ch12

Page 1

&+$37(5 62/87,21 0$18$/ )XQGDPHQWDOV RI 7KHUPRG\QDPLFV WK (GLWLRQ 6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

of

FUNDAMENTALS

Thermodynamics

&217(17

68%6(&7,21

&RUUHVSRQGHQFH WDEOH &RQFHSW 6WXG\ *XLGH 3UREOHPV 0L[WXUH FRPSRVLWLRQ DQG SURSHUWLHV 6LPSOH SURFHVVHV (QWURS\ JHQHUDWLRQ $LU ZDWHU YDSRU PL[WXUHV 7DEOHV DQG IRUPXODV RU SV\FKURPHWULF FKDUW 3V\FKURPHWULF FKDUW RQO\ $YDLODELOLW\ H[HUJ\ LQ PL[WXUHV 5HYLHZ 3UREOHPV

352% 12


&+$37(5

WK HGLWLRQ

6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

6RQQWDJ %RUJQDNNH :\OHQ

7KH FRUUHVSRQGHQFH EHWZHHQ WKLV SUREOHP VHW DQG WKH WK HGLWLRQ FKDSWHU SUREOHP VHW 1HZ

WK QHZ QHZ QHZ QHZ PRG QHZ QHZ PRG PRG QHZ QHZ PRG PRG QHZ PRG QHZ QHZ QHZ PRG

1HZ

WK QHZ QHZ PRG PRG QHZ PRG PRG QHZ QHZ PRG PRG PRG PRG QHZ QHZ QHZ PRG

1HZ

WK QHZ D F QHZ QHZ

1HZ

WK QHZ QHZ QHZ QHZ

7KH SUREOHPV WKDW DUH ODEHOHG SV\FKURPHWULF FKDUW RQO\ FDQ EH VROYHG ZLWKRXW WKH FKDUW XVLQJ WKH IRUPXODV DQG WKH VWHDP WDEOHV 0RVW RI WKH VROXWLRQV WKRXJK EHFRPH OHQJWK\ DQG WULDO DQG HUURU W\SH GXH WR WKH LPSOLFLW FRQQHFWLRQV EHWZHHQ WKH YDULDEOHV VKRZQ LQ WKH FKDUW $Q DOWHUQDWLYH WR WKLV SURFHGXUH LV WKH XVH RI WKH FRPSXWHU VRIWZDUH ZKLFK FDQ EH XVHG LQVWHDG RI WKH FKDUW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQFHSW 6WXG\ *XLGH 3UREOHPV

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

7KH WHPSHUDWXUH LV . 7KH SDUWLDO SUHVVXUH LV 3 \3WRW × N3D

$W WKLV 7 3 Y2 57 32 × P NJ

5HPDUN ,I ZH IRXQG WKH R[\JHQ PDVV WKHQ P2 Y2 9 P


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI R[\JHQ DQG RQH RI QLWURJHQ ERWK . DUH PL[HG WR SURGXFH NJ V DLU DW . N3D :KDW DUH WKH PDVV DQG YROXPH IORZ UDWHV RI HDFK OLQH" )RU WKH PL[WXUH 0 × × )RU 2 F × F × )RU 1 6LQFH WKH WRWDO IORZ RXW LV NJ V WKHVH DUH WKH FRPSRQHQW IORZV LQ NJ V 9ROXPH IORZ RI 2 LQ LV 57 9 FPY FP 3 × × P V 9ROXPH IORZ RI 1 LQ LV 57 9 FP Y FP 3 × × P V

$ IORZ RI JDV $ DQG D IORZ RI JDV % DUH PL[HG LQ D PROH UDWLR ZLWK VDPH 7 :KDW LV WKH HQWURS\ JHQHUDWLRQ SHU NPROH IORZ RXW"

)RU WKLV HDFK FRPSRQHQW WKH PROH IUDFWLRQ LV RQH KDOI VR B B (T ∆6 5 OQ OQ 5


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ ULJLG FRQWDLQHU KDV NJ DUJRQ DW . DQG NJ DUJRQ DW . ERWK DW N3D 1RZ WKH\ DUH DOORZHG WR PL[ ZLWKRXW DQ\ H[WHUQDO KHDW WUDQVIHU :KDW LV ILQDO 7 3" ,V DQ\ V JHQHUDWHG" (QHUJ\ (T 8 ± 8 PX PX D PX E P&Y 7 ± 7 D ± 7 E 7 7 D 7 E . 3URFHVV (T 9 FRQVWDQW ! 3 9 P57 P5 7 D 7 E 3 9 D 3 9 E 3 9 3 3 N3D ∆6 GXH WR WHPSHUDWXUH FKDQJHV RQO\ QRW 3 LQWHUQDOO\ ZH KDYH D 4 RYHU D ∆7

∆6 P V ± V D P V ± V E P&S > OQ 7 7 D OQ 7 7 E @ × >OQ OQ @ N- . Ar

Ar cb


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ ULJLG FRQWDLQHU KDV NJ &2 DW . DQG NJ DUJRQ DW . ERWK DW N3D 1RZ WKH\ DUH DOORZHG WR PL[ ZLWKRXW DQ\ KHDW WUDQVIHU :KDW LV ILQDO 7 3" 1R 4 1R : VR WKH HQHUJ\ HTXDWLRQ JLYHV FRQVWDQW 8 (QHUJ\ (T 8 ± 8 P&2 X ± X &2 P$U X ± X $U

P&2 &Y &2 7 ± 7 &2 P$U&Y $U 7 ± 7 $U

× × × 7 × × × ×

7 .

9 9 9&2 9$U P&2 5&2 7&2 3 P$U5$U7$U 3

× × × × P 3UHVVXUH IURP LGHDO JDV ODZ DQG (T IRU 5

3 × × × N3D

CO 2

Ar cb


$ IORZ RI NJ V DUJRQ DW . DQG DQRWKHU IORZ RI NJ V &2 DW . ERWK DW N3D DUH PL[HG ZLWKRXW DQ\ KHDW WUDQVIHU :KDW LV WKH H[LW 7 3" 1R ZRUN LPSOLHV QR SUHVVXUH FKDQJH IRU D VLPSOH IORZ 7KH HQHUJ\ HTXDWLRQ EHFRPHV PKL PKH PKL $U PKL &2 PKH $U PKH &2 ⇒ P&2 &S &2 7H ± 7L &2 P$U&S $U 7H ± 7L $U ⇒ P$U&S $U7L P&2 &S &2 7L >P$U&S $U P&2 &S &2 @ 7H

× × × × × × × 7H

7H . 3H N3D

$U 0,;,1*

&2

0L[

&+$0%(5 FE

6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

:KDW LV WKH UDWH RI HQWURS\ LQFUHDVH LQ SUREOHP " 8VLQJ (T WKH PROH IUDFWLRQ RI &2 LQ WKH PL[WXUH LV )URP (TV DQG IURP WKH WZR LQOHW VWDWHV WR VWDWH ×

@ ∆6 ×> OQ ± OQ × ×> OQ ± OQ

@ N: . ,I , ZDQW WR KHDW D IORZ RI D FRPSRQHQW PL[WXUH IURP WR . DW FRQVWDQW 3 KRZ PDQ\ SURSHUWLHV DQG ZKLFK RQHV GR , QHHG WR NQRZ WR ILQG WKH KHDW WUDQVIHU" <RX QHHG WR NQRZ WKH IORZ UDWH WKH IRXU PDVV IUDFWLRQV DQG WKH FRPSRQHQW VSHFLILF KHDW YDOXHV


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

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

Y &

V

:KDW KDSSHQV WR UHODWLYH DQG DEVROXWH KXPLGLW\ ZKHQ PRLVW DLU LV FRROHG" 5HODWLYH KXPLGLW\ LQFUHDVH ZKLOH DEVROXWH KXPLGLW\ UHPDLQV FRQVWDQW 6HH )LJV DQG


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

,I , KDYH DLU DW N3D DQG D ± R& E R& DQG F R& ZKDW LV WKH PD[LPXP DEVROXWH KXPLGLW\ , FDQ KDYH" +XPLGLW\ LV UHODWHG WR UHODWLYH KXPLGLW\ PD[ DQG WKH SUHVVXUHV DV LQ (T ZKHUH IURP (T 3Y Φ 3J DQG 3D 3WRW 3Y 3Y Φ 3J ω 3 3WRW Φ3J D D 3J N3D ω × E 3J N3D ω ×

F 3J N3D QR PD[ ω IRU 3WRW N3D


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&DQ PRLVW DLU EHORZ WKH IUHH]LQJ SRLQW VD\ ± R& KDYH D GHZ SRLQW"

<HV $W WKH GHZ SRLQW ZDWHU ZRXOG EHJLQ WR DSSHDU DV D VROLG ,W VQRZV

([SODLQ LQ ZRUGV ZKDW WKH DEVROXWH DQG UHODWLYH KXPLGLW\ H[SUHVVHV" $EVROXWH KXPLGLW\ LV WKH UDWLR RI WKH PDVV RI YDSRU WR WKH PDVV RI GU\ DLU ,W VD\V KRZ PXFK ZDWHU LV WKHUH SHU XQLW PDVV RI GU\ DLU 5HODWLYH KXPLGLW\ LV WKH UDWLR RI WKH PROH IUDFWLRQ RI YDSRU WR WKDW LQ D VDWXUDWHG PL[WXUH DW WKH VDPH 7 DQG 3 ,W H[SUHVVHV KRZ FORVH WR WKH VDWXUDWHG VWDWH WKH ZDWHU LV

$Q DGLDEDWLF VDWXUDWLRQ SURFHVV FKDQJHV Φ Ï‰ DQG 7 ,Q ZKLFK GLUHFWLRQ" 5HODWLYH KXPLGLW\ DQG DEVROXWH KXPLGLW\ LQFUHDVH DQG WHPSHUDWXUH GHFUHDVHV

, ZDQW WR EULQJ DLU DW R& Φ WR D VWDWH RI R& ω GR , QHHG WR DGG RU VXEWUDFW ZDWHU"

$VVXPLQJ 3 N3D

$W °& ω × ×

7R JHW WR ω LW LV QHFHVVDU\ WR VXEWUDFW ZDWHU


0L[WXUH FRPSRVLWLRQ DQG SURSHUWLHV

6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ JDV PL[WXUH DW °& N3D LV 1 + 2 DQG 2 RQ D PROH EDVLV )LQG WKH PDVV IUDFWLRQV WKH PL[WXUH JDV FRQVWDQW DQG WKH YROXPH IRU NJ RI PL[WXUH 6ROXWLRQ 7KH FRQYHUVLRQ IROORZV WKH GHILQLWLRQV DQG LGHQWLWLHV )URP (T FL \L 0L ∑ \M0M )URP (T

0PL[ ∑ \M0M × × ×

F1 F+ 2

F2 VXPV WR 2.

)URP (T − 5PL[ 5 0PL[ N- NJ .

9 P5PL[ 7 3 × × P


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI 1 $U DQG 2 RQ D PDVV EDVLV LV LQ D F\OLQGHU DW N3D . DQG YROXPH P )LQG WKH PROH IUDFWLRQV DQG WKH PDVV RI DUJRQ 6ROXWLRQ 7RWDO PL[WXUH

39 P 5PL[7

)URP (T

5PL[ ∑ FL5L × × ×

P 39 5PL[7 × × NJ PDU P NJ

N- NJ .

)URP (T

\L FL 0L ∑ FM 0M

FL

0L

FL 0L

\L

1

$U

2

URXQG XS


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI 1 $U DQG 2 RQ D PROH EDVLV LV LQ D F\OLQGHU DW N3D . DQG YROXPH P )LQG WKH PDVV IUDFWLRQV DQG WKH PDVV RI DUJRQ 6ROXWLRQ )URP (T FL \L 0L ∑ \M0M (T

0PL[ ∑ \M0M × × ×

F1 ×

F$U ×

F2 × VXPV WR 2.

)URP (T − 5PL[ 5 00,; N- NJ .

PPL[ 39 5PL[ 7 × × NJ

P$U F$U × PPL[ × NJ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ QHZ UHIULJHUDQW 5 LV D PL[WXUH RI 5 5 DQG 5 D RQ D PDVV EDVLV )LQG WKH PROH IUDFWLRQV WKH PL[WXUH JDV FRQVWDQW DQG WKH PL[WXUH KHDW FDSDFLWLHV IRU WKLV QHZ UHIULJHUDQW 6ROXWLRQ )URP WKH FRQYHUVLRQ LQ (T ZH JHW FL 0L FL 0L \L 5 5 5 D (T

5PL[ ∑ FL5L × × ×

N- NJ . (T

&3 PL[ ∑ FL &3 L × × ×

N- NJ . (T

&9 PL[ ∑ FL&9 L × × ×

N- NJ . &3 0,; 50,;


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ FDUEXUHWHG LQWHUQDO FRPEXVWLRQ HQJLQH LV FRQYHUWHG WR UXQ RQ PHWKDQH JDV QDWXUDO JDV 7KH DLU IXHO UDWLR LQ WKH F\OLQGHU LV WR EH WR RQ D PDVV EDVLV +RZ PDQ\ PROHV RI R[\JHQ SHU PROH RI PHWKDQH DUH WKHUH LQ WKH F\OLQGHU" 6ROXWLRQ 7KH PDVV UDWLR P$,5 P&+ VR UHODWH PDVV DQG PROH Q P 0

P$,5 Q$,5 Q&+ P&+ × 0&+ 0$,5 × Q2 Q2 Q$,5 → Q Q × Q × PROH 2 PROH &+ &+ &+ $,5


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

:HLJKLQJ RI PDVVHV JLYHV D PL[WXUH DW °& N3D ZLWK NJ 2 NJ 1 DQG NJ &+ )LQG WKH SDUWLDO SUHVVXUHV RI HDFK FRPSRQHQW WKH PL[WXUH VSHFLILF YROXPH PDVV EDVLV PL[WXUH PROHFXODU ZHLJKW DQG WKH WRWDO YROXPH 6ROXWLRQ )URP (T \L PL 0L ∑ PM 0M

QWRW ∑ PM 0M

\2

\1

\&+

)URP (T

32 \2 3WRW × N3D

31 \1 3WRW × N3D

3&+ \&+ 3WRW × N3D

− 9WRW QWRW 57 3 × × P

Y 9WRW PWRW P NJ )URP (T

0PL[ ∑ \M0M PWRW QWRW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ NJ PL[WXUH RI 1 2 DQG &2 E\ PDVV LV DW N3D DQG . )LQG WKH PL[WXUH JDV FRQVWDQW DQG WKH WRWDO YROXPH 6ROXWLRQ )URP (T

5PL[ ∑ FL5L × × ×

N- NJ . ,GHDO JDV ODZ 39 P5PL[7

9 P5PL[7 3 × × P


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ P VWRUDJH WDQN ZLWK IXHO JDVHV LV DW °& N3D FRQWDLQLQJ D PL[WXUH RI DFHW\OHQH & + SURSDQH & + DQG EXWDQH & + $ WHVW VKRZV WKH SDUWLDO SUHVVXUH RI WKH & + LV N3D DQG WKDW RI & + LV N3D +RZ PXFK PDVV LV WKHUH RI HDFK FRPSRQHQW" 6ROXWLRQ $VVXPH LGHDO JDVHV WKHQ WKH UDWLR RI SDUWLDO WR WRWDO SUHVVXUH LV WKH PROH IUDFWLRQ \ 3 3WRW \& + \& + \& + 39 × QWRW − NPROHV 57 × P& + Q0 & + \& + QWRW 0& + × × NJ P& + Q0 & + \& + QWRW 0& + × × NJ P& + Q0 & + \& + QWRW 0& +

× × NJ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ SLSH FURVV VHFWLRQDO DUHD P FDUULHV D IORZ RI 2 DQG 1 E\ PROH ZLWK D YHORFLW\ RI P V DW N3D . 7R LQVWDOO DQG RSHUDWH D PDVV IORZ PHWHU LW LV QHFHVVDU\ WR NQRZ WKH PL[WXUH GHQVLW\ DQG WKH JDV FRQVWDQW :KDW DUH WKH\" :KDW PDVV IORZ UDWH VKRXOG WKH PHWHU WKHQ VKRZ" 6ROXWLRQ )URP (T FL \L 0L ∑ \M0M )URP (T

0PL[ ∑ \M0M × ×

(T

− 5PL[ 5 0PL[ N- NJ .

Y 5PL[ 7 3 × P NJ

ρ Y NJ P P ρ$9 × × NJ V


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ QHZ UHIULJHUDQW 5 D LV D PL[WXUH RI 5 DQG 5 LQ D PDVV UDWLR :KDW DUH WKH RYHUDOO PROHFXODU ZHLJKW WKH JDV FRQVWDQW DQG WKH UDWLR RI VSHFLILF KHDWV IRU VXFK D PL[WXUH" (T

5PL[ ∑ FL5L × × N- NJ .

(T

&3 PL[ ∑ FL &3 L × × N- NJ .

(T

&9 PL[ ∑ FL&9 L × × N- NJ . &3 PL[ 5PL[

NPL[ &3 PL[ &9 PL[ 0 ∑ \M0M ∑ FM 0M


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

6LPSOH SURFHVVHV

$W D FHUWDLQ SRLQW LQ D FRDO JDVLILFDWLRQ SURFHVV D VDPSOH RI WKH JDV LV WDNHQ DQG VWRUHG LQ D / F\OLQGHU $Q DQDO\VLV RI WKH PL[WXUH \LHOGV WKH IROORZLQJ UHVXOWV &RPSRQHQW + &2 &2 1 3HUFHQW E\ PDVV 'HWHUPLQH WKH PROH IUDFWLRQV DQG WRWDO PDVV LQ WKH F\OLQGHU DW N3D °& +RZ PXFK KHDW WUDQVIHU PXVW EH WUDQVIHUUHG WR KHDW WKH VDPSOH DW FRQVWDQW YROXPH IURP WKH LQLWLDO VWDWH WR °&" 6ROXWLRQ

'HWHUPLQH PROH IUDFWLRQV IURP (T \L FL 0L ∑ FM 0M

∑ FM 0M

0PL[ ∑ FM 0M

)URP (T \+ ×

\&2 ×

\&2 ×

\1 ×

− 5PL[ 5 0PL[ N- NJ .

P 39 57 × × × NJ

&9 0,; ∑ FL &9 L × ×

× × N- NJ .

4 8 8 P&9 7 7 × × × N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH PL[WXUH LQ 3UREOHP LV KHDWHG WR . ZLWK FRQVWDQW YROXPH )LQG WKH ILQDO SUHVVXUH DQG WKH WRWDO KHDW WUDQVIHU QHHGHG XVLQJ 7DEOH $ 6ROXWLRQ & 9 0L[WXUH RI FRQVWDQW YROXPH 3URFHVV 9 FRQVWDQW

!

: ∫ 3 G9

(QHUJ\ (T

4 P X − X ≅ P &9PL[ 7 − 7

,GHDO JDV

39 P57

! 3 3 7 7 9 9

3 3 7 7 × N3D )URP (T

&9PL[ ∑ FL&9 L × × ×

4 × N-

N- NJ .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH PL[WXUH LQ 3UREOHP LV KHDWHG XS WR . LQ D FRQVWDQW SUHVVXUH SURFHVV )LQG WKH ILQDO YROXPH DQG WKH WRWDO KHDW WUDQVIHU XVLQJ 7DEOH $ 6ROXWLRQ & 9 0L[WXUH 3URFHVV 3 FRQVWDQW

!

: ∫ 3 G9 3 9 − 9

(QHUJ\ (T

4 P X − X : P X − X 3P Y − Y

P K − K ≅ P &3 PL[ 7 − 7

)URP (T

5PL[ ∑ FL5L × × ×

N- NJ . )URP (T

&3 PL[ ∑ FL &3 L × × ×

N- NJ . 9 P 5PL[ 7 3

4 × ± N-

× × P


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ SLSH IORZV NJ V RI D PL[WXUH ZLWK PDVV IUDFWLRQV RI &2 DQG 1 DW N3D . +HDWLQJ WDSH LV ZUDSSHG DURXQG D VHFWLRQ RI SLSH ZLWK LQVXODWLRQ DGGHG DQG N: HOHFWULFDO SRZHU LV KHDWLQJ WKH SLSH IORZ )LQG WKH PL[WXUH H[LW WHPSHUDWXUH 6ROXWLRQ & 9 3LSH KHDWLQJ VHFWLRQ $VVXPH QR KHDW ORVV WR WKH RXWVLGH LGHDO JDVHV (QHUJ\ (T 4 P KH − KL P&3 PL[ 7H − 7L )URP (T

&3 PL[ ∑ FL &3 L × × N- NJ .

6XEVWLWXWH LQWR HQHUJ\ HTXDWLRQ DQG VROYH IRU H[LW WHPSHUDWXUH 7H 7L 4 P&3 PL[ × .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ SLSH IORZV NPROH D VHFRQG PL[WXUH ZLWK PROH IUDFWLRQV RI &2 DQG 1 DW N3D . +HDWLQJ WDSH LV ZUDSSHG DURXQG D VHFWLRQ RI SLSH ZLWK LQVXODWLRQ DGGHG DQG N: HOHFWULFDO SRZHU LV KHDWLQJ WKH SLSH IORZ )LQG WKH PL[WXUH H[LW WHPSHUDWXUH 6ROXWLRQ & 9 3LSH KHDWLQJ VHFWLRQ $VVXPH QR KHDW ORVV WR WKH RXWVLGH LGHDO JDVHV − (QHUJ\ (T 4 P KH − KL Q KH − KL Q&3 PL[ 7H − 7L )URP (TV RQ SDJH WKH H[WHQVLRQ RI (T − − &3 PL[ ∑ \L &L × × × ×

N- NPRO 6XEVWLWXWH LQWR HQHUJ\ HTXDWLRQ DQG VROYH IRU H[LW WHPSHUDWXUH − 7H 7L 4 Q&3 PL[ × .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ ULJLG LQVXODWHG YHVVHO FRQWDLQV NJ RI R[\JHQ DW N3D . VHSDUDWHG E\ D PHPEUDQH IURP NJ FDUERQ GLR[LGH DW N3D . 7KH PHPEUDQH LV UHPRYHG DQG WKH PL[WXUH FRPHV WR D XQLIRUP VWDWH )LQG WKH ILQDO WHPSHUDWXUH DQG SUHVVXUH RI WKH PL[WXUH 6ROXWLRQ & 9 7RWDO YHVVHO &RQWURO PDVV ZLWK WZR GLIIHUHQW LQLWLDO VWDWHV 0DVV P P2 P&2 NJ 3URFHVV 9 FRQVWDQW ULJLG ! : LQVXODWHG ! 4 (QHUJ\ 8 8 P2 &9 2 7 7 2 P&2 &9 &2 7 7 &2 ,QLWLDO VWDWH IURP LGHDO JDV 7DEOH $ &9 2 N- NJ &9 &2 N- NJ . 2 92 P57 3 × × P &2 9&2 P57 3 × × P )LQDO VWDWH PL[WXUH

50,; ∑ FL5L > × × @ N- NJ .

7KH HQHUJ\ HTXDWLRQ EHFRPHV P2 &9 2 7 P&2 &9 &2 7

7 N-

3 P5PL[7 9 × × N3D

! 7 . )URP PL[WXUH JDV FRQVWDQW DQG WRWDO YROXPH

P2 &9 2 7 2 P&2 &9 &2 7 &2


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI ZDWHU DQG FDUERQ GLR[LGH E\ PDVV LV KHDWHG IURP . WR . DW FRQVWDQW SUHVVXUH N3D )LQG WKH WRWDO FKDQJH LQ HQWKDOS\ DQG HQWURS\ XVLQJ 7DEOH $ YDOXHV 6ROXWLRQ )URP (T

5PL[ ∑ FL5L × × N- NJ .

)URP (T

&3 PL[ × × N- NJ .

K − K ≅ &3 PL[ 7 − 7 × ± N- NJ

)URP (T

V − V &3 PL[ OQ 7 7 − 5 PL[ OQ 3 3

&3 PL[ OQ 7 7 OQ N- NJ .

$V WKH WZR WRWDO SUHVVXUHV DUH WKH VDPH WKH SUHVVXUH FRUUHFWLRQ WHUP GURSV RXW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

'R 3UREOHP EXW ZLWK YDULDEOH KHDW FDSDFLW\ XVLQJ YDOXHV IURP 7DEOH $ $ PL[WXUH RI ZDWHU DQG FDUERQ GLR[LGH E\ PDVV LV KHDWHG IURP . WR . DW FRQVWDQW SUHVVXUH N3D )LQG WKH WRWDO FKDQJH LQ HQWKDOS\ DQG HQWURS\ XVLQJ 7DEOH $ YDOXHV 6ROXWLRQ )URP (T

K − K F+ 2 K − K + 2 F&2 K − K &2 ± ±

N- NJ )URP (T DQG (T

V − V V°7 − V°7 − 50,; OQ 3 3

F+ 2 V°7 − V7° + 2 F&2 V°7 − V7° &2

± ±

N- NJ . $V WKH WZR WRWDO SUHVVXUHV DUH WKH VDPH WKH SUHVVXUH FRUUHFWLRQ WHUP GURSV RXW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$Q LQVXODWHG JDV WXUELQH UHFHLYHV D PL[WXUH RI &2 + 2 DQG 1

RQ D PDVV EDVLV DW . N3D 7KH YROXPH IORZ UDWH LV P V DQG LWV H[KDXVW LV DW . N3D )LQG WKH SRZHU RXWSXW LQ N: XVLQJ FRQVWDQW VSHFLILF KHDW IURP $ DW . 6ROXWLRQ & 9 7XUELQH 6WHDG\ LQOHW H[LW IORZ ZLWK DQ LGHDO JDV PL[WXUH T (QHUJ\ (T :7 P KL − KH P&3 PL[ 7L − 7H 3URSHUWLHV )URP (TV DQG

5PL[ ∑ FL5L × × × N- NJ .

&3 PL[ ∑ FL &3 L × × ×

N- NJ . 39 P5PL[7 ! P 39 5PL[7 P × × NJ V :7 × − N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

6ROYH 3UREOHP XVLQJ WKH YDOXHV RI HQWKDOS\ IURP 7DEOH $ $Q LQVXODWHG JDV WXUELQH UHFHLYHV D PL[WXUH RI &2 + 2 DQG 1

RQ D PDVV EDVLV DW . N3D 7KH YROXPH IORZ UDWH LV P V DQG LWV H[KDXVW LV DW . N3D )LQG WKH SRZHU RXWSXW LQ N: XVLQJ FRQVWDQW VSHFLILF KHDW IURP $ DW . 6ROXWLRQ & 9 7XUELQH 6WHDG\ LQOHW H[LW IORZ ZLWK DQ LGHDO JDV PL[WXUH T (QHUJ\ (T :7 P KL − KH P ∑FM KL − KH M 3URSHUWLHV )URP (TV DQG

5PL[ ∑ FL5L × × ×

N- NJ . 39 P5PL[7 ! P 39 5PL[7 P × × NJ V 1RZ JHW WKH K YDOXHV IURP 7DEOH $ DOO LQ N- NJ :7 >

@

N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$Q LQVXODWHG JDV WXUELQH UHFHLYHV D PL[WXUH RI &2 + 2 DQG 1

RQ D PROH EDVLV DW . N3D 7KH YROXPH IORZ UDWH LV P V DQG LWV H[KDXVW LV DW . N3D )LQG WKH SRZHU RXWSXW LQ N: XVLQJ FRQVWDQW VSHFLILF KHDW IURP $ DW . & 9 7XUELQH 6WHDG\ IORZ LQOHW H[LW IORZ ZLWK DQ LGHDO JDV PL[WXUH DQG QR KHDW WUDQVIHU VR T − (QHUJ\ (T :7 P KL − KH Q KL − KH Q &3 PL[ 7L − 7H − ,GHDO JDV ODZ 39 Q57 ! 39 × Q − NPROH V 57 ×

7KH PL[WXUH KHDW FDSDFLW\ EHFRPHV − − &3 PL[ ∑ \L &L × × × × × × N- NPRO . :7 × − N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

6ROYH 3UREOHP XVLQJ WKH YDOXHV RI HQWKDOS\ IURP 7DEOH $ & 9 7XUELQH 6WHDG\ IORZ LQOHW H[LW IORZ ZLWK DQ LGHDO JDV PL[WXUH DQG QR KHDW WUDQVIHU VR T (QHUJ\ (T :7 P KL − KH Q KL − KH Q > ∑\M KL − KH M @ − ,GHDO JDV ODZ 39 Q57 ! 39 × NPRO V Q − 57 × 5HDG WKH HQWKDOSLHV IURP 7DEOH $ WKH\ DUH DOO LQ N- NPRO :7 > @

N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ SLVWRQ F\OLQGHU GHYLFH FRQWDLQV NJ RI D PL[WXUH RI PHWKDQH DQG SURSDQH JDVHV E\ PDVV DW . DQG N3D 7KH JDV LV QRZ VORZO\ FRPSUHVVHG LQ DQ LVRWKHUPDO 7 FRQVWDQW SURFHVV WR D ILQDO SUHVVXUH RI N3D 6KRZ WKH SURFHVV LQ D 3 9 GLDJUDP DQG ILQG ERWK WKH ZRUN DQG KHDW WUDQVIHU LQ WKH SURFHVV 6ROXWLRQ & 9 0L[WXUH RI PHWKDQH DQG SURSDQH WKLV LV D FRQWURO PDVV $VVXPH PHWKDQH SURSDQH DUH LGHDO JDVHV DW WKHVH FRQGLWLRQV (QHUJ\ (T P X − X 4 : 3URSHUW\ IURP (T 5PL[ 5&+ 5& + × × N- NJ . 3URFHVV 7 FRQVWDQW LGHDO JDV !

: ∫ 3 G9 P5PL[7 ∫ 9 G9 P5PL[7 OQ 9 9

P5PL[7 OQ 3 3

× × OQ N- 1RZ KHDW WUDQVIHU IURP WKH HQHUJ\ HTXDWLRQ ZKHUH ZH QRWLFH WKDW X LV D FRQVWDQW LGHDO JDV DQG FRQVWDQW 7 VR

4 P X − X : : N-

3

3 & Y

7

Y

7 & V


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQVLGHU 3UREOHP DQG ILQG WKH YDOXH IRU WKH PL[WXUH KHDW FDSDFLW\ PDVV EDVLV DQG WKH PL[WXUH UDWLR RI VSHFLILF KHDWV NPL[ ERWK HVWLPDWHG DW . IURP YDOXHV GLIIHUHQFHV RI K LQ 7DEOH $ :LWK WKHVH YDOXHV PDNH DQ HVWLPDWH IRU WKH UHYHUVLEOH DGLDEDWLF H[LW WHPSHUDWXUH RI WKH WXUELQH DW N3D 6ROXWLRQ :H ZLOO ILQG WKH LQGLYLGXDO KHDW FDSDFLWLHV E\ ILQLWH GLIIHUHQFHV

&3 L GK G7 ∆K ∆7 K K 5HDG WKH K YDOXHV IURP 7DEOH $ &3 &2 N- NJ .

&3 + 2 N- NJ .

&3 1 N- NJ .

3URSHUWLHV )URP (TV DQG

5PL[ ∑ FL5L × × × N- NJ .

&3 PL[ ∑ FL &3 L × × ×

N- NJ . &9 PL[ &3 PL[ 5PL[ N- NJ .

N &3 PL[ &9 PL[

5HYHUVLEOH DGLDEDWLF WXUELQH ! 3URFHVV LV V FRQVWDQW $VVXPH FRQVWDQW DYHUDJH KHDW FDSDFLWLHV VR (T

N 7H 7L 3H 3L N .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQVLGHU 3UREOHP DQG ILQG WKH YDOXH IRU WKH PL[WXUH KHDW FDSDFLW\ PROH EDVLV DQG WKH PL[WXUH UDWLR RI VSHFLILF KHDWV NPL[ ERWK HVWLPDWHG DW . IURP YDOXHV GLIIHUHQFHV RI K LQ 7DEOH $ :LWK WKHVH YDOXHV PDNH DQ HVWLPDWH IRU WKH UHYHUVLEOH DGLDEDWLF H[LW WHPSHUDWXUH RI WKH WXUELQH DW N3D :H ZLOO ILQG WKH LQGLYLGXDO KHDW FDSDFLWLHV E\ ILQLWH GLIIHUHQFHV − &3 L GK G7 ∆K ∆7 K K 1RZ UHDG WKH K YDOXHV IURP 7DEOH $ DOO LQ N- NPRO − &3 &2 N- NPRO . − &3 + 2 N- NPRO . − &3 1 N- NPRO . − − &3 PL[ ∑ \L &3 L × × × N- NPRO . − − − &9 PL[ &3 PL[ 5 N- NPRO . − − N &3 PL[ &9 PL[ 5HYHUVLEOH DGLDEDWLF WXUELQH ! 3URFHVV LV V FRQVWDQW $VVXPH FRQVWDQW DYHUDJH KHDW FDSDFLWLHV N

7H 7L 3H 3L N .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI NJ QLWURJHQ DQG NJ R[\JHQ LV DW N3D . LQ D SLVWRQ F\OLQGHU NHHSLQJ FRQVWDQW SUHVVXUH 1RZ N- LV DGGHG E\ KHDWLQJ )LQG WKH ILQDO WHPSHUDWXUH DQG WKH LQFUHDVH LQ HQWURS\ RI WKH PL[WXUH XVLQJ 7DEOH $ YDOXHV 6ROXWLRQ & 9 0L[WXUH LQ WKH SLVWRQ F\OLQGHU (QHUJ\ (T

P X − X 4 :

3URFHVV 3 FRQVWDQW

!

: ∫ 3 G9 3 9 − 9

4 P X − X : P X − X P3 Y − Y P K − K

K − K 4 P ≅ &3 PL[ 7 − 7

)URP (T DQG 7DEOH $

&3 PL[ × × N- NJ .

7 7 4 P&3 PL[

× . )URP (T

P V − V P>&3 PL[ OQ 7 7 − 5 OQ 3 3 @

× × OQ N- .

Mixture

F=C


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI NJ QLWURJHQ DQG NJ R[\JHQ LV DW N3D . LQ D SLVWRQ F\OLQGHU NHHSLQJ FRQVWDQW SUHVVXUH 1RZ N- LV DGGHG E\ KHDWLQJ )LQG WKH ILQDO WHPSHUDWXUH DQG WKH LQFUHDVH LQ HQWURS\ RI WKH PL[WXUH XVLQJ 7DEOH $ YDOXHV 6ROXWLRQ & 9 0L[WXUH LQ WKH SLVWRQ F\OLQGHU (QHUJ\ (T

P X − X 4 :

3URFHVV 3 FRQVWDQW

!

: ∫ 3 G9 3 9 − 9

4 P X − X : P X − X P3 Y − Y

P K − K

K − K 4 P N- NJ 6LQFH 7 LV VR KLJK ZH XVH 7DEOH $ YDOXHV JXHVVLQJ D 7

K − K . ± ±

N- NJ WRR KLJK K − K . ± ±

7 > ± ± @ .

N- NJ

WRR ORZ

)URP (TV DQG V − V V°7 V°7 1 V°7 V°7 2

± ±

N- NJ .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

1HZ UHIULJHUDQW 5 D LV D PL[WXUH RI 5 DQG 5 LQ D PDVV UDWLR $ SURFHVV EULQJV NJ 5 D IURP . WR . DW D FRQVWDQW SUHVVXUH N3D LQ D SLVWRQ F\OLQGHU )LQG WKH ZRUN DQG KHDW WUDQVIHU 6ROXWLRQ & 9 5 D (QHUJ\ (T

P X − X 4 − : 4 3 9 − 9

3URFHVV 3 FRQVWDQW : 3 9 − 9 P5 7 − 7

4 P X − X : P K − K

)URP (T

5PL[ ∑ FL5L × × N- NJ .

)URP(T &3 PL[ × × N- NJ . )URP WKH SURFHVV HTXDWLRQ

: × ± N-

)URP WKH HQHUJ\ HTXDWLRQ

4 × ± N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ SLVWRQ F\OLQGHU FRQWDLQV NJ DUJRQ DQG NJ K\GURJHQ DW . N3D 7KH PL[WXUH LV FRPSUHVVHG LQ DQ DGLDEDWLF SURFHVV WR N3D E\ DQ H[WHUQDO IRUFH RQ WKH SLVWRQ )LQG WKH ILQDO WHPSHUDWXUH WKH ZRUN DQG WKH KHDW WUDQVIHU LQ WKH SURFHVV 6ROXWLRQ & 9 0L[WXUH LQ F\OLQGHU &RQWURO PDVV ZLWK DGLDEDWLF SURFHVV 4 &RQW (T P P P (QHUJ\ (T P X − X − : (QWURS\ (T P V − V ∫ G4 7 6 JHQ 3URFHVV DGLDEDWLF DQG DVVXPHG UHYHUVLEOH JLYHV LVHQWURSLF

5PL[ ∑ FL5L × × N- NJ .

&3 PL[ ∑ FL&3L × × N- NJ .

&9 &3 5 N- NJ . 5DWLR RI VSHFLILF KHDWV N &3 &9 7KH FRQVWDQW V LVHQWURSLF SURFHVV IURP (T V V ! 7 7 3 3 N N . 7KH HQHUJ\ HTXDWLRQ JLYHV WKH ZRUN DV

: P X − X P&9 7 − 7

× N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

1DWXUDO JDV DV D PL[WXUH RI PHWKDQH DQG HWKDQH E\ PDVV LV IORZLQJ WR D FRPSUHVVRU DW °& N3D 7KH UHYHUVLEOH DGLDEDWLF FRPSUHVVRU EULQJV WKH IORZ WR N3D )LQG WKH H[LW WHPSHUDWXUH DQG WKH QHHGHG ZRUN SHU NJ IORZ 6ROXWLRQ & 9 &RPSUHVVRU 6WHDG\ DGLDEDWLF T UHYHUVLEOH VJHQ (QHUJ\ (T Z KH[ KLQ (QWURS\ (T

VL VJHQ VH

3URFHVV UHYHUVLEOH ! VJHQ ! VH VL $VVXPH LGHDO JDV PL[WXUH DQG FRQVWDQW KHDW FDSDFLW\ VR ZH QHHG N DQG &3 )URP (T DQG

5PL[ ∑ FL5L × × N- NJ .

&3 PL[ ∑ FL&3L × × N- NJ .

&9 &3 PL[ 5PL[ N- NJ . 5DWLR RI VSHFLILF KHDWV N &S &Y 7KH LVHQWURSLF SURFHVV JLYHV (T 7H 7L 3H 3L N N . :RUN IURP WKH HQHUJ\ HTXDWLRQ

ZF LQ &3 7H 7L ± N- NJ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI NJ R[\JHQ DQG NJ RI DUJRQ LV LQ DQ LQVXODWHG SLVWRQ F\OLQGHU DUUDQJHPHQW DW N3D . 7KH SLVWRQ QRZ FRPSUHVVHV WKH PL[WXUH WR KDOI LWV LQLWLDO YROXPH )LQG WKH ILQDO SUHVVXUH WHPSHUDWXUH DQG WKH SLVWRQ ZRUN 6ROXWLRQ & 9 0L[WXUH &RQWURO PDVV ERXQGDU\ ZRUN DQG QR 4 DVVXPH UHYHUVLEOH (QHUJ\ (T X X T Z Z (QWURS\ (T V V 3URFHVV FRQVWDQW V ! 3YN FRQVWDQW Y Y $VVXPH LGHDO JDVHV 7 !! 7& DQG XVH NPL[ DQG &Y PL[ IRU SURSHUWLHV (T 5PL[ Σ FL5L × × N- NJ . (T &3PL[ Σ FL&3L × × N- NJ . &YPL[ &3PL[ 5PL[ N- NJ . 5DWLR RI VSHFLILF KHDWV

NPL[ &3PL[ &YPL[

7KH UHODWLRQV IRU WKH SRO\WURSLF SURFHVV (T 3 3 Y Y N 3 N N3D (T 7 7 Y Y N 7 N . :RUN IURP WKH HQHUJ\ HTXDWLRQ

: PWRW X X PWRW &Y 7 7 × N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH VXEVWDQFH 5 D VHH 3UREOHP LV DW N3D . ,W LV QRZ EURXJKW WR N3D . LQ D UHYHUVLEOH SRO\WURSLF SURFHVV )LQG WKH FKDQJH LQ VSHFLILF YROXPH VSHFLILF HQWKDOS\ DQG VSHFLILF HQWURS\ IRU WKH SURFHVV 6ROXWLRQ (T 5PL[ Σ FL5L × × N- NJ . (T &3PL[ Σ FL&3L × × N- NJ .

Y 57 3 × P NJ

Y 57 3 × P NJ

Y Y ± P NJ

K − K &3PL[ 7 − 7 ± N- NJ )URP (T

V − V &3PL[ OQ 7 7 − 5PL[ OQ 3 3 OQ ± OQ N- NJ .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7ZR LQVXODWHG WDQNV $ DQG % DUH FRQQHFWHG E\ D YDOYH 7DQN $ KDV D YROXPH RI P DQG LQLWLDOO\ FRQWDLQV DUJRQ DW N3D °& 7DQN % KDV D YROXPH RI P DQG LQLWLDOO\ FRQWDLQV HWKDQH DW N3D °& 7KH YDOYH LV RSHQHG DQG UHPDLQV RSHQ XQWLO WKH UHVXOWLQJ JDV PL[WXUH FRPHV WR D XQLIRUP VWDWH 'HWHUPLQH WKH ILQDO SUHVVXUH DQG WHPSHUDWXUH 6ROXWLRQ & 9 7DQNV $ % &RQWURO PDVV QR : QR 4 (QHUJ\ (T 8 8 P$U&9 7 7$ P& + &92 7 7%

P$U 3$ 9$ 57$ × × NJ

P& + 3% 9% 57% × × NJ

&RQWLQXLW\ (T

P P$U P& + NJ

(QHUJ\ (T × 7 × 7

6ROYLQJ 7 .

5PL[ Σ FL5L × × N- NJ .

3 P 57 9$ 9% × × N3D

B

cb

A


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ FRPSUHVVRU EULQJV 5 D VHH SUREOHP IURP ± R& N3D XS WR N3D LQ DQ DGLDEDWLF UHYHUVLEOH FRPSUHVVLRQ $VVXPH LGHDO JDV EHKDYLRU DQG ILQG WKH H[LW WHPSHUDWXUH DQG WKH VSHFLILF ZRUN 6ROXWLRQ & 9 &RPSUHVVRU 3URFHVV T DGLDEDWLF DQG UHYHUVLEOH (QHUJ\ (T Z KL KH (QWURS\ (T VH VL VJHQ ∫ GT 7 VL VL )URP (T 5PL[ ∑ FL5L × × N- NJ . )URP(T &3 PL[ × × N- NJ .

5PL[ &3 PL[

)RU FRQVWDQW V LGHDO JDV DQG XVH FRQVWDQW VSHFLILF KHDW DV LQ (T

5 &S

7H 7L 3H 3L

7H ×

Z ≅ &3 PL[ 7L 7H ±

N- NJ

.


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI FDUERQ GLR[LGH DQG ZDWHU E\ PDVV LV EURXJKW IURP . 03D WR . N3D LQ D SRO\WURSLF SURFHVV WKURXJK D VWHDG\ VWDWH GHYLFH )LQG WKH QHFHVVDU\ KHDW WUDQVIHU DQG ZRUN LQYROYHG XVLQJ YDOXHV IURP 7DEOH $ 6ROXWLRQ 3URFHVV 3YQ FRQVWDQW OHDGLQJ WR Q OQ Y Y OQ 3 3 Y 57 3

 ×    Q OQ   OQ      × 

(T 5PL[ Σ FL5L × × N- NJ . (T &3 PL[ Σ FL&3L × × N- NJ . :RUN LV IURP (T Q Q5 Z ⌠YG3 Q 3HYH 3LYL Q 7H 7L N- NJ ⌡ +HDW WUDQVIHU IURP WKH HQHUJ\ HTXDWLRQ

T KH KL Z &3 7H 7L Z N- NJ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

6ROYH 3UREOHP XVLQJ VSHFLILF KHDWV &3 ∆K ∆7 IURP 7DEOH $ DW . $ PL[WXUH RI FDUERQ GLR[LGH DQG ZDWHU E\ PDVV LV EURXJKW IURP . 03D WR . N3D LQ D SRO\WURSLF SURFHVV WKURXJK D VWHDG\ VWDWH GHYLFH )LQG WKH QHFHVVDU\ KHDW WUDQVIHU DQG ZRUN LQYROYHG XVLQJ YDOXHV IURP 7DEOH $ 6ROXWLRQ 8VLQJ YDOXHV IURP 7DEOH $ ZH HVWLPDWH WKH KHDW FDSDFLWLHV N- NJ .

&3 &2

&3 + 2

N- NJ .

(T &3 PL[ Σ FL&3L × × N- NJ . (T 5PL[ Σ FL5L × × N- NJ . 3URFHVV 3YQ & ! Q OQ 3 3 OQ Y Y DQG XVH 3Y 57

 ×    Q OQ   OQ      × 

:RUN LV IURP (T Q Q5 Z ⌠YG3 Q 3HYH 3LYL Q 7H 7L N- NJ ⌡ +HDW WUDQVIHU IURP HQHUJ\ HTXDWLRQ

T KH KL Z N- NJ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ E\ PDVV JDV PL[WXUH RI PHWKDQH &+ DQG HWK\OHQH & + LV FRQWDLQHG LQ D F\OLQGHU SLVWRQ DW WKH LQLWLDO VWDWH N3D . P 7KH SLVWRQ LV QRZ PRYHG FRPSUHVVLQJ WKH PL[WXUH LQ D UHYHUVLEOH SRO\WURSLF SURFHVV WR WKH ILQDO VWDWH . P &DOFXODWH WKH ILQDO SUHVVXUH WKH SRO\WURSLF H[SRQHQW WKH ZRUN DQG KHDW WUDQVIHU DQG HQWURS\ FKDQJH IRU WKH PL[WXUH 6ROXWLRQ ,GHDO JDV PL[WXUH &+ & + HDFK E\ PDVV ! F&+ F& +

5PL[ Σ FL5L × × N- NJ .

&Y PL[ ∑ FL &YL × × N- NJ .

6WDWH P 3 9 5PL[7 × × NJ 6WDWH 7 . 9 P ,GHDO JDV 39 P57 VR WDNH UDWLR

9 7 ! 3 3 9 7 N3D

7 9 Q 3URFHVV 39Q FRQVWDQW DQG 39 P57 JLYHV 7 9    7 9 OQ 7 Q OQ 9 ! Q DOVR IRU WKLV SURFHVV ZH JHW (T RU (T ! : ∫ 3 G9 Q 3 9 3 9 N- (QHUJ\ (T 4 8 8 : P &Y PL[ 7 7 : × N- &KDQJH RI HQWURS\ IURP (T V V &Y PL[ OQ 7 7 5PL[ OQ 9 9

OQ OQ

N- NJ . DQG 6 6 P V V N- .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH JDV PL[WXUH IURP 3UREOHP LV FRPSUHVVHG LQ D UHYHUVLEOH DGLDEDWLF SURFHVV IURP WKH LQLWLDO VWDWH LQ WKH VDPSOH F\OLQGHU WR D YROXPH RI / 'HWHUPLQH WKH ILQDO WHPSHUDWXUH RI WKH PL[WXUH DQG WKH ZRUN GRQH GXULQJ WKH SURFHVV 6ROXWLRQ )URP (T

5PL[ ∑ FL5L × × ×

P 39 5PL[7 × × × NJ

× N- NJ .

&9 0,; ∑ FL &9 L × ×

× × N- NJ . &3 0,; &9 0,; 5PL[ N- NJ .

→ N &3 &9

7KH SURFHVV DGLDEDWLF DQG UHYHUVLEOH LV LVHQWURSLF H[SUHVVHG LQ (T

9 → 7 7 9

: ∆8 P&9 7 7

× × × N-

N

.


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

(QWURS\ JHQHUDWLRQ

$ IORZ RI NJ V PL[WXUH RI &2 DQG 2 E\ PDVV LV KHDWHG LQ D FRQVWDQW SUHVVXUH KHDW H[FKDQJHU IURP . WR . E\ D UDGLDWLRQ VRXUFH DW . )LQG WKH UDWH RI KHDW WUDQVIHU DQG WKH HQWURS\ JHQHUDWLRQ LQ WKH SURFHVV 6ROXWLRQ & 9 +HDW H[FKDQJHU Z (QHUJ\ (T 4LQ P KH KL 9DOXHV IURP 7DEOH $ GXH WR WKH KLJK 7 4LQ > × ± × ± @ N: (QWURS\ (T PHVH PLVL 4 7V 6JHQ $V WKH SUHVVXUH LV FRQVWDQW WKH SUHVVXUH FRUUHFWLRQ LQ (T GURSV RXW WR JLYH WKH JHQHUDWLRQ DV 6JHQ P VH VL 4 7V

> × ± × ± @

N: . . 5DGLDWLRQ

L

H


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&DUERQ GLR[LGH JDV DW . LV PL[HG ZLWK QLWURJHQ DW . LQ DQ LQVXODWHG PL[LQJ FKDPEHU %RWK IORZV DUH DW N3D DQG WKH PDVV UDWLR RI FDUERQ GLR[LGH WR QLWURJHQ LV )LQG WKH H[LW WHPSHUDWXUH DQG WKH WRWDO HQWURS\ JHQHUDWLRQ SHU NJ RI WKH H[LW PL[WXUH 6ROXWLRQ &9 PL[LQJ FKDPEHU 7KH LQOHW UDWLR LV VR P&2 P1 DQG DVVXPH QR H[WHUQDO KHDW WUDQVIHU QR ZRUN LQYROYHG &RQWLQXLW\ (T P1 P1 PH[ P1 (QHUJ\ (T P1 K1 K&2 P1 KPL[ H[

7DNH . DV UHIHUHQFH DQG ZULWH K K &3PL[ 7

&3 1 7L 1 &3 &2 7L &2 &3 PL[ 7PL[ H[

&3 PL[ ∑ FL&3 L × × N- NJ .

&3 PL[7PL[ H[ &3 1 7L 1 &3 &2 7L &2 N- NJ

7PL[ H[ .

7R ILQG WKH HQWURSLHV ZH QHHG WKH SDUWLDO SUHVVXUHV ZKLFK DVVXPLQJ LGHDO JDV DUH HTXDO WR WKH PROH IUDFWLRQV WLPHV WKH WRWDO SUHVVXUH

\L >FL 0L@ ∑ FM 0M

\1 > @ > @

\&2 − \1 6JHQ PH[VH[ PV L&2 PV L1 P1 VH VL 1 P1 VH VL &2 6JHQ 7 7 >&31 OQ H[ ± 51 OQ \1 @ >&3&2 OQ H[ ± 5&2 OQ \&2 @ 7L1 7L&2 P1

> OQ ± OQ @ > OQ ± OQ @

N- NJ PL[ .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7DNH 3UREOHP ZLWK LQOHW WHPSHUDWXUH RI . IRU WKH FDUERQ GLR[LGH DQG . IRU WKH QLWURJHQ )LUVW HVWLPDWH WKH H[LW WHPSHUDWXUH ZLWK WKH VSHFLILF KHDWV IURP 7DEOH $ DQG XVH WKLV WR VWDUW LWHUDWLRQV XVLQJ $ WR ILQG WKH H[LW WHPSHUDWXUH 6ROXWLRQ &9 PL[LQJ FKDPEHU 7KH LQOHW UDWLR LV VR P&2 P1 DQG DVVXPH QR H[WHUQDO KHDW WUDQVIHU QR ZRUN LQYROYHG

&RQWLQXLW\ (T P1 P1 PH[ P1 (QHUJ\ (T P1 K1 K&2 P1 KPL[ H[

&3 PL[ ∑ FL&3 L × × N- NJ .

7DNH . DV UHIHUHQFH DQG ZULWH K K &3PL[ 7

&3 1 7L 1 &3 &2 7L &2 &3 PL[ 7PL[ H[

&3 PL[7PL[ H[ &3 1 7L 1 &3 &2 7L &2 × × × N- NJ

7PL[ H[ .

$ PRUH DFFXUDWH DQVZHU UHVXOWV IURP XVLQJ WKH LGHDO JDV 7DEOHV )URP 7DEOH $ ΣPLQ KLQ P1 > × × @ P1 × # . ΣPH[ KH[ P1 > × @ P1 × # . ΣPH[ KH[ P1 > × @ P1 ×

1RZ OLQHDU LQWHUSRODWLRQ EHWZHHQ . DQG . 7H[ × .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&DUERQ GLR[LGH JDV DW . LV PL[HG ZLWK QLWURJHQ DW . LQ DQ LQVXODWHG PL[LQJ FKDPEHU %RWK IORZV DUH FRPLQJ LQ DW N3D DQG WKH PROH UDWLR RI FDUERQ GLR[LGH WR QLWURJHQ LV )LQG WKH H[LW WHPSHUDWXUH DQG WKH WRWDO HQWURS\ JHQHUDWLRQ SHU NPROH RI WKH H[LW PL[WXUH &9 PL[LQJ FKDPEHU VWHDG\ IORZ 7KH LQOHW UDWLR LV Q&2 Q1 DQG

DVVXPH QR H[WHUQDO KHDW WUDQVIHU QR ZRUN LQYROYHG &RQWLQXLW\ Q&2 Q1 QH[ Q1 (QHUJ\ (T Q1 K1 K&2 Q1 KPL[ H[ 7DNH . DV UHIHUHQFH DQG ZULWH K K &3PL[ 7 &3 1 7L 1 &3 &2 7L &2 &3 PL[ 7PL[ H[ )LQG WKH VSHFLILF KHDWV LQ 7DEOH $ WR JHW &3 PL[ ∑ \L&3 L × × ×

N- NPRO . &3 PL[7PL[ H[ &3 1 7L 1 &3 &2 7L &2 N- NPRO

7PL[ H[ .

3DUWLDO SUHVVXUHV DUH WRWDO SUHVVXUH WLPHV PROHIUDFWLRQ 3H[ 1 3WRW 3H[ &2 3WRW 6JHQ QH[VH[ QV L&2 QV L1 Q1 VH VL 1 Q1 VH VL &2

7H[ − 7H[ − 6JHQ Q1 >&31 OQ 7 5OQ \1 &3&2 OQ 7 5OQ \&2 @

L1

L&2

> @ N- NPRO PL[ .

1 &2

0,;,1* &+$0%(5 FE

0L[

6 JHQ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7DNH 3UREOHP ZLWK LQOHW WHPSHUDWXUH RI . IRU WKH FDUERQ GLR[LGH DQG . IRU WKH QLWURJHQ )LUVW HVWLPDWH WKH H[LW WHPSHUDWXUH ZLWK WKH VSHFLILF KHDWV IURP 7DEOH $ DQG XVH WKLV WR VWDUW LWHUDWLRQV XVLQJ $ WR ILQG WKH H[LW WHPSHUDWXUH &9 PL[LQJ FKDPEHU VWHDG\ IORZ 7KH LQOHW UDWLR LV Q&2 Q1 DQG

DVVXPH QR H[WHUQDO KHDW WUDQVIHU QR ZRUN LQYROYHG &3 &2 × &3 1 × N- NPRO . &RQWLQXLW\ (TXDWLRQ ΣQLQ ΣQH[ (QHUJ\ (TXDWLRQ ΣQLQ KLQ ΣQH[ KH[ Q1 &3 &2 7LQ 7H[ &2 Q1 &3 1 7LQ 7H[ 1

× × 7H[ × 7H[

± 7H[

)URP 7DEOH $ ΣQLQ KLQ # . ΣQH[ KH[ # . ΣQH[ KH[ # . ΣQH[ KH[

Q1 > × × @ Q1 > × @ Q1 > × @ Q1 > × @

7H[ . Q1 × Q1 × Q1 × Q1 ×

1RZ OLQHDU LQWHUSRODWLRQ EHWZHHQ . DQG . 7H[ × .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH RQO\ NQRZQ VRXUFHV RI KHOLXP DUH WKH DWPRVSKHUH PROH IUDFWLRQ DSSUR[LPDWHO\ × − DQG QDWXUDO JDV $ ODUJH XQLW LV EHLQJ FRQVWUXFWHG WR VHSDUDWH P V RI QDWXUDO JDV DVVXPHG WR EH +H PROH IUDFWLRQ DQG &+ 7KH JDV HQWHUV WKH XQLW DW N3D °& 3XUH KHOLXP H[LWV DW N3D °& DQG SXUH PHWKDQH H[LWV DW N3D °& $Q\ KHDW WUDQVIHU LV ZLWK WKH VXUURXQGLQJV DW °& ,V DQ HOHFWULFDO SRZHU LQSXW RI N: VXIILFLHQW WR GULYH WKLV XQLW" 3 N3D 1 2 &+ He 7 R& +H CH4 DW N3D R& 3 N3D 3 9 P V 7 R& : N:

&9

Q 3 9 57 × × NPRO V ! Q Q Q NPRO V &3 +H × N- NPRO . &3 &+ × N- NPRO . 4&9 Q K Q K Q K :&9 Q &3 +H 7 7 Q &3 &+ 7 7 :&9

× × N: 6JHQ Q V Q V Q V 4&9 7

>

OQ OQ

N: . !

>

@

×

OQ OQ

@

×


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI NJ V FDUERQ GLR[LGH DW . N3D LV PL[HG ZLWK D IORZ RI NJ V ZDWHU DW . N3D DQG DIWHU WKH PL[LQJ LW JRHV WKURXJK D KHDW H[FKDQJHU ZKHUH LW LV FRROHG WR . E\ D . DPELHQW +RZ PXFK KHDW WUDQVIHU LV WDNHQ RXW LQ WKH KHDW H[FKDQJHU" :KDW LV WKH HQWURS\ JHQHUDWLRQ UDWH IRU WKH ZKROH SURFHVV" 6ROXWLRQ 1 3 4 Mixing chamber 2

& 9 7RWDO PL[LQJ VHFWLRQ DQG KHDW H[FKDQJHU 6WHDG\ IORZ DQG QR ZRUN 7R GR WKH HQWURS\ DW WKH SDUWLDO SUHVVXUHV ZH QHHG WKH PROH IUDFWLRQV

Q+ 2 P+ 2 0+ 2 NPRO V Q&2 P&2 0&2 NPRO V

\+ 2 \&2 ± \+ 2 (QHUJ\ (T P+ 2 K P&2 K 4FRRO P+ 2 K + 2 P&2 K &2 4FRRO (QWURS\ (T P+ 2 V P&2 V 6JHQ 7 P+ 2 V + 2 P&2 V &2 DPE

Q cool

$V 7 LV IDLUO\ KLJK ZH XVH 7DEOH $ IRU SURSHUWLHV RQ D PDVV EDVLV + 2 K >N- NJ@ R V >N- NJ .@ 7

4FRRO P+ 2 K ± K + 2 P&2 K ± K &2

&2

± ± N: 4FRRO 6JHQ P+ 2 V + 2 ± V P&2 V &2 ± V 7 DPE

> ± ± OQ @ > ± ± OQ @ ± N: .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI KHOLXP DQG QLWURJHQ E\ PDVV HQWHUV D WXUELQH DW 03D . DW D UDWH RI NJ V 7KH DGLDEDWLF WXUELQH KDV DQ H[LW SUHVVXUH RI N3D DQG DQ LVHQWURSLF HIILFLHQF\ RI )LQG WKH WXUELQH ZRUN 6ROXWLRQ $VVXPH LGHDO JDV PL[WXUH DQG WDNH &9 DV WXUELQH (QHUJ\ (T Z7 V KL KHV (QWURS\ (T VHV VL DGLDEDWLF DQG UHYHUVLEOH 3URFHVV (T 7 7 3 3 N N HV

L

H

L

3URSHUWLHV IURP (T DQG

&3 PL[ × × N- NJ .

5PL[ × × N- NJ .

N N 5 &3 PL[

7HV .

Z7V &3 7L 7HV N- NJ

Z7 DF ηZ7V N- NJ :7 DF PZ7 DF N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

5HSHDW 3UREOHP IRU DQ LVHQWURSLF FRPSUHVVRU HIILFLHQF\ RI 6ROXWLRQ & 9 &RPSUHVVRU 6WHDG\ DGLDEDWLF T UHYHUVLEOH VJHQ (QHUJ\ (T Z KH[ KLQ (QWURS\ (T 3URFHVV UHYHUVLEOH ! VJHQ ! VH VL

VL VJHQ VL VH

$VVXPH LGHDO JDV PL[WXUH DQG FRQVWDQW KHDW FDSDFLW\ VR ZH QHHG N DQG &3 )URP (T DQG

5PL[ ∑ FL5L × × N- NJ .

&3 PL[ ∑ FL&3L × × N- NJ .

&9 &3 PL[ 5PL[ N- NJ . 5DWLR RI VSHFLILF KHDWV N &S &Y 7KH LVHQWURSLF SURFHVV JLYHV (T 7H 7L 3H 3L N N . :RUN IURP WKH HQHUJ\ HTXDWLRQ ZF LQ &3 7H 7L ± N- NJ

7KH DFWXDO FRPSUHVVRU UHTXLUHV PRUH ZRUN

ZF DFWXDO ZF LQ η N- NJ &S 7H DFWXDO 7L ! 7H DFWXDO 7 ZF DFWXDO &3 .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ ODUJH DLU VHSDUDWLRQ SODQW WDNHV LQ DPELHQW DLU 1 2 E\ PROH DW N3D °& DW D UDWH RI NJ V ,W GLVFKDUJHV D VWUHDP RI SXUH 2 JDV DW N3D °& DQG D VWUHDP RI SXUH 1 JDV DW N3D °& 7KH SODQW RSHUDWHV RQ DQ HOHFWULFDO SRZHU LQSXW RI N: &DOFXODWH WKH QHW UDWH RI HQWURS\ FKDQJH IRU WKH SURFHVV $LU 1 3 N3D SXUH 2 2 7 R& 3 N3D SXUH 1 R 3 N3D 7 & 7 R& P NJ V :,1 N: 6ROXWLRQ 7R KDYH WKH IORZ WHUPV RQ D PDVV EDVLV OHW XV ILQG WKH PDVV IUDFWLRQV

)URP (T FL \L 0L ∑ \M0M

F2 × > × × @

F1 F2 P F2 P NJ V P F1 P NJ V 7KH HQHUJ\ HTXDWLRQ (T JLYHV WKH KHDW WUDQVIHU UDWH DV 4&9 Σ P∆KL :&9 P2 &3 2 7 −7 P1 &3 1 7 − 7 :&9 × × − − N: 7KH HQWURS\ HTXDWLRQ (T JLYHV WKH JHQHUDWLRQ UDWHV DV 6JHQ Σ PL∆VL − 4&9 7 P V P V − P V − 4&9 7 8VH (T IRU WKH HQWURS\ FKDQJH

>

@

Σ PL∆VL OQ − OQ

> − OQ @ − N: . 6JHQ − N: .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ VWHDG\ IORZ RI NJ V RI FDUERQ GLR[LGH DQG ZDWHU E\ PDVV DW . DQG N3D LV XVHG LQ D KHDW H[FKDQJHU ZKHUH N: LV H[WUDFWHG IURP WKH IORZ )LQG WKH IORZ H[LW WHPSHUDWXUH DQG WKH UDWH RI FKDQJH RI HQWURS\ XVLQJ 7DEOH $ 6ROXWLRQ & 9 +HDW H[FKDQJHU 6WHDG\ LQOHW H[LW QR ZRUN &RQWLQXLW\ (T F&2 F+ 2 (QHUJ\ (T 4 P KH − KL ! KH KL 4 P ,QOHW VWDWH 7DEOH $ KL × × N- NJ ([LW VWDWH KH KL 4 P N- NJ 7ULDO DQG HUURU IRU 7 ZLWK K YDOXHV IURP 7DEOH $ # . KH N- NJ

# . KH N- NJ

# . KH N- NJ

,QWHUSRODWH WR KDYH WKH ULJKW K 7 . (QWURS\ (T PVH PVL 4 7 6JHQ 7KH UDWH RI FKDQJH RI HQWURS\ IRU WKH IORZ LV 3 LV DVVXPHG FRQVWDQW R R P VH VL P V7H V7L

> @

N: . 7KH HQWURS\ JHQHUDWLRQ UDWH FDQQRW EH HVWLPDWHG XQOHVV WKH DYHUDJH 7 DW ZKLFK WKH KHDW WUDQVIHU OHDYHV WKH FRQWURO YROXPH LV NQRZQ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ VWHDG\ IORZ RI NPRO V RI FDUERQ GLR[LGH DQG ZDWHU DW . DQG N3D LV XVHG LQ D KHDW H[FKDQJHU ZKHUH N: LV H[WUDFWHG IURP WKH IORZ )LQG WKH IORZ H[LW WHPSHUDWXUH DQG WKH UDWH RI FKDQJH RI HQWURS\ XVLQJ 7DEOH $ & 9 +HDW H[FKDQJHU 6WHDG\ IORZ LQOHW H[LW QR ZRUN &RQWLQXLW\ (T \&2 \+ 2 (QHUJ\ (T 4 P KH − KL Q KH − KL ! KH KL 4 Q ,QOHW VWDWH 7DEOH $ KL × × N- NPRO ([LW VWDWH KH KL 4 Q N- NPRO 7ULDO DQG HUURU IRU 7 ZLWK K YDOXHV IURP 7DEOH $ # . KH N- NPRO # . KH N- NPRO

,QWHUSRODWH WR KDYH WKH ULJKW K 7 .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI NJ V VWHDP DW N3D R& LV PL[HG ZLWK NJ V R[\JHQ DW N3D . LQ D VWHDG\ IORZ PL[LQJ FKDPEHU ZLWKRXW DQ\ KHDW WUDQVIHU )LQG WKH H[LW WHPSHUDWXUH DQG WKH UDWH RI HQWURS\ JHQHUDWLRQ & 9 0L[LQJ FKDPEHU VWHDG\ IORZ QR ZRUN QR KHDW WUDQVIHU 7R GR WKH HQWURSLHV ZH QHHG WKH PROH IUDFWLRQV

P+ 2 P2 Q+ 2 0 NPRO V Q2 0 NPRO V + 2 2

\+ 2 \2 (QHUJ\ (T P+ 2 K P2 K P+ 2 K + 2 P2 K 2 (QWURS\ (T P+ 2 V P2 V 6JHQ P+ 2 V + 2 P2 V 2 6ROYH IRU 7 IURP WKH HQHUJ\ HTXDWLRQ P+ 2 K + 2 ± K P2 K 2 ± K P+ 2 &3 + 2 7 ± 7 P2 &3 2 7 ± 7

× 7 ± ± × 7 ±

7 . 6JHQ P+ 2 V + 2 ± V P2 V 2 ± V

7 7 P+ 2 > &3 + 2 OQ 7 5 OQ \+ 2 @ P2 > &3 2 OQ 7 5 OQ \2 @

> OQ ± OQ @

N: .

> OQ ± OQ @


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ WDQN KDV WZR VLGHV LQLWLDOO\ VHSDUDWHG E\ D GLDSKUDJP 6LGH $ FRQWDLQV NJ RI ZDWHU DQG VLGH % FRQWDLQV NJ RI DLU ERWK DW °& N3D 7KH GLDSKUDJP LV QRZ EURNHQ DQG WKH ZKROH WDQN LV KHDWHG WR °& E\ D °& UHVHUYRLU )LQG WKH ILQDO WRWDO SUHVVXUH KHDW WUDQVIHU DQG WRWDO HQWURS\ JHQHUDWLRQ & 9 7RWDO WDQN RXW WR UHVHUYRLU (QHUJ\ (T 8 8 PD X X D PY X X Y 4 (QWURS\ (T DQG 6 6 PD V V D PY V V Y 4 7UHV 6JHQ 9ROXPH 9 9$ 9% PYYY PDYD P

YY 9 PY 7 ! 3 Y N3D

YD 9 PD 7 ! 3 D P57 9 N3D 3 WRW 3 Y 3 D N3D

:DWHU WDEOH % X N- NJ X N- NJ V N- NJ . V N- NJ . $LU WDEOH $ X N- NJ X N- NJ V7 N- NJ . V7 N- NJ . )URP HQHUJ\ HTXDWLRQ ZH KDYH

4 N-

)URP WKH HQWURS\ HTXDWLRQ ZH KDYH

6JHQ > ×OQ @

N- .

$

4

%

&


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KUHH VWHDG\ IORZV DUH PL[HG LQ DQ DGLDEDWLF FKDPEHU DW N3D )ORZ RQH LV NJ V RI 2 DW . IORZ WZR LV NJ V RI 1 DW . DQG IORZ WKUHH LV NJ V RI &2 DW . $OO IORZV DUH DW N3D WKH VDPH DV WKH WRWDO H[LW SUHVVXUH )LQG WKH H[LW WHPSHUDWXUH DQG WKH UDWH RI HQWURS\ JHQHUDWLRQ LQ WKH SURFHVV 6ROXWLRQ & 9 0L[LQJ FKDPEHU QR KHDW WUDQVIHU QR ZRUN 2 &RQWLQXLW\ (T P P P P 0,; 1 (QHUJ\ (T P K P K P K P K PL[ &2 (QWURS\ (T P V P V P V 6JHQ P V

$VVXPH LGHDO JDVHV DQG VLQFH 7 LV FORVH WR . XVH KHDW FDSDFLW\ IURP $ LQ WKH HQHUJ\ HTXDWLRQ DV P &3 2 7 7 P &3 1 7 7 P &3 &2 7 7

× × × × × ×

× × × 7

! 7 ! 7 . 6WDWH LV D PL[WXUH VR WKH FRPSRQHQW H[LW SUHVVXUH LV WKH SDUWLDO SUHVVXUH )RU HDFK FRPSRQHQW VH − VL &3 OQ 7H 7L − 5 OQ 3H 3L DQG WKH SUHVVXUH UDWLR LV 3H 3L \ 3 3L \ IRU HDFK P Q ∑ 0

\2 \1 \&2 7KH HQWURS\ JHQHUDWLRQ EHFRPHV 6JHQ P V V P V V P V V

> OQ OQ @ > OQ OQ @ > OQ OQ @

N: .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

5HFRQVLGHU WKH 3UREOHP EXW OHW WKH WDQNV KDYH D VPDOO DPRXQW RI KHDW WUDQVIHU VR WKH ILQDO PL[WXUH LV DW . )LQG WKH ILQDO SUHVVXUH WKH KHDW WUDQVIHU DQG WKH HQWURS\ FKDQJH IRU WKH SURFHVV & 9 %RWK WDQNV &RQWURO PDVV ZLWK PL[LQJ DQG KHDWLQJ RI WZR LGHDO JDVHV × NPRO ×

− Q$U 3$ 9$ 57$

− Q& + 3% 9% 57%

&RQWLQXLW\ (T (QHUJ\ (T

× NPRO ×

Q Q$U Q& + NPRO 8 8 Q$U&9 7 7$ Q& + &92 7 7% 4

− 3 Q 57 9$ 9% × × N3D

4 × ×

× × N- ∆66855 4 76855 ∆66<6 Q$U∆6$U Q& + ∆6& +

\$U

7 − \$U3 ∆6$U &3 $U OQ 7 5 OQ 3

× × OQ OQ

N- NPRO .

7 − \ & + 3 ∆6& + && + OQ 7 5 OQ 3

$

$

%

%

× × OQ OQ

N- NPRO . $VVXPH WKH VXUURXQGLQJV DUH DW . LW KHDWV WKH JDV ∆61(7 Q$U∆6$U Q& + ∆6& + ∆66855

× ×

N- .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$LU ZDWHU YDSRU PL[WXUHV

$WPRVSKHULF DLU LV DW N3D R& DQG UHODWLYH KXPLGLW\ )LQG WKH DEVROXWH KXPLGLW\ DQG WKH GHZ SRLQW RI WKH PL[WXUH ,I WKH PL[WXUH LV KHDWHG WR R& ZKDW LV WKH QHZ UHODWLYH KXPLGLW\" 6ROXWLRQ (T 3Y φ 3J × N3D (T Z 3Y 3WRW 3Y × ± 7GHZ LV WKH 7 VXFK WKDW 3J 7 3Y N3D % +HDWLQJ

! !

7 ≅ °& Z LV FRQVWDQW

! 3Y LV FRQVWDQW

)URP 7DEOH % 3J °& N3D

φ = 3Y 3J RU


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQVLGHU P RI DWPRVSKHULF DLU ZKLFK LV DQ DLU±ZDWHU YDSRU PL[WXUH DW N3D °& DQG UHODWLYH KXPLGLW\ )LQG WKH PDVV RI ZDWHU DQG WKH KXPLGLW\ UDWLR :KDW LV WKH GHZ SRLQW RI WKH PL[WXUH" 6ROXWLRQ $LU YDSRU 3 N3D 7 R& φ 8VH 7DEOH % DQG WKHQ (T

3J 3VDW N3D ! 3Y φ 3J × N3D 3Y 9 × PY 5 7 NJ × Y

3D 3WRW 3Y ± N3D

3D9 × NJ PD 5 7 × D

PY Z P D

7GHZ LV 7 ZKHQ 3Y 3J N3D

7DEOH % JLYHV 7 R&


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH SURGXFWV RI FRPEXVWLRQ DUH IORZLQJ WKURXJK D KHDW H[FKDQJHU ZLWK &2 + 2 DQG 1 RQ D YROXPH EDVLV DW WKH UDWH NJ V DQG N3D :KDW LV WKH GHZ SRLQW WHPSHUDWXUH" ,I WKH PL[WXUH LV FRROHG °& EHORZ WKH GHZ SRLQW WHPSHUDWXUH KRZ ORQJ ZLOO LW WDNH WR FROOHFW NJ RI OLTXLG ZDWHU" 6ROXWLRQ 9ROXPH EDVLV LV WKH VDPH DV PROH IUDFWLRQ

\+ 2 3+ 2 × N3D

7DEOH % 7'(: R&

&RRO WR R& 7'(: VR VDWXUDWHG → 3* N3D

\+ 2 Q+ 2 Y Q+ 2 Y Q+ 2 Y SHU NPRO PL[ LQ

→ Q/,4

00,; ,1 × × × NJ NPRO Q0,; ,1 P727$/ 00,; ,1 NPRO V Q/,4 &21' × NPRO V RU P/,4 &21' × NJ V

)RU NJ WDNHV ~ PLQXWHV


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI NJ V VDWXUDWHG PRLVW DLU UHODWLYH KXPLGLW\ DW N3D R& JRHV WKURXJK D KHDW H[FKDQJHU DQG FRPHV RXW DW R& :KDW LV WKH H[LW UHODWLYH KXPLGLW\ DQG KRZ PXFK SRZHU LV QHHGHG" 6ROXWLRQ 6WDWH φ

3Y 3J

(T Z 3Y 3D × ± 6WDWH 1R ZDWHU DGGHG ! Z Z ! 3Y 3Y φ 3Y 3J RU (QHUJ\ (T 4 P K P K PD K K DLU ZPD K K YDSRU PWRW PD PY PD Z

(QHUJ\ HTXDWLRQ ZLWK &3 DLU IURP $ DQG K¶V IURP % PWRW PWRW 4 Z &3 DLU ± Z Z KJ KJ

× × ±

N:

. 4


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ QHZ KLJK HIILFLHQF\ KRPH KHDWLQJ V\VWHP LQFOXGHV DQ DLU WR DLU KHDW H[FKDQJHU ZKLFK XVHV HQHUJ\ IURP RXWJRLQJ VWDOH DLU WR KHDW WKH IUHVK LQFRPLQJ DLU ,I WKH RXWVLGH DPELHQW WHPSHUDWXUH LV − °& DQG WKH UHODWLYH KXPLGLW\ LV KRZ PXFK ZDWHU ZLOO KDYH WR EH DGGHG WR WKH LQFRPLQJ DLU LI LW IORZV LQ DW WKH UDWH RI P V DQG PXVW HYHQWXDOO\ EH FRQGLWLRQHG WR °& DQG UHODWLYH KXPLGLW\" 6ROXWLRQ 2XWVLGH DPELHQW DLU 39 φ 3* × N3D $VVXPLQJ 3 3 N3D ! 3$ N3D 3$ 9 × P$ 5 7 NJ V × $ )URP (T Z ×

&RQGLWLRQHG WR 7 R& φ

(T 39 φ 3* × ! (T Z × &RQWLQXLW\ HTXDWLRQ IRU ZDWHU P/,4 ,1 P$ Z Z

NJ V NJ K

,1 2876,'(

6,'(


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQVLGHU D P V IORZ RI DWPRVSKHULF DLU DW N3D °& DQG UHODWLYH KXPLGLW\ $VVXPH WKLV IORZV LQWR D EDVHPHQW URRP ZKHUH LW FRROV WR °& N3D +RZ PXFK OLTXLG ZDWHU ZLOO FRQGHQVH RXW" 6ROXWLRQ 6WDWH

3J 3VDW N3D ! 3Y φ 3J × N3D 3Y 9 × PY 5 7 NJ V × Y PY 3Y Z 3 P$ $ PY P$ Z NJ V P$ FRQWLQXLW\ IRU DLU

&KHFN IRU VWDWH 3J °& N3D 3Y VR OLTXLG ZDWHU RXW

. 4

/LTXLG

6WDWH LV VDWXUDWHG φ 3Y 3J N3D 3Y Z 3 $ PY Z P$ × NJ V POLT PY PY ± NJ V 1RWH WKDW WKH JLYHQ YROXPH IORZ UDWH DW WKH LQOHW LV QRW WKDW DW WKH H[LW 7KH PDVV IORZ UDWH RI GU\ DLU LV WKH TXDQWLW\ WKDW LV WKH VDPH DW WKH LQOHW DQG H[LW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI NJ V FRPSOHWHO\ GU\ DLU DW 7 N3D LV FRROHG GRZQ WR °& E\ VSUD\LQJ OLTXLG ZDWHU DW °& N3D LQWR LW VR LW EHFRPHV VDWXUDWHG PRLVW DLU DW °& 7KH SURFHVV LV VWHDG\ VWDWH ZLWK QR H[WHUQDO KHDW WUDQVIHU RU ZRUN )LQG WKH H[LW PRLVW DLU KXPLGLW\ UDWLR DQG WKH IORZ UDWH RI OLTXLG ZDWHU )LQG DOVR WKH GU\ DLU LQOHW WHPSHUDWXUH 7 6ROXWLRQ VDWXUDWHG

3Y 3J N3D DQG KIJ °& N- NJ

(T

Z ×

'U\ DLU

&RQWLQXLW\ (T

(QHUJ\ (T

& 9 %R[

/LTXLG ZDWHU

PD POLT P D Z ! POLT Z PD NJ V PD KD POLT KI PD KD Z KJ

KD KD &SD 7 7 Z KJ Z KI Z KIJ N- NJ GU\ DLU

! 7 °&


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ SLVWRQ F\OLQGHU KDV NJ RI VDWXUDWHG PRLVW DLU DW N3D °& ,I LW LV KHDWHG WR °& LQ DQ LVREDULF SURFHVV ILQG 4 DQG WKH ILQDO UHODWLYH KXPLGLW\ ,I LW LV FRPSUHVVHG IURP WKH LQLWLDO VWDWH WR N3D LQ DQ LVRWKHUPDO SURFHVV ILQG WKH PDVV RI ZDWHU FRQGHQVLQJ 6ROXWLRQ (QHUJ\ (T P X X 4 : ,QLWLDO VWDWH φ 7DEOH % 3Y N3D KY 3Y (T Z 3 3 WRW Y

(T ZLWK PD PWRW PY PWRW Z PD JLYHV

(T

PD PWRW Z NJ PY Z PD NJ

&DVH D 3 FRQVWDQW ! : P3 Y Y ! 4 P X X : P K K PD&S 7 7 P KY KY 6WDWH Z Z 7 ! 3Y 3Y DQG 7DEOH % KY N- NJ 3J N3D 3Y φ 3 RU φ J

(T

)URP WKH HQHUJ\ HTXDWLRQ

4 × N-

&DVH E 7 FRQVWDQW φ ! 3Y 3J N3D 3Y 3Y Z 3 3 3 D WRW Y PY Z PD NJ POLT PY PY NJ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ VDWXUDWHG DLU ZDWHU YDSRU PL[WXUH DW R& N3D LV FRQWDLQHG LQ D P FORVHG WDQN LQ HTXLOLEULXP ZLWK NJ RI OLTXLG ZDWHU 7KH WDQN LV KHDWHG WR R& ,V WKHUH DQ\ OLTXLG ZDWHU LQ WKH ILQDO VWDWH" )LQG WKH KHDW WUDQVIHU IRU WKH SURFHVV D 6LQFH 9/,4 P/,4Y) ≈ P 9*$6 ≈ 9 $,5 +

9$3 /,4

4

φ → 3Y 3* N3D

Z ×

3D 9 × PD 5 7 NJ ! PY Z PD NJ × D $W VWDWH 3D × × N3D

Z0$; ×

%XW Z $&78$/

Z0$; → 1R OLTXLG DW

4 PD XD XD PY XY PY XY POLT XOLT

× × × ×

N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$PELHQW PRLVW DLU HQWHUV D VWHDG\ IORZ DLU FRQGLWLRQLQJ XQLW DW N3D °& ZLWK D UHODWLYH KXPLGLW\ 7KH YROXPH IORZ UDWH HQWHULQJ WKH XQLW LV / V 7KH PRLVW DLU OHDYHV WKH XQLW DW N3D °& ZLWK D UHODWLYH KXPLGLW\ RI /LTXLG FRQGHQVDWH DOVR OHDYHV WKH XQLW DW °& 'HWHUPLQH WKH UDWH RI KHDW WUDQVIHU IRU WKLV SURFHVV 6ROXWLRQ 6WDWH

39 φ 3* ×

Z × 3$ 9 × P$ 5 7 NJ V × $

39 3* Z × (QHUJ\ (T 4&9 P$K$ P9 K9 P$K$ P9 K$ P K/ 4&9 P$ &3 $ 7 7 Z K9 Z K9 Z Z K/

× ×

× N- NJ DLU 4&9 N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQVLGHU D / ULJLG WDQN FRQWDLQLQJ DQ DLU±ZDWHU YDSRU PL[WXUH DW N3D °& ZLWK D UHODWLYH KXPLGLW\ 7KH V\VWHP LV FRROHG XQWLO WKH ZDWHU MXVW EHJLQV WR FRQGHQVH 'HWHUPLQH WKH ILQDO WHPSHUDWXUH LQ WKH WDQN DQG WKH KHDW WUDQVIHU IRU WKH SURFHVV 6ROXWLRQ

3Y φ3* × N3D 6LQFH PY FRQVW 9 FRQVW DOVR 3Y 3*

3* 3Y × 7 7 × 7 7 $VVXPH 7 R& × 3* & $VVXPH 7 R& × 3* &

Z Z

LQWHUSRODWLQJ → 7 R&

PD 3D 9 5D7 × × NJ VW ODZ (T 4 8 8 PD XD XD PY XY XY

N- NJ

→ 4 N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$LU LQ D SLVWRQ F\OLQGHU LV DW °& N3D DQG D UHODWLYH KXPLGLW\ RI ,W LV QRZ FRPSUHVVHG WR D SUHVVXUH RI N3D LQ D FRQVWDQW WHPSHUDWXUH SURFHVV )LQG WKH ILQDO UHODWLYH DQG VSHFLILF KXPLGLW\ DQG WKH YROXPH UDWLR 9 9 6ROXWLRQ &KHFN WR VHH LI WKH VHFRQG VWDWH LV VDWXUDWHG RU QRW )LUVW DVVXPH QR ZDWHU LV FRQGHQVHG Z Z 3Y 3 3Y

Z Z ! 3Y ! 3J N3D &RQFOXVLRQ LV VWDWH LV VDWXUDWHG

φ Z 3J 3 3J

7R JHW WKH YROXPH UDWLR ZULWH WKH LGHDO JDV ODZ IRU WKH YDSRU SKDVHV 9 9D 9Y 9I PD5D PY 5Y 7 3 POLT YI 9 9D 9Y PD5D PY 5Y 7 3 7DNH WKH UDWLR DQG GLYLGH WKURXJK ZLWK PD5D7 3 WR JHW 9 9 3 3 > Z Z Z 3 YI 5D7@ > Z @

7KH OLTXLG FRQWULEXWLRQ LV QHDUO\ ]HUR LQ WKH QXPHUDWRU


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ / ULJLG YHVVHO LQLWLDOO\ FRQWDLQV PRLVW DLU DW N3D °& ZLWK D UHODWLYH KXPLGLW\ RI $ VXSSO\ OLQH FRQQHFWHG WR WKLV YHVVHO E\ D YDOYH FDUULHV VWHDP DW N3D °& 7KH YDOYH LV RSHQHG DQG VWHDP IORZV LQWR WKH YHVVHO XQWLO WKH UHODWLYH KXPLGLW\ RI WKH UHVXOWDQW PRLVW DLU PL[WXUH LV 7KHQ WKH YDOYH LV FORVHG 6XIILFLHQW KHDW LV WUDQVIHUUHG IURP WKH YHVVHO VR WKH WHPSHUDWXUH UHPDLQV DW °& GXULQJ WKH SURFHVV 'HWHUPLQH WKH KHDW WUDQVIHU IRU WKH SURFHVV WKH PDVV RI VWHDP HQWHULQJ WKH YHVVHO DQG WKH ILQDO SUHVVXUH LQVLGH WKH YHVVHO 6ROXWLRQ 3 φ 3 × N3D Y

L

+ 2

$,5 + 2

*

3Y × N3D 3D 3D N3D

Z × Z ×

PD × × NJ

3 N3D

PYL NJ

XY XY ≈ X* DW R& DQG XD XD (QHUJ\ (T 4&9 PD XD XD PY XY PY XY PYLKL

PYL X* DW 7 KL N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ ULJLG FRQWDLQHU P LQ YROXPH FRQWDLQV PRLVW DLU DW °& N3D φ = 7KH FRQWDLQHU LV QRZ FRROHG WR °& 1HJOHFW WKH YROXPH RI DQ\ OLTXLG WKDW PLJKW EH SUHVHQW DQG ILQG WKH ILQDO PDVV RI ZDWHU YDSRU ILQDO WRWDO SUHVVXUH DQG WKH KHDW WUDQVIHU 6ROXWLRQ &9 FRQWDLQHU P P P X P X 4 6WDWH °& φ ! Z 7GHZ °&

)LQDO VWDWH 7 7GHZ VR FRQGHQVDWLRQ φ

3Y 3J × N3D 3D 3WRW 3Y N3D PD 3D 9 57 NJ PY Z PD NJ 3Y 3J N3D 3D 3D 7 7 N3D

3 3D 3Y N3D PY 3Y 9 5Y7 NJ 9 YJ VWHDP WDEOH PI PY PY NJ

7KH KHDW WUDQVIHU IURP WKH HQHUJ\ HTXDWLRQ EHFRPHV 4 PD X X D PY XJ PI XI PY XJ PD &Y 7 7 PY PI PY N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ ZDWHU ILOOHG UHDFWRU RI P LV DW 03D °& DQG ORFDWHG LQVLGH DQ LQVXODWHG FRQWDLQPHQW URRP RI P WKDW FRQWDLQV DLU DW N3D DQG °& 'XH WR D IDLOXUH WKH UHDFWRU UXSWXUHV DQG WKH ZDWHU ILOOV WKH FRQWDLQPHQW URRP )LQG WKH ILQDO SUHVVXUH &9 7RWDO FRQWDLQHU PY X X PD X X 4 : ,QLWLDO ZDWHU Y X PY 9 Y NJ ,QLWLDO DLU PD 39 57 × × NJ 6XEVWLWXWH LQWR HQHUJ\ HTXDWLRQ X × 7 X 7 N- NJ Y 9 PY P NJ 7ULDO DQG HUURU SKDVH 7JXHVV Y ! [ ! X ! /+6 7 /+6 7 /+6 7 /+6 ! 7 °& /+6 2. [ 3VDW N3D

3D 3D 9 7 9 7 × × × N3D ! 3 3D 3VDW N3D

100 m3

3

1m


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7DEOHV DQG IRUPXODV RU SV\FKURPHWULF FKDUW

$ IORZ PRLVW DLU DW N3D °& UHODWLYH KXPLGLW\ LV FRROHG WR °& LQ D FRQVWDQW SUHVVXUH GHYLFH )LQG WKH KXPLGLW\ UDWLR RI WKH LQOHW DQG WKH H[LW IORZ DQG WKH KHDW WUDQVIHU LQ WKH GHYLFH SHU NJ GU\ DLU 6ROXWLRQ & 9 &RROHU PY POLT PY 7DEOHV 3J N3D 3Y N3D ω 3Y N3D 3J ! ω KY N- NJ KY N- NJ KI N- NJ

T RXW &3 7 7 ω KY ω KY ω ω KI

× × × N- NJ GU\ DLU

3V\FKURPHWULF FKDUW 6WDWH 7 7GHZ °& ! φ a PY PD ω K PY PD ω K× POLT PD ω ω KI N- NJ PD T RXW PDK× POLT KI PD K× !

T RXW K× ω ω KI K× ± × ±

N- NJ GU\ DLU Z

Φ = Φ = 'HZ SRLQW

Φ = Φ =

7GHZ

7 GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ NJ V GU\ DLU RI PRLVW DLU DW °& UHODWLYH KXPLGLW\ IORZV IURP WKH RXWVLGH VWDWH GRZQ LQWR D EDVHPHQW ZKHUH LW FRROV WR °& VWDWH 7KHQ LW IORZV XS WR WKH OLYLQJ URRP ZKHUH LW LV KHDWHG WR °& VWDWH )LQG WKH GHZ SRLQW IRU VWDWH DQ\ DPRXQW RI OLTXLG WKDW PD\ DSSHDU WKH KHDW WUDQVIHU WKDW WDNHV SODFH LQ WKH EDVHPHQW DQG WKH UHODWLYH KXPLGLW\ LQ WKH OLYLQJ URRP DW VWDWH 6ROYH XVLQJ SV\FKURPHWULF FKDUW D 7GHZ Z Z φ Z K× N- NJ DLU E 7 7GHZ VR ZH KDYH φ OLTXLG ZDWHU DSSHDU LQ WKH EDVHPHQW ! Z K× DQG IURP VWHDP WEO KI POLT PDLU Z Z NJ V F (QHUJ\ HTXDWLRQ PDLU K× POLT KI PDLU K× 4RXW

4RXW > × @ N:

G Z Z °& ! φ ,I \RX VROYH E\ WKH IRUPXODV DQG WKH WDEOHV WKH QXPEHUV DUH

3J 3Y × N3D

Z ×

3Y 3J 7GHZ ! 7GHZ °&

φ 3Y 3J N3D Z × POLT PDLU Z Z × NJ V

Z Z ! 3Y 3Y 3J

φ 3Y 3J Z

Φ =

'HZ SRLQW

Φ = Φ =

Φ = 7GHZ

7 GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7ZR PRLVW DLU VWUHDPV ZLWK UHODWLYH KXPLGLW\ ERWK IORZLQJ DW D UDWH RI NJ V RI GU\ DLU DUH PL[HG LQ D VWHDG\ VHWXS 2QH LQOHW IORZVWUHDP LV DW °& DQG WKH RWKHU DW °& )LQG WKH H[LW UHODWLYH KXPLGLW\ 6ROXWLRQ &9 PL[LQJ FKDPEHU &RQWLQXLW\ (T ZDWHU PDLU Z PDLU Z PDLU ZH[ (QHUJ\ (T PDLU K× PDLU K× PDLU K×H[ 3URSHUWLHV IURP WKH WDEOHV DQG IRUPXODV

3J 3Y × N3D

Z ×

3J 3Y × N3D

Z ×

&RQWLQXLW\ (T ZDWHU ZH[ Z Z )RU WKH HQHUJ\ HTXDWLRQ ZH KDYH K× KD ZKY VR

K×H[ K× K× KD H[ KD KD ZH[KY H[ Z KY ZKY

ZH ZLOO XVH FRQVWDQW KHDW FDSDFLW\ WR DYRLG DQ LWHUDWLRQ RQ 7H[

&S DLU 7H[ 7 7 &S + 2 ZH[7H[ Z 7 Z 7

7H[ > &S DLU 7 7 &S + 2 Z 7 Z 7 @ > &S DLU ZH[&S + 2@

> × × @

°& ZH[ 3Y H[ Z 3WRW N3D H[

3J H[ N3D ! φ RU

3URSHUWLHV WDNHQ IURP WKH SV\FKURPHWULF FKDUW 6WDWH Z K× 6WDWH Z K× &RQWLQXLW\ (T ZDWHU ZH[ Z Z (QHUJ\ (T K×H[ K× K× N- NJ GU\ DLU

H[LW ZH[ K×H[ ! 7H[ °& φ

1RWLFH KRZ WKH HQHUJ\ LQ WHUPV RI WHPSHUDWXUH LV FORVH WR WKH DYHUDJH RI WKH WZR IORZV EXW WKH UHODWLYH KXPLGLW\ LV QRW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH GLVFKDUJH PRLVW DLU IURP D FORWKHV GU\HU LV DW R& UHODWLYH KXPLGLW\ 7KH IORZ LV JXLGHG WKURXJK D SLSH XS WKURXJK WKH URRI DQG D YHQW WR WKH DWPRVSKHUH 'XH WR KHDW WUDQVIHU LQ WKH SLSH WKH IORZ LV FRROHG WR R& E\ WKH WLPH LW UHDFKHV WKH YHQW )LQG WKH KXPLGLW\ UDWLR LQ WKH IORZ RXW RI WKH FORWKHV GU\HU DQG DW WKH YHQW )LQG WKH KHDW WUDQVIHU DQG DQ\ DPRXQW RI OLTXLG WKDW PD\ EH IRUPLQJ SHU NJ GU\ DLU IRU WKH IORZ 6ROXWLRQ 6WDWH Z K× 7GHZ R& 6WDWH R& 7GHZ VR LW LV VDWXUDWHG Z K× N- NJ DLU POLT PD ω ω NJ NJ GU\ DLU (QHUJ\ (T 4 PD K× K× ± ω ω KI ± ± × N- NJ GU\ DLU

2

1


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ VWHDG\ VXSSO\ RI P V DLU DW °& N3D UHODWLYH KXPLGLW\ LV QHHGHG WR KHDW D EXLOGLQJ LQ WKH ZLQWHU 7KH RXWGRRU DPELHQW LV DW °& N3D UHODWLYH KXPLGLW\ :KDW DUH WKH UHTXLUHG OLTXLG ZDWHU LQSXW DQG KHDW WUDQVIHU UDWHV IRU WKLV SXUSRVH" 6ROXWLRQ $LU 5D N- NJ . &S N- NJ . 6WDWH 7 °& φ 3 N3D 3J N3D 3Y φ 3J N3D 3D 3 3Y N3D ! ω 3Y 3D 6WDWH 7 °& 3 N3D φ 9 P V 3J N3D 3Y φ 3J N3D 3D 3 3Y N3D ω 3Y 3D PD 3D 9 5D7 × × NJ V 6WHDP WDEOHV % KY N- NJ KY N- NJ

6WDWH $VVXPH /LT :DWHU DW 7 °& KI N- NJ &RQVHUYDWLRQ RI 0DVV PD PD PI PY PY PI PD ω ω × NJ V VW/DZ 4 PD KD PY KY PI KI PD KD PY KY PI 4 &S 7 7 ω KY ω KY KI ! 4 N: PD PD


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ FRPELQDWLRQ DLU FRROHU DQG GHKXPLGLILFDWLRQ XQLW UHFHLYHV RXWVLGH DPELHQW DLU DW °& N3D UHODWLYH KXPLGLW\ 7KH PRLVW DLU LV ILUVW FRROHG WR D ORZ WHPSHUDWXUH 7 WR FRQGHQVH WKH SURSHU DPRXQW RI ZDWHU DVVXPH DOO WKH OLTXLG OHDYHV DW 7 7KH PRLVW DLU LV WKHQ KHDWHG DQG OHDYHV WKH XQLW DW °& N3D

UHODWLYH KXPLGLW\ ZLWK YROXPH IORZ UDWH RI P V )LQG WKH WHPSHUDWXUH 7 WKH PDVV RI OLTXLG SHU NLORJUDP RI GU\ DLU DQG WKH RYHUDOO KHDW WUDQVIHU UDWH 6ROXWLRQ 0,; ,1

&22/

+($7

' . 4 & /,4 287

0,; 287

&9 . 4 +

D 3Y φ 3* × N3D Z × 3Y φ 3* × N3D Z Z × P/,4 ′ PD Z Z NJ NJ DLU 3* 3Y N3D → 7 R& E )RU D & 9 DURXQG WKH HQWLUH XQLW 4&9 4+ 4& 1HW KHDW WUDQVIHU VW ODZ 4&9 PD KD KD Z KY Z KY P/ ′ K/ ′ PD × × × N- NJ DLU 3D 9 × PD 5 7 NJ V × D 4&9 N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

8VH WKH IRUPXODV DQG WKH VWHDP WDEOHV WR ILQG WKH PLVVLQJ SURSHUW\ RI φ, ω DQG 7GU\ WRWDO SUHVVXUH LV N3D UHSHDW WKH DQVZHUV XVLQJ WKH SV\FKURPHWULF FKDUW D φ ω E 7GU\ & 7ZHW & 6ROXWLRQ D )URP (T ZLWK 3D 3 3Y VROYH IRU 3Y

3Y 3 ω ω × N3D

)URP (T 3J 3Y φ N3D ! 7 °& E $W °& 3J ! ω ×

)URP WKH VWHDP WDEOHV % KI DQG KIJ N- NJ KY

)URP (T

ω >&S 7 7 ω KIJ @ KY KI

)URP (T ZLWK 3D 3 3Y VROYH IRU 3Y

3Y 3 ω ω

)URP (T φ 8VLQJ WKH SV\FKURPHWULF FKDUW (

D 7GU\ °& E ω φ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$Q LQVXODWHG WDQN KDV DQ DLU LQOHW ω = DQG DQ RXWOHW 7 = °& φ ERWK DW N3D $ WKLUG OLQH VSUD\V NJ V RI ZDWHU DW °& N3D )RU VWHDG\ RSHUDWLRQ ILQG WKH RXWOHW VSHFLILF KXPLGLW\ WKH PDVV IORZ UDWH RI DLU QHHGHG DQG WKH UHTXLUHG DLU LQOHW WHPSHUDWXUH 7 6ROXWLRQ 7DNH &9 WDQN LQ VWHDG\ VWDWH &RQWLQXLW\ DQG HQHUJ\ HTXDWLRQV DUH &RQWLQXLW\ (T ZDWHU P PD Z PD Z (QHUJ\ (T P KI PD K× PD K× $OO VWDWH SURSHUWLHV DUH NQRZQ H[FHSW 7 )URP WKH SV\FKURPHWULF FKDUW ZH JHW 6WDWH Z K× 6WDWH KI VWHDP WEO PD P Z Z NJ V

K× K× Z Z KI × &KDUW Z K× ! 7 °& 8VLQJ WKH WDEOHV DQG IRUPXODV ZH JHW 6WDWH

3J 3Y × N3D

Z ×

PD P Z Z NJ V

7R DYRLG LWHUDWLRQV RQ 7 ZH XVH VSHFLILF KHDW YDOXHV DOVR IRU ZDWHU YDSRU E\ ZULWLQJ KY KY &S K R 7 7 VR WKH HQHUJ\ HTXDWLRQ LV

&S D 7 Z &S K R 7 7 Z KY &S D 7 Z KY Z Z KI 7KH HTXDWLRQ QRZ EHFRPHV

× 7 ×

7 °&


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI PRLVW DLU IURP D GRPHVWLF IXUQDFH VWDWH LV DW R& UHODWLYH KXPLGLW\ ZLWK D IORZ UDWH RI NJ V GU\ DLU $ VPDOO HOHFWULF KHDWHU DGGV VWHDP DW R& N3D JHQHUDWHG IURP WDS ZDWHU DW R& 8S LQ WKH OLYLQJ URRP WKH IORZ FRPHV RXW DW VWDWH R& UHODWLYH KXPLGLW\ )LQG WKH SRZHU QHHGHG IRU WKH HOHFWULF KHDWHU DQG WKH KHDW WUDQVIHU WR WKH IORZ IURP VWDWH WR VWDWH Liquid 2 3 4 cb

1

6WDWH Z K× N- NJ GU\ DLU 6WDWH Z K× N- NJ GU\ DLU POLT PD ω ω ± NJ V (QHUJ\ (T IRU KHDWHU 4KHDWHU POLT KRXW ± KLQ ± N: (QHUJ\ (T IRU OLQH 4OLQH PD K× ± K× ± POLT KYDS ± ± × ± N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ ZDWHU FRROLQJ WRZHU IRU D SRZHU SODQW FRROV °& OLTXLG ZDWHU E\ HYDSRUDWLRQ 7KH WRZHU UHFHLYHV DLU DW °& φ = N3D WKDW LV EORZQ WKURXJK RYHU WKH ZDWHU VXFK WKDW LW OHDYHV WKH WRZHU DW °& φ = 7KH UHPDLQLQJ OLTXLG ZDWHU IORZV EDFN WR WKH FRQGHQVHU DW °& KDYLQJ JLYHQ RII 0: )LQG WKH PDVV IORZ UDWH RI DLU DQG WKH DPRXQW RI ZDWHU WKDW HYDSRUDWHV 6ROXWLRQ &9 7RWDO FRROLQJ WRZHU VWHDG\ VWDWH &RQWLQXLW\ (T IRU ZDWHU LQ DLU ZLQ PHYDS PD ZH[ (QHUJ\ (T PD K×LQ P K PD K×H[ P PHYDS K ,QOHW °& UHO KXP ! ZLQ K×LQ ([LW °& UHO KXP ! ZH[ K×H[

7DNH WKH WZR ZDWHU IORZ GLIIHUHQFH WR PHDQ WKH 0: 4 P K P PHYDS K 0: PD K×H[ K×LQ PD N: ! PD NJ V PHYDS ZH[ ZLQ PD × NJ V 7KH QHHGHG PDNH XS ZDWHU IORZ FRXOG EH DGGHG WR JLYH D VOLJKWO\ GLIIHUHQW PHDQLQJ WR WKH 0:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI DLU DW °& φ = LV EURXJKW LQWR D KRXVH ZKHUH LW LV FRQGLWLRQHG WR °& UHODWLYH KXPLGLW\ 7KLV LV GRQH ZLWK D FRPELQHG KHDWHU HYDSRUDWRU ZKHUH DQ\ OLTXLG ZDWHU LV DW °& )LQG DQ\ IORZ RI OLTXLG DQG WKH QHFHVVDU\ KHDW WUDQVIHU ERWK SHU NLORJUDP GU\ DLU IORZLQJ )LQG WKH GHZ SRLQW IRU WKH ILQDO PL[WXUH &9 KHDWHU DQG HYDSRUDWRU 8VH SV\FKURPHWULF FKDUW ,QOHW Z K× N- NJ GU\ DLU KI N- NJ ([LW Z K× N- NJ GU\ DLU 7GHZ °&

)URP WKHVH QXPEHUV ZH VHH WKDW ZDWHU DQG KHDW PXVW EH DGGHG &RQWLQXLW\ HT DQG HQHUJ\ HTXDWLRQ JLYH P/,4 ,1 P$ Z Z NJ NJ GU\ DLU T K× K× Z Z KI N- NJ GU\ DLU


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

,Q D FDU¶V GHIURVW GHIRJ V\VWHP DWPRVSKHULF DLU °& UHODWLYH KXPLGLW\ LV WDNHQ LQ DQG FRROHG VXFK WKDW OLTXLG ZDWHU GULSV RXW 7KH QRZ GU\HU DLU LV KHDWHG WR °& DQG WKHQ EORZQ RQWR WKH ZLQGVKLHOG ZKHUH LW VKRXOG KDYH D PD[LPXP RI UHODWLYH KXPLGLW\ WR UHPRYH ZDWHU IURP WKH ZLQGVKLHOG )LQG WKH GHZ SRLQW RI WKH DWPRVSKHULF DLU VSHFLILF KXPLGLW\ RI DLU RQWR WKH ZLQGVKLHOG WKH ORZHVW WHPSHUDWXUH DQG WKH VSHFLILF KHDW WUDQVIHU LQ WKH FRROHU 6ROXWLRQ 4 FRRO 4 KHDW /LTXLG

6ROYH XVLQJ WKH SV\FKURPHWULF FKDUW Z Φ = 'HZ SRLQW Φ = IRU Φ =

$LU LQOHW °& φ ! Z 7GHZ °& K× $LU H[LW °& φ ! Z 7GHZ °&

7 GU\ 7 GHZ 7 GHZ 7R UHPRYH HQRXJK ZDWHU ZH PXVW FRRO WR WKH H[LW 7GHZ IROORZHG E\ KHDWLQJ WR 7H[ 7KH HQWKDOS\ IURP FKDUW K× DQG IURP % KI °& &9 FRROHU POLT PDLU Z Z NJ OLT NJ DLU T 4&9 PDLU K× Z Z KI K× × N- NJ GU\ DLU ,I WKH VWHDP DQG DLU WDEOHV DUH XVHG WKH QXPEHUV DUH 6WDWH 3J 3Y ! Z KJ KD ! K× 6WDWH 3J 3Y ! Z

6WDWH ZJ Z ! 7 7 GHZ °& KI KJ KD ! K× POLT PDLU T × N- NJ DLU


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$WPRVSKHULF DLU DW °& UHODWLYH KXPLGLW\ RI LV WRR ZDUP DQG DOVR WRR GU\ $Q DLU FRQGLWLRQHU VKRXOG GHOLYHU DLU DW °& DQG UHODWLYH KXPLGLW\ LQ WKH DPRXQW RI P SHU KRXU 6NHWFK D VHWXS WR DFFRPSOLVK WKLV ILQG DQ\ DPRXQW RI OLTXLG DW °& WKDW LV QHHGHG RU GLVFDUGHG DQG DQ\ KHDW WUDQVIHU 6ROXWLRQ &9 DLU FRQGLWLRQHU )LUVW ZH PXVW FKHFN LI ZDWHU VKRXOG EH DGGHG RU VXEWUDFWHG :H FDQ NQRZ WKLV IURP WKH DEVROXWH KXPLGLW\ UDWLR 3URSHUWLHV IURP WKH WDEOHV DQG IRUPXODV 6WDWH

3J 3Y × N3D

Z ×

6WDWH

3J 3Y × N3D

Z ×

$V Z JRHV XS ZH PXVW DGG OLTXLG ZDWHU 1RZ ZH JHW &RQWLQXLW\ (T P$ Z POLT P$ Z (QHUJ\ (T P$K× PL[ POLTKI 4&9 P$K× PL[ )RU WKH OLTXLG IORZ ZH QHHG WKH DLU PDVV IORZUDWH RXW P K P V P$ 3D 9 57 × NJ V POLT P$ Z Z NJ V NJ K 4&9 P$>&S D 7 7 Z KY Z KY @ POLTKI

> × × @

× N: ,I IURP SV\FKURPHWULF FKDUW ,QOHW Z K×PL[ KI N- NJ ([LW Z K×PL[ N- NJ GU\ DLU 3Y DQG P$ 3D 9 57 VDPH DV DERYH 4&9 P$ K× PL[ K× PL[ POLTKI ×

N:

/LTXLG ZDWHU ,QOHW

&RROHU ([LW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

2QH PHDQV RI DLU FRQGLWLRQLQJ KRW VXPPHU DLU LV E\ HYDSRUDWLYH FRROLQJ ZKLFK LV D SURFHVV VLPLODU WR WKH DGLDEDWLF VDWXUDWLRQ SURFHVV &RQVLGHU RXWGRRU DPELHQW DLU DW °& N3D UHODWLYH KXPLGLW\ :KDW LV WKH PD[LPXP DPRXQW RI FRROLQJ WKDW FDQ EH DFKLHYHG E\ VXFK D WHFKQLTXH" :KDW GLVDGYDQWDJH LV WKHUH WR WKLV DSSURDFK" 6ROYH WKH SUREOHP XVLQJ D ILUVW ODZ DQDO\VLV DQG UHSHDW LW XVLQJ WKH SV\FKURPHWULF FKDUW )LJ ( Cooled Ambient 3 3 N3D Air

2

1 3

Liquid

Air

7 R& φ 3Y φ 3J × ω ×

)RU DGLDEDWLF VDWXUDWLRQ 0D[ FRROLQJ LV IRU φ VW ODZ (T

ω KY KI &S 7 7 ω KIJ

φ ω × 3* 3 3*

2QO\ RQH XQNQRZQ 7 7ULDO DQG HUURU RQ HQHUJ\ HTXDWLRQ

&S7 ω KIJ ω KI &S7 ω KY

× ×

7 R& 3* KI KIJ

! ω ×

/+6 × × ×

7 R& 3* KI KIJ

! ω ×

/+6 × × ×

OLQHDU LQWHUSRODWLRQ 7 R&

7KLV PHWKRG GRHV ORZHU WKH WHPSHUDWXUH EXW WKH UHODWLYH DQG DEVROXWH KXPLGLW\ EHFRPHV YHU\ KLJK DQG WKH VOLJKWHVW FRROLQJ OLNH RQ D ZDOO ZLOO UHVXOW LQ FRQGHQVDWLRQ

E FKDUW ( $GLDEDWLF VDWXUDWLRQ 7 ≈ :HW%XOE7HPSHUDWXUH ≈ R&


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI PRLVW DLU DW R& UHODWLYH KXPLGLW\ ZLWK D IORZ UDWH RI NJ V GU\ DLU LV PL[HG ZLWK D IORZ RI PRLVW DLU DW R& DQG DEVROXWH KXPLGLW\ RI Z ZLWK D UDWH RI NJ V GU\ DLU 7KH PL[LQJ WDNHV SODFH LQ DQ DLU GXFW DW N3D DQG WKHUH LV QR VLJQLILFDQW KHDW WUDQVIHU $IWHU WKH PL[LQJ WKHUH LV KHDW WUDQVIHU WR D ILQDO WHPSHUDWXUH RI R& )LQG WKH WHPSHUDWXUH DQG UHODWLYH KXPLGLW\ DIWHU PL[LQJ )LQG WKH KHDW WUDQVIHU DQG WKH ILQDO H[LW UHODWLYH KXPLGLW\ 6ROXWLRQ & 9 7RWDO 6HWXS VWDWH LV LQWHUQDO WR &9

4 KHDW

FRRO

PD Z PD Z PD PD Z PD PD Z (QHUJ\ (T PD K× PD K× PD PD K× 6WDWH )URP 3V\FKURPHWULF FKDUW Z K× N- NJ GU\ DLU &RQWLQXLW\ (T

6WDWH )URP 3V\FKURPHWULF FKDUW Φ K× N- NJ GU\ DLU PD Z PD Z Z Z PD PD PD K× PD K× K× N- NJ GU\ DLU PD PD 6WDWH )URP 3V\FKURPHWULF FKDUW 7 R& Φ 6WDWH R& Z 5HDG IURP 3V\FKURPHWULF FKDUW

K× Φ

1RZ GR WKH HQHUJ\ HTXDWLRQ IRU WKH ZKROH VHWXS (QHUJ\ (T PD K× PD K× 4 PD PD K× 4 PD PD K× PD K× PD K×

± ± N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$Q LQGRRU SRRO HYDSRUDWHV NJ K RI ZDWHU ZKLFK LV UHPRYHG E\ D GHKXPLGLILHU WR PDLQWDLQ °& φ = LQ WKH URRP 7KH GHKXPLGLILHU VKRZQ LQ )LJ 3 LV D UHIULJHUDWLRQ F\FOH LQ ZKLFK DLU IORZLQJ RYHU WKH HYDSRUDWRU FRROV VXFK WKDW OLTXLG ZDWHU GURSV RXW DQG WKH DLU FRQWLQXHV IORZLQJ RYHU WKH FRQGHQVHU )RU DQ DLU IORZ UDWH RI NJ V WKH XQLW UHTXLUHV N: LQSXW WR D PRWRU GULYLQJ D IDQ DQG WKH FRPSUHVVRU DQG LW KDV D FRHIILFLHQW RI SHUIRUPDQFH β = 4/ :F = )LQG WKH VWDWH RI WKH DLU DV LW UHWXUQV WR WKH URRP DQG WKH FRPSUHVVRU ZRUN LQSXW 6ROXWLRQ 7KH XQLW PXVW UHPRYH NJ K OLTXLG WR NHHS VWHDG\ VWDWH LQ WKH URRP $V ZDWHU FRQGHQVHV RXW VWDWH LV VDWXUDWHG 6WDWH °& ! Z K× &9 WR POLT PD Z Z ! Z Z POLT PD T/ K× K× Z Z KI Z × 6WDWH Z ! 7 °& K× KI T/ × N- NJ GU\ DLU &9 7RWDO V\VWHP K× K× :HO PD Z Z KI

6WDWH Z :F

N- NJ GU\ DLU Z K× ! 7 °& φ3 PD T/ β N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

3V\FKURPHWULF FKDUW RQO\

8VH WKH SV\FKURPHWULF FKDUW WR ILQG WKH PLVVLQJ SURSHUW\ RI φ, ω, 7ZHW 7GU\ E 7GU\ & φ D 7GU\ & φ F 7GU\ & DQG ω G 7GU\ & 7ZHW &

6ROXWLRQ D °& φ ! ω 7ZHW °& E °& φ ! ω 7ZHW °& F °& ω ! φ 7ZHW °&

G °& 7ZHW °& ! ω φ Z

Φ =

Φ = E

F

G D

Φ = Φ = 7 GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

8VH WKH SV\FKURPHWULF FKDUW WR ILQG WKH PLVVLQJ SURSHUW\ RI φ, ω, 7ZHW 7GU\ D φ ω E 7ZHW & φ G 7GU\ & ω F ω DQG 7ZHW &

6ROXWLRQ D φ ω ! 7GU\ °& 7ZHW °& E 7ZHW °& φ ! 7GU\ °& ω F ω 7ZHW °& ! 7GU\ °& φ

G 7GU\ °& ω ! φ 7ZHW °& Z Φ =

Φ = E G

D

F

Φ = Φ = 7 GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

8VH WKH IRUPXODV DQG WKH VWHDP WDEOHV WR ILQG WKH PLVVLQJ SURSHUW\ RI φ, ω, DQG 7GU\ WRWDO SUHVVXUH LV N3D UHSHDW WKH DQVZHUV XVLQJ WKH SV\FKURPHWULF FKDUW D φ ω E 7ZHW & φ F 7GU\ & 7ZHW & D )URP (T 3Y 3 ω ω × N3D )URP (T 3J 3Y φ N3D ! 7 °& E $VVXPH 7ZHW LV DGLDEDWLF VDWXUDWLRQ 7 DQG XVH HQHUJ\ (T $W °& 3J ! ω ×

/+6 ω KY KI &S7 5+6 &S7 ω KIJ

5+6 × × N- NJ

ω φ3J φ3J ZKHUH 3J LV DW 7 7ULDO DQG HUURU

/+6 & /+6 & ! 7 °& ω F $W °& 3J ! ω ×

KI DQG KIJ N- NJ KY

)URP (T ω >&S 7 7 ω KIJ @ KY KI 3Y 3 ω ω φ 8VLQJ WKH SV\FKURPHWULF FKDUW (

D 7GU\ °& E 7GU\ °& ω

F ω φ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

)RU HDFK RI WKH VWDWHV LQ 3UREOHP ILQG WKH GHZ SRLQW WHPSHUDWXUH 6ROXWLRQ 7KH GHZ SRLQW LV WKH VWDWH ZLWK WKH VDPH KXPLGLW\ UDWLR DEV KXPLGLW\ ω DQG FRPSOHWHO\ VDWXUDWHG φ )URP SV\FKURPHWULF FKDUW

D

7GHZ °&

F 7GHZ °&

E

7GHZ °&

G 7GHZ °&

)LQGLQJ WKH VROXWLRQ IURP WKH WDEOHV LV GRQH IRU FDVHV D F DQG G DV

(T VROYH 3Y 3J ω3WRW >ω @ 3VDW 7GHZ LQ %

)RU FDVH E XVH HQHUJ\ (T WR ILQG ω ILUVW IURP 7DG VDW 7ZHW Φ =

Z

Φ =

G

E

D

F

Φ = Φ = 7 GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RPSDUH WKH ZHDWKHU WZR SODFHV ZKHUH LW LV FORXG\ DQG EUHH]\ $W EHDFK $ LW LV °& N3D UHODWLYH KXPLGLW\ DQG EHDFK % KDV °& N3D UHODWLYH KXPLGLW\ 6XSSRVH \RX MXVW WRRN D VZLP DQG FDPH RXW RI WKH ZDWHU :KHUH ZRXOG \RX IHHO PRUH FRPIRUWDEOH DQG ZK\" 6ROXWLRQ <RXU VNLQ EHLQJ ZHW DQG DLU LV IORZLQJ RYHU LW ZLOO IHHO 7ZHW :LWK WKH VPDOO GLIIHUHQFH LQ SUHVVXUH IURP N3D XVH WKH SV\FKURPHWULF FKDUW $ °& φ ! 7ZHW °&

% °& φ ! 7ZHW °& $W EHDFK $ LW LV FRPIRUWDEOH DW % LW IHHOV FKLOO\ Z

Φ = Φ = $

Φ = Φ = %

7 GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$PELHQW DLU DW N3D °& UHODWLYH KXPLGLW\ JRHV WKURXJK D FRQVWDQW SUHVVXUH KHDW H[FKDQJHU DV D VWHDG\ IORZ ,Q RQH FDVH LW LV KHDWHG WR °& DQG LQ DQRWKHU FDVH LW LV FRROHG XQWLO LW UHDFKHV VDWXUDWLRQ )RU ERWK FDVHV ILQG WKH H[LW UHODWLYH KXPLGLW\ DQG WKH DPRXQW RI KHDW WUDQVIHU SHU NLORJUDP GU\ DLU 6ROXWLRQ &9 KHDW H[FKDQJHU P$L P$H PYL PYH ZH ZL KD ZKY L T KD ZKY H K×H T K×H K×L 8VLQJ WKH SV\FKURPHWULF FKDUW L ZL K×L

&DVH , H 7H R& ZH ZL ! K×H

φH T N- NJ GU\ DLU

&DVH ,, H ZH ZL φH ! K×H 7H R& T N- NJ GU\ DLU

Φ =

Z

Φ = &$6( ,, H

L

H &$6( ,

Φ = Φ = 7GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ IORZ RI PRLVW DLU DW °& UHODWLYH KXPLGLW\ VKRXOG EH SURGXFHG IURP PL[LQJ RI WZR GLIIHUHQW PRLVW DLU IORZV )ORZ LV DW °& UHODWLYH KXPLGLW\ DQG IORZ LV DW °& DQG KDV 7ZHW °& 7KH PL[LQJ FKDPEHU FDQ EH IROORZHG E\ D KHDWHU RU D FRROHU 1R OLTXLG ZDWHU LV DGGHG DQG 3 N3D )LQG WKH WZR FRQWUROV RQH LV WKH UDWLR RI WKH WZR PDVV IORZ UDWHV PD PD DQG WKH RWKHU LV WKH KHDW WUDQVIHU LQ WKH KHDWHU FRROHU SHU NJ GU\ DLU 6ROXWLRQ 4 KHDW FRRO & 9 7RWDO 6HWXS VWDWH LV LQWHUQDO WR &9

PD Z PD Z PD PD Z (QHUJ\ (T PD K× PD K× 4D PD PD K× 'HILQH [ PD PD DQG VXEVWLWXWH LQWR FRQWLQXLW\ HTXDWLRQ Z Z ! [ Z Z [ Z ! [ Z Z &RQWLQXLW\ (T

(QHUJ\ HTXDWLRQ VFDOHG WR WRWDO IORZ RI GU\ DLU T× 4D PD PD K× >[ [ @ K× > [ @ K×

± × − ×

N- NJ GU\ DLU Z

Φ =

Φ = Φ =

7GU\

6WDWH Z K× 6WDWH Z K× 6WDWH Z K× 7GHZ °&


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

,Q D KRW DQG GU\ FOLPDWH DLU HQWHUV DQ DLU FRQGLWLRQHU XQLW DW N3D °& DQG UHODWLYH KXPLGLW\ DW WKH VWHDG\ UDWH RI P V /LTXLG ZDWHU DW °& LV VSUD\HG LQWR WKH DLU LQ WKH $& XQLW DW WKH UDWH NJ KRXU DQG KHDW LV UHMHFWHG IURP WKH XQLW DW DW WKH UDWH N: 7KH H[LW SUHVVXUH LV N3D :KDW DUH WKH H[LW WHPSHUDWXUH DQG UHODWLYH KXPLGLW\" 6WDWH 7 °& 3 N3D φ 9D P V 3J N3D 3Y φ 3J N3D 3D 3 3Y N3D 3Y 3D 9D ω 3 PD 57 NJ V KY N- NJ D D 6WDWH /LT :DWHU °& PI NJ KU NJ V KI N- NJ &RQVHUYDWLRQ RI 0DVV PD PD PY P PY ω PI PD ω

6WDWH 3 N3D DQG 3Y 3 ω ω N3D VW/DZ 4 PD KD PY KY PI KI PD KD PY KY 4 N: KD KD ω KY ω KY PI KI 4 PD 8QNQRZQV KD KY LPSOLFLWO\ JLYHQ EH D VLQJOH XQNQRZQ 7 3Y 7ULDO DQG (UURU IRU 7 7 & 3J N3D φ 3 J ,I ZH VROYHG ZLWK WKH SV\FKURPHWULF FKDUW ZH ZRXOG JHW a 6WDWH PY PD ω K 6WDWH ω PI PD ω 1RZ WKH HQHUJ\ HTXDWLRQ EHFRPHV a a K K PI KI 4 PD *LYHQ ω ZH ILQG WKH VWDWH DURXQG & DQG φ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQVLGHU WZR VWDWHV RI DWPRVSKHULF DLU °& 7ZHW = °& DQG °& φ 6XJJHVW D V\VWHP RI GHYLFHV WKDW ZLOO DOORZ DLU LQ D VWHDG\ IORZ SURFHVV WR FKDQJH IURP WR DQG IURP WR +HDWHUV FRROHUV GH KXPLGLILHUV OLTXLG WUDSV HWF DUH DYDLODEOH DQG DQ\ OLTXLG VROLG IORZLQJ LV DVVXPHG WR EH DW WKH ORZHVW WHPSHUDWXUH VHHQ LQ WKH SURFHVV )LQG WKH VSHFLILF DQG UHODWLYH KXPLGLW\ IRU VWDWH GHZ SRLQW IRU VWDWH DQG WKH KHDW WUDQVIHU SHU NLORJUDP GU\ DLU LQ HDFK FRPSRQHQW LQ WKH V\VWHPV 8VH WKH SV\FKURPHWULF FKDUW ( Z K× φ 7GHZ °& K×GHZ

Z K× φ 7GHZ °& K×GHZ

6LQFH Z ! Z ZDWHU PXVW EH DGGHG LQ SURFHVV , WR ,, DQG UHPRYHG LQ WKH SURFHVV ,, WR , :DWHU FDQ RQO\ EH UHPRYHG E\ FRROLQJ EHORZ GHZ SRLQW WHPSHUDWXUH VR , WR ,, $GLDE VDW , WR 'HZ ,, WKHQ KHDWHU IURP 'HZ ,, WR ,, ,, WR , &RRO WR 'HZ , WKHQ KHDW 'HZ , WR , 7KH ILUVW RQH FDQ EH GRQH EHFDXVH 7GHZ ,, 7DG VDW , , WR ,, T K×,, K×GHZ,, N- NJ DLU ,, WR , TFRRO K×,, K×GHZ, Z Z KI DW 7GHZ,

× N- NJ DLU TKHDW K×, K×GHZ, N- NJ DLU


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7R UHIUHVK DLU LQ D URRP D FRXQWHUIORZ KHDW H[FKDQJHU VHH )LJ 3 LV PRXQWHG LQ WKH ZDOO GUDZLQJ LQ RXWVLGH DLU DW °& UHODWLYH KXPLGLW\ DQG SXVKLQJ RXW URRP DLU °& UHODWLYH KXPLGLW\ $VVXPH DQ H[FKDQJH RI NJ PLQ GU\ DLU LQ D VWHDG\ IORZ GHYLFH DQG DOVR WKDW WKH URRP DLU H[LWV WKH KHDW H[FKDQJHU WR WKH DWPRVSKHUH DW °& )LQG WKH QHW DPRXQW RI ZDWHU UHPRYHG IURP WKH URRP DQ\ OLTXLG IORZ LQ WKH KHDW H[FKDQJHU DQG 7 φ IRU WKH IUHVK DLU HQWHULQJ WKH URRP 6WDWH Z K× 7GHZ °& 7KH URRP DLU LV FRROHG WR °& 7GHZ VR OLTXLG ZLOO IRUP LQ WKH H[LW IORZ FKDQQHO DQG VWDWH LV VDWXUDWHG °& φ ! Z K× KI N- NJ °& φ ! Z K× N- NJ GU\ DLU &9 WR POLT PD Z Z NJ PLQ &9 URRP PY RXW PD Z Z PD Z Z NJ PLQ &9 +HDW H[FKDQJHU PD K× K× PD K× K× POLTKI K× K× K× K× Z Z KI × N- NJ GU\ DLU Z Z K× ! 7 °& φ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$YDLODELOLW\ H[HUJ\ LQ PL[WXUHV

&RQVLGHU WKH PL[LQJ RI D VWHDP IORZ ZLWK DQ R[\JHQ IORZ LQ 3UREOHP )LQG WKH UDWH RI WRWDO LQIORZLQJ DYDLODELOLW\ DQG WKH UDWH RI H[HUJ\ GHVWUXFWLRQ LQ WKH SURFHVV $ IORZ RI NJ V VWHDP DW N3D R& LV PL[HG ZLWK NJ V R[\JHQ DW N3D . LQ D VWHDG\ IORZ PL[LQJ FKDPEHU ZLWKRXW DQ\ KHDW WUDQVIHU )LQG WKH H[LW WHPSHUDWXUH DQG WKH UDWH RI HQWURS\ JHQHUDWLRQ ([HUJ\ )ORZ ΦLQ P ψLQ P+ 2 ψ P2 ψ

ψ K ± KR ± 7R V ± VR

&3 + 2 7 ± 7R ± 7R >&3 + 2 OQ 7 7R ± 5 OQ 3 3R @

± ± > OQ OQ @

± ± N- NJ

ψ K ± KR ± 7R V ± VR

&3 2 7 ± 7R ± 7R >&3 2 OQ 7 7R ± 5 OQ 3 3R @

± ± > OQ OQ @

± ± N- NJ ΦLQ P+ 2 ψ P2 ψ N:

& 9 0L[LQJ FKDPEHU VWHDG\ IORZ QR ZRUN QR KHDW WUDQVIHU 7R GR WKH HQWURSLHV ZH QHHG WKH PROH IUDFWLRQV P+ 2 P2 Q+ 2 0 NPRO V Q2 0 NPRO V + 2 2

\+ 2 \2 (QHUJ\ (T P+ 2 K P2 K P+ 2 K + 2 P2 K 2 (QWURS\ (T P+ 2 V P2 V 6JHQ P+ 2 V + 2 P2 V 2 6ROYH IRU 7 IURP WKH HQHUJ\ HTXDWLRQ P+ 2 K + 2 ± K P2 K 2 ± K P+ 2 &3 + 2 7 ± 7 P2 &3 2 7 ± 7

× 7 ± ± × 7 ±

7 .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

6JHQ P+ 2 V + 2 ± V P2 V 2 ± V

7 7 P+ 2 > &3 + 2 OQ 7 5 OQ \+ 2 @ P2 > &3 2 OQ 7 5 OQ \2 @

> OQ ± OQ @

N: .

> OQ ± OQ @

7KH H[HUJ\ GHVWUXFWLRQ LV SURSRUWLRQDO WR WKH HQWURS\ JHQHUDWLRQ

ΦLQ 7R 6JHQ × N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ PL[WXUH RI FDUERQ GLR[LGH DQG ZDWHU E\ PDVV LV IORZLQJ DW . N3D LQWR D KHDW H[FKDQJHU ZKHUH LW LV XVHG WR GHOLYHU HQHUJ\ WR D KHDW HQJLQH 7KH PL[WXUH OHDYH WKH KHDW H[FKDQJHU DW . ZLWK D PDVV IORZ UDWH RI NJ PLQ )LQG WKH UDWH RI HQHUJ\ DQG WKH UDWH RI H[HUJ\ GHOLYHUHG WR WKH KHDW HQJLQH & 9 +HDW H[FKDQJHU VWHDG\ IORZ DQG QR ZRUN )URP 7DEOH $

R

&2 KLQ N- NJ V7 LQ N- NJ .

&2 KH[ N- NJ V7 H[ N- NJ .

+ 2 KLQ N- NJ V7 LQ N- NJ .

+ 2 KH[ N- NJ V7 H[ N- NJ .

R

R

R

(QHUJ\ (T 4 P KLQ ± KH[ P ∑ \L KLQ KH[ L

> ± ± @ > @ N: (QWURS\ FKDQJH VLQ ± VH[ ± ± N- NJ . ([HUJ\ )OX[ Φ P ψLQ ± ψH[ 4 ± 7R P VLQ ± VH[

± × ×

N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

)LQG WKH VHFRQG ODZ HIILFLHQF\ RI WKH KHDW H[FKDQJHU LQ 3UREOHP $ IORZ RI NJ V PL[WXUH RI &2 DQG 2 E\ PDVV LV KHDWHG LQ D FRQVWDQW SUHVVXUH KHDW H[FKDQJHU IURP . WR . E\ D UDGLDWLRQ VRXUFH DW . )LQG WKH UDWH RI KHDW WUDQVIHU DQG WKH HQWURS\ JHQHUDWLRQ LQ WKH SURFHVV 6ROXWLRQ 7KH VHFRQG ODZ HIILFLHQF\ IROORZV (T ZKHUH WKH ZDQWHG WHUP LV WKH IORZ LQFUHDVH RI H[HUJ\ DQG WKH VRXUFH LV WKH UDGLDWLRQ

ΦIORZ P ψH[ ± ψLQ

+HDW H[FKDQJHU (QHUJ\ (T

7R ΦVRXUFH 4LQ ± 7

VRXUFH 4LQ P KH KL

9DOXHV IURP 7DEOH $ GXH WR WKH KLJK 7 4LQ > × ± × ± @ N: 7R

N: ΦVRXUFH 4LQ ± 7 VRXUFH (QWURS\ (T PHVH PLVL 4 7V 6JHQ

$V 3 & WKH SUHVVXUH FRUUHFWLRQ LQ (T GURSV RXW WR JLYH JHQHUDWLRQ DV 6JHQ P VH VL 4 7V

> × ± × ± @ N: .

ΦIORZ ΦVRXUFH ± ΦGHVWUXFWLRQ ΦVRXUFH ± 7 6JHQ ± × ΦIORZ η ΦVRXUFH

5HPDUN :H FRXOG DOVR H[SOLFLWO\ KDYH IRXQG WKH IORZ H[HUJ\ LQFUHDVH

. 5DGLDWLRQ L

H


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

5HYLHZ 3UREOHPV

$ SLVWRQ F\OLQGHU FRQWDLQV KHOLXP DW N3D DW DPELHQW WHPSHUDWXUH °& DQG LQLWLDO YROXPH RI / DV VKRZQ LQ )LJ 3 7KH VWRSV DUH PRXQWHG WR JLYH D PD[LPXP YROXPH RI / DQG WKH QLWURJHQ OLQH FRQGLWLRQV DUH N3D °& 7KH YDOYH LV QRZ RSHQHG ZKLFK DOORZV QLWURJHQ WR IORZ LQ DQG PL[ ZLWK WKH KHOLXP 7KH YDOYH LV FORVHG ZKHQ WKH SUHVVXUH LQVLGH UHDFKHV N3D DW ZKLFK SRLQW WKH WHPSHUDWXUH LQVLGH LV °& ,V WKLV SURFHVV FRQVLVWHQW ZLWK WKH VHFRQG ODZ RI WKHUPRG\QDPLFV"

3 N3D 7 R& 9 / 9PD[ / 9

3 N3D 7 R& 3L N3D 7L R&

&RQVWDQW 3 WR VWRSV WKHQ FRQVWDQW 9 9PD[ ! :FY 3 9 9 4FY 8 8 :FY QLKL Q K Q K QLKL 3 3 9 Q$ K$ K$L Q% K% K% 3 3 9 Q% Q 3 9 57 × × NPRO Q Q$ Q% 3 9 57 × × NPRO

Q$ Q Q% NPRO

0ROH IUDFWLRQV \$ \%

4FY × × × ×

N- 6JHQ Q V Q V QLVL 4FY 7 Q$ V$ V$L Q% V% V% 4FY 7

V$ V$L OQ 5 OQ

V% V% OQ 5 OQ

6JHQ × ×

N- . !

6DWLVILHV QG ODZ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ VSKHULFDO EDOORRQ KDV DQ LQLWLDO GLDPHWHU RI P DQG FRQWDLQV DUJRQ JDV DW N3D °& 7KH EDOORRQ LV FRQQHFWHG E\ D YDOYH WR D / ULJLG WDQN FRQWDLQLQJ FDUERQ GLR[LGH DW N3D °& 7KH YDOYH LV RSHQHG DQG HYHQWXDOO\ WKH EDOORRQ DQG WDQN UHDFK D XQLIRUP VWDWH LQ ZKLFK WKH SUHVVXUH LV N3D 7KH EDOORRQ SUHVVXUH LV GLUHFWO\ SURSRUWLRQDO WR LWV GLDPHWHU 7DNH WKH EDOORRQ DQG WDQN DV D FRQWURO YROXPH DQG FDOFXODWH WKH ILQDO WHPSHUDWXUH DQG WKH KHDW WUDQVIHU IRU WKH SURFHVV Ï€ 9$ 3$ 9$ × $ P$ 57 NJ × $ P% 3% 9% 57% × × NJ

3 9$ 3$ 9$ →

%

CO 2

3 P 9$ 9$ 3 $ 8QLIRUP LGHDO JDV PL[WXUH 3 9$ 9% P$5$ P%5% 7

7 × × .

:

4 P$&9 $ 7 7$ P%&9 % 7 7% :

× ×

N-

3 9$ 3$ 9$ × × N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$Q LQVXODWHG YHUWLFDO F\OLQGHU LV ILWWHG ZLWK D IULFWLRQOHVV FRQVWDQW ORDGHG SLVWRQ RI FURVV VHFWLRQDO DUHD P DQG WKH LQLWLDO F\OLQGHU KHLJKW RI P 7KH F\OLQGHU FRQWDLQV PHWKDQH JDV DW . N3D DQG DOVR LQVLGH LV D / FDSVXOH FRQWDLQLQJ QHRQ JDV DW . N3D 7KH FDSVXOH QRZ EUHDNV DQG WKH WZR JDVHV PL[ WRJHWKHU LQ D FRQVWDQW SUHVVXUH SURFHVV :KDW LV WKH ILQDO WHPSHUDWXUH ILQDO F\OLQGHU KHLJKW DQG WKH QHW HQWURS\ FKDQJH IRU WKH SURFHVV $S P K P !

9WRW 9D 9E P

0HWKDQH 0 NJ NPRO &S N- NJ . 5 N- NJ . 1HRQ 0 NJ NPRO &S N- NJ . 5 N- NJ . 6WDWH 0HWKDQH 7DO . 3D N3D 9D 9WRW 9E P

1HRQ 7EO . 3E N3D 9E /

3DO9DO PD PD 5 7 NJ QD 0 NPRO D DO D

3EO9EO PE PE 5 7 NJ QE 0 NPRO E EO E

6WDWH 0L[ 3 PL[ 3D N3D (QHUJ\ (T 4 PD XD XD PE XE XE : 4 : ∫ 3 G9 3 9 9 WRW 3 9 PWRW5PL[7 PD5D PE5E 7 $VVXPH &RQVWDQW 6SHFLILF +HDW PD&SD 7 7D PE&SE 7 7E PD5D PE5E 7 3 9 WRW 6ROYLQJ IRU 7 . 9

PD5D PE5E 7 9 P K $ P 3 S

4 (QWURS\ (T ∆6QHW PD VD VD PE VE VE 7 4

QD \D Q Q \E \D D E 7 \ D 3 VD VD &SD OQ7 5DOQ 3 N- NJ . D

D

7 \ E 3 VE VE &SE OQ7 5E OQ 3 N- NJ . E

∆6QHW N- .

E


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$Q LQVXODWHG ULJLG P WDQN $ FRQWDLQV &2 JDV DW °& 03D $Q XQLQVXODWHG

ULJLG P WDQN % FRQWDLQV HWKDQH & + JDV DW N3D URRP WHPSHUDWXUH °& 7KH WZR DUH FRQQHFWHG E\ D RQH ZD\ FKHFN YDOYH WKDW ZLOO DOORZ JDV IURP $ WR % EXW QRW IURP % WR $ 7KH YDOYH LV RSHQHG DQG JDV IORZV IURP $ WR % XQWLO WKH SUHVVXUH LQ % UHDFKHV N3D DQG WKH YDOYH LV FORVHG 7KH PL[WXUH LQ % LV NHSW DW URRP WHPSHUDWXUH GXH WR KHDW WUDQVIHU )LQG WKH WRWDO QXPEHU RI PROHV DQG WKH HWKDQH PROH IUDFWLRQ DW WKH ILQDO VWDWH LQ % )LQG WKH ILQDO WHPSHUDWXUH DQG SUHVVXUH LQ WDQN $ DQG WKH KHDW WUDQVIHU WR IURP WDQN % 6ROXWLRQ B A CO 2

C 2H 6

7DQN $ 9$ P VWDWH $ &2 7$ °& . 3$ 03D &Y &2 × &3 &2 × N- NPRO . 7DQN % 9% P VWDWH % & + 7% °& . 3% N3D 6ORZ )ORZ $ WR % WR 3% N3D DQG DVVXPH 7% 7% 7 7RWDO PROHV VFDOHV WR SUHVVXUH VR ZLWK VDPH 9 DQG 7 ZH KDYH 3% 9% Q% 57% 3% 9% Q% PL[ 57% 0ROH IUDFWLRQ

Q% 3% \& + % Q 3 % %

3% 9% × Q% 57 NPRO 5 × % 3% 9% × Q% PL[ 57 NPRO 5 × % Q&2 % ± NPRO 1RZ ZH FDQ ZRUN EDFNZDUGV WR ILQDO VWDWH LQ $ 3$ 9$ × Q$ 57 NPRO Q$ Q$ Q&2 % NPRO 5× $ & 9 $ $OO &2 7UDQVLHQW ZLWK IORZ RXW DQG DGLDEDWLF (QHUJ\ (T 4&9 $ Q$ X $ Q$ X $ QDYH K DYH


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

Q$ &Y &2 7$ Q$ &Y &2 7$ QDYH &3 &2 7$ 7$ × 7$ × × 7$

3$

!

7$ .

Q$ 57$ × 5 × N3D 9$

& 9 % 7UDQVLHQW ZLWK IORZ LQ DQG QRQ DGLDEDWLF 4&9 % Q%L K %L DYH QX % QX % QX &2 % QX & + % QX & + % 4&9 % × × ± × N-


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ P LQVXODWHG ULJLG YHVVHO LV GLYLGHG LQWR WZR HTXDO SDUWV $ DQG % E\ DQ LQVXODWHG SDUWLWLRQ DV VKRZQ LQ )LJ 3 7KH SDUWLWLRQ ZLOO VXSSRUW D SUHVVXUH GLIIHUHQFH RI N3D EHIRUH EUHDNLQJ 6LGH $ FRQWDLQV PHWKDQH DQG VLGH % FRQWDLQV FDUERQ GLR[LGH %RWK VLGHV DUH LQLWLDOO\ DW 03D °& $ YDOYH RQ VLGH % LV RSHQHG DQG FDUERQ GLR[LGH IORZV RXW 7KH FDUERQ GLR[LGH WKDW UHPDLQV LQ % LV DVVXPHG WR XQGHUJR D UHYHUVLEOH DGLDEDWLF H[SDQVLRQ ZKLOH WKHUH LV IORZ RXW (YHQWXDOO\ WKH SDUWLWLRQ EUHDNV DQG WKH YDOYH LV FORVHG &DOFXODWH WKH QHW HQWURS\ FKDQJH IRU WKH SURFHVV WKDW EHJLQV ZKHQ WKH YDOYH LV FORVHG ∆30$; N3D 3$ 3% 03D % $ 9$ 9% P &+ &2 7$ 7% R& . &2 LQVLGH % V% V% WR 3% N3D 3$ N3D

)RU &2 N ! 7% . Q% 3% 9% 57% × × Q$ Q$ × × NPRO 4 Q X QL X L Q$ &9 $ 7 7$ Q% &9 % 7 7% ∑

L

× × 7 × × 7

6ROYH 7 .

3 × × N3D

3$ × N3D

3% 3 3$ N3D

V $ V $ × OQ OQ N- NPRO . V % V % × OQ OQ N- NPRO .

∆61(7 × × N- .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$Q DLU ZDWHU YDSRU PL[WXUH HQWHUV D VWHDG\ IORZ KHDWHU KXPLGLILHU XQLW DW VWDWH °& UHODWLYH KXPLGLW\ DW WKH UDWH RI P V $ VHFRQG DLU YDSRU VWUHDP HQWHUV WKH XQLW DW VWDWH °& UHODWLYH KXPLGLW\ DW WKH UDWH RI P V /LTXLG ZDWHU HQWHUV DW VWDWH °& DW WKH UDWH RI NJ SHU KRXU $ VLQJOH DLU YDSRU IORZ H[LWV WKH XQLW DW VWDWH °& &DOFXODWH WKH UHODWLYH KXPLGLW\ RI WKH H[LW IORZ DQG WKH UDWH RI KHDW WUDQVIHU WR WKH XQLW $VVXPH 3 N3D 6WDWH 7 °& φ 9D P V 3J N3D 3Y φ 3J N3D 3D 3 3Y N3D 3Y 3D 9D ω 3 PD 57 NJ V D D PY ω PD NJ V KY KJ N- NJ 6WDWH 7 °& φ 9D P V 3J N3D 3Y φ 3J N3D 3D 3 3Y N3D 3Y 3D 9 D ω 3 PD 57 NJ V D D PY ω PD NJ V KY KJ N- NJ 6WDWH /LTXLG 7 °& PI NJ KU NJ V KI N- NJ

6WDWH 7 °&

&RQWLQXLW\ (T DLU PD PD PD NJ V &RQWLQXLW\ (T ZDWHU PY PY PY PI NJ V P Y 3Y ω 3 3 → 3Y N3D P D Y 3Y 3J N3D φ 3 KY KJ N- NJ J VW/DZ 4 PD KD PY KY PD KD PY KY PI KI PD KD PY KY 4 × × × ×

× × ×

N:


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

<RX KDYH MXVW ZDVKHG \RXU KDLU DQG QRZ EORZ GU\ LW LQ D URRP ZLWK °& φ 7KH GU\HU : KHDWV WKH DLU WR °& EORZV LW WKURXJK \RXU KDLU ZKHUH WKH DLU EHFRPHV VDWXUDWHG DQG WKHQ IORZV RQ WR KLW D ZLQGRZ ZKHUH LW FRROV WR °& )LQG WKH UHODWLYH KXPLGLW\ DW VWDWH WKH KHDW WUDQVIHU SHU NLORJUDP RI GU\ DLU LQ WKH GU\HU WKH DLU IORZ UDWH DQG WKH DPRXQW RI ZDWHU FRQGHQVHG RQ WKH ZLQGRZ LI DQ\ 7KH EORZGU\HU KHDWV WKH DLU DW FRQVWDQW VSHFLILF KXPLGLW\ WR DQG LW WKHQ JRHV WKURXJK DQ DGLDEDWLF VDWXUDWLRQ SURFHVV WR VWDWH ILQDOO\ FRROLQJ WR °& UHO KXP ! Z K× N- NJ GU\ DLU

Z Z 7 ! φ2 K× N- NJ GU\ DLU

&9 WR

Z Z T K× K× N- NJ GU\ DLU PD 4 T NJ V &9 WR Z Z POLT PD PD K× POLT KI PD K×

φ ! 7 7ZHW °& Z φ 7 ! Z POLT Z Z PD × J V ,I WKH VWHDP WDEOHV DQG IRUPXOD V DUH XVHG WKHQ ZH JHW KJ KJ 3J 3Y

3J Z Z Z 3Y 3Y

φ 3Y 3J KI

7ULDO DQG HUURU IRU DGLDEDWLF VDWXUDWLRQ WHPSHUDWXUH

7 °& Z 3Y 3J N3D

Z ×


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

6WHDP SRZHU SODQWV RIWHQ XWLOL]H ODUJH FRROLQJ WRZHUV WR FRRO WKH FRQGHQVHU FRROLQJ ZDWHU VR LW FDQ EH UHFLUFXODWHG VHH )LJ 3 7KH SURFHVV LV HVVHQWLDOO\ HYDSRUDWLYH DGLDEDWLF FRROLQJ LQ ZKLFK SDUW RI WKH ZDWHU LV ORVW DQG PXVW WKHUHIRUH EH UHSOHQLVKHG &RQVLGHU WKH VHWXS VKRZQ LQ )LJ 3 LQ ZKLFK NJ V RI ZDUP ZDWHU DW °& IURP WKH FRQGHQVHU HQWHUV WKH WRS RI WKH FRROLQJ WRZHU DQG WKH FRROHG ZDWHU OHDYHV WKH ERWWRP DW °& 7KH PRLVW DPELHQW DLU HQWHUV WKH ERWWRP DW N3D GU\ EXOE WHPSHUDWXUH RI °& DQG D ZHW EXOE WHPSHUDWXUH RI °& 7KH PRLVW DLU OHDYHV WKH WRZHU DW N3D °& DQG UHODWLYH KXPLGLW\ RI 'HWHUPLQH WKH UHTXLUHG PDVV IORZ UDWH RI GU\ DLU DQG WKH IUDFWLRQ RI WKH LQFRPLQJ ZDWHU WKDW HYDSRUDWHV DQG LV ORVW $LU YDS 3 N3D R 4 7 R& 1 7 & φ P NJ V LIQ H 2O 3 N3D 7 R&

:%7 R&

7 R&

2

$LU YDS 2' 7 ′ R& φ ′ Z ′ × /,4 ,1 KD ′ KD Z ′ K)* ′ × Z KY K) ′ R&

2

3Y × Z × &RQV PDVV PD PD PD P PY P PY RU P P PD Z Z DQG VHW U P PD VW ODZ P K PDKD PY KY P K PDKD PY KY

RU U K KD KD Z KY U Z Z K Z KY

U K K KD KD Z KY Z KY Z Z K

U ×

3

× × U P PD → PD NJ V P P PD Z Z × ∆P P


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$PELHQW DLU LV DW D FRQGLWLRQ RI N3D °& UHODWLYH KXPLGLW\ $ VWHDG\ VWUHDP RI DLU DW N3D °& UHODWLYH KXPLGLW\ LV WR EH SURGXFHG E\ ILUVW FRROLQJ RQH VWUHDP WR DQ DSSURSULDWH WHPSHUDWXUH WR FRQGHQVH RXW WKH SURSHU DPRXQW RI ZDWHU DQG WKHQ PL[ WKLV VWUHDP DGLDEDWLFDOO\ ZLWK WKH VHFRQG RQH DW DPELHQW FRQGLWLRQV :KDW LV WKH UDWLR RI WKH WZR IORZ UDWHV" 7R ZKDW WHPSHUDWXUH PXVW WKH ILUVW VWUHDP EH FRROHG" 3 3 N3D 1 5 7 7 R& 2 MIX COOL φ φ φ . 4 . 3 3 7 R& -Q MIX= 0 -Q COOL LIQ H 2O φ 3Y 3Y × N3D ! Z Z ×

3Y × N3D ! Z ×

& 9 0L[LQJ FKDPEHU &DOO WKH PDVV IORZ UDWLR U PD PD

FRQV PDVV Z U Z U Z

(QHUJ\ (T KD Z KY UKD UZ KY U KD U Z KY

→ UZ U

PD 3* RU U P Z ZLWK Z × 3 D *

× × U× ×7 U Z KY

RU U ×7 Z K*

$VVXPH 7 R& → 3* K*

Z ×

> × × @ ≈ 2.

U × × U × ×

>

@

U PD PD


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ VHPLSHUPHDEOH PHPEUDQH LV XVHG IRU WKH SDUWLDO UHPRYDO RI R[\JHQ IURP DLU WKDW LV EORZQ WKURXJK D JUDLQ HOHYDWRU VWRUDJH IDFLOLW\ $PELHQW DLU QLWURJHQ R[\JHQ RQ D PROH EDVLV LV FRPSUHVVHG WR DQ DSSURSULDWH SUHVVXUH FRROHG WR DPELHQW WHPSHUDWXUH °& DQG WKHQ IHG WKURXJK D EXQGOH RI KROORZ SRO\PHU ILEHUV WKDW VHOHFWLYHO\ DEVRUE R[\JHQ VR WKH PL[WXUH OHDYLQJ DW N3D °& FRQWDLQV RQO\ R[\JHQ 7KH DEVRUEHG R[\JHQ LV EOHG RII WKURXJK WKH ILEHU ZDOOV DW N3D °& WR D YDFXXP SXPS $VVXPH WKH SURFHVV WR EH UHYHUVLEOH DQG DGLDEDWLF DQG GHWHUPLQH WKH PLQLPXP LQOHW DLU SUHVVXUH WR WKH ILEHU EXQGOH $ 1 % 2 3 N3D 0.79 A 1 2 0.79 A MIX + + 5 % 3 N3D 0.21 B 0.0416 B O2 $OO 7 R& 3 P1 = ? (0.79/0.95) 0.1684 B

/HW V$ V% DW 7 R& 3 V 0,; \$ 5 OQ \$ \% 5 OQ \% V 0,; 5 OQ 3 3 \$ 5 OQ \$ \% 5 OQ \% 3XUH % V 5 OQ 3 3 )RU Q V Q V Q V

5 OQ 3 3 OQ OQ

OQ 3 3 OQ 3 3 OQ 3

3 PLQ N3D

>

OQ 3 3 OQ OQ

)RU 3 ! 3 PLQ ZH ZRXOG KDYH HQWURS\ JHQHUDWLRQ ∆6 !

@


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

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

6WDWH V V K N- NJ 7 R&

6WDWH [ K KI N- NJ 7 °&

6WDWH K K DQG 3 3

R-22 Evaporator

Air

4

3

7

R-22 Condenser

Air

o

Air sat. 20 C

in

ex 1

2 C

WC

T+ K ± K ± N- NJ

T/ K ± K ± N- NJ

V

R 7 & 3 N3D

1RZ ZH JHW Z& K ± K ± N- NJ

)RU WKH DLU SURFHVVHV OHW XV XVH WKH SV\FKURPHWULF FKDUW


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$LU LQOHW ZLQ K×LQ N- NJ GU\ DLU 7GHZ R& ! R& $LU φ Z K× KI %

1RZ GR WKH FRQWLQXLW\ IRU ZDWHU DQG HQHUJ\ HTXDWLRQV IRU WKH FRROLQJ SURFHVV POLT PDLU ZLQ Z ± NJ V 4FRRO PDLU K×LQ K× POLTKI ± ± N: 1RZ WKH KHDWHU IURP WKH 5 F\FOH KDV 4KHDW 4FRRO T+ T/ N: VR WKH FRPSUHVVRU ZRUN LV WKH EDODQFH RI WKH WZR :& 4KHDW 4FRRO ± N: (QHUJ\ HT IRU WKH DLU IORZ EHLQJ KHDWHG 4KHDW PDLU K×H[ K× ⇒ K×H[ K× 4KHDW PDLU K×H[ N- NJ GU\ DLU DQG ZH[ Z /RFDWH VWDWH LQ WKH SV\FKURPHWULF FKDUW

7 R& DQG φ

Z

Φ =

Φ = 7GU\


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ / LQVXODWHG WDQN FRQWDLQV 1 JDV DW N3D DQG DPELHQW WHPSHUDWXUH °& 7KH WDQN LV FRQQHFWHG E\ D YDOYH WR D VXSSO\ OLQH IORZLQJ &2 DW 03D °& $ PL[WXUH RI 1 &2 E\ PROH VKRXOG EH REWDLQHG E\ RSHQLQJ WKH YDOYH DQG DOORZLQJ &2 IORZ LQ WR DQ DSSURSULDWH SUHVVXUH LV UHDFKHG DQG FORVH WKH YDOYH :KDW LV WKH SUHVVXUH" 7KH WDQN HYHQWXDOO\ FRROV WR DPELHQW WHPSHUDWXUH )LQG WKH QHW HQWURS\ FKDQJH IRU WKH RYHUDOO SURFHVV CO2 i 9 / 3 N3D 7 7 R&

N2

NPRO Q NPRO

VW ODZ QLKL Q X Q X IRU FRQVW VSHFLILF KHDWV QL&3RL7L QL&9RL Q &9R 7 Q &9R 7 %XW QL Q → &3RL7L &9RL7 &9R 7 7

× × × 7 × 7

7 . 3 Q 57 9 × × N3D

&RRO WR 7 7 .

3 3 × 7 7 × N3D 4 Q &9R 7 7 × ×

× × N- ∆61(7 Q V Q V QLV L 4 7 QL> V &2 V L@ Q > V 1 V @ 4 7

× × OQ OQ

N- .

3L 03D 7L R& $W VWDWH \1 \&2 Q &2 Q 1 Q 1 3 9 57

× × OQ


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

$ F\OLQGHU SLVWRQ ORDGHG ZLWK D OLQHDU VSULQJ FRQWDLQV VDWXUDWHG PRLVW DLU DW N3D P YROXPH DQG DOVR NJ RI OLTXLG ZDWHU DOO DW DPELHQW WHPSHUDWXUH °& 7KH SLVWRQ DUHD LV P DQG WKH VSULQJ FRQVWDQW LV N1 P 7KLV F\OLQGHU LV DWWDFKHG E\ D YDOYH WR D OLQH IORZLQJ GU\ DLU DW N3D °& 7KH YDOYH LV RSHQHG DQG DLU IORZV LQWR WKH F\OLQGHU XQWLO WKH SUHVVXUH UHDFKHV N3D DW ZKLFK SRLQW WKH WHPSHUDWXUH LV °& 'HWHUPLQH WKH UHODWLYH KXPLGLW\ DW WKH ILQDO VWDWH WKH PDVV RI DLU HQWHULQJ WKH F\OLQGHU DQG WKH ZRUN GRQH GXULQJ WKH SURFHVV 3 N3D 7 R& 7 9 P P/,4 NJ $3 P NV N1 P 3L N3D 3 N3D 7L R& 7 R&

A+V

3 3 NV $S 9 9

i

DRY AIR

9 → 9 P

φ RU Z ×

3Y 9 × PY 5 7 Z P$ × Y

$VVXPH QR OLTXLG DW VWDWH PY PY P/ NJ

→ 3Y

D φ

3$ 9 × E P$ 5 7 × $

P$

P$L P$ P$ NJ

F :&9 ⌠ ⌡ 3G9 3 3 9 9 N-

PY5Y7 × × N3D 9

× ×


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

&RQVLGHU WKH SUHYLRXV SUREOHP DQG DGGLWLRQDOO\ GHWHUPLQH WKH KHDW WUDQVIHU 6KRZ WKDW WKH SURFHVV GRHV QRW YLRODWH WKH VHFRQG ODZ

D 4&9 P$ K$ P$ K$ P$LK$L PY KY PY KY P/ K/ 3 9 3 9 :&9

× × × × × × ×

× ± × × ×

N-

E ∆6&9 6 6 P$ V$ P$ V$ PYVY PY VY P/ V/

∆66855 4&9 7 P$LV$L

∆61(7 P$ V$ P$ V$ P$LV$L PYVY PY VY P/ V/ 4&9 7

P$ V$ V$L P$ V$L V$ PYVY PY VY P/ V/ 4&9 7

V$ V$L OQ OQ

V$L V$ OQ OQ

VY V* 5Y OQ φ OQ

VY V* 5Y OQ φ ±

V/ V) → ∆61(7 × ±

× ± ×

N- .


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

7KH DLU FRQGLWLRQLQJ E\ HYDSRUDWLYH FRROLQJ LQ 3UREOHP LV PRGLILHG E\ DGGLQJ D GHKXPLGLILFDWLRQ SURFHVV EHIRUH WKH ZDWHU VSUD\ FRROLQJ SURFHVV 7KLV GHKXPLGLILFDWLRQ LV DFKLHYHG DV VKRZQ LQ )LJ 3 E\ XVLQJ D GHVLFFDQW PDWHULDO ZKLFK DEVRUEV ZDWHU RQ RQH VLGH RI D URWDWLQJ GUXP KHDW H[FKDQJHU 7KH GHVLFFDQW LV UHJHQHUDWHG E\ KHDWLQJ RQ WKH RWKHU VLGH RI WKH GUXP WR GULYH WKH ZDWHU RXW 7KH SUHVVXUH LV N3D HYHU\ZKHUH DQG RWKHU SURSHUWLHV DUH RQ WKH GLDJUDP &DOFXODWH WKH UHODWLYH KXPLGLW\ RI WKH FRRO DLU VXSSOLHG WR WKH URRP DW VWDWH DQG WKH KHDW WUDQVIHU SHU XQLW PDVV RI DLU WKDW QHHGV WR EH VXSSOLHG WR WKH KHDWHU XQLW 6WDWHV DV QRWHG RQ )LJ 3 WH[W SDJH

$W VWDWH R& 39 φ 3* ×

Z ×

$W 7 R& Z Z Z

(YDSRUDWLYH FRROLQJ SURFHVV WR VWDWH ZKHUH 7 R& $V LQ (T Z KY KI &3 $ 7 7 Z KIJ

Z ×

Z × 3Y 3Y

39 N3D φ

)ROORZLQJ QRZ WKH IORZ EDFN ZH KDYH

$W 7 R& Z Z

(YDSRUDWLYH FRROLQJ SURFHVV WR VWDWH ZKHUH 7 R&

Z KY KI &3 $ 7 7 Z KIJ

Z × ! Z

)RU DGLDEDWLF KHDW H[FKDQJHU P$ P$ P$ P$ P$ $OVR Z Z Z Z 6R QRZ RQO\ 7 LV XQNQRZQ LQ WKH HQHUJ\ HTXDWLRQ

K$ Z KY K$ Z KY K$ Z KY K$ Z KY

RU &3 $7 Z KY KY &3 $ 7 7 7 Z KY KY

7 KY


6RQQWDJ %RUJQDNNH DQG YDQ :\OHQ

%\ WULDO DQG HUURU 7 R& KY N- NJ )RU WKH KHDWHU Z Z 4 P$ &3 $ 7 7 Z KY KY

N- NJ GU\ DLU


Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.