handbook of modern sensors

364 11 Flow Sensors
(A)
(B)
Fig. 11.4. Thermoanemometer. (A) a basic two-sensor design; (B) cross-sectional view of a
temperature detector.
closer to the R s detector, that detector will register a higher temperature. When the
medium flows, heat dissipation increases due to forced convection. The higher the
rate of flow, the higher the heat dissipation and the lower the temperature that will be
registered by the R s detector. Heat loss is measured and converted into the flow rate
of the medium.
A fundamental relationship of thermoanemometry is based on King’s law [2]
( √ )
2πρc dv
Q = kl 1 +
(T s − T 0 ) , (11.12)
k
where k and c are the thermal conductivity and specific heat of a medium at a given
pressure, ρ is the density of the medium, l and d are the length and diameter of the
sensor, respectively, T s is the surface temperature of the sensor, T 0 is the temperature
of the medium away from the sensor, and v is the velocity of the medium. Collis and
Williams experimentally proved [3] that King’s theoretical law needs some correction.
For a cylindrical sensor with l/d ≫ 1, a modified King’s equation yields the velocity
of the medium:
v = K ( ) dQ 1 1.87
, (11.13)
ρ dt T s − T 0
where K is the calibration constant. It follows from the above that to measure a flow,
a temperature gradient between the sensor and the moving medium and the dissipated
heat must be measured. Then, velocity of the fluid or gas becomes, although nonlinear,
a quite definitive function of thermal loss (Fig. 11.5A).
To maintain the R s detector at T s and to assure a sufficient thermal gradient with
respect to T 0 , heat loss must be compensated for by supplying the appropriate power
to the heating element.Also, we may consider a flow sensor without a separate heating
element. In such a sensor, the R s detector operates in a self-heating mode; that is, the
electric current passing through its resistance generates enough Joule heat to elevate
its temperature to T s . At that temperature, the second detector has resistance R s .
Assuming that conductive heat loss through connecting wires and sensor’s enveloping