Nonlinear theory of a hot-wire anemometer

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Title:
Nonlinear theory of a hot-wire anemometer
Series Title:
NACA TM
Physical Description:
23 p. : ill ; 27 cm.
Language:
English
Creator:
Betchov, R
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Airplanes -- Wings   ( lcsh )
Hot-wire anemometer   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
A theoretical analysis is presented for the hot-wire anemometer to determine the differences in resistance characteristics as given by King's equation for an infinite wire length and those given by the additional considerations of (a) a finite length of wire with heat loss through its ends and (b) heat loss due to a nonlinear function of the temperature difference between the wire and the air.
Bibliography:
Includes bibliographic references (p. 19).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by R. Betchov.
General Note:
"Report date July 1952."
General Note:
"Translation of "Theórie non-linéaire de l̕ anémomètre à fil chaud." Koninklijke Nederlandsche Akademie van Wetenschappen. Mededeling No. 61 uit het Laboratorium voor Aero- en Hydrodynamica der Technische Hogeshcool te Delft. Reprinted from Proceedings Vol. LII, No. 3, 1949."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778597
oclc - 86076296
sobekcm - AA00006181_00001
System ID:
AA00006181:00001

Full Text
AMcA TMI 1







e re V(; .77 -7


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 13h6


NONLINEAR THEORY OF A HOT-WIRE ANEMOMETER*

By R. Betchov


We study here the properties of a hot-wire 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 non-lineaire 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. 195-207.







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 cross-sectional 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 steady-state 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


0-2 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 -
1-M 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 VE-t


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 10-2


p = 3.75


Assuming y = 1.2 x 10-3 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 p-Ey\
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"* = 1.1-El = 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 10-percent 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 Hot-Wire Anemometry.
Phil. Trans. of the Royal Society of London. Series A, Vol. 214,
1914, pp. 373-432.

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. 1134-1141.

4. Betchov, R.: L'inertie Thermique des Anemometres a Fils Chauds.
Med. 54, ibid. 51, 1948, pp. 224-233.

5. Betchov, R.: L'influence de la Conduction Thermique sur les
Anemometres a Fils Chauds. Med. 55, ibid. 51, 1948, pp. 721-730.










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



do--o
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


I-f-t


04
0.4',


020.2-

.4


...0.6


14 .9 0-8 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*.


NACA-Lmgley 7-7-52 1000


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P 1 1 111r 11117 m,


~II rr II I I I r i


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CLL W 2 -S
o d CU 01U v. bi uQ M a)c .
S o aj 0 oE
IV IU CU z< wa,





S-
r=u






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( CcM l r 3 Co a



00.0 C4 Cua~U ed
*0 m LO
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o 01W E CL....5uv
S r- a c S =

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wE -rncn a>. 0 ri L 33 g
0 0
LS^4W S 3WS .0i.g I
0" r- o w r"." -2
a* ^; .- ; = i B -j;( l ii




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,-' >WC 'aE a) C
5-0 w m-




01w E W. W 0 CD5 a
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Cu w^ i- t E us os o C



Lml' p A" 0

M)O

ci s 0 s3c r sc
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