ª Œª Œ 29ƒ 4B Á 2009 7œ pp. 357 ~ 363 ª pv w ü s w A Study on The Velocity Distribution in Closed Conduit by Using The Entropy Concept kyáeá½á Choo, Tai HoÁOk, Chi YoulÁKim, Jin WonÁMaeng, Seung Jin Abstract When yields the mean velocity of the closed conduit which is used generally, it is available to use Darcy Weisbach Friction Loss Head equation. But, it is inconvenient very because Friction Loss coefficient f is the function of Reynolds Number and Relative roughness (ε/d). So, it is demanded more convenient equation to estimate. In order to prove the reliability and an accuracy of Chiu s velocity equation from the research which sees hereupon, proved agreement very well about measured velocity measurement data by using Laser velocimeter which is a non-insertion velocity measuring equipment from the closed conduit (Laser Doppler Velocimeter: LDV) and an insertion velocity measuring equipment and the Pitot tube which is a supersonic flow meter (Transit-Time Flowmeters). By proving theoretical linear-relation between maximum velocity and mean velocity in laboratory flume without increase and decrease of discharge, the equilibrium state of velocity in the closed conduit which reachs to equilibrium state corresponding to entropy parameter M value has a trend maintaining consistently this state. If entropy M value which is representing one section is determinated, mean velocity can be gotten only by measuring the velocity in the point appearing the maximum velocity. So, it has been proved to estimate simply discharge and it indicates that this method can be a theoretical way, which is the most important in the future, when designing, managing and operating the closed conduit. Keywords : Chiu's equation, mean velocity, entropy M, conduit s³ w Darcy-Weisbach œ w ù, ù f Reynolds (ε/d) w w» rw rw. Chiu œ y w» w d e (Laser Doppler Velocimeter: LDV) q (Ultrasonic Flowmeter: U/F), de vm (Pitot Tube) w dw d Chiu œ w sƒ ew w. x s³ x w ü sxk, pv ql M w sxk w wš sxk wš w w w. w, w tw pv ql M w d s³ w š l w w, z ƒ w s³ dw w. w : Chiu œ, s³, pv ql M, 1. kz(2002) w, w x ƒ, t š w. x x mù x x ƒ w. x ü (free surface)» ƒ w w ƒ t. ü x w j» d(laminar flow) ù(turbulent flow) w w w. d w w(analytical solutions)ƒ ƒwù ù zw zy lœw mœwœ (E-mail : thchoo@pusan.ac.kr) zw zy lœw mœwœ (E-mail : cyok@pusan.ac.kr) w œ (E-mail : kjwon100@kwater.or.kr) w œw (E-mail : maeng@cbnu.ac.kr) 29ƒ 4B 2009 7œ 357
w wƒ ƒw xù x mw w w w w. x¾ w s³ w» w ƒ z š œ Darcy-Weisbach œ w ù, ù f Reynolds (ε/d) w w» rw rw. w w š w tx x w. h L = f --- l V 2 Darcy Weisbach d 2g ( )», h L, f, l ¼, d, V s³, g ƒ. w» ƒ w j., ùkü š ƒw m s³ œ(mean Velocity Formula)š š. x w wxz(2007) wš Hazen-Williams(, 1910), Manning, Ganguillet-Kutter (, 1869), Weston(1890) ù, s³ (C) (n) y w j ù. x¾ w w w. w» w x q w ù d ù w. ¾ s³ d d» mw d mw d s³ s³ wù, x xœ mw s³ w š x w, ù yw š w w.» œ 2 s ùkü 2 s txw œ. w w» w w ƒ Chiu œ w. Chiu» w s w w ym pv w s wš, y w eü w y j Chiu œ wš. 2. ü Chiu œ w» w (Laser Doppler Velocimeter: LDV), q (Ultrasonic Flowmeter: U/F), vm (Pitot Tube) w x mw ü d wš w. x x 120 l/min 170 l/min¾ 6z yw dw (1) 9 w q yƒ w w dw e (Laser Doppler Velocimeter: LDV) w dw l qƒ mw nq w w w w q ng w ƒƒ qw ùkù w q (Ultrasonic Flowmeter: U/F) d w l ywš w. w, w x dwš w e vm(pitot Tube) w d w l w x mw d w Chiu œ w wš w. w de (Laser Doppler Velocimeter: LDV) q (Ultrasonic Flowmeter: U/F) R e 29,000~42,000, vm (pitot tube) R e 470,000 d Data ù wš w, w l w Chiu pv œ w dw Chiu œ w sƒ ew w š, yw s³ ƒ ³e, x w w p v ql(m)ƒ w sxk w wš w. 3. Chiu s œ(x qv ) Chiu œ y m pv w 2 œ, pv wƒ pv y w k y w w, w ƒ w, vw 2 œ s³ œ, Dr. Chiuƒ w» Chiu œ š. qk ƒš qv s»w, d ù š ñù eù. Chao-Lin Chiu(1987, 1989, 1993, 2006) w w s w. u = u max ln 1 + e M ( 1)Fu M [ ( )]», u œ s s³, u max, F(u) ƒsw š M pv. Chao-Lin Chiu(2006) w x s w (2) w» w z w ƒ œ F(u) ww. R x qv e (2) 358 ª Œª Œ
w ƒ qv ü u =0, u max qv, š qv l 0, qv ww. Monte Carlo Simulation w q v z ü v ü mw w v šw w. j z N w z n w, v sp qv 0 r R w r w. š» u, u ù y. Fu ( ) = n N --- = πr2 πr 2 - = 1 πr 2 r R 2 (3) ùkù F(u) w x qv w s. ü s ùkü. u = u max --ln 1 + e M ( 1) 1 M r R 2 w, s³ w (5). e M u = φ u max», φ = - (5) e M 1 M», u 2 s³, u max, M p v š φ 2 s³ x ùkü. (4) qv s w y e. y u max ƒ qv w. š qv u 0. ƒ (4). du - = u max -- dr M 2r - e M 1 R 2 ( ) --- 1 e M 2 r + ( 1) 1 R (6) y qv ƒ 0 w w. š qv (7) š, ww ƒ. du -- dr r=r = 2u ( max em 1) MRe M Fig. 1 Velocity Distribution in Circular Pipe Base on Eq. (4) 1 (3) (4) (6) (7) (4) Fig. 1 u/u max w 1 r/r w ùkù qv x ƒw š, qv w w. 4. d Data d Chiu s x l w» w vw. w ƒ y dw w w., Ÿ w w vz w qƒ yw, q yƒ w w dw e (Laser Doppler Velocimeter: LDV) w dw l w., qƒ mw nq w w w w q ng w ƒƒ qw ùkù w q (Ultrasonic Flowmeter: U/F) w x l w., dwš w e vm (pitot tube) w d w l w. w w l (Laser Doppler Velocimeter: LDV) q (Ultrasonic Flowmeter: U/F) Õq LDV w d w (2000, w w»)ö yw, vm(pitot tube) w d l NEL(National Engineering Laboratory East Kilbride, Glasgow) ÕInsertion Meters and Velocity Measuring ProbesÖš yw Chiu s yw. Table 1~3 LDV U/F x x 120 l/min 170 l/min¾ 6 yw dw, vm w d. 5. 5.1. pv M ƒƒ d w dzj M, u max w. pv qk M u max ƒš dzj s³ u w. ƒƒ s³ v wì wù wì w tw sxk pv M w. pv ql M u max w Table 4. Table 5~7 LDV q 120 l/min~170 l/min¾ 6 yw vm w 29ƒ 4B 2009 7œ 359
Table 1. LDV w d (D90 mm) (LPM) Position(r/R) -0.8333-0.6666-0.3333 0 +0.3333 +0.6666 +0.8333 120 Velocity(m/s) 0.245 0.291 0.316 0.321 0.316 0.294 0.25 130 Velocity(m/s) 0.279 0.330 0.352 0.347 0.353 0.332 0.283 140 Velocity(m/s) 0.297 0.350 0.380 0.375 0.383 0.354 0.303 150 Velocity(m/s) 0.325 0.381 0.410 0.404 0.412 0.385 0.332 160 Velocity(m/s) 0.348 0.406 0.442 0.440 0.444 0.410 0.353 170 Velocity(m/s) 0.370 0.426 0.468 0.461 0.470 0.435 0.380 Table 2. U/F w d (D90 mm) (LPM) Position(r/R) -0.6666-0.3333 0 +0.3333 +0.6666 120 Velocity(m/s) 0.212 0.259 0.230 0.255 0.208 130 Velocity(m/s) 0.245 0.293 0.284 0.302 0.243 140 Velocity(m/s) 0.281 0.325 0.328 0.330 0.279 150 Velocity(m/s) 0.306 0.350 0.354 0.349 0.303 160 Velocity(m/s) 0.333 0.376 0.388 0.374 0.328 170 Velocity(m/s) 0.359 0.398 0.402 0.399 0.342 Table 3. vm w d (D100 mm) Position(r/R) -0.952-0.800-0.612-0.331 +0.331 +0.612 +0.800 +0.952 1z Velocity m/s 4.00 4.63 5.04 5.30 5.26 5.02 4.58 3.87 2z Velocity m/s 3.96 4.67 5.07 5.29 5.39 5.12 4.76 3.95 Table 4. pv ql M u max d w M, u max d M, u max w s³( u ) U e s³ M, φ = = φ(m) ƒš tw sxk p v M (LPM) D(m) Table 5. (LDV) R(m) u max (m/sec) U max - 1 e M M 1 M (m/sec) φ (M) R 2 120 0.09 0.045 0.3244 5.1104 0.2629 0.8104 0.9993 130 0.09 0.045 0.3583 5.8031 0.2976 0.8307 0.9980 140 0.09 0.045 0.3868 5.4642 0.3176 0.8212 0.9985 150 0.09 0.045 0.4167 5.8516 0.3467 0.8320 0.9984 160 0.09 0.045 0.4502 5.4858 0.3700 0.8219 0.9992 170 0.09 0.045 0.4739 5.8066 0.3937 0.8308 0.9988 d d z w ùw, (u max ) pv ql M d ƒš (4) t m v SYSTAT w dz j w. u (LPM) D(m) Table 6. (U/F) R(m) u max (m/sec) M (m/sec) φ (M) R 2 120 0.09 0.045 0.2513 3.7061 0.1898 0.7554 0.9894 130 0.09 0.045 0.2988 3.1297 0.2170 0.7262 0.9959 140 0.09 0.045 0.3343 3.5280 0.2497 0.7468 0.9994 150 0.09 0.045 0.3578 3.8715 0.2730 0.7630 0.9998 160 0.09 0.045 0.3870 3.9049 0.2958 0.7645 0.9999 170 0.09 0.045 0.4067 4.2113 0.3162 0.7776 0.9991 Traverse D(m) Table 7. (vm) R(m) u max (m/sec) M (m/sec) φ (M) R 2 1z 0.1 0.05 5.3006 8.9528 4.7092 0.8884 0.9991 2z 0.1 0.05 5.3833 8.8395 4.7751 0.8870 0.9996 5.2. d Chiu œ s Fig. 2~7 t d w Chiuœ d š, x LDV q 120 l/min~170 l/min¾ yw œ d vm w d Chiuœ s ƒ ew š. 5.3. sxk Fig. 8~9 Chiu s³ l sxk M yw s ³ w sxk w w u u 360 ª Œª Œ
Fig. 2 Chiuœ d (LDV 120 l/min) Fig. 3 Chiuœ d (LDV 160 l/min) Fig. 4 Chiuœ d (U/F 140 l/min) Fig. 5 Chiuœ d (U/F 170 l/min) Fig. 6 Chiuœ d (Pitot tube-1) Fig. 7 Chiuœ d (Pitot tube-2) š. x ù x w w z pv q l w sxk wwš, w sx k wš w» pv q l M w d s³ w š l dw. w s³ x z w φ (M) w. M pv s w w s³ ùkü. w w yw w (Chiu Said, 1995). 5.4. sxk M œ s d y qw» w (Laser Doppler Velocimeter: LDV) q (Ultrasonic Flowmeter: U/F) d w z wš w., w l M œ s txw d w. w w sxk, s³ x z φ (M) = 0.8548, 0.8275 (5) w M 6.8365, 5.6851 w. w œ ù x w w y wš pv q 29ƒ 4B 2009 7œ 361
Fig. 8 sxk (LDV) Fig. 9 sxk (U/F) Fig. 10 Chiuœ(sxk M) d (LDV 120 l/min) Fig. 11 Chiuœ(sxk M) d (LDV 160 l/min) Fig. 12 Chiuœ(sxk M) d (U/F 140 l/min) l M w sxk w wš, w sxk wš w w Fig. 10~13¾ š. 6. Chiu œ y w» w de (Laser Doppler velocimeter: LDV) q Fig. 13 Chiuœ(sxk M) d (U/F 170 l/min) (Ultrasonic Flowmeter: U/F), de vm (Pitot tube) w x d d Chiu œ w sƒ ew w, x s³ x w ü sxk, pv ql M w s xk wwš, w sxk wš w w y w. w x ƒw d 362 ª Œª Œ
w w, w. 1. d Chiu œ s w» w (Laser Doppler Velocimeter: LDV), q (Ultrasonic Flowmeter: U/F), vm(pitot tube) yw w, x dw d Chiu œ w sƒ ew w. 2.» œ y pv w œ, pv qk M u max œ s ùký w. 3. 120 l/min~170 l/min yw x M s³ x š, p v ql M w sxk w w š sxk wš w w wš, (4) d w s³ x w. 4. (conduit) w tw pv q l M w d s³ w š l w. z ƒ w s³ d w w. w w (2) w. šx (1996) w.. ½(2003) qq p w ü s³ d. w, w w». (2008) Chiu sœ w. w, w. (2008) pv w sxk w. w, w. (2000) A Study Flow Measurement Methods using Ultrasonic Flowmeter and LDV. w, w w». kz œ(2002) w. xq. (2008) Manning Chiu œ w œ s w. w, w. ky(2002) pv w d»(ii) - t -, wmwz, wmwz, 22«4By, pp. 507-515. w wxz(2007). q. y(2004) q. ()lj. Chiu, C.-L. (1986) Structure of 3-D flow in rectangular open channels, Journal of Hydraulic Engineering, ASCE, Vol. 112, No. 11, pp. 1050-1068. Chiu, C.-L. (1987) Entropy and probability concepts in Hydraulics, Journal of Hydraulic Engineering, ASCE, Vol. 113, No. 5, pp. 583-599. Chiu, C.-L. (1988) Entropy and 2-D velocity distribution in open channels, Journal of Hydraulic Engineering, ASCE, Vol. 114, No. 10, pp. 738-756. Chiu, C.-L. (1989) Velocity distribution in open channel flow, Journal of Hydraulic Engineering, ASCE, Vol. 115, No. 5, pp. 576-594. Choo, T.H. (1998) Efficient Method of Discharge Measurement in Sandys Rivers. Ph. D. Thesis, Dep. of Civ. & Env. Engrg. Univ. of Pittsburgh, Pittsburgh, 1998. Choo, T.H. (1990) Estimation of energy and momentum coefficients in open channel flow by Chiu's velocity distribution equation. M.S. Thesis, Dep. of Civ. Engrg. Univ. of Pittsburgh, Pittsburgh, 1990. Chiu, C.-L. and Murray, D.W., and Choo, T.H. (1992) Variation of velocity distribution along nonuniform open-channel flow, Journal of Hydraulic Engineering, ASCE, Vol. 118, No. 7, pp. 989-1001. Chao-Lin Chiu (1993) Application of probability and entropy concepts in pipe-flow sturdy. Journal of Hydraulic Engineering, ASCE, Vol. 119, No. 6, pp. 742. Chiu, C.-L. and Said, C.A.A. (1995) Maximum and mean velocities and entropy in open-channel flow, Journal of Hydraulic Engineering, ASCE, Vol. 121, No. 1, pp. 26-35. Chiu, C.-L. and Tung, N.-C. (2002) Maximum velocity and regularities in open-channel flow. Journal of Hydraulic Engineering, ASCE, Vol. 128, No. 4, pp. 390-398. Chiu, C.-L. and Hsu, S.-M. (2006) Probabilistic approach to modeling of velocity distributions in fluids flows. Journal of Hydraulic Engineering, ASCE, Vol. 316, pp. 28-42. Mr D Boam, Insertion Meters and Velocity Measuring Probes, National Engineering Laboratory. Moramarco et al. (2004) Estimation of Mean Velocity in Natural Channels Based in Chiu's Velocity Distribution Equation, Journal of Hydraulic Engineering, ASCE, Vol. 9, No. 1, pp. 42-50. (: 2009.3.31/: 2009.5.8/: 2009.6.22) 29ƒ 4B 2009 7œ 363