LHP Stability TheoryLHP Stability Theory

Triem T. HoangTriem T. HoangTTH Research Inc.TTH Research Inc.

Robert W. BaldauffRobert W. BaldauffU S Naval Research LaboratoryU S Naval Research Laboratory

GSFC· 2015U.S. Naval Research LaboratoryU.S. Naval Research Laboratory

Outline

• Background of Loop Heat Pipe (LHP) Technology

• Nonlinear Dynamics in LHPs

O ill t B h i i LHP O ti• Oscillatory Behaviors in LHP Operation

• Linear Stability Theory

• Instability Criteria

• Discussion / Summary

• Path Forward

Loop Heat Pipe

(E)T

EG

(E)WT

EQ

EPMc

Heat Source

(E)

Attached Payload

( )SATT

(R)WT

)W(RPMc

Evaporator Pump

ReservoirTSAT, PE

(E)

TL(IN)

EPMc

1

VQm

2Q

PrimaryWick

(R)SATT

(L)RG

(W)RG

W

(R) )W(

Evaporator Pump

VapoLiqu

id m

L

TSAT, PR(R)

PLOOP PE PR Non-wickFlow PathSecondary

Wi k(R)LT )W(

RPMcCondenser

or mV

Wick

Metal Casing

Secondary

Heat Sink

TSAT, PLVI(LVI)

TSAT(C)

Secondary Wick

Not Shown

LHP is a capillary-pumped device having no mechanical moving part

LHP Steady State Operation

18

20

45

50

C) (W

/K)TSINK = 10oC and TAMB = 23oC

Capillary PumpReservoir

)IN(LT

TSAT

RLHPQ12

14

16

30

35

40

pera

ture

(o

ce 1

/RLH

P

1/R

p y p

Vapor

Liqu

id

Condenser/Radiator

QmV

TSINK

Q

QRTT 6

8

10

15

20

25

tion

Tem

p

ondu

ctan

c

TSAT

1/RLHPPoint 10

24

QRTT LHPSINKSAT

0

2

4

0

5

10

0 100 200 300 400 500 600 700

Satu

rat

Ther

mal

C

Point 3

0 100 200 300 400 500 600 700 T

Power Input (Watts)

LHP is a nonlinear heat transport system

NRL LHP (113g/300W/273K) 01/17/2003

Temperature Oscillations in LHPsNRL LHP (113g/300W/273K) 01/17/2003

TC7

TC3

op (P

a)

ure

(K)

TC 9,10,11TC 6,7,8

13

18 24 30

292517

16 26 2827

15

14

12

9

10

11

8

7

6VAPOR LINE

EVAPORATOR

NRL LHP

TC36

TC33

DP

Pres

sure

Dro

Tem

pera

tu

DP

TC 1,3,5

35

13

37

21

20 22 32 38

33

312319

12

42

5

3

136

CONDENSER

LIQUID LINE

COMPENSATIONCHAMBER

TC33

Time

TC 4TC2AP34

12/8/1999 25W start, -163K sink

TC08 TC23 TC31 TC22 TC24 TC32 TC06 Q1JPL TES LHP

Ku, J., “High Frequency Low Amplitude Temperature Oscillations in Loop Heat Pipe Operation,” Paper No. 2003-01-2387, SAE 33rd International Conference on Environmental Sciences (ICES), Vancouver, British Columbia, Canada, July 7-10, 2003.

298

308

(K)

15

20

25

30

tts)

TC08 TC23 TC31 TC22 TC24 TC32 TC06 Q1

POWER

TC8TC6

JPL TES LHP

268

278

288

Tem

pera

ture

0

5

10

15

Pow

er (W

at

TC31

TC22

TC23 TC24

25860 120 180 240 300 360 420 480 540 600 660

Elapsed Time (min)

-5

TC32

Ku, J., “Low Frequency High Amplitude Temperature Oscillations in Loop Heat Pipe Operation,” Paper No. 2003-01-2388, SAE 33rd International Conference on Environmental Sciences (ICES), Vancouver, British Columbia, Canada, July 7-10, 2003.

LHP Nonlinear DynamicsHeat Input QIN

)IN(T

QIN

Heat Transfer AcrossEvaporator Wall

(E)Capillary Pump

Vapor

Liqu

idCondenser/Radiator

ReservoirLT

)E(dTTSAT

GE

TW(E) (McP)E

(E)rCondenser/Radiator

EIN

)E(W

EP QQdt

dTMc

LHP Nonlinear DynamicsHeat Input QIN

)IN(T

QIN

Heat Transfer AcrossEvaporator Wall

(E)Capillary Pump

Vapor

Liqu

idCondenser/Radiator

ReservoirLT

)E(dTTSAT

GE

TW(E) (McP)E

(E)rCondenser/Radiator

EIN

)E(W

EP QQdt

dTMc

VVM

dtd

dtd

)2(

)VAPOR(VV

C

)2(C

SINKSAT)MAX(

C)(2

C VV)TT(GQ

dtdV

VVQQ1

VVdtdt)2(

CV)2(

CVL

)2(C1

)2(CVL

LHP Nonlinear DynamicsHeat Input QIN

)IN(T

QIN

Heat Transfer AcrossEvaporator Wall

(E)Capillary Pump

Vapor

Liqu

idCondenser/Radiator

ReservoirLT

)E(dTTSAT

GE

TW(E) (McP)E

(E)rCondenser/Radiator

EIN

)E(W

EP QQdt

dTMc

VVM

dtd

dtd

)2(

)VAPOR(VV

C

)2(C

SINKSAT)MAX(

C)(2

C VV)TT(GQ

dtdV

VVQQ1

VVdtdt)2(

CV)2(

CVL

)2(C1

)2(CVL

h4Tm4T

LC

)2(C)L(

CL)L(

LL)L(

BRLVIL

mV

V1RRR)PP(md

)IN(

LT

1V

Qm

0)TT(cD

h4Tm4t

TL)L(

PL

L

L

L

L

LL

V)2(

CV

C

)2(C

CL

CL

LL

LL

B

B

VV1

AL

AL

ALdt

V

)2(C

LL mdt

dVm

LHP Nonlinear DynamicsHeat Input QIN

)IN(T

QIN

Heat Transfer AcrossEvaporator Wall

(E))TT(cmQ )IN(

LSATPLSC

dtdV

VQQQQ

dtd )2(

CV)VAPOR(

R

)L(R

)W(R2SCV

Capillary Pump

Vapor

Liqu

idCondenser/Radiator

ReservoirLT

)E(dTTSAT

GE

TW(E) (McP)E

(E)rCondenser/Radiator

EIN

)E(W

EP QQdt

dTMc

VVM

dtd

dtd

)2(

)VAPOR(VV

C

)2(C

SINKSAT)MAX(

C)(2

C VV)TT(GQ

dtdV

VVQQ1

VVdtdt)2(

CV)2(

CVL

)2(C1

)2(CVL

h4Tm4T

LC

)2(C)L(

CL)L(

LL)L(

BRLVIL

mV

V1RRR)PP(md

)IN(

LT

1V

Qm

0)TT(cD

h4Tm4t

TL)L(

PL

L

L

L

L

LL

V)2(

CV

C

)2(C

CL

CL

LL

LL

B

B

VV1

AL

AL

ALdt

V

)2(C

LL mdt

dVm

LHP Nonlinear DynamicsHeat Input QIN

)IN(T

QIN

Heat Transfer AcrossEvaporator Wall

(E))TT(cmQ )IN(

LSATPLSC

dtdV

VQQQQ

dtd )2(

CV)VAPOR(

R

)L(R

)W(R2SCV

Capillary Pump

Vapor

Liqu

idCondenser/Radiator

ReservoirLT

)E(dTTSAT

GE

TW(E) (McP)E

(E)

Vapor-LiquidHeat Transfer

Vapor-Res. WallHeat Transfer

TW(R) (McP)R

(W) TL(R) (McP)R

(L)rCondenser/Radiator

EIN

)E(W

EP QQdt

dTMc

VVM

dtd

dtd

)2(

)VAPOR(VV

)W(

R)R(

W QdT

TSAT

GR(R)

(W)

TSAT(R)

GR(L)

C

)2(C

SINKSAT)MAX(

C)(2

C VV)TT(GQ

dtdV

VVQQ1

VVdtdt)2(

CV)2(

CVL

)2(C1

)2(CVL

h4Tm4T )L(

RP

)MAX(SC

)L(R

)R(L

McQ)1(Q

dtdT

)W(RP

RW

McQ

dt

LC

)2(C)L(

CL)L(

LL)L(

BRLVIL

mV

V1RRR)PP(md

)IN(

LT

1V

Qm

0)TT(cD

h4Tm4t

TL)L(

PL

L

L

L

L

LL

V)2(

CV

C

)2(C

CL

CL

LL

LL

B

B

VV1

AL

AL

ALdt

V

)2(C

LL mdt

dVm

LHP Governing Equations

L

)2(C

L

)2(C mQ1

dtdV

)2(V

Fluid Dynamics:

)2(

CCLLLB

LC

C)L(CL

)L(LL

)L(BRLVI

L

VV1

AL

AL

AL

mV

V1RRR)PP(

dtmd

CCLLLB VAAA

dtdVQQ

VV

1dt

dT )2(C)V(

E

)2(C1

)2(CVL

V

)E(SAT

Thermodynamics:

VVT CVL

SAT

dtdVQQQQ

VVV

1dt

dT )2(C)V(

R

)L(R

)W(R2

)MAX(SC

)2()V(V

)R(SAT

VVVT

)2(CVL

)V(LHP

SAT

V

)MAX()L()R(

)L()W()R(

)W()E( dTdTdT

Heat Transfer:

)MAX(SC

)L(R

L)L(RP

)W(R

W)W(RPEIN

WEP Q)1(Q

dtdTMc Q

dtdTMc QQ

dtdTMc

LHP Linear Stability Theory

)(dtd xfx

T )R()R()E()R()E()2( TTTTTV

Governing Equations:

where )R(L

)R(W

)E(W

)R(SAT

)E(SATL

)2(C TTTTTmV x

T 7654321 ffffffff

where,

Let xSS be the equilibrium (steady) state and |xxSS| <

Is LHP state x capable of reaching steady state xSS ?

)()()(dtd

SSSSSSSSSS xxJxx

xfxx

Linearized Equations:

k

tk

ke(t) Axx SS xSS x xSS

x

where k’s are eigenvalues of JSS.

LHP Linear Stability Theory

Characteristic Equation: 0)( det SSJI

2 0 cb 2

2

00c4bb

• Real parts of all k’s are negative solution decays to zero

65432,1 0 0 2

Real parts of all k s are negative solution decays to zero steady state

• At least one real part of k’s is positive solution grows exponentially unstable operation

• At least one real part is zero and the rest are negative i l iinconclusive

LHP Instability Criteria

)R(SAT

)R(L)L(

RRSSE

2

SSE

2

V

LCV

SSE

2

V

L)E(

SAT

EV4 T

T1GQQ1

QQ11G

QQ21

TQ

SSSSSS

INP)L(

RP)W(

RPSS)E(

W

E

EP6

QcMcMc

1TQ

Mc1

)R(L)L(R22L

)2(C)MAX( T1GQ1Q11VG

)CRIT(HFLA

SSE

2

V

L

)R(SAT

)(R

VSSESSEVC

)(C

SSSAT

E G

QQ21

T1G

Q1

Q11

VG

TQ

SSEV Q

)CRIT(

LFHAINPEP

)L(RP

)W(RP

E

)E(W

G1

Q1

cMcMcMc

QT

LFHAINPEPSSE GQccQ

Multi Time Scale Dynamical System

• Fluid Dynamic Subsystem 1 and 2 time constant 1 = 0.11.0 seconds1 perturbations decay with time, i.e. Re(1,2) < 0

• Thermodynamic Subsystem 3 and 4Thermodynamic Subsystem 3 and 4 time constant 2 = 110 minutes 3 = 0 and 4 = condensation of reservoir vapor causes instability

• Heat Transfer Subsystem 5 and 6y 5 6 time constant 3 = 110 hours 3 = 0 and 4 =

l tt h d th l i t bilit large attached thermal mass causes instability

Very High Frequency Oscillations

Pressure Oscillation Frequency f > 1Hz

NRL LHPJan 17 200331

30

29

2,000

1,900

1,800

Evap.

Res

NRL LHPJan 17, 2003

C) Pa)

28

27

26

1,700

1,600

1 500

Res.

Perat

ure

(o C

re D

rop

(P

26

25

24

1,500

1,400

1,300Return

P

Tem

pe

Pres

sur

23

22

1,200

1,10013:00 13:02 13:04 13:06 13:08 13:10

ReturnLiquid

TimeTime

Ku, J., “High Frequency Low Amplitude Temperature Oscillations in Loop Heat Pipe Operation,” Paper No. 2003-01-2387, SAE 33rd International Conference on Environmental Sciences (ICES), Vancouver, British Columbia, Canada, July 7-10, 2003.

High Frequency Oscillations

)CRIT(HFLA

SSSAT

E GTQ

For Unstable LHP Operation:

1SS,EQ

e

Zone 1 Zone 2Zone 3

(E)WT

SATTAMBT

0 SSSATT

ondu

ctan

c W

)CRIT(HFLAG

)MAX(CGat

ure

or C

o

3’1’

3SS,EQ

CG

)3(SS,EQ)1(

SS,EQ

Tem

pera

)'3(SS,EQ

)'1(SS,EQ

Low Frequency Oscillations

)CRIT(

LFHAINPEP

)L(RP

)W(RP

SSE

)E(W

G1

Q1

cMcMcMc

QT

For Unstable LHP Operation:ESS

0e

Zone 1 Zone 2Zone 3

(E)WT

0

SSIN

)E(W

QT

nduc

tanc

e

13

)CRIT(G/1ture

or C

on

)3(INQ)1(

INQ

for large attachedLFHAG/1

Tem

pera

t for large attached thermal mass

Low Frequency Oscillations

)CRIT(

LFHAINPEP

)L(RP

)W(RP

SSE

)E(W

G1

Q1

cMcMcMc

QT

For Unstable LHP Operation:ESS

0e

Zone 1 Zone 2Zone 3

(E)WT

0

SSIN

)E(W

QT

nduc

tanc

e

1 3

ture

or C

o

)3(INQ)1(

INQ

1’ 3’3”1”

)CRIT(G/1 f d i tt h d

Tem

pera

t

4)CRIT(

LFHAG/1 for decreasing attached thermal mass

Discussion

Effects of Sink Temperature on LHP Stability

25

30

35re

(o C) Predictions

JPL TES/LHP Data

15

20

Tem

pera

tur

0

5

10

Satu

ratio

n T

‐5

0

0 20 40 60 80 100 120 140 160 180 200

S

Power Input (W)

INQ

Power Input (W)

Ku, J., “Low Frequency High Amplitude Temperature Oscillations in Loop Heat Pipe Operation,” Paper No. 2003-01-2388, SAE 33rd International Conference on Environmental Sciences (ICES), Vancouver, British Columbia, Canada, July 7-10, 2003.

Discussion

Effects of Bayonet Heat Exchanger onHigh-Frequency/Low-Amplitude Oscillations

Vapor Control Volume

Reservoir Evaporator

Lm

Liquid Control Volume

)R(SATT

Vapor Temperature in Reservoir & Pump Core

)IN(LSATPL

)MAX(SC TTcmQ

(OUT)BT

s

(IN)LT

Liquid Temperature in Bayonet TL

Reservoir Wick Core

)MAX(

SC

)IN(L

)OUT(BP

QTTcm

sDistance Along Bayonet

Summary

• LHP as Dynamical System– three (3) synergistically interacting subsystems: fluid dynamics,

thermodynamics heat transfer processesthermodynamics, heat transfer processes– three (3) corresponding time scales: 1 (0.11 seconds), 2 (1

10 minutes), 3 (110 hours)

• LHP Linear Stability Theory– Is LHP state x capable of reaching steady state xSS ?

I bili C i i• Instability Criteria

)CRIT(

)R(SAT

)R(L)L(

RV

R

SSE

2

SSE

2

V

L

C

)2(C)MAX(

C

E GTT1G

QQ1

QQ11

VVG

Q

)CRIT(

HFLA

SSE

2

V

L

SSSS

SSSAT

E G

QQ21

TQ

)L()W()E( 11McMcT )CRIT(

LFHAINPEP

)(RP

)(RP

SSE

)(W

G1

Q1

cMcMcMc

QT

Path Forward

Verify LHP stability theory against available test data