Thermal significance and optimal transfer in vessels bundles is influenced by vascular density

(1)

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Dichamp, Jules and De Gournay, Frédéric and Plouraboué, Franck Thermal significance

and optimal transfer in vessels bundles is influenced by vascular density. (2019) International

Journal of Heat and Mass Transfer, 138. 1-10. ISSN 0017-9310

OATAO

(2)

Thermal

s

ignificance and optimal tran

s

fer in

v

e

ss

el

s

bundle

s

i

s

influenced b

y v

a

s

cular den

s

ity

J

.

D

ic

h

a

mp

a

,

F. D

e

G

o

urna

y

b.

F

.

Pl

o

u

r

ab

o

u

e

a

,

*

'Institut de Mecanique des Fluides de Toulouse, UMR CNRS-INPT/UPS No. 5502, France

• lnstitut de Mathematiques de Toulouse, CNRS and Universite Paul Sabatier, Toulouse, France

ARTICL E I NFO A BSTRACT

Keywords: Thermal significance Counter-<:urrent Blood flow Heat transfer Bio-heat

A semi analytic method is used in order to systematically compute stationary 3 D coupled convection diffusion in various parallel counter current configurations and evaluate their thermal significance.

This semi analytic method permits a complete exploration of physiologically relevant parameter space associated with the bio heat transfer of parallel vessels bundles. We analyze thermal significance with various previously proposed criteria Optimal transfer configurations are found to depend on the vascular

density and Peclet numbers. The relevance of these findings for bio heat modeling in tissues is discussed.

1 .

I

ntroduction

B

i

o

h

eat tra

n

sfe

r

is

r

eleva

n

t

in m

a

n

y

ph

ys

i

ologica

l

co

nt

exts

suc

h

as

t

he

rm

o

r

egu

l

at

i

o

n

,

l

y

pol

ysis, o

r

t

h

e

rm

o

th

e

r

ap

i

es a

m

o

n

g

ot

h

e

r

s

.

O

n

t

h

e co

n

t

r

ary to

m

ost

h

eteroge

n

eous

m

ed

i

u

m

, b

i

olog

i

ca

l

tissues a

r

e

li

v

in

g

d

y

n

a

m

ic st

ru

ctures

. Th

ey ex

p

e

ri

e

n

ce bot

h

l

oca

l

co

n

vect

i

o

n in

s

i

de vessels, d

iff

us

i

o

n in

co

mp

l

ex structures

in

add

i

t

i

o

n

w

i

t

h r

eact

i

o

n r

ates

in

s

i

de

t

issue

. Th

e co

m

p

l

ex

i

ty o

f m

i

cr

o

vascu

l

a

t

ure

i

s a serious

hin

d

r

a

n

ce

for

ex

h

aust

i

vely

t

ak

in

g

int

o

accou

n

t t

h

e role o

f

t

h

e vascu

l

a

r

excha

n

ges

i

n b

i

o

h

eat t

r

a

n

s

fer.

Th

is is w

h

y va

ri

ous

m

ode

l

s

h

ave bee

n pro

posed

t

o app

ro

x

im

ate

l

y

desc

ri

be b

i

o heat exc

h

a

n

ges

[1 5)

. Th

is is es

p

ec

i

a

l

ly t

ru

e fo

r

co

n

vec

t

i

o

n

w

i

t

hin p

ara

ll

e

l

vesse

l

s,

th

e

r

eleva

n

ce o

f

w

h

i

ch ca

n

be fou

nd

in m

a

n

y

p

hys

i

olog

i

ca

l

co

n

texts (e

.

g

. m

uscles, bo

n

es, etc

...

)

[6 9)

.

Quo

tin

g

[9)

, 's

in

ce a co

n

s

i

de

r

ab

l

e

fr

act

i

o

n

o

f

b

l

o

od

vesse

l

s are

fou

nd in

pa

irs

vesse

l

vesse

l

heat t

r

a

n

sfe

r

has ge

n

era

ll

y bee

n p

ostu

l

ate

d

as one o

f

t

h

e

m

ost

imp

o

r

ta

n

t

h

eat

tr

a

ns

fe

r m

ec

h

a

n

is

m

s

in

vo

l

ved

i

n dete

rminin

g t

h

e t

i

ssue te

mp

e

r

ature dist

ri

but

i

o

n

s'

[10 12)

. Th

is

w

h

y t

h

e case o

f p

a

r

a

ll

e

l

vesse

l

s bu

nd

l

es

h

as

receive

d

so

m

e atte

n

t

i

o

n

[9,6 8)

.

In

a broade

r

co

nt

ex

t h

eat t

ran

sfe

r in t

ube bu

nd

l

es

h

as also bee

n

p

r

evious

l

y

in

vestigated

in

t

h

e sea

r

c

h

fo

r

o

p

t

im

a

l

t

r

a

n

sfe

r

co

nfi

gu

rat

i

o

n

s

. F

o

r n

atu

r

a

l

co

n

vect

i

o

n

[13)

fou

nd in

te

r

es

tin

g sca

lin

g

l

aws

*

Corresponding author.

E-mail addresses: jules.dichamp@imft.fr

U

-Dichamp

), frederic.de-gournay@insa

-toulouse.fr (F. De Goumay), franck.plouraboue@imft.fr (F. Plouraboue).

fo

r

o

p

t

i

ma

l

s

p

a

cin

g betwee

n t

ubes

. F

urt

h

e

rm

ore, w

h

e

n

so

m

e

ex

t

e

r

na

l

for

ced

fl

ows is ap

pli

ed a

ro

u

nd th

e

t

ube bu

nd

l

es (co

n

s

i

d

e

r

ed has soli

d

sou

rc

es) a

r

at

h

e

r imp

o

rt

a

n

t body o

f li

te

r

atu

r

e ca

n

be

fo

u

nd

, e

.

g.

[14,15)

. O

the

r m

o

r

e co

m

p

l

ex co

nfi

gu

r

at

i

o

n

fo

r

o

p

t

i

m

a

l

t

r

a

n

s

fer

has a

l

so bee

n in

vestigate

d

such as trees

[16)

o

r for

p

u

l

sated

fl

ows

in

tubes

[17)

.

An

a

lt

e

r

nat

i

ve a

pp

r

oac

h

to add

r

ess a

n

d s

implif

y heat t

r

a

n

s

fer in

tissue, is

t

o

r

ea

li

ze t

h

at

n

ot eve

ry

vesse

l

pa

r

t

i

cip

ates to

h

eat

excha

n

ges

. Thi

s is

in

dee

d

k

n

ow

n

t

h

at,

f

ro

m

co

n

vect

i

o

n

do

min

ate

d

a

rt

e

r

i

a

l

in

l

ets, w

h

e

n

p

ro

g

r

ess

in

g dow

n in

to

th

e vascu

l

a

r

t

h

ree, t

h

e

h

eat

fl

ux (vesse

l

to tissue) g

ra

dua

ll

y s

hrin

ks dow

n

wa

rd

so as to

r

each equ

ili

b

ri

u

m

w

i

t

h

t

h

e su

rr

ou

nd

i

ng t

i

ssue

. Th

is s

imp

li

fled p

ictu

r

e,

m

i

g

ht n

o

t

e

x

ac

tl

y tu

m

out to be

tru

e is so

m

e ex

tr

e

m

e

case, w

h

e

n t

he

m

etabolic

pr

oduct

i

o

n

a

n

d/o

r

co

n

su

mp

t

i

o

n

i

ns

i

de

t

h

e tissue

prod

uces te

mp

e

r

ature g

r

a

d

i

e

n

ts associated w

ith

'

h

ot'

h

eat sou

rc

es o

r

s

in

ks

. H

oweve

r

, we w

ill

see

in

t

h

e

foll

ow

in

g

t

hat,

in m

ost

ph

ys

i

o

l

ogica

ll

y

r

eleva

n

t s

i

tuat

i

o

n

s,

h

eat p

r

oduct

i

o

n

/co

n

su

mp

t

i

o

n from

t

h

e t

i

ssue

m

etabolis

m h

as a

n

e

gli

g

i

ble

infl

ue

n

ce

o

n h

eat

fl

uxes

. H

e

n

ce,

in m

ost

t

issue

h

eat t

r

a

n

s

fer

is

m

os

tl

y geo

m

etrica

ll

y (e

.

g

.

, re

l

ated to the size, s

h

ape, le

n

gt

h

a

nd

dista

n

ces

betwee

n

vesse

l

s) a

nd th

e

rm

a

ll

y(e

.

g

.

, associated w

i

t

h th

e

in

l

et, ou

t

let a

n

d

r

e

l

at

i

ve blood

fl

ows) co

n

t

roll

ed

. In

t

h

is co

n

text,

i

t

i

s

in

te

r

est

in

g

t

o be ab

l

e to k

n

ow, a p

ri

o

ry

, w

h

e

n

a vesse

l

w

ill m

a

inl

y

eq

u

ili

b

r

ate

i

ts

h

eat excha

n

ges w

i

t

h

t

h

e su

r

rou

ndin

g tissue so as

to be able to

in

fe

r

, w

h

ic

h

vesse

l

s are t

h

e

rm

a

ll

y s

i

gn

ifi

ca

n

t.

Th

e

rm

a

l

s

i

g

nifi

ca

n

ce,

h

as bee

n

discussed

q

uite abu

nd

a

n

t

l

y

in

t

h

e

li

te

r

atu

r

e

[18,4,19 23)

.

S

in

ce, ge

n

e

r

ica

ll

y, t

h

e loca

l

t

ran

s

fer

(3)

rate from the fluid into the solid is found to abruptly decay from

the inlet along the longitudinal direction[24,25], it has been con

sidered that some exponential decay with typical decaying length provide the typical length associated with thermal relaxation. This phenomenological choice is indeed justified in the case of parallel tube exchangers for which it is proven that generalized Graetz decomposition with an infinite set of exponentially decaying

modes in the longitudinal direction holds[26 30]. Since thermal

significance has also been investigated with other criteria than

relaxation length, i. e arterial efficiency (Cf Section2.4) we wish

to analyze and compare them so as to get a more comprehensive analysis of the interest and validity of thermal significance.

Hence, in this paper, we analyze various distinct features of heat transfer in vessels bundles, associated with thermal significance, and optimal transfer using a quasi analytic approach previously

described in[29,30]. In Section2 the set of governing equations

and their dimensionless formulation are provided. Some insights about the parameters used and their physiological relevance is dis

cussed in Section 2.4. In Section 3 we first analyze unbalanced

counter current configuration for transfer. We first investigate

the influence of vessels radius in Section3.1so as to show that,

when vessels are few diameter apart, this parameter is intrinsically irrelevant, being mainly taken care of by Péclet numbers ratio. Then, we pursue the analysis of thermal significance through the

analysis of arterial effectiveness in Section3.2as well as thermal

equilibrium length in Section3.3 versus relevant dimensionless

parameters. Finally we analyze heat transfer in Section 3.4 for

which optimal configurations are brought to the fore.

2. Governing problem and dimensionless formulation 2.1. Configuration under study

In this paper we wish to analyze the vessel/tissue transfer in

idealized bundles of parallel vessels, as depicted inFigs. 1 and 2.

The tissue/vessel system is modeled as a transversely infinite exchanger into which a pattern of periodic parallel vessels trans port heat. This choice has been made so as to gain insight into the exchanges independently of the transverse boundary condi

tions as opposed to[30]. To be more precise on the terminology

that we will use in the following we will refer to the exchanger region as the finite (but transversely periodic) domain within

0; Ls

½ 0; Ls½ 0; Le½ depicted in Fig. 1. Hence, the tissue model under study has a finite extent in the longitudinal direction, but needs boundary conditions for the heat at tubes inlets and outlets. In order to circumvent the problem of arbitrarily setting such

boundary conditions at z 0 and z Le, we connect the finite tis

sue exchanger to infinite inlet and outlet reservoirs connecting

tubes, as in [28,30]. Hence three compartments are considered.

The inlets and outlets where convection diffusion arises in the ves sels only, and the tissue exchanger.

As in[30], we consider laminar convection diffusion arising in a

fluid having constant properties. In the following we consider tubular vessels having circular sections. The artery is considered as the hot reference whilst the vein outlet provide the cold one (most results hereby presented do not depend on this arbitrary choice). Fully developed Poiseuille longitudinal velocity is pre

scribed in each vessels

v

a

v

að Þezr 2 Va 1 ðr=RaÞ2

ez and

v

vð Þezr 2 Vv1 ðr=RvÞ2ezwhere Va respectively Vv

and Ra respectively Rv stands for the average velocity and

radius of the artery respectively vein and ezis the unit vector

along z direction. The ’hot’ arterial vessel refers to the tube with

homogeneous inlet temperature at plus infinity Tþ1_a and the ’cold’

vessel tube refers to the tube with homogeneous inlet temperature at minus infinity T_v1as represented onFig. 1.

In most of the following we will consider unbalanced counter current configuration where longitudinal velocities are different

such that VaP Vvsince this is mostly what arises in most physio

logical contexts. In this work we investigate three family of geo metrical parameters: the arterial and vein radius, their length and their distances. Keeping with dimensionless geometrical

parameters this leads to radius ratio RH Rv=Ra, vessel’s aspect

ratio LH_Le=Ra and the dimensionless distance between vessels

a

H_{Ra=R0. In the chosen configuration associated with periodic}

bundles of vessels, the last parameter is directly related with the vessels surface density also equals to the volumic vascular density, i.e. vessel’s volume over total volume, denoted/ 4pRH 2

LHs

2 (note that

this volumic vascular density also equals the transverse surfacic vascular density, since the hereby considered configuration is

invariant along z direction). The relation between_{/ &}

a

H _reads

/_{ð Þ}

a

H p aH2

2 and is represented inFig. 2.

Nomenclature

Le tissue exchanger length

Ls period of the transverse tissue exchanger domain length

LH Le=Ra dimensionless vessels length

Ra arterial radius

Rv venous radius

R0 distance from the center of R a the bundle to the center

of vessels

a

H ratio Ra=R0

/

pa

2_{=2 vascular density}

Xa arterial transverse fluid domain

Xv venous transverse fluid domain

Xt tissue transverse solid domain

X0

t tissue transverse sold domain at z 0

XLe

t tissue transverse sold domain at z Le

@Xext

a external arterial boundary (z 0 and z Le)

@Xext

v external venous boundary (z 0 and z Le)

kF fluid thermal conductivity

kT tissue thermal conductivity

c heat capacity

P metabolic heat production

Nua Nusselt number in artery

Nuv Nusselt number in vein

a arterial effectiveness

‘eq thermal equilibrium length

q

fluid density

Va arterial averaged blood velocity

Vv venous averaged blood velocity

Qa arterial blood flow

Q v venous blood flow

Pea Péclet number in artery

Pev Péclet number in vein

Ti temperature field of phase i

Ti average transverse temperature of phase i

T1_i homogeneous temperature at infinity of phase i

i either arterial a (fluid), venous

v

(fluid) or tissue t

(4)

X

L;

Fig. 1. Schematic layout of the elementary tissue geometry unit defining the tissue per-fused by vessels bundles and the various important parameters associated with it: 4 is the tissue/exchanger length, whilst I., is its period in transverse directions, both x and y.

o

?,

d,'

denote the tissues sectional areas on exchanger's edges respectively on z 0 and z L,.

8't;'

and

&rr,

denote the vessels sectional area outside the exchanger, i.e. within z e [-oo, OJ u [4, +ooJ. T;"' and T

.®

are the temperature imposed in some 'far field' reservoir in arteries and veins.

r.

® and r

;»

are the temper-ature resulting ( and thus numeriG11ly evaluated) from the exchange inside the tissue/exchanger and are not specified. r • • •• ... '"· , ... •• • • • • • •• •• • • • ·• •r • • • •• •• • • ' ' I , ••· . . , • • - .. I \

....

I \

·-·

I

• ---

-

--

--- ~ - - - -~

{

;

-

-. -.

.

:

-,···

: I

...

'

..

.

, I : ,

-.

-• I

-

-,

-

-:

;

0 . 8 ~ - - - ~

0.7 ~ c 0.6

'g/

0.5 (I) -0

_....

0.4 ... .

ro

"S 0.3

u

~ 0.2

>

0.1 ...

;

0.3

0 .

4 a

*

0.7 0.8

Fig. 2. Schematic layout of the transverse section of the tissue with periodic bundle of vessels. Ra and R. are the arterial and venous radius, Ro is the center-to-center distance between vessels and tissue. O. stands for the arterial,

o

.

for the vein and

0, for tissue domain's sectional area inside the tissue exchanger. I., is the period of the transverse domain in both x and y directions. The graph below shows the relation between transverse normalized distance ex and the vascular density ,J,

=*'

/2.

2.2.

Go

v

erning problem

Th

en, we w

ri

te t

h

e stat

i

o

n

ary t

h

e

r

ma

l

e

n

e

r

gy ba

l

a

n

ce

i

n t

hr

ee

dimens

i

ons as

pc

v

• VT

It

,

H

0 in

vesse

l

s

k

7

!l.T

P

in

soli

d

(1)

with

k

F

being the fl

u

i

d conductivity, k

7

_{the tissue one,}

_p

_{the fluid}

density, c the heat capacity and

P

the metabolic heat production

inside the tissue

[9]

. T

he above problem reads more explicitly for

longitudinally invarian

t

velocity convection

v

(

x

,

y

)

e,.,

p

cv

(

x,y

)a

,T k

F(a;+a;+ti,)

T

0 in

vesse

l

s

k

7

(a;

+a; +a;)

T P

in

soli

d

(2)

Th

en co

n

t

in

u

i

ty o

f

tempera

t

ure a

nd

fluxes at vesse

l

s

/

tissue

in

te

r

faces

an.

(

fo

r

a

r

tery su

r

face

)

a

n

d

an

.

(

fo

r

ve

i

n su

r

face

)

, t

h

at

we

n

o

t

e here

a

n.

,

v

read as

(3)

Tlao.,. Tlao,

k

F

a

.T

i

an.

,

.

k

7

a

.T

i

an,

On the external part of the exchanger config

u

ration, i.e

.

for

z

~

0 or

z

~ L,

we consider homogeneous Neumann conditions as in

[3

0 ]

(4)

F

o

r

t

h

e edges o

f

the tissue doma

in

o

n

z

0 and

z

Le

we co

n

s

i

der homogeneous ad

i

abatic Neuma

nn

co

n

d

i

t

i

ons,

(5)

At infinity, in the artery t

u

be section

0a

and in the vein section

n

v

,

temperat

u

res are imposed

Tl0.(x,y,z-+ +oo) Tl0.(x,y,z-+ oo)

T

:'°

T

_"",

2.3. Dimensionless fonnulation

(6)

F

o

ll

ow

i

ng p

r

ev

i

ous stud

i

es

(

e

.

g

.

[31

])

,

J

et us

n

ow use d

i

men

s

i

on

l

ess va

r

i

ab

l

e w

i

t

h

*

in

dex defined as x

R,,x*

,

Y

R.y*

,

z

R,,z*

,

L

,

LR,,*

,

Va

2 V

0

v:

,

Vv

2 V

.v!, a

n

d

_

T

__

T

~

_,

-

T T

v'"'

T. +

00

T

00

+

T

_,

T

00_•

V a V

(7)

Now, using the d

i

mensionless position ~*

(

x*

,

y*

)

in the

(

x

,

y

)

p

l

ane

(2)

now reads

( a;.

+

a;

..

+

a;.)

r*

0 ,

in

Vessels

(a;.+

a;

..

+

a;.

)

r*

P

*

,

in

T

issue

(8)

with,

P

ecle

t

n

u

mber being

Pe

a.,,

2pcV

a

,

vR

a,v/

t,

and dimensionless

metabolic so

u

rce term

(5)

PH P R 2

kt Tþ1a Tv1

: ð9Þ

At the interface between vessels_@Xvand tissue edge_@Xtinside the exchangers one has continuity of temperature and fluxes

THj_@XH a;v T H_j @XH t kF@nTHj_@XH a;v k T @nTHj_@XH t: ð10Þ Due to the choice of dimensionless temperature the imposed cold and hot sources verify

THj_XH a n H_{; z}H_{! þ1} THþ1a 1 in arterial inlet THj_XH v nH; zH! 1 TH 1_v 1 in venous inlet ð11Þ

2.4. Dimensionless parameters and thermally significant numbers Dimensionless formulation thus brings seven different dimen

sionless parameters: two hot and cold Péclet numbers

Pea;v 2

q

cVa;vRa;v=kF, the conductivity ratio between the tissue and the blood kT=kF

, vessels aspect ratio LH Le=Ra, vessels radius

ratio RH R_v=Ra, dimensionless distance

a

H _Ra=R0_{and finally the}

dimensionless metabolic rate PH P R2=kt Tþ1a Tv1

. In order to more closely delimit physiologically relevant parameter space, we prescribe a conductivity ratio kT=kF

1, since it is generally very

close to one in most tissues[32]. Furthermore, using data obtained

from [9] for muscle where kt 0:5 W K 1

m 1_{; R} ₅₀₀

l

_m,

Tþ1a 37C, assuming outlet venous temperature Tþ1v 35C, as

well as P 675 W m 3_{for hypothermia and P} _{97000 W m} 3

for hyperthermia, we found the following range: PH

1:7 10 4 ₂_{:4 10} 2_.

Hence, in the following we neglect the influence of the meta

bolic source term PH. The significant parameter space dimension

is thus five, rather than seven, associated with Pea; Pev; LH;

a

H and

RH. Hence, we mainly focus our interest toward physiologically rel

evant parameters for which Pevis fixed at various value between 5

to 20, whilst varying Peabetween 5 to several hundreds. We explore the effect of LH_;

a

H_{and R}H_{on the vessels thermally signifi} cance as well as heat transfer.

Three additional dimensionless parameter of interest are also resulting from the considered configuration for evaluation of thermal significance. Let us first define spatial averaging (more

precisely dimensionless surface averaging) using notation

Ti R_X

iTdXi= R

XidXi, with index i a;

v

; t respectively associated

withXathe arterial,Xvvenous andXttissue domain’s transverse

sections, as depicted inFig. 2. Then, the arterial efficiency

ahas

been defined and used[18,4,19 21],

a TaHðzH 0Þ TtHðzH 0Þ TH;þ1a TH; 1v TaHðzH 0Þ TtHðzH 0Þ 2 ð12Þ Secondly, thermal significance has also been related to the expo

nential decay of the temperature along vessels [22,23] through

the vessels thermal equilibrium length ‘H

eq. It is here defined as the length for which the average temperature decays by an expo nential factor from the arterial entrance (here in zH _LH₎ THt ‘Heq 1 e 1 THt LH : ð13Þ

For this parameter to be independent of longitudinal length it must be expressed as a fraction of the exchanger’s length, we will thus consider the ratio‘H_eq=LH

Thirdly, we consider the vessel’s surface heat flux through the

Nusselt number derived from dimensionless formulation as in[30]

Nua;v 2 Z LH 0 dzH Z @Xa;v@n HTHdXa;v ð14Þ

where@Xa;v is the edge of arterial and vein (which are circles) and d@Xa;v is the integration element along those edges.

2.4.1. Biological relevance of the parameter choices

Since most studies related to bioheat do not consider dimen sionless numbers, it is interesting to evaluate values arterial and

venous Péclet on different physiological contexts.Table 1provides

these figures evaluated from thermal measurements performed on two species in different organs. Péclet values range from 0.23 to 263.66 for the arterial part of the circulation and from 0.07 to 118.68 for the venous one. Typical ratio between arterial and venous range from 2.22 to 3.29. In most of the results presented

in forthcoming sections, we considered venous Péclet Pev 5

and 10 associated with the small/intermediate part of the circula tion. This choice is also motivated by trying to infer the influence of venous Péclet on other parameters by doubling its value. Further more, the arterial to venous Péclet range is chosen between 1 to 20.

This is higher than in the reported values ofTable 1but one has to

born in mind that these measurements are obviously limited, and do not cover the entire range of physiologically relevant situations (e.g. arterioles and arteries vasodilate so as to increase blood flow leading to higher Péclet ratios). In any case, studying the limit of high Péclet ratio is interesting to understand the asymptotic behavior of vascular bundles.

Finally it is interesting to mention that vessels density/ in most

tissues rarely reach values larger than 20%, except for special con

texts. Nevertheless, it can be locally very large in the proximity of arterial and venule pairs. For being able to describe both generic situations and special cases, the values chosen for/ and

a

are illus tratedFig. 2.

2.5. General comments on the formulation and numerical method The details of the numerical method used in this contribution

has already been detailed in[28], as well as briefly described and

used in[30].[30] also provides several numerical validations of

the method. The main idea of the numerical approach is to sepa rate the whole problem into three distinct Graetz problems respec tively associated with inlet tubes, outlet tubes and the exchanger (containing both tissue and fluid domains). Each Graetz problem follows a different expression of the dimensionless temperature field since there is a different spectral decomposition. The ampli tudes of Graetz are then coupled together into a linear system which is set so as to ’match’ the continuity of temperature and fluxes at the exchanger inlets and outlets. A detailed description of the method can be found in[28]. Let us just recall the theoretical

Table 1

Physiological values used for Péclet computation assuming that c 3651 J/(kg _C), kF _{0:51 W/(m}_{C) and}_q _{1046 kg/m}3

[32].

Physiological context Dð m)l V (mm/s) Pe

Human retina (venous bifurcations) [33] 125.9 ± 26.44 8.63 ± 2.33 8.5 ± 3.81 Dog mesenteric[34]

Main venous branches 2511.89 6.31 118.68

Terminal veins 1445.44 5.58 60.4

Venules 32.11 0.29 0.07

Main artery branches 1063.33 33.11 263.66

Terminal branches 611.88 35.21 161.33

(6)

ex

pr

ess

i

o

n

s o

f

t

h

e d

im

ens

i

o

nl

ess te

m

pe

r

atu

r

e fie

l

ds

in

eac

h

do

m

a

in.

F

o

r

sake o

f

s

im

plicity we w

ill

j

ust w

ri

te

th

e express

i

o

n

s

fo

r

o

n

e arte

r

i

a

l

a

n

d o

n

e ve

n

ous do

m

a

in

s a

l

t

h

oug

h

t

h

e

r

e a

r

e fou

r

o

f

eac

h in

ou

r

co

nfi

gurat

i

o

n.

T"'W

,

z

*

)

I:

xtT~

(

t

)e-t

tz*

+

xnrn

(

{

*

)

e

J.,,(

z*

L*

)

z

*

€

[

O

,

L

*

]

N'

T"'

(

{

*

,

z

*

)

x;;

+

I:

x

~

t

~(

{

*

)

e

µ

t

(

z*

t*

)

v

e

i

n

z

*

;;;,:

L

*

N•

T"'

(

~

*,

z

*

)

~

+

I::

x.u.W

)

e

Y

,

z* a

rt

e

ry

z

*

,;;;

O

N•

T"'

(

~

*,

z

*

)

r:

00

+

L

}ntn

(

~

*

)

e

11n

1 "

ve

i

n

z

*

,;;;

O

N'

T"'

(

~

*

,

z

*

)

r;

+

00

₊

_I::

_x

_~

_u

_~

₍

_~

_*

₎

_e

_Yt

₍

_z*

_t*

₎

_a

_rtery

_z

_*

_;;;,:

_L*

N'

(

15 )

wi

t

h

x

;

amplitudes of modes,

T

;

,

t

;

and u

;

eigenvectors,

,1,

;

,

µ

;

and

Y

;

eigenva

l

ues respective

l

y of exchanger doma

i

n, venous domain

and arterial doma

i

n

.

A

s

p

r

evious

l

y

m

e

n

t

i

o

n

ed, d

im

e

ns

i

o

n

l

ess

inl

et te

mp

e

r

atures

r:

+oo

a

nd

r;

00

va

l

ues a

r

e res

p

ect

i

ve

l

y

1 et

1 a

n

d u

n

k

n

ow

n

s

x

₀

,

~

a

r

e co

m

puted by t

h

e

m

et

h

od as

r

espec

ti

ve

l

y t

h

e

h

o

m

o

ge

n

eous te

m

pe

ra

tu

r

e

fi

e

l

ds a

t min

us

in

fi

ni

ty fo

r

t

h

e a

r

te

r

i

a

l

vessels

a

nd pl

us

in

fi

ni

ty

for

t

h

e ve

n

ous vessels

.

3. R

e

su

l

ts

3.1. Influence of

venous radius

H

e

r

e, we fi

r

st expose

h

ow vesse

l

's rad

i

us

infl

ue

n

ces

h

eat

tr

a

n

s

fe

r.

Fig. 3

p

rov

i

des t

h

e a

r

te

ri

a

l

t

r

a

n

sfe

r

rate de

p

e

nd

e

n

ce ve

rs

us t

h

e

a

rt

e

ri

a

l

t

o ve

n

ous

P

e

cl

e

t r

at

i

o Pea

/

Pe

v

, w

hi

l

st ot

h

e

r

pa

r

a

m

ete

r

s

be

in

g

p

r

esc

ri

bed

.

We

in

vestigate two di

ff

ere

n

t

p

a

r

a

m

eters t

h

at

a

r

e fixed t

hro

ug

h

c

h

a

n

ge o

f r

ad

i

us va

r

i

at

i

o

n.

We fi

r

st fix t

h

e ve

n

ous

P

eclet

n

u

m

be

r

Pe

v

as co

mp

a

r

ed to a refere

n

ce s

im

ulat

i

o

n

w

i

t

h

R!

"

'

K:,

1

_/

_Ra

₁

_in

_{Fig. 3}

_a

_.

_{We t}

_h

_{en fix, t}

_h

_e

_fl

_ux

_r

_at

_i

_o

Q

*

=

Q

./

Q

':

1

₍

_Pe

_v

_R

_v)!

_{( Pe}

_':

1

_R

_':

1₎

_{is kept co}

_ns

_ta

_n

_t

_i

_n

_{Fig. 3}

_b

_{. In}

t

h

e

l

ate

r

, a

m

aste

r

cu

rv

e

for

tra

n

s

fer

is obse

rv

e

d

ove

r

a

r

at

h

e

r l

arge

ra

n

ge o

f P

eclet

r

at

i

o Pea

/

Pe

v

,

r

each

in

g a

l

ate

pl

ateau

for l

arge

Pe,,

/

Pe

v

va

l

ues,

h

av

in

g a weak de

p

e

nd

e

n

ce o

n

R

*

(

8% va

r

i

at

i

o

n

s

a

r

e observed

in

Fig. 3

b w

h

e

n

va

ryin

g R

*

by 100%

)

. A

s expected,

fo

r

a g

i

ve

n

vesse

l

e

n

gt

h

as

p

ect

r

at

i

o

L

*

,

tr

a

n

sfe

r

e

ff

ec

ti

ve

n

ess sat

ura

t

es w

h

e

n in

creas

in

g t

h

e arte

ri

a

l

P

eclet

n

u

m

be

r

, so as to

r

each a

p

la

t

eau

for

w

h

ic

h

t

h

ere is

n

o ga

in

fo

r

t

r

a

n

s

fer in in

c

r

eas

in

g b

l

o

od

fl

ow fu

rth

e

r. Th

e va

l

ue o

f

t

h

is

p

la

t

eau is re

l

ated to L

*

, but is a

l

so

70,---,---,---,---;::::=::;:::==:::;-,

n

,.=

1.0 G5 ~ R,.

=

I.I 60

R

,.=

L.2 • • R,.=1.4 55 ; . , , - , · ,,. •· ·• /4,

=

2.0

i.

50

t

r

~

__

....

_

____

.,,

45

;

!'

~·

..

· ..

~

.. •

~

...

11

· '

,

.·

.

...---40 -•~_.: : . - · 1:1'

3s

r

_.

.

_,

,

30 ₀

_\---~5--~

₁

_~0--~

₁

_{~5----='20~---='}

₂

₅

_·

a

)

Pe,.f Pev

re

l

ated

t

o

t

he t

h

e

rm

a

l

equ

ili

br

i

u

m

le

n

gt

h

ieq

subsequent

l

y stud

i

ed

in

Sect

i

o

n

3.3 . Thi

s obse

rv

at

i

o

n in

d

ic

ates t

h

at t

h

e

infl

ue

n

ce o

f

R*

ca

n

be

r

ecasted

i

nto

th

e

P

e

cl

et ra

t

i

o Pea

/

Pe

.

va

ri

at

i

o

n

s, as

l

o

n

g as

Q

*

is p

r

es

cri

bed

. H

e

n

ce, o

n

e ca

n r

educe pa

r

a

m

ete

r

s

p

ace d

im

e

n

s

i

o

n

by o

n

e,

fr

o

m

fi

f

t

h

to fou

r

s

in

ce t

h

e de

p

e

n

da

n

ce w

ith

R*

is s

l

ave

to a given va

l

ue Q

*

=

Q

./

Q. Pea

/(

Pe

.

R*

)

.

We

n

ow fu

r

t

h

e

r

exa

m

i

n

ate the

rm

a

l

s

i

g

nifi

ca

n

ce a

nd

heat t

r

a

n

sfe

r

e

ff

ectiveness

de

p

e

n

da

n

ce w

i

t

h

Pea

,

Pe

.

,

rx

*

a

n

d L

*.

3.2. Arterial effectiveness evaluation

Co

n

s

i

de

rin

g t

h

at ave

r

age

t

e

mp

e

r

atu

r

e is a

fi

gure o

f in

terest, t

h

e

a

rt

e

ri

a

l

e

ff

ec

ti

ve

n

ess

€a

defined

in

(12)

p

ro

v

i

des a

n

est

im

ate o

f th

e

re

l

axat

i

o

n

towa

rd eq

ui

li

b

r

i

u

m

between vesse

l

s a

nd

t

i

ssues

. Th

e

greate

r

€a

,

th

e

l

a

r

ge

r

the exc

h

a

n

ges, a

nd t

he bette

r

t

h

e

rm

a

l

e

ff

ec

t

i

ve

n

ess

.

He

n

ce, s

im

ila

rl

y

wi

t

h

t

h

e

h

eat t

r

a

ns

fe

r p

e

rf

o

rm

a

n

ces, t

h

is

average e

ff

ect

i

veness

€a

m

o

n

oto

n

ica

ll

y levels o

ff

fo

r in

c

r

eas

in

g

a

rt

e

ri

a

l

co

n

vect

i

o

n. In

Fig. 4

,

fo

r

fixed Pe

.

, as Pea

in

creases ove

r

a

w

i

de

r

a

n

ge,

€a

g

ro

ws towa

r

d a syste

m

atic saturat

i

o

n for h

ig

hl

y

co

n

vect

i

ve

tr

a

n

sfe

r r

eg

im

e

. N

ot su

r

p

r

is

in

g

l

y t

h

e p

r

ecise va

l

ue o

f

venous

P

eclet

n

u

m

be

r

Pe

.

does

n

ot

infl

ue

n

ce

m

uch t

h

e satu

r

ated

va

l

ues o

f

€

a,

s

in

ce a

n

uppe

rm

ost ga

in

o

f

20% is obse

rv

ed co

m

pa

rin

g

Fig. 4

a b fo

r t

wice a va

l

ue

in

Pe

v

(

sa

m

e co

mm

ent app

li

es to t

h

e

co

m

pa

r

iso

n

o

f

Fig. 4

c & d

. Th

is resu

l

t ca

n

be exp

l

a

in

e

d

by t

h

e fact

t

h

at t

i

ssue

/

vesse

l

s excha

n

ges are do

min

ated by arte

r

i

a

l

/

tissue

fl

uxes

r

at

h

e

r th

a

n

ve

n

ous/

t

issues o

n

es w

h

e

n th

e a

rt

e

ri

a

l

P

eclet

Pe,,

i

s

m

uch

l

arge

r

t

h

a

n

ve

n

ous o

n

e

(

i.

e

.

w

h

e

n

Pea

/

Pe

v

»

1 )

. If

t

ru

e,

t

h

is

in

te

rp

r

e

t

at

i

o

n im

plies t

h

a

t

,

for

a

lm

ost equ

ili

b

r

ated s

i

tuat

i

o

ns

w

h

e

n

Pea

/

Pe

v

is

cl

ose

t

o o

n

e, t

h

e p

r

ecise va

l

ue o

f

Pe

v

s

h

ould

m

at

te

r. Thi

s is co

nfirm

e

d

by t

h

e observat

i

o

n

t

h

at t

h

e co

r

res

p

o

n

d

in

g

€a

a

lm

ost doub

l

es

for

twice a va

l

ue

in

Pe

.

w

h

e

n

Pea

/

Pe

v

~

1 co

m

pa

r

in

g

Fig. 4

a b as well as

Fig.

4c

& d

. N

ow co

n

s

i

de

rin

g t

h

e se

n

sit

i

v

i

ty

o

f

t

h

e a

rt

e

r

i

a

l

e

ff

ect

i

veness

€a

t

o t

h

e vessels as

p

ect

r

at

i

o

L

*

,

one ca

n

ote

t

hat w

h

e

n

L

*

is

m

u

l

t

i

p

li

ed by fou

r

co

mp

a

rin

g

Fig.

4a

& c o

r

s

im

ila

rl

y

Fig. 4

b & d,

€a

ba

r

e

l

y cha

n

ges by

m

o

r

e t

han

20% ove

r

t

h

e a

ll

ra

n

ge o

f P

e

cl

et

r

at

i

o

. I

t

i

s

int

e

r

est

in

g to

n

ote t

h

at, t

h

e u

n

sen

s

i

t

i

v

i

ty o

f

€a

t

o L

*

d

rastica

ll

y

in

c

r

eases as rx

*

aug

m

e

n

ts

. In

deed,

in

t

h

e

m

ost '

dil

ut

ed

' case,

wh

e

n

rx

*

0.2, o

n

e ca

n

clea

rl

y obse

rv

e a

cha

n

ge between bu

ll

et cu

rv

es o

f

Fig.

4a

& c

. H

oweve

r

, fo

r l

a

r

ge

r

va

l

ues o

f

rx

*

, cha

n

ges

in

€

a

f

ro

m

Fig.

4a

c a

n

d

Fig. 4

b d are very

weak.

Th

is ca

n

be ex

p

l

a

in

ed by

th

e existe

n

ce o

f h

eat

t

ra

n

sfe

r

"

scree

nin

g lengt

h

" associated w

i

t

h th

e t

r

a

n

sve

r

se d

i

sta

n

ce

between vessels, give

n

by d

im

e

n

s

i

o

n

l

ess

n

u

m

be

r

rx

*.

As

rx

*

decreases,

th

e

t

ra

n

sve

r

se

l

e

n

gt

h

betwee

n

vessels beco

m

es

t

he p

r

e

do

min

a

n

t le

n

gt

h

sca

l

e fo

r

tra

n

sfe

r

, w

h

i

ch 's

cr

eens' the

infl

uence o

f

l

o

n

g

i

tud

in

a

l

e

n

g

th.

5 0 ~ - - - ~ - - ~

•··• R.,

=

1.0 R.,= I.I R., -1.2 • • R., = 1.4 I?., - 2.0

3%

!-

~5-~

l

~

0 - ~

1 ~5-

~

2

0 ;,---2~5~,.:.=::::::;-=~4

0 :;::::~4

5 b

)

Pe«/Pe.

Fig. 3. Arterial Nusselt number versus Peclet ratio Pea/Pe. compared to reference simulation (R';1

Ra

,

i.e. R" 1} and L" 10,a:* 02, for {a) fixed venous Peclet

(Pe.

w

:t

10), {b) fixed dimensionless venous 0ux Q*"' Q./Q':' (Pe.R.)/(W:'K:') and

w

:t

10. For fixed dimensionless 0ux ratio Q* in {b) a master curve is

(7)

0.30 0.40 0.25 0.35

.

. . . =

•• ••...,~ II 111 I IIIEO.-.. 0.30

.-·

~

·

0.20

"

!'

.

l

,

• ""

₀_.₁₅

:

.

'i'>

..

a,•- 0.2 ·b

.

·•

a• - 0.2 o• = 0.3 a' =0.3 0.10

.

ex" - 0.5 _0.₁₅ o." = 0.5

• _{• •}

_o'₌_0.6

• _{• •}

_a'₌_0.6

o.oq

₁ ₅ ₁₀ ₁₅ ₂₀ 0.100 ₅ ₁₀ ₁₅ ₂₀ ₂₅

a)

P

ea/

P

ev

_b

₎

P

ea/

Pe

.,

0.30 0.35

...

~""....,.,

0.25

...

···•

...

0.20

...

,,,

...

•. ••

..J

0.15

.

_~

.

,.

0.10 a,•= 0.2 a"'= 0.2 a,'- 0.3 a' - 0.3 0.05

.

O'= 0.5 a'= 0.5

• _{• •}

a,•= 0.6

•

a• =0.6 0.000 ₅ ₁₀ ₁₅ ₂₀ 0.050

•

₅ ₁₀ ₁₅ ₂₀ ₂₅

c

)

P

e«/

P

ev

_d)

P

e,,/

P

ev

Fig. 4. Arterial effectiveness~ versus Peclet ratio Pe0/Pe. for several distances between vessels a* 0.2,0.3,0.5,0.6, venous Peclet Pe. and vessels aspect ratio L* : {a) Pe. 5,L* 5, {b) Pe. 10,L* 5, {c) Pe. 5,L* 20, {d) Pe. 10,L* 20.

0.30

...

0.40

...

0.35

.,,...

...

0.25

..

.

..

~

,.

0.30

.

0.20 !'

_.,

.

...

.

0.Z::,

..

.

,

.

.:J

015

_"

""

,

.

,

..

0. HJ L· - 5.0

i

+

L' -5.0 0.15

!

•

L' = 10.0 'fo L• = IO.O 0.05 L·

=

20.0 0.10 ,;,. L'

=

20.o

•

L' = 50.0

•

L' = 50.0 0.000 ₅ ₁₀ ₁₅ ₂₀ ₅ ₁₀ ₁₅ ₂₀ ₂₅

a

)

Pe

a/

Pe

v

P

ea/

Pe

.,

0.30 0.35 0.25 0.30 0.20 0.25 0.20

_;J

0.15 d

""

0.15 0.10 L· = 5.0 L* = 5.0 L' = 10.0 0.10 ~ L' = LO.O 0.05

.

L· - 20.0

.

L' - 20.0 0.05

• • •

L· = 50.0

• • •

L' = 50.0 0.000 ₅ ₁₀ ₁₅ ₂₀ 0.000 ₅ _JO ₁₅ ₂₀

₂

₅

c

)

P

ea/

P

ev

_d)

P

ea/

P

ev

Fig. 5. Arterial effectiveness €a versus Peclet ratio Pea/Pe. for several vessels length L* 5, 10,20,50, and several venous Peclet and distance between vessels: {a) Pe. 5,rx 0.2, {b) Pe. 10,rx 0.2, {c) Pe. 5,rx 0.5, {d) Pe. 10,rx 0.5.