At 1atm = 101,325Pa, one mole of gas at 20 0 C = 293K has volume V = 2.40 x10-2 m 3 = 24 litres

Transcription

1 Ideal Gases and eat Enges he oeration, and theoretial effiieny, of ombustion driven iston enges (e.g. diesel or etrol fuelled) an be analysed by onsiderg harges to ressure, temerature and volume of the gaseous omonents. his roeeds by aountg for the energy hanges the gas as a result of heat alied and work done ombed with the ideal gas equation, whih relates the hysial roerties of the gas. Ideal Gas Equation n ressure asals (a) olume / ubi metres Number of moles of gas emerature sales 00 o is equivalent to o F. R olar gas onstant 8. Jmol - K - emerature /Kelv F An ideal gas assumes a large number of ot artiles ollidg elastially. It neglets any short-range termoleular fores resultg from reulsion or attration due to moleular harges, and the fat that moleules have a fite volume i.e. are not fitely small! his means a real gas is not fitely omressible whereas an ideal gas has no suh limits. At atm = 0,5a, one mole of gas at 0 0 = 9K has volume =.0 x0 - m = litres he Kelv temerature sale (or absolute sale) is roortional to the mean keti energy of moleules. Ideal Gas Equation ratial units n atm litres K n / atm.87 / litres Seial ases of the ideal gas equation: Boyles s Law. At onstant temerature, gas ressure is versely roortional to volume. onstant harles Law. At onstant ressure, gas volume is roortional to temerature. / atm o / 7 n / atm.87 / litres o / 7 onstant / K U -0 o F = -0 o Internal energy of n moles of gas mol = 6.0 x 0 moleules. So energy of a moleule is Boltzmann's onstant u nkb k R/ JK B Number of degrees of freedom of moleular motion (e.g = for x,y,z translation) o 0 o 500 / litres Fahrenheit is a temerature sale, where o F is the freezg ot of water and o F is the boilg ot of water, defed at sea level at standard atmosheri ressure (0,5a). It was roosed 7 by Daniel Gabriel Fahrenheit. 0 o F orresonded to the lowest temerature he ould ool bre (salt water) and 00 o F was the average human body temerature (7 o ). A more oular sale is the elsius sale, with 0 o and 00 o reresentg the freezg and boilg ots of water at standard atmosheri ressure. Anders elsius 70-7 Daniel Fahrenheit illiam homson (Lord Kelv) Robert Boyle Jaques harles 76-8 hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE

2 eat, work and ternal energy of an ideal gas x dx A F onsider a ylder of gas beg omressed by a fore F. he work done on the gas by the fore is: d Fdx he ressure atg uon the gas is: F A and the volume hange is: d Adx d d A A d d If heat d is sulied to the gas then the First Law of hermodynamis (that Energy a losed system is onserved) means the ternal energy hange is du d d d du d First Law he ternal energy for n moles of an ideal gas is U onstant volume roess (isohori) d d d d d 0 d du d he Ideal Gas Equation is U U du onstant volume heat aaity for n moles d i.e. no work done on gas First Law d So for a onstant volume hange, the heat aaity is a onstant for an ideal gas. R onstant volume seifi heat aaity. is the molar volume /kg Gas /gmol - Aetylene 6.0 Air Ammonia 7.0 Argon 9.98 Benzene 78. Butane 58. arbon dioxide.0 arbon onoxide 8.0 hlore Ethyl Alohol 6.07 Fluore elium.00 ydrogen hloride ydrogen Sulhide Kryton 8.80 ethane 6.0 Natural Gas 9.00 Nitrogen 8.0 Neon 0.79 Oxygen.9988 Ozone roane.097 Sulhur dioxide 6.06 onstant ressure roess (isobari) d d d d U du Se d onstant d d d du d d d d onstant ressure heat aaity for n moles of ideal gas Ideal Gas Equation First Law ayer Relationshi Exerimentally it is very hard to mata a onstant volume as heat is added, so onstant volume heat aaity is diffiult to measure diretly. owever, onstant ressure heat aaity is muh easier to measure, as one an allow volumes to hange order to mata equilibrium with the ambient ressure. otal amount of heat sulied to m kg of gas is therefore: m oluene 9. Xenon.0 ater vaour 8.0 he ayer relationshi is therefore very useful workg out the onstant volume heat aaity from the onstant ressure heat aaity. htt:// hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE

3 eat sulied and work done for an isobari roess d d he ayer Relationshi and m n n n R( ) 0 d n n n R 0 Se is onstant R R R R Adiabati (or isentroi) roess i.e. no heat added. ork done is the sole ause of hanges ternal energy U d 0 ln ln ln U du d( ) du d d du d du d d d d d d d d d onstant d k 0 0 k onstant his defes an adiabati hange onstant temerature (isothermal) roess du 0 onstant d d d d d 0 ln 0 ork done on gas for an adiabati hange d d d d First Law Usg the Ideal Gas Equation eat sulied to gas ork done on gas d d d d So work done by gas on the surroundgs is: ln hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE

4 eat Enges arnot yle U ositions to Isothermal exansion of an ideal gas at the hot reservoir temerature. Se gas temerature, and therefore ternal energy is onstant, the work done by the gas on the surroundgs must exatly equate to the heat absorbed by the gas. ositions to Isentroi (i.e. adiabati or no heat added or lost ) exansion of the gas. he work done by the gas on the surroundgs is owered by the loss of ternal energy of the gas as it ools from the temerature of the hot reservoir to the temerature of the old reservoir. ositions to Isothermal omression of the gas. In order for the temerature, and hene the ternal energy, to rema onstant, the heat lost by the gas to the old reservoir must equate to the work done on it by the surroundgs. ositions to Isentroi omression of the gas, heatg it from the temerature of the old reservoir to the temerature of the old reservoir. onstant ln 0 Assume we have n moles of ideal gas, none of whih are lost the roess. Inut arameters are: ot reservoir temerature (Kelv) old reservoir temerature (Kelv) Ideal gas equation olume of gas at osition the yle olume of gas at osition the yle hese two volumes are derived from the other uts: olume of gas at osition the yle olume of gas at osition the yle ayer relation onstant ork done by the ideal gas on the surroundgs Between ositions and ln, ln Between ositions and Between ositions and out ln, ln Between ositions and,, (,) o omlete the yle, d out, ork done by the gas on the surroundgs is the area enlosed by the yle, Niolas Léonard Sadi arnot (796-8), hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE

5 arnot enge ont... otal work done is: d,,,, ln ln ln ln ln ln ln ln Defe heat enge effiieny as the ratio of ork done by the gas to the heat ut ln ln he arnot Enge effiieny deends only on the reservoir temeratures. Se the temerature range annot go beyond the range of reservoir temeratures, the arnot yle reresents the most effiient way of extratg work given an amount of heat ut.* Any other roess would ouy less area the S, diagram. ln he effiieny of a arnot heat enge an be more simly derived by onsideration of Entroy S. his is a measure of disorder a substane. he Seond Law of hermodynamis states for any hange, the total amount of Entroy the Universe must rease. d If heat is added a reversible roess: ds For the arnot yle, the isentroi stages have no heat hange hene they are at onstant Entroy. (Note this alies to the ideal gas, the surroundgs will hange entroy due to the exhange of work with the ideal gas). e annot reate entroy the yle for the gas, as the yle returns to the origal state (,, ), and Entroy is a salar funtion of state (i.e. like otential energy the ath does not matter). he (S,) urve for the yle is therefore a retangle. In the isothermal stages, temerature is a onstant, so both ases out S ln From the First Law of hermodynamis du d d du ds d Over the whole yle the ternal energy doesn t hange, so the work done by the gas is: d ds S ln ln ln *here is a better justifiation on the next age! hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE 5

6 General notes regardg the Seond Law of hermodynamis, and the maximum ossible effiieny of heat enges It is ossible to bound the effiieny of any heat enge usg (i) the law of onservation of energy and (ii) the Seond Law of hermodynamis. hat the uer bound of effiieny equates with the effiieny of the arnot enge is erhas an even stronger justifiation of the statement that a arnot enge is the most effiient sheme ossible, if deed it ould be ratially realized. Any heat enge is essentially flow of heat from a hot reservoir to a older one. By a reservoir we mean a thermal mass that is so large that it will not hange temerature when the heat we assoiate with our enge is taken from or added to it. he differene heat taken from the hot reservoir, and the heat transferred to the old reservoir, is the maximum ossible work done by the enge. his must be true to satisfy the law of energy onservation, or the First Law of hermodynamis. ot reservoir First Law: out he Seond Law of hermodynamis states that for every hange there an never be an overall derease Entroy. For our idealized system, this means the loss of entroy of the hot reservoir must at least be omensated for by the ga entroy of the old reservoir. eat Enge out old reservoir S S out 0 out 0 Stotal S S S total Entroy hanges of hot and old reservoirs Seond Law: Notes on reversibility A reversible heat enge is one whih is assumed to oerate at thermodynami equilibrium at all times. he ideal gas equations, and assoiated relationshis, hold and there are no losses due to frition et. In other words, the differential form of the First Law holds at all times; i.e. where hanges du ternal energy are fully aounted for by heat hange d and work done d = -d. his means that there is no net ternal energy and deed entroy hange over the omlete yle. his means the ideal yle ould be run reverse without breakg the Seond Law, se a zero net entroy hange is ermitted. out ombg with the First Law exression: out 0 out Defe enge effiieny: 0 So se the arnot enge has effiieny this is as effiient as thermodynamis allows, so the arnot yle is (one examle*) of the most effiient heat enge ossible. *A Brayton Enge (adiabati omression, isobari heatg, adiabati exansion, isobari oolg) has a similar theoretial effiieny as a arnot yle. hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE 6

7 eat Enges - Retangular, yle Assume we have n moles of ideal gas, none of whih are lost the roess. Inut arameters are: ressure at osition the yle ressure at osition the yle, (, ) (, ),, (, ) Gas temerature at osition the yle olume of gas at osition, the yle olume of gas at osition, the yle, (, ) Note arnot effiieny for this heat enge would be % 87 7 Between ositions and n n, ( ), d R S, n n ln Between ositions and n,, 0 d eat ut to gas ork done by gas eat outut from gas ork done by gas S, n n ln Net heat ut,, n ( ) n ( ) Between ositions and n, ( ), d S, n n ln Between ositions and n n ( ) ( ) ( ) ( ) R ( ) ( ) n,, 0 d hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE 7 eat outut from gas ork done by gas eat ut to gas ork done by gas S, n n ln Net work done by gas,,,, ( - ) - ( - ) - - Enge effiieny - - ( ) ( ) - - R R R

8 eat Enges he Otto yle he Otto yle is the basis of sark-ignition iston enge, whih is essentially how a tyial etrol driven enge oerates. ositions 0-: Air is drawn to iston/ylder arrangement at onstant ressure. roess Adiabati (isentroi) omression of the air via a iston. Between ositions and Between ositions and n 0, S 0,,, R n,, 0 d S, n n ln -ve se work is beg done on the gas eat outut via exhaust o omlete the yle roess onstant-volume heat transfer to the workg gas from an external soure while the iston is at maximum omression. his roess is tended to reresent the ignition of the fuel-air mixture and the subsequent raid burng. roess Adiabati (isentroi) exansion (ower stroke). roess onstant-volume roess whih heat is rejeted from the air while the iston is at maximum exansion. roess 0 Air is released to the atmoshere at onstant ressure. Assume we have n moles of ideal gas, none of whih are lost the roess. Inut arameters are: ressure of gas at osition the yle ressure of gas at osition the yle olume of gas at osition the yle olume of gas at osition the yle emerature of gas at osition the yle R R R Between ositions and n,, 0 d S, n n ln Between ositions and 0, S 0,,, eat ut to gas via sark ignition ork done by gas (, ) ork done by the gas on the surroundgs is the area enlosed by the yle d (, ) n, n out, Nikolaus Otto (8-89) (, ) out (, ) hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE 8

9 Otto enge ont... otal work done is: d,, r r r r r r r r r r Defe the omression ratio r Defe heat enge effiieny as the ratio of ork done by the gas to the heat ut r r r r So the Otto Enge effiieny deends only on the omression ratio, and the ratio of seifi heats (, ) ork done by the gas on the surroundgs is the area enlosed by the yle d (, ) (, ) out (, ) From the revious age: n n R R r R R R hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE 9

10 eat Enges he Diesel yle he Diesel yle is the basis of a diesel enge, whih is ubiquitous transort aliations. Unlike etrol-driven enges, diesel variants are more suited to heavy mahery. hey an be found owerg most shis as well as truks, buses and ars. ositions 0-: Air is drawn to iston/ylder arrangement at onstant ressure. roess Adiabati (isentroi) omression of the air via a iston. roess onstant-ressure (isobari) heat transfer to the workg gas from an external soure while the iston is at maximum omression. his roess is tended to reresent the ignition of the fuel-air mixture and the subsequent raid burng. his is different from the Otto yle, whih is onstant volume (isohori) heatg durg this stage. In the Diesel yle, the heat generated from air omression is suffiient to ignite trodued fuel vaours. In the Otto yle a sark lug is used stead to ignite the fuel. roess Adiabati (isentroi) exansion (ower stroke). roess onstant-volume roess whih heat is rejeted from the air while the iston is at maximum exansion. roess 0 Air is released to the atmoshere at onstant ressure. Assume we have n moles of ideal gas, none of whih are lost the roess. Inut arameters are: ressure of gas at osition the yle olume of gas at osition the yle olume of gas at osition the yle olume of gas at osition the yle emerature of gas at osition the yle R R R Between ositions and Between ositions and n R eat outut, n via exhaust, 0 d S, n n ln, -ve se work is beg done, 0, S, 0 on the gas Between ositions and n,, Between ositions and d S, n nln 0, S 0,,, eat ut to gas durg ombustion ork done by gas (, ) (, ) o omlete the yle ork done by the gas on the surroundgs is the area enlosed by the yle n, n out, d Rudolf Diesel (858-9) out (, ) (, ) hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE 0

11 Diesel enge ont... otal work done is: d,,, r s r r r r s s s r r r s r s r r r r s r s s r ( ) ( ) r s r s r s ( ) r s r s Defe the omression ratios r s r s Defe heat enge effiieny as the ratio of ork done by the gas to the heat ut r s r s ( ) r s From the revious age: r r( s ) r s r s r s r s (, ) (, ) ork done by the gas on the surroundgs is the area enlosed by the yle d (, ) out (, ) n R n R R r s s r s he Diesel enge is tyially more effiient than a etrol (Otto) enge se the former works on the basis of self ignition due to high omression. his knokg is undesirable for etrol enges, so a lower r value is required. hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE

12 omarg Otto and Diesel heat enges r s R (, ) r s R R ork done by the gas on the surroundgs is the area enlosed by the yle d (, ) (, ) out (, ) (, ) (, ) s r s r diesel otto Diesel r( s ) r s r s Otto r r r ork done by the gas on the surroundgs is the area enlosed by the yle d (, ) out (, ) hysis toi handout - Ideal gases & heat enges Dr Andrew Frenh. AGE