07/03/2017 Lectue 21 Consevation of Linea Momentum using RTT Befoe mi-semeste exam, we have seen the 1. Deivation of Reynols Tanspot Theoem (RTT), 2. Application of RTT in the Consevation of Mass pinciple 3. RTT was witten in the fom: DB If the contol volume is moving, then DB t U ( v. nˆ ) A system CV t U ( v. ˆ n) A system CV Whee v is elative velocity = v v s In inex notation, fo a efomable an moving contol volume, we can wite: DB ij... system t CV U v n A ij... ij... k k RTT can be applie fo the pinciple of consevation of linea momentum. In this case, the extensive popety will be the linea momentum ( mv ) an the coesponing intensive popety will be v i.e., B = mv an β = v D( mv) t system CV vu v( v. nˆ ) A In physics, you have stuie Newton s secon law, which states that the ate of change of linea momentum in a system is the net foce acting on it. So, D( mv) F vu v( v. ˆ n) A t system CV
We shoul note that: v is velocity elative to an inetial non-acceleating co-oinate system. F is the vecto sum of all foces acting on the system mateial (consists of suface foces an boy foces). The boy foces can be gavity foce, electomagnetic foce, etc. The suface foces ae pessue foce, viscous foce, etc. The Pessue Foce Recall, in the chapte on hyostatics, we escibe about the pessue foce. The foce ue to pessue is always pepenicula an inwas to the plane, that is consiee. So, if you have a contol volume of any shape, then the net pessue foce acting on the suface of the contol volume will be: F p( nˆ ) A pes Fig.1: Unifom pessue foce on a boy (Souce: Flui Mechanics by F.M. White) On a close suface, if the pessue has unifom value (o magnitue) pa, then the net pessue foce F p( nˆ) A p na ˆ 0 pes (This esult is inepenent of the shape of the contol suface).
Fo example, a contol volume fome by enclose suface, whee outsie unifom atmospheic pessue patm acts, Then F ( ˆ ) 0 pes patm n A Example: (Aopte fom FM White s Flui Mechanics) A nozzle is use to contol the wate exit an is of the following shape. The wate pessue at the entance of the nozzle is 280 kpa. The atmospheic pessue is 103 kpa. Diamete at section 1 is D1=7 cm, an at section 2 is D2=2.5 cm. Estimate the net pessue foce. Fig. 2: Poblem Statement an efeence cooinate axes (Souce: Flui Mechanics by F.M. White) Solution: We nee to fist fame a suitable contol volume (as goo as the fee boy iagam you have stuie in soli mechanics). The contol volume is shown:
Fig. 2: Repesentation of the contol volume (Souce: Flui Mechanics by F.M. White) You have inlet at section 1 an outlet at section 2. The atmospheic pessue of 103 kpa acts thoughout all sufaces of the contol volume. Then we can use gage pessue Pgage=p - patm Acting on specific potions of the contol sufaces. An F ( ˆ ) pes pgage n A Now, pgage=280-103=177 kpa F pessue 17710 ( nˆ ) A 3 Net pessue foce 17710 0.07 F 681e ˆ 0eˆ 0eˆ pessue 1 2 3 681eˆ N In the application of RTT to linea momentum pinciple 1 3 4 2 i.e., D( mv) t system CV vu v( v. nˆ ) A If the openings on the sufaces pemit one-imensional inflow an outflow an if the flow is steay, then F moutletvoutlet minletvinlet
Example: (Aopte fom FM White s Flui Mechanics) A fixe vane tuns a wate jet of coss sectional aea A though an angle θ without changing its velocity magnitue. The flow is steay, pessue is pa eveywhee an fiction on the vane is negligible. a) Fin the components of vane foce in x1 an x2 iections. Fig. 3: Poblem figue (Souce: Flui Mechanics by F.M. White) Soln: The fixe vane is shown above. Now you nee to visualize appopiate contol volume (simila to fee boy iagam). The otte line shows the contol volume. As the vane leg is cut by the contol suface, you nee to povie a eaction vane foce. As the vane is fixe, (it is not moving), the net foce acting on the contol volume shoul be the eaction vane foce. i.e., F Fvane m( v2 v1) m Av The components ae: Fig. 4: Foce iagam (Souce: Flui Mechanics by F.M. White)
F m( v v ) x1 2x 1 1 x1 F m( v v ) F F x2 2x 1 2 x2 x1 x2 Avv(cos 1) Avv sin [(cos 1) ˆ sin ˆ ] 2 Fvane Av e1 e2