4th International Energy Conversion Engineering Conference and Exhibit (IECEC) 26 - 29 June 2006, San Diego, California

AIAA 2006-4003

An Initial Non-Equilibrium Porous-Media Model for CFD Simulation of Stirling Regenerators

Roy ~ e w * NASA Glenn Research Center, Cleveland, OH, 44135

Terry simont Universi@ of Minnesota, MinneapoIis, h4N, 55455

David cedeon: Gedeon Associates, Athens, OH, 45701

Mounir 1brahim4 and Wei on^*' Cleveland State Universiry, CleveJmd, OH, 44115

The objective of this paper is to define empirical parameters (or closwre models) for an initial thermai non-equilibrium porous-media model for use in Computational Fluid Dynamics (CFD) codes for simulation of Stirling regenerators. The two CFD codes currently being used at Glenn Research Center (GRC) for Stirling engine modeling are Fluent and CFD-ACE. The porous-media models available in each of these codes are equilibrium models, which assmne that the solid matrix and the fluid are in thermal equilibrium at each spatial location within the porous medium. This is believed to be a poor assumption for the oscillating-flow environment within Stirling regenerators; Stirling 1-D regenerator models, used in Stirling design, w e non-equilibrium regenerator models and suggest regenerator matrix and gas average temperatures can differ by several degrees at a given axial location end time during the cycle. A NASA regenerator research grant has been providing experimental and computational results to support definition of various empirical coefficients needed in defining a noa-equilibrium, macroscopic, porous-media model (i-e., to define "closure" relations). The grant effort is being led by Cleveland State University, with subcontractor assistance from the University of Minnesota, Gedeon Associates, and Sunpower, Inc. Friction-factor and heat-transfer correlations based on data taken with the NASAlSunpower oscillating-flow test rig also provide experimentally based correlations that are useful in defining parameters for the porous-media model; these correlations are documented in Gedeon Associates' Sage Stirling-Code Manuals. These sources of experimentally based information were used to define the following terms and parameters needed in the non-equilibrium porous-media model: hydrodynamic dispersion, permeability, inertial coefficient, fluid effective thermal conductivity (including themal dispersion and estimate of tortuosity effects}, and fluid-solid heat transfer coefficient. Solid effective thermal conductivity (including the effect of tortuosity) was aIso estimated. Determination of the porous-media model parameters was based on planned use in a CFD model of Jnfinia's Stirling Technology Demonstration Convertor (TDC), which uses a random-fiber regenerator matrix. The non-equilibrium porous-media model presented is considered to be an initial, o r "draft," model for possible incorporation in commercial CFD codes, with the expectation that the empirical parameters will likely need to be updated once resulting Stirling CFD model regenerator and engine results have been analyzed. The emphasis of the paper is on use of available data to define empirical parameters (and closure models) needed in a thermal non-equilibrium porous-media model for Stirling regenerator simulation. Such a model has not yet been implemented by the authors or their associates. However, it is anticipated that a thermal non-equilibrium model such as that presented here, when iacorporated in the CFD codes, will improve our ability to accurately model Stirling regenerators with CFD relative to current thermal-equilibrium porous-media models.

Research Engineer, Thermal Energy Conversion Branch, 2 1000 Brookpark Rd.lMS 30 1-2, AIAA Member ' Professor, Department of Mechanical Engineering, 1 1 1 Church St., S.E., AlAA Member

Proprietor of Gedeon AssociatesJStirlEng Software Developer, 16922 South Canaan Road, Non-Member AIAA $ Professor, Mechanical Engineering Department, 1960 E. 24Ih Street, AIAA Associate Fellow *1

Research Associate, Mechanical Engineering Department, 1960 E. 24Ih Street, AIAA Member

https://ntrs.nasa.gov/search.jsp?R=20070018065 2018-07-30T08:56:36+00:00Z

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

An

Initi

al N

on-E

quili

briu

m P

orou

s-M

edia

Mod

el

for C

FD S

imul

atio

n of

Stir

ling

Reg

ener

ator

sby

Roy

Tew

, NA

SA G

RC

Terr

y Si

mon

, U. o

f Min

neso

taD

avid

Ged

eon,

Ged

eon

Ass

ocia

tes

Mou

nirI

brah

iman

d W

ei R

ong,

Cle

vela

nd S

tate

Uni

vers

ity

Pres

ente

d by

Roy

Tew

for

IEC

EC 2

006

June

26-

29, 2

006

San

Die

go, C

A

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Ack

now

ledg

men

ts

The

wor

k de

scrib

ed in

this

pap

er w

as p

erfo

rmed

for t

he N

ASA

Sc

ienc

e M

issi

on D

irect

orat

e (S

MD

) and

the

Rad

iois

otop

e Po

wer

Sy

stem

(RPS

) Pro

gram

. Any

opi

nion

s, fi

ndin

gs a

nd c

oncl

usio

ns

or re

com

men

datio

ns e

xpre

ssed

in th

is re

port

, are

thos

e of

the

auth

ors

and

do n

ot n

eces

saril

y re

flect

the

view

s of

the

Nat

iona

lA

eron

autic

s an

d Sp

ace

Adm

inis

trat

ion.

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

•Ill

ustr

atio

ns o

f Stir

ling

engi

ne, 2

-D s

imul

atio

n re

sults

, reg

en.

mat

eria

l

•Ev

iden

ce th

at a

ssum

ptio

n of

gas

/sol

id th

erm

al e

quili

briu

m in

re

gene

rato

r not

val

id

•Fo

r ref

eren

ce, t

herm

al n

on-e

quili

briu

m p

orou

s-m

edia

co

nser

vatio

n eq

uatio

ns

•Po

rous

-med

ia m

odel

qua

ntiti

es in

thes

e eq

uatio

ns n

eedi

ng

defin

ition

or “

clos

ure”

•Su

mm

ary

of a

vaila

ble

info

rmat

ion

for d

efin

ing

thes

e qu

antit

ies

•C

oncl

udin

g re

mar

ks

Pres

enta

tion

Out

line

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Sche

mat

ic o

f Inf

inia

’sTD

C 5

5 W

e St

irlin

gEn

gine

Pist

ons

have

Cle

aran

ce S

eals

=>N

o C

onta

ct w

ith C

ylin

ders

Use

s H

eliu

m “

Wor

king

”G

as

Coo

ler R

egen

erat

or

Hea

ter

-sup

plie

dB

yR

adio

isto

pes

Dis

plac

erPi

ston

Dis

plac

er F

lexu

res

Pow

erPi

ston

Pow

er P

isto

n Fl

exur

es

Line

ar A

ltern

ator

Stat

iona

ry M

agne

ts

M

ovin

g Iro

n

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

GR

C T

empe

ratu

re C

onto

urs:

Gen

erat

ed w

ith D

yson

’s F

luen

t2-

D A

xisy

mm

etric

CFD

Mod

el o

f Inf

inia

’sTD

C S

tirlin

gEn

gine

(Usi

ng

Ther

mal

-Equ

ilibr

ium

Por

ous-

Med

ia M

odel

for R

egen

erat

or S

imul

atio

n)

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Fi

gure

1 B

ekae

rt n

omin

al 1

2 m

icro

n ro

und

fiber

s 316

stai

nles

s ste

el. M

easu

red

mea

n ef

fect

ive

diam

eter

13.

4 m

icro

ns. F

rom

90%

por

osity

rege

nera

tor m

atri

x m

ade

and

test

ed u

nder

cur

rent

D

OE

rege

nera

tor r

esea

rch

prog

ram

. Mic

rogr

aph

cour

tesy

of N

ASA

GRC

.

Cur

rent

ly U

sed

Reg

ener

ator

Ran

dom

Fib

er M

ater

ial

Bek

aert

316

Stai

nles

s St

eel

Nom

inal

12

Mic

ron

Rou

nd F

iber

sM

ean

Eff.

D.=

13.4

mic

rons

& F

iber

s A

re N

ot R

ound

90 %

Por

osity

---fr

om D

avid

Ged

eon

mem

o.re

port

ing

on D

OE

Reg

ener

ator

Res

earc

h C

ontr

act W

ork,

Pho

toB

y G

RC

.

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Evid

ence

that

Gas

/Sol

id T

herm

al E

quili

briu

m A

ssum

ptio

n is

not

Val

id•

UM

N T

ests

with

Eng

ine

Valu

es o

f Dim

ensi

onle

ss V

aria

bles

sho

wSi

gnifi

cant

Tem

pera

ture

Diff

eren

ces

betw

een

Gas

and

Sol

id-

To b

e sh

own

•Sa

ge 1

-D M

odel

of I

nfin

ia’s

TDC

55

We

Engi

ne s

how

s

sign

ifica

nt G

as/S

olid

Tem

pera

ture

Diff

eren

ces

in R

egen

erat

or-

To b

e sh

own

•En

thal

py F

lux

Loss

es th

roug

h th

e R

egen

erat

or fr

om th

e H

ot to

th

e C

old

calc

ulat

ed b

y Sa

ge a

re s

igni

fican

t (to

be

show

n) a

ndar

e ex

pect

ed to

be

sens

itive

to th

erm

al e

quili

briu

m/n

on-

equi

libriu

m a

ssum

ptio

n•

Enth

alpy

Flu

x at

Poi

nt a

long

1-D

Flo

w A

xis

is:

dtT

cm

gas

p∫

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Reg

ener

ator

Res

earc

h Te

st S

ectio

n at

UM

N

FIGU

RE 9.

The

Sch

emati

c of t

he U

MN

Expe

rimen

tal F

acili

ty an

d the

Test

Secti

on.

1----

oscil

lator

y flo

w ge

nerat

or, 2

----p

iston

, 3---

-flow

distr

ibuto

r, 4-

---co

oler,

5----

plen

um,

6----

regen

erato

r, 7-

---sc

reen m

atrix,

8----

electr

ical h

ating

coil,

9----

isolat

ion d

uct,

10---

-hot

-wire

, 11-

---th

ermoc

ouple

, 12-

---co

oling

wate

r in,

13---

-cool

ing w

ater o

ut

Plen

um

Freq

uenc

y =

0.4

Hz

(24

RPM

), S

trok

e =

356

mm

, Pi

ston

Dia

met

er =

356

mm

Reg

en. P

oros

ity =

90%

, D

imen

sion

less

Par

amet

ers

Sim

ilar t

o an

Infin

iaEn

gine

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

UM

N W

ire

Scre

en R

egen

erat

or(C

onsi

dere

d to

Yie

ld a

n A

ppro

xim

atio

n of

Ran

dom

Fib

er)

•90

% P

oros

ity•

Stai

nles

s Ste

el 3

04 W

elde

d Sc

reen

s•

200

Lay

ers o

f 6.3

mm

x 6

.3 m

m M

esh

•W

ire

Dia

met

er =

0.8

1 m

m•

Eac

h Sc

reen

Rot

ated

45

Deg

. Rel

. to

Nex

t

•R

epre

sent

ativ

e St

irlin

gE

ngin

eC

hose

n to

def

ine

Tes

t Sec

tion

Dim

ensi

onle

ss P

aram

eter

s(M

ax. R

eyno

lds #

& V

alen

si#)

•H

ot-W

ire

Ane

mom

etry

Mea

sure

men

ts

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

UM

N T

est D

ata:

Tem

pera

ture

Diff

eren

ce b

etw

een

Gas

/Sol

id in

Reg

ener

ator

Mea

sure

men

ts m

ade

with

Hot

Wir

e in

Axi

al C

ente

rat

80th

Lay

er o

f Scr

een

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Flui

d an

d S

olid

Tem

pera

ture

Dur

ing

one

cycl

eM

icro

& M

acro

CFD

resu

lts V

S E

xper

imen

t

297.

5

298

298.

5

299

299.

5

300

300.

5

301

301.

5

302

302.

5

303

090

180

270

360

Cra

nk A

ngle

Temperature

Ts-C

FD M

icro

Tf C

FD M

icro

Ts E

XPTf

EXP

T C

FDR

C-M

AC

RO

CSU

CFD

“M

icro

-”an

d “M

acro

-”C

alcu

latio

ns &

UM

N T

est D

ata

(Mic

ro-C

alcu

latio

ns m

ade

usin

g R

EV

of a

ctua

l geo

met

ry)

(Mac

ro-C

alcu

latio

ns m

ade

with

CFD

-AC

E T

herm

al E

quili

briu

m P

orou

s Med

ia M

odel

)

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Reg

ener

ator

Gas

and

Sol

id T

empe

ratu

res a

t Hot

-End

, Mid

-Poi

nt a

nd C

old-

End

as P

redi

cted

by

1-D

Sag

e C

ode—

for

Infin

ia’s

TD

C 5

5 W

e E

ngin

e (U

sing

1-D

The

rmal

Non

-Equ

ilibr

ium

Reg

ener

ator

Mod

el)

Re

gene

rato

r Gas

and

Mat

rix T

empe

ratu

res

at H

ot E

nd

770

775

780

785

790

795

800

12

34

56

7

Equa

lly S

pace

d Ti

mes

ove

r the

Cyc

le

Temperature (Deg. K)

Gas

Matri

x

Reg

ener

ator

Gas

and

Mat

rix

Tem

pera

ture

s at

C

old

End

380

385

390

395

400

12

34

56

7Eq

ually

Spa

ced

Tim

es o

ver t

he C

ycle

Temperature (Deg. K)

Gas

Mat

rix

Rege

nera

tor G

as a

nd M

atrix

Tem

pera

ture

s at

M

id-P

oint

570

575

580

585

590

12

34

56

7

Equa

lly S

pace

d Ti

mes

ove

r the

Cyc

le

Temperature (Deg. K)

Gas

Mat

rix

TDC

Ope

ratin

g C

ondi

tions

& P

redi

ctio

ns

•O

pera

ting

Con

ditio

ns (N

ot D

esig

n) w

ere

•H

ot-E

nd T

empe

ratu

re =

823

K (5

50 C

)•

Col

d-En

d Te

mpe

ratu

re =

363

K (9

0 C

)•

Freq

uenc

y of

Ope

ratio

n =

82.5

Hz

•H

eliu

m M

ean

Pres

sure

= 2

.59

MPa

(376

psi

)•

Perf

orm

ance

Pre

dict

ions

wer

e•

Engi

ne E

lect

rical

Pow

er =

55.

9 W

e•

Engi

ne E

ffici

ency

= 1

9.4%

•H

eat I

nto

Engi

ne =

288

W•

Reg

ener

ator

Ent

halp

y Fl

ux L

oss

= 22

.4 W

Reg

en. E

ntha

lpy

Flux

Los

s ≡

dtT

cm

gas

p∫

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Non

-Equ

ilibr

ium

Por

ous-

Med

ia C

onse

rvat

ion

Equa

tions

[]

01

=•

∇+

∂

∂u

ff

tρ

βρ

()

[]

uu

u

uuu

uu

u

KfC

fK

ff

ff

pf

tf

f

eff

ρμ

βρρ

βν

ρβ

ρ

β

−−

−∇

•∇

+−∇

=•

∇+

∂

∂⎟ ⎠⎞

⎜ ⎝⎛~~

211

()

[]

[] (

)dt

fp

df

Ts

Tf

dVsf

dA

sfh

KCK

Tk

ht

fh

ff

ffe

ff

f

+−

+⋅

⎟⎟ ⎠⎞⎜⎜ ⎝⎛

++

•∇•

∇=

•∇

+∂

∂u

uu

uρ

μρ

β

ρ1

()

[]

()

fT

sT

sdV

sfdA

sfhT

kt

sT

sC s

sse

−−

∇••

∇=

∂

∂ρ

Con

tinui

ty:

Mom

entu

m:

Flui

dE

nerg

y:

Solid

Ene

rgy:

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Poro

us-M

edia

Qua

ntiti

es N

eedi

ng D

efin

ition

or “

Clo

sure

”

()

[]

uu

u

uuu

uu

u

KfC

fK

ff

ff

pf

tf

f

eff

ρμ

βρρ

βν

ρβ

ρ

β

−−

−∇

•∇

+−∇

=•

∇+

∂

∂⎟ ⎠⎞

⎜ ⎝⎛~~

211

Mom

entu

m E

quat

ion:

()

[]

[] (

)dt

fp

df

Ts

Tf

dVsf

dA

sfh

KCK

Tk

ht

fh

ff

ffe

ff

f

+−

+⋅

⎟⎟ ⎠⎞⎜⎜ ⎝⎛

++

•∇•

∇=

•∇

+∂

∂u

uu

uρ

μρ

β

ρ1

Flui

dE

nerg

y:

()

[]

()

fT

sT

sdV

sfdA

sfhT

kt

sT

sC s

sse

−−

∇••

∇=

∂

∂ρ

Solid

Ene

rgy:

Poro

sity

Perm

eabi

lity

Iner

tial C

oeffi

cien

t

Hyd

rody

nam

ic D

ispe

rsio

n

Flui

d Ef

fect

ive

Ther

mal

Con

duct

ivity Fl

uid-

Solid

Hea

t Tra

nsfe

r Coe

ffici

ent

Solid

Effe

ctiv

e Th

erm

al C

ondu

ctiv

ity

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Det

erm

inat

ion

of P

erm

eabi

lity

and

Iner

tial C

oeffi

cien

t(1

)U

. of M

inn.

Lar

ge-S

cale

Wire

Scr

een

Mea

sure

men

ts

(2)

Ass

ume

Qua

si-S

tead

y Fl

ow, E

quat

e D

arcy

-For

chhe

imer

Stea

dy 1

-DM

omen

tum

Equ

atio

n w

ith D

arcy

Fric

tion

Fact

or M

omen

tum

Eq.

&

Use

NA

SA/S

unpo

wer

Osc

illat

ing-

Flow

Rig

Dar

cy F

rictio

n-Fa

ctor

Dat

a

2u

KfC

uK

Lpf

ρμ

+=

∇2

21u

hdDf

Lpf

ρ=

∇

Coe

ffic

ient

U

MN

Lar

ge-S

cale

Scr

eens

(dw=8

.1E

-4 m

) T

DC

Ran

dom

Fib

er

U

MN

Old

, Ex

perim

enta

l U

MN

New

, Ex

perim

enta

l C

SU C

alcs

. Sa

ge C

or.

Sage

Cor

. U

nidi

rect

iona

l Fl

ow T

ests

K (m

2 ) 1.

07E-

7 1.

86E-

7 8.

9E-7

8.

24E-

7 4.

08E-

10

3.52

E-10

2W

K/d

0.

163

0.28

3 1.

36

1.26

-

-

Cf

0.04

9 0.

052

0.14

0.

13-0

.11

Re=

25-1

00

0.19

-0.1

7 R

e=25

-100

0.

154-

0.09

5 R

e=25

-100

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Flui

d &

Sol

id E

ffect

ive

Ther

mal

Con

duct

ivity

Ten

sors

•A

ssum

e on

ly d

iago

nal e

lem

ents

of t

enso

rs a

re n

on-z

ero

•Th

en, i

n te

rms

of 3

-D c

ylin

dric

al c

oord

inat

es--

⎥⎥ ⎦⎤

⎢⎢ ⎣⎡

⎥⎥ ⎦⎤

⎢⎢ ⎣⎡

⎥⎥ ⎦⎤

⎢⎢ ⎣⎡

+

+

+

=

++

++

++

=≡

xxdi

sk

xxst

agf

kdi

sk

stag

fk

rrdi

sk

rrst

agf

k

xxdi

sk

xxto

rf

kf

kdi

sk

tor

fk

fk

rrdi

sk

rrto

rf

kf

k

xxfe

kfe

krr

fek

fek

,,

,0

0

0,

,,

0

00

,,

,

,,

,0

0

0,

,,

0

00

,,

,

,0

0

0,

0

00

,

ϑϑϑϑ

ϑϑϑϑ

ϑϑ

⎥⎥ ⎦⎤

⎢⎢ ⎣⎡

⎥⎥ ⎦⎤

⎢⎢ ⎣⎡

⎟ ⎠⎞⎜ ⎝⎛

⎟ ⎠⎞⎜ ⎝⎛

⎟ ⎠⎞⎜ ⎝⎛

=≡

xxto

rsk

skf

tor

sksk

f

rrto

rsk

skf

xxsek

sekrr

sek

sek

,,

,0

0

0,

,,

0

00

,,

,

,0

00

,0

00

,ϑϑ

ϑϑ

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Expe

rimen

tal V

alue

s of

Flu

id T

herm

al D

ispe

rsio

n C

ondu

ctiv

ity

•In

Tab

le b

elow

,an

d ax

ial t

herm

al d

ispe

rsio

n co

nduc

tiviti

es d

eter

min

ed b

y va

rious

expe

rimen

tal m

easu

rem

ents

xxdi

syy

dis

kk

,,

,, r

espe

ctiv

ely,

are

the

radi

al

Es

timat

ed T

herm

al D

isper

sion

Poro

us m

edia

Cu

rren

t Dir

ect

Mea

sure

men

ts at

UM

N,

Niu7

Pe.

kkU

d.

cρk

M,e

ddy

εf

dis,y

yh

pfdis,y

y02

0or

020

==

=

Weld

ed S

cree

n

Hun

t and

Tien

22

Pe

kk

fyydi

s00

11.0

,=

Fi

brou

s Med

ia

Met

zger

, Did

ierje

an,

and

Mai

llet23

59.1,

,07

3.0

and

)05.0

03.0(Pe

kkPe

kk

fxxdi

s

fyydi

s=

−=

Pa

cked

Sph

eres

Ged

eon9

560

0.9,

for

06.0or

50.0,

91.262.0

,=

=≈

=−

PePe

kkPe

kk

fxxdi

s

fxxdi

sβ

βW

oven

Scr

een

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Estim

ates

of F

luid

-Sta

gnan

t and

Sol

id-E

ffect

ive

Ther

mal

Con

duct

iviti

es--f

or 9

0% p

oros

ity w

ire s

cree

n or

rand

om fi

ber m

atrix

---#1

•B

ase

calc

ulat

ions

bel

ow o

n ai

r,=

fk

0.02

6 W

/m-K

and

sta

inle

ss s

teel

,=

sk13

.4 W

/m-K

•Fo

r rad

ial &

azi

mut

hald

irect

ions

, ass

ume

appr

oxim

atel

y pa

ralle

l sol

id&

flui

d flo

w p

aths

& th

us b

ased

on

the

para

llel m

odel

:

==

βf

kst

agf

k,

)90.0()

/3

1026(

mK

Wx

−=

mK

W/

0234

.0

=sek

()

β−1sk

=)1.0()

/4.

13(m

KW

=m

KW

/34.1

•M

cFad

den’

s (U

MN

) cal

cula

tions

bas

ed o

n ac

tual

wel

ded

scre

en g

eom

etry

sugg

est p

aral

lel m

odel

val

ue o

f sol

id e

ffect

ive

ther

mal

con

duct

ivity

shou

ld b

e m

ultip

lied

by 0

.625

()

838

.062

5.0

/34.1

625

.01

==

−=

∴x

mK

Wk

seks

β

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Estim

ates

of F

luid

-Sta

gnan

t and

Sol

id-E

ffect

ive

Ther

mal

Con

duct

iviti

es--f

or 9

0% p

oros

ity w

ire s

cree

n or

rand

om fi

ber m

atrix

---#2

•Fo

r the

axi

al d

irect

ion

use

a m

odifi

catio

n of

the

serie

s m

odel

•A

lum

ped

effe

ctiv

e so

lid +

flui

d ef

fect

ive

ther

mal

con

duct

ivity

(not

incl

udin

g th

erm

al d

ispe

rsio

n), b

ased

on

the

serie

s m

odel

is --

•O

nly

slig

htly

larg

er th

an m

olec

ular

flui

d co

nd. o

f 0.0

26 W

/m-K

& li

kely

too

smal

l sin

ce s

erie

s m

odel

impl

ies

wire

s no

t tou

chin

g in

axi

al d

irect

ion

•3-

D C

FD m

icro

scop

ic s

imul

atio

n of

REV

of t

he U

MN

wel

ded

scre

en b

yby

Ron

gof

CSU

sug

gest

s ab

ove

serie

s va

lue

shou

ld b

e m

ultip

lied

by 2

.157

(

) ()

mK

W

xsk

fk

fs

eff

k/

0289

.0

4.13

1.03

1026

90.01

11,

=+

−

=−

+=

+β

β

()(

)m

KW

mK

Wf

sef

fk

/06

23.0

157

.2/

0289

.0,

==

+∴

•Th

e ab

ove

wou

ld b

e ap

prop

riate

for a

n eq

uilib

rium

mod

el.

Seei

ng n

oob

viou

s w

ay to

sep

arat

e va

lues

for a

flui

d &

sol

id in

the

axia

l dire

ctio

n—pr

opos

e us

ing

the

sam

e va

lue

for f

luid

-sta

gnan

t and

sol

id-e

ffect

ive

cond

.in

the

axia

l dire

ctio

n, h

opin

g th

at o

vera

ll ef

fect

will

be

reas

onab

le.

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Hea

t Tra

nsfe

r Coe

ffici

ents

Bet

wee

n Fl

uid

& S

olid

Mat

rix

•G

ood

sour

ces

of h

eat t

rans

fer c

oeffi

cien

t cor

rela

tions

for w

ire s

cree

nan

d ra

ndom

fibe

r mat

rices

are

Ged

eon’

sSa

ge m

anua

ls a

nd N

ASA

Tech

nica

l Mem

oran

dum

s co

ntai

ning

NA

SA/S

unpo

wer

osci

llatin

g-flo

wrig

dat

a.

•Th

e fo

llow

ing

corr

elat

ions

are

in te

rms

of N

usse

ltN

o., P

ecle

tNum

ber

( Rey

nold

s N

o. x

Pra

ndtl

No.

), an

d po

rosi

ty:

•Fo

r wire

scr

een:

•Fo

r ran

dom

fibe

r:

•w

here

--

79.1)

66.099.0

.1(β

PeN

u+

=

61.2)

66.016.1

.1(β

PeN

u+

=

kpchd

uPr

RePe

khhd

Nu

μ

μ

ρ=

==

,

Osc

illat

ing-

Flow

Rig

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Hyd

rody

nam

ic D

ispe

rsio

n Te

rm (

Poss

ibly

Neg

ligib

leEf

fect

for t

his

App

licat

ion—

This

Nee

ds to

be

Che

cked

)•

Hyd

rody

nam

ic d

ispe

rsio

n te

rm--f

luid

mom

entu

m e

quat

ion:

poro

sity

;

a

vera

ge-c

hann

el fl

uid,

or l

ocal

, vel

ocity

insi

de m

atrix

spat

ial-v

aria

tion

of a

vera

ge-c

hann

el fl

uid

velo

city

insi

de m

atrix

;

•M

ost i

mpo

rtan

t ter

ms

from

the

abov

e te

nsor

exp

ress

ion

are

for t

rans

port

norm

al to

the

flow

and

are

:

•N

iuof

UM

N—

base

d on

rege

nera

tor e

xper

imen

ts--s

how

ed th

at:

•Th

is in

form

atio

n co

uld

be u

sed

to c

heck

the

sign

ifica

nce

of th

e te

rm—

but

has

not b

een

done

, yet

.

uuuu

~~~~

1≅

β=

β=

u=

u~

ww

vw

uw

wv

vv

ju

vw

uv

uu

u~

~~

~~

~~

~~

~~

~~

~~

~~

~~

~kk

kjki

jkj

jiik

ijii

uu

++

++

++

++

=

uv

vu

~~

or~

~

rUU hd

rUU hd

rUM

vu

uvvu

∂∂−

=∂∂

−=

∂∂−

=′

′≈

=λ

ββ

εβ

ββ

β21

02.021

212

2~~

2~~

Gle

nn R

esea

rch

Cen

ter

at L

ewis

Fie

ld

Con

clud

ing

Rem

arks

•A

ther

mal

-non

-equ

ilibr

ium

por

ous-

med

ia m

odel

is n

eede

d to

re

plac

e th

e ex

istin

g th

erm

al e

quili

briu

m m

odel

s in

CFD

cod

es fo

rac

cura

te s

imul

atio

n of

Stir

ling

rege

nera

tors

•Tr

ansi

ent,

com

pres

sibl

e-flo

w, c

onse

rvat

ion

equa

tions

are

su

mm

ariz

ed fo

r ref

eren

ce in

dis

cuss

ing

poro

us-m

edia

mod

el

para

met

ers

need

ing

defin

ition

for a

mac

rosc

opic

, the

rmal

-non

-eq

uilib

rium

por

ous-

med

ia m

odel

•A

vaila

ble

expe

rimen

tal i

nfor

mat

ion

is d

iscu

ssed

for d

efin

ition

of

the

hydr

odyn

amic

-dis

pers

ion

term

and

the

perm

eabi

lity

& in

ertia

l co

effic

ient

s in

the

mom

entu

m e

quat

ion,

and

the

heat

tran

sfer

co

effic

ient

& th

erm

al d

ispe

rsio

n te

rms

in th

e en

ergy

equ

atio

ns•

Met

hods

are

out

lined

for e

stim

atin

g flu

id-s

tagn

ant a

nd s

olid

-ef

fect

ive

ther

mal

con

duct

iviti

es•

Ade

quat

e in

form

atio

n is

giv

en fo

r def

initi

on o

f an

initi

al th

erm

al

non-

equi

libriu

m p

orou

s-m

edia

mod

el fo

r use

in a

CFD

mod

el o

f th

e re

gene

rato

r of a

TD

C S

tirlin

gen

gine

. St

ill n

eed

to im

plem

ent.

•A

pplic

atio

n of

this

mod

el in

Stir

ling

CFD

cod

es m

ay d

emon

stra

te

that

furt

her r

efin

emen

t of t

hese

par

amet

ers,

or o

f the

mod

el it

self

may

be

requ

ired