Recovery corrections for butt-welded, straight-wire thermocouples in high-velocity, high-temperature gas streams

MISSING IMAGE

Material Information

Title:
Recovery corrections for butt-welded, straight-wire thermocouples in high-velocity, high-temperature gas streams
Series Title:
NACA RM
Physical Description:
19 p. : ill. ; 28 cm.
Language:
English
Creator:
Simmons, Frederick S
Lewis Research Center
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Thermocouples   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: Experimental measurements show that a reasonable correlation among recovery corrections at various pressures and temperatures for butt-welded straight-wire thermocouples is given by an empirical equation in which the correction is seen to be proportional to the fifth root of the pressure and inversely proportional to the fourth root of the temperature. Resultant probable errors in temperature measurements are presented and discussed.
Bibliography:
Includes bibliographic references (p. 8).
Statement of Responsibility:
by Frederick S. Simmons.
General Note:
"Report date July 23, 1954."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003810742
oclc - 135105995
sobekcm - AA00006161_00001
System ID:
AA00006161:00001

Full Text
~~.. .. I L ADON /


RESEARCH
iSi":" .c. FA H
;- g==R =


Rh,? WA4fl7a


MEMORANDUM


R RECOVERY CORRECTIONS FOR BUTT -WELDED, STRAIGHT -WIRE

S ,THERMOCOUPLES IN HIGH-VELOCITY, HIGH-TEMPERATURE

GAS STREAMS

By Frederick S. Simmons
;':, '*,'*..


Lewis Flight Propulsion Laboratory
Cleveland, Ohio


a in ,--


urilWsfy OF FLORIDA
DOCLUM IS DEPARTMENT
120 MARSTN SCIENCE USRARY
:. RPO. EBOX 117011
SGAWIFM I F, R. ?2 11.7011 USA
v:.: ... :..


S NATIONAL ADVISORY COMMITTEE

1 iFOR AERONAUTICS
F." -::'^!'":" FO R
:i,;'.":. WASHINGTON
September 24, 1954
4I WS NT
i ;,'' ,:f .." .: .
i ,f ,. '!. ... ..E~lE: :E. "::f ? 'E:: .#: .. .


1U"E'`;'";


RM E54G22a


'"


. i .. .. : "


W A..
ri ...
.. .: .
.. ..











NACA RM E54G22a


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


RESEARCH MEMORANDUM


RECOVERY CORRECTIONS FOR BUTT-WELDED, STRAIGHT-WIRE THERMOCOUPLES

IN HIGH-VELOCITY, HIGH-TEMPERATURE GAS STREAMS

By Frederick S. Simmons


SUMM.IARY

Recovery corrections were experimentally determined for several
diameters of chromel-alumel and platinum 13 percent rhodium platinum
butt-welded thermocouples in a gas stream at temperatures from ambient
to 20000 R and Mach numbers from 0.2 to 1.0. The recovery corrections
at various temperatures and pressures are reasonably correlated with an
empirical equation in which the correction is seen to be proportional
to the fifth root of the pressure and inversely proportional to the
fourth root of the temperature.

Probable errors in temperature measurement due to resultant uncer-
tainties in the corrections are presented and discussed.


INTRODUCTION

It is a familiar fact that a body immersed in a gas may attain
thermal equilibrium at a temperature other than that of the gas. For a
gas at rest or moving at a low velocity, this temperature is the result-
ant of a balance of convective heat transfer between the body and the
gas with the radiant and conductive heat transfer between the body and
the external surroundings. For a gas moving at a high velocity, how-
ever, an additional factor becomes important: the aerodynamic heating
effect, which is the result of friction and stagnation of the gas near
the body.

For temperature measurements involving the immersion of instruments
such as thermocouples in gas streams, corrections to the indicated tem-
peratures are frequently necessary, and usually the errors in the meas-
urements consist mainly of the uncertainty of the magnitudes of these
corrections. A bibliography of the subject is given in reference 1.

For conditions of high temperatures and high velocities such as
those encountered in jet-engine exhaust gases, the corrections for the
heat-transfer effects and for the aerodynamic effects are of the same








NACA RM E54G22a


order of magnitude. Methods for the calculation of the former are given
in reference 2. The present work is concerned with the magnitudes and
probable errors of the latter and is part of a program of high-temperature
measurements being conducted at the NACA Lewis laboratory.


SYMBOLS

The following symbols are used in this report:

Cp specific heat at constant pressure

D diameter of wires

L length of wires

M Mach number

Pr Prandtl number

p static pressure

R gas constant

Re Reynolds number

r recovery factor

T total temperature

t static temperature

tad adiabatic temperature

v velocity of gas

T tad
TA


y ratio of specific heats

p density of gas

P viscosity of gas







NACA RM E54G22a


ANALYSIS

The total temperature of a gas stream is defined by

v2
T=t+ ()

The relation between the total temperature and the temperature of a
body immersed in the gas, for the case of zero heat transfer to exter-
nal surroundings, is usually given in terms of a recovery factor r,
which is defined by

tad t (2)
r =(2)

or

tad t + r 2
2Cp
where the adiabatic temperature tad is the temperature attained by the
body in the absence of external heat transfer. The recovery factor is,
in general, a function of the geometric configuration of the body and,
in order of decreasing importance, of the Prandtl, Mach, and Reynolds
numbers of the gas stream. .A detailed discussion is given in refer-
ence 3.

For purposes of temperature measurement, it is more convenient to
deal with a ratio A defined by

T t(3)
T

The relation of A and r is given by

A = (1 r)(l t/T) (4)

Since the ratio t/T is primarily a function of the Mach number, A is
a function of the Prandtl, Mach, and Reynolds numbers and of the geome-
try of the body.

Experience has shown that for the case of cylindrical wires in cross
flow, the Mach number becomes an important parameter and the function
may conveniently be assumed to have the form

A = fl(M, Pr)x(Re)n (5)







NACA RM E54G22a


Substitution in the Reynolds number of the relations (ref. 4)

P = p/RT
v = MVyT (6)

Soc T0O.7

shows that

MDp
Re oc T.2 (7)


and A may be written
n
S= f2(M, Pr) Dn (8)
T1.2n

If the Prandtl number is independent of temperature and pressure, equa-
tion (8) may be written

\n /(T 1.2n n


where
Don pn
a0 = f3(M) O1.2n
TO

the subscript zero denoting reference conditions of pressure and tempera-
ture and a reference diameter. It may then be expected that for a given
wire diameter

A = ( n / 1.2n
A= ') (9)

Similarly, at a given pressure and temperature the variation of
A with diameter should be represented by

A = Ao ( n (10)
70-)







NACA RM E54G22a


From the definition of A, the total temperature T is equal to
the adiabatic temperature tad multiplied by a factor i/(l A).
Since in practice A is much less than unity, this factor can also be
expressed as (1 + A) so that A represents a fractional correction
factor.

The present work was directed toward determination of whether for-
mulas like equations (9) and (10) could adequately describe the depend-
ence of A upon pressure, temperature, and diameter and, if so, toward
evaluation of the exponent n. Considerable experimental data, accu-
mulated over a period of years in the course of other research work,
were analyzed for that purpose.


EXPERI'IENTAL PROCEDURE

The recovery characteristics of thermocouples at elevated gas tem-
peratures were obtained from tests made in the high-temperature tunnel
described in reference 5 with a modified test section, shown schemati-
cally in figure 1. The modified test section consisted of three con-
centric Inconel cylinders 24 inches long, the outer one 6 inches in
diameter, the inner one forming a nozzle approximately 2 inches in di-
ameter. Gas flowed between the cylinders, and 12 thermocouples welded
to the inner surface of the inner liner indicated that this surface was
within a few degrees of the gas temperature under all conditions, thus
establishing that net radiation between the test thermocouples and the
surroundings was negligible. The gas temperature was measured with a
0.020-inch-diameter chromel-alumel thermocouple in the stagnation region.
Preliminary tests showed this thermocouple to indicate the same as a
high-recovery probe with a chromel-alumel thermocouple and as a plati-
num 13 percent rhodium platinum thermocouple 0.020 inch in diameter
placed nearby, to within wire calibration accuracies. The temperature
was steady to within 50 R, and no measurable gradients existed in the
stagnation region or in the nozzle exit. The pressure ratio across the
nozzle was measured with tubes in the stagnation region and in the plane
of the nozzle exit as shown. The Mach number was calculated by using
values for the specific heat ratio at the various temperatures obtained
from reference 6. The test thermocouples were butt-welded and the junc-
tion reduced to wire diameter; they were installed with the wires ex-
tending across the jet as shown in figure 1. In all cases, a differen-
tial in temperature was measured between identical thermocouples in the
stagnation region and nozzle exit with a sensitive recording potentiom-
eter. The sizes -f the wires tested were 0.032-inch- and 0.020-inch-
diameter chromel-alumel and 0.020-inch- and 0.010-inch-diameter platinum
13 percent rhodium platinum. In these tests the variation of the
ratio A with Mach numbers from 0.2 to 1.0 was obtained at four ap-
proximate temperature levels: ambient, 10000, 15000, and 20000 R. In
all tests for a given wire, the gas temperature was controlled to with-
in 150 R.







NACA RM E54G22a


To determine the effect of pressure on recovery characteristics,
chromel-alumel and platinum 13 percent rhodium platinum thermocouples
of various diameters were installed in a similar manner and tested at
ambient temperature in a 3-inch air jet discharging into a receiver of
controlled pressure. In these tests, the Mach number was varied from
0.2 to 1.0 at the approximate pressure levels of 15, 30, and 50 inches
mercury absolute.


RESULTS AND CONCLUSIONS

Effect of Pressure and Temperature

The experimentally determined values of A at various temperatures
are shown in figures 2 to 5 and at various pressures in figures 6 and 7;
in each case, the values of A are plotted against the Mach number of
the stream. Analysis of the data gives an average value of 0.2 for the
exponent n in the expression obtained for A in the preceding section.
Upon substitution of this value into equation (9), the equation becomes

A = A(p/po)1/5 (TO/T)1/4 (11)

The degree of correlation may be seen in figures 2 to 7 where the data
are presented as plots of %0 against M, A0 for this purpose being
taken as


)P/5 0L/4 x(measured A)


The correlation appears sufficiently good to justify use of equation (11)
for purposes of temperature measurements.


Effect of Wire Diameter

The diameter effect is noted in figures 8 and 9, where the results
in tests on carefully welded and machined chromel-alumel and platinum
13 percent rhodium platinum wires of various diameters at ambient
pressure and temperature are presented. Analysis of the data similarly
gives an average value of 0.2 for the exponent n in equation (10),
which becomes


A = A (D/DO)1/5


(12)







NACA RM E54G22a


The correlation represented by equation (12) appears strictly applicable
only at Mach numbers below 0.8. However, use of the equation at higher
Mach numbers will not introduce errors greater than errors arising from
other causes.


Effect of Fabrication Quality

Figure 10 shows the extent of variations of A with M from tests
on several 0.040-inch and 0.020-inch chromel-alumel thermocouples which
had not been precisely welded and had been hand-filed to approximate
wire diameter. From these and similar tests it has been qualitatively
observed that, in general, the poorer the construction of the junction,
the lower the values of A throughout the range of Mach number; the
variations, however, are quite unpredictable. This fact offers an
explanation for the difference in the values of A against M at
T = 5400 R in the tests (figs. 2 to 5) performed in the high-temperature
apparatus as compared with those (figs. 6 and 7) performed in the air
jet. The former tests were performed at an earlier date when means were
not available for accurate welding and machining of thermocouple wires;
the latter tests were made when these means had become available but
the high-temperature test facility no longer existed.

In the case of machined wires at ambient temperatures, there appears
a pronounced maximum in the LA rinst M curve at Mach numbers
around 0.75. This effect presumably is the result of changes in the
flow pattern around the wires and is related to similar effects observed
in drag measurements on circular cylinders (ref. 7). The effect is
noticeably reduced on the unmachined wires at low temperatures and ap-
pears to vanish at the higher temperatures. The reasons for this are
not obvious.


Probable Value of 60 and Its Probable Error

For the pressure range of 0.5 to 2 atmospheres, temperature range
of 5000 to 20000 R, and diameter range of 0.01 to 0.04 inch, the most
probable value of 60 and the probable error in this value are given
in figure ll(a) for carefully machined wires and in figure ll(b) for
all wires, regardless of character of fabrication.

If the value of A is experimentally determined for a particular
thermocouple at room temperature and pressure, regardless of character
of fabrication, the probable error to be expected when the wire is used
at other temperatures and pressures and when correction equation (ll)
is applied is substantially the same as the probable error shown in
figure ll(a). The probable errors shown in figures 11(a) and (b) rep-
resent probable errors on the order of 1/4 and 1/2 percent, respec-
tively, in temperature measurements.







NACA RM E54G22a


It should be pointed out that the,results reported herein are
strictly applicable only for butt-welded wires of great length compared
with their diameters (L/D > 50), perpendicular to the gas stream, and
in a region free from interfering bodies. Should thermocouple wires
be mounted in a holder of comparatively large diameter, large differences
may be observed in the relation of A to M as a result of the inter-
ference effects; such a configuration would require calibration. The
effect of the interference of the support is illustrated by the data
published in reference 8 for a particular design of bare-wire type
thermocouple probe. These data are also indicated in figure ll(a) for
comparison with the data on very long isolated wires.


Lewis Flight Propulsion Laboratory
National Advisory Committee for Aeronautics
Cleveland, Ohio, July 23, 1954


REFEREE NC ES

1. Freeze, Paul D.: Bibliography on the Measurement of Gas Tempera-
tures. Circular 513, Nat. Bur. Standards, Aug. 20, 1951.

2. Scadron, Marvin D., and Warshawsky, Isidore: Experimental Determi-
nation of Time Constants and Nusselt Numbers for Bare-Wire Thermo-
couples in High-Velocity Air Streams and Analytic Approximation
of Conduction and Radiation Errors. NACA TN 2599, 1952.

3. Johnson, H. A., and Rubesin, M. W.: Aerodynamic Heating and Convec-
tive Heat Transfer Summary of Literature Survey. Trans. A.S.M.E.,
vol. 71, no. 5, July 1949, pp. 447-456.

4. Kennard, Earl H.: Kinetic Theory of Gases. First ed., McGraw-Hill
Book Co., Inc., 1938.

5. Scadron, Marvin D.: Analysis of a Pneumatic Probe for Measuring
Exhaust-Gas Temperatures with Some Preliminary Experimental Results.
NACA RM-E52All, 1952.

6. Pinkel, Benjamin, and Turner, L. Richard: Thermodynamic Data for the
Computation of the Performance of Exhaust-Gas Turbines. NACA WR
E-23, 1945. (Supersedes NACA ARR 4B25.)

7. Gowen, Forrest E., and Perkins, Edward W.: Drag of Circular Cylin-
ders for a Wide Range of Reynolds and Mach Numbers. NACA TN 2960,
1953. (Supersedes NACA RM AKSC20.)

8. Scadron, Marvin D., Warshawsky, Isidore, and Gettelman, Clarence C.:
Thermocouples for Jet-Engine Gas Temperature Measurement. Proc.
Inst. Soc. Am., Paper No. 52-12-3, vol. 7, .1:2, pp. 142-148.









NACA RM E54G22a


To atmosphere


Typical wall thermocouple


Pressure tubes


Insulating firebrick













hromel


hromel









'erence thermocouple


38From combustor
From combustor '------


Figure 1. Schematic diagram of test apparatus showing typical thermocouple installation.
(Pressure tubes and reference thermocouple are shown in same plane as test
thermocouples for convenience. Actual installation was perpendicular.)









NACA RM E54G22a


.04



CO
03 -O 1- --




O Total
.02 temperature,
T,
<> OR

0 0 540
S960
.01 1460
0 1960




0
.2 .3 .4 .5 .6 .7 .8 .9 1.


Mach number, M


FP-ire 2. Unmachined 0.040-inch chromel-alumel at various temperatures.
,Ic pressure, 30 inches mercury absolute; reference total tempera-
ture, 5400 R.


0









NACA RM E54G22a


.U4 -0- -E-

0 .


00

.032
_A A
.02




00







.05
oil
0---- -- --- --- -- --- -- ------- -- --- -- --__ --__ _


.05 r-- ---- -- -- ---j-- --i ---- ]--i-- --i ----i--
<












Total
.02 temperature,
T
o o








0 540
ID 2 1010
.01 0 1510
SA 2010




.2 .3 .4 .5 .6 .7 .8 .9 1.0
Mach number, M

Figure 3. Unmachined 0.020-inch chromel-alumel at various temperatures.
Static pressure, 30 inches mercury absolute; reference total tempera-
ture, 5400 R.
.01 -- -- -'- -- -- -- --- 0 1510 --









ture, 540 R.


)









NACA RM E54G22a


0 0 0 0


0 0


o 0
o A

0



0
0









0 <
-- -- -- -- -- -- -- -- ^ ^ -



0 0 0 0


S960
o

QTotal


146OR
0" 0 525
1 960
-_-- -- 0 1460
A 1960




2 .3 .4 .5 .6 .7 .8 .9 1.
Mach number, M

Figure 4. Unmachined 0.020-inch platinum 13 percent rhodium platinum
at various temperatures. Static pressure, 30 inches mercury absolute;
reference total temperature, 540 R.


0


.01




0


.05




.04




.03

AO


.02




.01









NACA RM E54G22a 13


.06














<0 0
.04.





.03--




Total
.02- -i OL temperature,
T,


0 525
0> i 960

U O A 1960



C __

.2 .3 .4 .5 .6 .7 .8 .9 1.0
Mach number, M

Figure 5. Unmachined 0.010-inch platinum 13 percent rhodium platinum
at various temperatures. Static pressure, 30 inches mercury absolute;
reference total temperature, 5400 R.








NACA RM E54G22a


Ub -- 1- 1-|--

Static pressure,
P,
in. Hg abs
05-
0 15 [
O 30
O 5o O

00


04 ---- --- ---- ---- --- -- --- -- -- -- --- -- --- -- --- --


04 -0



0




.02




015-- -- -- --- -- ------ -- -- -- -- --- -- -- -
030







.04 --------





03--
O
.05 --










.01
,02------ -- --- -- --- --- -- -- -- -- -- -- -- ---




.0
.2 .3 .4 .5 .6 .7 .8 .9 i.


Mach


number, M


Figure 6. Mvihlned 0.025-inch chromel-alumel at various pressures. Total
temperature, t.V4' R; reference static pressure, 30 inches mercury absolute.









NACA RM E54G22a


Static ____<>
pressure, <-
P,
.04-- n. Hg abs_ 0 0a
0 15 C
o 30 q

00
.05


ao

.02 ---





.01




0


.05 --




.04-- ----- -;0 '-^t
.04 _-___










.02 o--




.01




01 __
0 -- -- -- _-- ---- ---- ---- ----
.2 .3 .4 .5 .6 .7 .8 .9 i.
Mach number, M


Figure 7. Machined 0.019-inch platinum 13 percent
at various pressures. Total temperature, 5400 R;
pressure, 30 inches mercury absolute.


rhodium platinum
reference static









NACA RM E54G22a


.2 .3 .4 .5 .6 .7
Mach number, M


.9 1.0


Figure 8. Machined chromel-alumel of various diameters. Static pressure,
60 inches mercury absolute; total temperature, 5400 R; reference diameter,
0.020 inch.







NACA RM E54G22a


.06- I I
Diameter,
D,
in.

0 0.040 0
.05 .032
.025
.019
x
.017 -Ph

.04-----




.03 -





.02




.01




0-


.6
Mach number, M


1.0


Figure 9. Machined platinum 13 percent rhodium platinum of various diameters.
Static pressure, 30 inches mercury absolute; total temperature, 5400 R; reference
diameter, 0.020 inch.








NACA RM E54G22a


(a) Diameter, 0.040 inch.


.02





.01





0


.05





.04-





.03





.02





.01





0
.2


.9 1.0


Figure 10. Chromel-alumel wires of various character of fabrication. Static
pressure, 30 inches mercury absolute; total temperature, 5400 R.


.3 .4 .5 .6 .7 .8
Mach number, M

(b) Diameter, 0.020 inch.
















.05





04 / \


7--- -_._, .1 -i \ -

i/ *" ^-


.03 '


A


.0o -- --- -







(a) Machined wires.
.05- ------ -- -





.04 -


t r I


- -- --


2 .3 .4 .5 .6 .7 .8 .9 1.0
Mach number, M

(b) All wires regardless of character of fabrication.

Figure 11. Mean curve and probable error at reference static pressure of 30 inches
mercury absolute and reference total temperature of 5400 R. Pressure range, 0.5
to 2 atmospheres; temperature range, 5000 to 20000 R; diameter range, 0.01 to 0.04
inch; reference diameter, 0.020 inch.


NACA-Langley 9-24-54 350


NACA RM E54G22a


------~


- I













Su) .. .. | 2
v' :3-$42 2 0" o








0 E0
oFoU)='2
0, >. C) 4) g m w








02E 002 CC
a a a) (v









ca z
a.. 0 0d
U) >5
w *1.- i-l -lC;< c~i2
S, Qi O a0












0U 0 o
D I IU) n S CciC
.P.C-U
iPa -< g a --








0Z 0 r- ,,-s o
> 'S :03 w a
o- SO L5 ID~ 5)^ s- w 'd
MUW- E:%, 1.~s s 41"
O .E!
>, = = C bl) Q)
r. Ei Q) a 5 =, E
> cd r-0. 0 m)~

> a 1,Id C

z~ mmipk.-a W 06 0.
s&Sg 8?St^as| I
il 41
nlUW 0)000 i
OgO O^ fe' 3^3 OS in ,,
tdCI~ 9


4 C 6 CUi
h- CUc i
t -o

C4C
IU I o
>, E -. r-u i
0 k d "
E. 0 a E 44
-4 zO~e
gg| 24 :1-
s N i gig g


N? F; N Ryj

C-U E- S US
N Su .~ |S
,^/ 2|| p ll


14) Q)
V Sj C-0U)OE
as4 = .4Mz
C4 ': .4 .4i-


M
Ud,

= H < to
a);Dww w ~
g9r


cm~m v
dIZ 04 C -0
0 : E- <*
0 <



0 u 2=Pr
WL)WH E :
^3gs


CQ = = =mm



M P 4)1 C

;> wnFc~.Era
i|l|
4 f-Ct C
ZZiullCC


0 0
0 o 0












U.au2 a00 ^
BlSx u n
W H" WS 3 L.!
z z3 lz w 5 [-C


Mo 3 4) 1
CU > I

u 02U)
0 in .a C
C> 3
4jCd C
CU $ e 0 .4 4
( CUV 0 CU,
-g&oS32m



$4 a E -C5 :s 4,
Ioo~agSS=




M bD fc .-- Wc B
0 0
0 .0 w t

Ci 0 )0 -1 C U c

0 0 =o
*~.2 0-.M
3C- "I ~ s *


to) Im C.wa. 00 Z M,
Bb S- U)5.
(ucEB-eO)F
02 s o tg
Cl.f 2 l
C dr:- wsl
Hagi-Ow



' as sgads
gr : CU02a
m CU U ~0 CU ~
C..~9 :
00 ;" 0.
ar CU CU CU


I-


-I 00 00 C U
-u ~- N CUy



to CIS
r U
c c
w mc m, u



C4 C4.
">g.S .2 -aG^ >

*-01 1 0111 V 8
Ijl gisle *








Q 0 E C.i C 0 OU

.Sa l-i' rl>a (L., a


'2~ 0 (WEC..
C.. 0
i~ll, S-^^IJ~








0 2 d.E.0
0 02
ZZO 0 5Z





0 02 1:4 0 4) "C.
EC.U E 1'6 :'
SMw. EE



0 Ow~U OQ '..4

0 0.0
2 ^ 1 u u :s 00 a)







4,d -r ,
t8^ in v2e!c1



W .0 >4 C m 0 .



0
W W ow
CC9~~ B ~~8~ ac CC
> 0 ",!4 41 -1 m


tko~g w M g = 6 =
Cd WE. W W 0 6t Z 0 :, (1



l |Z II Do >4 v6 =
CC
u ~~:-:
0.02)O. U ~c










f








UNIVERSITY OF FLORIDA


3 126208106 608 5







UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
-. S`X 117011 !
'*" 1 1f .7l 1 .USA "

10.

F ::...


.l :


















"Pi


::
...:.=.:

".. t
.!;;