Macroscopic Momentum Balances

Transcription

1 Lecture 13 F. Morrson CM /22/2013 CM3110 Transport I Part I: Flud Mechancs Macroscopc Momentum Balances Professor Fath Morrson Department of Chemcal Engneerng Mchgan Technologcal Unersty 1 Macroscopc Balances Mass Momentum Energy (on mechancal systems) Plan of Attack: Descrbe Dere (mass, momentum) Apply to example problems (See Morrson, An Introducton to Flud Mechancs, secton 9.1.2) Calculate forces on walls, sze pumps, calculate flud frcton, pressure drops, etc. 1

2 Lecture 13 F. Morrson CM /22/2013 Macroscopc Balances Use when we do not need the detals of the elocty profle 3 types: -mass -momentum -energy (as appled to mechancal systems) Macroscopc Mass Balance: process V, olumetrc flow rate unt 1 V 2, densty Arbtrary, sngle-nput, sngle-output system: specal case of elocty perpendcular to areas A 1 ˆn (1) V pont 2 ˆn 2 (2) A 2 pont 1 Specal case: Assumptons: steady state sngle-nput, sngle output () perpendcular to A constant across surface 2

3 Lecture 13 F. Morrson CM /22/2013 Macroscopc Mass Balance: Mass n = Mass out aerage elocty through surface A 1 Assumptons: steady state sngle-nput, sngle output () perpendcular to A constant across surface cross-sectonal area, n aerage elocty through surface A 2 cross-sectonal area, out Arbtrary, sngle-nput, sngle-output system: elocty s NOT perpendcular to cross-sectonal areas plane 1 A 1 ˆn (1) V plane 2 ˆn 2 2 A 2 (2) 3

4 Lecture 13 F. Morrson CM /22/2013 da nˆ ( ) cos () ( ) sn = outwardly pontng unt normal nˆ Macroscopc Mass Balance: Ths takes care of out or n 0= net mass out cos cos Assumptons: steady state sngle-nput, sngle output () NOT perpendcular to A constant across surface da ( ) nˆ cos () ( ) = outwardly pontng unt normal nˆ sn 4

5 Lecture 13 F. Morrson CM /22/2013 Remnder: relates to the orentaton of nlet and outlet surfaces n the chosen coordnate system da nˆ ( ) cos () ( ) sn = outwardly pontng unt normal nˆ Macroscopc Momentum Balance: steady state dp F ˆ on CV nds dt CS Momentum balance on a control olume We can specalzed the conecte term for macroscopc control olumes, net momentum conected out n ds N n ds N boundng surfaces ˆ ˆ CS 1 CS A nˆ () 5

6 Lecture 13 F. Morrson CM /22/2013 Macroscopc Momentum Balance: steady state dp F ˆ on CV nds dt See Chapter 9 CS net momentum for detaled conected out olumes, We can specalzed the conecte term for macroscopc control deraton N nˆds nˆds CS 1 CS A N boundng surfaces Momentum balance on a control olume nˆ () Macroscopc Momentum Balance (contnued) N net momentum nˆ ds nˆ ds out of CV CS 1 CS N 1 CS () () nˆds ˆ n da A Input, output surfaces We can now specfy for each A ( ) ( ) ( ) ( ) ˆ cos Separate elocty magntude from the drecton 6

7 Lecture 13 F. Morrson CM /22/2013 da Macroscopc Momentum Balance (contnued) net momentum conected out A We hae assumed that the drecton of () does not ary across A. A nˆ () ( ) () () ˆ n da ( ) ( ) ( ) ˆ cos () () () ˆ cos da A group the elocty magntudes together 2 () () ˆ cos da A only the elocty magntudes ary across ; they appear as Assumptons: steady state sngle-nput, sngle output NOT perpendcular to A constant across surfaces constant across surfaces Macroscopc Momentum Balance, contnued 2 2 (1) (1) (2) (2) 01cos ˆ 1 da ˆ 2cos2 da F on, A 1 A2 7

8 Lecture 13 F. Morrson CM /22/2013 We can wrte these terms compactly as as we now show Macroscopc Momentum Balance, contnued 2 2 (1) (1) (2) (2) 01cos ˆ 1 da ˆ 2cos2 da F on, A 1 A2 Recall that the aerage of a functon f s calculated from: f ( x, y) A A f da da 1 A A f da Macroscopc Momentum Balance, contnued 2 2 (1) (1) (2) (2) 01cos ˆ 1 da ˆ 2cos2 da F on, A 1 A2 (1) 2 (2) 2 A1 A2 (1) 2 (1) (2) 2 (2) cos ˆ 1 2 A2 cos 2 0 A ˆ 1 1 F, on But what s ths? We can make ths look more lke other conecte terms we hae seen by ntroducng a factor relatng to aerage elocty squared. 8

9 Lecture 13 F. Morrson CM /22/2013 quantfes the araton of the true elocty profle from plug flow (flat profle). Velocty Correcton Factor cos ˆ (1) 2 (1) (2) 2 (2) ˆ on, cos2 F A A defne: 2 2 expermental result turbulent = lamnar = 0.75 Result: Steady State Macroscopc Momentum Balance (conecte terms) (1) 2 (2) 2 1A1 cos 1 (1) 2A2 cos 2 (2) F ˆ ˆ on, 1 2 ector equaton Steady State Macroscopc Momentum Balance (force terms) (1) 2 (2) 2 1A1 cos 1 (1) 2A2 cos 2 (2) F ˆ ˆ on, 1 2 F contact noncontact on, F ˆ contact n ds S surface Molecular forces (scosty and pressure) MCV g graty 9

10 Lecture 13 F. Morrson CM /22/2013 Contact Forces = pressure + scous Vscous: R Ths s the force on the flud (force on walls s ) Pressure: F ˆ contact n ds S surface n ˆ ( pi) ds S ( pn ) ˆ ds A ( pna ) ˆ Steady State Macroscopc Momentum Balance # # See nsde front Coer of Morrson, ~1 10

11 Lecture 13 F. Morrson CM /22/2013 Compare wth the (more famlar) Naer-Stokes Macroscopc Momentum Balance # # Mcroscopc Momentum Balance p Macroscopc Momentum Balance # # Mcroscopc Momentum Balance Rate of change of momentum wth tme p 11

12 Lecture 13 F. Morrson CM /22/2013 Conecte Macroscopc Momentum Balance terms # # Mcroscopc Momentum Balance Rate of change of momentum wth tme p Conecte Macroscopc Momentum Balance Pressure terms forces # # Mcroscopc Momentum Balance Rate of change of momentum wth tme p 12

13 Lecture 13 F. Morrson CM /22/2013 Conecte Macroscopc Momentum Balance Pressure terms forces Vscous forces # # Mcroscopc Momentum Balance Rate of change of momentum wth tme p Conecte Macroscopc Momentum Balance Pressure terms forces Vscous forces Graty force # # Mcroscopc Momentum Balance Rate of change of momentum wth tme p 13

14 Lecture 13 F. Morrson CM /22/2013 Macroscopc Momentum Balance # # Mcroscopc Momentum Balance p We know how to apply ths Macroscopc Momentum Balance Now we need to learn when and how to apply ths # # Mcroscopc Momentum Balance p We know how to apply ths 14

15 Lecture 13 F. Morrson CM /22/2013 Macroscopc Momentum Balance Example: Calculate the force a ppe Assume: steady state 1 r z flud =force on wall = -force on flud 29 Macroscopc Momentum Balance Example: Calculate the force on a reducng bend Assume: steady state 1 neglect graty ˆn 2 (2) 2 flud densty, flud densty, y ˆn 1 (1) x 15

16 Lecture 13 F. Morrson CM /22/2013 Types of Momentum Transfer Mcroscopc conecton (momentum flows n) pressure forces scous forces (or scous flux) body forces (graty) Macroscopc conecton (momentum flows n) pressure forces wall forces (due to scosty) body forces (graty) After calculatng the flow feld wth mcroscopc balances you can calculate wall forces Wth macroscopc balances you can often calculate wall forces drectly Problem-Solng Procedure - Steady State Macroscopc Momentum Problems # # 1. sketch system; choose system on whch you wll perform balance 2. choose coordnate system 3. perform macroscopc mass balance Consder angles carefully 4. perform macroscopc momentum balance (ector equaton; forces are pressure, graty, force on the wall; all forces ON the system) 5. sole (usually for force on the wall) 16

17 Lecture 13 F. Morrson CM /22/2013 Soluton to force on a reducng bend: Many useful handouts: 17