Temperature Distribution in Blood Perfused Tissue'

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Temperature Distribution in Blood Perfused Tissue.
Introduction of 1D Steady State(SS) Heat
Conduction 1D SS w/ Const. Heat Generation 1D
SS w/ Heat Generation f(T) Plane Wall and Fin
Equations.
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Differential Control Volume (Cartesian)
Incropera and DeWitt. 3rd Edition, 1996.
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Differential Control Volume in Cylindrical
Coordinates.
Incropera and DeWitt. 3rd Edition, 1996.
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Differential Control Volume in Spherical
Coordinates
Incropera and DeWitt. 3rd Edition, 1996.
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Incropera and DeWitt. 3rd Edition, 1996.
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Energy Equation in 1-D Cartesian
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Energy Equation in 1-D Cartesian SS, k const.
qm
x0
xL
x
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1D SS Plane Wall with constant Heat Generation.
Integrate twice and apply boundary conditions.
The maximum temperature will be found at xL/2
if Tc Ts. An Increase of qmL2/8k. In
addition, one can solve for the heat transfer
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SS Plane Wall with Temperature Dependent Heat
Generation - i.e. Blood Perfusion.
mbperfusion (g/m3s) Cbspecific heat(J/g) T
temperature () A surface area (m3) xdistance
into tissue.
x
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SS Plane Wall with Temperature Dependent Heat
Generation - i.e. Blood Perfusion.
With rearrangement
x
Use exp/sinh/cosh fin solutions.
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Fin Solutions
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Energy Balance on an extended surface
(Fin). P perimeter(m), k A and h have their
usual meanings.
dx
x
If k and A are const. then
General govern. Eqn.
If only T is a function of x then Solution is
exp, sinh, cosh
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Incropera and DeWitt. 3rd Edition, 1996.
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Fins of Nonuniform Cross-Sectional Area
r1
Ac 2prt, P 2pr and convection occurs on top
and bottom.
t
r2
where
General Solution
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Fins of Nonuniform Cross-Sectional Area Contd
Assuming adiabatic tip and T(r1)Tb.
Total Fin Heat Transfer
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Energy Balance on Perfused Vessel
h2pRdx(Tb,x-T0 )
mbcbTb,in
mbcbTb,out
dx
x
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SS Plane with Temperature Dependent Heat
Generation - i.e. Blood Perfusion.
Temperature Distribution
A Tc ? Ta
B Tc Ta
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SS Plane with Temperature Dependent Heat
Generation - i.e. Blood Perfusion.
Heat Flux Solutions
A Tc ? Ta
B Tc Ta
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Comparison of 1-D SS Solutions (TaTc)
1. Tqm0 Tc - x/L(Tc - Ts) 2. Tqmcnst Tc -
x/L(Tc - Ts) (qmL2)/(2k)((x/L) -
(x/L)2) 3. TqmfTac Tc - (Tc -
Ts)(SinhGMAx/SinhGMAL) 4. TqmfT Ta
(Ts - Ta - (Tc - Ta)Exp-GMAL)(SinhGMAx/
SinhGMAL) (Tc - Ta)Exp-GMAx GMA
2/L Nondimensional blood flow.
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3 and 4
TcTa 37
Ts34
1
Ta Tc in this case so 3 and 4 overlap.
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Shape Factor AnalysisHeat Transfer Between
Parallel Blood Vessels or Between Blood Vessel
and Surface of Skin
T2
R2
d2
a
l
y 0
a
d1
R1
T1
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Solutions
Assume vessel walls are constant T, and To is the
surface temperature at y0, q1 is the source
strength per unit length.
The T difference can be found by evaluating the
equation at two convenient points, such as x10,
y1-d1R1 and x20, y2d2-R2. The final results
with bd/R is
What we are really doing is solving this by the
method of shape factors. The usual shape factor
involves cosh-1(b) which Chato notes is
a Multivalued function.
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Solutions (contd)
If the blood vessels are the same size, then the
solution becomes even Simpler
For b d/R gt1.9, which is satisfied in most
practical cases, the Solution reduces to
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Wisslers Approach
Finally, since one usually knows the blood
temperature, not the Wall of the vessel, Wissler
made the following proposal
Where these are the mixed cup temperatures of
artery and vein Countercurrent (parralel) pair.
Clearly, if the temperatures are Known the heat
transfer can be calculated as well.
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Incropera and DeWitt. 3rd Edition, 1996.
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Incropera and DeWitt. 3rd Edition, 1996.
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Blood Vessel Equilibration - Chato 1980/1990
T
R1
To
Tb
T1
R1 Tb
T1
R2
To
x
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2 Main Questions

• At what distance from the entrance will the
• blood reach tissue temperature?
• 2. What is the blood temperature at nay given
• length of the blood vessel?

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Solution
Where
Further Solutions depend on the x dependence of
Lambda.
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Two Extreme Solutions
For R2 R1, Nue 4, Gz RePrD/L 18
For R2 gtgt R1, Nue 0.32
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Thermal Distribution in 1-D Cartesian k
const., heat generation0, Constant temperature
BCs Uniform initial condition
T(x,0) F(x)
T(0,t)T1
T(L,t)T2
x0
xL
a thermal diff b eigenvalue
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Dimensionless Length WhereEffectiveness is 0.95
or above.
For R2 R1, Nue 4, Gz RePrD/L 18
Can show 1/Gz0.1737 OR L
RePrD0.1737
For R2 gtgt R1, Nue 0.32
Can show 1/Gz2.34 OR L RePrD2.34
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Chato 1980 and 1990- Which Different Vessel
Classes Equilibrate?
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Chato 1980 and 1990- What is the Equilibration
Length of these Different Vessel Classes?
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Transient Solutions
Lumped Semi-Infinite Examples Separation of
Variable Solutions.
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Lumped Transient Solutions
Valid for Bi hLc/k ltlt1 Allows assumption
that Tb const.
Thermal time constant. Decay time to 1/e initial
q? q?.
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Semi-Infinite Domain Problems. Temperature and
Heat Flux for 3 Cases
Incropera and DeWitt. 3rd Edition, 1996.
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Temperature and Heat Flux Solutions.
Incropera and DeWitt. 3rd Edition, 1996.
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Error Function Solution. Application to Burn
Injury
a 0.4/(9994200) Ts 70 Ti 37 Plot3DTs
(Ti - Ts)Erfx/((4at)(0.5)), x, 0, 0.04,
t, 0.1, 1000, PlotRange -gt 37, 70
PlotTs (Ti - Ts)Erf0.04/((4at)(0.5)),
t, 0, 100000, PlotRange -gt 37, 70
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Graphical Temperature Solution for Convective
Case 3.
Incropera and DeWitt. 3rd Edition, 1996.
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Interfacial Temperature Solution for 2
Semi-Infinite Bodies w/ Different Ti
Incropera and DeWitt. 3rd Edition, 1996.
Safe Touch. Subramanian and Chato, ASME JBME
120(5)727-736 (1998).
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Thermal Distribution in 1-D Cartesian k
const., heat generation0, Constant temperature
BCs Uniform initial condition
T(x,0) F(x)
T(0,t)T1
T(L,t)T2
x0
xL
a thermal diff b eigenvalue
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Osizik 1993 - Table 2.2
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Thermal History T(x,t) in 1-D Cartesian k
const., qm0, convective BC, uniform initial
condition
1 Term Approximations Or Heissler Charts can be
Used for Fo gt0.2
x0
xL
Ti
x
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Plane Wall
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Cylinder
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Sphere
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2D Transient Conduction Solution in Short
Cylinder. Product Solution
Incropera and DeWitt. 3rd Edition, 1996.
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Examples of Product Solutions
P - Plane Wall C - Cylinder S - Semi-Infinite
Domain
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Perfusion Models. Early Work - keff
f(perfusion) gt k conduction only Seagrave(1971)
- Simple derivation of same.
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Counter Current Models
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Combined Perfusion and Counter Current Models
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