A General Set of Bioheat Transfer Equations Based on the Volume Averaging Theory - Porous Media: Applications in Biological Systems and Biotechnology - page 34

which forms a cubic equation for R i / ( R m − R i ). The root of the cubic equation

is quite complex. However, it is found that the following explicit expression
based on Newton's shooting method gives a quite accurate value for the root:

⎛
⎝

⎞
⎠

ω eff
α c

S m
k c ( T 0 −

1
6

ω eff
α c

+ 1
8

+6

R i
R m −

S m
k c ( T 0 −

T i )

R i = R i

+3

ω eff
α c

S m
k c ( T 0 −

T i )

+3

T i )

(1.92)

where α c = k c /ρ c c c is the thermal diffusivity of the unfrozen tumor. Thus, the
second on the right-hand side of equation (1.86) may be estimated as

1
6

T i ) R i

r = R i

ω eff
α c

∂T
∂r

S m
k c ( T 0 −

R i k c =2 k c ( T 0 − +3

T i )

⎛ ⎞

ω eff
α c

S m
k c ( T 0 −

+6

⎝ ⎠

+ 1
8

T i )

(1.93)

ω eff
α c

S m
k c ( T 0 −

+3

T i )

Upon substituting (1.88) and (1.93) into (1.86), we have

T i )) R i dR i
dt

( ρ i h sf + ρ c c c ( T 0 −

1
6

T i ) R i

ω eff
α c

= k i T i −

T p
ln R R −

S m
k c ( T 0 −

2 k c ( T 0 − +3

T i )

⎛
⎝

⎞
⎠

ω eff
α c

S m
k c ( T 0 −

+6

+ 1
8

T i )

(1.94)

ω eff
α c

S m
k c ( T 0 −

+3

T i )

which reduces to
dt ∗ = 1+ Sr
Ste

R i ln R i

2 dR i

ln R i
Cr

ω ∗ +6 Met
4( ω ∗ +3 Met ) 3 ( ω ∗ +3 Met ) 1 / 2 R i +

1 −

(1.95)

where

R i = R i /R p

(1.96a)

and

t ∗ = α i t/R p (1.96b)