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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 HotWire 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 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 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 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 nonlinearite 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. 432439. IBetchov, R.: Th6orie nonlineaire de l'anemometre L fil chaud, (M.eded. 61). Proc. Kon. lied. Adad. .. Wetensch. Amsterdami, 52, 1949, pp. 195207. NACA TM 1223 order to remove the silver, we used a jet of an acid solution (50 percent distilled water, 50percent 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 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 = RI2 (74) R Ro 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 oA (75) H = 1 tanh io/To NACA TM 1223 with oo = (7 6) and we plotted the theoretical curves corresponding to the values 3 2 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 oneday 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 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 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 the points. 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 lowfrequency oscillator; the alternating intensity i is indicated by a special apparatus. We had R1 = 100 and R2 = 1,000 ohms; the selfinduction 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 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 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 manner: 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 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 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 constant 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 jcR = 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 phasedisplacement 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 BI2 1 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 for Aeronautics NACA TM 1223 Flask 300 cm3 Microscope 3 Electrolytic circuit Support and hot wire Liquid jet 2 (1) Start (2, The platiurm appears (3) The jet forms a point (4) Finished wire Figure 1. Preparation of the Wollaston wires. NACA TM 1223 0 \ \ _ \O o \ \ c o so o n 0t COD tO = v E o n b 0 o a I 0 0 CO" 0 c), NACA TM 1223 Figure 3. Effect of conduction at low temperature. 700 (m a) 600 500 400 o First day xSecond day 1.5 2 R/Ro Figure 4. Effect of nonlinearity. Ho 1000 2 (ma) 750 500 x X   10 NACA TM 1223 E E 0 o o S 1, 0 0 ,Lj o aS S d <, ^ ^ ^ h IIACA TM 1223 11 .c C C 0 II * Jo   oo 0 _rli o  oo C) Ca /=2 r"^ 1^i tc^ / r d) , Q: a, >^ ~\ ' ^ g" "i r.n ^ Q, ^ p~i^1 ~! ^ S 58 s eb 0 s fc V4 b01 'd D I 1 'E 9 0 a c aa L " r 2 i 1.0 M 4n L. 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