Force = F Piston area = A


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1 CHAPTER III Ths chapter s an mportant transton between the propertes o pure substances and the most mportant chapter whch s: the rst law o thermodynamcs In ths chapter, we wll ntroduce the notons o heat, work and conservaton o mass. III.1. ork ork s bascally dened as any transer o energy (except heat) nto or out o the system. In the next part, we wll dene several orms o work. But, rst we wll ocus our attenton on a partcular knd o work called: compressve/expansve work. hy s ths mportant? Because t s the man orm o work ound n gases and t s vtally mportant to many useul thermodynamc applcatons such as engnes, rergerators, ree expansons, lqueactons, etc. By denton, an appled orce F causes an nntesmal dsplacement ds then, the work done d s gven by: d = F. ds and as that orce keep actng, those nntesmal work contrbutons add up such that: = d = F. ds Ths s the general denton o work, however, or a gas t s more convenent to wrte ths expresson under an other orm. Consder rst the pstoncylnder arrangement: Force = F Pston area = A dx Here we can apply a orce F to the pston and cause t to be dsplaced by some amount dx. But, n thermodynamcs, t s better to talk about the pressure P = F/A rather than the orce because the pressure s szendependent. Makng ths sht gves a key result: ( ) = PA dx = Pd 49
2 Note that the pston moves n, then d s negatve, so s negatve whch means work s done on the system and ts nternal energy s ncreased. I the pston moves out, then d s postve, so s postve and the system does work on ts envronment and ts nternal energy s reduced. Ths s a general expresson o work or a gas, t sn t pston and cylnder specc. For example, n a balloon you use the same equaton, but d s just calculated slghtly derently (or a sphercal balloon, t would be 4πr 2 dr). As you may notce rom the expresson above, work s related to pressure and volume. As a consequence, work can be represented usng a P dagram. Furthermore to compute the work, or any process we are nterested n what the ntal volume and the nal volume are snce d =. As shown n Fg.3.3, the work done s just the area underneath the process on a Pcurve. ork = Area = P (  ) Area = Pd P olume olume Fgure.3.3. P dagram and work denton. An mportant thng to realze s that ths has sgncant mpact on how much work s done by a partcular process between a gven (P, ) and (P, ). I you look at Fg.3.4, you ll see just three o many possble Pprocesses between (P, ) and (P, ), the areas under these curves are derent, whch means that each has a derent. Ths s known as a path dependent process. In contrast, a path ndependent process depends only on the start and end pont and not how you get between them an example s gravtatonal potental energy t only depends on the change n heght, not the path you take n changng that heght. 50
3 P Fnal pont ork () P Intal pont olume P Fnal pont ork () P Intal pont olume Fnal pont P ork () P Intal pont olume Fgure.3.4. Several P dagrams or the same ntal and nal condtons. 51
4 III.1.1. Some Common works Constant olume: In a constant volume process d=0, and so the work must be 0 also. There s no work n a gas unless t changes ts volume. Constant : Here P s constant, so we can take t out the ront o the ntegral. Hence: = P d = P ( ) Isothermal Expanson: we use the deal gas law as P=nRT, we obtan: = nrt d Here, R s a constant; n and T (sothermal) are constant, thereore: = nrt d = nrt ( ln ln ) III.1.3. Polytropc process Example A gas n a pstoncylnder assembly undergoes an expanson process or whch the relatonshp between pressure and volume s gven by a n = 1.5 b n = 1.0 c n = 0 P n = ct The ntal pressure s 3 bar, the ntal volume s 0.1 m 3, and the nal volume s 0.2 m 3. Determne the work or the process, n kj : 52
5 III.2. Several orms o work III.2.1. Electrcal ork I electrons cross the boundares o the system a work s generated. Ths work can be computed as: 2 = I dt 1 here (I) s the current and s the voltage. III.2.2. Shat ork In a large majorty o engneerng devces, the work s transmtted by a rotatng shat. Ths knd o work can be computed as ollow: = 2πNT & here, N& torque. s the number o tours per unt o tme (tours/mn ; tours/second, ) and T s the III.2.3. Sprng ork For a lnear elastc sprng the work can be computed as: 1 = k ( x x ) 2 1 here; x 1 and x 2 are the ntal and nal dsplacements o the sprng, and k s the sprng constant. Example [Schaum s page 48] The ar n a crcular cylnder s heated untl the sprng s compressed 50 mm. Fnd the work done by the ar on the rctonless pston. The sprng s ntally unstretched. K = 2500 N/m 50 kg 10 cm 53
6 III.3. Heat Heat can be transmtted through the boundares o the system only durng a nonthermal equlbrum state. Heat s transmtted, thereore, solely due to the temperature derence. The net heat transerred to a system s dened as: Q = net Qn Q out Here, Q n and Q out are the magntudes o the heat transer values. In most thermodynamcs texts, the quantty Q s meant to be the net heat transerred to the system, Q net. e oten thnk about the heat transer per unt mass o the system, q. q = Q m Heat transer has the unts o energy measured n joules (we wll use klojoules, kj) or the unts o energy per unt mass, kj/kg. Snce heat transer s energy n transton across the system boundary due to a temperature derence, there are three modes o heat transer at the boundary that depend on the temperature derence between the boundary surace and the surroundngs. These are conducton, convecton, and radaton. However, when solvng problems n thermodynamcs nvolvng heat transer to a system, the heat transer s usually gven or s calculated by applyng the rst law, or the conservaton o energy, to the system. An adabatc process s one n whch the system s perectly nsulated and the heat transer s zero. III.4. Summary  Heat s dened as the spontaneous transer o energy across the boundary o a system due to a temperature derence between the system and ts surroundngs. There s no external orce medatng ths process.  ork s bascally dened as any other transer o energy nto or out o the system. The most mportant orm o work n thermodynamcs s compressve work, whch s due to a change n volume aganst or due to an external orce (or pressure) on a gas. III.5. The mechancal equvalent o heat (Joule s experment) In the 1800s Joule spent a lot o tme ponderng the quanttatve relatonshp between derent orms o energy, lookng to see how much s lost n convertng rom one orm to another. As you ll already know, when rcton s present n some mechancal system we always end up losng some o the mechancal energy, and n 1843 Joule dd a amous experment showng that ths lost mechancal energy s converted to heat. As shown n the gure below, Joule s apparatus conssts o water n a thermally nsulated vessel. Heavy blocks allng at a constant speed (mechancal energy) are connected to a paddle mmersed n the lqud. Some o the mechancal energy s lost to the water as rcton between the water and the paddles. Ths results n an ncrease n the temperature o the water, as measured by a thermometer mmersed n the water. I we gnore the energy lost n the bearngs and through the walls, then the loss n gravtatonal potental energy assocated wth the blocks equals the work done by the paddles on the water. By varyng the condtons o the experment, he notced that the loss n mechancal energy 2mgh was proportonal to the ncrease n water temperature T, wth a proportonalty constant 4.18J/ C. Ths was one o the key experments leadng up to the dscovery o the 1 st law 54
7 o thermodynamcs. James Prescott Joule, (December 24, 1818 October 11, 1889) was an Englsh physcst, born n Sale, near Manchester. Joule studed the nature o heat, and dscovered ts relatonshp to mechancal work. Ths led to the theory o conservaton o energy, whch led to the development o the rst law o thermodynamcs. The SI unt o work, the joule, s named ater hm. He worked wth Lord Kelvn to develop the absolute scale o temperature, made observatons on magnetostrcton, and ound the relatonshp between the low o current through a resstance and the heat dsspated, now called Joule's law. 55
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