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NATIONAL ADVISORY COITIT'ITEE FOR AERONAUTICS
TECHNICAL 4EMIORANDUI, 1223
SOME EXPERIENCES REGARDING THE NONLINEARITY OF HOT WIRES*
By R. Betchov and W. Welling
We compare here the results of some experiences with the formulas
established in our preceding report "Nonlinear Theory of a Hot-Wire
Anemometer." We shall show that the nonlinear term plays a role as
important as the thermal conduction in the calculation of the thermal
inertia of the hot wire.
According to our nonlinear theory the equation of the hot wire
must contain the terms expressing:
(a) The heat transfer from the wire to the air in proportion to
the temperature T
(b) The heat transfer in proportion to T2 (nonlinearity)
(c) The heat conduction at the ends of the wire
(d) The thermal inertia due to the specific heat and the mass of
We shall study first the effects (a), (b), and (c) in treating the
case of 11 wires of small diameter (2 microns), and then turn to the
effect of inertia. We refer without further specifications to the
formulas of the nonlinear theory, numbered from 1 to 73, and shall
continue with the number 74.
II. PREPARATION OF THE WIRES
We prepared our wires by utilizing Wollaston wire of platinum,
bent in U form and soldered to a support before being cleaned. In
"Quelques experiences sur la non-linearite des fils chauds."
Koninklijke Iiederlandse Amkdemie van Wetenschappen. Mlededeling io. 66
uit het Laboratorium voor Aero- en Hydroddynamica der Technische Hogeschool
te Delft. Reprinted from Proceedings Vol. LIII, lio. 4, 1950, pp. 432-439.
IBetchov, R.: Th6orie non-lineaire de l'anemometre L fil chaud,
(M.eded. 61). Proc. Kon. lied. Adad. -.. Wetensch. Amsterdami, 52, 1949,
NACA TM 1223
order to remove the silver, we used a jet of an acid solution (50-
percent distilled water, 50-percent HN03) and electrolysis with a cur-
rent of 5 to 20 milliamperes. Figure 1 indicates the arrangement used.
The jet does not break the wire. Nevertheless, if the diameter is
smaller than 5 microns, the dust particles entrained by the liquid are
dangerous and the solution must be filtered each time before using.
The flask is mounted on a support which can be precision adjusted. Fig-
ure 1 shows the sequence of operations viewed under the microscope.
One can obtain extremely short wires by displacing the jet perpen-
dicularly to the wire, so as to remove the silver, sometimes in front,
sometimes at the rear. The cleaned wire is rinsed with ordinary water
and brought to a faint incandescence so as to permit microscopic examina-
tion. Only wires that redden in a regular and symmetrical manner are
III. STATIC CALIBRATION
We give here the results obtained with 11 wires of platinum with
lengths between 0.25 and 1.6 millimeters and with diameters of about
2 microns. Figure 2 indicates the cold resistances and the lengths
and shows the order of magnitude of the individual variations.
Every wire had been calibrated with air streams of 2 to 10 meters
per second and we studied the magnitude
H = RI2 (74)
as a function of the ratio R/Ro. Extrapolating starting from the
measured values of H, we determined the limiting value Ho, corresponding
to the case R/Ro = 1.
Our wires gave 11 values of Ho for V = 2 meters per second which
we indicated as functions of the wire lengths in figure 3. The theory
yields for I tending toward zero
H = 1 tanh io/To
NACA TM 1223
oo = (7 6)
and we plotted the theoretical curves corresponding to the values
Z = 6 x 102 millimeter and x 10i millimeter. One can see that,
if I tends toward infinity, Ho tends toward the value 500 milli-
amperes2.e King's formula gives us, for d = 2.1 microns, A = 500 mil-
liamperes and we thus can see that the effect of conduction at low
temperature corresponds to the theory.
When the resistance increases, the effect of thermal conduction
tends to make H diminish. lie take as example the wire No. 8, of a
length of 0.75 millimeter. Figure 4 gives us the experimental values
of H, measured twice, with a one-day interval. Taking into account
only the conduction, one obtains for H the dotted curve, according to
H A 1 (I"/A) tanh R/ (77)
1 tanh /E
The solidly drawn curve was calculated taking into consideration
the nonlinearity according to formula (36) with the coefficients
7 = 1.14 x 10 A = 450 milliamperes", Zo = 7 x 10 millimeters.
We see that it corresponds to the experience of the first day, and that
one has about Ho = 555. The values of H measured the next day are
Probably the differences are caused by dust particles which have
settled on the wire during that time interval and produce an enlarge-
ment of the region of immobile air around the wire, thus reducing the
transport of heat by the air stream.
We studied the increase of H with the temperature and for every
wire treated we measured the ratio
S_ R =2\ -Ho
H_\Ro H ) -0 (7 8)
This ratio can be calculated and figure 5 shows the experimental
and theoretical results.
The theory seems satisfactory to us, in spite of the deviations of
NACA TM 1223
IV. DYNAMIC CALIBRATION
We measured the response of a hot wire to fluctuation of the elec-
tric current. For that purpose, the wire was placed in a bridge
(fig. 6) fed by the plate flow of a pentode. The heating current can
be modulated with the aid of a low-frequency oscillator; the alternating
intensity i is indicated by a special apparatus. We had R1 = 100
and R2 = 1,000 ohms; the self-induction L compensated both the self-
induction of the line leading to 'the hot wire and its ohmic resistance.
The bridge electromotive force was applied to an analyzer which trans-
mitted only the signal of the frequency of the oscillator, permitting
operation without impediment by turbulence. The filtered signal was
transmitted to a cathodic oscillograph which enabled us to balance the
bridge for the frequency used.
Since analyzer and oscillograph were grounded, it was necessary to
especially insulate the feeding system.
Actually, the rectifier and the oscillator represent, normally,
with respect to the alternating network, a capacity of about 5,000 ppf;
this network is always grounded at some point which introduces an unde-
sirable element into the circuit. We eliminated this inconvenience by
using a transformer which has a weak capacity between the primary and
In order to eliminate the skin effect, we had to employ a special
line leading to the hot wire. In this manner, the bridge proved satis-
factory from 0 to 75 kilocycles. The impedance of the circuit R'C'
has the purpose of compensating the fluctuations of resistance of the
wire and the calculation shows that when the bridge is balanced the
electromotive force rl is proportional to the electromotive force at
the boundaries of R'. The measurements were made in the following
1. The wire is placed in the tunnel and subjected to the air stream,
with i = 0 and R' = 0. One then adjusts R3 so as to balance the
bridge. A galvanometer (not represented in the figure) is used for that
2. In modulating the current, with i of the order of 3 percent
of I, and at a low frequency, one adjusts R' and C' in such a
manner as to balance the bridge for alternating current. The values
of R' and C' as well as the frequency are noted.
3. The same procedure is followed with increasing frequencies f,
up to about 10,000 periods.
NACA TM 1223
The product 2nfR'C' gives the tangent of the angle of phase
displacement between the alternating current traversing the wire and
the variation of its resistance. This phase displacement amounts to
450 for a certain value f49o of the frequency which we compare (a)
with the theoretical value f* for the linear and infinite wires (51),
(b) with the value f*- of our nonlinear theory, and (c) with the
value given by Dryden
1 o RO12 (79)
Dryden 2x mc R Ro
deduced from equation (51) by replacing A by RI2/(R Ro). We have
here the general relation w = 2if.
In order to calculate these theoretical values, one must know the
-I 16ao (eo)
It can be seen that an error of 5 percent concerning the diameter of
the wire results in an error of 20 percent concerning that constant;
thus it is preferable to determine it empirically. The calculation of
that constant gave us, in the case of the wire No. 8, values that were
too low; we multiplied it arbitrarily by a factor 1.26, so as to make
theory and experience coincide if the temperature of the wire is low.
Figure 7 indicates the measured quantities f45o as functions of the
ratio R/Ro, as well as the calculated curves.
One can see that the empirical results are intermediate between
f and fDryden" In order to explain this, one must take into account
the variation with the temperature, the specific heat of the metal, the
product ap, and the density. With the aid of the International Critical
Tables, we estimated that the constant ap/mc diminishes according to
the approximate formula
jc-R = I[ o 1R (81)
with e = 0.06 in the case of platinum. This correction reduces the
calculated frequencies. We showed on figure 7 the curve obtained from
f** and indicated "f** corrected." This correction should be applied
equally to fDryden and could still increase the differences stated
before. Other wires give analogous results, and we estimate that our
nonlinear theory corresponds to experience.
NACA TM 1223
After having studied the frequency giving a phase displacement of
450 we must examine the behavior of the wire when the frequency is lower
or higher. The tangent of the phase-displacement angle q equal to the
ratio of the imaginary and the real part of the electromotive force rl
is given by the product 2nR'C'f, and the product R'C' should be
independent of the frequency in absence of thermal conduction if the
formula (67) were exact.
When f is smaller than f45, the amplitude corresponds to the
calculated values (formula 66) which is normal since this result depends
only on the derivative of R with respect to I, and the theory gives
the correct values of H.
The product R'C' is constant as shown in figure 8 as long as
f is smaller than f45, but the calculated values are slightly lower
than the measured ones.
Beyond f45, the amplitude diminishes according to the theory and
the formula (64) is verified, but the product R'C' decreases more
rapidly than was foreseen in the theory.
The dotted curves of figure 8 were calculated from the formula (62)
and it can be demonstrated that, for frequencies tending toward infinity,
the phase displacement tends toward
tan A i2 + BI2 iE f (82)
1 B2 f
A 12 + BI2
It seems, therefore, that this abnormal phase displacement is even
more pronounced than was expected according to experience; however, the
experimental errors may considerably affect our results, particularly
the error concerning R3. Also, it must be noted that, when the tangent
varies, for instance from 10 to 20, the angle varies only from 840 to
870 which reduces the importance of this effect.
Finally, we have carried out a few preliminary tests with tungsten
wires and have found the quantity H to be remarkably constant,
regardless of the length of the wire.
Translated by Mary L. Mahler
National Advisory Committee
NACA TM 1223
Support and hot wire
(1) Start (2, The plati-urm appears
(3) The jet forms a point (4) Finished wire
Figure 1.- Preparation of the Wollaston wires.
NACA TM 1223
tO = v
NACA TM 1223
Figure 3.- Effect of conduction at low temperature.
o First day
Figure 4.- Effect of nonlinearity.
10 NACA TM 1223
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