The wire can be considered as a cylinder of diameter d and length l, with the ratio d/l << 1 (1/240 for the reference probe), placed perpendicular to the direction of the velocity U, which is considered uniform over the entire length of the wire. The thermal power exchanged by convection between the wire at temperature Tw, and the fluid at temperature Ta can be calculated from the equation
Q = hA(Tw – Ta) = nlNudA(Tw – Ta) (3.3)
where h is the film coefficient of heat transfer, A is the lateral surface of the wire, Nud is the Nusselt number referred to the diameter of the wire (Nud = hd/l) and l is the coefficient of thermal conductivity of the fluid. In Equation (3.3) the following assumptions are made:
■ the wire is so thin that the wall temperature is equal to the body temperature
■ flow is incompressible (M << 1), it is indeed assumed that the adiabatic wall temperature and the static temperature, Ta, coincide.
For an indefinite cylinder (l/d ^ ^) the Nusselt number can be expressed as a power function of the Reynolds number (referred to the diameter of the cylinder) and the Prandtl number whose exponents depend on whether the motion is laminar or turbulent. If the motion is laminar (Red < Redcr), which is always true for the hot wire anemometer, as the
diameter of the wire is very small (of the order of mm), it can be considered valid for the empirical equation:
Nud = 0.42Pr0’2 + 0.57Pr0’33 Re/5 (3.4)
after Kramers and van der Hegge Zijnen. For the reference probe immersed in a stream of air at standard temperature and pressure and velocity U = 30 [m/s], Red = 10 and Nud = 2.
In practice, even an indefinite cylinder does not follow Equation (3.4) with sufficient accuracy: the exponent of the Reynolds number is 0.48 at lower speeds and 0.51 at higher speeds. The variation of the exponent is still larger for sensors of finite length such as those used in anemometers. The sensors shaped as cones, wedges or domes show exponents significantly different from 0.5.
Because it is only relevant to have a functional relationship between the Nusselt number and velocity of the stream, as a calibration must be carried out for each probe, one can write:
Nud = a + b^Red
Equation (3.3) becomes:
Q=+ bn/Red)(Tw – Ta) = (a+bJU)) – T (3.5)
where a and b are two constants to be determined by calibration, which include the geometric properties of the wire and the fluid parameters. Since the response of the hot wire anemometer depends on many fluid dynamics parameters, the probe could be used to measure temperature, thermal conductivity, pressure, heat flux in addition to speed. For the same reason, when a single variable is measured, care must be taken to keep constant all the other variables: in particular, when measuring speed, it is important to compensate for any temperature difference between test and calibration or any change in fluid temperature during the test.