chapter 10 heat transfer in living tissue 10.1 introduction examples hyperthermia cryosurgery ...
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CHAPTER 10
HEAT TRANSFER IN LIVING TISSUE
10.1 Introduction Examples Hyperthermia Cryosurgery Skin burns Frost bite Body thermal regulation Modeling
Modeling heat transfer in living tissue requiresthe formulation of a special heat equation
1
VesselsArtery/veinAorta/vena cavaSupply artery/veinPrimary vesselsSecondary vesselsArterioles/venulesCapillaries
Key features
(1) Blood perfused tissue
(2) Vascular architecture
(3) Variation in blood flow rate and tissue properties
10.2 Vascular Architecture and Blood Flow
10.1 Fig.
vein
arterySAV
arteriolevenule s
s
P
tissue
c
dia. m 15-5 ,capillarie c
dia. m 300-100 vein, and arery primary P
dia. m 1000-003 vein, and arery supply main AVS
scapillarie
2
Tb
Blo
od te
mpe
ratu
re
temp. blood temp. tissue
temp. tissuetemp. blood
path flow blood
mixing
0aT
aortaprimaryarteries
scapillarieprimaryveins
cavavena
10.2 Fig. variation etemperatur blood of Schematic vessels in
10.3 Blood Temperature Variation
• Blood mixing from various sources brings temperature to 0aT
Equilibration with tissue: prior to arterioles and capillaries Metabolic heat is removed from blood near skin
• Blood leaves heart at 0aT
3
(4) Blood Temperature:
(1) Equilibration Site: Arterioles, capillaries & venules(2) Blood Perfusion:
10.4 Mathematical Modeling of Vessels-Tissue Heat Transfer
10.4.1 Pennes Bioheat Equation (1948)
(a) Formulation
Assumptions:
(3) Vascular Architecture: No influence
4
Blood reaches capillary bed at
body temperature 0aT, leaves at tissue temperature T
Neglects flow directionality. i.e.isotropic blood flow
10.3 Fig.
capillarybed
arteriole
venule
x
y
mq
Let bq net rate of energy added by the blood per unit
volume of tissue mq rate of metabolic energy production per unit
volume of tissue
dzdydxqqdzdydxqE mbg )( (a)
5
inE gE outE E (1.6)
Treat energy exchange due to blood perfusion as energy
generation
Conservation of energy for the
element shown in Fig. 10.3:
Formulation of bq: Blood enters at body temperature 0aT and exists at the tissue temperature T
)( 0 TTwcq abbbb (10.1)
bc specific heat of blood
Eq. (10.1) into (a)
bw blood volumetric flow rate per unit tissue volume b density of blood
)( 0 TTwcqq abbbm (10.2)
Eq. (1.6) leads to (1.7). Modify eq. (1.7): set 0 WVU , and use eq. (10.2),
t
TcqTTwcTk mabbb
)( 0 (10.3)
c specific heat of tissue 6
k thermal conductivity of tissue density of tissue Tk conduction terms, form depends on coordinates:
Cartesian coordinates:
)()()(z
Tk
zy
Tk
yx
Tk
xTk
(10.3a)
cylindrical coordinates:
)()()(2
11
z
Tk
z
Tk
rr
Trk
rrTk
(10.3b)
spherical coordinates:
)()()(222
22 sin
1sin
sin
11
T
kr
Tk
rr
Trk
rrTk
( 1 0 .3 c )
7
Notes on eq. (10.3):
t
TcqTTwcTk mabbb
)( 0 (10.3)
(1) This is known as the Pennes Bioheat equation
(2) The blood perfusion term is mathematically identical to surface convection in fins, eqs. (2.5), (2.19), (2.23) and (2.24)
(3) The same effect is observed in porous fins with coolant flow (see problems 5.12, 5.17, and 5.18)
(b) Shortcomings of the Pennes equation
Equilibration Site:(1)
• Does not occur in the capillaries
8
Occurs in the thermally significant pre-arteriole and post-venule vessels (dia. 70-500 m )
Thermally significant vessels: 1L
Le
eL = Equilibration length: distance blood travels for its temperature to equilibrate with tissue
Blood Perfusion:(2)
Perfusion in not isotropic Directionality is important in energy interchange
Vascular Architecture :(3)
•Local vascular geometry not accounted for•Neglects artery-vein countercurrent heat
exchange•Neglects influence of nearby large vessels 9
Blood Temperature:(4)
(c) Applicability
Surprisingly successful, wide applications
Reasonable agreement with some experiments
10
Blood does not leave tissue at local temperature T
0aTBlood does not reach tissue at body core temperature
Example 10.1: Temperature Distribution in the Forearm
0
rR
modelbw mq
forearm
bw mq
10.4 Fig.
Th,
Model forearm as a cylinder
Blood perfusion rate bw
Metabolic heat production mq
Convection at the surface
Heat transfer coefficient is h
Ambient temperature is T
Use Pennes bioheat equation to determine the 1-D temperature distribution
(1) Observations
11
• Arm is modeled as a cylinder with uniform energy generation Heat is conduction to skin and removed by convection In general, temperature distribution is 3-D
(2) Origin and Coordinates. See Fig. 10.4
(3) Formulation
(i) Assumptions
(1) Steady state (2) Forearm is modeled as a constant radius cylinder (3) Bone and tissue have the same uniform properties (4) Uniform metabolic heat (5) Uniform blood perfusion(6) No variation in the angular direction(7) Negligible axial conduction
12
(8) Skin layer is neglected(9) Pennes bioheat equation is applicable
(ii) Governing Equations
Pennes equation (10.3) for 1-D steady state radial heat transfer
0)(1
0)(
k
qTT
k
wc
dr
dTr
dr
d
rm
abbb
(a)
(iii) Boundary Conditions:
,0)0(
dr
dT or T(0) = finite (b)
TRThdr
RdTk )(
)( (c)
(4) Solution
13
Rewrite (a) in dimensionless form. Define
0
0
a
a
TT
TT
,
R
r (d)
(d) into (a)
0)(
1
0
22
)(
TTk
Rq
k
Rwc
d
d
d
d
a
mbbb
(e)
Define
k
Rwc bbb2
(f)
)( 0
2
TTk
Rq
a
m (g)
(f) and (g) into (e)14
01
)(
d
d
d
d (h)
The boundary conditions become
,0)0(
d
d or )0( finite (i)
]1)1([)1(
Bi
d
d (j)
Bi is the Biot number
k
hRBi
Homogeneous part of (h) is a Bessel differential equation. The solution is
)()()( 0201 KCIC (k)
15
Boundary conditions give
)()(
])/(1[
011
IBiI
BiC
, 02C (m)
(m) into (k)
)/(])/(1[)(
)( 0010
0
)()(
RrIIBiI
Bi
TT
TrTr
a
a
( n )
(5) Checking
Dimensional check: Bi,and are dimensionless. The arguments of the Bessel functions are dimensionless.
16
Limiting check: If no heat is removed (),arm reaches a uniform temperature . All metabolic heat is transferred to the blood. Conservation of energy for the blood:
Limiting check:
Dimensional check:
)( 0aobbbm TTwcq Solve for oT
bbb
mao wc
qTT
0 (o)
Set 0Bih
bbb
ma wc
qTrT
0)( (p)
which agrees with (o)
(6) Comments(i) Solution depends on 3 parameters: Bi, metabolic heat , and blood perfusion parameter (ii) Setting 0r and Rr in (n) gives center and surface temperatures
17
(iii) The solution for zero metabolic heat production is obtained by setting 0 mq
(iv) The solution for zero blood perfusion can not be deduced from (n). Setting 0 in (n) gives . Solution is obtained by setting 0 in (h) and then solving for T:
22
)/(14
1
2
1
)/(Rr
BikqR
TT
m
(q)
10.4.2 Chen-Holmes Equation
• First to show that equilibration occurs prior to reaching the arterioles• Accounts for blood directionality• Accounts for vascular geometry• The Pennes equation is modified to:
18
t
TcqTkTucTTcwTk mpbbabbb
)( **
( 1 0 . 4 )
NOTE:
(1) *bw = local perfusion rate (2) *aT= blood temperature upstream of the arterioles 0aT
(3) u blood velocity vector, accounts for directionality (4) Tucbb . energy convected by equilibrated blood.
Note similarity with convection term in moving fins (eq.2.19) and with flow through porous media (eq. 5.6)
(5) Tk p conduction due to temperature fluctuations in
equilibrated blood
19
(6) pk “perfusion conductivity”, depends on blood velocity
and inclination relative to temperature gradient, vessel radius and number density
Limitations(1) Vessel diameter m300
(2) 6.0L
Le
(3) Requires detailed knowledge of the vascular network and blood perfusion
10.4.3 Three-Temperature Model for Peripheral Tissue
Rigorous Approach
• Accounts for vasculature and blood flow directionality20
10.5 Fig.
artery vein
aT vT
x
0
tissueTlayer deep
teintermedialayer
cutaneous
layer
cbw skinsT plexus
cutaneous• Assign three temperature variables:
aT(1) Arterial temperature
vT(2)Venous temperature(3) Tissue temperature T
• Identify three layers:(1) Deep layer: thermally
significant counter-
current artery-vein pairs
(2) Intermediate layer:porous media
(3) Cutaneous layer: thin,independently supplied by counter-current artery-veinvessels called cutaneous plexus
• Regulates surface heat flux
21
• Consists of two regions: (i) Thin layer near skin with negligible blood flow(ii) Uniformly blood perfused layer (Pennes model)
FormulationSeven equation: 3 for the deep layer 2 for the intermediate layer 2 for the cutaneous layer • Model is complex• Simplified form for the deep layer is presented in the next section• Attention is focused on the cutaneous layer: (i) Region 1, blood perfused. For 1-D steady state:
22
0)( 1021
2
TTk
wc
dx
Tdc
cbbb (10.5)
1T temperature variable in the lower layer
0cT temperature of blood supplying the cutaneous pelxus
cbw cutaneous layer blood perfusion rate x coordinate normal to skin surface
(ii) Region 2, pure conduction , for 1-D steady state:
022
2
dx
Td (10.6)
The 3 eqs. for ,aT vT and T are replaced by one equation
10.4.3 Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue
23
10.6 Fig.
tissueT
aTartery
vTvein volume control
•Effect of vasculature and heat exchange between artery, vein, and tissue are retained
•Added simplification narrows applicability of result
(a) Assumptions
Control VolumeContains artery-vein pairs Countercurrent flow, va TT Includes capillaries, arterioles
and venules
(1) Uniformly distributed blood bleed-off leaving artery is
equal to that returning to vein (2) Bleed-off blood leaves artery at aT and enters the vein at vT
24
(3) Artery and vein have the same radius
(4) Negligible axial conduction through vessels (5) Equilibration length ratio 1/ LLe
(6) Tissue temperatureT is approximated by
2/)( va TTT (10.7)
(7) One-dimensional: blood vessels and temperature gradient are in the same direction
(b) FormulationConservation of energy for tissue in control volume takes intoconsideration:
(1) Conduction through tissue(2) Energy exchange between vessels and tissue due to capillary blood bleed-off from artery to vein
25
(3) Conduction between vessel pairs and tissue Note: Conduction from artery to tissue not equal to conduction from the tissue to the vein (incomplete countercurrent exchange)
Conservation of energy for the artery, vein and tissue and conservation of mass for the artery and vein give
meff qx
Tk
xt
Tc
)( (10.8)
effk= effective conductivity, defined as
22
2 )(1 uack
nkk bbeff
( 10.9)
a vessel radius
n number of vessel pairs crossing surface of control volume per unit area
u average blood velocity in countercurrent artery or vein 26
shape factor, defined as )2/cosh(al
(10.10)
l center to center spacing between two parallel and isothermal vessels
NOTE effk accounts for the effect of vascular geometry and blood
perfusion a, , n and u depend on the vascular geometry
Conservation of mass gives u in terms of inlet velocity ou to tissue layer and the vascular geometry. Eq. (10.9) becomes
)(
)2(1
2
2
V
k
uackk
b
oobbeff (10.11)
27
oa vessel radius at inlet to tissue layer, 0x
)(V dimensionless vascular geometry function (independent of blood flow)
Lx / dimensionless distance L tissue layer thickness ou blood velocity at inlet to tissue layer, 0x
NOTE: )/2( boobb kuac is independent of vascular geometry.
It represents the inlet Peclet number:
b
oobbo k
uacPe
2 (10.12)
Eq. (10.12) into eq. (10.11)
][ )(1 2 VPekk oeff (10.13)
Notes on effk : 28
(1) For the 3-D case, orientation of vessel pairs relative to the direction of local tissue temperature gradient gives rise to a tensor conductivity
(2) The second term on the right hand side of eqs. (10.11) and (10.13) represents the enhancement in tissue conductivity due to blood perfusion
Cutaneous layer: Use eqs. (10.5) and (10.6)
0)( 1021
2
TTk
wc
dx
Tdc
cbbb (10.12)
022
2
dx
Td (10.13)
Rewrite eq. (10.5) in terms of the Peclet number:0Pe
29
L
uanw ooob
2 (10.14)
on = number of arteries entering tissue layer per unit area
Eq. (10.12) into eq. (10.14)
obb
boob Pe
cL
kanw
2
(10.15)
Define R
b
bc
wL
wLR
1 (10.16)
R = total rate of blood to the cutaneous layer to the total rate of blood to the tissue layer
1L = is the thickness of the cutaneous layer
Eqs. (10.15) and (10.16) into (10.5) 30
0)(2 100
121
2
TTRPekL
kan
dx
Tdc
boo (10.17)
(c) Limitation and Applicability
Accurate tissue temperature prediction for: (1) Vessel diameter < μm200
(2) Equilibration length ratio 2.0/ LLe
(3) Peripheral tissue thickness < 2mm
31
Results are compared with 3- temperature model of Section 10.4.3
Example 10.2: Temperature Distribution in Peripheral Tissue
Peripheral tissue Skin surface at sT Blood supply temperature 0aT
Neglect blood flow through cutaneous layer
vascular geometry is described by )(V
2)( CBAV
555 1010109.15,1032.6 CandBA
.
5107
0 1
)(V
10.7 Fig.
effk)](1[ 2 VPek o
(i) Use the Weinbaum-Jiji equation determine temperature distribution
(ii) Express results in dimensionless form:
32
Lx/ , sa
s
TT
TT
0
, b
oobb
k
uacPe
20 ,
)(0
2
sa
m
TTk
Lq
mq
)( xkeff
10.8 Fig.
00aT
skin sT
x
(iii) Plot showing effect of blood flow & metabolic heat
(1) ObservationsVariation of k with distance is known Tissue can be modeled as a single
layer with variable effk
• Metabolic heat is uniform • Temperature increases as blood perfusion and/or metabolic heat are increased
(2) Origin and Coordinates. See Fig. 10.8
(3) Formulation33
(i) Assumptions(1) All assumptions leading to eqs. (10.8) and (10.9) are applicable(2) Steady state(3) One-dimensional
(5) Skin is maintained at uniform temperature
(6) Negligible blood perfusion in the cutaneous layer.
(ii) Governing Equations. Obtained from eq. (10.8)
)](1[ 20 VPekkeff (b)
34
(4) Tissue temperature at the base x = 0 is equal to 0aT
(a)0)( meff qdx
dTk
dx
d
2)( CBAV (c)
(iii) Boundary Conditions
(d)0)0( aTT
sTLT )( (e) (4) Solution
Define
L
x , ,
0 sa
s
TT
TT
)(0
2
sa
m
TTk
Lq
(f)
Substituting (b), (c) and (f) into (a)
0)(1 220
d
dCBAPe
d
d (g)
Boundary conditions
35
1)0( (h)
0)1( (i)
Integrating (g) once
1
220 )(1 C
d
dCBAPe
integrating again
2220
220
1)(1)(1
CCBAPe
d
CBAPe
dC
(j) integrals (j) are of the form
2
cba
d and 2
cba
d (k)
where 36
,1 20APea ,2
0BPeb 20CPec (m)
Evaluate integrals, substitute into (j)
212
11
2tan)(ln
2
1
2tan
2
Cd
cb
d
bcba
c
d
cb
dC
(n)
24 bacd (o)
Boundary conditions (h) and (i) give the constants
1C and
2C
37
where
Note:
(1) a, b, c and d depend on .0Pe Listed in Table 10.1
(2) 1C depends on both oPe and :
38
d
cb
d
cb
d
b
cba
cba
c
d
cb
d
cb
d
C
2tan
2tanln
2
1
2tan
2tan2
112
111
(p )
d
cb
d
b
d
d
cb
d
b
d
b
a
cba
cC
2tantan
2
2tantanln
2
11
11
11
1
(q)
oPe
10.1 Table
a 1.2275 3.0477b -0.5724 -5.1516 c 0.36 3.24d 1.44 12.96
60 180
0 0.5
0.5
1.0
1.0
02.0
10.9 Fig.
60oPe 180oPe
0.20
0.60.81.0
1.441.131.060.41.011.001.02
3.052.151.511.121.021.14
10.2 Table
kkeff /6.0
180oPe
60oPe
600Pe and 02.0 : 047.11 C
1800Pe and 6.0 : 0176.11 C
(3) Table 10.2 lists enhancement in k
(4) Fig. 10.9 shows )(
39
(5) Checking
Boundary conditions (h) and (i)are satisfied
Tissue temperature increases as bloodperfusion and metabolic heat are increased
(6) Comments
(i) Enhancement in effk due to blood perfusion
(ii) Temperature distribution for 600 Pe and 02.0 is nearly linear. At 1800Pe and 6.0 the temperature is higher
40
, , 0Pe and the arguments of 1tan and ln are dimensionless Dimensional check:
Boundary conditions check:
Qualitative check:
(iii) The governing parameters are 0Pe and . The two are physiologically related
(iv) Neglecting blood perfusion in the cutaneous layer during vigorous exercise is not reasonable
10.4. 5 The s-Vessel Tissue Cylinder Model
Model Motivation
• Shortcomings of the Pennes equation• The Chen-Holmes equation and the Weinbaum-Jiji equation are complex and require vascular geometry data
(a) Basic Vascular Unit
Vascular geometry of skeletal muscles has common features
• Main supply artery and vein, SAV
41
t
t
tdia. mm1
cylindermuscle
mm5.0P
PP
c
tc
t
t
ts
s
s
s
sdia. m10050
dia. m5020
dia. m300100 SAV
dia. m1000300
tarrangemenvascular tiverepresenta a of Schematic 10.10 Fig.
• Primary pairs, P
• Secondary pairs, s
• Terminal arterioles and venules, t
• Capillary beds, c42
NOTE: Blood flow in the SAV, P and s is countercurrent
Each countercurrent s pair is surrounded by a cylindrical tissue which is approximately 1 mm
Diameter and typically 10-15 mm long
• The tissue cylinder is a repetitive unit consisting of arterioles, venules and capillary beds• This basic unit is found in most skeletal muscles• A bioheat equation for the cylinder represents the governing equation for the aggregate of all muscle cylinders
(b) Assumptions
(1) Uniformly distributed blood bleed-off leaving artery is equal to that returning to vein of the s vessel pair
43
(2) Negligible axial conduction through vessels and cylinder
(3) Radii of the s vessels do not vary along cylinder (4) Negligible temperature change between inlet to P vessels and inlet to the tissue cylinder (5) Temperature field in cylinder is based on conduction with a heat-source pair representing the s vessels(6) Outer surface of cylinder is at uniform temperature
(c) Formulation
• Capillaries, arterioles and venules are essentially in local thermal equilibrium with the surrounding tissue
• Three temperature variables are needed: T, aT and vT
• Three governing equations are formulated
44
s vessels within the cylinder are thermally significant: va TTT
artery
vein
vein
arterytissue tissue
cylinder tissue (a)
section-cross enlarged (b)
10.11 Fig.
al
vll
R
localT
L L
x
• Navier-Stokes equations of motion give the velocity field in the s vessels (axially changing Poiseuille flow)
Boundary Conditions
(1) Continuity of temperature at the surfaces of the vessels (2) Continuity of radial flux at the surfaces of the vessels
45
(3) Tissue temperature at cylinder radius R is assumed uniform equal to localT (4) Symmetry at the mid-plane Lx gives 0
x
T
(5) Inlet artery bulk temperature at 0x is specified as 0abT
(6) At Lx the flow in the s vessels vanishes and the artery, vein and tissue are in thermal equilibrium at the local tissue temperature localT
(d) Solution
• The three eqs. for T, aT and vT are solved analytically • Solution gives 0vbT , the outlet bulk vein temperature at
0x
Simplified Case
Assume: 46
localT
case simplified
lvl
al
vein
artery
tissue
R
(1) Artery and vein are equal in size
(2) Symmetrically positioned relative to center of cylinder, i.e., va ll
Results
Tat 0x is given by
11212
211
12
11
0
00
A
A
A
A
TT
TTT
localab
vbab (10.18)
24
111ln
4
1)(
2
2
11R
lRA a
(10.19)
47
4
4
2
2
12cos2
1ln4
1
R
l
R
l
l
RA aa
(10.20)
(e) Modification of Pennes Perfusion Term
Eq. (10.18) gives
)( 000 localabvbab TTTTT (a)
Conservation of energy for blood at 0x gives the total energybqdelivered by blood to cylinder
)( 002
vbabaabbb TTuacq (10.21)
(a) into (10.21)
)( 02
localabaabbb TTTuacq
Dividing by the volume of cylinder
48
)( 02
2
2 localabaa
bbb TTT
LR
uac
LR
q
(10.22)
Blood flow energy generation per unit tissue volume: bq
LR
qq bb 2 (10.23)
Blood flow per unit volume bw:
LR
uaw aab 2
2
(10.24)
(10.23) and 10.24) into (10.22)
)( 0 localabbbbb TTTwcq (10.25)
Since ,lRit follows that TTlocal
(10.25) becomes
)( 0 TTTwcq abbbbb (10.26)
49
Artery supply temperature (1) body core temperature 0aT
bqin Pennes equation is given by
)( 0 TTwcq abbbb ( 1 0 .1 )
Comparing (10.26) with (10.1):
(2) A correction factor, T, is added in (10.26)
Use (10.26) to replace the blood perfusion term in thePennes equation (10.3)
t
TcqTTTwcTk mabbbb
)( 0 (10.27)
NOTE:
(1) This is the bioheat equation for the s-vessel cylinder model
50
(2) is a correction coefficient defined in (10.18)T(a) It depends only on the vascular geometry of the tissue
cylinder(b) It is independent of blood flow rate (c) Its value for most muscle tissues ranges from 0.6 to 0.8 (d) This vascular structure parameter is much simpler
than that required by Chen-Holmes and Weinbaum- Jiji equations
(3) The model analytically determines the venous return temperature
(4) Accounts for contribution of countercurrent heat exchange in the thermally significant vessels.
(5) The artery temperature0abT appearing in eq. (10.27) is unknown
51
(a) It is approximated by the body core temperature in the Pennes bioheat equation(b) Its determination involves countercurrent heat exchange in SAV vessels
(6) While equations (10.5) and (10.6) apply to the cutaneous layer of peripheral tissue, eq. 10.23 applies to the region below the cutaneous layer.
Example 10.3: Surface Heat Loss from Peripheral Tissue sT
xbw
cbw
0abT 0cbT0
10.12 Fig.
1L
L tissue
cutaneos
Peripheral tissue of thickness LCutaneous layer of thickness 1L Blood perfusion bw Primary vessel supply temperature0abT
52
Cutaneous plexus:Perfusion rate bcw (uniformly distributed),
blood supply temperature 0cbT
Skin temperature sT
Specified correction coefficient T
Use the s-vessel tissue cylinder model, determine surfaceflux
(1) Observations
• Temperature distribution gives surface flux• This is a two layer problem: tissue and cutaneous
(2) Origin and Coordinates. See Fig. 10.12
Metabolic heat mq
53
((3) Formulation (i) Assumptions (1) Apply all assumptions leading to (10.5) and (10.27) (2) Steady state (2) One-dimensional (3) Constant properties (4) Uniform metabolic heat in tissue layer (5) Negligible metabolic heat in cutaneous layer (7) Uniform blood perfusion in cutaneous layer (8) Tissue temperature at x = 0 is0abT (9) Specified surface temperature
(ii) Governing EquationsFourier’s law at surface:
54
(a)
x
LLTkqs
)( 11
k tissue conductivity sqsurface heat flux 1T temperature distribution in the cutaneous layer
Tissue layer temperature T: eq. (10.27):
0)( 0
*
2
2
k
qTT
k
wc
dx
Td mab
bbb T, Lx0 (b)
0)( 101
2
TTk
wc
dx
Tdcb
cbbb , 1LLxL (c)
(iii) Boundary Conditions55
Need 2 equations: one for tissue layer and one for cutaneous
Cutaneous layer temperature 1T: eq. (10.5):
0)0( abTT (d)
))( 1LTLT (e)
dx
LdT
dx
LdT )()( 1 (f)
sTLLT )( 11 (g)
(4) Solution
Let,0
0
abs
ab
TT
TT
,0
01
abs
ab
TT
TT
L
x (h)
(b) and (c) become
0)( 0
2*
2
2
2
sab
bbbb
TTkLq
Tk
Lwc
d
d
, 10 (i)
56
00
002
2
2
sTT
TT
k
Lwc
d
d
ab
cbabcbbb
, 011 (j)
Dimensionless parameters:
k
Lwc bbb2
,
k
Lwc cbbbc
2 , ,)( 0
2
sab
b
TTk
Lq
sTT
TT
ab
cboab
0
0
( k ) (k) into (i) and (j)
0*2
2
Td
d , 10 (m)
02
2
ccd
d, 011 (n)
L
L10 (o)
57
Boundary conditions
0)0( (p)
)1()1( (q)
dx
d
dx
d )1()1( (r)
1)1( 0 (s)
Solutions to (m) and (n):
*
** coshsinhT
TBTA
( t )
cc DC coshsinh ( u )
Boundary conditions (p)-(s) give constants
58
**
32*
321*
*
0
coshsinh
cosh1
)1(cosh
cosh)1(
TTCCT
CCCT
T
A c
c
(v)
*TB
(w)
21** cosh CCTTAC ( x )
)1(tanhcosh)1(cosh
101
**2
0
CTTACD
c
( y )
21,CCand3Care given by
)1(cosh
sinh)1(sinh
0
*
*1
c
ccTT
C
59
102 1(tanhsinhcosh
ccccC
)1(tanhcoshsinh 03 cccC
Surface heat flux:
)1(sinh)1(cosh)( 00
0
ccc
sab
DCTTk
Lqs
( z ) (5) Checking
Limiting check:
01)()( abTxTxT and0sq
60
Parameters 0,,,, c are dimensionless
Special case: 00 acabs TTT and ,0bq solutions (t), (u) and (z) reduce to the expected results
Dimensional check:
(5) Comments
(i) Five governing parameters: ,,,,cand0 (ii) Use solution (z) to examine the effect of cutaneous
blood perfusion on surface heat flux
(iii) Changing blood flow rate through the cutaneous layer is a mechanism for regulating body temperature
(iv) The solution does not apply to the special case of zero blood perfusion rate sinceandcappear in the differential equations as coefficients of the variables and
61