EGN 3353C Fluid Mechanics

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Brice Wells
5 years ago
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Lectue Look at V ρv na at inlet: only component is in x iection: ( ( ρ u ˆ ˆ ˆ ˆ in, avgi uin, avgi Aini uin, avgmi ρu A m in, avg in at exit: only component is in z iection: ( ( ρ w ˆ ˆ ˆ ˆ out,

1 Lectue Look at V ρv na at inlet: only component is in x iection: ( ( ρ u ˆ ˆ ˆ ˆ in, avgi uin, avgi Aini uin, avgmi ρu A m in, avg in at exit: only component is in z iection: ( ( ρ w ˆ ˆ ˆ ˆ out, avgk wout, avgk Aoutk wout, avgmk ρw A m out, avg out Note: We sometimes inclue β mom. flux coection facto to make V ρv na exact. ecommenation: Choose so velocity vecto is n V ρv na x : ± umi ˆ, y: ± vmj ˆ, z: ± wmk ˆ Usually, only one of uv,, o w is not zeo! See example at left. Choice of fo this poblem? Impotant to consie: o boy foces (weight of stuff insie often neglecte since this is static an we e inteeste in the exta foces to the flui motion o Gey : cuts the bolts on the flange so thee will be a eaction foce on the pessue foces Pout P so only nee P. atm in gage usually aw eaction foce(s in positive iections if foce <, foce is actually in othe iection o Magenta : oesn t cut the bolts so no eaction foce BUT now have to consie the pessue an shea (viscous foces along insie walls (Ba choice! A in

2 emembe Newton s thi law with egas to eaction foces. You e solving fo the foce on the (e.g., Example in this case the foce of the bolts on the. This is equal an opposite to the foce of the on the bolts! Given: P, A, A, Q Wate flows though a nozzle at the en of the pipe as shown above. What is the hoizontal eaction foce exete by the suppot? Assume: ( steay, ( incompessible flow, (3 unifom at inlet an exit β Daw fixe as shown. Denote as inlet an as exit. Mass Consevation: ρ + ρv na V na Momentum x Q + Q Q u A Q u A Q F Vρ + VρV na + na uρ P, gage A + x u ρu A + u ρu A aw in + iection Q Q ( P A + ρq u u foce by suppot on., gage

3 Example Assume: ( steay, ( incompessible, (3 unifom flow at inlet β an jet oesn t spea appeciably, (4 jet leaves in vetical iection. Daw fixe as shown. Denote as inlet an as exit. Mass Consevation: Just like in example, Q const Given: V Vi ˆ, An Wate flows though a nozzle an impacts to the plate. What is the hoizontal eaction foce equie to hol the plate stationay? F QV AV + na uρ P, gage An F V( ρq + u ( ρ Q awn in - iection jet exhaust is P jet leaves in atm vetical iection ρ ρ eaction foce >. If you ae holing the plate in place, you will have to exet a foce place. F ρ AV in x iection to hol it in

Example 3 A eucing elbow is use to eflect wate flow at a ate of 4 kg/s in a hoizontal pipe upwa θ 3 while acceleating it. The elbow ischages wate into the atmosphee.

4 Example 3 A eucing elbow is use to eflect wate flow at a ate of 4 kg/s in a hoizontal pipe upwa θ 3 while acceleating it. The elbow ischages wate into the atmosphee. The coss-sectional aea of the elbow is A 3 cm at the inlet an A 7 cm at the outlet. The elevation iffeence between the centes of the outlet an the inlet is 3 cm. The weight of the elbow an the wate in it is consiee to be negligible. Detemine (a the gage pessue at the cente of the inlet of the elbow an (b the anchoing foce neee to hol the elbow in place. Assume: ( steay, ( incompessible, (3 unifom flow at inlet an exit, (4 neglect boy foces Daw fixe as shown. Note it is outsie (not insie to avoi shea foces at wall. Denote as inlet an as exit. Continuity m ρ V A ρva ρva + na uρ ( θ ( P A F V m V m ( cos, gage + x + unknowns nee anothe equation Benoulli: If we follow a steamline fom the cente of inlet to cente of the outlet an neglect fiction (which is not eally tue hee, then P + ρv + ρgz P + ρv + ρgz o

5 z-momentum + na Fz wρ wρv F boy, z neglect + F u ( m + V sinθ ( m z F mv sin θ z P, gage P P ρ( V V + ρg( z z ( Will the tue P, gage be highe o lowe than fom (? Substitute ( into ( an solve x ( cosθ F m V V P A, gage

Conservation of Linear Momentum using RTT

Conservation of Linear Momentum using RTT 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