5.5 Further Issues
361
1
R point
=
1
2
π k f d(2 ln(κ) + K − 3.9),
(5.226)
where κ is the thermal conductivity ratio and κ =
k s
k f
. K is of the order of unity, and
it is determined by the outer field. In practice, it is difficult to calculate the value
of K . A new approach is developed here to calculate the contact point resistance.
By adding the fluid conduction contribution, the total conductive effective thermal
conductivity in Eq. (5.223) for a packed bed is written as
k point
k f
= ε r + N (1 − ε r )
1
π k f d R point
(5.227)
On the other hand, using the Kunii–Smith equation [14], the effective thermal
conductivity is given as
k point
k f
= ε r +
β
Ψ +
γ
κ
(1 − ε r )
(5.228)
where β, γ , and Ψ are the dimensionless geometry parameters. For the close packing
of spheres, the evaluated parameters are γ =
2
3
, β =
1+
√
3
2 +
√
2
3
3
, and
Ψ =
1
2
κ−1
κ
2 sin
2
(θ 0 )
ln [κ − (κ − 1) cos(θ 0 )] −
κ−1
κ
[1 − cos(θ 0 )]
−
2
3κ
(5.229)
where angle θ 0 is determined by the packing arrangement of the bed. It is difficult to
calculate the value of θ 0 for all possible cases. The evaluated parameter is available for
certain packing. In the densest packing (ε r = 0.260), the parameters are sin
2
(θ 0 ) =
1
4
√
3
and N = 12.
Therefore, by solving Eqs. (5.227)–(5.227), the contact thermal resistance can be
obtained as
R point =
√
3
2πβk f d
κ−1
κ
2
ln [κ − (κ − 1) cos(θ 0 )] −
κ−1
κ
[1 − cos(θ 0 )]
(5.230)
When the conductivity ratio is larger than 100, i.e., κ > 100, the thermal resistance
of the contact point (PPP mechanism) is shown in Fig. 5.102, which can be reduced
to
1
R
point k f d
=
2πβ
√
3
(ln(κ) − ln[1 − cos(θ 0 )] − [1 − cos(θ 0 )]).
(5.231)
Lastly, with the combination of the conduction through the contact area (PPA
mechanism) and the fluid near the contact point (PFP mechanism), the total effective
thermal conductivity of the fluid–particle system can be formulated as