Matter and Interactions - Supplement 1 - Gases and Heat Engines

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" I JT UIF WPMVNF 7 PG UIF CPY BOE O = //7 JT UIF OVNCFS PG NPMFDVMFT QFS N JO UIF CPY TP XIBU EPFT UIJT FRVBUJPO SFEVDF UP

26&45*0/

8F IBWF UIF TUSJLJOH SFTVMU UIBU UIF BJS JOTJEF UIF CPY QVTIFT EPXO PO UIF CPY XJUI B GPSDF FRVBM UP UIF DPNCJOFE XFJHIU .H PG BMM / NPMFDVMFT JO UIF CPY 'Z = −/NH = −.H 8F IBWF TIPXO UIBU UIF EJGGFSFODF JO UIF UJNF BOE TQBDFBWFSBHFE NPNFOUVN USBOTGFST CZ NPMFDVMBS DPMMJTJPOT UP UIF UPQ BOE CPUUPN PG UIF CPY JT FRVBM UP UIF XFJHIU PG UIF HBT JO UIF CPY BT QSFEJDUFE CZ UIF .PNFOUVN 1SJODJQMF GPS B NVMUJQBSUJDMF TZTUFN 5IF QSFTTVSF EJGGFSFODF JT WFSZ TMJHIU CVU UIFO UIF XFJHIU PG UIF HBT JT WFSZ TNBMM GPS UIBU NBUUFS 8IBU JT TVSQSJTJOH JT UIBU BU BOZ QBSUJDVMBS JOTUBOU SFMBUJWFMZ GFX PG UIF NPMFDVMFT BSF BDUVBMMZ JO DPOUBDU XJUI UIF CPY ZFU UIF FGGFDUT PG UIFTF SFMBUJWFMZ GFX NPMFDVMFT JT UIF TBNF BT UIPVHI UIFZ XFSF BMM TJUUJOH PO UIF CPUUPN PG UIF CPY :PV DPVME UIJOL PG B CPY GVMM PG XBUFS JO UIF TBNF XBZ 5IF XBUFS QSFTTVSF JT MBSHFS BU UIF CPUUPN UIBO BU UIF UPQ CVU XJUI XBUFS UIJT EJGGFSFODF JT RVJUF MBSHF DPSSFTQPOEJOH UP UIF NVDI IJHIFS EFOTJUZ PG XBUFS *O GBDU B DPMVNO PG XBUFS POMZ  N IJHI NBLFT B QSFTTVSF FRVBM UP UIBU QSPEVDFE CZ UIF NBOZ LJMPNFUFST PG BUNPTQIFSF $POTJEFS B DPMVNO PG XBUFS  N IJHI XJUI DSPTTTFDUJPOBM BSFB " 5IF EFOTJUZ PG XBUFS JT  HDN XIJDI JT  ×  LHN  5IF UPUBM NBTT PG UIF DPMVNO JT UIF EFOTJUZ LHN UJNFT UIF WPMVNF "  N

TP UIF QSFTTVSF PO UIF CPUUPN PG UIF DPMVNO JT  ×  LH/N "  N  // LH

≈  ×  //N " 5IJT JT UIF TBNF SFTVMU XF GPVOE GPS BUNPTQIFSJD QSFTTVSF BU 451 *O UIF QSFDFEJOH EJTDVTTJPO UIF CPY XBT JO WBDVVN CVU VOEFS OPSNBM DPOEJUJPOT UIF CPY XF XFSF XFJHIJOH XPVME IBWF CFFO TVSSPVOEFE CZ BJS 4VQQPTF UIBU UIF CPY IBT UIJO XBMMT BOE JT JOJUJBMMZ PQFO UP UIF BJS 5IFO XF DMPTF UIF MJE USBQQJOH BJS JOTJEF JU $POTJEFS BMM GPSDFT JODMVEJOH CVPZBODZ GPSDFT BOE EFUFSNJOF XIBU UIF TDBMFT XJMM SFBE 8JMM JU CF UIF XFJHIU PG UIF CPY BMPOF PS UIF CPY QMVT BJS

26&45*0/

5IF TDBMFT XJMM NFBTVSF UIF XFJHIU PG UIF CPY BMPOF CFDBVTF UIF XFJHIU PG UIF BJS JT .BJS H BOE UIF CVPZBODZ GPSDF JT .BJS H VQXBSE

4 &OFSHZ 5SBOTGFST Maximum height

v

h

m Scale 'JHVSF 4 5IF CBMM IBT KVTU IJU UIF TDBMF BOE IBT SFCPVOEFE VQXBSE

4

"QQMJDBUJPO 8FJHIU PG B #PVODJOH .PMFDVMF 5IFSF JT B SFMBUFE DBMDVMBUJPO UIBU JT BNVTJOH :PV NBZ IBWF EPOF B SFMBUFE IPNFXPSL QSPCMFN FBSMJFS PO DPMMJTJPOT JO XIJDI DBTF ZPV BMSFBEZ XFOU UISPVHI B TJNJMBS DBMDVMBUJPO $POTJEFS B TJOHMF NPMFDVMF PUIFSXJTF JO WBDVVN UIBU CPVODFT VQ BOE EPXO PO B TDBMF XJUIPVU MPTJOH TJHOJàDBOU FOFSHZ 'JHVSF 4 TIPXT B TOBQTIPU BU UIF JOTUBOU UIBU UIF NPMFDVMF JT KVTU CPVODJOH VQ BGUFS IJUUJOH UIF TDBMF 8IBU EP UIF TDBMFT SFBE *G UIF TDBMFT DBO SFTQPOE RVJDLMZ XF XJMM TFF CSJFG TQJLFT FBDI UJNF UIF NPMFDVMF TUSJLFT JU 'JHVSF 4  -FUT EFUFSNJOF XIBU UIF UJNFBWFSBHFE GPSDF JT )PX MPOH EPFT JU UBLF GPS UIF NPMFDVMF UP SFBDI UIF UPQ PG JUT USBKFDUPSZ I TUBSUJOH XJUI B TQFFE W 8IBU UIFO JT UIF UJNF ΔU CFUXFFO JNQBDUT )PX NVDI NPNFOUVN USBOTGFS JT UIFSF UP UIF TDBMF PO FBDI JNQBDU

26&45*0/

*U UBLFT B UJNF JOUFSWBM W/H UP HP VQ GPS UIF TQFFE UP EFDSFBTF GSPN W UP  XJUI BDDFMFSBUJPO −H BOE BOPUIFS UJNF JOUFSWBM W/H UP DPNF EPXO TP UIF UJNF CFUXFFO JNQBDUT JT ΔU = W/H &BDI JNQBDU USBOTGFST BO BNPVOU PG NPNFOUVN Δ1 = NW 5IFSFGPSF UIF UJNFBWFSBHFE GPSDF JT Δ1 NW = = NH ΔU W/H

Δt 'JHVSF 4 *NQBDUT PG UIF CBMM PO UIF TDBMF

*G UIF TDBMF JT TMVHHJTI BOE DBOU SFTQPOE JO B UJNF BT TIPSU BT ΔU = W/H UIF TDBMF XJMM TJNQMZ SFHJTUFS UIF WBMVF NH KVTU BT UIPVHI UIF NPMFDVMF XFSF TJUUJOH RVJFUMZ PO UIF TDBMF

4 &/&3(: 53"/4'&34

Piston

Cylinder filled with gas 'JHVSF 4 " DZMJOEFS XJUI B QJTUPO UIBU DBO NPWF WFSUJDBMMZ

Sand

Gas

*O UIJT TFDUJPO XF PGGFS BO BOBMZTJT PG FOFSHZ USBOTGFST CFUXFFO B HBT BOE JUT TVSSPVOEJOHT 8F XJMM BEESFTT TVDI RVFTUJPOT BT UIFTF )PX NVDI FOFSHZ USBOTGFS 2 EVF UP B UFNQFSBUVSF EJGGFSFODF JT SFRVJSFE UP SBJTF UIF UFNQFSBUVSF PG B HBT CZ POF EFHSFF UIF IFBU DBQBDJUZ PG B HBT )PX EPFT UIF UFNQFSBUVSF PG B HBT DIBOHF XIFO ZPV DPNQSFTT JU RVJDLMZ %P UIF BOTXFST UP UIFTF RVFTUJPOT EFQFOE PO XIBU LJOE PG HBT JT JOWPMWFE 8F OFFE B EFWJDF UIBU MFUT VT DPOUSPM UIF áPX PG FOFSHZ JOUP BOE PVU PG B HBT JO UIF GPSN PG XPSL 8 PS FOFSHZ USBOTGFS 2 8F XJMM VTF B TZTUFN DPOTJTUJOH PG B DZMJOESJDBM DPOUBJOFS DPOUBJOJOH HBT UIBU JT FODMPTFE CZ B QJTUPO UIBU DBO NPWF JO BOE PVU PG UIF DZMJOEFS XJUI MJUUMF GSJDUJPO CVU XIJDI àUT UJHIUMZ FOPVHI UP LFFQ UIF HBT GSPN MFBLJOH PVU 'JHVSF 4  5IJT JT TJNJMBS UP B DZMJOEFS JO BO BVUPNPCJMF FOHJOF JOUP XIJDI JT TQSBZFE B NJYUVSF PG HBTPMJOF WBQPS BOE BJS 5IF NJYUVSF JT JHOJUFE CZ B TQBSL BOE UIF DIFNJDBM SFBDUJPOT SBJTF UIF UFNQFSBUVSF BOE QSFTTVSF WFSZ IJHI WFSZ RVJDLMZ 5IF QJTUPO JT QVTIFE PVUXBSE XIJDI UVSOT B TIBGU UIBU VMUJNBUFMZ ESJWFT UIF XIFFMT

'PSDF BOE 1SFTTVSF 'JHVSF 4 TIPXT B DZMJOEFS XJUI B WFSUJDBMSVOOJOH QJTUPO PO XIJDI XF DBO MPBE WBSZJOH BNPVOUT PG TBOE JO PSEFS UP CF BCMF UP DPOUSPM UIF QSFTTVSF PG UIF DPOUBJOFE HBT BOE UP CF BCMF UP EP DPOUSPMMFE BNPVOUT PG XPSL PO UIF HBT 8F XJMM BMTP QVU UIF DZMJOEFS JO DPOUBDU XJUI IPU PS DPME PCKFDUT BOE BMMPX FOFSHZ USBOTGFS JOUP PS PVU PG UIF DZMJOEFS $POTJEFS UIF QJTUPO QMVT TBOE BT UIF TZTUFN PG JOUFSFTU GPS B NPNFOU BOE UIJOL BCPVU XIBU GPSDFT BDU PO UIJT TZTUFN

26&45*0/

'JHVSF 4 5IF QJTUPO DBO CF MPBEFE XJUI WBSZJOH BNPVOUT PG TBOE

" GSFFCPEZ EJBHSBN GPS UIF QJTUPO + TBOE TZTUFN 'JHVSF 4 JODMVEFT UIF EPXOXBSE HSBWJUBUJPOBM GPSDFT PO UIF QJTUPO .H BOE PO UIF TBOE NH

BO

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT

PA

mg Mg

Pair A

'JHVSF 4 'PSDFT PO UIF QJTUPO

VQXBSE UJNF BOE TQBDFBWFSBHFE GPSDF EVF UP UIF QSFTTVSF PG UIF FODMPTFE HBT PO UIF MPXFS TVSGBDF PG UIF QJTUPO BOE B EPXOXBSE UJNF BOE TQBDFBWFSBHFE GPSDF EVF UP UIF QSFTTVSF PG UIF PVUTJEF BUNPTQIFSF PO UIF VQQFS TVSGBDF PG UIF TZTUFN 4JODF QSFTTVSF JT GPSDF QFS VOJU BSFB UIF VQXBSE GPSDF JT 1" XIFSF " JT UIF TVSGBDF BSFB PG UIF CPUUPN PG UIF QJTUPO 4JNJMBSMZ UIF EPXOXBSE GPSDF DPOUSJCVUFE CZ UIF PVUTJEF BUNPTQIFSF JT 1BJS "  5IFTF QSFTTVSFSFMBUFE GPSDFT BSF OPU BDUVBMMZ DPOUJOVPVT TJODF UIFZ BSF UIF SFTVMU PG DPMMJTJPOT PG JOEJWJEVBM HBT NPMFDVMFT XJUI UIF QJTUPO BOE TBOE )PXFWFS UIF SBUF PG DPMMJTJPOT JT TP FYUSFNFMZ IJHI PWFS UIF BSFB PG UIF QJTUPO UIBU UIF GPSDF TFFNT FTTFOUJBMMZ DPOTUBOU 'PS FYBNQMF JO $IFDLQPJOU  ZPV GPVOE UIBU PO BWFSBHF BU 451 TUBOEBSE UFNQFSBUVSF BOE QSFTTVSF BCPVU  ×  IFMJVN BUPNT TUSJLF B UJOZ NNEJBNFUFS TFDUJPO FWFSZ TFDPOE *O NFDIBOJDBM FRVJMJCSJVN WFMPDJUZ PG QJTUPO OPU DIBOHJOH

TPMWF GPS UIF HBT QSFTTVSF JOTJEF UIF DZMJOEFS

26&45*0/

5IF .PNFOUVN 1SJODJQMF UFMMT VT UIBU JO FRVJMJCSJVN UIF OFU GPSDF PO UIF QJTUPO + TBOE TZTUFN NVTU CF [FSP GSPN XIJDI XF BSF BCMF UP EFEVDF UIBU UIF QSFTTVSF PG UIF HBT JOTJEF UIF DZMJOEFS JT 1 = 1BJS +

.H + NH "

" 4VEEFO $IBOHF #Z WBSZJOH UIF BNPVOU PG TBOE N XF DBO WBSZ UIF QSFTTVSF PG UIF HBT VOEFS TUVEZ 26&45*0/

*G ZPV TVEEFOMZ BEE B MPU PG TBOE XIBU IBQQFOT

*G ZPV TVEEFOMZ BEE B MPU PG TBOE UP UIF QJTUPO UIFSF JT TVEEFOMZ B TJ[BCMF OPO[FSP OFU EPXOXBSE GPSDF PO UIF QJTUPO + TBOE TZTUFN 'OFU = "1BJS + .H + NH − "1 = /  5IF QJTUPO TUBSUT UP QJDL VQ TQFFE EPXOXBSE "T JU EPFT TP JU SVOT JOUP HBT NPMFDVMFT BOE UFOET UP JODSFBTF UIFJS TQFFET 26&45*0/

8IBU IBQQFOT UP UIF UFNQFSBUVSF PG UIF HBT JO UIF

DZMJOEFS

Add or remove one grain at a time

Gas

4JODF IJHIFS BWFSBHF TQFFE NFBOT IJHIFS UFNQFSBUVSF UIF HBT UFNQFSBUVSF TUBSUT UP JODSFBTF BU UIF TBNF UJNF UIBU UIF WPMVNF PG UIF HBT JT EFDSFBTJOH 5IJT JT B EPVCMF XIBNNZ CPUI JODSFBTFE UFNQFSBUVSF BOE EFDSFBTFE WPMVNF DPOUSJCVUF UP JODSFBTFE QSFTTVSF TJODF 1 = //7 L# 5 5IFSFGPSF UIF QSFTTVSF JO UIF HBT RVJDLMZ SJTFT BOE FWFOUVBMMZ UIFSF XJMM CF B OFX FRVJMJCSJVN XJUI B MPXFS QJTUPO TVQQPSUJOH NPSF TBOE BOE B IJHIFS HBT QSFTTVSF JO UIF DZMJOEFS )PXFWFS HFUUJOH UP UIBU OFX FRVJMJCSJVN JT QSFUUZ DPNQMJDBUFE *G UIFSF JT OP GSJDUJPO PS PUIFS FOFSHZ EJTTJQBUJPO UIF QJTUPO XJMM PTDJMMBUF EPXO BOE VQ XJUI UIF HBT QSFTTVSF HPJOH VQ BOE EPXO *U JT FWFO QPTTJCMF UP EFUFSNJOF BO FGGFDUJWF iTQSJOH TUJGGOFTTu GPS UIF HBT BOE DBMDVMBUF UIF GSFRVFODZ PG UIF PTDJMMBUJPO )PXFWFS JO BOZ SFBM TZTUFN UIFSF XJMM CF TPNF GSJDUJPO TP XF LOPX UIBU UIF TZTUFN XJMM FWFOUVBMMZ TFUUMF EPXO UP B OFX FRVJMJCSJVN DPOàHVSBUJPO

2VBTJTUBUJD 1SPDFTTFT 'JHVSF 4 #Z BEEJOH PS SFNPWJOH POF HSBJO PG TBOE BU B UJNF XF FGGFDU B iRVBTJTUBUJDu DPNQSFTTJPO PS FYQBOTJPO

5P BWPJE UIFTF DPNQMJDBUFE UIPVHI JOUFSFTUJOH USBOTJFOU FGGFDUT XF XJMM TUVEZ XIBU IBQQFOT XIFO XF BEE TBOE WFSZ DBSFGVMMZ BOE WFSZ TMPXMZ POF HSBJO BU B UJNF BOE XF BTTVNF UIBU UIF OFX FRVJMJCSJVN JT FTUBCMJTIFE BMNPTU JNNFEJBUFMZ 5IJT JT DBMMFE B iRVBTJTUBUJD DPNQSFTTJPOu CFDBVTF UIF TZTUFN JT BU BMM UJNFT WFSZ OFBSMZ JO FRVJMJCSJVN 'JHVSF 4  4JNJMBSMZ JG XF TMPXMZ SFNPWF POF HSBJO BU B UJNF XF DBO DBSSZ PVU B iRVBTJTUBUJD FYQBOTJPOu /PUF

4 &OFSHZ 5SBOTGFST

4

JO QBSUJDVMBS UIBU BU OP UJNF EPFT UIF QJTUPO IBWF BOZ TJHOJàDBOU BNPVOU PG LJOFUJD FOFSHZ BOE NBDSPTDPQJD LJOFUJD FOFSHZ JT FTTFOUJBMMZ [FSP BU BMM UJNFT 0G DPVSTF UIFSF JT QMFOUZ PG NJDSPTDPQJD LJOFUJD FOFSHZ JO UIF HBT NPMFDVMFT BOE UIF PVUTJEF BJS NPMFDVMFT BOE JO UIF UIFSNBM NPUJPO PG BUPNT JO UIF DZMJOEFS XBMMT QJTUPO BOE TBOE 4VQQPTF XF DBSSZ PVU B MFOHUIZ UJNFDPOTVNJOH RVBTJTUBUJD DPNQSFTTJPO CZ BEEJOH MPUT PG TBOE POF HSBJO BU B UJNF WFSZ TMPXMZ 5IF QJTUPO HPFT EPXO RVJUF B XBZT UIF QSFTTVSF JO UIF HBT JT B MPU IJHIFS BOE UIF WPMVNF PG UIF HBT JT B MPU TNBMMFS *G XF LOPX UIF QSFTTVSF BOE WPMVNF XF DBO DBMDVMBUF UIF UFNQFSBUVSF CZ VTJOH UIF JEFBM HBT MBX 1 = //7 L# 5 BTTVNJOH UIF HBT EFOTJUZ JTOU UPP IJHI UP NBLF UIJT JOWBMJE  $BO XF QSFEJDU IPX MPX UIF QJTUPO XJMM HP GPS B HJWFO BNPVOU PG BEEFE TBOE 0EEMZ FOPVHI OP‡OPU XJUIPVU LOPXJOH TPNFUIJOH NPSF BCPVU UIJT EFWJDF 5IFSF BSF UXP FYUSFNF DBTFT UIBU BSF CPUI JNQPSUBOU JO QSBDUJDF BOE DBMDVMBCMF GPS BO JEFBM HBT *G UIF BQQBSBUVT JT NBEF PG NFUBM B WFSZ HPPE UIFSNBM DPOEVDUPS BOE JT JO HPPE UIFSNBM DPOUBDU XJUI B MBSHF PCKFDU BU UFNQFSBUVSF 5 UIF QSPDFTT XJMM QSPDFFE BU OFBSMZ DPOTUBOU UFNQFSBUVSF XJUI FOFSHZ USBOTGFS GSPN UIF HBT JOUP UIF TVSSPVOEJOHT *G UIF BQQBSBUVT JT NBEF PG HMBTT B WFSZ QPPS UIFSNBM DPOEVDUPS

UIF UFNQFSBUVSF PG UIF HBT JOTJEF UIF DZMJOEFS XJMM SJTF JO B QSFEJDUBCMF XBZ XJUI OFHMJHJCMF FOFSHZ USBOTGFS UP UIF TVSSPVOEJOHT JO UIF GPSN PG FOFSHZ USBOTGFS 2 8F XJMM BOBMZ[F CPUI LJOET PG QSPDFTTFT DPOTUBOU UFNQFSBUVSF QSPDFTTFT BOE OP2 QSPDFTTFT

Add sand

$POTUBOU5FNQFSBUVSF *TPUIFSNBM $PNQSFTTJPO Compress gas Q Energy transfer due to temperature difference

Water at temperature T 'JHVSF 4 " DZMJOEFS PG HBT JNNFSTFE JO B MBSHF WPMVNF PG XBUFS

4VQQPTF UIBU UIF DZMJOEFS JT NBEF PG NFUBM XIJDI JT B WFSZ HPPE UIFSNBM DPOEVDUPS BOE JT TJUUJOH JO B WFSZ CJH UVC PG XBUFS XIPTF UFNQFSBUVSF JT 5 BT TIPXO JO 'JHVSF 4 "T XF DPNQSFTT UIF HBT UIF UFNQFSBUVSF JO UIF HBT TUBSUT UP JODSFBTF )PXFWFS UIJT XJMM MFBE UP FOFSHZ áPXJOH PVU PG UIF HBT JOUP UIF XBUFS CFDBVTF XIFOFWFS UIF UFNQFSBUVSFT EJGGFS JO UXP PCKFDUT UIBU BSF JO UIFSNBM DPOUBDU XJUI FBDI PUIFS XF IBWF TFFO UIBU UIFSF JT B USBOTGFS PG FOFSHZ GSPN UIF IPUUFS PCKFDU JOUP UIF DPMEFS PCKFDU *O GBDU GPS NBOZ NBUFSJBMT UIF SBUF PG FOFSHZ USBOTGFS JT QSPQPSUJPOBM UP UIF UFNQFSBUVSF EJGGFSFODF‡EPVCMF UIF UFNQFSBUVSF EJGGFSFODF EPVCMF UIF SBUF BU XIJDI FOFSHZ USBOTGFST GSPN UIF IPUUFS PCKFDU JOUP UIF DPMEFS POF 5IF NFDIBOJTN GPS FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF JT UIBU BUPNT JO UIF IPUUFS PCKFDU BSF PO BWFSBHF NPWJOH GBTUFS UIBO BUPNT JO UIF DPMEFS PCKFDU TP JO DPMMJTJPOT CFUXFFO BUPNT BU UIF CPVOEBSZ CFUXFFO UIF UXP TZTUFNT JU JT MJLFMZ UIBU FOFSHZ XJMM CF HBJOFE CZ UIF DPMEFS PCKFDU BOE MPTU CZ UIF IPUUFS PCKFDU &OFSHZ USBOTGFS PVU PG UIF HBT XJMM MPXFS UIF UFNQFSBUVSF PG UIF HBT TJODF UIF UPUBM FOFSHZ PG UIF HBT JT QSPQPSUJPOBM UP UIF UFNQFSBUVSF 2VJDLMZ UIF UFNQFSBUVSF PG UIF HBT XJMM GBMM CBDL UP UIF UFNQFSBUVSF PG UIF XBUFS 5IF UFNQFSBUVSF PG UIF CJH UVC PG XBUFS PO UIF PUIFS IBOE XJMM IBSEMZ DIBOHF BT B SFTVMU PG UIF FOFSHZ BEEFE UP JU GSPN UIF HBT CFDBVTF UIFSF JT TVDI B MBSHF NBTT PG XBUFS UP XBSN VQ 5IFSFGPSF UIF FOUJSF RVBTJTUBUJD DPNQSFTTJPO UBLFT QMBDF FTTFOUJBMMZ BU UIF UFNQFSBUVSF PG UIF XBUFS BOE UIF àOBM UFNQFSBUVSF PG UIF HBT JT OFBSMZ UIF TBNF BT UIF JOJUJBM UFNQFSBUVSF PG UIF HBT 5IJT JT B DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO BMTP DBMMFE BO iJTPUIFSNBMu DPNQSFTTJPO XIJDI KVTU NFBOT DPOTUBOU UFNQFSBUVSF  4JNJMBSMZ XF DBO TMPXMZ SFNPWF TBOE GSPN UIF QJTUPO BOE DBSSZ PVU B RVBTJTUBUJD DPOTUBOUUFNQFSBUVSF FYQBOTJPO 4VQQPTF UIBU UIF PSJHJOBM HBT QSFTTVSF BOE WPMVNF XFSF 1 BOE 7 BOE BT UIF SFTVMU PG B DPOTUBOUUFNQFSBUVSF QSPDFTT UIF àOBM WPMVNF JT 7  8IBU JT UIF àOBM QSFTTVSF 1

26&45*0/

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT 4JODF UIF UFNQFSBUVSF IBTOU DIBOHFE GSPN UIF JEFBM HBT MBX 1 = //7 L# 5 XF DBO EFEVDF UIBU 1 7 = 1 7 BOE UIFSFGPSF 1 = 1 7 /7 

&OFSHZ JO B $POTUBOU5FNQFSBUVSF $PNQSFTTJPO " NPSF EJGàDVMU RVFTUJPO XF DBO BTL BOE BOTXFS JT UIJT )PX NVDI FOFSHZ XBT BEEFE UP UIF XBUFS JO UIF DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO 8IBU FOFSHZ JOQVUT BOE PVUQVUT XFSF NBEF UP UIF HBT 8IBU FOFSHZ DIBOHF PDDVSSFE JO UIF HBT

26&45*0/

5IF QJTUPO EJE XPSL PO UIF HBT BOE UIFSF XBT FOFSHZ USBOTGFS 2 PVU PG UIF HBT BOE JOUP UIF XBUFS  5IF &BSUIT HSBWJUBUJPOBM GPSDF EJE XPSL PO UIF HBT TJODF UIF DFOUFS PG NBTT PG UIF HBT XFOU EPXO

CVU UIJT JT OFHMJHJCMZ TNBMM DPNQBSFE UP UIF XPSL EPOF CZ UIF QJTUPO UIF MPXFSJOH PG UIF IFBWZ QJTUPO JOWPMWFT NVDI NPSF HSBWJUBUJPOBM FOFSHZ UIBO UIF MPXFSJOH PG UIF MPXNBTT HBT  %JE UIF UPUBM FOFSHZ PG UIF HBT DIBOHF JHOPSJOH UIF TNBMM HSBWJUBUJPOBM FOFSHZ DIBOHF

26&45*0/

4JODF UIF UPUBM FOFSHZ PG BO JEFBM HBT JT QSPQPSUJPOBM UP UFNQFSBUVSF JODMVEJOH SPUBUJPOBM BOE WJCSBUJPOBM FOFSHZ JG UIF HBT JT OPU NPOBUPNJD

BOE XF NBEF TVSF UIBU XF LFQU UIF UFNQFSBUVSF DPOTUBOU UIF UPUBM FOFSHZ PG UIF HBT EJE OPU DIBOHF 5IFSFGPSF XF IBWF UIJT FOFSHZ FRVBUJPO GPS UIF PQFO TZTUFN UIBU JT UIF HBT Δ FOFSHZ PG HBT = 8 CZ QJTUPO − |2| UIBU áPXFE JOUP XBUFS =  4ZNCPMJDBMMZ Δ&HBT = 8 + 2 =  XIFSF 2 JT OFHBUJWF USBOTGFS PVU PG TZTUFN  *O UIFSNBM QSPDFTTFT PG UIJT LJOE UIF FOFSHZ FRVBUJPO JT DBMMFE iUIF àSTU MBX PG UIFSNPEZOBNJDTu *G XF DBO DBMDVMBUF UIF XPSL EPOF CZ UIF QJTUPO XF DBO FRVBUF UIBU SFTVMU UP UIF BNPVOU PG FOFSHZ USBOTGFS 2 JOUP UIF XBUFS 5IF QJTUPO FYFSUT B UJNF BOE TQBDFBWFSBHFE GPSDF 1" PO UIF HBT XIFSF 1 JT UIF QSFTTVSF JO UIF HBT BOE " JT UIF DSPTTTFDUJPOBM BSFB PG UIF QJTUPO 'JHVSF 4  Fon gas = PA 'JHVSF 4 'PSDF PO UIF HBT CZ UIF QJTUPO

26&45*0/

*G UIF QJTUPO ESPQT B EJTUBODF I JT UIF XPSL JU EPFT 1"I

"T UIF QJTUPO ESPQT UIF QSFTTVSF JO UIF HBT JODSFBTFT TP UIF GPSDF JT OPU DPOTUBOU 8F IBWF UP JOUFHSBUF UIF WBSJBCMF GPSDF UISPVHI UIF EJTUBODF I JO PSEFS UP EFUFSNJOF UIF XPSL *G XF NFBTVSF Y EPXOXBSE GSPN UIF JOJUJBM QJTUPO QPTJUJPO BU FBDI TUFQ EY PG UIF XBZ UIF JODSFNFOU PG XPSL EPOF JT 1" EY )PXFWFS " EY JT BO JODSFNFOU PG WPMVNF CBTF BSFB " UJNFT BMUJUVEF EY

BOE UIF DIBOHF JO UIF WPMVNF JT OFHBUJWF 1VUUJOH UIJT BMM UPHFUIFS XF IBWF UIJT 803, %0/& #: " 1*450/ 0/ " ("4  7 8=− 1E7 7

$IFDL UIF TJHO *G UIF WPMVNF EFDSFBTFT UIF JOUFHSBM XJMM CF OFHBUJWF BOE UIF NJOVT TJHO JO GSPOU PG UIF JOUFHSBM NBLFT UIF XPSL EPOF PO UIF HBT CF QPTJUJWF XIJDI JT DPSSFDU $POWFSTFMZ JO BO FYQBOTJPO PG B HBT UIF JOUFHSBM JT QPTJUJWF BOE UIF XPSL JT OFHBUJWF CFDBVTF UIF HBT JT EPJOH XPSL PO UIF QJTUPO SBUIFS UIBO UIF PUIFS XBZ SPVOE *O B DPOTUBOUUFNQFSBUVSF JTPUIFSNBM DPNQSFTTJPO XPSL JT EPOF PO UIF HBT 8IFSF EPFT UIJT FOFSHZ áPX HP )PX EPFT UIF UFNQFSBUVSF TUBZ DPOTUBOU

26&45*0/

4 &OFSHZ 5SBOTGFST

Initial volume V1

More sand

Final volume V2 'JHVSF 4 5IF HBT JT DPNQSFTTFE BU B DPOTUBOU UFNQFSBUVSF 5 UIF DZMJOEFS JT JNNFSTFE JO B MBSHF UVC PG XBUFS BU UFNQFSBUVSF 5 

4

"T UIF QJTUPO NPWFT EPXO JU JODSFBTFT UIF BWFSBHF FOFSHZ PG UIF HBT NPMFDVMFT UIBU SVO JOUP JU 5IFSFGPSF UIF JOQVU XPSL TUBSUT UP SBJTF UIF UFNQFSBUVSF PG UIF HBT IJHIFS TQFFET

CVU UIJT IJHIFS UFNQFSBUVSF DBVTFT FOFSHZ USBOTGFS 2 PVU PG UIF DZMJOEFS JOUP UIF WFSZ TMJHIUMZ MPXFSUFNQFSBUVSF XBUFS 5IF OFU FGGFDU JT GPS FOFSHZ UP áPX JOUP UIF HBT JO UIF GPSN PG NFDIBOJDBM XPSL BOE PVU PG UIF HBT JOUP UIF XBUFS JO UIF GPSN PG FOFSHZ USBOTGFS 2 5IFSF JT OP DIBOHF JO UIF UPUBM FOFSHZ PG UIF HBT CFDBVTF UIF UFNQFSBUVSF PG UIF HBT EJEOU DIBOHF 5IF PVUXBSE FOFSHZ USBOTGFS CSJOHT UIF HBT UFNQFSBUVSF CBDL UP XIBU JU XBT CFGPSF UIF GBMMJOH QJTUPO USJFE UP JODSFBTF UIF UFNQFSBUVSF /PX XF DBO DBMDVMBUF RVBOUJUBUJWFMZ UIF XPSL EPOF CZ UIF QJTUPO JO UIF DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO XIJDI JT FRVBM UP UIF FOFSHZ USBOTGFS GSPN UIF HBT UP UIF TVSSPVOEJOH XBUFS 8F BEE MPUT PG TBOE POF HSBJO BU B UJNF UP NBJOUBJO RVBTJTUBUJD FRVJMJCSJVN

BOE XF DPNQSFTT UIF HBT GSPN BO JOJUJBM WPMVNF 7 UP B àOBM WPMVNF 7 'JHVSF 4  3FQMBDF 1 CZ /L# 5/7 XIFSF 5 JT B DPOTUBOU  7  7 /L# 5 8 =− 1E7 = − E7 7 7 7    7 E7 7 7 8 = − /L# 5 = −/L# 5< MO 7 >7 = −/L# 5 MO  7 7 7 4JODF UIF QSPDFTT XBT B DPNQSFTTJPO 7 JT TNBMMFS UIBO 7  4JODF − MO 7 /7 = MO 7 /7

XF IBWF   7 8CZ QJTUPO = |2JOUP XBUFS | = /L# 5 MO 7

P

+VTU BT UIF TZNCPM 8 JT OPSNBMMZ VTFE GPS iXPSL u TP UIF TZNCPM 2 JT OPSNBMMZ VTFE GPS iFOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODFu )FSF 2 JT OFHBUJWF FOFSHZ PVUQVU  V Compression

Expansion

'JHVSF 4 *G 5 JT LFQU DPOTUBOU UIFO UIF QSPEVDU 17 JT BMTP DPOTUBOU

(SBQIJDBM 3FQSFTFOUBUJPO 5IFSF JT B VTFGVM HSBQIJDBM SFQSFTFOUBUJPO PG UIJT DPOTUBOUUFNQFSBUVSF QSPDFTT 8F QMPU UIF QSFTTVSF 1 WT UIF WPMVNF 7 BT TIPXO JO 'JHVSF 4 BOE UIF QSPDFTT JT SFQSFTFOUFE CZ B DVSWF PG DPOTUBOU 5 XIJDI JT BMTP B DVSWF PG DPOTUBOU 17 TJODF 17 = /L# 5  5IF BSFB VOEFS UIF QSPDFTT DVSWF JT UIF JOUFHSBM PG 1E7 *O B DPNQSFTTJPO UIF WPMVNF EFDSFBTFT TP XF NPWF UP UIF MFGU PO UIF HSBQI BOE UIF JOUFHSBM JT OFHBUJWF 8 JT QPTJUJWF BOE 2 JT OFHBUJWF *O BO FYQBOTJPO UIF WPMVNF JODSFBTFT BOE XF NPWF UP UIF SJHIU PO UIF HSBQI BOE UIF JOUFHSBM JT QPTJUJWF 8 JT OFHBUJWF BOE 2 JT QPTJUJWF  7 8=− 1E7 7

5IF 'JSTU -BX PG 5IFSNPEZOBNJDT 'PS IJTUPSJDBM SFBTPOT JO UIFSNBM QSPDFTTFT UIF &OFSHZ 1SJODJQMF JT DBMMFE iUIF àSTU MBX PG UIFSNPEZOBNJDTu 5)& '*345 -"8 0' 5)&3.0%:/".*$4 Δ&TZT = 8 + 2 *O UIF QBSUJDVMBS DBTF PG DPOTUBOUUFNQFSBUVSF JTPUIFSNBM DPNQSFTTJPO PG BO JEFBM HBT XF IBWF Δ&JOU = 8 + 2 =  XIFSF &JOU JT UIF iJOUFSOBMu FOFSHZ PG UIF HBT UIF TVN PG UIF USBOTMBUJPOBM SPUBUJPOBM WJCSBUJPOBM BOE PUIFS FOFSHZ UFSNT PG BMM UIF NPMFDVMFT 

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT 'PS IJTUPSJDBM SFBTPOT NBOZ UFYUCPPLT EFBMJOH XJUI UIFSNBM QIZTJDT EFàOF XPSL 8 BT UIF XPSL EPOF CZ B TZTUFN SBUIFS UIBO XPSL EPOF PO B TZTUFN TP UIF TJHO PG 8 JT DIBOHFE 8F EFàOF 8 BT UIF XPSL EPOF PO UIF TZTUFN UP CF DPOTJTUFOU XJUI PUIFS OPOUIFSNBM VTFT PG UIF &OFSHZ 1SJODJQMF

)FBU $BQBDJUZ

Lock piston V = constant Temperature rise ΔT

Q

8F XFSF BCMF UP DBMDVMBUF UIF JNQPSUBOU QSPQFSUJFT PG B DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO XIFSF UIF BQQBSBUVT XBT JO HPPE UIFSNBM DPOUBDU XJUI JUT TVSSPVOEJOHT #FGPSF BOBMZ[JOH UIF PQQPTJUF FYUSFNF XIFSF UIF BQQBSBUVT JT UIFSNBMMZ JOTVMBUFE GSPN JUT TVSSPVOEJOHT BOE OP FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF JT JOWPMWFE XF XJMM SFWJFX UIF DPODFQU PG IFBU DBQBDJUZ JO UIJT DPOUFYU -PDL UIF QJTUPO JO QPTJUJPO TP UIBU UIF WPMVNF PG UIF HBT XPOU DIBOHF 'JHVSF 4

BOE TP OP XPSL JT EPOF 8 =   1VU UIF BQQBSBUVT JO B CJH UVC PG XBUFS XIPTF UFNQFSBUVSF JT IJHIFS UIBO UIF HBT UFNQFSBUVSF TP UIBU FOFSHZ USBOTGFS XJMM HP GSPN UIF XBUFS UP UIF HBT "MMPX BO BNPVOU PG FOFSHZ USBOTGFS 2 EVF UP UIF UFNQFSBUVSF EJGGFSFODF UP FOUFS UIF HBT BOE PCTFSWF IPX NVDI UFNQFSBUVSF SJTF Δ5 IBT PDDVSSFE JO UIF HBT 8F EFàOF UIF iTQFDJàD IFBU DBQBDJUZ $7 BU DPOTUBOU WPMVNFu PO B QFSNPMFDVMF CBTJT JO UIF GPMMPXJOH XBZ XIFSF / BT VTVBM JT UIF UPUBM OVNCFS PG NPMFDVMFT JO UIF HBT Δ&UIFSNBM = /$7 Δ5

Water at high temperature T

4JODF 8 =  UIJT JT BMTP FRVBM UP 2 5IF MBSHFS UIF TQFDJàD IFBU DBQBDJUZ UIF TNBMMFS UIF UFNQFSBUVSF SJTF Δ5 GPS B HJWFO FOFSHZ JOQVU 2 26&45*0/

'JHVSF 4 *G UIF QJTUPO JT MPDLFE TP UIF WPMVNF PG UIF HBT DBOOPU DIBOHF XF DBO NFBTVSF $7 PG UIF HBT

Piston rises P = constant

Temperature rise ΔT

Q

Water at high temperature T

DPOTUBOU WPMVNF

8IBU JT $7 GPS B NPOBUPNJD HBT TVDI BT IFMJVN

4JODF UIF UPUBM FOFSHZ JO B NPOBUPNJD HBT JT /,USBOT =  /L# 5 UIF FOFSHZ USBOTGFS 2 XJMM JODSFBTF UIF UPUBM FOFSHZ PG UIF HBT CZ BO BNPVOU  /L# Δ5 TP $7 =  L#  &YQFSJNFOUBM NFBTVSFNFOUT PG UIF TQFDJàD IFBU DBQBDJUZ PG NPOBUPNJD HBTFT BHSFF XJUI UIJT QSFEJDUJPO 'PS HBTFT UIBU BSF OPU NPOBUPNJD TVDI BT OJUSPHFO /

$7 DBO CF MBSHFS UIBO  L# CFDBVTF UIF UPUBM FOFSHZ DBO CF MBSHFS UIBO  /L# 5 EVF UP SPUBUJPOBM BOE WJCSBUJPOBM FOFSHZ SFMBUJWF UP UIF DFOUFS PG NBTT PG UIF NPMFDVMF #FGPSF RVBOUVN NFDIBOJDT UIFPSFUJDBM QSFEJDUJPOT GPS UIF DPOUSJCVUJPO PG UIF SPUBUJPOBM BOE WJCSBUJPOBM FOFSHJFT UP UIF TQFDJàD IFBU DBQBDJUZ PG HBTFT EJE OPU BHSFF XJUI FYQFSJNFOUBM NFBTVSFNFOUT 5IJT XBT POF PG UIF QV[[MFT UIBU XBT FWFOUVBMMZ SFTPMWFE CZ RVBOUVN NFDIBOJDT BT XF GPVOE JO $IBQUFS  41&$*'*$ )&"5 $"1"$*5: "5 $0/45"/5 70-6.& 0/ " 1&3.0-&$6-& #"4*4  $7 = L# GPS B NPOBUPNJD HBT )F FUD

  $7 ≥ L# GPS PUIFS HBTFT / FUD



)FBU $BQBDJUZ BU $POTUBOU 1SFTTVSF 'JHVSF 4 *G UIF QJTUPO JT GSFF UP NPWF UIF QSFTTVSF JOTJEF UIF DZMJOEFS SFNBJOT UIF TBNF BOE XF DBO NFBTVSF $Q PG UIF HBT

*G XF EPOU MPDL UIF QJTUPO CVU MFU UIF HBT FYQBOE BU DPOTUBOU QSFTTVSF UIF JODPNJOH FOFSHZ USBOTGFS 2 EVF UP UIF UFNQFSBUVSF EJGGFSFODF OPU POMZ SBJTFT UIF FOFSHZ PG UIF HBT CZ BO BNPVOU Δ&UIFSNBM = /$7 Δ5 CVU BMTP SBJTFT UIF QJTUPO XIJDI JOWPMWFT BO BNPVOU PG XPSL 8 'JHVSF 4  8F EFàOF UIF

4 &OFSHZ 5SBOTGFST

4

iTQFDJàD IFBU DBQBDJUZ $1 BU DPOTUBOU QSFTTVSFu PO B QFSNPMFDVMF CBTJT BT GPMMPXT 2 = /$1 Δ5

DPOTUBOU QSFTTVSF

&WJEFOUMZ $1 JT CJHHFS UIBO $7 CFDBVTF /$1 Δ5 = /$7 Δ5 + 8 8F DBO DBMDVMBUF UIF XPSL EPOF PO UIF QJTUPO CZ UIF HBT BT UIF HBT FYQBOET XIJDI JT UIF OFHBUJWF PG UIF XPSL EPOF CZ UIF QJTUPO PO UIF HBT  8=

7

7

 1E7 = 1

7 7

E7 = 17 − 17 = /L# 5 − /L# 5 = /L# Δ5

5IFSFGPSF GPS BO JEFBM HBT UIFSF JT B TJNQMF SFMBUJPOTIJQ CFUXFFO $7 BOE $1  /$1 Δ5 = /$7 Δ5 + /L# Δ5 $1 = $7 + L# $IFDLQPJOU  8IBU JT UIF TQFDJàD IFBU DBQBDJUZ BU DPOTUBOU QSFTTVSF PO B QFSNPMFDVMF CBTJT $1 GPS IFMJVN

.PMBS 4QFDJàD )FBU $BQBDJUZ 4QFDJàD IFBU DBQBDJUZ JT PGUFO HJWFO PO B QFSNPMF CBTJT SBUIFS UIBO B QFSNPMFDVMF CBTJT 5IF NPMBS TQFDJàD IFBU DBQBDJUZ BU DPOTUBOU WPMVNF GPS BO JEFBM HBT JT $7 =  3 XIFSF 3 JT UIF NPMBS HBT DPOTUBOU  +,

XIJDI JT × L#  5IF NPMBS TQFDJàD IFBU DBQBDJUZ BU DPOTUBOU QSFTTVSF JT $1 = $7 +3 GPS BO JEFBM HBT Initial volume V1

More sand

Final volume V2 'JHVSF 4 *G UIF DZMJOEFS BOE QJTUPO BSF NBEF PG JOTVMBUJOH NBUFSJBM OP FOFSHZ USBOTGFS 2 DBO PDDVS CFUXFFO UIF HBT BOE JUT TVSSPVOEJOHT 5IJT JT DBMMFE BO BEJBCBUJD DPNQSFTTJPO

/P2 "EJBCBUJD $PNQSFTTJPO 8JUI UIF BQQBSBUVT NBEF PG NFUBM BOE TJUUJOH JO B CJH UVC PG XBUFS UIF UFNQFSBUVSF EVSJOH UIF DPNQSFTTJPO EJEOU DIBOHF /PX XF BOBMZ[F UIF PQQPTJUF FYUSFNF 8F NBLF UIF DZMJOEFS BOE QJTUPO PVU PG HMBTT PS DFSBNJD XIJDI BSF QPPS DPOEVDUPST GPS FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF UP NJOJNJ[F FOFSHZ USBOTGFS PVU PG UIF HBT BT UIF HBT HFUT IPUUFS EVSJOH UIF DPNQSFTTJPO *O GBDU XF NBLF UIF BQQSPYJNBUJPO UIBU UIFSF JT OP FOFSHZ USBOTGFS BU BMM 'JHVSF 4  4VDI B OP2 QSPDFTT JT BMTP DBMMFE iBEJBCBUJDu XIJDI NFBOT iOP FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODFu  )PX SFBMJTUJD JT TVDI B OP2 BQQSPYJNBUJPO 'PS NBOZ SFBM TJUVBUJPOT UIJT DBO CF B SBUIFS HPPE BQQSPYJNBUJPO JG UIF DPNQSFTTJPO PS FYQBOTJPO JT GBTU TP UIBU UIFSF JTOU FOPVHI UJNF GPS TJHOJàDBOU FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF 4VDI B áPX PG FOFSHZ GSPN POF PCKFDU UP BOPUIFS JT B SBUIFS TMPX QSPDFTT 'PS FYBNQMF B DVQ PG DPGGFF TJUUJOH PO B UBCMF NBZ TUBZ RVJUF IPU GPS UFO NJOVUFT PS NPSF )PXFWFS EJEOU XF TBZ UIBU XF XFSF HPJOH UP DBSSZ PVU DPNQSFTTJPOT BOE FYQBOTJPOT WFSZ TMPXMZ iRVBTJTUBUJDBMMZu :FT TP UIFSF JT B DPOUSBEJDUJPO )PXFWFS JU JT PGUFO UIF DBTF UIBU NPUJPO NBZ CF TMPX FOPVHI UP CF B HPPE BQQSPYJNBUJPO UP B RVBTJTUBUJD QSPDFTT BOE OFWFSUIFMFTT NBZ BMTP CF GBTU DPNQBSFE UP UIF UJNF SFRVJSFE GPS TJHOJàDBOU FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF "HBJO UIJOL PG IPX MPOH JU UBLFT B DVQ PG DPGGFF UP DPPM PGG 8F BHBJO DPOTJEFS UIF HBT JO UIF DZMJOEFS BT UIF TZTUFN PG JOUFSFTU BOE UIF XPSL EPOF CZ UIF QJTUPO JT FRVBM UP UIF JODSFBTF JO FOFSHZ PG UIF HBT  7 8=− 1E7 = /$7 Δ5 7

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT 8IZ EJE XF VTF $7 JO UIJT FRVBUJPO 5IF WPMVNF JTOU DPOTUBOU JO UIJT DPNQSFTTJPO

26&45*0/

"O JEFBM HBT JT VOJRVF BNPOH NBUFSJBMT JO UIBU JUT UPUBM FOFSHZ JT FOUJSFMZ EFUFSNJOFE CZ UIF UFNQFSBUVSF JOEFQFOEFOU PG WPMVNF PS QSFTTVSF 4P UIF DIBOHF PG FOFSHZ PG UIF HBT JUTFMG JT KVTU /$7 Δ5 FWFO XIFO UIF WPMVNF JT OPU DPOTUBOU "T B QSBDUJDBM NBUUFS SFBM HBTFT CFIBWF UIJT XBZ BT MPOH BT UIF EFOTJUZ JT OPU UPP IJHI "U IJHI EFOTJUJFT UIF FOFSHZ PG UIF HBT JODMVEFT B TJHOJàDBOU UFSN BTTPDJBUFE XJUI UIF JOUFSBUPNJD GPSDFT 8F DBO XPSL UISPVHI UIF JOUFHSBUJPO TUBSUJOH GSPN B iEJGGFSFOUJBMu GPSN PG UIF FRVBUJPO TIPXO BCPWF −1E7 = /$7 E5 6TF UIF HBT MBX 1 = //7 L# 5 UP TVCTUJUVUF GPS UIF QSFTTVSF −/L# 5

E7 = /$7 E5 7

%JWJEF UISPVHI CZ L# 5 BOE SFBSSBOHF   $7 E5 E7 + = L# 5 7    $7 E5 E7 + = L# 5 7 /PX JOUFHSBUF



$7 L#

 MO 5 + MO 7 = DPOTUBOU

3FXSJUF VTJOH UIF QSPQFSUJFT PG MPHBSJUINT   MO 5 $7 /L# + MO 7 = MO 5 $7 /L# 7 = DPOTUBOU 5IFSFGPSF XF IBWF 5 $7 /L# 7 = DPOTUBOU $IFDLQPJOU  6TF UIF JEFBM HBT MBX UP FMJNJOBUF 5 GSPN UIJT FYQSFTTJPO BOE TIPX UIBU 17 γ = DPOTUBOU JO B OP2 QSPDFTT XIFSF UIF QBSBNFUFS γ JT EFàOFE BT UIF SBUJP PG UIF DPOTUBOUQSFTTVSF TQFDJàD IFBU DBQBDJUZ UP UIF DPOTUBOUWPMVNF TQFDJàD IFBU DBQBDJUZ γ ≡ $1 /$7  4JODF $1 JT HSFBUFS UIBO $7 γ JT HSFBUFS UIBO  *O QBSUJDVMBS JG UIF JOJUJBM QSFTTVSF BOE WPMVNF BSF 1 BOE 7 BOE UIF àOBM QSFTTVSF BOE WPMVNF BSF 1 BOE 7 UIFO $ /L

$ /L

1 7γ = 1 7γ  "MUFSOBUJWFMZ XF BMTP IBWF 5 7 # 7 = 5 7 # 7 

P (Q = 0; adiabatic) (PVγ = constant)

(SBQIJDBM 3FQSFTFOUBUJPO V (T, PV constant; isothermal) (PV = constant) 'JHVSF 4 17 DVSWFT GPS B OP2 BEJBCBUJD QSPDFTT BOE B DPOTUBOU UFNQFSBUVSF JTPUIFSNBM QSPDFTT

"HBJO UIFSF JT B VTFGVM HSBQIJDBM SFQSFTFOUBUJPO PG UIJT OP2 QSPDFTT *O 'JHVSF 4 XF QMPU UIF QSFTTVSF 1 WT UIF WPMVNF 7 BOE UIF QSPDFTT JT SFQSFTFOUFE CZ B DVSWF BMPOH XIJDI 17γ = DPOTUBOU 5IF BSFB VOEFS UIF QSPDFTT DVSWF SFQSFTFOUT UIF XPSL 8 'PS DPNQBSJTPO XF BMTP TIPX UIF DVSWF GPS B DPOTUBOUUFNQFSBUVSF QSPDFTT 5 BOE 17 DPOTUBOU  5IF OP2 DVSWF JT NVDI TUFFQFS UIBO UIF DPOTUBOUUFNQFSBUVSF DVSWF

8PSL WT 2 8F IBWF TUVEJFE UIF SFTQPOTF PG B HBT UP FOFSHZ JOQVUT BOE PVUQVUT CPUI NFDIBOJDBM 8 XPSL BOE UIFSNBM 2 FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF  8PSL SFQSFTFOUT PSHBOJ[FE PSEFSMZ NBDSPTDPQJD FOFSHZ JOQVU

4 'VOEBNFOUBM -JNJUBUJPOT PO &GàDJFODZ

4

&OFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF SFQSFTFOUT EJTPSHBOJ[FE EJTPSEFSMZ NJDSPTDPQJD FOFSHZ JOQVU 5IFSF JT SBOEPNOFTT BU UIF BUPNJD MFWFM JO UIF DPMMJTJPOT PG BUPNT XIJDI JT UIF CBTJD NFDIBOJTN GPS FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF *O $IBQUFS  XF TBX EFFQ DPOTFRVFODFT PG UIF EJTUJODUJPO CFUXFFO XPSL 8 BOE FOFSHZ USBOTGFS 2

4 '6/%".&/5"- -*.*5"5*0/4 0/ &''*$*&/$:

Net thermal energy transfer Q

Engine Net work done on surroundings 'JHVSF 4 "O FOHJOF DPOWFSUT OFU FOFSHZ USBOTGFS 2 JOUP VTFGVM XPSL EPOF CZ UIF FOHJOF PO UIF TVSSPVOEJOHT

"O JNQPSUBOU UFDIOPMPHZ JOWPMWFT UIF DPOWFSTJPO PG FOFSHZ USBOTGFS 2 EVF UP B UFNQFSBUVSF EJGGFSFODF JOUP VTFGVM XPSL EPOF CZ BO FOHJOF PO TPNF PCKFDU JO UIF TVSSPVOEJOHT 'JHVSF 4  4VDI BO FOHJOF JT DBMMFE B iIFBU FOHJOFu 'PS FYBNQMF JO B TUFBNQPXFSFE FMFDUSJDJUZ HFOFSBUJOH QMBOU DPBM JT CVSOFE UP XBSN VQ XBUFS UIBU ESJWFT B TUFBN FOHJOF UP UVSO B HFOFSBUPS XIJDI DPOWFSUT UIF XPSL EPOF CZ UIF TUFBN FOHJOF JOUP FMFDUSJD FOFSHZ &OFSHZ DPOTFSWBUJPO PG DPVSTF QVUT MJNJUT PO IPX NVDI VTFGVM XPSL ZPV DBO HFU GSPN UIF CVSOJOH PG UIF DPBM‡CVU UIFSF JT B GVSUIFS MJNJUBUJPO EVF UP UIF TFDPOE MBX PG UIFSNPEZOBNJDT *U UVSOT PVU UIBU JO B QSBDUJDBM HFOFSBUJOH QMBOU POMZ BCPVU POFUIJSE PG UIF FOFSHZ PG UIF DPBM DBO CF UVSOFE JOUP VTFGVM XPSL 5IF GVOEBNFOUBM QSPCMFN JT UIBU FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF JT B EJTPSEFSMZ FOFSHZ USBOTGFS BOE UIF TFDPOE MBX PG UIFSNPEZOBNJDT UBLFT UIBU JOUP BDDPVOU 8F XJMM àOE UIBU UIF NPTU FGàDJFOU QSPDFTTFT BSF iSFWFSTJCMFu QSPDFTTFT‡ UIBU JT QSPDFTTFT UIBU QSPEVDF OP DIBOHF JO UIF FOUSPQZ PG UIF 6OJWFSTF B SFWFSTFE NPWJF PG TVDI B QSPDFTT MPPLT QPTTJCMF  *O UIF OFYU TFDUJPOT XF EJTDVTT UXP QSPDFTTFT NFDIBOJDBM GSJDUJPO BOE àOJUFUFNQFSBUVSFEJGGFSFODF FOFSHZ USBOTGFS XIJDI BSF NBKPS DPOUSJCVUPST UP FOUSPQZ QSPEVDUJPO BOE XIPTF FGGFDUT NVTU CF NJOJNJ[FE JO PSEFS UP PCUBJO UIF NPTU FGàDJFOU QFSGPSNBODF JO B NFDIBOJDBM TZTUFN 8IFSF XF BSF IFBEFE JT UP FTUBCMJTI MJNJUT PO IPX FGàDJFOUMZ UIFSNBM FOFSHZ DBO CF UVSOFE JOUP NFDIBOJDBM FOFSHZ JO B IFBU FOHJOF BT B DPOTFRVFODF PG UIF TFDPOE MBX PG UIFSNPEZOBNJDT 5IJT JT B SFWFBMJOH FYBNQMF PG UIF QPXFS PG UIF TFDPOE MBX PG UIFSNPEZOBNJDT UP TFU MJNJUT PO QPTTJCMF QIFOPNFOB

'SJDUJPO BOE &OUSPQZ 1SPEVDUJPO

TH

Cross-sectional area A

. Q

L . Q

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TL

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4

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QSPQPSUJPOBM UP UIF UFNQFSBUVSF EJGGFSFODF UXJDF UIF UFNQFSBUVSF EJGGFSFODF UXJDF UIF SBUF PG FOFSHZ USBOTGFS

QSPQPSUJPOBM UP UIF DSPTTTFDUJPOBM BSFB " PG UIF CBS UXJDF UIF DSPTTTFDUJPOBM BSFB JT MJLF IBWJOH UXP CBST

BOE JOWFSTFMZ QSPQPSUJPOBM UP UIF MFOHUI - PG UIF CBS UXJDF UIF MFOHUI PG UIF CBS IBMG UIF FOFSHZ USBOTGFS SBUF  3"5& 0' &/&3(: 53"/4'&3 %6& 50 " 5&.1&3"563& %*''&3&/$& 5) − 5-

2˙ = σ" 6OJUT BSF +T PS 8 2˙ JT B QPTJUJWF RVBOUJUZ BOE JT UIF BCTPMVUF WBMVF PG UIF SBUF PG FOFSHZ áPX JOUP PS PVU PG B TZTUFN EVF UP B UFNQFSBUVSF EJGGFSFODF 5IF RVBOUJUZ 5) − 5- /- JT DBMMFE UIF iUFNQFSBUVSF HSBEJFOUu 5IF MBSHFS UIF HSBEJFOU UIF NPSF SBQJEMZ UIF UFNQFSBUVSF DIBOHFT XJUI EJTUBODF BMPOH UIF CBS

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2 5

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4 " .BYJNBMMZ &GàDJFOU 1SPDFTT

4

5IFSFGPSF UIF DIBOHF JO UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF JO B UJNF JOUFSWBM ΔU JT   ˙ ˙ 2ΔU 2ΔU   ˙ + +  = 2ΔU − Δ4TPVSDF + Δ4TJOL + Δ4CBS = − 5) 55- 5) *T UIJT RVBOUJUZ QPTJUJWF PS OFHBUJWF *T UIJT DPOTJTUFOU XJUI UIF TFDPOE MBX PG UIFSNPEZOBNJDT

26&45*0/

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BU XIBU SBUF JT FOFSHZ USBOTGFSSFE GSPN UIF TMJHIUMZ IPUUFS TPVSDF UP UIF TMJHIUMZ DPPMFS TJOL

26&45*0/

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4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT ZPVOH 'SFODI FOHJOFFS 4BEJ $BSOPU JO  )JT BDIJFWFNFOU JT BMM UIF NPSF SFNBSLBCMF CFDBVTF JU DBNF CFGPSF UIF QSJODJQMF PG FOFSHZ DPOTFSWBUJPO XBT FTUBCMJTIFE "GUFS VTJOH UIF TFDPOE MBX PG UIFSNPEZOBNJDT UP EFUFSNJOF UIF NBYJNVN QPTTJCMF FGàDJFODZ PG JOàOJUFMZ TMPX IFBU FOHJOFT JO UIF MBTU QBSU PG UIF TVQQMFNFOU XF XJMM BOBMZ[F FOHJOFT UIBU SVO BU VTFGVM TQFFET 8F XJMM àOE BT FYQFDUFE UIBU UIFZ BSF FWFO MFTT FGàDJFOU UIBO UIF JOàOJUFMZ TMPX FOHJOFT

" $ZDMJD 1SPDFTT PG B 3FWFSTJCMF &OHJOF M

Gas isothermally expands

Step 1: Expand from V1 to V2 T2 = T1 TH

QH

Source at TH

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4 " .BYJNBMMZ &GàDJFOU 1SPDFTT

4

&OUSPQZ BOE &OFSHZ JO -JGUJOH UIF 1JTUPO 3FNFNCFS UIBU UIF FOUSPQZ DIBOHF PG B TZTUFN XIFO UIFSF JT FOFSHZ USBOTGFS 2 JT Δ4 = 2/5 *O MJGUJOH UIF QJTUPO UIF FOUSPQZ PG UIF IJHIUFNQFSBUVSF TPVSDF IBT DIBOHFE CZ BO BNPVOU Δ4) = −2) /5) XIFSF 2) JT B QPTJUJWF RVBOUJUZ 5IF FOUSPQZ PG UIF HBT IBT DIBOHFE CZ BO BNPVOU Δ4HBT = +2) /5)  5IF FOUSPQZ PG UIF 6OJWFSTF IBTOU DIBOHFE BU BMM 5IF JODSFBTFE FOUSPQZ PG UIF HBT JT BTTPDJBUFE XJUI UIF GBDU UIBU UIFSF BSF NPSF XBZT UP BSSBOHF UIF NPMFDVMFT JO B MBSHF WPMVNF UIBO JO B TNBMM WPMVNF

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5IF /FFE GPS B $ZDMF

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Gas adiabatically expands

Step 2: Expand from V2 to V3 T3 TL

5IF QSPCMFN JT UIJT 8F OFFE UP CF BCMF UP EP UIJT BHBJO BOE BHBJO BOE BHBJO 'PS FYBNQMF XF XBOU UP LFFQ QVNQJOH XBUFS PVU PG UIF EFFQ NJOF PWFS BOE PWFS )PXFWFS UP SFQFBU UIF MJGUJOH QSPDFTT XF IBWF UP SFUVSO UP UIF PSJHJOBM TUBUF XJUI UIF HBT DPNQSFTTFE 8F DPVME TJNQMZ SFWFSTF UIF QSPDFTT EPJOH XPSL PO UIF HBT DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO

XJUI FOFSHZ USBOTGFS GSPN UIF HBT JOUP UIF IJHIUFNQFSBUVSF TPVSDF 5IF OFU FGGFDU UIPVHI XPVME CF UIBU UIFSF XBT [FSP OFU FOFSHZ USBOTGFS UP UIF HBT BOE [FSP OFU XPSL EPOF PO UIF TVSSPVOEJOHT 8F OFFE UP SVO PVS IFBU FOHJOF JO B OPOUSJWJBM SFQFUJUJWF DZDMF XIFSF XF DBO MJGU UIF QJTUPO SFQFBUFEMZ &BDI UJNF XF MJGU UIF QJTUPO BOE EP VTFGVM XPSL XF OFFE UP HFU UIF QJTUPO CBDL EPXO BHBJO XJUIPVU IBWJOH UP EP UIF TBNF BNPVOU PG XPSL UP QVTI JU EPXO *G XF DBO HFU UIF HBT CBDL UP JUT PSJHJOBM TUBUF XJUI OFU XPSL EPOF PO UIF TVSSPVOEJOHT XF XJMM IBWF B VTFGVM EFWJDF 0OF QPTTJCJMJUZ GPS CSJOHJOH UIF QJTUPO EPXO XPVME CF UP MFU UIF HBT DPPM EPXO )PXFWFS UP MPXFS UIF UFNQFSBUVSF XF XPVME IBWF UP QMBDF UIF HBT DZMJOEFS JO DPOUBDU XJUI B MBSHF PCKFDU BU TPNF MPX UFNQFSBUVSF 5- DBMMFE UIF iTJOLu CFDBVTF BT XFMM TFF XF XJMM EVNQ TPNF FOFSHZ JOUP JU 8F DBOU QMBDF UIF HBT DZMJOEFS JO DPOUBDU XJUI UIF TJOL JNNFEJBUFMZ CFDBVTF UIF HBT JT BU B IJHI UFNQFSBUVSF 5) BOE QMBDJOH UIF IPU HBT JO DPOUBDU XJUI UIF DPME TJOL XPVME NFBO UIBU UIFSF XPVME CF B MBSHF UFNQFSBUVSF EJGGFSFODF BOE UIFSF XPVME CF B MBSHF JODSFBTF JO UIF FOUSPQZ PG UIF 6OJWFSTF BTTPDJBUFE XJUI UIF FOFSHZ USBOTGFS GSPN UIF IPU HBT UP UIF DPME TJOL

" /P2 "EJBCBUJD &YQBOTJPO 'JHVSF 4 /P2 FYQBOTJPO UIF HBT UFNQFSBUVSF ESPQT UP 5- 

5P BWPJE UIJT MBSHF QSPEVDUJPO PG FOUSPQZ XF OFFE UP CSJOH UIF UFNQFSBUVSF PG UIF HBT EPXO BMNPTU UP 5- CFGPSF NBLJOH DPOUBDU XJUI UIF TJOL 5P EP UIJT XF EJTDPOOFDU UIF HBT DZMJOEFS GSPN UIF IJHIUFNQFSBUVSF TPVSDF BOE QFSGPSN B SFWFSTJCMF OP2 BEJBCBUJD FYQBOTJPO XIJDI EPFT TPNF NPSF XPSL PO UIF TVSSPVOEJOHT BOE MPXFST UIF UFNQFSBUVSF PG UIF FOHJOF 'JHVSF 4  5IJT JT BDDPNQMJTIFE CZ TMPXMZ SFNPWJOH TPNF XFJHIU GSPN UIF QJTUPO BMMPXJOH UIF HBT UP FYQBOE BOE UIF UFNQFSBUVSF UP GBMM 8F TUPQ UIF FYQBOTJPO XIFO UIF UFNQFSBUVSF PG UIF HBT JT KVTU TMJHIUMZ IJHIFS UIBO UIF MPX UFNQFSBUVSF 5- PG UIF TJOL

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT Step 3: Compress from V3 to V4 T4 = T3 TL

M Gas isothermally compresses

QL

Sink at TL

'JHVSF 4 $POTUBOUUFNQFSBUVSF DPNQSFTTJPO UIF HBT UFNQFSBUVSF JT KVTU TMJHIUMZ IJHIFS UIBO 5- 

M

Step 4: Compress from V4 to V1 T1 TH

Gas adiabatically compresses

'JHVSF 4 /P2 DPNQSFTTJPO HBT UFNQFSBUVSF SJTFT UP 5)  5IF FOHJOF JT OPX CBDL UP JUT PSJHJOBM TUBUF SFBEZ UP CFHJO BOPUIFS DZDMF

" $POTUBOU5FNQFSBUVSF *TPUIFSNBM $PNQSFTTJPO 8F DBO OPX TBGFMZ QMBDF UIF HBT DZMJOEFS JO DPOUBDU XJUI UIF MPXUFNQFSBUVSF TJOL 5IJT JT PLBZ BT GBS BT FOUSPQZ QSPEVDUJPO JT DPODFSOFE CFDBVTF XF NBEF TVSF UIBU UIF UFNQFSBUVSF PG UIF HBT JT POMZ WFSZ TMJHIUMZ IJHIFS UIBO UIF TJOL UFNQFSBUVSF 5-  5IF QJTUPO JT OPX RVJUF IJHI BOE XF OFFE UP CSJOH JU EPXO 8F EP UIJT CZ TMPXMZ BEEJOH TPNF XFJHIU UP UIF QJTUPO QFSGPSNJOH B DPOTUBOUUFNQFSBUVSF JTPUIFSNBM DPNQSFTTJPO 'JHVSF 4  5IFSF JT FOFSHZ USBOTGFS JOUP UIF TJOL #FDBVTF UIF UFNQFSBUVSF PG UIF JEFBM HBT JT OFBSMZ 5- BU BMM UJNFT UIFSF JT OP FOFSHZ DIBOHF JO UIF HBT 5IFSFGPSF UIF XPSL UIBU XF EP JT FRVBM UP UIF FOFSHZ USBOTGFS 2- JOUP UIF TJOL 5IFSF JT BO JODSFBTF JO UIF FOUSPQZ PG UIF TJOL Δ4- = +2- /5- BOE UIFSF JT BO FOUSPQZ EFDSFBTF JO UIF FOUSPQZ PG UIF HBT Δ4HBT = −2- /5-  5IF FOUSPQZ PG UIF 6OJWFSTF EPFTOU DIBOHF 5IF EFDSFBTFE FOUSPQZ PG UIF HBT JT BTTPDJBUFE XJUI UIF GBDU UIBU UIFSF BSF GFXFS XBZT UP BSSBOHF UIF NPMFDVMFT JO B TNBMM WPMVNF UIBO JO B MBSHF WPMVNF

" /P2 "EJBCBUJD $PNQSFTTJPO 8F TUPQ UIF DPNQSFTTJPO BU B QBSUJDVMBS TUBUF PG UIF HBT DIPTFO XJUI DBSF TP UIBU XF DBO EP UIF GPMMPXJOH 8F EJTDPOOFDU UIF FOHJOF GSPN UIF MPXUFNQFSBUVSF TJOL BOE QFSGPSN B OP2 DPNQSFTTJPO UIBU SBJTFT UIF UFNQFSBUVSF PG UIF HBT UP B UFNQFSBUVSF KVTU CBSFMZ MFTT UIBO 5) UIF UFNQFSBUVSF PG UIF IJHIUFNQFSBUVSF TPVSDF 8F EP TPNF XPSL UP QFSGPSN UIJT DPNQSFTTJPO 'JHVSF 4  5IJT DPNQSFTTJPO CSJOHT UIF TZTUFN CBDL UP JUT PSJHJOBM TUBUF EFOTJUZ QSFTTVSF UFNQFSBUVSF

TP XF DBO SFQFBU UIF DZDMF PG GPVS QSPDFTTFT BMM PWFS BHBJO 5IF QPTTJCMF VTFGVMOFTT PG UIJT FOHJOF JT UIBU XF DBO SVO JU SFQFBUFEMZ PWFS BOE PWFS 8IBU SFNBJOT UP CF BOBMZ[FE JT IPX NVDI XPSL JT EPOF PO UIF TVSSPVOEJOHT JO POF DZDMF BOE IPX NVDI FOFSHZ JOQVU XF OFFE UP TVQQMZ 0G DPVSTF JG XF àOE UIBU UIF OFU XPSL JT [FSP UIF FOHJOF XPOU CF VTFGVM )PXFWFS FYQMJDJU DBMDVMBUJPOT GPS BO JEFBM HBT TIPX UIBU BU MFBTU JO UIBU DBTF UIFSF JT OFU XPSL EPOF PO UIF TVSSPVOEJOHT 8F XJMM àOE UIBU UIJT JT USVF GPS BOZ SFWFSTJCMF FOHJOF SVO JO TVDI B DZDMF 5IJT SFWFSTJCMF DZDMF PG UXP DPOTUBOUUFNQFSBUVSF JTPUIFSNBM QSPDFTTFT BOE UXP OP2 BEJBCBUJD QSPDFTTFT SVOOJOH CFUXFFO B IJHIUFNQFSBUVSF TPVSDF BOE B MPXUFNQFSBUVSF TJOL JT DBMMFE B i$BSOPU DZDMFu

4VNNBSZ PG UIF $ZDMF 'JHVSFT 4 BOE 4 TVNNBSJ[F UIF GPVS QSPDFTTFT PG UIF FOHJOF DZDMF M P Step 1: T2 = T1 TH

Step 1: Expansion at constant TH Step 4: Adiabatic compression

M

V2

Step 2: T3 TL M

M

Step 2: Adiabatic expansion V1

Step 3: T4 = T3 TL

Step 4: T1 TH

V3

M Step 3: Compression at constant TL V1 V4

V2

V3

'JHVSF 4 1 WT 7 GPS UIF FOHJOF DZDMF 5IF TIBEFE BSFB SFQSFTFOUT UIF OFU XPSL EPOF CZ UIF FOHJOF PO UIF TVSSPVOEJOHT

QH Source at TH

QL V4

'JHVSF 4 5IF IFBU FOHJOF DZDMF

Sink at TL

4 " .BYJNBMMZ &GàDJFOU 1SPDFTT

4

4UFQ  *TPUIFSNBM FYQBOTJPO JO DPOUBDU XJUI 5) 4UFQ  "EJBCBUJD FYQBOTJPO OP DPOUBDU  UFNQFSBUVSF GBMMT UP 54UFQ  *TPUIFSNBM DPNQSFTTJPO JO DPOUBDU XJUI 54UFQ  "EJBCBUJD DPNQSFTTJPO OP DPOUBDU  UFNQFSBUVSF SJTFT UP 5) 5IF GPVS QSPDFTTFT SFUVSO UIF HBT UP JUT PSJHJOBM UFNQFSBUVSF BOE WPMVNF 5IF RVFTUJPO JT EJE XF HFU BOZ OFU XPSL PVU PG UIJT DZDMF 'JHVSF 4 TIPXT UIF QSFTTVSF WT WPMVNF EVSJOH UIF DZDMF &BSMJFS JO UIJT TVQQMFNFOU XF TBX UIBU 7 UIF XPSL EPOF PO UIF HBT JT − 7  1E7 TP UIF XPSL EPOF CZ UIF HBT PO UIF  7 TVSSPVOEJOHT JT + 7 1E7 XIJDI JT UIF BSFB VOEFS POF PG UIFTF DVSWFT #FDBVTF  7 JODSFBTFT EVSJOH TUFQT  BOE  UIF XPSL EPOF PO UIF TVSSPVOEJOHT JT QPTJUJWF CVU UIJT XPSL JT OFHBUJWF JO TUFQT  BOE  CFDBVTF 7 EFDSFBTFT 5IF OFU XPSL EPOF PO UIF TVSSPVOEJOHT JT UIFSFGPSF SFQSFTFOUFE CZ UIF TIBEFE BSFB TIPXO JO 'JHVSF 4 5IFSF JT OFU FOFSHZ PVUQVU . SFMBUFE UP UIF GBDU UIBU UIF 17 DVSWFT GPS UIF BEJBCBUJD QSPDFTTFT TIPXO JO CMVF BOE HJWFO CZ 17γ  DPOTUBOU

BSF TUFFQFS UIBO UIF 17 DVSWFT GPS UIF JTPUIFSNBM QSPDFTTFT TIPXO JO SFE BOE HJWFO CZ 17  DPOTUBOU 

&OUSPQZ JO B $ZDMF PG B 3FWFSTJCMF &OHJOF -FU 2) CF UIF BCTPMVUF WBMVF PG UIF FOFSHZ FYUSBDUFE GSPN UIF TPVSDF BU UFNQFSBUVSF 5) EVSJOH UIF DPOTUBOUUFNQFSBUVSF FYQBOTJPO BOE MFU 2- CF UIF BCTPMVUF WBMVF PG UIF FOFSHZ EVNQFE JOUP UIF TJOL BU UFNQFSBUVSF 5- EVSJOH UIF DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO 3FNFNCFS UIBU UIF FOUSPQZ DIBOHF PG B TZTUFN XIFO UIFSF JT FOFSHZ USBOTGFS 2 JT Δ4 = 2/5 2) 5) 25IF FOUSPQZ DIBOHF PG UIF MPXUFNQFSBUVSF TJOL JO POF DZDMF = + 5-

5IF FOUSPQZ DIBOHF PG UIF IJHIUFNQFSBUVSF TPVSDF JO POF DZDMF = −

26&45*0/

*T UIFSF BOZ FOUSPQZ DIBOHF JO UIF FOHJOF JO POF DZDMF

5IF FOUSPQZ DIBOHF PG UIF FOHJOF JO POF DZDMF JT [FSP CFDBVTF UIF HBT JT CSPVHIU CBDL UP JUT PSJHJOBM TUBUF 5IFSFGPSF JO POF DZDMF IPX NVDI FOUSPQZ DIBOHF JT UIFSF JO UIF TVSSPVOEJOHT BT B SFTVMU PG UIF XPSL EPOF PO BOE CZ UIF TVSSPVOEJOHT

26&45*0/

0OMZ UIF FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF BGGFDUT UIF FOUSPQZ EVSJOH POF DZDMF XPSL EPFTOU DPOUSJCVUF

BOE XF DBMDVMBUFE UIFTF UISPVHI UIF FRVBUJPO Δ4 = 2/5 8F XFSF DBSFGVM UP SVO UIF FOHJOF JO B SFWFSTJCMF XBZ BWPJEJOH NBLJOH BOZ JODSFBTF JO UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF TP JO POF DZDMF XF IBWF 2 ) 2− + + =  5) 55IFSFGPSF XF DBO XSJUF UIJT JNQPSUBOU SFMBUJPOTIJQ GPS POF DZDMF PG UIF FOHJOF 0/& $:$-& 0' " 3&7&34*#-& &/(*/& 22) = 5) 55IJT JT B TVSQSJTJOHMZ TJNQMF SFTVMU GPS TVDI B DPNQMJDBUFE QSPDFTT 8IBU EPFT UIJT TJNQMF SFTVMU EFQFOE PO *U EFQFOET TPMFMZ PO UIF TFDPOE MBX PG UIFSNPEZOBNJDT UIBU TUBUJTUJDBMMZ UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF XJMM FTTFOUJBMMZ OFWFS EFDSFBTF 8F JMMVTUSBUFE UIF QSPDFTTFT JO UIF DZDMF XJUI B DZMJOEFS àMMFE XJUI BO JEFBM HBT CVU UIF SFTVMU BCPWF EPFTOU BDUVBMMZ EFQFOE BU BMM PO UIF EFUBJMT PG IPX

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT UIF FOHJOF JT DPOTUSVDUFE *O QBSUJDVMBS JU EPFTOU NBUUFS XIFUIFS UIF FOHJOF DPOUBJOT IFMJVN HBT PS B TPMJE CMPDL PG SVCCFS PS B MJUFS PG MJRVJE BMDPIPM 5IFSF XPVME CF BO FBTJMZ WJTJCMF EJGGFSFODF CFUXFFO VTJOH B TPMJE PS MJRVJE SBUIFS UIBO B HBT JO UIF FOHJOF CFDBVTF UIF EJTUBODF UISPVHI XIJDI UIF QJTUPO XPVME NPWF XPVME CF NVDI TNBMMFS UIBO XJUI B HBT )PXFWFS UIF TJNQMF SFTVMU TIPXO BCPWF XPVME SFNBJO UIF TBNF

&OFSHZ JO B $ZDMF PG B 3FWFSTJCMF &OHJOF

Source at TH

-FUT SFWJFX UIF FOFSHZ DIBOHFT JO UIF FOHJOF JO POF DZDMF 5IF TPVSDF JOQVUT BO BNPVOU PG FOFSHZ 2)  5IF TJOL SFNPWFT BO BNPVOU PG FOFSHZ 2-  5IF XPSLJOH TVCTUBODF SFUVSOT UP JUT PSJHJOBM TUBUF TP JU VOEFSHPFT OP DIBOHF PG FOFSHZ *T UIFSF BOZUIJOH FMTF :FT JU NBZ CF UIBU UIFSF IBT CFFO OFU XPSL EPOF PO UIF TVSSPVOEJOHT )PX DBO XF UFMM 8F DBO VTF UIF GBDU UIBU UIF JOUFSOBM FOFSHZ PG UIF FOHJOF EPFT OPU DIBOHF JO POF DPNQMFUF DZDMF Δ&FOHJOF = 2) − 2- + 8 = 

QH

6TF UIF SFTVMU UIBU 2) /5) = 2- /5- CFDBVTF UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF EPFTOU DIBOHF

UP EFUFSNJOF XIFUIFS 8 JT QPTJUJWF PS OFHBUJWF

26&45*0/

Engine QL

M = QH – QL

Sink at TL

'JHVSF 4 $POTFSWBUJPO PG FOFSHZ MFUT VT EFUFSNJOF UIF OFU NFDIBOJDBM FOFSHZ PVUQVU GSPN UIF FOHJOF UP UIF TVSSPVOEJOHT

#FDBVTF 2) = 5) /5- 2- > 2- XF IBWF 8 = 2- −2) <  5IF GBDU UIBU 8 JT OFHBUJWF NFBOT UIBU PVS IFBU FOHJOF EPFT OFU XPSL PO UIF TVSSPVOEJOHT JO POF DZDMF 'JHVSF 4 PVUMJOFT UIF CBTJD TDIFNF *O POF DZDMF UIF OFU FGGFDU JT UIBU UIF FOHJOF UBLFT JO FOFSHZ 2) GSPN B IPU TPVSDF EPFT TPNF XPSL PO TPNFUIJOH GPS FYBNQMF UVSOT BO FMFDUSJD HFOFSBUPS

BOE FYIBVTUT UIF SFNBJOJOH FOFSHZ 2- JOUP B DPME TJOL 5IF NBHOJUVEF PG UIF XPSL EPOF PO UIF TVSSPVOEJOHT JT . = |8| = 2) − 2-  5IF DSVDJBM JTTVF JT UIBU UIF IFBU FOHJOF XJMM OPU SVO JO SFQFBUBCMF DZDMFT XJUIPVU FYIBVTUJOH TPNF FOFSHZ UP UIF MPXUFNQFSBUVSF TJOL :PV DBOOPU DPOWFSU BMM PG UIF FOFSHZ 2) JOUP VTFGVM FOFSHZ PVUQVU B QPSUJPO PG 2) HPFT JOUP UIF MPXUFNQFSBUVSF TJOL 2-  8F OFFE UIF MPXUFNQFSBUVSF CMPDL UP BMMPX VT UP CFHJO UIF DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO QIBTF PG UIF DZDMF JO B XBZ UIBU BWPJET BOZ FOFSHZ USBOTGFS XJUI TJHOJàDBOU UFNQFSBUVSF EJGGFSFODF XIJDI XPVME JODSFBTF UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF 8F IBWF UP QBZ GPS UIF IJHIUFNQFSBUVSF FOFSHZ 2) UIBU XF VTF BOE UIF MPTT PG TPNF PG UIF JOQVU FOFSHZ JO UIF GPSN PG 2- JT BO VOGPSUVOBUF GBDU PG MJGF 8F QBZ GPS DPBM PS GVFM PJM PS FMFDUSJDJUZ UP XBSN TPNFUIJOH VQ UP UIF IJHI UFNQFSBUVSF 5) GSPN XIJDI XF DBO ESBX FOFSHZ UP EP XPSL GPS VT 'PS FYBNQMF JO BO PME SBJMSPBE TUFBN FOHJOF B àSF NBJOUBJOFE XBUFS BU B IJHI UFNQFSBUVSF UP DPOTUJUVUF UIF IJHIUFNQFSBUVSF TPVSDF *O BO BVUPNPCJMF FOHJOF XF CVSO HBTPMJOF UP DSFBUF B IJHI UFNQFSBUVSF BOE QVTI UIF QJTUPOT *O BO FMFDUSJDJUZ HFOFSBUJOH TUBUJPO XF CVSO DPBM PS GVFM PJM PS VTF OVDMFBS àTTJPO SFBDUJPOT UP DSFBUF B IJHIUFNQFSBUVSF TPVSDF GSPN XIJDI UP ESJWF UIF HFOFSBUPST 8FE MJLF UP HFU PVS GVMM NPOFZT XPSUI 2)

CVU XF EPOU 8F POMZ HFU . = |8| = 2) − 2- IBWJOH iMPTUu TPNF PG UIF FOFSHZ UP B MPXUFNQFSBUVSF TJOL XIJDI UZQJDBMMZ JT PVS PSEJOBSZ TVSSPVOEJOHT BU BSPVOE  ◦ $  , 

&GàDJFODZ 5IJT MFBET UP UIF RVFTUJPO XIBU GSBDUJPO PG UIF FOFSHZ 2) UIBU XF QBZ GPS UVSOT JOUP VTFGVM FOFSHZ PVUQVU . 5IJT GSBDUJPO JT DBMMFE UIF iFGàDJFODZu PG UIF IFBU FOHJOF . FGàDJFODZ ≡ 2) 26&45*0/

SFTVMU

(JWFO UIJT EFàOJUJPO PG FGàDJFODZ QSPWF UIF GPMMPXJOH

4 " .BYJNBMMZ &GàDJFOU 1SPDFTT

4

5)& &''*$*&/$: 0' " 3&7&34*#-& &/(*/& &GàDJFODZ =  −

55)

8F IBWF ./2) = 2) − 2- /2) =  − 2- /2) =  − 5- /5)  5IJT NBZ TFFN TVSQSJTJOH 5IF FGàDJFODZ PG B SFWFSTJCMF FOHJOF EFQFOET TPMFMZ PO UIF SBUJP PG UIF BCTPMVUF UFNQFSBUVSFT PG UIF TPVSDF BOE TJOL 8F FNQIBTJ[F UIBU JU EPFTOU NBUUFS IPX UIF FOHJOF JT EFTJHOFE PS XIBU LJOE PG NBUFSJBMT JU JT NBEF PG 5IJT JT UIF IJHIFTU FGàDJFODZ XF DBO FWFS BDIJFWF GPS DPOWFSUJOH UIFSNBM FOFSHZ JOUP VTFGVM XPSL JO B IFBU FOHJOF $IFDLQPJOU  "O BDUVBM FMFDUSJDJUZ HFOFSBUJOH QMBOU JT QPXFSFE CZ B OVDMFBS SFBDUPS XJUI B IJHI UFNQFSBUVSF PG  ◦ $ BOE MPX UFNQFSBUVSF PG  ◦ $ OFBS SPPN UFNQFSBUVSF  8IBU XPVME CF UIF FGàDJFODZ JG XF DBO USFBU UIF QSPDFTTFT BT CFJOH SFWFSTJCMF

/P 0UIFS &OHJOF $BO #F .PSF &GàDJFOU 8F DBO TIPX UIBU OP IFBU FOHJOF SVOOJOH CFUXFFO UFNQFSBUVSFT 5) BOE 5DBO CF NPSF FGàDJFOU UIBO B SFWFSTJCMF FOHJOF 5IF QSPPG JT B iQSPPG CZ DPOUSBEJDUJPOu 4VQQPTF UIBU BO JOWFOUPS DMBJNT UP IBWF JOWFOUFE TPNF DMFWFSMZ EFTJHOFE OFX LJOE PG DZDMJD FOHJOF XJUI B IJHIFS FGàDJFODZ UIBO  − 5- /5) XIFO SVO CFUXFFO UIFTF TBNF UXP UFNQFSBUVSFT *O UIBU DBTF GPS B HJWFO 2) UIF OFX FOHJOF XPVME FYIBVTU MFTT 2- UIBO JT FYIBVTUFE GSPN B SFWFSTJCMF FOHJOF BOE UIF FOUSPQZ DIBOHF PG UIF 6OJWFSTF XJUI UIJT OFX FOHJOF XPVME CF OFHBUJWF JOTUFBE PG [FSP −

2) 2+ <  *NQPTTJCMF

5) 5-

5IJT XPVME CF B WJPMBUJPO PG UIF TFDPOE MBX PG UIFSNPEZOBNJDT TP JU JT JNQPTTJCMF 8F DBO CF RVJUF TVSF UIBU UIF JOWFOUPST DMBJNT BSF OPU WBMJE /PUF DBSFGVMMZ UIBU UIF JOWFOUPST DMBJNT EPOU WJPMBUF FOFSHZ DPOTFSWBUJPO /PUIJOH BCPVU FOFSHZ DPOTFSWBUJPO GPSCJET DPOWFSUJOH  PG UIF JOQVU FOFSHZ 2) JOUP VTFGVM FOFSHZ PVUQVU . JO B DZDMF 5IF JNQPTTJCJMJUZ MJFT SBUIFS JO UIF NBTTJWF JNQSPCBCJMJUZ PG TFFJOH UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF EFDSFBTF 8F UPPL HSFBU QBJOT UP NBLF B SFWFSTJCMF IFBU FOHJOF GPS XIJDI UIF UPUBM FOUSPQZ DIBOHF PG UIF 6OJWFSTF XBT [FSP /P PUIFS FOHJOF DBO CF NPSF FGàDJFOU *O GBDU BOZ SFBM FOHJOF XJMM IBWF TPNF GSJDUJPO BOE TPNF FOFSHZ USBOTGFS BDSPTT EJGGFSJOH UFNQFSBUVSFT JO XIJDI DBTF UIF FOUSPQZ DIBOHF PG UIF 6OJWFSTF XJMM CF QPTJUJWF 4P UIF TFDPOE MBX PG UIFSNPEZOBNJDT MFBET UP UIF GPMMPXJOH XIFSF i= u BQQMJFT POMZ UP JEFBM SFWFSTJCMF FOHJOFT BOE i> u BQQMJFT UP SFBM FOHJOFT &/5301: $)"/(& 0' 5)& 6/*7&34& '03 3&"- )&"5 &/(*/&4 Δ46OJWFSTF = −

2) 2+ ≥ 5) 5-

8F TIPVME OPU VOEFSFTUJNBUF UIF OFFE GPS UIF JOHFOVJUZ PG JOWFOUPST IPXFWFS 5IF SFWFSTJCMF IFBU FOHJOF NBEF XJUI BO JEFBM HBT DZMJOEFS JT QSFUUZ VTFMFTT GPS QSBDUJDBM QVSQPTFT 5IF DIBMMFOHF UP JOHFOVJUZ JT UP EFTJHO FOHJOFT UIBU BSF QSBDUJDBM BOE UIJT JOWPMWFT NBOZ LJOET PG FOHJOFFSJOH EFTJHO EFDJTJPOT BOE USBEFPGGT )PXFWFS OP NBUUFS IPX JOHFOJPVT UIF EFTJHO UIF TFDPOE MBX PG UIFSNPEZOBNJDT QVUT B SJHJE MJNJUBUJPO PO IPX FGàDJFOU BOZ IFBU FOHJOF DBO CF 5IFSF BSF OPOIFBU FOHJOFT UIBU IBWF IJHI FGàDJFODZ "MNPTU BMM PG UIF FMFDUSJD FOFSHZ áPXJOH JOUP BO FMFDUSJD NPUPS JT DPOWFSUFE JOUP NFDIBOJDBM FOFSHZ 'VFM DFMMT DPOWFSU DIFNJDBM FOFSHZ JOUP NFDIBOJDBM FOFSHZ XJUI IJHI

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT FGàDJFODZ 5IF MJNJUBUJPO XF IBWF GPVOE BQQMJFT POMZ UP IFBU FOHJOFT JO XIJDI B IJHIUFNQFSBUVSF TPVSDF ESJWFT UIF FOHJOF BOE OFDFTTBSJMZ FYIBVTUT FOFSHZ UP B MPXUFNQFSBUVSF TJOL *G Δ46OJWFSTF >  JO POF DZDMF PG B IFBU FOHJOF TIPX UIBU UIF FGàDJFODZ JT MFTT UIBO UIF JEFBM FGàDJFODZ  − 5- /5) PCUBJOFE XJUI B SFWFSTJCMF FOHJOF Δ46OJWFSTF >  22) > 55) 52> 2) 5)     25< 2)  − . = 2) − 2- = 2)  − 2) 5)   5. < − FGàDJFODZ = 2) 5)

26&45*0/

3VOOJOH UIF &OHJOF #BDLXBSE " 3FGSJHFSBUPS PS )FBU 1VNQ

Refrigerator Room at TH

8F DPVME SVO PVS FOHJOF CBDLXBSE TJODF XF UPPL DBSF UP NBLF BMM BTQFDUT PG UIF DZDMF SFWFSTJCMF 4UBSUJOH GSPN PVS PSJHJOBM TUBSUJOH QPJOU XF XPVME EJTDPOOFDU GSPN UIF IPU CMPDL QFSGPSN B OP2 FYQBOTJPO UP MPXFS UIF UFNQFSBUVSF UP UIBU PG UIF DPME CMPDL DPOOFDU UP UIF DPME CMPDL EP B DPOTUBOUUFNQFSBUVSF FYQBOTJPO UIFO EJTDPOOFDU GSPN UIF DPME CMPDL /FYU XF EP B OP2 DPNQSFTTJPO UP SBJTF UIF UFNQFSBUVSF UP UIBU PG UIF IPU CMPDL DPOOFDU UP UIF IPU CMPDL BOE EP B DPOTUBOUUFNQFSBUVSF DPNQSFTTJPO CBDL UP UIF PSJHJOBM TUBUF 5IF OFU FGGFDU JO POF DZDMF JT UIBU UIF MPXUFNQFSBUVSF CMPDL HJWFT TPNF FOFSHZ 2- UP UIF FOHJOF UIF IJHIUFNQFSBUVSF CMPDL BCTPSCT TPNF FOFSHZ 2) GSPN UIF FOHJOF BOE XF EP TPNF QPTJUJWF XPSL PO UIF TZTUFN JOTUFBE PG UIF FOHJOF EPJOH XPSL PO UIF TVSSPVOEJOHT  )FSF JT UIF &OFSHZ 1SJODJQMF XIFO XF SVO UIJT SFWFSTFE FOHJOF

QH = M + QL

Δ&FOHJOF = 2- − 2) + 8 = 

Engine QL

6TF UIF SFTVMU UIBU 2) /5) = 2- /5- XIJDI JT TUJMM WBMJE GPS PVS SFWFSTFE FOHJOF BT ZPV DBO DPOWJODF ZPVSTFMG UP EFUFSNJOF XIFUIFS 8 JT QPTJUJWF PS OFHBUJWF

26&45*0/

M

Leakage QL Food at TL 'JHVSF 4 3VO UIF FOHJOF CBDLXBSE BOE ZPV IBWF B SFGSJHFSBUPS

4JODF 2) = 5) /5- 2- > 2- XF IBWF 8 = 2) − 2- >  5IJT TFFNT BO PEE LJOE PG iFOHJOFu JU EPFTOU EP BOZ XPSL GPS VT‡XF IBWF UP EP XPSL PO JU *T UIBU PG BOZ VTF :FT "U UIF DPTU PG EPJOH TPNF XPSL XF FYUSBDU FOFSHZ GSPN B MPXUFNQFSBUVSF CMPDL BOE FYIBVTU JU JOUP B IJHIUFNQFSBUVSF CMPDL 5IJT JT B SFGSJHFSBUPS 'JHVSF 4  $POTJEFS IPX XF LFFQ GPPE DPME JO BO PSEJOBSZ IPNF SFGSJHFSBUPS " SFWFSTFE FOHJOF NBJOUBJOT UIF JOTJEF PG UIF SFGSJHFSBUPS BU B MPX UFNQFSBUVSF 5- XIJMF UIF SPPN JT BU B IJHIFS UFNQFSBUVSF 5)  "MUIPVHI UIF EPPS BOE XBMMT BSF IFBWJMZ JOTVMBUFE TPNF FOFSHZ EPFT MFBL UISPVHI GSPN UIF SPPN BOE UIJT FOFSHZ 2- NVTU CF SFNPWFE UP LFFQ UIF GPPE BU B DPOTUBOU MPX UFNQFSBUVSF 5-  8F FYIBVTU BO BNPVOU PG FOFSHZ 2) JOUP UIF SPPN BU UFNQFSBUVSF 5)  *U UBLFT BO BNPVOU PG NFDIBOJDBM FOFSHZ JOQVU . UP BDIJFWF UIJT FGGFDU PG NPWJOH FOFSHZ PVU PG UIF DPME SFHJPO BOE JOUP UIF XBSN SFHJPO *O UIF NPTU GBWPSBCMF DBTF OP JODSFBTF JO UPUBM FOUSPQZ XF IBWF UIF GPMMPXJOH UXP DPOEJUJPOT TUFNNJOH GSPN FOUSPQZ BOE FOFSHZ DPOTJEFSBUJPOT UIF IJHIUFNQFSBUVSF FYIBVTU FOFSHZ NVTU FRVBM UIF TVN PG UIF MPXUFNQFSBUVSF FOFSHZ QMVT UIF XPSL XF QVU JO UP ESJWF UIF DZDMF  2) 2= 5) 5-

BOE 2) = 2- + .

4 8IZ %POU 8F "UUBJO UIF 5IFPSFUJDBM &GàDJFODZ

4

)FSF PVS WJFX PG iFGàDJFODZu TIJGUT B CJU 8IBU XF DBSF BCPVU JT IPX NVDI NFDIBOJDBM FOFSHZ JOQVU . XF IBWF UP EP UP SFNPWF BO BNPVOU PG MFBLBHF FOFSHZ 2- GSPN UIF JOTJEF PG UIF SFGSJHFSBUPS 4IPX UIBU UIF FOFSHZ SFNPWBM XF HFU QFS BNPVOU PG XPSL EPOF JT BT GPMMPXT

26&45*0/

2 = > . 5) /5- − 

Heat pump

*TOU UIJT SFTVMU TPNFXIBU TVSQSJTJOH "SF XF HFUUJOH TPNFUIJOH GPS OPUIJOH XIFO 2- JT CJHHFS UIBO UIF XPSL . UIBU XF EP /P UIFSFT OP WJPMBUJPO PG FOFSHZ DPOTFSWBUJPO *UT KVTU UIBU 2- QMVT . FRVBMT UIF PVUQVU FOFSHZ 2) BOE XF POMZ IBWF UP BEE B NPEFTU BNPVOU PG XPSL UP ESJWF UIF FOFSHZ iVQIJMMu

House at TH

Leakage QH

QH = M + Q L Engine QL

M

Ground at TL 'JHVSF 4 3VO UIF FOHJOF CBDLXBSE BOE ZPV IBWF B IFBU QVNQ

)FBU 1VNQ " SFMBUFE EFWJDF JT UIF iIFBU QVNQu VTFE JO TPNF BSFBT UP XBSN IPNFT CZ ESJWJOH MPXUFNQFSBUVSF FOFSHZ iVQIJMMu JOUP B IJHIFSUFNQFSBUVSF IPVTF 'JHVSF 4 TIPXT B IPVTF XIPTF JOUFSJPS JT NBJOUBJOFE BU B UFNQFSBUVSF 5) EFTQJUF DPOUJOVBM MFBLBHF 2) JOUP UIF PVUTJEF BJS &OFSHZ 2- BU MPX UFNQFSBUVSF 5- JT QVNQFE PVU PG UIF HSPVOE JOUP UIF IPVTF CZ UIF BEEJUJPO PG TPNF XPSL . UIBU XF EP JO BO FOHJOF DBMMFE B IFBU QVNQ 5IJT JT TPNFXIBU TJNJMBS UP UIF SFGSJHFSBUPS TJUVBUJPO FYDFQU UIBU OPX XIBU XF DBSF BCPVU JT IPX NVDI IJHIUFNQFSBUVSF FOFSHZ 2) XF HFU QFS BNPVOU PG XPSL 8 EPOF CZ UIF IFBU QVNQ 4IPX UIBU UIJT SBUJP JT HJWFO CZ UIF GPMMPXJOH FRVBUJPO GPS UIF IFBU QVNQ

26&45*0/

2)  = > .  − 5- /5)

"HBJO UIJT GFFMT MJLF XFSF HFUUJOH TPNFUIJOH GPS OPUIJOH TJODF 2) JT HSFBUFS UIBO UIF XPSL 8 UIBU XF EP "GUFS BMM JG ZPV XBSN UIF IPVTF EJSFDUMZ XJUI HBT PS PJM PS FMFDUSJD FOFSHZ ZPV IBWF UP QBZ GPS FWFSZ KPVMF PG MFBLBHF OPU TPNF TNBMM GSBDUJPO PG UIF MFBLBHF 4P XIZ BSFOU IFBU QVNQT NVDI NPSF DPNNPOMZ VTFE UIBO UIFZ BSF #BTJDBMMZ CFDBVTF XFWF CFFO DBMDVMBUJOH UIF CFTU EFBM XF DBO QPTTJCMZ PCUBJO UIF DBTF PG UIF SFWFSTJCMF FOHJOF‡UIBU JT POF XIPTF PQFSBUJPO MFBET UP OP JODSFBTF JO UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF  3FBM FOHJOFT SVOOJOH FJUIFS GPSXBSE PS SFWFSTFE EPOU BUUBJO UIF UIFPSFUJDBM NBYJNVN QFSGPSNBODF BT XF XJMM TFF JO UIF OFYU TFDUJPOT

4 8): %0/5 8& "55"*/ 5)& 5)&03&5*$"- &''*$*&/$: *O $IFDLQPJOU  ZPV DBMDVMBUFE UIF SFWFSTJCMFFOHJOF FGàDJFODZ PG BO BDUVBM OVDMFBSQPXFSFE FMFDUSJDJUZ HFOFSBUJOH QMBOU UP CF  CVU UIF PCTFSWFE FGàDJFODZ PG UIJT QMBOU JT POMZ  3FBM IFBU FOHJOFT EP OPU BUUBJO UIF FGàDJFODZ QSFEJDUFE CZ UIF TFDPOE MBX PG UIFSNPEZOBNJDT GPS JEFBM SFWFSTJCMF FOHJOFT *O GBDU UIF SFBM FGàDJFODZ JT PGUFO POMZ BCPVU IBMG PG UIF UIFPSFUJDBM FGàDJFODZ 5IF SFBTPO GPS MFTT UIBO PQUJNVN QFSGPSNBODF JO SFBM FOHJOFT JT QBSUMZ EVF UP NFDIBOJDBM GSJDUJPO CVU UIJT FGGFDU DBO CF NJOJNJ[FE CZ HPPE EFTJHO BOE QSPQFS MVCSJDBUJPO 5IF NBJO MJNJUBUJPO PO QFSGPSNBODF DPNFT GSPN UIF QSBDUJDBM OFDFTTJUZ PG JODPSQPSBUJOH FOFSHZ USBOTGFST CFUXFFO QBSUT PG UIF TZTUFN UIBU BSF BU TJHOJàDBOUMZ EJGGFSFOU UFNQFSBUVSFT XIJDI MFBET UP TJ[BCMF JODSFBTFT JO UIF UPUBM FOUSPQZ PG UIF 6OJWFSTF BOE UP NVDI SFEVDFE FGàDJFODZ

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT

5IF QSPCMFN JT TQFFE "T XF TBX FBSMJFS JO UIF TVQQMFNFOU UIF SBUF PG FOFSHZ USBOTGFS 2˙ JO KPVMFT QFS TFDPOE CFUXFFO UXP PCKFDUT TVDI BT UIF IPU PS DPME CMPDL BOE UIF FOHJOF BU UFNQFSBUVSFT 5) BOE 5- DPOOFDUFE CZ TPNF DPOEVDUJOH NBUFSJBM PG MFOHUI - BOE DSPTTTFDUJPOBM BSFB " XJUI UIFSNBM DPOEVDUJWJUZ σ JT UIJT 5) − 5-

2˙ = σ" 26&45*0/ 8IBU JT UIF SBUF PG FOFSHZ USBOTGFS JO B SFWFSTJCMF FOHJOF XIFO JO DPOUBDU XJUI UIF IPU CMPDL PS UIF DPME CMPDL "MBT B SFWFSTJCMF IFBU FOHJOF JT UPUBMMZ JNQSBDUJDBM XIFO JU DPNFT UP HFUUJOH BOZUIJOH EPOF JO B àOJUF BNPVOU PG UJNF CFDBVTF UIF SBUF PG FOFSHZ USBOTGFS JT [FSP *G BO FOHJOF JT QFSGFDUMZ SFWFSTJCMF B DZDMF UBLFT BO JOàOJUF BNPVOU PG UJNF $POTJEFS UIF QPSUJPO PG B IFBU FOHJOFT DZDMF XIFSF UIF XPSLJOH TVCTUBODF JT JO UIFSNBM DPOUBDU XJUI UIF IJHIUFNQFSBUVSF TPVSDF *O PSEFS UP DBSSZ PVU UIF FYQBOTJPO RVJDLMZ UIFSF NVTU CF B IJHI SBUF PG FOFSHZ USBOTGFS GSPN UIF TPVSDF JOUP UIF XPSLJOH TVCTUBODF 5IBU NFBOT XF OFFE B MBSHF DPOUBDU BSFB " B TIPSU EJTUBODF - BOE B IJHI UIFSNBM DPOEVDUJWJUZ σ .PTU TJHOJàDBOU PG BMM UIF UFNQFSBUVSF PG UIF TPVSDF 5) NVTU CF DPOTJEFSBCMZ IJHIFS UIBO UIF UFNQFSBUVSF PG UIF FOHJOF MFBEJOH UP JSSFWFSTJCJMJUZ BOE JODSFBTF PG UIF FOUSPQZ PG UIF 6OJWFSTF

5IF &GàDJFODZ PG B /POSFWFSTJCMF &OHJOF Source at TH

. QH = b(TH – T1) T1 Intermediate temperatures

Engine . . . M = QH – QL

T2

. QL = b(T1 – TL)

Sink at TL 'JHVSF 4 " IFBU FOHJOF UIBU SVOT BU B OPO[FSP SBUF

4VQQPTF UIBU XIFO UIF FOHJOF JT JO DPOUBDU XJUI UIF IJHIUFNQFSBUVSF CMPDL BU UFNQFSBUVSF 5) UIF FOHJOF JT BU B MPXFS UFNQFSBUVSF 5 TP UIBU UIFSF JT B àOJUF FOFSHZ USBOTGFS SBUF JOUP UIF FOHJOF PG 2˙ ) = C 5) − 5 KPVMFT QFS TFDPOE XIFSF C JT B DPOTUBOU UIBU MVNQT UPHFUIFS UIF GBDUPST PG UIFSNBM DPOEVDUJWJUZ DSPTTTFDUJPOBM BSFB BOE MFOHUI σ"/-  4JNJMBSMZ BTTVNF UIBU XIFO UIF FOHJOF JT JO DPOUBDU XJUI UIF MPXUFNQFSBUVSF CMPDL BU UFNQFSBUVSF 5- UIF FOHJOF JT BU B IJHIFS UFNQFSBUVSF 5 TP UIBU UIFSF JT B àOJUF FOFSHZ USBOTGFS SBUF JOUP UIF XPSLJOH TVCTUBODF PG 2˙ - = C 5 −5- KPVMFT QFS TFDPOE /PX UIF FOFSHZ áPX EJBHSBN MPPLT MJLF 'JHVSF 4 XJUI BO JEFBM SFWFSTJCMF FOHJOF SVOOJOH CFUXFFO UIF UFNQFSBUVSFT 5 BOE 5 CVU XJUI JSSFWFSTJCMF àOJUFSBUF FOFSHZ USBOTGFS CFUXFFO UIJT SFWFSTJCMF FOHJOF BOE UIF TPVSDF BOE TJOL BU UFNQFSBUVSFT 5) BOE 5-  &OFSHZ FOUFST UIF FOHJOF GSPN UIF IJHIUFNQFSBUVSF TPVSDF BU B SBUF 2˙ ) ˙ BOE FOFSHZ JT EVNQFE JOUP UIF XPSL JT EPOF PO UIF TVSSPVOEJOHT BU B SBUF . MPXUFNQFSBUVSF TJOL BU B SBUF 2˙ -  "MM PG UIFTF RVBOUJUJFT BSF NFBTVSFE JO XBUUT KPVMFT QFS TFDPOE  'SPN FOFSHZ DPOTFSWBUJPO XF IBWF ˙ + 2˙ 2˙ ) = . 5IJT FOHJOF XJMM OFDFTTBSJMZ CF MFTT FGàDJFOU UIBO B SFWFSTJCMF FOHJOF EVF UP UIF JSSFWFSTJCMF FOFSHZ USBOTGFST BT DBO CF TFFO CZ OPUJOH UIBU UIF FOHJOF JT FGGFDUJWFMZ PQFSBUJOH CFUXFFO B IJHI UFNQFSBUVSF UIBU JT MPXFS UIBO 5) BOE B MPX UFNQFSBUVSF UIBU JT IJHIFS UIBO 5- 

3FGSJHFSBUPST BOE )FBU 1VNQT 4JNJMBS JTTVFT PG FGàDJFODZ BQQMZ UP SFGSJHFSBUPST BOE IFBU QVNQT &OFSHZ MFBLT JOUP B SFGSJHFSBUPS BU TPNF SBUF KPVMFT QFS TFDPOE BOE NVTU CF SFNPWFE CZ iQVTIJOH JU UIFSNBMMZ VQIJMMu UP B IJHIFS UFNQFSBUVSF UIF UFNQFSBUVSF PG UIF SPPN )PXFWFS JO PSEFS UP IBWF B OPO[FSP SBUF PG FOFSHZ áPX GSPN UIF GPPE UP UIF FOHJOF UIF FOHJOF NVTU SFBDI BO FWFO MPXFS UFNQFSBUVSF UIBO UIF GPPE XJUI FOUSPQZQSPEVDJOH FOFSHZ USBOTGFS GSPN UIF GPPE JOUP UIF FOHJOF "MTP

4 "QQMJDBUJPO " 3BOEPN 8BML

Refrigerator Heat exchanger at TH > Troom Troom . . . QH = M + Q L . Leakage QL

Engine . M

. QL

Tfood TL < Tfood 'JHVSF 4 " SFGSJHFSBUPS XJUI OPO[FSP DPPMJOH SBUF

Heat pump Radiator at TH > Troom Troom

Engine . QL

JO PSEFS GPS UIFSF UP CF B OPO[FSP SBUF PG FOFSHZ áPX GSPN UIF FOHJOF UP UIF SPPN BJS UIF FOHJOF NVTU SFBDI BO FWFO IJHIFS UFNQFSBUVSF UIBO UIF BJS XJUI FOUSPQZQSPEVDJOH FOFSHZ USBOTGFS GSPN UIF FOHJOF JOUP UIF BJS 4DIFNBUJDBMMZ UIF TJUVBUJPO MPPLT MJLF 'JHVSF 4 0GUFO B SFGSJHFSBUPS PS GSFF[FS IBT FYQPTFE DPJMT B iIFBU FYDIBOHFSu

XIFSF UIF FOFSHZ USBOTGFS PDDVST CFUXFFO UIF XPSLJOH TVCTUBODF DJSDVMBUJOH JO UIF DPJMT BOE UIF BJS *G ZPV UPVDI UIF DPJMT ZPV àOE UIBU UIFZ BSF JOEFFE NVDI IPUUFS UIBO SPPN UFNQFSBUVSF JO PSEFS UP ESJWF B TVGàDJFOUMZ IJHI SBUF PG FOFSHZ USBOTGFS JOUP UIF BJS /PUJDF OPU POMZ UIBU UIF IFBU FYDIBOHFS NVTU CF IPUUFS UIBO UIF BJS CVU BMTP UIBU JU JT B SBUIFS MBSHF BOE DPTUMZ EFWJDF CFDBVTF JU IBT UP IBWF B MBSHF TVSGBDF BSFB UP HFU B MBSHF SBUF PG FOFSHZ USBOTGFS *O UIF DBTF PG IFBU QVNQT XIFSF UIF IFBU QVNQ QJDLT VQ MPXUFNQFSBUVSF FOFSHZ VOEFSHSPVOE UIF IFBU QVNQ NVTU SFBDI BO FWFO MPXFS UFNQFSBUVSF JO PSEFS UP HFU B OPO[FSP SBUF PG FOFSHZ áPX GSPN UIF HSPVOE JOUP UIF FOHJOF "MTP JO PSEFS GPS UIFSF UP CF B OPO[FSP SBUF PG FOFSHZ áPX JOUP UIF IPVTF UIF FOHJOF NVTU SFBDI BO FWFO IJHIFS UFNQFSBUVSF UIBO UIF BJS XJUI FOUSPQZQSPEVDJOH FOFSHZ USBOTGFS GSPN UIF FOHJOF JOUP UIF BJS 5IF SBEJBUPST IFBU FYDIBOHFST JO UIF IPVTF NVTU CF DPOTJEFSBCMZ IPUUFS UIBO UIF BJS 5IFSF JT DPOTJEFSBCMF FYQFOTF JO BMM UIF NFUBM JO UIF HSPVOE BOE JO UIF IPVTF UIBU FOBCMFT BEFRVBUF FOFSHZ USBOTGFS SBUF 4DIFNBUJDBMMZ UIF TJUVBUJPO MPPLT MJLF 'JHVSF 4 4P BMUIPVHI B IFBU QVNQ EPFT IBWF B UIFPSFUJDBM BEWBOUBHF JO XBSNJOH B IPVTF JO QBSU GSPN FOFSHZ JO UIF DPME HSPVOE UIF FYQFOTF PG UIF IFBU FYDIBOHFST BOE UIF QSPCMFNT PG HPJOH UP SBUIFS MPX UFNQFSBUVSFT JO UIF HSPVOE BSF QSBDUJDBM MJNJUBUJPOT FTQFDJBMMZ JO WFSZ DPME DMJNBUFT )FBU QVNQT BSF NPSF VTFGVM JO DMJNBUFT XIFSF UIF XJOUFST BSF OPU UPP TFWFSF

4 "11-*$"5*0/ " 3"/%0. 8"-,

. . . QH = M + QL . Leakage QH

4

. M

Tground TL < Tground 'JHVSF 4 " IFBU QVNQ XJUI OPO[FSP FOFSHZ USBOTGFS SBUF EVF UP B UFNQFSBUVSF EJGGFSFODF

*G ZPV XPVME àOE JU VTFGVM UP TUVEZ POF NPSF BQQMJDBUJPO PG UIF CBTJD TUBUJTUJDBM DPODFQUT VOEFSMZJOH PVS BOBMZTJT PG B HBT IFSF JT BO JOUFSFTUJOH POF *G XF DPVME XBUDI POF TQFDJBM NPMFDVMF XBOEFSJOH BSPVOE JO UIF BJS DPMMJEJOH GSFRVFOUMZ XJUI BJS NPMFDVMFT JUT QBUI XPVME MPPL TPNFUIJOH MJLF 'JHVSF 4 5IJT LJOE PG NPUJPO JT DBMMFE B iSBOEPN XBMLu *U JT TPNFXIBU TVSQSJTJOH UP àOE UIBU EFTQJUF UIF SBOEPN OBUVSF PG UIJT NPUJPO XF DBO DBMDVMBUF TPNFUIJOH TJHOJàDBOU BCPVU UIF NPUJPO VTJOH TJNQMF TUBUJTUJDBM SFBTPOJOH

0OF %JNFOTJPO 'PS TJNQMJDJUZ àSTU XFMM DPOTJEFS KVTU UIF Y DPNQPOFOU PG UIF NPUJPO 1JDL UIF PSJHJO PG UIF Y BYJT UP CF BU UIF PSJHJOBM QPTJUJPO PG UIF TQFDJBM NPMFDVMF 5IF àSTU UIJOH UIBU IBQQFOT JT UIBU UIF TQFDJBM NPMFDVMF NPWFT VOUJM JU DPMMJEFT XJUI BO BJS NPMFDVMF 8F DBMM UIJT àSTU Y DPNQPOFOU PG UIF EJTQMBDFNFOU ΔY  5IJT DPNQPOFOU PG UIF EJTQMBDFNFOU NBZ CF UP UIF SJHIU +Y EJSFDUJPO PS UP UIF MFGU −Y EJSFDUJPO  "T B SFTVMU PG UIF DPMMJTJPO UIF TQFDJBM NPMFDVMF NBZ DIBOHF EJSFDUJPO BOE DIBOHF TQFFE 5IF OFYU EJTQMBDFNFOU CFGPSF BOPUIFS DPMMJTJPO XF DBMM ΔY BOE TP PO "GUFS / DPMMJTJPOT UIF OFU EJTQMBDFNFOU ΔY BXBZ GSPN UIF PSJHJO JT ΔY = ΔY + ΔY + ΔY + ΔY + · · · + ΔY/

'JHVSF 4 " % SBOEPN XBML HFOFSBUFE CZ B DPNQVUFS QSPHSBN VTJOH SBOEPN OVNCFST BOE NBOZ TUFQT

"GUFS / DPMMJTJPOT XIBU JT UIF BWFSBHF NPTU QSPCBCMF QPTJUJPO PG UIF TQFDJBM NPMFDVMF 0O BWFSBHF JU JT KVTU BT MJLFMZ UIBU JU IBT NPWFE UP UIF SJHIU BT NPWFE UP UIF MFGU TP UIF BWFSBHF Y DPNQPOFOU PG EJTQMBDFNFOU PVHIU UP CF [FSP *U JT FBTZ UP TFF UIBU UIJT XJMM CF UIF DBTF CZ UBLJOH UIF BWFSBHF WBMVF ΔY = ΔY + ΔY + ΔY + · · · + ΔY/

4

4VQQMFNFOU 4 (BTFT BOE )FBU &OHJOFT

&BDI PG UIF JOEJWJEVBM Y EJTQMBDFNFOUT BSF FRVBMMZ MJLFMZ UP CF UP UIF MFGU PS UIF SJHIU TP UIF BWFSBHF WBMVF PG UIF OUI Y EJTQMBDFNFOU GPS O =    . . . /

JT [FSP 5IFSFGPSF UIF BWFSBHF WBMVF PG UIF OFU Y DPNQPOFOU PG EJTQMBDFNFOU JT BMTP [FSP 0O BWFSBHF BGUFS / DPMMJTJPOT UIF TQFDJBM NPMFDVMF FOET VQ UP UIF MFGU PG UIF PSJHJO BT PGUFO BT JU FOET VQ UP UIF SJHIU )PXFWFS XF DBO BTL UIF RVFTUJPO i0O BWFSBHF IPX GBS BXBZ GSPN UIF PSJHJO BMPOH UIF Y BYJT EPFT UIF TQFDJBM NPMFDVMF HFU OP NBUUFS XIFUIFS JU FOET VQ UP UIF MFGU PS UIF SJHIU u " HPPE JOEJDBUPS PG UIJT EJTUBODF JT UIF BWFSBHF WBMVF PG UIF TRVBSF PG UIF OFU Y EJTQMBDFNFOU ΔY  CFDBVTF UIBUT BMXBZT B QPTJUJWF OVNCFS 8F DBO DBMDVMBUF UIJT BWFSBHF 4JODF ΔY = ΔY + ΔY + ΔY + ΔY + · · · + ΔY/ XF IBWF ΔY  = ΔY  + ΔY  + · · · + ΔY/  + ΔY ΔY + ΔY ΔY + ΔY ΔY + · · · *O UIJT TRVBSF PG UIF OFU Y EJTQMBDFNFOU UIFSF BSF UXP LJOET PG UFSNT TRVBSFT PG JOEJWJEVBM Y EJTQMBDFNFOUT TVDI BT ΔY  BOE iDSPTT UFSNTu MJLF ΔY ΔY  (JWFO UIF SBOEPN OBUVSF PG UIF QSPDFTT XF FYQFDU UIBU POFGPVSUI PG UIF UJNF CPUI ΔY BOE ΔY BSF QPTJUJWF QSPEVDU JT QPTJUJWF

POFGPVSUI PG UIF UJNF UIFZSF CPUI OFHBUJWF QSPEVDU JT QPTJUJWF

POFGPVSUI PG UIF UJNF ΔY JT QPTJUJWF XJUI ΔY OFHBUJWF QSPEVDU JT OFHBUJWF

BOE POFGPVSUI PG UIF UJNF ΔY JT OFHBUJWF XJUI ΔY QPTJUJWF QSPEVDU JT OFHBUJWF  5IFSFGPSF UIF BWFSBHF WBMVF PG FBDI DSPTT UFSN ΔYJ ΔYK J = / K JT [FSP "T GPS UIF PUIFS UFSNT UIF TRVBSFT PG JOEJWJEVBM Y EJTQMBDFNFOUT TVDI BT ΔY  UIF BWFSBHF WBMVF PG FBDI PG UIFTF UFSNT JT TPNF OVNCFS EY  SFMBUFE UP E UIF TRVBSF PG UIF NFBO GSFF QBUI UIBU XF EJTDVTTFE JO 4FDUJPO 4 $PVOU VQ IPX NBOZ PG UIFTF TRVBSF UFSNT UIFSF BSF GPS / DPMMJTJPOT BOE DBMDVMBUF UIF BWFSBHF WBMVF PG UIF TRVBSF PG UIF OFU EJTQMBDFNFOU

26&45*0/

ΔY  = ? 5IFSF BSF / PG UIFTF UFSNT BOE UIF TRVBSF SPPU PG UIJT RVBOUJUZ JT UIF iSPPUNFBOTRVBSFu PS iSNTu WBMVF PG UIF OFU EJTQMBDFNFOU  √ ΔYSNT = ΔY  = / EY 8F DBO BMTP XSJUF UIJT FRVBUJPO JO UFSNT PG UJNF *G XF MFU W CF UIF BWFSBHF TQFFE PG UIF TQFDJBM NPMFDVMF CFUXFFO DPMMJTJPOT BOE XF MFU 5 CF UIF BWFSBHF UJNF CFUXFFO DPMMJTJPOT XF IBWF W = E/5 "MTP UIF UPUBM UJNF U GPS UIF / DPMMJTJPOT JT U = /5 3FXSJUF UIF FRVBUJPO GPS UIF SNT EJTQMBDFNFOU JO UFSNT PG KVTU EY W BOE U UIBU JT FMJNJOBUF / BOE 5  ΔYSNT = ?

√ √ √ 8F àOE UIBU ΔYSNT = /EY EY = / W5 EY = WEY U 5IJT JT B TPNFXIBU DVSJPVT SFTVMU 'PS PSEJOBSZ NPUJPO BU DPOTUBOU TQFFE EJTQMBDFNFOU JODSFBTFT QSPQPSUJPOBM UP UJNF EPVCMF UIF UJNF EPVCMF UIF EJTQMBDFNFOU *O DPOUSBTU UIF SNT EJTQMBDFNFOU JO B SBOEPN XBML HSPXT XJUI UIF TRVBSF SPPU PG UIF UJNF PO BWFSBHF UIF SNT EJTQMBDFNFOU EPVCMFT XIFO UIF UJNF RVBESVQMFT 5IJT DBMDVMBUJPO XBT EPOF GPS UIF Y DPNQPOFOU PG UIF NPUJPO CVU XF DBO HFOFSBMJ[F PVS SFTVMU UP SFBM UISFFEJNFOTJPOBM NPUJPO JO B HBT 'JHVSF 4 TIPXT B UISFFEJNFOTJPOBM EJTQMBDFNFOU JOWPMWJOH ΔY ΔZ BOE Δ[ 26&45*0/

2

2

(Δx (Δx 2

2 + z) +Δ

Δy

Δy

+Δ 2 z)

Δx 'JHVSF 4 1ZUIBHPSFBO UIFPSFN JO UISFF EJNFOTJPOT

Δz

4VNNBSZ

4

:PV DBO TFF JO 'JHVSF 4 B UISFFEJNFOTJPOBM WFSTJPO PG UIF 1ZUIBHPSFBO UIFPSFN GPS USJBOHMFT 8F DBO BTTVNF UIBU UIF NPUJPO JO FBDI EJNFOTJPO JT JOEFQFOEFOU PG UIF NPUJPOT JO UIF PUIFS UXP EJNFOTJPOT BOE XF IBWF UIF GPMMPXJOH SFTVMU   √ ΔSSNT = ΔY  + ΔZ  + Δ[  = /EY + /EZ + /E[ = / E )FSF EY JT UIF DPNQPOFOU JO UIF Y EJSFDUJPO PG UIF UISFFEJNFOTJPOBM NFBO GSFF QBUI E

8SJUJOH UIF SNT EJTQMBDFNFOU JO UFSNT PG UIF BWFSBHF TQFFE W UIF UJNF U BOE UIF NFBO GSFF QBUI E XF IBWF √ √ ΔSSNT = WE U 5IJT SFTVMU JT TPNFXIBU VOVTVBM CFDBVTF JU QSFEJDUT DPSSFDUMZ JU UVSOT PVU

UIBU GPS UIJT SBOEPN QSPDFTT UIF EJTQMBDFNFOU GSPN UIF TUBSUJOH MPDBUJPO JT QSPQPSUJPOBM UP UIF TRVBSF SPPU PG UIF UJNF SBUIFS UIBO UP UIF UJNF

46.."3: ,&: *%&"4

'PS B NVMUJBUPN HBT NPMFDVMF

8F DBO NPEFM B HBT BT UJOZ CBMMT JO SBQJE SBOEPN NPUJPO "QQMZJOH TUBUJTUJDBM JEFBT UP UIJT NJDSPTDPQJD NPEFM BMMPXT VT UP QSFEJDU NBDSPTDPQJD SFMBUJPOTIJQT BNPOH QSFTTVSF WPMVNF BOE UFNQFSBUVSF JODMVEJOH UIF JEFBM HBT MBX 5IF EFàOJUJPO PG UFNQFSBUVSF JO UFSNT PG FOUSPQZ JT DPOTJTUFOU XJUI UIF FWFSZEBZ UFNQFSBUVSF TDBMF #PUI XPSL BOE FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF DBO DIBOHF UIF UFNQFSBUVSF BOEPS WPMVNF PG B HBT JO B DIBNCFS XJUI B NPWBCMF QJTUPO 5IF BUUBJOBCMF FGàDJFODZ PG FOHJOFT UIBU DPOWFSU UIFSNBM FOFSHZ JOUP VTFGVM XPSL JT DPOTUSBJOFE CZ UIF TFDPOE MBX PG UIFSNPEZOBNJDT .FBO GSFF QBUI E O ≈ 

3PPUNFBOTRVBSF TQFFE WSNT ≡ W 7 8PSL EPOF PO B HBT 8 = − 7  1E7

 Δ&UPU = Δ

 L# 5 + &SPU + &WJC 



$7 =  L# GPS B NPOBUPNJD HBT )F FUD

$7 ≥  L# GPS PUIFS HBTFT / FUD

$1 = $7 + L# NPMFDVMBS TQFDJàD IFBU DBQBDJUZ

$1 = $7 + 3 NPMBS TQFDJàD IFBU DBQBDJUZ

*O B DPOTUBOUUFNQFSBUVSF JTPUIFSNBM DPNQSFTTJPO 

7 8CZ QJTUPO = 2JOUP TVSSPVOEJOHT = /L# 5 MO 7



*O B OP2 BEJBCBUJD DPNQSFTTJPO 17γ = DPOTUBOU XIFSF γ = $1 /$7 BOE BMTP 5 $7 /L# 7 = DPOTUBOU



'JSTU MBX PG UIFSNPEZOBNJDT Δ&TZT = 8 + 2 .PMFDVMBS TQFDJàD IFBU DBQBDJUZ BU DPOTUBOU WPMVNF $7  2 = /$7 Δ5 .PMFDVMBS TQFDJàD IFBU DBQBDJUZ BU DPOTUBOU QSFTTVSF $1  2 = /$1 Δ5 /VNCFS PG HBT NPMFDVMFT IJUUJOH BO BSFB " QFS TFDPOE    O"W % WBSJPVT TQFFET

 1= O 



Q N

 = OL# 5

XIFSF O = //7 OVNCFS PG NPMFDVMFT QFS DVCJD NFUFS

L# =  × − +, 3 =  ×  L# =  +, "U iTUBOEBSE UFNQFSBUVSF BOE QSFTTVSFu 451 XIJDI NFBOT B UFNQFSBUVSF PG  ◦ $FMTJVT PS  , BOE B QSFTTVSF PG  BUN

POF NPMF PG B HBT PDDVQJFT  - B MJUFS JT  DVCJD DFOUJNFUFST  4FBMFWFM QSFTTVSF  BUN JT  ×  //N  3BUF PG FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF 5) − 5-

2˙ = σ" +T

-

4

4VQQMFNFOU  (BTFT BOE )FBU &OHJOFT

'PS B DZDMJD FOHJOF SVOOJOH CFUXFFO IJHI UFNQFSBUVSF 5) BOE MPX UFNQFSBUVSF 5- XF IBWF

5IF FGàDJFODZ PG B SFBM IFBU FOHJOF &GàDJFODZ =

&OUSPQZ DIBOHF PG UIF VOJWFSTF GPS SFBM IFBU FOHJOFT

Δ46OJWFSTF = −

2) 2+ ≥ 5) 5-

8 5≤ − 2) 5)

5IF = TJHO BQQMJFT POMZ JG UIF IFBU FOHJOF JT SFWFSTJCMF FYUSFNFMZ TMPX QSPDFTTFT OP GSJDUJPO  " IFBU FOHJOF SVO JO SFWFSTF JT B SFGSJHFSBUPS PS B IFBU QVNQ 

26&45*0/ 2 )PX EPFT UIF NFBO GSFF QBUI PG BO BUPN JO B HBT DIBOHF JG UIF UFNQFSBUVSF JT JODSFBTFE XJUI UIF WPMVNF LFQU DPOTUBOU &YQMBJO CSJFáZ

130#-&.4 4FDUJPO 4 rr1 (BT MFBLT BU B SBUF - JO NPMFDVMFT QFS TFDPOE UISPVHI B TNBMM DJSDVMBS IPMF *G UIF EFOTJUZ PG UIF HBT JT EPVCMFE UIF ,FMWJO UFNQFSBUVSF JT EPVCMFE BOE UIF SBEJVT PG UIF IPMF JT EPVCMFE XIBU JT UIF OFX MFBL SBUF rr1 4VQQPTF UIBU XF NBLF B DJSDVMBS IPMF  NN JO EJBNFUFS JO B CBMMPPO $BMDVMBUF UIF JOJUJBM SBUF BU XIJDI DIMPSJOF FTDBQFT UISPVHI UIF IPMF BU  ◦ $

JO OVNCFS PG DIMPSJOF BUPNT MFBWJOH UIF CBMMPPO QFS TFDPOE rr1 8IBU JT UIF BQQSPYJNBUF UJNF CFUXFFO DPMMJTJPOT GPS POF QBSUJDVMBS BJS NPMFDVMF rr1 *O BO FYBNQMF QSPCMFN JO 4FDUJPO 4 XF DPOTJEFSFE MFBLBHF GSPN B áFYJCMF QBSUZ CBMMPPO *G UIF MFBLBHF JT GSPN B SJHJE DPOUBJOFS B NFUBM TUPSBHF UBOL GPS FYBNQMF

UIF OVNCFS PG BUPNT QFS DVCJD NFUFS O XJMM EFDSFBTF XJUI UJNF U *G UIF UPUBM WPMVNF PG UIF UBOL JT 7 UIFSF BSF / = O7 BUPNT JO UIF UBOL BU BOZ JOTUBOU TP XF DBO XSJUF UIF GPMMPXJOH iEJGGFSFOUJBMu FRVBUJPO UIBU JT BO FRVBUJPO UIBU JOWPMWFT EFSJWBUJWFT   E O7 = − O"W EU  5IJT TBZT iUIF SBUF PG DIBOHF PG UIF OVNCFS PG BUPNT JO UIF UBOL JT FRVBM UP UIF OFHBUJWF PG UIF SBUF BU XIJDI BUPNT BSF MFBWJOH UIF UBOLu B 4IPX UIBU UIF EJGGFSFOUJBM FRVBUJPO JT TBUJTàFE JG UIF OVNCFS EFOTJUZ

O = OJOJUJBM F− "W/7 U XIFSF U JT UIF UJNF FMBQTFE TJODF UIF IPMF XBT NBEF +VTU QMVH UIJT GVODUJPO PG O BOE JUT EFSJWBUJWF JOUP UIF FRVBUJPO BOE TIPX UIBU UIF UXP TJEFT PG UIF FRVBUJPO BSF FRVBM GPS BMM WBMVFT PG UIF UJNF U "MTP TIPX UIBU UIF JOJUJBM QBSUJDMF EFOTJUZ O JT FRVBM UP OJOJUJBM  C %FTQJUF UIF GBODZ NBUI UIJT TPMVUJPO JT SFBMMZ POMZ BQQSPYJNBUF CFDBVTF UIF BWFSBHF TQFFE JTOU B DPOTUBOU CVU JT EFDSFBTJOH 4VQQPTF IPXFWFS UIBU XF VTF B IFBUFS UP LFFQ UIF DPOUBJOFS BOE UIF HBT BU B OFBSMZ DPOTUBOU UFNQFSBUVSF TP UIBU UIF BWFSBHF TQFFE EPFT SFNBJO OFBSMZ DPOTUBOU 4VQQPTF UIBU UIF SJHJE DPOUBJOFS JT BHBJO B TQIFSF  DN JO EJBNFUFS XJUI B DJSDVMBS IPMF  NN JO EJBNFUFS "CPVU IPX MPOH XPVME JU UBLF GPS NPTU PG UIF IFMJVN UP MFBL PVU &YQMBJO ZPVS DIPJDF PG XIBU ZPV NFBO CZ iNPTUu

rr1 /BUVSBM VSBOJVN PSF DPOTJTUT NPTUMZ PG UIF JTPUPQF 6  QSPUPOT BOE  OFVUSPOT

CVU  PG UIF PSF DPOTJTUT PG UIF JTPUPQF 6  QSPUPOT BOE  OFVUSPOT  #FDBVTF POMZ 6 àTTJPOT JO B SFBDUPS JOEVTUSJBM QSPDFTTFT BSF VTFE UP FOSJDI UIF VSBOJVN CZ FOIBODJOH UIF 6 DPOUFOU 0OF PG UIF FOSJDINFOU NFUIPET JT iHBTFPVT EJGGVTJPOu 5IF HBT 6' VSBOJVN IFYBáVPSJEF JT NBOVGBDUVSFE GSPN TVQQMJFT PG OBUVSBM VSBOJVN BOE áVPSJOF FBDI PG UIF  áVPSJOF BUPNT IBT  QSPUPOT BOE  OFVUSPOT  " DPOUBJOFS JT à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àSTU MBSHF HBTFPVT EJGGVTJPO QMBOU XBT DPOTUSVDUFE EVSJOH 8PSME 8BS ** BU 0BL 3JEHF 5FOOFTTFF BOE VTFE JOFYQFOTJWF 5FOOFTTFF 7BMMFZ "VUIPSJUZ FMFDUSJDJUZ rr1 :PV BSF PO B TQBDFDSBGU NFBTVSJOH  N CZ  N CZ  N XIFO JU JT TUSVDL CZ B QJFDF PG TQBDF KVOL MFBWJOH B DJSDVMBS IPMF PG SBEJVT  NN VOGPSUVOBUFMZ JO B QMBDF UIBU DBO CF SFBDIFE POMZ CZ NBLJOH B UJNFDPOTVNJOH TQBDFXBML "CPVU IPX NVDI UJNF EP ZPV IBWF UP QBUDI UIF MFBL &YQMBJO XIBU BQQSPYJNBUJPOT ZPV NBLF JO BTTFTTJOH UIF TFSJPVTOFTT PG UIF TJUVBUJPO 4FDUJPO 4 rr1 " SPVHIMZ TQIFSJDBM NFUFPS NBEF NBJOMZ PG JSPO EFOTJUZ BCPVU  HDN JT IVSUMJOH EPXOXBSE UISPVHI UIF BJS BU MPX BMUJUVEF "U BO JOTUBOU XIFO JUT TQFFE JT  ×  N/ T DBMDVMBUF UIF BQQSPYJNBUF SBUF PG DIBOHF PG UIF NFUFPST TQFFE %P UIF BOBMZTJT GPS UXP EJGGFSFOU NFUFPST‡POF XJUI B SBEJVT PG  N BOE POF XJUI B SBEJVT PG  N

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àMMFE FYQMPTJWFMZ CZ BJS SVTIJOH JO BGUFS UIF NFUFPS IBT QBTTFE 5IFSF JT HPPE FWJEFODF UIBU B WFSZ MBSHF NFUFPS QFSIBQT  LN JO EJBNFUFS IJU UIF &BSUI OFBS UIF :VDBUBO 1FOJOTVMB JO TPVUIFSO .FYJDP  NJMMJPO ZFBST BHP BOE DBVTFE TP NVDI EBNBHF UIBU UIF EJOPTBVST CFDBNF FYUJODU 4FF UIF FYDFMMFOU BDDPVOU JO 5 SFY BOE UIF $SBUFS PG %PPN CZ 8BMUFS "MWBSF[ 1SJODFUPO 6OJWFSTJUZ 1SFTT   "MWBSF[ JT UIF HFPMPHJTU XIP NBEF UIF àSTU EJTDPWFSJFT MFBEJOH UP PVS DVSSFOU VOEFSTUBOEJOH PG UIJT DBUBDMZTNJD FWFOU rr1 " SJHJE UIFSNBMMZ JOTVMBUFE DPOUBJOFS XJUI B WPMVNF PG  - JT àMMFE XJUI POF NPMF PG IFMJVN HBT  H QFS NPMF BU B UFNQFSBUVSF PG  $  ,  5IF DPOUBJOFS JT TJUUJOH JO B SPPN TVSSPVOEFE CZ BJS BU 451 B $BMDVMBUF UIF QSFTTVSF JOTJEF UIF DPOUBJOFS JO //N  C $BMDVMBUF UIF SPPUNFBOTRVBSF BWFSBHF TQFFE PG UIF IFMJVN BUPNT D /PX PQFO B UJOZ TRVBSF IPMF JO UIF DPOUBJOFS XJUI BSFB  × − N UIF IPMF JT  NN PO B TJEF  "GUFS  T IPX NBOZ IFMJVN BUPNT IBWF MFGU UIF DPOUBJOFS E "JS NPMFDVMFT GSPN UIF SPPN FOUFS UIF DPOUBJOFS UISPVHI UIF IPMF EVSJOH UIFTF  T 8IJDI JT HSFBUFS UIF OVNCFS PG BJS NPMFDVMFT UIBU FOUFS UIF DPOUBJOFS PS UIF OVNCFS PG IFMJVN BUPNT UIBU MFBWF UIF DPOUBJOFS &YQMBJO CSJFáZ F %PFT UIF QSFTTVSF JOTJEF UIF DPOUBJOFS JODSFBTF TMJHIUMZ TUBZ UIF TBNF PS EFDSFBTF TMJHIUMZ EVSJOH UIFTF  T &YQMBJO CSJFáZ *G ZPV IBWF UP NBLF BOZ TJNQMJGZJOH BTTVNQUJPOT TUBUF UIFN DMFBSMZ 4FDUJPO 4 r1 *G ZPV FYQBOE UIF WPMVNF PG B HBT DPOUBJOJOH / NPMFDVMFT UP UXJDF JUT PSJHJOBM WPMVNF XIJMF NBJOUBJOJOH B DPOTUBOU UFNQFSBUVSF IPX NVDI FOFSHZ USBOTGFS EVF UP B UFNQFSBUVSF EJGGFSFODF JT UIFSF GSPN UIF TVSSPVOEJOHT r1 *G ZPV DPNQSFTT B WPMVNF PG IFMJVN DPOUBJOJOH / BUPNT UP IBMG UIF PSJHJOBM WPMVNF JO B XFMMJOTVMBUFE DZMJOEFS XIBU JT UIF SBUJP PG UIF àOBM QSFTTVSF UP UIF PSJHJOBM QSFTTVSF rr1 " NPMF PG OJUSPHFO JT DPNQSFTTFE CZ QJMJOH MPUT PG TBOE PO UIF QJTUPO UP B WPMVNF PG  - BU SPPN UFNQFSBUVSF  ,  5IF DZMJOEFS JT QMBDFE PO BO FMFDUSJD IFBUJOH FMFNFOU XIPTF UFNQFSBUVSF JT NBJOUBJOFE BU  , " RVBTJTUBUJD FYQBOTJPO JT DBSSJFE PVU BU DPOTUBOU UFNQFSBUVSF CZ WFSZ TMPXMZ SFNPWJOH HSBJOT PG TBOE GSPN UIF UPQ PG UIF QJTUPO XJUI UIF UFNQFSBUVSF PG UIF HBT TUBZJOH DPOTUBOU BU  , 8IFO UIF WPMVNF JT  - IPX NVDI FOFSHZ USBOTGFS 2 IBT HPOF GSPN UIF IFBUJOH FMFNFOU JOUP UIF HBT )PX NVDI XPSL 8 IBT CFFO EPOF PO UIF QJTUPO CZ UIF HBT )PX NVDI IBT UIF FOFSHZ PG UIF HBT DIBOHFE :PV NVTU BTTVNF UIBU UIFSF JT OP FOFSHZ USBOTGFS GSPN UIF HBT UP UIF TVSSPVOEJOH BJS BOE OP GSJDUJPO JO UIF NPUJPO PG UIF QJTUPO BMM PG XIJDI JT QSFUUZ VOSFBMJTUJD JO UIF SFBM XPSME /FWFSUIFMFTT UIFSF BSF QSPDFTTFT UIBU DBO CF BQQSPYJNBUFE CZ B DPOTUBOUUFNQFSBUVSF FYQBOTJPO 5IJT QSPCMFN JT BO JEFBMJ[BUJPO PG B SFBM QSPDFTT

4

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5 EFDSFBTFT CFDBVTF BWFSBHF W EFDSFBTFT  ×  NPMFDVMFT QFS TFDPOE - ≈  × − N  ×  /N    L# + L# =  L#



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