ABSTRACT 
An analysis is made of the unsteady heat flow through a solid cylindri- 
cal wire containing heat sources and a coating in the form of a cylindrical shell. 
The differential equations are linearized and the solutions are found by the 
method of Laplace transforms. The response of a constant-current coated wire 
is obtained for two step-like initiating conditions: (1) a change in convective 
cooling and (2) a change in power or current input. The response of a constant- 
temperature coated wire is obtained for a step-like change in convective cooling. 
An equivalent time constant for each of the three cases is computed. The fre- 
quency responses of the wire to sinusoidal initiating disturbances are derived 
in each case by applying Duhamel’s theorem. Bare wire responses are obtained 
by letting the coating thickness shrink to zero. 
INTRODUCTION 
Hot-wire sensing elements have been used for many years in wind tunnels to measure 
fluctuations in velocity and temperature of the air stream. The wire is usually mounted in 
either aconstant-current or a constant-resistance electronic circuit. In the former case the 
current in the wire is kept constant and the wire temperature and resistance change as the 
rate of convective cooling changes. In the latter case the response consists of a change in 
current which keeps the temperature and resistance of the wire constant. Although there is 
no time lag or distortion in the response of a constant temperature wire it requires a finite 
time to set up temperature changes in a constant-current wire. The distortion in the response 
of a constant-current wire may be corrected by a simple electronic differentiating circuit. 
The elements of this circuit may be adjusted by obtaining a satisfactory frequency response 
to a change in current input. 
Many difficulties arise when attempts are made to use a hot-wire turbulence-sensing 
element in water. One of the most persistent is the formation of a film on the wire, even in 
relatively clean water.! For use in sea water or in other corrosive liquids it would probably 
be necessary to coat the wire with an insulating material. The presence of a film or coating 
not only reduces the wire sensitivity but also increases the time lag in the response, as it 
requires a finite time for temperature changes to penetrate the coating. Therefore, the re- 
sponse of a constant-current coated wire will show more attenuation and distortion than that 
of a bare wire and even the response of a constant-temperature coated wire will show some 
distortion. 
lReferences are listed on page 48. 
