Improving the refrigeration and gas liquefaction performance of Gifford-McMahon and active magnetic regenerative cryocoolers: a study of flow maldistribution, unbalance, and asymmetry

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Improving the Refrigeration and Gas Liquefaction Performance of

Gifford-McMahon and Active Magnetic Regenerative Cryocoolers:

A Study of Flow Maldistribution, Unbalance, and Asymmetry

by

Ian Gregory Spearing B .A .S c., University o f Toronto. 1990 M .A .Sc.. University o f Victoria, 1995

A Dissertation Submitted in Partial Fulfilment o f the Requirements for the Degree o f

DOCTOR OF PHILOSOPHY

in the Department o f Mechanical Engineering

W e accept this dissertation as conforming to the required standard

r J.A. Barclay, Supervisor (Department qc Mechanical Engineering)

ember (Department o f Mechanical Engineering)

er (Department o f Mechanical Engineering)

Dr. J.D. Jones. Outside Member (Simon Fraser University)

Dr. T. Bose. External Examiner (Université du Québec à Trois-Rivières)

0 IAN GREGORY SPEARING. 2000

University o f Victoria

A ll rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or

other means, without the express written permission o f the author.

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Supervisor; Dr. John A . Barclay

Abstract

C o st-effectiv e liquefaction o f gases at cry o g en ic tem peratures requires a com bined approach of desig n in g efficien t refrigeration cycles to generate cooling capacity and designing efficient liquefaction processes to utilize that capacity. T his d issertatio n addresses both approaches for im provem ent of the liquefaction process. M agnetic refrigeration em p lo y in g the m agnetocaloric effect o f ferrom agnetic m aterials has been identified as potentially m ore efficient and c o st-effectiv e than conventional refrigeration system s. O ne m agnetic cycle that show s prom ise for efficiently achieving cooling over large tem p eratu re spans is active m agnetic regenerative refrigeration (.AMRR). In this cycle the m agnetic m aterial serves the dual functions o f w ork input and therm al reg eneration, how ever the operation is comple.x with coupled fluid, therm al, and magnetic phenom ena and a clearer understanding o f the reg en erativ e operation is required. M odels to elucidate the How ch aracteristics o f the regenerative heat ex ch an g ers o f rotary A M R R and G ifford-M cM ahon (G.M) system s using a com m ercially av ailab le co m p u tatio n al fluid dynam ics (C FD ) softw are package are described. T h eo retical results are presented to q u alify and quantify the effect o f m aldistributed tlow within regenerators. E xperim ental results o f an im proved regenerator fo r the G.M system based on the CFD flow sim u latio n s are presented. E fforts to d evelop an im proved A M R R therm al model using the com m ercial package are also d escribed.

T he second ap proach for the design o f effic ie n t liquefiers addresses a draw back o f the usual em bodim ent o f the A M R R cycle, nam ely, the provision o f cooling at a single tem perature w hich necessitates that cry o g en ic d esigns have m ultiple stages providing c o o lin g o v er a range o f discrete tem p eratu res for an efficien t liquefaction process. U se o f m ultiple stages leads to increased expense and co m p lex ity . A sim ple, inexpensive plum bing change o f the flow through the regenerator o f a single-stage d ev ice can significantly increase the overall liquefaction cap acity com pared to the usual fiow co n fig u ratio n m aking additional staging unnecessary. This dissertation describ es the alternative flow arran g em en t, know n variously as “bypass flow ," “perm anent flow ." o r "D C flow .” w hich is suitable fo r all passive and active regenerative refrigeration cycles used as liquefiers. Theoretical

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Ill

results showing increased liquefaction capacity when b ypass flow is employed are given for active magnetic regenerative and Gifford-M cMahon systems. Experimental results are presented for a single- stage GM refrigerator modified for bypass flow which demonstrates increased liquefaction capacity .

Examiners:

D M . A. Barclay, Supervisor (Department o f M echanical Engineering)

D r N. ujilanTM ember (Department o f Mechanical Engineering)

Dr. S. D ost, Member (Department o f Mechanical Engineering)

Dr. J.D. Jones. Outside Member (Simon Fraser University)

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w

A

and

I I ...

2 .Applications o f R egen erators: G iffo rd -M cM a h o n and .Active M agnetic R egen erative R efrigerators 26 2.1 Introduction ...26

2.2 T he G iffo rd -M cM ah o n C y c l e ... 26

2.2.1 B asis o f R efrigeration ... 26

2.2.2 GM S chem atic and C ycle D e s c r ip tio n ... 27

2.2.3 GM R eg en erato r B alance and Sym m etry ...30

2.3 A ctive M agnetic R egenerative R e f r i g e r a t i o n ... 30

2.3.1 B asis o f M agnetic R e f r ig e r a tio n ... '0

2.3.2 Fundam ental R e la tio n s ...32

2.3.3 T he A ctive M agnetic R egenerative C y c l e ... 35

2.3.4 A M R R eg en erato r Solid Energy B alance E q u a t i o n ... 3X 2.3.5 A M R R eg en erato r B alance and S y m m e try ... 39

2.3.6 .AMR R egenerator E ffe c tiv e n e s s ...40

2.3.7 Rotary A ctive M agnetic R e g e n e r a tio n ... 40

3 L iteratu re R eview 42 3.1 Introduction ... 42

3.2 R eg en erato r M athem atical M odels and S olution M e t h o d s ...42

3.2.1 E m pirical M o d e ls ...43

3.2.2 Energy B alance M odels ... 45

3.2.3 C losed M eth o d s ...46

3.2.4 O pen M e t h o d s ...5 1 3.3 R eg en erato r F low M a ld is trib u tio n ...53

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3.5 R egenerators A pplied in R efrig eratio n C ycles ...6Ü

3.5.1 R egenerators in G M System s ...60

3.5.2 R egenerators in A M R S y s te m s ... 63

3.5.3 M agnetic R eg en erato r M o d e l l i n g ...63

3.6 A p plications o f U nbalanced R e g e n e ra to rs ...6V 3.6.1 B oreas C r y o c o o l e r ... 6V 3.6.2 Bypass A ctive M agnetic R e g e n e ra to rs ...7(j 4 R eg en era to r Flow M ald istrib u tion " I 4.1 M aldistribution in C ryogenic G as E.xpansion S y s t e m s ... ' I 4.1.1 P e n n e y 's M ethod E x ten d ed ... 71

4.1.2 C om pensating for M aldistribution by Increasing R educed L e n g t h ..."5 4.2 N um erical Sim ulation o f F low w ithin a R e g e n e r a t o r ... '’6

4.2.1 M o t i v a t i o n ... 76

4.2.2 C onventional I "-S tag e G ifford-M cM ahon R e g e n e r a to r ...77

4.2.3 M odified 1 "-S tage G iffo rd -M cM ah o n R e g e n e r a t o r ... S2 4.3 C onventional and M odified R egenerator P e r f o r m a n c e ...S6 4.3.1 C om parison o f E.xperim ental R esults ...n6 5 R efrigeration and L iq u efaction P rin cip les 40 5.1 L iquefaction o f G a s e s ...40 5.1.1 R efrigerator V ersus L i q u e f i e r ... 41 5.2 Ideal P e r f o r m a n c e ... 41 5.2.1 C arnot C y c l e ... 41 5.2.2 C old G as R e f r i g e r a t o r ... 43 5.2.3 Ideal L i q u e f i e r ...44

5.3 C om parison o f the Ideal C y cles A pplied to G as L iquefaction ... 96

5.3.1 L atent and S ensible C o o lin g R e q u i r e m e n t s ... 46

5.3.2 Figure o f M e r i t ...4S 5.3.3 Separation o f S en sib le and L atent C ooling R e q u ir e m e n ts ... 100

5.4 U nbalanced, A sym m etrical R eg en erato r L iq u e f ie r s ... 102

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V I I

5.5 E ffect o f B ypass F lo w on G M S y s t e m s ... 105

5.5.1 T h erm al P erform ance ... 105

5.5.2 E ffect o f U n b alance on the T herm al R a t i o ... 105

5.5.3 E ffect o f U n b alance on the G M C old H ead M inim um T em p eratu re . . . . 106

5.6 B ypass A M R R e g e n e r a t o r ... 100

5.6.1 B ypass A M R S i m u l a t i o n s ... 110

5.6.2 E ffect o f B ypass Flow on the M agnetic W ork R a t e ... 115

5.6.3 L atent and S en sib le Load M a t c h i n g ... 11»

5.6.4 T o tal E q u iv alen t C ooling C apacity ... 119

5.6.5 C ase Study: E ffect o f B ypass Flow on the L iquefaction o f E thane at 225 K ... 121

6 S im u la tio n o f A c tiv e M a g n e tic R e g e n e ra to rs U sin g C F X -T A S C F lo w 12" 6.1 Introduction ... 127

6.1.1 P roblem w ith E xisting M o d e l s ... 12"^ 6.2 R otary R eg en erato r F luids Sim ulation U sing C F X -T A S C F lo w ... 12»

6.2.1 C o n serv atio n o f Fluid M ass and M om entum ... 12»

6.2.2 M om en tu m S ource T e r m ... 12»

6.2.3 Scope o f M odel and M e th o d o lo g y ... 129

6.2.4 B oundary C o n d i t i o n s ... 1.^2 6.3 R otary R eg en erato r F low Sim ulation R e s u l t s ... 13.’

6.3.1 Basic F lo w P a t t e r n ... 133

6.3.2 F luid V elo city D e c o m p o s itio n ... 135

6.4 R eg en erato r T h e rm a l M odelling in C F X - T A S C F lo w ... 13S 6.4.1 C o n serv atio n o f Energy ... 13»

6.4 .2 Energy S o u rce T erm s and S olid-F luid C o u p lin g ... 139

6.4.3 E nergy E q u atio n S o l u t i o n ... 143

6.4.4 E nergy S o u rce T erm s and C oupling. R e v i s i t e d ... 145

6.5 T h erm al M o d ellin g U sing a C ustom T A S C F lo w R e l e a s e ... 147

6.5.1 F eatures o f the C ustom R e l e a s e ... 147

6.5.2 C o n v e r g e n c e ... 147

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Vlll 6.5.4 T herm al S olution G rid D e p e n d e n c y ... 15 I

6.5.5 Source o f the T herm al S olution G rid D e p e n d e n c y ... 15?

6.5.6 Im plem entation I s s u e s ... 154

6.5.7 C ontrol V olum e D e f in itio n ... 154

6.5.8 G rid B l o c k - O f f ... 155

6.5.9 C artesian G rid R e p r e s e n ta tio n ... 157

6.6 .Active M agnetic R egenerator S i m u l a t i o n ... 158

6.7 C o n c l u s i o n ... 161 7 B y p a ss G iffo rd -M c M a h o n E x p e rim e n ta l A p p a r a tu s 16? 7.1 Introduction ... 16? 7.2 Bypass GM A p p a r a t u s ... 16? 7.2.1 C ycle S c h e m a t i c ... 16? 7.2.2 E xperim ental A p p a r a t u s ... 166

7.3 Key Elem ents o f the A pparatus D e s i g n ... 169

7.3.1 C old H ead and C old B uffer ... 169

7.3.2 W arm B uffer V olum e and M ake-up G a s ... 17 j 7.3.3 Bypass H e a t e r ... 172 7.3.4 Bypass M etering V a l v e ... 174 7.3.5 I n s u la tio n ... 1"5 7.3.6 C oldbox V a c u u m ... 1~5 7.3.7 M ass Flow M e a s u re m e n t... 175 7.3.8 T em perature M e a s u re m e n t... 176

8 B y p a ss G iffo rd -M c M a h o n C y cle E x p e rim e n ta l P r o c e d u r e a n d R e su lts 178 8.1 P ro c e d u re ... 178

8.2 R e s u l t s ... 180 8.2.1 E xperim ents 1 and 2:

O peration o f the B ypass A pparatus as a Standard GM C r y o c o o l e r 1 SO

8.2.2 E xperim ents 3 and 4:

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IX

8.2.3 E.xperim ent 5:

O p eratio n o f the Bypass A pparatus as a C om bined G M C ry o co o ler . . . . 187

8.3 C o m p ariso n o f the C ooling Perform ance o f the Standard G M . Bypass G M , and C om bined G M C y c l e s ... 189

8.3.1 Basis o f C o m p a r i s o n ... 189

8.3.2 C om parison o f Perform ance as a R e f r ig e r a to r ... 189

8.3.3 C om parison o f Perform ance as a L i q u e f i e r ... 192

8.4 L iq u efactio n Y i e l d s ... 195

8.4.1 L iquefaction Y ield for the Bypass G M C y c l e ... 195

8.4.2 L iquefaction Yield for the C om bined G M C ycle ... 198

8.4.3 C ycle Input P o w e r ... 204

8.4.4 E ffect on Y ield o f E qualizing Input P ow er ... 206

8.5 E ffect o f B ypass Flow on the G M C ycle ... 207

8.5.1 E xperim ent 6: Investigation o f the P ressure-V olum e D iagram s for the Bypass .apparatus . 207 8.5.2 Pressure T r a n s d u c e r ... 208

8.5.3 Initial C o o l d o w n ... 212

8.5.4 W arm B uffer V olum e Plus M ake-up G a s ... 215

8.5.5 Standard G M C ycle versus C om bined G M C ycle O p e r a t i o n ... 216

8.5.6 V alve T i m i n g ... 217

8.6 S um m ary and C o n c lu s io n s ... 219

9 C o n clu sio n s and R ecom m en d ation s 2 2 1 9.1 C o n clu sio n s ... 221 9.1.1 T h eo retical S t u d i e s ... 221 9.1.2 E xperim ental S t u d i e s ... 224 9.2 R e c o m m e n d a tio n s ... 226 9.2.1 T h eo retical S t u d i e s ... 226 9.2.2 E xp erim en tal S t u d i e s ... 227 R eferen ces 230

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A p p en d ix A - G M R egen erator D im en sion s 23.S

A p p en d ix B - S am p le A M R Input F ile 240

A ppendix C - M agnetic F ield P rofile 244

A p p en d ix D - C h arge A m p lifier C ircu it 246

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List of Tables

T able 4.1 C om parison o f the m axim um gas speed and ideal uniform flow speed

w ithin the conventional G M regenerator in the first o f the

reg en erato r len g th ...

T ab le 4.2 C o m p ariso n o f the m axim um gas speed and ideal uniform tlow speed w ithin the first 22% o f the active conical region o f a non-uniform area

G M reg en erato r... S5

T ab le 5.1 Sum m ary o f key param eters o f the bypass A M R sim u latio n s... 110

T ab le 7.1 C allen d ar-V an D usen coefficients corresponding to com m on platinum

resistance th erm o m eter stan d ard s... 17ft

T able 7.2 C oefficients for an im proved tem perature-resistance correlation for DIN

4 3 7 6 0 Standard. 100 Q P R T 's ... 177

T ab le 8.1 Bypass G M cycle o p eratio n flow states and corresp o n d in g optim al

liquefaction states o f d euterium and n eo n ... ! 9ft

T able 8.2 C om parison o f d eu teriu m liquefaction yields under B ypass GM cycle

and S tandard G M cycle o p eratio n ... 197

T ab le 8.3 C o m p ariso n o f neon liquefaction yields u n d er B ypass GM cycle and

S tandard G M cycle o p e ratio n ... 197

T able 8.4 C om bined G M cycle operation flow states and corresp o n d in g optim al

liquefaction states o f various g ases... 199

T able 8.5 C oefficients for tem perature dependent cold head piezoelectric pressure

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T a b le 8.6 C old head pressure sen so r calibration d ata from the addition o f w arm

b u ffer m ake-up gas to the co m p resso r sy stem ... 211

T ab le 8.7 C om parison o f o p eratio n d a ta fo r a S tan d ard GM cycle and a C om bined G M cycle w ith approxim ately the same cold head

tem p eratu re ... 216

T ab le A .l C onventional G M reg en erato r T A S C F low m odel d im en sio n s... 23s

T ab le A .2 C onical GM regenerator T A S C F low m odel d im en sio n s... 22V

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List of Figures

F igu re 1.1 S c h em atic d iag ram o f a fixed-bed co u n tertlo w re g e n e rativ e heat

e x c h a n g e r...5

F igu re 1.2 T em p o ral tem perature variation o f fluid and m atrix in a therm al

reg en erato r du rin g hot and cold b lo w s...6

Figure 1.3 S p a tia l te m p eratu re variation o f fluid and m atrix in a therm al

reg en erato r at the instants o f flow rev ersal... 6

F igu re 1.4 V olum etric heat cap acity o f phosphor-bronze, stain less steel, and lead

as a function o f tem p eratu re... V

F igu re 1.5 V olum etric heat cap acity o f helium as a function o f pressu re and

tem perature near the lam bda p o in t...

9

F igure 1.6 Schem atic d iag ram o f tandem fixed-bed co unterflow regenerative heat

ex ch an g ers allow ing sim ultaneous hot and cold blow p e rio d s... 12

F igu re 1.7 S ch em atic d iag ram o f an axial flow rotary re g e n e rato r... 12

F igu re 1.8 S ch em atic d iag ram o f a radial flow rotary re g e n e rato r... 13

F igu re 1.9 S y m m etric-b alan ced regenerator therm al ratio.

T]„,.

as a function of

red u ced length

A

and reduced period

H.

a fter H ausen [1 9 ]...22

F igu re 2.1 S ch em atic d iag ram o f a single-stage G ifford-M cM ahon re frig e rato r... 2S

F igu re 2.2 T e m p e rature-entropy diagram o f the G ifford-M cM ahon refrigeration

c y c le ... 29

F igure 2.3 Ideal pressure-volum e d iagram fo r the co ld ex p a n sio n space o f a

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F igu re 2 .4 T em perature-entropy diag ram o f a one-shot m agnetic cooling process.

w here

H

gives the applied m agnetic field ...? I

F igure 2.5 T e m p eratu re-en tro p y d ia g ra m o f a m agnetic C arnot cycle for a

param agnetic sa lt...12

F igure 2.6 .Adiabatic tem p eratu re ch ange, o f gad o lin iu m as a function of

tem perature and applied fie ld ...14

F igure 2.7 C o m p ariso n o f m agnetic re frig e ratio n and co n v en tio n al gas

com pression re frig eratio n ...11

F igure 2.8 S ch em atic d iag ram o f a fix ed -b ed active m agnetic regenerative

refrig erato r... 16

F igure 2.9 Schem atic representation o fo v erlap p in g tem perature-entropy cycles of ad jacent .AMR solid elem ents defining the envelope o f the overall .AMR

refrigeration c y c le ...I s

F igu re 2.10 S chem atic diagram o f a single-stage rotary A M R refrigerator show ing

key system co m p o n en ts...4 1

F igure 3.1 R elative error in T ip le r’s em pirical correlation o f regenerator therm al

ratio com pared to H au sen ’s therm al ratio ... 44

F igure 3.2 R elative error in D a tsk o v sk ii's em pirical correlatio n o f reg en erato r

therm al ratio com pared to H au sen ’s therm al ra tio ...45

F igure 3.3 R egenerator ineffectiv en ess as a function o f m inim um reduced length

(after Kays and L ondon [5 4 ])...5”

F igure 3 .4 Schem atic diagram o f the B oreas c ry o co o ler... 69

F igure 4.1 M inim um thermal ratio required to m aintain regenerator lim its betw een 300 K and from reg en erato r flow gas ex p an sio n having A T = 0.3T

and zero external therm al lo a d s...~2

F igure 4.2 R eg en erato r th erm al ra tio p erfo rm an ce red u ctio n from tlow m aldistribution as a function o f uniform flow reduced length and

red u ced p erio d ... 74

F igu re 4.3 P ercen tag e increase in re g e n e rato r red u ced length required to com pensate for effects o f m aldistributed flow (m * = l ) to give equal

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F igure 4 .4 S ch em atic d iagram o f a conventional GM 1 ’‘-stage reg en erato r seen in

c ro ss-se c tio n ...

F igure 4.5 H elium fluid flow streaklines from inlet to outlet in a 45'^ m odel section o f a conventional T '-stag eG M reg en erato rsh o w in g recirculation eddies

near the in let r e g io n ...SU

F igure 4.6 R e p resen tatio n o f gas speed w ithin the first

20%

o f the length o f a

co n v en tio n al G M re g e n e rato r... Si

Figure 4.7 Sch em atic d iagram o f a conical taper GM 1 ’'-stage reg en erato r as seen

in c ro ss-sectio n ... S3

Figure 4.8 H elium fluid tlow streaklines from inlet to outlet in a 45" m odel section o f a conical P '-stag e G M regenerator show ing sm ooth tlow w ithout

recirculation throughout the entire regenerator m atrix...S4

F igure 4.9 R ep resen tatio n o f gas speed w ithin the first

20%

o f the activ e conical

region o f a n o n-uniform area G M reg en erato r...S4

Figure 4 .1 0 C om parison o f the frequency dependent no-load minim um tem peratures o f the co n v en tio n al and conical regenerators in the sin g le-stag e GM

refrig e rato r... 87

Figure 4.11 C om parison o f the P ‘ and 2"‘‘-stage no-load m inim um tem p eratu res as a function o f operation frequency for the conventional and conical

regenerators in the tw o-stage G M refrig erato r... Ss

Figure 4.12 C om parison o f the coo lin g perform ance curves for the conventional and

conical regenerators in the tw o-stage GM refrig erato r... S9

Figure 5.1 T em p eratu re-en tro p y d iagram o f the C arnot refrigeration c y c le ... 92

Figure 5.2 T em p eratu re-en tro p y d iagram o f the reversible cold-gas refrigeration

c y c le ...93

F igure 5.3 T em p eratu re-en tro p y d iagram o f the ideal liquefaction p ro c e ss...95

F igure 5.4 S atu ratio n tem p eratu re from trip le point to c ritic a l p oint saturation p ressu res vs. the ratio o f latent heat to sensible heat, /i,, for a num ber

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F igure 5.5 R atio o f ideal refrigerator to ideal liquefier liquefaction yield for equal

input pow er. as a function o f saturation tem perature from the

triple point to the critical point for a num ber o f gases cooled from

300 K... '■W

F igu re 5.6 R atio o f the ideal refrig erato rco efficien to f perform ance to the cold-gas

refrig erato r coefficient o f perform ance.

T](/T]ccr

^

as a function of saturation tem perature for the sensible cooling requirem ents from 300 K

to for a num ber o f g ases... 101

F igure 5.7 Schem atic diagram o f a single-stage G ifford-M cM ahon refrigerator m odified fo r bypass regenerator o p eratio n ... 104

F igu re 5.8 Schem atic diagram o f a single-stage A M R refrigerator m odified for bypass reg en erato r o p eratio n ... 104

F igu re 5.9 R egenerator tem p eratu re span efficiency as a function of unbalance ratio for various values o f A ^ „, 11. and A Tr ... 10" F igure 5.10 P redicted B ypass G M m inim um cold head tem perature as a function of bypass ra tio ... lOS F igure 5.11 C om parison o f dim ensionless tem perature profiles w ithin balanced sym m etric and unbalanced asym m etric reg en erato rs... 100

F igure 5 .12 Solid tem perature profile fo r an A M R with equal heat transfer tluid flow for hot and cold blow p erio d s... I l l F igure 5.13 U tilization factor for an A M R with equal heat transfer fiuid fiow for hot and cold blow p erio d s... 112

F igure 5.1 4 Solid tem perature profile o f a bypass A M R with 4% bypass fio w ... 113

F igure 5.15 U tilization factor o f a bypass A M R with 4% bypass flow ... 114

F igu re 5.1 6 Solid tem perature profile o f a bypass A M R with 87c bypass fiow ... 115

F igure 5.1 7 M agnetization pow er per unit volum e for an A M R w ithout bypass fiow ... 116

F igure 5.1 8 C o m p ariso n o f the d istribution o f radial w ork input rate betw een standard and bypass A M R 's ... 11" F igu re 5.1 9 Bypass A M R latent cooling capacity and ratio o f latent to sensible cooling capacity as a function o f bypass flow ratio ... 119

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\ M 1

F igu re 5.2 0 A M R e q u iv alen t coo lin g load co efficien t o f perform ance as a function

o f bypass flow ratio for applied fields o f 4 T esla and 5 T e s la ... 120

F igu re 5.21 .AMR e q u iv alen t cooling load coefficient o f perform ance as a function o f latent to sen sib le co o lin g capacity ratio for an applied field o f

4 T e sla ... 121

F igure 5.22 R atio o f the liquefaction capacity o f ethane at 225 K o f the 4 T esla bypass .AMR to an ideal liquefier as a function o f the bypass .AMR

sensible co o lin g heat e.xchanger effectiveness and bypass fiow ra tio ... 124

Figure 6.1 S ch em atic rep resen tatio n o f the A M R m odel w ith fiow ducts for the

co m p u tatio n al fiuid dynam ics fiow stu d ies... I 2(J

F igure 6.2 G eneral representation o f the com putational grid used in the .AMR fiow

sim u la tio n s... I ,'(J

F igure 6.3 Fluid fiow streak lin es for an A M R rotating counter-clockw ise having

a fiuid to ring relative velocity ratio at the outer rim o f 0 .4 7 ... 134

F igu re 6.4 S urface rep resen tin g the local ang u lar rotation rate o f the fiuid w ithin

an AM R ro tatin g at 10 radian s ' ... 136

Figure 6.5 Surface rep resen tin g the local radial velocity o f the fiuid w iihin an

A M R ro tatin g at 10 radian s ' ... 137

F igure 6.6 R epresentation o f the fiuid (upper) and solid (low er) unattached grids allo w in g representation o f conjugate heat transfer over a volum e using

the stan d ard release o f T A S C F lo w ... 13h

F igure 6.7 T ypical e rro r residual reduction as a function o f iteration n u m b er for

the d u al-g rid T A S C F lo w rotary regenerator sim u latio n ... 144

F igu re 6.8 T ypical e rro r residual reduction as a function o f iteration n u m b er for

the cu sto m T A S C F lo w release rotary regenerator sim u latio n ... 14>i

F igu re 6.9 T em perature profile o f a passive regenerator o f reduced length A = 300 and re d u c e d p eriod

II

= 10 as calcu lated in the custom T.ASCFlow

release on a n o n-uniform g rid ... 149

F igu re 6 .1 0 C o m p ariso n o f the solid tem perature versus radial position w ithin the passive re g e n e rato r ring as solved by the S pearing M a ste r's thesis

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X \ I I I

F igu re 6.11 E qually spaced snapshots o f the hot blow tem perature profile versus

radial position for the custom T A S C F lo w re lease ... 151

F igu re 6.12 C om parison o f the tem perature profile across the reg en erato r bed at the

end o f the hot blow for d ifferen t grid node spacing sch em es... 152

F igu re 6.13 C om p ariso n o f face-centred control volum e and node-centred control

volum e d e fin itio n s... 155

F igu re 6.1 4 T A S C F lo w c o n tro l volum e defin itio n at the intersection o f duct.

re g en erato r ring, and blo ck -o ff reg io n s... 156

F igu re 6.15 N orm alized m agnetic field profile and its derivative used in the custom

T A S C F lo w release .AMR sim u latio n s... 15V

F igu re 6.16 S olid m atrix tem perature profile o f a g adolinium reg en erato r operating

b etw een inlet tem peratures o f 200 K and 273 K un d er a 5 T esla applied

field as calcu lated using the custom T A S C F low re le a se ... 160

F igu re 7.1 S chem atic diagram o f the B ypass G iffo rd -M cM ah o n c y c le ... 164

F igu re 7.2 Schem atic diagram o f the B ypass G M test apparatus show ing key

system com ponents and in stru m en tatio n ... 16S

F igu re 7.3 Photo o f the B ypass G M ex p erim en tal ap p aratu s show ing room

-tem p eratu re com ponents and d ata acquisition sy s-tem ... I6X

F igu re 7.4 Photo o f the cold head tlan g e w ith H all probe, piezoelectric pressure tran sd u cer, check valve, and zeolite ad so rb en t filter com ponents

m ou n ted ... I'O

F igu re 7.5 Photo o f the first heater design consisting o f fibreglass coated nichrom e w ire w rap p ed .aro u n d and epoxied to stainless steel tubing, then

w rapped w ith alum inized m ylar su p erin su latio n ... ! ~3

F igu re 7.6 Photo o f the second heater design consisting o f stainless tubing w rapped

around and sold ered to cartridge h eaters... 174

F igu re 8.1 C o ld head tem perature and cold head tem perature rate o f change as a function o f elap sed tim e for the B ypass G M ap p aratu s u n d er Standard

G M cy cle o p e ra tio n ... 1 SO

F igu re 8.2 G M co m p re sso r return line m ass flow rate as a function o f elapsed tim e d u rin g initial cooldow n o f the B ypass G M ap p aratu s u n d er Standard

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\ l \

Figure 8.3 C old head tem p eratu re as a function o f applied cold head cooling load for S tandard G M cy cle o p eration, w ith and w ithout use o f the w arm

buffer and m ake-up g a s... IS3

F igure 8.4 M easured B ypass G M apparatus bypass m ass flow rate as a function of the n u m b er o f valve turns, w ith and w ithout use o f the w arm b u ffer

and m ake-up g a s ... IS4

F igure 8.5 C old head tem p eratu re and cold bu ffer exit tem perature as a function o f bypass m ass flow rate, w ith and w ithout use o f the w arm b u ffer and

m ake-up g a s ... IS5

Figure 8.6 C old head tem p eratu re and co ld b u ffer exit tem perature versus bypass co o lin g load, w ith and w ithout use o f the w arm buffer and m ake-up

g a s... Is;'

Figure 8.7 C old head tem p eratu re versus bypass and cold head co o lin g loads for

the C om bined G M c y c le ... 1 SS

F igure 8.8 C old b u ffe r ex it tem perature versus bypass and cold head cooling loads

for the C o m b in ed G M cy c le ... I s s

F igure 8.9 B lack box c o n v erter concept to com pare S tandard G M . Bypass G.M. and C om bined GM perform ance o f the Bypass G.M apparatus when

operating as a refrigerator... I VI)

F igure 8.10 C old head tem perature as a function of eq uivalent refrigerator operation cold head load for the S tandard G M . Bypass G M . and C om bined GM

c y c le s... IVI

Figure 8.11 Interpolated Standard GM cycle and C om bined GM cycle d euterium liquefaction yields for the co n d itio n s o f flow case 2 as a fu n ctio n o f

bypass h eat e x ch an g er effe c tiv en e ss... 200

Figure 8.12 R atio o f C o m b in ed G M cycle liquefaction yield to Standard G M cycle

liquefaction yield for flow cases 1 through 4 ... 202

F igure 8.13 Ratio o f C o m b in ed G M cycle liquefaction yield to Standard GM cycle

liquefaction yield fo r flow cases 8 and 9 ... 20?

F igure 8.14 Isotherm al c o m p resso r input p o w er as a function o f cold head load for

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w

F igure 8.15 R atio o f C om bined GM cycle liquefaction yield to S tandard G M cycle liquefaction yield for flow cases 8 and 9 a fter scaling for equal ideal

c o m p resso r input p ow er... 206

F igure 8.16 M easured and curve fit cold head pressure sensor response as a function

o f co ld head tem p eratu re... 209

F igure 8.17 C o ld head expansion space pressure-volum e diagram s recorded during

initial cooldow n o f the Bypass GM ap p a ra tu s... 213

F igu re 8.18 T em p eratu re dependent phase shift o f cold head p ressu re extrem a relative to pressure extrem a o f PV diag ram recorded at 282 K during

initial co o ld o w n ... 213

F igure 8.19 PV diagram cold head w ork rate and cold head p ressu re sw ing as a

function o f elapsed tim e during initial c o o ld o w n ... 214

Figure 8.20 C om parison o f the PV diagram s at a m inim um no-load tem perature of 23.6 K w ith and w ithout the use o f the w arm buffer volum e and m ake

up g a s... 215

F igure 8.21 C om parison o f PV diagram s for a Standard G M cycle and a Bypass G M cycle presented in Table 8.7 having approxim ately equal cold head

te m p eratu res... 21"

F igure 8.22 M odified PV diagram s for the Standard G M flow case presented in T able 8.7 having theoretical phase shifts o f 0, -10.3, -20.6, and -36.9

degrees in valve cycle tim ing relative to existing tim in g ... 218

F igure 8.23 E xpected Standard G M cycle PV w ork rate for a phase shift in valve

tim ing from 0° to -36.9° relative to e x istin g tim in g ... 219

F igure D .l C ustom high im pedance charge am p lifier circu it used w ith the K istler

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\ \ l

Nomenclature

Symbols

A • reg en erato r m atrix surface area [m ’]

C • tlu id heat capacity [J ’kg ''K 'l

C, • tlu id heat capacity [J-kg ' ’K ']

C,| • solid heat cap acity at constant field [J’kg '-K 'l

C, • solid heat capacity [J ‘kg ' ‘K 'l

C \ • d im en sio n less param eter, m etering valve flow coefficient

d. • particle d iam eter [m]

D • reg en erato r bed d iam eter [m]

Dp • particle d iam eter [m]

f • d im en sio n less param eter, friction factor

f^T • d im en sio n less param eter, gas expansion tem perature change proportionality constant

G • m ass flux per unit free flow area [kg-s '"m -|

h • bulk heat tran sfer co efficien t [W 'm " K ']

h • fluid en th alp y [J ’kg ']

Hr • d im en sio n less param eter, ratio o f fluid latent heat to sensible heat

H • applied m agnetic field [A-m ']

H, • tlu id en th alp y [J-kg ']

L • d im en sio n less param eter, regenerator ineffectiveness

k, • fluid therm al co n d u ctiv ity [W -m '"K ']

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W I l M m M , Nu P P Pr PRch

Q

q

R R Re % S Sn S.„ Sc Sp t Te Te.p T„ T, m aterial m agnetization [A ’m '] tlu id m ass tlow rate [kg-s '] reg e n e rato r m atrix m ass [kg]

d im en sio n less param eter, N usselt num ber reg e n e rato r blow period (C hapter 1 only) [s] p ressu re [N -m ‘]

d im en sio n less param eter, Prandtl num ber

dim en sio n less param eter, cold head pressure ratio heat ab so rp tio n /rejectio n rate [W]

heat flux v ector [\V-m ’] radius [m]

electrical resistance [ohm]

d im en sio n less param eter, R eynold’s n um ber gas c o n stan t [J-kg '-K ‘]

entro p y [J-kg '-K ']

energy eq u atio n volum etric source term [W -m '] m om entum equation volum etric source term [kg-m '-s'-J

d iscretizatio n eq uation source term constant co efficien t [units vary] d iscretizatio n eq uation source term active c o efficien t [units vary ] tim e [s]

co ld (reserv o ir, inlet) tem perature [K]

m inim um co ld tem perature with u n balance [K] hot (reserv o ir, inlet) tem perature [K]

tlu id tem p eratu re [K]

d im en sio n less tluid tem perature

Ts

t

T,f,0

solid tem p eratu re [K]

d im en sio n less solid tem perature

average fluid outlet tem perature [K]

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w i l l U, U V X V, V ,

w

• co m p o n en t o f velocity in the i^-coordinate direction [m-s 'l • d im en sio n less param eter, utilization factor

• com p o n en t o f velocity in the y-coordinate direction [m-s ' ] • velocity v ector [m-s ']

• distance [m]

• volum e o f fluid en train ed w ithin a reg en erato r [m '] • sensor voltage [V] • w ork rate [W]

Greek

R iitfi

a

P

Y Ô

A

a t,„ a t, Gr E n c 'HrGR n,h ^th.P * ipm

0 A

V ,

d im en sio n less param eter, m aterial porosity

d im en sio n less param eter, regenerator unbalance factor dim en sio n less param eter, ratio o f gas specific heats infinitesim al change

d ifference

adiabatic tem perature change [K]

d im ensionless constant, net expansion tem perature change proportionality, factor e rro r in indicated tem perature [K]

d im en sio n less param eter, heat ex ch an g er effectiveness

d im en sio n less param eter, refrig erato r coefficient o f perform ance

d im en sio n less param eter, cold gas refrig erato r coefficient o f perform ance d im en sio n less param eter, regenerator therm al ratio

d im en sio n less param eter, regenerator therm al ratio with unbalance d im en sio n less param eter, tem perature span efficiency

an g u lar position [radian]

d im en sio n less param eter, reg en erato r reduced length fluid viscosity [kg-m ‘‘s ']

co n stan t, perm eability o f free space [4% x 10 ' H m 'j initial co n d itio n specific gas volum e [m ^'kg 'j

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\ \ 1\

n

p. p , o 4) 4) w

d im en sio n less param eter, regenerator reduced period fluid density [kg-m'"]

solid d en sity [kg-m ’^]

volum etric energy generation term [W -m ']

d im en sio n less param eter, regenerator dim ensionless tim e G iffo rd -M cM ah o n cycle period

viscous stress ten so r [J’m ^] an in dependent variable

an g u lar w idth (o f a rotary regenerator duct region) [radian] an g u lar rotation rate [rad ian 's ']

Subscripts

ad B B,EQ C CH C H T cold com p exp f f f f,o fg initial, or at co ndition I final, o r at co ndition 2 adiabatic bypass

therm odynam ically eq uivalent to the bypass value cold

(heat exch an g er) cold inlet stream (heat exch an g er) cold oulet stream cold head

co n ju g ate heat transfer com ponent d u rin g the co ld blow

co m p resso r ex p an d er final state

fluid

fluid satu rated liquid state fluid outlet

change o f state betw een saturated liquid and vapour states fluid vapour state

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xxv H h, h: har hot J mal min o P R Qeq R:L s sat sf T uniform CO hot

(heat exchanger) hot inlet stream (h eat exch an g er) hot ou let stream h arm onic m ean value

d uring the hot blow ideal

initial state inner

coordinate direction index coordinate direction index under m aldistributed flow m inim um value

o u ter

for the process gas for the return line

for the therm odynam ically eq u iv alen t co o lin g load ideal refrig erato r relative to ideal liquefier solid

saturated fluid state betw een solid and fluid at tem perature T u n d er uniform flow advection com ponent

Superscr

pts

(single prim e) hot blow (double prim e) cold blow

(double prim e) p er unit area (C h ap ter 5 only) (triple prim e) per unit volum e

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W V l

Overstrike

• (overbar) average • (overdot) rate * _{• (star) dimensionless} (arrow) vector

Acronyms

A C Alternating C urrent

A M R Active M agnetic R egenerator

A M R R • Active M agnetic Regenerative Refrigerator

AVVG A m erican W ire Gauge

C G R C old G as Refrigerator

C H T Conjugate Heat T ran sfer

C N G C o m p ressed Natural Gas

C O P Coefficient o f Performance

DC Direct Current

FOM Figure o f Merit

G M G iffo rd -M cM ah o n

ILL: Incomplete L ow er-U pper

L N G Liquefied Natural Gas

L V D T Linear Voltage Displacement Transducer

P R T Platinum Resistance T herm om eter

PV Pressure-V olum e

RMS Root M ean Squared

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XWl l

Acknowledgements

M \ enduring stay within the Cryofuel Systems G roup at the University o f Victoria, completing t'ir\i my M a ste r's degree and herein my Doctorate, was filled with many challenges and rewards. Over the last 8 Vi years I have learned a great deal, and I w ould like to express my thanks to those who provided innumerable exam ples for academ ic and personal betterment. In particular. 1 would like to e.xpres\ my gratitude to:

• My Supervisor. Dr. John Barclay, for patiently awaiting the final draft o f my dissertation

as 1 toiled away at home, missing the expected deadlines and disregarding the encouragem ent to com plete expeditiously.

• Research Associate. Dr. J e f f Hall, who went on to bigger and better things during m\ early years as a Doctoral student. His generous spirit and dow n-to-earth sensibility found no match.

• Research Associate. Dr. Edgar Nelson, w ho went on to bigger and better things during

my later years as a Doctoral student. His advice and insight were instrumental m understanding "the big picture."

• Fellow graduate student. A ndrew Rowe, w ho gave bountiful suggestions and advice and

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\ \ \ I I I

• Finally, an d most o f all, my lovely wife, M oira, and o ur beautiful daughter, Madeleine,

who both bring smiles to my face and jo y to my life with each m om ent we spend together. T h e diagram s were still pesky, but the m ornings are no longer cranky. 1 c o u ld n 't have done it w ithout you two.

1 would like to acknowledge Natural Sciences and Engineering Research Council o f Canada i NSHRC i. and Centra Gas, Inc.. a division o f W estcoast Energy, for their support o f projects o f which this work forms a part. I am also grateful o f the generous Fellowship awarded to me by the Universitv of Victoria.

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Chapter 1 Introduction

1.1 Objective

This dissertation was performed within the C ryofuel Systems Group at the University o f Victoria. Broadly, the aim o f Cryofuel Systems and its original founding group, the Institute for Integrated Energy Systems (lESU/c). can be stated as the developm ent o f energy systems that simultancuu\l\

• O ffer a foundation for economic growth and industrial diversification;

• Cause minimal environmental intrusion, and especially, reduce climate destabilizing

em issions, and;

• Provide flexibility and resilience in response to technical, geopolitical, and environmental

change.

T o w a rd s these goals. Cryofuel Systems is currently engaged in the developm ent o f systems for gas purification, cryo g en ic fuel handling, and a dvanced refrigeration and liquefaction, with a major emphasis on the developm ent of magnetic refrigeration for the liquefaction o f natural gas for vehicular transportation markets.

.Aside from one-shot cooling systems used in physics experim ents, magnetic refrigeration, based on

the m agnetic-field-induced temperature changes,

or magnetocaloric effect,

o f certain materials, is an

im m ature technology. Development o f solutions to practical im plementation details, as well as a d evelopm ent o f a fundamental theoretical understanding o f the active magnetic re g e n e r a te c

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refrigerator and the requirem ents o f the gas liquefaction process are required before cost-effective systems can be built. Towards these developm ent requirements, the work em bodied in this dissertation had the following objective:

The o b je c tiv e o f th is stu d y w a s to a d v a n c e the u n d e rsta n d in g o f th e j l i i i d m e ch a n ic s a n d th e rm o d y n a m ic s d e s c r ib in g th e p r o c e s s e s within p a s s iv e a n d a c tiv e m a g n e tic re g e n e ra tiv e r e fr ig e r a to r s th rough th e o r e tic a l d e r iv a tio n , m odellin g, a n d sim u la tio n o f th e re g e n e ra tiv e h e a t exch an gers, a n d to su g g est, bu ild, test, a n d p r o v e p r a c tic a l im p r o v e m e n ts to th ese s y s te m s b y d ir e c t experim en t.

The scope o f the objective was limited to three specific goals:

1 ) Determine thee.xtentand im pact of flow maldistribution within the I''-stage regenerator o f a com m ercial G iffo rd -M cM ah o n refrigerator and develop an im proved design: 2) Extend the results o f e x istin g thermal models and develop an im proved thermo-tliiid

model o f an active magnetic regenerative refrigerator in the context o f gas liquefaction, and;

3) Verify by experim entation key results o f the above two goals.

.A study o f the extent and impact o f flow maldistribution within the 1''-stage regenerator o f a com m ercial G M refrigerator was carried out by:

1 ) extension o f existing theoretical models o f the thermal performance o f maldistributed flow regenerators;

2) developm ent o f com putational fluid mechanics simulations o f the 3 -dim ensional flow patterns within a 1 '"-stage G M regenerator, for an existing com m ercial design and a novel design, and;

3) experim ental com parison o f the perform ance o f the commercial design and the novel design under a range o f typical operating conditions.

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An extension o f the existing thermal models for active magnetic regenerative refrigerators and experim ental verification o f key liquefier theoretical and simulation results was carried out b\ :

1) d e v elo p m en t o f theoretical foundations for im proved performance o f regenerati\e cry o co o lers in gas liquefaction applications;

2) study o f the characteristics o f magnetic regenerative systems under various operating conditions using existing theoretical thermal simulation c om puter models:

3) d e v elo p m en t o f com putational fluid mechanics simulations o f the How patterns w ithin a 2-dim ensional rotary regenerator model using com m ercially available software with additional user source code;

4) d e v elo p m en t o f a new thermal simulation com puter model in conjunction with the fluid m echanics sim ulations using commercially available softw are, custom modification'^ to that softw are, and additional user source code, and;

5 ) experimental verification o f the theoretical foundations for improved performance of G,\l and ,AMR systems in gas liquefaction applications by im plementation of the resultant liquefier design modifications to a G M system operating as a simulated gas liquefaction system.

1.2 Motivation

Rapid and c onvenient transportation has become a mainstay o f ourdeveloped societies, using vehicles pow ered prim arily by high carbon content fuels. Several researchers [ 11, [2|, [3|, [4], [5|, [b| have identified eco n o m ic, geo-political, and environmental benefits to a society that shifts its energv currencies to low carbon content fuels, such as natural gas ( NO), and ultimately to carbon-free fuels, such as hydrogen (H;), Natural gas, as provided by pipeline distribution systems, is composed primarily o f methane, which has four atoms o fh y d ro g e n for each ato m o f carbon, the highest ratio of hydrogen to carbon o f the hydrocarbon fuels. The exact com position o f pipeline natural gas vanes, but it has been m easured to contain 82,2 - 97,87c methane, with ethane, nitrogen, carbon dioxide, heavier h ydrocarbons, and other trace gases and odourants m aking up the remainder [7],

Recent efforts to introduce NO fuelled vehicles have been h ampered, in part, by the limited range of these vehicles due prim arily to the low volumetric storage capacity o f compressed gas fuel storage

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tanks. Fuel storage, and hence vehicle range, can be increased by storing the fuel as a liquid rather than as a co m pressed gas. Liquefied natural gas (LN G ) at atm ospheric pressure has 2.6 times the volumetric energy density o f co m pressed natural gas (CNG) at 3000 psig. .At atm ospheric pressure natural gas liquefies at 112 K, considerably below room temperature o f 293 K. Liquids below 123 K

are known as cryogens, and LN G can thus be term ed a cryogenic fuel or

crwfuel.

To supply L N G to a vehicular fuel market, a supply o f natural gas and an ine.xpensive and efficient refrigeration system for liquefaction are required. Natural gas distribution systems e.xist in most major North .American cities; however, current refrigeration system s in the L N G tem perature range are expensive o r inefficient, or both. New liquefaction technologies that are less expensive and more efficient are required to reduce the refuelling system cost com ponent o f LNG.

The Cryofuel Systems G roup at the University o f Victoria has identified active m agnetic regenerati\e refrigeration (.AMRR) as a viable technique to achieve refrigeration at high efficiencies and competitive capital expense c om pared to current technologies. .A thorough understanding of the thermal and fluid processes within the regenerator is critical to designing and achieving optimal perform ance at the lowest possible cost for a regenerative magnetic refrigerator.

Refrigeration system im provem ents and capital cost reductions can additionally be achieved h\ improvements in any ancillary equipm ent. For the A M R liquefier proposed by Cryofuel Systems, this includes the regenerator within the G ifford-M cM ahon refrigerator that is em ployed for conduction cooling o f the superconducting magnet systems required by the magnetic refrigerator. Within the scope o f the com plete liquefier system, this dissertation focuses on the role of regenerators and their

operational impact on the overall liquefaction process under various design conditions. An

understanding o f the construction and thermal function of the regenerator provides the foundation for this work.

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1.3 Regenerative Heat Exchangers

Regenerative heat exchangers, or

regenerators,

are co m p a c t heat exchangers in which heat is

alternately stored a n d removed using a heat storage matrix. During the storage phase, commonK called the

hot blow,

hot gas is passed through and gives up its heat to the regenerator matri.x. .-\t the end o f the hot blow period, the flow of hot gas stops and cold gas flow begins, typically in the opposite

direction to the hot gas flow. During the

cold blow,

the cold gas picks up heat previously stored in the

regenerator matrix. .At the end o f the cold blow period, the flow o f cold gas stops and the next cxcle of hot and cold b lo w s begins.

Figure 1.1 show s a schematic diagram of a single Fi.xed-bed regenerator under periodic hot and cold blow operation as described above. Valves are used to coordinate the hot blow and cold blou gas llovs streams, here arbitrarily labelled as "w aste" and "p ro cess" streams. D istinction o f what constitutes a "process" stream or "w aste" stream is entirely d ependent on the regenerator application.

_ Hot Blow; Ch Valve Closed

Oh Valve Open Cold Blow: Cc Valve Closed

Oc Valve Open

Cool Cold Hot Warm

Waste Process Waste Process

Stream Stream Stream Stream

F igure 1.1 Schematic diagram o f a fixed-bed counterflow regenerative heat exchanger.

Various modes o f regenerator operation can be realized, but a pseudo-steady-state cyclic operation is

commonly utilized. This is achieved a fte ra n u m b e r o f repeated, fixed-period hot and cold blow s where ultimately a fter one hot and one cold blow, the tem perature at any one point in the matrix or tluid within the re g en erato r is identical to its value one full cycle earlier. Figure 1.2 show s the variation with time o f the matrix temperature and the fluid tem perature at som e station within the regenerator o f Figure 1.1 u n d erg o in g cyclic flow reversals after achieving pseudo-steady-state.

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C old B low Hot B low

I

m atrix i~ fluid Time

F i g u r e 1.2 T em poral tem perature variation o f fluid and matrix in a thermal regenerator during hot and cold blows.

fluid hot blow

I

Regenerator Passage Length

c o ld blow

F igure 1.3 Spatial tem perature variation o f fluid and matrix in a thermal regenerator at the instants o f flow reversal.

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Figure 1.3 shows the spatial variation of the matrix temperature and fluid temperature at the instants o f flow reversal. T he u p p er curves represent the temperature o f the fluid and matrix at the end of the hot blow and the start o f the cold blow, while the lower curves represent the end o f the cold blow and the start o f the hot blow. Each station in the matrix, such as the position designated by the vertical line in Figure 1.3, fluctuates between these two sets o f curves in a tim e-dependent m anner similar to that shown in Figure 1.2.

1.4 The Ideal Regenerator

1.4.1 Ideal Thermal and Mechanical Properties

The ideal counterflow regenerator can be viewed as a black box device that accepts gas at T,, and cools it to T c After som e period of time, the flow is reversed and gas enters at T and is instantly warmed to T„. Regardless o f the length o f time taken for the blow period, gas only leaves at the constant temperature o f T f O r T ,,. The ideal regenerator is a perfect thermal insulator between the limits ofT,, and T , so that there is no heat leak longitudinally through the device.

The ideal regenerator is also described by its mechanical requirements. T he regenerator is perfectly sound structurally to withstand any loads applied to it. including loads applied by the action of the gas flow through it. by external mechanical loading, by internal thermal stresses, or by internal loads trom gravitational, magnetic, o relectrical fields, for example. T he ideal regenerator has no fluid entrained within it because it has zero void volume, and has zero pressure drop associated w ith the fluid flow mg through it.

1.4.2 Implications of Regenerator Ideality

W ith these lofty design criteria and exotic materials, the designer might conclude that the ideal regenerator will be an expensive component. Fortunately, that is not the case, since the ideal regenerator would cost nothing to produce.

The ideal regenerator is impossible to achieve. T o achieve zero pressure drop, the flow would have to be entirely frictionless. A chieving zero void volume would preclude the possibility o f flow channels

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through the matrix in which the fluid w o u ld traverse. Achieving instant w arm ing orco o lin g fora finite gas flow would require the heat transfer coefficient or heat transfer area to be infinite, or the heat capacity o f the fluid to be zero. T o achieve a constant outlet temperature would require the heat capacity o f the matrix to be infinite ( w hich then effectively precludes setting the necessary temperature gradient within the matrix) or the heat capacity o f the fluid to be zero. T o achieve perfect thermal insulation across the regenerator w ould require materials o f zero thermal conductivity to be used in the construction.

1.5 Practical Regenerators

1.5.1 Practical Thermal and Mechanical Properties

Practical regenerators balance the various conflicting requirements o f the ideal regenerator according to their application. Heat transfer and heat transfer area is maximized by use o f fine geometries w ith very high specific areas, such as packed particle beds, parallel plates, perforated plates, wire screens, etched foils, packed wires, and hollow tubes.

T h e th erm al mass o f the regenerator matrix fo ra fix e d -v o lu m e regenerator is maxim ized b\ selection o f m aterials having high volumetric heat capacity. As the regenerator operation temperature dips below - 2 0 - 3 0 K, conventional regenerator materials such as phosphor-bronze, stainless steel, and lead lose th eir effectiveness as regenerator materials because o f a rapid decline in their heat capacities. Figure 1.4 show s the heat capacities o f these conventional materials as a function of temperature

T h e d ecline in the heat capacity o f conventional regenerator materials at low temperatures is particularly problematic because the heat transfer gas used at these low temperatures, namely helium, exhibits a rise in heat capacity. Figure 1.5 shows the pressure dependent heat capacity o f helium at low temperature.

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P h o s p h o r B ro n z e S ta in le s s S te e l L ead B

■s

t o ' 100 150 T e m p c ia tu ic |K | 250 200 200

F igu re 1.4 Volum etric heat capacity o f phosphor-bronze, stainless steel, and lead as a function o f temperature.

Pressure IMPa) - 2.0 0.4 QJ 0.2

F ig u re 1.5 Volum etric heat capacity o f helium as a function o f pressure and tem perature near the lambda point.

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10

Nearly 25 years ago, Buschow

et al.

[8] pointed out the potential use o f lanthanide m a t e r i a l as replacements to conventional regenerator materials. In the last decade, replacement o f the conventional regenerator matrix material with rare-earth/transition metal com pounds such as Er-,Ni, ErNi,, .Co , HoCu;. and elemental Nd, which exhibit a heat capacity spike at low temperatures due to magnetic

ordering, has yielded im proved regenerator performance [9.10], In practice, many o f these

intermetallic com pounds have proven brittle and can experience chipping from thermal and tluid stresses. C hipping and abrasion can lead to a fine dust that clogs the How channels o f the bed. or worse, contaminates the heat transfergas leading to failure of any associated compression equipment.

Researchers have com e up with som e novel means to prevent these regenerator materials from s l o w K

pulverizing during operation [11].

T he porosity o f a regenerator is primarily a function o f the material geometry selected, with some options offering greater control o f the final porosity, which can be used to find a balance between the thermal and pressure drop losses. Som e geometries, such as packed particle beds, tend to p n m d e essentially isotropic porosity, whereas others, such as parallel plates, can be considered to have zero porosity perpendicular to the plates.

Some porosity perpendicular to the fluid flow is desirable to allow the How to redistribute itself e\ en K across the bed and to counteract the effects o f flow channel dimensional variations o r c o n s tn c tio n \ a s

well as inherent How instabilities. If the heat transfer gas flowing through the regenerator has a temperature dependent viscosity decreasing with decreasing temperature, then a positive feedback tlow instability can develop during the cold blow. In regions where flow is for some reason initialK higher than adjacent regions during the cold blow, there will be increased local cooling, leading to decreased local viscosity, w hich in turn leads to increased flow. Periodic flow interruptions and channels perpendicular to flow help to redistribute the flow and to minim ize flow instability, maldistribution [12],

Thermal conduction longitudinally through the regenerator is minimized by use of discontinuous media or periodic thermal breaks. Regenerators m ade by stacking layers of wire screens, for example, w here flow is perpendicular to the screens, allow thermal conduction to be m inimized in the direction o f fluid flow because o f the nearly point contacts betw een successive screens. The relatively high thermal conductivity o f screens perpendicular to the fluid flow direction has the secondary benefit of

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minimizing thermal gradients that might develop across the width o f the regenerator due to flow maldistribution, for exam ple.

The heat transfer gas also contributes to longitudinal thermal conduction, and as such. an> real regenerator will have a non-zero longitudinal thermal conductivity because o f the flow channels. Determining the effective thermal conductivity o f a packed bed can become a com plicated function of many parameters. T so tsa s and Martin [13| consider a packed b e d ’s thermal conductivity to depend on the thermal conductivity o f the solid and gas phases, the porosity, the pressure and temperature of the gas, the particle d ia m e te r and size distribution, the mechanical properties o f the solid ( for contact point area and surface heat transfer properties), the optical properties o f the solid ( for radiation i. and the therm odynam ic and optical properties o f the gas.

Edwards and R ichardson [ 14] studied the thermal conductivity augmentation from the effect of ga\ dispersion during tlow th rough the bed. Vortmeyer and Adam [15] reviewed several researchers' work and found that most researchers agree that the effective thermal conductivity consists o f an effective conductivity without flow plus a contribution depending on the flow rate, but that there is considerable difference in the o b se rv e d magnitude o f the latter term. Duncan

er al.

[16] provided a number of analytical and empirical relations for the effective thermal conductivity for packed beds.

1.6 Regenerator Configurations

R egenerators may be categorized as either fixed bed or rotary configurations. In fixed bed

regenerators, the matrix material is stationary and valves (or some means o f gas displacem ent) are used to alternately direct the hot and cold fluid streams. The regenerator o f Figure 1.1 is an example o f a fixed bed regenerator. In processes where hot and cold streams must flow at the same instant, tw o I or more ) fixed bed regenerators are used in tandem. W hile hot gas flows through the first regenerator acting as a thermal store, cold gas flows through the second regenerator acting as a thermal source. .\fte r some time, the flow control valves change their open/closed state so that the regenerators reverse their roles. Figure 1.6 show s a schematic diagram o f a tandem fixed bed regenerator arrangement. The system can be ex ten d ed to multiple beds to accom m odate unequal hot and cold blow periods and to limit the outlet fluid tem perature swing.

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Cold Process Stream Cool ~i~ Waste Stream

g ]c

Matrix 1

(Heat Storage) _Warm

Process Stream À Hot Waste Stream Matrix 2 (Heat Recovery) C Valve Closed O Valve Open

F igu re 1.6 Schematic diagram o f tandem fixed-bed counterflow regenerative heat exchangers allow ing sim ultaneous hot and cold blow periods.

In contrast to the fixed bed regenerator, the rotary regenerator moves the matrix material past stationary hot and cold blow flow ducts to effect the periodic tlow reversal within the regenerator matrix. As a result, the pseudo-steady-state rotary regenerator has a spatial, rather than temporal, variation o f fluid outlet temperature. Figure 1.7 shows a schematic diagram o f a rotary regenerator w here tluid flows axially through the regenerator matrix, while Figure 1.8 shows an arrangement with radial fluid tlow through the matrix.

Hot Waste Stream Warm Process Stream n? Cool I Waste i Stream i Cold Process Stream

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Warm Process Stream Cold Process ^ , ! Stream ^ool I Waste — Stream, Hot Waste - - Stream ,

F igure 1.8 Schematic diagram o f a radial flow rotary regenerator.

1.7 Regenerator Modelling

1.7.1 General Considerations

re g e n e rato r's function is fairly straightforward; however, its operation is cited as one of the most difficult to analyze mathem atically [17]. A complete description o f the operation o f a regenerator is a com plex function o f many coupled parameters, including:

internal matrix geom etry (particles, wire screens, thin plates, tubes, wires, etc.i; external matrix geom etry (rectangular, cylindrical, etc.):

re g en erato r configuration (fixed bed. rotary):

mass a n d heat capacities o f matrix material and heat transfer fiuid: effective thermal conductivity o f matrix material and heat transfer fiuid: effective heat transfer betw een matrix material and heat transfer fluid: radiative properties o f matrix material and heat transfer fluid: