handbook of modern sensors

100 3 Physical Principles of Sensing
Fig. 3.39. Temperature profile in laminated materials.
For instance, heat passage through a rod is governed by a law which is similar to Ohm’s
law. The heat flow rate (thermal “current”) is proportional to the thermal gradient
(thermal “voltage”) across the material (dT /dx) and the cross-sectional area A:
H = dQ
dt
=−kA dT
dx , (3.118)
where k is called thermal conductivity. The minus sign indicates that heat flows in
the direction of temperature decrease (a negative derivative is required to cancel the
minus sign). A good thermal conductor has a high k (most of metals), whereas thermal
insulators (most of dielectrics) have a low k. Thermal conductivity is considered
constant; however, it increases somewhat with temperature. To calculate a heat conduction
through, say, an electric wire, temperatures at both ends (T 1 and T 2 ) must be
used in equation
H = kA T 1 − T 2
L , (3.119)
where L is the length of the wire. Quite often, thermal resistance is used instead of
thermal conductivity:
R = L k , (3.120)
Then, Eq. (3.119) can be rewritten
H = A T 1 − T 2
R . (3.121)
Values of thermal conductivities for some materials are shown in Table A.17.
Figure 3.40 shows an idealized temperature profile within the layers of laminated
materials having different thermal conductivities. In the real world, heat transfer