handbook of modern sensors

3.14 Dynamic Models of Sensor Elements 117
3.14.2 Thermal Elements
Thermal elements include such things as heat sinks, heating elements, insulators,
heat reflectors, and absorbers. If heat is of concern, a sensor should be regarded as a
component of a larger device. In other words, heat conduction through the housing
and the mounting elements, air convection, and radiative heat exchange with other
objects should not be discounted.
Heat may be transferred by three mechanisms: conduction, natural and forced convection,
and thermal radiation (Section 3.12). For simple lumped parameter models,
the first law of thermodynamics may be used to determine the temperature changes in
a body. The rate of change of a body’s internal energy is equal to the flow of heat into
the body minus the flow of heat out of the body, very much like fluid moves through
pipes into and out of a tank. This balance may be expressed as
C dT = Q, (3.157)
dt
where C = Mc is the thermal capacity of a body (J/K), T is the temperature (K), Q
is the heat flow rate (W), M is the mass of the body (kg), and c is the specific heat of
the material (J/kg K). The heat flow rate through a body is a function of the thermal
resistance of the body. This is normally assumed to be linear, and, therefore,
Q = T 1 − T 2
R , (3.158)
where R is the thermal resistance (K/W) and T 1 − T 2 is a temperature gradient across
the element, where heat conduction is considered.
For illustration, we analyze a heating element (Fig. 3.50A) having temperature
T h . The element is coated with insulation. The temperature of the surrounding air is
T a . Q 1 is the rate of heat supply to the element, and Q 0 is the rate of heat loss. From
Eq. (3.157),
C dT h
= Q 1 − Q 0 , (3.159)
dt
but, from Eq. (3.158),
Q 0 = T h − T a
R , (3.160)
(A)
(B)
Fig. 3.50. Thermal model of a heating element (A); an electrical circuit diagram (B) with resistive,
capacitive, and inductive components.