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AMcA TMI 1 e re V(; .77 7 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 13h6 NONLINEAR THEORY OF A HOTWIRE ANEMOMETER* By R. Betchov We study here the properties of a hotwire anemometer under the supposition that the heat transfer from the wire to the air depends, first, on the difference in temperature and, second, on the square of that difference. This latter hypothesis is confirmed by experience, and the consequences might be of great importance, that effect of non linearity is stronger than the effect of thermal conduction. I. THE NONLINEAR LAW OF KING The heat quantity Q removed per second by an air stream V from a wire of the diameter d and unit length is given by King in the form Q = (a + bV3d)T. (1) where T denotes the temperature difference between wire and air. King's calculation, approximately confirmed by experience, yields a = X' b = V2W'5'c' (2) with x = thermal conductivity of the air, 6' = density of the air, and c' = specific heat of the air for constant volume. These quantities may vary with T, and experience shows that a increases while b remains practically constant. Intuitively, one may interpret this effect by saying that the air in contact with the wire is heated which increases its conductivity. In compensation, its density decreases because the pressure varies only very slightly. Obviously, the effects on x' and 5' compensate one another, and only a varies. *"Theorie nonlineaire de l'anemometre a fil chaud." Koninklijke Nederlandsche Akademie van Wetenschappen. Mededeling No. 61 uit het Laboratorium voor Aero en Hydrodynamica der Technische Hogeschool te Delft. Reprinted from Proceedings Vol. LII, No. 3, 1949, pp. 195207. 2 NACA TM 1346 King states in his original report (ref. 1) that a increases by 0.114 percent per degree; he also describes there an effect of the diameter on that term a which we shall not discuss here. Thus it is advisable to write Q = fa(l + 7T) + bVd}T .(3) where y takes the nonlinearity into account. One must not forget the hypotheses on which King bases his calcu lation: he contends that the air flow is without viscosity and that the heat flow in the immediate proximity of the wire is constant. He uses the specific heat at constant volume although the pressure is certainly more constant than the density. For that reason, we consider equation (3) as an empirical relation, valid for the wire unit length, and would wish to see King's problem made the subject of a more thorough investigation. Here we intend to study the effect of the term 7 on the properties of the hot wire; we simplify the notation by introducing P so that P LVd (4) Q = a(l + P + ?T)T (5) II. GENERAL EQUATION OF THE HOT WIRE We shall use the following symbols: S resistance of the wire, per unit length, at operating temperature So resistance of the wire, per unit length, at ambient temperature I intensity of the electric current heating the wire a coefficient of the variation of S according to the temperature weight of the wire, per unit length NACA TM 1346 specific heat of the wire, in joule/gram and degree coefficient of the thermal conductivity of the wire in watt/cm and degree wire crosssectional area semilength of the wire time coordinate of position, varying from 2 to Z We put .a a + bVVd A = (1 + P) = aS, aSo The equation of the hot wire must express the equilibrium between the heat supplied per second, the heat removed by the air stream, the heat required to raise the temperature of the wire, and the heat trans mitted by conduction. One obtains 2 a) 2 mc OS yo 32S SI = A(S So) + (S So) + 5 (7) cL2S 2 aSo at aSo x2 For the steadystate case, and introducing the parameters x y = Z* aS A I2 A I  Z= (S  S= 2 ay 12/A2 2 1 12/A 3 a2So (1 I2/A)2 3 a 1 + P (1 I2/A)2 one obtains the equation (7) in the form 6y2 2 3Y2 So) NACA TM 1346 III. EXACT INTEGRATION OF THE STATIC CASE Multiplying equation (9) by /Z/by and integrating, one obtains, with a constant S VGZ3 + Z2 2Z + const (10) One notes that oZ/oy is zero for a negative value of Z and may be zero for two positive values of Z. At the ends of the wire, one has S = So and Z = 0; at the center, Z must be positive and 6Z/by zero. The range of interest for us lies, therefore, between Z = 0 and az the first positive root which gives = 0 which we shall denote by Zy Z = B. We put Z(y) = B X2(y) (11) By virtue of the relation GB3 + B2 2B + const = 0 (12) one obtains h(6X/oy)2 = GX + EX2 + D (13) with E = 1 + 3GB Following, we shall consider indicating the temperature in the the wire, one has X = 0 and D = 2(1 B) 3GB2 (14) B as a new integration constant middle of the wire. At the center of 4(3X/6y)2 = D (15) NACA TM 1346 which shows that D is positive and generally small. the wire, Z = 0 and X = B. At the ends of The roots of equation (13) are X2 E /E2 + 4GD 2G Introducing the parameter 1 so that sinh P = sh p = 2'D E one can write equation (13) in the form 4(oX/oy)2 = GC [(ch 0 + 1) X2~ Ech 1) + X We define the angle cp, function of y, so that S D sin r E ch V l k2sin2' with k2 ch P + 1 2 ch 1 Equation (18) then becomes d =E ch f dy 1 k2sin~ (16) (17) (18) (19) (20) (21) 6 and we obtain the elliptic integral of the first kind U(k;cp) = dq) 1 ch y J0 i k2sin2 2 The variable y varies from S to 6, with 6 = /Z* and we have, for NACA TM 1346 (22) (23) y = t and X2 = B SE ch = 2 Jrmax JO d(p 1\ k2sin2 with = _sin Pmax Ech B 1 k2sinmax This last equation may be written 1 = k2 + D sinmax EB ch B From equation (20) one may deduce 131 + k2 tanh2(p/2) (27) With P ranging from 0 to +., k2 varies from 1 to 0.5. (24) (25) (26) NACA TM 1346 With G and B known, one may calculate successively D, E, P, k, and cmax. A table of U(k,p) then permits us to calculate t. Figure 1 gives the results obtained by this procedure and allows  starting out from a prescribed wire and with ? known to determine B from G and i. The temperature distribution over the wire is given by Z as a function of y, or by X as a function of T and p a function of y. The relation cp(y) is given by the quotient of equations (22) and'(24), namely :' 0 1 k2sin2_ J max d "0 Vi k2sin2T (28) From equations (19) namely and (25), one obtains a relation between cp and X, sin c sin2(max S k2sinmmax 1 ksin2,' The total resistance R of the wire is given by +1 R =  z 2So12 S ix = 2S A I I 1/ Z dy + 2Sol We introduce the cold resistance Ro and the function X RI 2 B R Ro = A 12 02 dy (29) (30) (31) 8 NACA TM 1346 We replace X according to equation (19) and dy according to equa tion (21), namely R 12 ( o 2D R Ro 0 B 2/ A _I t(E ch 0)7 2 max O1 sin2 ( k2sinc 3/2 dc (1 k2sin2p)3 The integral is equal to 26U/Ok2 and we obtain S Ro2( 2D U/ak2) R Ro = (33 A I2 E ch 0 U The values of dU/1k2 can be deduced from a good table of U(k,q) with sufficient approximation. If the wire were perfect, the expression in parentheses in equa tion (33) would have to be replaced by unity; therefore, we shall intro duce the quantity M so that M = 1 B + 2D 3U/k2 (34 E ch 0 U The formula (33) then gives us S= R 1 MI/A (3 1 I2/A and the important relation RI2  2 A R R, 1 M(I  1 M J (36) This last equation permits easy determination of the wire charac teristics because R, Ro, and I can be measured accurately and because the curves obtained as functions of R/RQ, for instance, indicate directly (32) ) ) ) NACA TM 1346 the effects of conduction and of nonlinearity. of M as a function of G and k; figure 1 M We calculated the values 2 shows our results. A good approximation is given by M 1 1 M B  1M B,1 where Bo corresponds to B, in the case t = c Bo + l + 60 1 B, =  (37) (38) IV. A FEW USEFUL APPROXIMATIONS In performing the calculations necessary for the plotting of fig ure 1, we have noted that one may assign to k the value unity without introducing large errors. This implies 0 = 0 and equation (19) then gives (39) The integral (22) becomes d i 1 + sin 1 Ey U = = L y cos : 2 1 sin T 2 whence, one deduces ch(2U) = + = ch(Ey) 1 sincT (40) (41) X = rtan : NACA TM 1346 X2 = (ch Ey 1) At the limits, one has B (ch (ES 1) 2E which gives Z= B / 1 1/ch VEt In order to calculate M, one must put lim aU/ak2 = I k>l 2 0 sin2 dc cos3cp which gives M= 1 1/ B 1 1/ch /Ei\ STh ~ (46) One can see that, due to the nonlinearity, 6 is replaced by C/E, and the central temperature is lowered. If one takes equation (44) as solution of equation (9), one sees that the equation is satisfied for a term of approximately (ch /Ey/ch \/E)2, and that B is given approximately by B = r 6 (1 l/ch FE) (47) 30 (42) (43) _ ch Ey\ ch ch E/ (44) 1 isin .p 4kcos2w  U(P)) (45) NACA TM 1346 E = F/1 +6G (48) We shall take an example that represents an extreme case. We choose a platinum wire with 10 percent of iridium, a diameter of 7 microns, and a length of 22 = 1.14 mm. Exposed to an air stream of 5 m/sec and heated with 75 mA, it gives us P = 2.9 * = 0.15 mm A = 1.2 x 102 p = 3.75 Assuming y = 1.2 x 103 and with the aid of figures 1 and 2, we determined G = 0.25 B = 0.76 E = 1.56 M = 0.4 The other parameters have the following calculated values: k = 0.99756 sh B = 0.14 :Pmax = 79.50 D = 0.0477 In figure 3, we show the profile of the temperatures calculated exactly, the profile calculated with the approximation k = 1, and the profile calculated with 7 = 0. It can be seen that the nonlinearity offers a more uniform temperature distribution, and that the approxi mation is sufficient. R By means of equation (35) one calculates = 1.5, whereas the calculation with = 0 would give the result 1.9. The mean tempera ture giving = 1.5 would be 3800. Ro As to the term RI2/R Ro, it changes from the value 1.24 A when the current is very weak to the value 1.35 A when I attains 75 mA. I2/A = 0.47 U = 2.34 Ro It varies therefore by about 10 percent between o 1 which is of the same order as the variations observed. NACA TM 1346 ard = 1.5 Ro Thus this magnitude is constant at 10 percent and King's law is verified; however, we shall see later on that the thermal inertia is very different from the expected value. V. THE DYNAMIC EQUATION OF THE HOT WIRE In order to calculate the variation of the total resistance of the wire when the current or the air stream fluctuate, one must go back to the equation (7). Replacing in this equation I, S, and V (contained in the term A) by I + iejwt, S + sejut, and V + vejwt, one obtains after suppressing the terms of the order zero, two, and more as well as the factor ejwt 2Sli + 2s a 2aS. v 2a ( me (S So) + As + (S So)s +  Sa2S2 aS xa 62S js So aSo br2 (49) We introduce A 12 z = s So 2 (50) and, identical to the formula (9) of our Mededeling No. 55 (ref. 5) a* = 2o(A 12) mc (51) One then obtains, after introduction of Z 2 + 2 A 1 P= z(1 + 3Z + jw/c*) 2 (52) NACA TM 1346 The function Z(y) appears twice must utilize the approximation k cations. Writing the expression obtains  +z 1 5y2 \ in this differential equation, and we = 1 in order to avoid great compli Z according to equation (44), one + 3GB + jw/* 3GB ch yEy\ = Ch + C ch 1 1/ch V Eti ch REl 1 ch V/E (53) Cl= (l I/A B I~ i 12/A 1 1/ch VEV 1 v P 1 B 2 V \ + P 1 12/A 1 1/ch V* (54) Ci I2/A B 1 v P 1 B 2 = 2 1 I2/A 1 1/ch /E 2 V 1 + P 1 2/A 1 1/ch VEt 2' 1 (55) The solution without the second member of equation (53) is a Mathieu function taken for a purely imaginary value of the argument, but if r/E is large enough, for instance, more than 4, one may neglect the term with ch VEy of the member on the left side of equation (53). Actually it is important only when y is close to E; however, we shall take as a limiting condition z(E) = 0, and the product z ch\ (Ey will never be important. We shall also neglect the term 1/ch VEF in the denominator of one of the terms on the left side of equation (53) which amounts to taking Z = B in the factor term of z. Taking the definition of E into account, one then has ,2 ch \IEy 2z + z(E + j/mw*) = Cl + C2 ch Ey 6?y ch v/E' the solution of which null for y = t is C2 /ch \Ey ch pEy\ jw/w*\ch E ch ptE (56) El ( h pEy Ep2 ch pIF / (57) NACA TM 1346 with p = 1 + ja/Ea)* (58) It can be seen that due to the nonlinearity the characteristic frequency w* is replaced by a new frequency which is higher by a fac tor E. (Compare with equation (8) of ref. 5.) The amplitudes according to Ci and C2 also are modified by the nonlinearity. The variation r of the total resistance R will be r 1 r = J s dx =  A I2 5 0 The integration gives Ro,2 Cl R012 1 tanh pEt\ P ,/ FpfE c+ (tmanh t tanh p ) (60) JC/W* C t p We assumed above that /Et is sufficiently large; also, we may assign to the hyperbolic tangents the value unity (the presence of a complex argument is here not of importance). One then obtains oI2 C' (1  A I Ep 1/pFEE) + C2/ 1 p  j/W*W (pE\ p VI. RESPONSE TO A FLUCTUATION OF THE CURRENT If one assumes v = 0 in equations (54) and (55), one may trans form equation (61), suppressing the terms containing 1/ch rE r = 2iRoI2/A (1 I/A( B))( 1/pt) BIR/A w* 1 S(1 I2/A)2 Ep2 F t J p (62) z dy (59) (61) NACA TM 1346 This formula gives the alternating electromotive force produced at the boundaries of the hot wire by the modulation current i in addition to the normal electromotive force Ri. Although this formula seems to be complicated, it can be adapted to the needs of practice. If the frequency w/21 tends toward infinity, equation (62) becomes rI = 2iRI2/A (1 12/A( B) I2ABE) (63) (1 I2/A)2 jw/lw If one takes into account that equation (35) gives R, that equation (46) gives M, and that equation (51) gives cn*, one finds rl =  CRI2 with C = ao (64) (65) w/21r mc The constant C corresponds to that of our former publications, and equation (64) shows that the electromotive force rI taken at high frequency permits to measure C without being impeded by either con duction or nonlinearity. The method described in reference 4 is there fore indicated rather than the one consisting of measuring the phase displacements, with w being of the order of w*. One may immediately verify this point by assuming w in the equa tion (49) as very large, thus making the effect of the terms of conduction and of nonlinearity negligible. When w tends toward zero, the electromotive force becomes 2iRoI2/A 1 r J2 1 11 / (66) rIl M I/A l B) (66) (1 I2/A)2 E A /E\ \ 2 / Thus the complex function rI according to equation (62) will change from equation (64) into equation (66) when w varies from 0 to a large value. The complex trace of this function gives practically a semicircle which permits to put approximately rI = rI = 0) (67) 1 + jWlW** NACA TM 1346 where w** denotes the effective characteristic frequency. In fig ure 4, we plotted the semicircle and indicated a few values of mo/w*. From equation (62), and in the case of the preceding example, we calcu lated the electromotive forces rl for a few values of w/Eu*. One can see that the two functions blend, at low frequencies, if one assumes W"* = 1.1El = 1.72i*. At high frequency, equations (64) and (67) will be equal which permits calculation of a satisfactory value of W** when the effect of thermal inertia is important. One obtains w** (68) i/ 1 A 1 B) 1 1/2EA 2 1 I2/AM In the case treated one finds w** = 1.23ELu*, that is, (u* = 1.9ao*: the effective characteristic frequency is almost twice the expected value; therefore, the approximation (67) gives a correct plot of the function, but the phases according to equation (68) will be only within a 10percent accuracy. The denominator of equation (68) depends chiefly on E, and it increases the characteristic frequency. Instead of compensating each other as in the static case, the two effects reduce the thermal inertia. Intuitively, one may say that the conduction shortens the hot part of the wire and thus reduces the heat required for modifying the central temperature; the nonlinearity depends on the presence of hot air around the wire, and the thermal inertia of the air is negligible which improves the spherical response of the anemometer. When the wire is dusty, the quantity of immobile air is greater, and the experience shows that the term a in equation (3) is increased while b remains unchanged. One must therefore expect a dynamic action of the dust of the wire to the extent that E is modified. The dynamic effect may be more important than the static effect. The wire in the quoted example demonstrates that, with a term RI2/R Ro constant at 10 percent, the characteristic frequency may be almost twice the normally foreseen value. NACA TM 1346 17 VII. RESPONSE TO A FLUCTUATION OF THE AIR STREAM Assuming i = 0 in the formulas (54) and (55), and maintaining v one may transform equation (61) into r 1 v P Rol3/A B 2 V 1 + P (1 I2/A)2 1 1/ch /E1 (69) Jl 1/p /E w* p 1 1 L Ep2 jW P VEi With u tending toward zero, one has 1 v P rl = 1 P 2V1 + P RI/A B (1 12/A)2 1 1/ch A/E If uj tends toward infinity, one has r 1 v P rl + 2 V 1 + P ^A1 Rol3/A B /TEil (1 12/A)2 1 1/ch aEL j,'w* and one may approximately replace equation (69) by the semicircle rI = rI(c = O) rl = 1 + ja /,li** with * = E ,re 1  1 1/ (71) (72) (73) which, in the case of the example treated above, gives w** = 1.81m*. (70) 18 NACA TM 1346 Thus there is, on principle, no equality between the dynamic reac tion to a variation i and to a variation v. This difficulty arises due to the term /ES, namely to the conduction; the nonlinearity tends toward diminishing its importance (factor V/E). We hope to publish some empirical results, and the calculation of the differences indicated by different authors, in the near future. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM 1346 REFERENCES 1. King, L. V.: On the Convection of Heat From Small Cylinders in a Stream of Fluid: Determination of the Convection Constants of Small Platinum Wires With Applications to HotWire Anemometry. Phil. Trans. of the Royal Society of London. Series A, Vol. 214, 1914, pp. 373432. 2. Simmons, L. F. G.: Note on Errors Arising in Measurements of Turbu lence. Aeron. Research Committee, Technical Report, R. & M. 1919 (4042), 1939. 3. Betchov, R., and Kuyper, E.: Un Amplificateur pour l'Etude de la Turbulence. Med. 50, Proc. Kon. Ned. Akad. v. Wetensch., Amsterdam, 50, 1947, pp. 11341141. 4. Betchov, R.: L'inertie Thermique des Anemometres a Fils Chauds. Med. 54, ibid. 51, 1948, pp. 224233. 5. Betchov, R.: L'influence de la Conduction Thermique sur les Anemometres a Fils Chauds. Med. 55, ibid. 51, 1948, pp. 721730. 20 NACA TM 1346 in 0 d 0 C'l a n. .. 0 a, d o L *  j d    + 6  U A  H Ohi 'ft bu NACA TM 1346 M/I M 1.4 1.2 1.0 .4 ____ ___   0.6      0.6 S.0 0.5 41_ 0.4 0 .2 e 2G G= 0 %, 11 1  0o t\ I I I I 2 3 4 5 7 10 15 20 30 50 I1 Figure 2. Values of M,'1 .I according to G and t. 22 NACA TM 1346 o bfl 0 0 oi Cd      o g' cd S /0 /O "=  n   __ a I . x.  0 I Ii c AM N 0 o0 a doo 0 C fM~Q ___ __ ~ S 0 C 0 0 0 tf hf *^ oob Fl NACA TM 1346 Imaginary 0,2 part of r 0.4 +6.2 I +o. Values of W/ u** Real part of rI 41  +o0.6 o0 Ift 04 0.4', 020.2 .4 ...0.6 14 .9 08 Values of W/Ewc I"I T I" " Figure 4. Study of the tension rI. The semicircle corresponds to formula (67); the points give rI, according to (62), for several values of w/Ew*. NACALmgley 7752 1000  I l . ,2T P 1 1 111r 11117 m, ~II rr II I I I r i t S3" ~ S? 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