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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1223 
 
 SOME EXPERIENCES REGARDING THE NONLINEARITY OF HOT WIRES 
 
 By R. Betchov and W. Welling 
 
 Translation of "Quelques experiences sur la non-linearite des fils chauds." 
 
 Koninklijke Nederlandse Akademie van Wetenschappen. Mededeling 
 
 No. 66 uit het Laboratorium voor Aero- en Hydrod3niamica 
 
 der Technische Hogeschool te Delft. Reprinted from 
 
 Proceedings Vol. LIII, No. 4, 1950. 
 
 NACA 
 
 Washington 
 June 1952 
 
 1^ OF FLORIDA 
 ^JTS DEPARTMENT 
 .^,,v,.:,6T0N SCIENCE LIBR/fY 
 
 PC. BOX 117011 
 GAINESVILLE. FL 32611-7011 
 
 JSA 
 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEM0RANDUI4 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 . 
 
 I. INTRODUCTION 
 
 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 T'^ (nonlinearity) 
 
 (c) The heat conduction at the ends of the wire 
 
 (d) The thermal inertia due to the specific heat and the mass of 
 
 the metal 
 
 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 7^. 
 
 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 
 
 "Quelq_ues experiences sur la non-linearite des lils chauds." 
 Koninklijke Nederlandse Akademie van Wetenschappen. Mededeling No. GG 
 uit het Laboratorium voor Aero- en Hydrodynamica der Technische Hoge school 
 te Delft. Reprinted from Proceedings Vol. LIII, No. h, 1950, pp. 432-U39. 
 
 -'-Betchov, R. : Theorie non-lineaire de I'anemometre a fil chaud, 
 (Meded. 6l) . Proc. Kon. Ned. Adad. v. Wetensch. Amsterdam, 52, 19^9, 
 pp. 195-207. 
 
2 MCA TM 1223 
 
 order to remove the silver, we used a jet of an acid solution f 50- 
 percent distilled water, 50-percent HNO3J and electrolysis with a cur- 
 rent of 5 to 20 mllliamperes . 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 sh6rt 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 
 used. 
 
 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 = -^li_ ilk) 
 
 R - Ro 
 
 as a function of the ratio R/Rq- Extrapolating starting from the 
 measured values of H, we determined the limiting value Hq, corresponding 
 to the case R/Rq = 1- 
 
 Our wires gave 11 values of Hq 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 
 
 ^° == 1 - tant ^0/^0 ^'^^^ 
 
NACA TM 1223 
 
 vith 
 
 and we plotted the theoretical curves corresponding to the values 
 
 * -2 -2 
 
 Zq = 6 X 10" millimeter and 8 x 10 millimeter. One can see that, 
 
 if I tends toward Infinity, Hq tends toward the value 500 milli- 
 
 amperes^.^ 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. We take as example the wire No. 8, of a 
 length of 0.75 millimeter. Figure k 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 I A (77) 
 
 1 - tanh 1/5 
 
 The solidly drawn curve was calculated taking into consideration 
 the nonlinearity according to formula (36) with the coefficients 
 
 7 = l.lif X 10"-*, A = 450 milliamperes , Zq = 7 x 10~ millimeters. 
 We see that it corresponds to the experience of the first day, and that 
 one has about Hq = 555. The values of H measured the next day are 
 lower. 
 
 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 
 
 LRq = 
 
 H VKq ^ ^/ - ^o 
 So 
 
 (78) 
 
 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 
 the points . 
 
NA.CA TM 1223 
 
 IV, DYNAI-lIC CALIBRATION 
 
 We measured the response of a hot wire to fluctuation of the elec-. 
 trie 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 
 he modulated with the aid of a low-frequency oscillator; the alternating 
 intensity i is indicated by a special apparatus. We had Rx = 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 [i\if; 
 this net^^jork 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 
 the secondary. 
 
 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 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 
 manner: 
 
 1. The wire is placed in the tunnel and subjected to the air stream, 
 with i = and r' =0. One then adjusts Ro so as to balance the 
 
 bridge. A galvanometer (not represented in the figure) is used for that 
 purpose . 
 
 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 5 
 
 The product 2jtfR'C gives the tangent of the angle of phase 
 displacement betveen the alternating current traversing the wire and 
 the variation of its resistance. This phase displacement amounts to 
 il-5° for a certain value fi^.'^o 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 
 
 Dryden 2it mc R - Rq ^'^' 
 
 deduced from equation (51) hy replacing A by RI^/('r - Rq) . We have 
 here the general relation o) = 23tf . 
 
 In order to calculate these theoretical values, one must know the 
 constant 
 
 oSp _ l6ap 
 Jt2d^c5 
 
 ST - 3S- (80) 
 
 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 fl^.50 as functions of the 
 
 ratio R/Rq, as well as the calculated curves. 
 
 One can see that the empirical results are intermediate between 
 f and ^Dryden' "^^ 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 (xp/mc diminishes according to 
 the approximate formula 
 
 m^-m)J^-<k-^] («^' 
 
 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 ^Dryden ^^^ 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 
 45° we must examine the behavior of the wire when the frequency is lower 
 or higher. The tangent of the phase -displacement angle cp equal to the 
 ratio of the imaginary and the real part of the electromotive force rl 
 is given by the product 2itR'C'f, and the product R'C should be 
 independent of the frequency - in absence of thermal conduction - if the 
 formula (6?) were exact. 
 
 When f is smaller than fl^.^^ the amplitude corresponds to the 
 
 calculated values (formula G(}) 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 ^l^.^^ hut the calculated values are slightly lower 
 
 than the measured ones. 
 
 Beyond ^14.5^ 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 
 
 Big _1_ 
 
 tan , = ilA^jLt^iE 5 J2 jX (82) 
 
 1 - 
 
 l2 + Bl2 
 
 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 Ro. Also, it must be noted that, when the tangent 
 
 varies, for instance from 10 to 20, the angle varies only from 84 to 
 87° 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. I^hler 
 National Advisory Committee 
 for Aeronautics 
 
NACA TM 1223 
 
 Flask 
 300 cm^ 
 
 Microscope 
 
 Electrolytic circuit 
 
 Support and hot wire 
 Liquid jet 
 
 (1) Start (2) The platinum appears 
 
 (3) The jet forms a point (4) Finished wire 
 
 Figure 1.- Preparation of the Wollaston wires. 
 
MCA TM 1223 
 
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MCA TM 1223 
 
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