46 Y Y Y Y 3.1 R Y Figures mm Nylon Glass Y (X > X ) X Y X Figure 5-1 X min Y Y d Figure 5-3 X =X min Y X =10 Y Y Y Y Figure 5-

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1 Y 3.2 Eq. (3) 1 R [s -1 ] ideal [s -1 ] Y [-] Y [-] ideal * [-] S [-] 3 R * ( ω S ) = ω Y = ω 3-1a ideal ideal X X R X R (X > X ) ideal * X S Eq. (3-1a) ( X X ) = Y ( X ) R > > θ ω ideal X θ = ideal * { ω ( X > X ) S( X > ) } ω 3-1b θ X θ Y 3

2 46 Y Y Y Y 3.1 R Y Figures mm Nylon Glass Y (X > X ) X Y X Figure 5-1 X min Y Y d Figure 5-3 X =X min Y X =10 Y Y Y Y Figure 5-4(a) 3.2 mm Al 18.5 s -1 Y

3 47 x X < x + x Y X Y (X > X ) Y Y (X > X ) X (a) Figure 5-4(b) X min X min X X Figure mm Nylon 13.5s -1 Y X 5.2 X [a.u.] E [J] 1 π 3 = d ρ v E 5-1 v X E X v E max, v

4 48 E max v v,max v tip E max X max Figure 5-4, 5 Z 2.58 X max Z X max X max max = k v,max d ρ ρf µ X x 5-2a f Eq. (5-2a) Eq. (5-1) v tip E max E max X max max = k X max d ρ ρ f µ E e 5-2b E X Eq. (5-2b) X E k x k e k x = k e = k x = k e = Hidaka et al., 1991; Hirajima et al., 2001 Hidaka et al.(1991) X I d v

5 49 X I v d 5-3 Hirajima et al.(2001) X I 1.27 v d ρ ξ ε X Hidaka et al.(1991) Eqs. (5-3) (5-4) Eqs. (5-3) (5-4) Eq. (5-2a) Eq. (5-2a) Eq. (5-2b) X E X E X E X E 5.3 Figure 5-6 E X E Figure 5-4(b) f(lne) lne lne < lne + (lne) Y (lne)

6 50 f ( ln e) Y = ( ln e ln E < ln e + ( ln e) ) ( ln e) 5-5 f(lne) (lne) lne Y (lne > lne ) f(lne) Y (E >E f(lne) lne Y ( E E ) = Y ( ln E > ln E ) = f ( ln e) d( ln e) θ + ln > θ 5-6 E θ f(lne) lne f(lne) g(lne) f(lne) g(lne) 1 f(lne) 1 f(lne) Y all g(lne) Y all f(lne) f ( ln e) Y g( ln e) 2 ( ln e ln E ) ( ) mean 2 ln ESD 1 = all = Yall exp 2 2π ln ESD 5-7 ( ln e) d( ln e) = Y ( E 0) Y = all f > E mean E SD f(lne) Y all E 0 J Y (E > 0) Y (E > 0) Y all Y all Y (E 0) Y (E > 0) 0 Y all 1 Y (X > X ) X Figures 5-1, 3 Y X > 0 Y all 3.2 mm TFE

7 51 X mm Glass 1 Y * S Y 1 Y 1 3 E mean E SD Y all f(lne) Eq. (5-6) Y E >E 3 Eq.(5-6) g(lne) Excel ln E mean ln E SD lne Z ln E ln E mean = 5-9 ln E SD Z 0 1 Z = (lne ln E mean ) / ln E SD G Z > Z Excel G Z > Z Y all Y E >E ( E E ) = Y G( Z ) Y > all > 5-10 θ Z θ 5.4 E mean E SD E mean E SD Y all Figure 5-6 Eq. (5-7) 3

8 52 E mean Re Eq. (5-1) v v tip E tip Re v tip E E = k Re 5-11 mean tip ρ f vtip d Re = 5-12 µ Figure 5-7 Re E mean /E tip E SD ln E SD Re ln 0.3 E SD = k 2 Re 5-13 Figure 5-8 Re ln E SD k 1 k 2 k 1 = 64 k 2 = Re E mean / E tip ln E SD Kee and Rielly(2004) Takahashi et al.(1992) N Takahashi et al.(1992) d Kee and Rielly(2004) d Takahashi et al.(1992) Kee and Rielly(2004) d N d N d Re

9 53 Re E tip E mean ln E SD 5.5 Y all Re E mean E SD Y all Y all 0 J Y (E > 0) Y Y Y all Gahn and Mersmann, 1999; Kee and Rielly, 2004; loß and Mersmann, 1989; Yokota et al., 1999 c c Y all St 2 * ρ d v = µ D r v D r f Stokes

10 54 Stokes Grootscholten et al., 1982a, 1982b; He et al., 1995; Kee and Rielly, 2004; Nienow, 1976; Takahashi et al., 1992, 1993; Yokota et al., 1999 Kee and Rielly (2004) St τ St = τ f = v D b g v tip 5-15 g v D b D w D h D b = ( D w + D h ) / 2 [m] f v Stokes Eq. (5-15) St f Y all St Equation (5-15) St Y all Figures 5-9, 10 Eqs. (5-15) (5-21) Y St S c St * St * c erry and Chiltion, 1973 Iinoya,1963 St * c St * c

11 55 Y all St Kee and Rielly (2004) St Y all St Kee and Rielly (2004) R ideal Y St Kee and Rielly (2004) Kee and Rielly (2004) Figure 5-9 Y all St Y all Y all all 3 ( St α) ( St α) γ + β Y 5-16 = 3 St Y all S St Y all Y all L Figure 5-11

12 56 D b d L = d D b 5-17 St JS St JS v g = 5-18 D b ( π N D) JS N JS Eq. (4-1) St JS St Y all N JS = N JS D St St JS v N JS St JS St JS L St JS Eq. (5-16) α k + = L k4st JS JS β = k St 5-20 γ = lim Y St ( k + k L) ( + k St ) all = lim S = N 8 JS 5-21 k 3 k 8 k 3 = 2.0 k 4 = 0.80 k 5 = 0.40 k 6 = 0.25 k 7 = 2.0 k 8 = 0.20 Y all Eqs. (5-15) (5-21)

13 57 St Y all Figures 5-9, 10 d Y all 0 St Y all ideal * 3.2 * 0 * 0 N JS St N N JS St JS Y all c c Figure 5-11 c = D x / D b L c L c Figure 5-12 Iinoya,1963 c Eq. (5-16) Y all c L Y all c L L St JS Eq. (5-19) St JS 0.8 1

14 58 S St Y all Figures 5-9, 10 d S St Y all Y all * St Y all St N N * Y all St JS St JS Eq. (5-20) St Y all Y all St N N S Y all 1 * = lim = k + k N 6 7 L 5-22 k 6 N k 6 1 Figure 5-9 Y all 1 k 6 1 k 7 L L Figure 5-12 L

15 59 c 4.3 S 2 St L S S N S St JS St JS Eq. (5-22) Eq. (5-21) 5.6 R (E >E ) N A d Re St L St JS E mean E SD Y all f(lne) Eq. (5-6) Y E >E Eq. (1-2) ideal R (E >E ) Figure 5-13 ideal K qd [-] Satake Kagaku Kikai Kogyo Kabushikigaisha (1995) N JS

16 60 K S [-] 9 6 K qd K S k 1 = 20 k 2 = 0.11 k 3 = 2.0 k 4 = 1.0 k 5 = 0.40 k 6 = 0.25 k 7 = 2.0 k 8 = 0.20

17 Y Y Y 2 1 Y X J E 3 2 E X Y(E > E ) E mean E SD Y all 3 4 E mean E SD Re 5 Y all St L

18 62 St JS 3 6 R (E > E )

19 63 Y ( X > X ) [-] Nylon OM TFE Glass Al X [a.u.] Fig. 5-1 Y X d = 3.2mm, N = 18.5s -1

20 64 Y ( X > X ) [-] X [a.u.] d 2.4mm 3.2mm 4.0mm 4.8mm Fig. 5-2 Y X Nylon N = 8.5s -1

21 65 Y ( X > X ) [-] d 1.4mm 1.7mm 1.9mm 2.5mm 3.2mm 4.0mm X [a.u.] Fig. 5-3 Y X Glass N = 18.5s -1

22 Y ( x X x + x )[-] X min X [a.u.] Fig. 5-4 (a) X x = 1 a.u.; Al, d = 3.2 mm; N = 18.5 s Y ( lnx lnx lnx + (lnx) )[-] lnx min ln X max lnx [a.u.] Fig. 5-4 (b) lnx (lnx) = 0.1 a.u.; Al, d = 3.2 mm; N = 18.5 s -1

23 67 Y ( lnx lnx lnx + (lnx) )[-] lnx min ln X max lnx [a.u.] Fig. 5-5 lnx 6 (lnx) = 0.11 a.u. Nylon, d = 3.2 mm N = 13.5 s -1

24 f ( ln e) 2 ( ln e 11.9) 2( 1.18) 1 + = exp 2 2π1.18 f (lne) [-] lne min ln E max lne [J] Fig. 5-6 lne (lne) = 0.2 Al, d = 3.2 mm N = 18.5 s -1

25 E mean / E tip [-] 10-1 E E tip = Re mean Nylon OM TFE Glass Al Re [-] Fig. 5-7 E mean / E tip Re

26 ln E SD [J] ln E SD = Re 0.3 Nylon OM TFE Glass Al Re [-] Fig. 5-8 E SD Re

27 71 Y all [-] Y all 3 ( St α) ( St α) γ + β = 3 d 2.4 mm 3.2 mm 4.0 mm 4.8 mm St [-] Fig. 5-9 Y all St Nylon

28 72 Y all [-] [kg/m 3 ] 140 (Nylon) 400 (OM) 1200 (TFE) 1500 (Glass) 1700 (Al) Y all 3 ( St α) ( St α) γ + β = St [-] Fig Y all St 3.2mm

29 73 η c = D D x b D x D b L = d D b Fig. 5-11

30 74 c [-] ( St α) ( St α) γ + β η C = L = d D b L = 1.0 L = 0.8 L = 0.6 L = 0.4 L = 0.2 L = St [-] Fig L c

31 5 75 Re St L St JS f lne f lne = St JS E mean E SD Y all E mean mean E SD SD Y all all Y E Eθ Y E Eθ = R E Eθ lneθ f lne d lne R E Eθ = Y E Eθ Fig (a)

32 76 R (E > E ) = ideal Y (E > E ) Eq. (3-1) 3 ω = K ND V Eq. (1-2) ideal qd Y ( E > E ) = f ( lne) d( lne) θ + lne θ Eq. (5-6) f ( lne) ( lne ln E ) 1 = Yall exp 2 2π lnesd 2 mean 2 ( lne ) SD Eq. (5-7) E mean E tip = k 1 Re 0. 8 Eq. (5-11) lne = k SD 2 Re ( St α) ( St α) γ + β all = 3 Eq. (5-13) Y Eq. (5-16) α = k 3 L + β = k γ = 5 St k JS St JS ( k 7 ) ( 8 ) 6 + k L 1 + k St JS Eq. (5-19) Eq. (5-20) Eq. (5-21) ρf vtip d Re = µ Eq. (5-12) v g St = D b v tip Eq. (5-15) L = d D b St v JS = Db g ( π N D) JS Eq. (5-17) Eq. (5-18) d ρ ρ f A N D b K qd etc Fig (b)

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,,

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,, 2012 10 13 1,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ).

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