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1 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels Chter 5. Incomressile Flo in Pies nd Chnnels Sher Stress nd Skin Friction in Pies ( 전단응력및표면마찰 ) * Sher-stress distrition For flly develoed flo, V V & F 0 β β from Eq. (4.51): F m& ( β V β V ) From Eq. (4.5), F S S F Fg 0 F s

2 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels F πr πr ( d) (πrd) 0 d 0 d r --- Eq. (5.1) πr d 로나누면 & r 과는무관 ( 관의단면적방향으로의압력은일정 ) t r r d d r Eq. (5.) Eq. (5.) 에서 Eq. (5.1) 을빼면, r r or r r

3 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Reltion eteen skin friction & ll sher 펌프에의한일이없고마찰을고려할경우의 Bernolli 방정식은 f h V gz V gz α ρ α ρ 일반적으로 > 이므로로표시할 --- Eq. (4.71) 수있고 flly develoed flo 인수평관을대상으로하며마찰은유체와관벽사이의 skin friction h fs 만존재하므로 s Z Z V V &,,, α α ( 압력강하는표면마찰에의한것이므로 ) 이경우 Bernolli 식은 fs s fs s h h ρ ρ ρ 즉, --- Eq. (5.4) From Eq. (5.), 0 s r D r h fs ρ ρ 4 를소거하면 s

4 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Friction fctor ( 마찰계수 ), f 여기서정의하는마찰계수 f 는 Fnning friction fctor 또다른마찰계수로 Blsis or Drcy friction fctor 가있는데이는 4f 에해당 f --- Eq. (5.6) ρ V / ρv ll sher stress density velocity hed 즉, ( 단위면적당전단력 ) ( 단위부피당운동에너지 ) skin friction h fs 와 friction fctor f 와의관계 : s h fs ρ r ρ 4 f V D --- Eq. (5.7) f s D ρv or s fρv D --- Eqs. (5.8)-(5.9)

5 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Flo in noncirclr chnnels In evlting the dimeter in noncirclr chnnels, n eqivlent dimeter ( 등가지름 ) D eq is sed. Deq 4r H r H : hydrlic rdis ( 수력학적반지름 ) r H S S : cross-sectionl re of chnnel : etted erimeter 1) Circlr te: ) Annlr ies: 3) Sqre dct: πd / 4 r H π D D 4 r H πd o / 4 πd i πd πd i o / 4 D o D 4 i r H 4 4 D eq D D eq D o -D i D eq 단면이원형이아닌관의경우 Reynolds nmer Re 또는 friction fctor f 등의계산시에 D 대신 D eq 혹은 r 대신 r H 를대입하여계산가능함을의미.

6 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels minr Flo in Pies nd Chnnels * minr flo of Netonin flids 원형단면을갖는흐름을대상, 속도분포는 centerline에대해대칭 deends only on r d µ dr d dr µ r r µ Eq. (5.3) 적용 r z ll centerline 적분하면 ( 경계조건 : 0 t r ) r 0 d r r µ r rdr r µ ( r r ) --- Eq. (5.15) r mx (t r 0) µ mx r 1 r --- Eq. (5.17)

7 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels Averge velocity V 1 ds S --- Eq. (4.11) ds πrdr Eq. (5.15) 대입한후적분 ( ) r r r r V 3 0 rdr r µ 4µ --- Eq. (5.18) V 이식을 r mx 와비교하면, 0. 5 µ In minr flo, mx --- Eq. (5.19) Kinetic energy correction fctor, α.0 Momentm correction fctor, β 4 3 Eq. (4.70) 에 (5.15) 와 (5.18) 을대입해계산 Eq. (4.50) 에 (5.15) 와 (5.18) 을대입해계산

8 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels Hgen-Poiseille eqtion Eq. (5.7) 과 Eq. (5.18) 을이용하여 대신보다실제적인 로변환하면, V sd 3µ or 3Vµ s D s --- Eq. (5.0) πd 여기서 q V 이므로 q 와 측정으로부터점도계산가능 : 4 s s D π 18q 4 µ : Hgen-Poiseille eqtion 또한 Eq. (5.7) 에서 4 /( D) 이므로, 8V µ D s --- Eq. (5.1) Eq. (5.1) 을 Eq. (5.7) 에대입하면 f 와 Re 사이의관계가유도됨 : 16µ DVρ 16 Re f --- Eq. (5.)

9 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * minr flo of non-netonin liqids -Poer l flids K d dr n 반지름 r 에따른 velocity rofile: r K 1/ n r 1 1/ n r 1 1/ n 1 1/ n Fig Velocity rofiles in the lminr flo of Netonin nd non-netonin liqids.

10 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels -Binghm model K o d dr 0 d dr t > t < o o 반지름 r 에따른 velocity rofile: 1 ( r r r) 1 0 K r Fig () Velocity rofile nd () Sher digrm for Binghm lstic flo - Some non-netonin mixtres t high sher violte the zero-velocity (no-sli). c. ex) mltihse flids (ssensions, fier-filled olymers) sli t the ll

11 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels Trlent Flo in Pies nd Chnnels viscos slyer ( 점성하층 ): viscos sher, eddy ffer lyer ( 완충층 ) or trnsition lyer ( 전이층 ): viscos sher & eddy 공존 trlent core ( 난류중심부 ): viscos sher, eddy diffsion viscos slyer C.. trlent core ll ffer lyer Velocity rofile for trlent flo: mch fltter thn tht for lminr flo Eddies in the trlent core: lrge t lo intensity in the ffer lyer: smll t high intensity Re C.. lminr flo trlent flo ll Most of the kinetic-energy content of the eddies lies in the ffer zone.

12 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Velocity distrition for trlent flo In terms of dimensionless rmeters y * V * f * y ρ y µ µ ρ : friction velocity : velocity qotient ( 무차원 ) ρ Re sed on * & y : distnce ( 무차원 ) y : distnce from te ll ( r r y) * Universl velocity distrition eqtions i) viscos slyer: ii) ffer lyer: iii) trlent core: y y 5.00ln y 3.05 intersection 으로부터 < 5 5 < y < 30 y > 30.5ln y 5.5 for viscos slyer for ffer zone for trlent core Re > 10,000 이상에서적용가능

Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Reltions eteen mximm velocity & verge velocity V the rtio V chnges ridly / mx mx For lminr flo, V / mx is exctly 0.5. from Eq. (5.

13 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Reltions eteen mximm velocity & verge velocity V the rtio V chnges ridly / mx mx For lminr flo, V / mx is exctly 0.5. from Eq. (5.19) When lminr flo chnges to trlent, from 0.5 to ot 0.7, & increses grdlly to 0.87 hen Re10 6. * Effect of roghness Rogh ie lrger friction fctor f ft'n of Re & k / D k : roghness rmeter k/d : reltive roghness For lminr flo, roghness hs no effect on f nless k is so lrge. Tyes of roghness

14 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Friction fctor chrt k / D Friction fctor lot for circlr ies (log-log lot)

Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Friction fctor for smooth te f 0.046 Re 0. 0.15 f 0.014 0.

15 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Friction fctor for smooth te f Re f Re * Non-Netonin flids for 50,000 < Re < 10 for 3,000 < Re < 3 10 (ide rnge) 6 6 * Drg redction Dilte olymer soltions in ter drg redction in trlent flo Aliction: fire hose ( fe m of PEO in ter cn dole the ccity of fire hose) n m

16 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Friction loss from sdden exnsion h K fe e V V : verge velocity of smller or strem section) K e : exnsion loss coefficient K e cn e clclted theoreticlly from the momentm lnce eqtion (4.51) nd the Bernolli eqtion (4.71). S 1 Ke for trlent flo ( α 1 & β 1) S minr flo인경우에는 α & β 4/3을사용하면 K e 를구할수있다.

17 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Friction loss from sdden contrction cross section of minimm re h K fc c V V : verge velocity of smller or donstrem section) K c : contrction loss coefficient K c < 0.1 for lminr flo h fc is negligile. S Kc for trlent flo (emiricl eqtion) S

18 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Friction loss from fittings h K ff * Totl friction f V V : verge velocity in ie leding to fitting K f : fitting loss coefficient Tle 5.1 oss coefficients for stndrd ie fittings h f 4 f D K c K e K skin friction loss coeff. contrction loss coeff. f V fitting loss coeff. exnsion loss coeff. Bernolli eqtion ithot m: ( ) f Ex. 5.) Homeork ρ g Z Z h 대입

19 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Minimizing exnsion nd contrction losses. Contrction loss cn e nerly eliminted y redcing the cross section grdlly. K c 0.05 In this cse, sertion & ven contrct do not occr.. Exnsion loss cn lso e minimized y enlrging the cross section grdlly To minimize exnsion loss, the ngle eteen the diverging lls of conicl exnder mst e less thn 7 o. For ngles > 35 o The loss throgh this exnder cn ecome greter thn tht throgh sdden exnsion. sertion oint ( 분리점 ) Sertion of ondry lyer in diverging chnnel

20 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels * Flo throgh rllel ltes (Pro. 5.1 & 5.3 과연관 ) In lminr flo eteen infinite rllel ltes, er lte flo y W y x loer lte 1µ V 임을보이고 / 를구하시오. mx, V / mx ( 풀이 ) Force lnce: yw yw W from Eq. (4.5) y 대입 d µ dy

21 Unit Oertions Chter 5. Incomressile Flo in Pies nd Chnnels y d y dy 0 / ) ( µ 적분하면, ( ) mx µ.c. 대입 ( mx t y0) ) ( y µ ( ) W dy W ds S V 1µ 1 1 / 0 1 V µ mx ) / ( 1 y 3 mx V Relted rolems: (Pros.) 5.4, 5.8, 5.10, 5.1, 5.13, 5.17, 5.0 nd 5.1

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