M E 320 Professor John M. Cimbala Lecture 16

Transcription

1 M E 320 Professor John M. Cimbala Lecture 16 Toay, we will: Do some more example problems lear CV momentum equation Discuss the control volume equation for angular momentum E. The Lear Momentum Equation for a Control Volume (contue) 4. Examples (contue) Example: Tension a cable Given: A cart with frictionless wheels an a large tank shoots water at a eflector plate, turng it by angle θ as sketche. The cart tries to move to the left, but a cable prevents it from og so. At the exit of the eflector, the Tank z water jet area A jet, its average velocity V jet, an its momentum flux correction factor β jet are Cart known. x V jet, A jet, β jet θ Cable To o: Calculate the tension T the cable. Solution: First step: Secon step: Use the approximate, most useful form of the lear momentum equation, F = F gravity + F pressure + F viscous + F other = rv + βmv βmv t V CV

2 Example: Force to hol a cart place Given: Water shoots of a large tank sittg on a cart. The water jet velocity is V j = 7.00 m/s, its cross-sectional area is A j = 20.0 mm 2, an the momentum flux correction factor of the jet is The water is eflecte 135 o as shown (θ = 45 o ), an all θ of the water flows back to the tank. The ensity of Tank the water is 1000 kg/m 3. To o: Neglectg friction on the wheels, calculate V j, A j, β j F the horizontal force F ( units of N) require to hol the cart place. Cart Solution: First step: Secon step: Use the approximate, most useful form of the lear momentum equation, F = F gravity + F pressure + F viscous + F other = rv + βmv βmv t V CV

3 Example: Force imparte by a water jet hittg a stationary plate Given: A horizontal water jet of area A j, average velocity V j, an momentum flux correction factor β j impges normal to a stationary vertical flat plate. z x To o: Calculate the horizontal force F require to keep the plate from movg. Solution: First step: Secon step: Use the approximate, most useful form of the lear momentum equation, F = F gravity + F pressure + F viscous + F other = rv + βmv βmv t V CV Nozzle V j, A j, β j F

4 Example: Force imparte by a water jet hittg a movg plate Given: A horizontal water jet of area A j, average velocity V j, an momentum flux correction factor β j impges normal to a movg vertical flat plate. The plate moves to the right at constant spee V p. z x To o: Calculate the horizontal force F require to keep the plate movg at constant spee V p. Nozzle V j, A j, β j Solution: First step: Secon step: Use the approximate, most useful form of the lear momentum equation, the x-irection, for a movg CV, but steay: F = F + F + F + F = βmu βmu x x, gravity x, pressure x, viscous x, other r r V p F

5 Example: Force on a bucket of a Pelton-type (impulse) hyroturbe Given: An impulse turbe is riven by a high-spee water jet (average ω jet velocity V j over jet area A j, with momentum flux correction factor β j) that impges on turng buckets attache to a turbe wheel as shown. The turbe wheel rotates at angular velocity ω, an is horizontal; therefore, gravity effects R are not important this problem. (The view the sketch is from the A top.) The turng buckets turn the e β e s water approximately 180 egrees, A an the water exits the bucket over j V j β j exit cross-sectional area A e with exit momentum flux correction factor β e. For simplicity, we approximate that the bucket imension s is much smaller than turbe wheel raius R (s << R). (a) To o: Calculate the force of the bucket on the turbe wheel, F bucket on wheel, at the stant time when the bucket is the position shown. (b) To o: Calculate the power elivere to the turbe F bucket on wheel wheel. F wheel on bucket Solution: We choose a control volume surroung the bucket, cuttg through the water jet at the let to the 2 bucket, an cuttg through the water exitg the bucket. A e β e Note that this is a movg control volume, movg to the ω R right at spee ω R. We also cut through the wele jot between the bucket an the turbe wheel, where the force F bucket on wheel is to be calculate. Because of Newton s thir law, the force actg on the control V j 1 CV volume at this location is equal magnitue, but A j β j opposite irection, an we call it F wheel on bucket. Sce the pressure through an compressible jet expose to atmospheric air is equal to P atm, the pressure at the let (1) is equal to P atm, an the pressure at the exit (2) is also equal to P atm. Solution to be complete class.

6 Use the x-component of the steay lear momentum equation for a movg CV, F = F + F + F + F = βmu βmu x x, gravity x, pressure x, viscous x, other r r

7 Angular Momentum Control Volume Analysis (Section 6-6, Çengel an Cimbala) 1. Equations an efitions See the erivation the book, usg the Reynols transport theorem. The result is: (Relative velocity) We simplify the control surface tegral for cases which there are well-efe lets an lets, just as we i previously for mass, energy, an momentum. The result is: Note that we cannot efe an angular momentum flux correction factor like we i previously for the ketic energy an momentum flux terms. Furthermore, many problems we consier this course are steay. For steay flow, Eq reuces to: Net moment or torque actg on the control volume by external means Rate of flow of = angular momentum of the control volume by mass flow Rate of flow of angular momentum to the control volume by mass flow Fally, many cases, we are concerne ab only one axis of rotation, an we simplify Eq to a scalar equation, Equation 6-52 is the form of the angular momentum control volume equation that we will most often use, notg that r is the shortest istance (i.e., the normal istance) between the pot ab which moments are taken an the le of action of the force or velocity beg consiere. By convention, counterclockwise moments are positive.

8 2. Examples See Examples 6-8 an 6-9 the book. Example 6-8 is iscusse more etail here. These moments are moments actg on the CV. These moments are moments ue to angular momentum.

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