damete.. The eynlds numbe. Cmment n the valdty the assumtn that the lw s lamna. NON-NEWTONIAN FLUIDS 8

29 A Newtnan lud as dscussed s a n ths tutal s a lud that beys the law A Nn Newtnan lud s geneally descbed by the nn-ea law n τ τ kγ& y du τ & γ dy τ y s knwn as the yeld shea stess and γ& s the ate shea stan. Fgue. shws the ncle ms ths equatn. Gah A shws an deal lud that has n vscsty and hence has n shea stess at any nt. Ths s ten used n theetcal mdels lud lw. Gah B shws a Newtnan Flud. Ths s the tye lud wth whch ths bk s mstly cncened, luds such as wate and l. The gah s hence a staght e and the gadent s the vscsty. Thee s a ange the lqud sem-lqud mateals that d nt bey ths law and duce stange lw chaactestcs. Such mateals nclude vaus dstus, ants, cements and s n. Many these ae n act sld atcles susended n a lqud wth vaus cncentatns. Gah C shws the elatnsh a Dlatent lud. The gadent and hence vscsty nceases wth γ& and such luds ae als called shea-thckenng. Ths henmenn ccus wth sme slutns suga and staches. Gah D shws the elatnsh a Pseud-lastc. The gadent and hence vscsty educes wth γ& and they ae called shea-thnnng. Mst dstus ae lke ths as well as clay and lqud cement.. Othe luds behave lke a lastc and eque a mnmum stess bee t sheas τ y. Ths s lastc behavu but unlke lastcs, thee may be n elastcty t sheang. Gah E shws the elatnsh a Bngham lastc. Ths s the secal case whee the behavu s the same as a Newtnan lud ecet the estence the yeld stess. Fdstus cntanng hgh level ats amate t ths mdel (butte, magane, chclate and Maynnase). Gah F shws the elatnsh a lastc lud that ehbts shea thckenng chaactestcs. Gah G shws the elatnsh a Cassn lud. Ths s a lastc lud that ehbts sheathnnng chaactestcs. Ths mdel was develed luds cntanng d lke slds and s ten aled t mlten chclate and bld. Fg.. 9

30 MATHEMATICAL MODELS The gahs that elate shea stess τ and ate shea stan γ ae based n mdels equatns. Mst ae mathematcal equatns ceated t eesent emcal data. Hschel and Bulkeley develed the we law nn-newtnan equatns. Ths s as llws. n τ τ Kγ& K s called the cnsstency cecent and n s a we. y In the case a Newtnan lud n and τ y 0 and K (the dynamc vscsty) τ γ& F a Bngham lastc, n and K s als called the lastc vscsty. The elatnsh educes t τ τ y γ& F a dlatent lud, τ y 0 and n> F a seud-lastc, τ y 0 and n< The mdel bth s τ Kγ& n The Hechel-Bulkeley mdel s as llws. τ τ y Kγ& n Ths may be develed as llws. n τ τ K & γ τ τ n K & γ dvdng by & γ n τ τ y & γ n K K & γ & γ & γ & γ τ τ y n K & γ The at s called the aaent vscsty & γ & γ a y y τ τ y K & γ & γ & γ n smetmes wtten asτ τ & γ n F a Bngham lastc n s a τ y K & γ F a Flud wth n yeld shea value τ 0 s y y a whee s called K & γ a n the lastc vscsty. The Cassn lud mdel s qute deent n m m the thes and s as llws. y τ τ Kγ& 30

31 THE FLOW OF A PLASTIC FLUID Nte that luds wth a shea yeld stess wll lw n a e as a lug. Wthn a cetan adus, the shea stess wll be nsucent t duce sheang s nsde that adus the lud lws as a sld lug. Fg.. shws a tycal stuatn a Bngham Plastc. Fg.. MINIMUM PESSUE The shea stess actng n the suace the lug s the yeld value. Let the lug be damete d. The essue ce actng n the lug s πd / The shea ce actng n the suace the lug s τ y π d L Equatng we nd πd / τ y π d L d τ y L/ τ y L/d The mnmum essue equed t duce lw must ccu when d s lagest and equal t the be the e. (mnmum) τ y L/D The damete the lug at any geate essue must be gven by d τ y L/ F a Bngham Plastc, the bunday laye between the lug and the wall must be lamna and the velcty must be elated t adus by the mula deved eale. u ( ) ( D d ) L 6 L FLOW ATE The lw ate shuld be calculated n tw stages. The lug mves at a cnstant velcty s the lw ate the lug s smly Q u css sectnal aea u πd / The lw wthn the bunday laye s und n the usual way as llws. Cnsde an elementay ng adus and wdth d. dq u π d ( ) π d L π Q L Q π L 3 ( ) d π L π Q L The mean velcty as always s dened as u m Q/Css sectnal aea. 3

32 WOKED EXAMPLE. n The Hechel-Bulkeley mdel a nn-newtnan lud s as llws. τ τ Kγ&. Deve an equatn the mnmum essue equed d e mete length n a staght hzntal e that wll duce lw. Gven that the essue d e mete length n the e s 60 Pa/m and the yeld shea stess s 0. Pa, calculate the adus the slug sldng thugh the mddle. SOLUTION y Fg. 3.3 The essue deence actng n the css sectnal aea must duce sucent ce t vecme the shea stess τ actng n the suace aea the cydcal slug. F the slug t mve, the shea stess must be at least equal t the yeld value τy. Balancng the ces gves the llwng. π τ y πl /L τ y / 60 0./ 0./ m 6.6 mm 3

33 WOKED EXAMPLE. A Bngham lastc lws n a e and t s bseved that the cental lug s 30 mm damete when the essue d s 00 Pa/m. Calculate the yeld shea stess. Gven that at a lage adus the ate shea stan s 0 s - and the cnsstency cecent s 0.6 Pa s, calculate the shea stess. SOLUTION F a Bngham lastc, the same they as n the last eamle ales. /L τ y / 00 τ y /0.05 τ y / 0.75 Pa A mathematcal mdel a Bngham lastc s τ τ Kγ& y Pa 33

34 ASSIGNMENT. eseach has shwn that tmat ketchu has the llwng vscus etes at 5 C. Cnsstency cecent K 8.7 Pa s n Pwe n 0.7 Shea yeld stess 3 Pa Calculate the aaent vscsty when the ate shea s, 0, 00 and 000 s - and cnclude n the eect the shea ate n the aaent vscsty. Answes γ a 50.7 γ 0 a 6.68 γ 00 a γ 000 a A Bngham lastc lud has a vscsty 0.05 N s/m and yeld stess 0.6 N/m. It lws n a tube 5 mm be damete and 3 m g. () Evaluate the mnmum essue d equed t duce lw. (80 N/m ) The actual essue d s twce the mnmum value. Sketch the velcty le and calculate the llwng. () The adus the sld ce. (3.75 mm) () The velcty the ce. (67.5 mm/s) (v) The vlumetc lw ate. (7.6 cm 3 /s) 3. A nn-newtnan lud s mdelled by the equatn du τ K d n whee n 0.8 and K 0.05 N s 0.8 /m. It lws thugh a tube 6 mm be damete unde the nluence a essue d 600 N/m e mete length. Obtan an eessn the velcty le and evaluate the llwng. () The cente e velcty. (0.953 m/s) () The mean velcty. (0.5 m/s) 3