Outline. Unit Two Work and Paths. Unit Two Goals. Unit Two Goals. More Unit Two Goals. Analysis of Work. Unit 2 Work and paths September 2, 2010

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1 Unit Work and paths Septemer, 00 Outle Unit Two Work and aths Mechanical Engeerg 70 Thermodynamics Larry Caretto Septemer, 00 Solution to quiz Assessment results Unit two work and paths Work, W = path d and w = W/m = path dv ath is equation givg () or a process Integral is area under path Work is positive, zero, negative i volume creases, does not change, decreases Work done y the system is positive Unit Two Goals Unit Two Goals As a result o studyg this unit you should e ale to d properties more Initial state easily than you were ale to do aer completg unit one ath use the path or a process, which is the equation that gives () or the process Fal state know the deition o work: dw = d d the work as the tegral o d over the path d the work as the area under the path on a - diagram W = path d Initial state ath Fal state 4 More Unit Two Goals use the description o the path as an equation recognize the dierence etween the path equation and the equation o state d values o itial pressure (or volume) and al pressure (or volume) rom the path equation i necessary use the path equation and the equation o state a trial-and-error procedure to d the al state Analysis o Work Start with dw = Fdx dw = (A)d(/A) = d Note that dimensions are (F/L ) times L a m = (N/m ) m = N-m = J ka m = kj and Ma m = MJ psia times 44 / gives l ; divided y.409 psia /Btu gives Btu W = path d = d over path Work depends on path 6 ME 70 Thermodynamics

2 Unit Work and paths Septemer, 00 Simple ath Integratg a Le Here we have an itial pot (), a al pot () path = trapezoid area = ( ) ( + )/ Work is positive Area calculation starts at = 0 What is work or = ka, = m, = ka, = m? W = ( ) ( + )/ = ( m m ) path = trapezoid area = ( ) ( + )/ Work is positive ecause i > (and pressure is always positive) Work is positive when system expands (system does work on surroundgs) Area calculation starts at = 0 7 ( ka + ka)/ = ka m 8 = kj Here the itial pot () and al pot () are reversed path = trapezoid area = ( ) ( + )/ Work is negative Reverse ath What is work or = ka, = m, = ka, = m? W = ( ) ( + )/ = ( m m ) ( ka + ka)/ = ka m = kj 9 Integratg a Reverse Le The prolem on the last chart reverses the itial pot () and al pot (), compared to the previous example path = trapezoid area = ( ) ( + )/ Work is negative ecause < Work is negative when system is compressed (work done on system) 0 Actual Integration o ath I we are given a mathematical equation or (), we can perorm the tegration Next slides gives tegration details or a simple lear path: = a + Get same results as area o trapezoid ( ) a W Actual Integration o ath II Integrate path rom last chart: = a + W ( ) a d a d a ME 70 Thermodynamics

3 Unit Work and paths Septemer, 00 Actual Integration o ath III W Factor last result and solve or W to get same ormula as area o trapezoid W This result gives W or = a + An Easier Way to Integrate Equation with ( c): = + ( ) Change tegration variale to y = so that d = dy, y = 0 when = and y = when = y y W d d ydy y y0 4 y0 More Complex ath Here we have an itial pot (), a al pot (4) and two termediate pots ( and ) path = two trapezoid areas plus rectangle area W = ( + )( )/ + - ( ) + ( + 4 ) ( 4 )/ = = - 4 = 4 Lkg ath and State To tegrate d, we need path equation, () ath equation is tegrated along path terms o and Can speciy states terms o T Can to use equation-o-state or tales to get v rom (,T) or rom (v,t) May have to use = m v 6 Example Calculation Given: 0 kg o H O at 00 o C and 0% quality is expanded to 400 o C at constant pressure Fd: Work Equation: W = path d W = - ( ) or constant pressure = = - How do we d -, and rom data given? Use property tales to d ressure (Ma) Example rolem 4.0 Saturation. 00 C C ath Speciic olume (m /kg) these rom given data 7 8 ME 70 Thermodynamics

4 Unit Work and paths Septemer, 00 Example Contued Use tales or H O to d and v Get total volume as = m v State mixed region; constant pressure, - = = sat (T = 00 o C) = 4.9 ka (page 94) Get v rom x = 0. usg the equation that v = v + x (v g v ) = ( x)v + x v g At 00 o C, v and v g, respectively, = and 0.7 m /kg Example Contued o o v ( x) v ( T 00 C) xv ( T 00 C) m 0.7 m m v ( 0.) 0. kg kg kg = m v = m or m = 0 kg ot has T = 400 o C (given) and = =.49 Ma Fd v rom property tales g 9 0 Example Concluded Interpolate etween =.40 and.60 Ma to d v at (, T ) = (.49 ka, 400 o C) roperty data on page m.78 m.78 m kg kg v kg.60 Ma.40 Ma.96 m.49 Ma.40 Ma kg Sce m = 0 kg, =.96 m W = (4.9 ka)( ) m Units or Work W (4.9 ka)( ) m kj kam Basic idea is that a m gives J, ka m gives kj, Ma m gives MJ, For other pressure units need to convert to kj; here W =,0 kj Ma would give work =.0 Ma m =.0 MJ Do We Use or v or Work? I we know the total volume (m ) we can d the total work J (or kj or MJ) I we know v and m we can d = mv I we only know v (and not m) we can d W/m = { path d}/m = path d(/m) = path dv J/kg (or kj/kg or MJ/kg) Use the symol w or W/m the work per unit mass Exercise Two pounds ( l m ) o water ills an itial volume, =, at an itial temperature o 00 o F A process is then executed along the path = + ( ) where = 00 l / and is the itial pressure The al pressure is 00 psia Fd the work 4 ME 70 Thermodynamics 4

5 Unit Work and paths Septemer, 00 Exercise Solution Given: Initial state: = and T = 00 o F Fal state: =00 psia System mass: m = l m ath: = + ( ) with = 00 l / Sustance: water Solution elements: Work equation W = path d roperty data: water tales ath equation relates and at all states along path cludg itial and al state Exercise Solution II Can use area under trapezoid or tegrate path equation or d to get an equation or the work W d d Let y = so that dy = d, y = 0 at = and y = at = y y W y dy y y0 y0 6 data on p 964 Exercise Solution III We now have an equation or W, ut we do not know or Fd rom itial state o T = 00 o F and v = /m = ( )/( l m ) =. /l m Fd this saturation region so = sat (00 o F) = psia = l / Fd al volume,, rom path equation, = + ( ) Solve = + ( ) or 7 Exercise Solution I Solvg or gives =.8 00 l l l Fd work W l 44.8 l.8.904x0 l 8 Exercise Solution Can d work as trapezoid area = ( ) ( + )/ Rememer that and were given Fd and as done previously 00 l l W rom state data 44 rom path equation x0 l 9 ME 70 Thermodynamics