Examples (Chapter )

Transcription

1 7 C Exales (Chater ) F x F y A A cs60 + A cs60 F Q ρq + ρ( ) cs60 Q + ρ( )( x )cs60 0 (atsheric) then Fx ρ Q -767 N In the y-directin: ( 0) A sin60 A sin60 + Fy Q Q ρq(0) + ρ( ) sin60 + ρ( )( sin60 Then, F y 0 Therefre, the frce n the late is -767 N ( ) Nted that the frce acting n the fluid is +767 N ( ) )

2 7 C Z Z 0 F ( A ) ( A ) ( +) ( +)

3 74 C (exsed t atshere)

4 75 C

5 76 C

6 77 C

7 78 C A shallw circular dish has a shared-edged rifice at its center. A hrizntal water jet, with a seed f, strikes the dish cncentrically. Evaluate the hrizntal frce (in Newtns) t hld the dish in lace. /s, D 6 c, d c Q F Q Q Q Slutin A D π(0.0) 0.008, A d π(0.0) Q ()(0.008) / s, Q ()( ) / s, Q Q Q F ρq + ρq ( sin 45 ) ρq FX / s ( 000)[ ( )() (0.0050)()(0.707) ( )()] 7.6 N Thus F 7.6 N ( )

8 79 C y y n+ (y n+ ) y

9 The flw arund a tw-diensinal bluff bdy is investigated. It is knwn that vrtices are 80 C generated behind the bdy at a shedding frequency ( ω), which deends n D, H, velcity (), fluid density (ρ ), and abslute viscsity (µ ). Related quantities are listed as fllws: ariables rttye Mdel D 0 c c H K/hr ρ. Kg/ (air) 998 Kg/ (water) µ 4 x 0-7 lbf sec/ft.4 x 0-5 lbf sec/ft ω ω Hz () By selecting "D" as ne f the reeated variables (yu need t rerly chse the ther tw), resent the relatinshi in a diensinless fr. (5%) () Fr siilarity, deterine (in K/hr) fr the del. (8%) () Cute ω (in Hz). (7%) Slutin f ( ω, D, H,, ρ, µ ) 0 n 6, k (M, L, t), 故有 (6 ) 個無因次參數 ωd H ρ D f (, D µ 取 ρ D 為重覆變數, 得, ) 0 ωd f ( H D ρ D, µ 或 )..() L 本題中, 已具幾何相似 ( λ L ) 為達到完全相似, 雷諾數應相等 ( 動力相似 ) L 0 ρ D 故 ( ) ρ D ( ), () r D. 0.4 ( ) ( ) ( ) ( ) ( ) ( ) (0) 7.8 K / hr.() r D 當完全相似時, ωd ( ) ωd ( ), 故 D 0 ω ( ) ( ) ω ( ) ( ) () 0.55 Hz.() D 0 7.8

10 8 C A enturi eter is a device fr the easureent f the flw rate in a ie. It is knwn that the ressure difference (D) between ints A and B deends n the ie diaeter (D), the ie seed (), the fluid density (ρ), the dynaic viscsity (µ) and the gravitatinal acceleratin (g). rttye Mdel D () (ft/s) 0? ρ (slug/ ).94.0 µ (lbf s/ft ) x x 0-5 g (/s ) D (si)? () Find the diensinless fr f the relatinshi between D and the related variables. (0%) () Assue that the gravitatinal frce is caratively nt irtant, what is in del ( ) equal t? (8%) () Again, as the gravitatinal frce is nt irtant, calculate (D). (7%) Slutin n 6 (6 個變數 ); k (M, L, t); 故有 (n-k) 個 π nubers. f ( D, D,, ρ, µ, g) 0, Chse reeated variables: ρ,, D f ( D, ρ,, D), 得 ρ ρ D π f ( µ, ρ,, D), 得 π µ D ρ D π f (g, ρ,, D), 得 π ; > F(, ) g D ρ µ g D () ρd ρd T get clete siilarity, Reynlds nubers are the sae. Thus ( ) ( ), ρ D.94 4 ( )( )( ) ( )( )( )(0) 8.8 ft / s () ρ D Then ( ρ ) ( ρ ) ρ.94 0, ( ) ( )( ) ( ) ( )( ) () si ρ 8.8

11 8 C The wer () needed fr the eratin f an axial u deends n the fluid density (ρ ), the rtating seed (N), the blade diaeter (D), the head lss (H) and vluetric flw rate (Q). Related quantities are listed belw: ariables rttye Mdel ρ (Kg/ ) N (turn/in) D (c) 5 5 H (ft) 0 H Q (ft /s) 7 Q (H) () By selecting "N" and D as tw f the reeated variables (yu need t rerly chse the third), resent the relatinshi in a diensinless fr. (%) () Fr siilarity, deterine Q fr the del. (6%) () redict. (6%) Slutin n 6 (6 個變數 ); k (M, L, t); 故有 (n-k) 個 π nubers. Reeated variables: ρ N D H π f ( ρ, N, D, H), 得 π D Q π f ( ρ, N, D, Q), 得 π N D H Q π f ( ρ, N, D, ), 得 π > f (, ) 5 5 ρ N D ρ N D D N D () Q Q N D [ ] [ ], Q ( ) Q ( ) (7) ft / s.() N D N D N D 00 5 當達到完全相似時,[ ] ρ N D [ ρ N D ] 5 5 成立 故, ρ N D ( ) ( ) ( ) ( ) ().5 H () ρ N D