7. Momentum balances Partly based on Chapter 7 of the De Nevers textbook (sections ).

Transcription

1 Lecture Notes CHE 31 Fluid Mechanics (Fall 010) 7 Moentu balances Partly based on Chater 7 of the De Neers tetbook (sections 71-73) Introduction Net to ass and energy oentu is an iortant quantity fluid echanics Moentu is ass ties elocity Sce elocity is a ector quantity oentu also is a ector quantity A oentu balance therefore is a ector equation Howeer any cases this course we will only consider oentu is one secific (and secified) direction and (thus) one coonent of the oentu balance Recall the general for of a balance for a scalar quantity q a well-secified olue with flow and flow as discussed LN03: dq = φ φ + B D q q q q Generalizg this to the ector quantity oentu (note the notation: ectors are bold face): d = φ + φ B D (1) Consider one-by-one each of the ters Eq 1 d 1) is the rate of change of oentu the entire olue In general the total oentu is an tegral oer the entire olue V: = d with the elocity ector Usually we consider all ass hag the sae elocity and we write = so that d = d ( ) Note that a change oentu can be the result of a change elocity and/or a change ass φ is the flu of oentu Usually this flu of oentu is the result of a ) ass flow φ that carries with it oentu In this resect it ay be sightful to consider elocity as concentration of oentu or secific oentu ie oentu er unit ass Then the oentu flu can be written as ass flu ties secific oentu at the flow boundary: φ = φ 3) In the sae sirit as under ite (): φ = φ ) B D The birth and death ter we consider together Recall Newton s second law: d d F = a = Physicists like to write Newton s second law as F = which is the sae if is constant The latter equation ay hel realizg that forces are able to create or kill oentu In a oentu balance the ter B D is the su of the forces actg on the olue: B D = F Includg all this Eq 1 and assug constant elocity through the syste gies V

2 d ( ) φ φ = + F () The rest of these Lecture Notes (and Chater 7 of the De Neers tet book) focuses on eales of alications of Eq Most of these eales consider a steady state: φ φ + F = 0 (3) (note that sce Eq 3 is a ector equation its right hand side has a zero ector 0) Eales Jet on a surface Figure 1 We estiate the forces that surfaces eerience when hit by a water jet see Figure 1 Here the ector character of the oentu balance becoes aarent What is the force the jet eerts on the surface? We ake soe silifyg assutions here The ajor one is that the liquid flows sideways (ie the radial (r) direction) on the surface In the real situation the jet would break u dros that oe a sectru of directions Settg u a steady-state oentu balance the -direction should gie us the force on the surface that (gien the syetry of the roble) we eect to be -direction The general steady state for is φ φ + F = 0 In -direction this gets φ φ + F = 0 (only one force F with -coonent F ) The essential ste is to realize that = 0 ; the flow has no elocity (and thus oentu) the -direction only the radial direction Then the eression for the force is F = φ The ass flow is φ = 0 kg/s the nozzle that roduces the jet has a diaeter d=35 φ so that the elocity with which the water eits the nozzle is =08 /s If we ρπ d assue the to be the sae elocity with which the water eits the nozzle (08 /s) then F = 16 N What is the eang of the us sign? It eans that to kee the surface at its lace we hae to eert a force of 16 N the negatie -direction In other words the jet eerts a force on the surface of 16 N the ositie -direction If we would not hold the late it (obiously) would be blown away the ositie -direction The force would be higher if the water would (artly) bounce back fro the surface as scheatically shown Figure ; thk through for yourself why this is

3 Figure Pie bend Consider a horizontally laced ie bend with an angle θ as defed Figure 3 The ie has a constant diaeter D If the liquid flowg through the bend has a constant density ρ the aerage elocity agnitude of the liquid flowg through the ie is the sae eery cross section of the bend The direction of the elocity howeer changes and we eect that we need forces ( and y direction) to kee the bend lace Assue a steady ass flu φ The elocity agnitude the ie we call U; U = φ π D ρ Figure 3 The steady state oentu balance oer the syste with the dashed rectangle - direction reads: φ U φ U cosθ + F = 0 In y-direction it reads φ U sθ + F = 0 (the us sign the latter equation is there sce only the flow contas ositie y- oentu) The forces to hold the ie bend lace therefore are F = φ U cosθ 1 F = φ U sθ The force -direction is negatie for any θ ( ) y Calculation of such forces lays a ajor role when designg iele systes Suose we ut a bend of 90 o the oil iele fro Lecture Notes 06b It transorted 08 3 oil er φv second through a ie with diaeter 0 c The elocity the ie was U = = 6 /s π D The density of the oil was ρ=90 kg/ 3 so that φ =736 kg/s The forces then are F = 7 kn F = 7 kn (kn = kilo Newton = 1000 Newton) U-bend y y

4 Figure 3b In any cases forces due to ressure (and ressure differences) are art of oentu balances Consider the followg eale: A steady-state turbulent oil flow is sent through the horizontally laced 180 o ie bend shown the figure The cross sectional area at the let (1) is A 1 =1 c at the eit () it is A =05 c The oil has constant density ρ=800 kg/ 3 The oluetric flow rate is φ V =0 litre/s; the gauge ressure at cross section (1) is 1 =05 bar (A) Detere the oil elocity at cross sections (1) and () (B) If we assue frictionless flow what is the gauge ressure at cross section ()? (C) Detere the force - direction to kee the ie bend lace φv φv (A) Mass balance: 1 = = /s = = 8 /s A1 A 1 (B) Fro Bernoulli s equation it follows = 1 + ρ ( 1 ) = 0058 bar (C) The steady-state oentu balance -direction has contribution fro oentu flu and flu ressure forces actg on the control olue and the force to kee the U- φ φ A A F F = φ + A A = 663 bend lace: = ( ) ; ( ) N Design of a water rocket Figure Water rockets are relatiely easy to ake launchg deices They usually consist of a (uside down) soda bottle filled with water and ued to a certa ressure with eg a bicycle u Once a hole the ca (which now is at the botto of the uside down bottle) is oened a water jet eerges which ushes the rocket to the air (see Figure ; also check

5 YouTube and search for water rocket ) With the tools we hae been discussg so far the course we should be able to (at least) coe u with soe design rules related to the desired ressure geoetrical constrats etc Alication of the Bernoulli equation between a ot at the liquid surface the bottle and a ot at the botto where the water flows learns that the flow elocity is = gh + Note that we (ilicitly) assued here that the liquid surface elocity ρ was uch less than which is OK if d D Also we alied the steady state Bernoulli equation to an unsteady roble (sce eg the liquid leel H is a function of tie) You hae seen this quasi steady state aroach before The oentu balance z-direction gets a bit colicated: the elocity U of the bottle is a function of tie (the bottle is suosed to accelerate) as is the ass of the bottle d π sce water is ourg contuously: ( U ) = ρd g The ass of the rocket is aly the water the rocket (the lastic bottle itself has not uch ass) so that π d π = ρ D H A ass balance relates H and : dh π = ρd = ρd Fally (if we consider the air the rocket eg as an ideal gas) the water leel and the ressure the rocket are related (thks this through for yourself) All these equations (Bernoulli z- oentu balance ass balance (ideal) gas law) are couled and can be used to detere U H and the trajectory of the rocket as a function of tie This is quite a colicated eercise that can not be done analytically (coutational ethods required) We will liit ourseles here to answerg the question what ressure we would need to actually get the rocket off the ground Assue D=15 c d=1 c Initially assue a liquid leel H=30 c Gien the d π d oentu balance ( U ) = ρd g for ( U ) to be ositie (then at least π itially the rocket goes u) we need ρ d > g Sce = gh + and ρ π = ρ D H the equality gets d > D Hg d Hg Sce usually D d we can ρ D silify this to > Hg d ρ With our nubers = 33 bar A good bicycle u can delier such ressure; I a not sure if a soda bottle can withstand it Notice this terestg design question: Usually we cannot do uch ab D and H (largely dictated by the soda bottle geoetry) We hae soe freedo d howeer A sall d is beneficial for it roides less water flow and thus (otentially) a longer flight It also requires ore ressure Howeer is liited by the strength of the bottle and/or the u we hae aailable