handbook of modern sensors

3.12 Heat Transfer 101
(A)
(B)
Fig. 3.40. Temperature profile in a joint (A) and a microscopic view of a surface contact (B).
through an interface of two adjacent materials may be different from the idealized
case. If we join together two materials and observe the heat propagation through the
assembly, a temperature profile may look like the one shown in Fig. 3.40A. If the sides
of the materials are well insulated, under steady-state conditions, the heat flux must
be the same through both materials. The sudden temperature drop at the interface,
having surface area, a, is the result of a thermal contact resistance. Heat transfer
through the assembly can be described as
T 1 − T 3
H =
, (3.122)
R A + R c + R B
where R A and R B are thermal resistances of two materials and R c is the contact
resistance,
R c = 1
h c a . (3.123)
The quantity h c is called the contact coefficient. This factor can be very important
in a number of sensor applications because many heat-transfer situations involve the
mechanical joining of two materials. Microscopically, the joint may look like the
one shown in Fig. 3.40B. No real surface is perfectly smooth, and the actual surface
roughness is believed to play a central role in determining the contact resistance.
There are two principal contributions to the heat transfer at the joint:
1. The material-to-material conduction through the actual physical contact
2. The conduction through trapped gases (air) in the void spaces created by the
rough surfaces
Because the thermal conductivity of gases is very small compared with many solids,
the trapped gas creates the most resistance to heat transfer. Then, the contact coefficient
can be defined as
h c = 1 L g
(
ac
a
2k A k B
k A + k B
+ a v
a k f
)
, (3.124)
where L g is the thickness of the void space, k f is the thermal conductivity of the fluid
(e.g., air) filling the void space, a c and a v are areas of the contact and void, respectively,