The American Society of Mechanical Engineers

A NG US-IM PR O V ED TECHNIQUE FOR CENTRIFUGAL-PUMP-EFFICIENCY MEASUREMENTS 23
104, tangent respectively to the two menisci, and is read by
means of a traveling microscope, as explained by the author. If
this method of reading is to be valid, however, the “rectangle”
abdc must be reproduced as such in the camera, which requires
(1) that the scale be located in the same plane as the center
lines of the tubes, and (2) that the camera back be parallel to this
plane. Departure from either or both of these conditions results
in a distortion of the form displayed in Fig. 10B, with obvious
inaccuracies in the results. (If the conditions producing this
distortion were in the opposite direction, then side bd, Fig. 10B,
would be less than oc.) It is generally much simpler to treat
these conditions separately as follows:
(а) Focal plane of camera parallel to plane of manometer tubes,
but scale either in front of or behind this plane. This is the condition
depicted in Fig. 10C, and gives rise to the obvious relation:
True differential U' z
---------------- ------------------------------------ _ --- = J d= —
Apparent differential (as read from photograph) U U
where, U‘ = distance from lens to tube axes
U — distance from lens to scale (which may be greater
or less than [/')
z = axial distance between tube axes and scale
If the venturi differential is very large, it is often impossible or
unwise to set the camera up midway between the two extremes
of the differential, and in such circumstances it is necessary to
raise the lens with respect to the camera back, as shown in Fig.
10C. Tilting of the entire camera is of course undesirable. If
needed for the refractive correction mentioned under section (c),
the partial angles of view, di (top) and di (bottom), may be computed
from the respective elevations of the lens and the two menisci,
easily obtained from measurements above floor level.
(б) Scale in same plane as manometer tubes, but camera back
not parallel to this plane. While simple enough, by means of a
spirit level, to make the camera back vertical, it is not so easy to
eliminate swing in the horizontal direction (i.e., in azimuth).
If tp is the amount of this swing, and 6i (6i‘ or 6 i) the angular
elevation of the ray passing through the meniscus, the amount
of the discrepancy resulting due to these conditions is given,
in magnitude only, by the expression S sin tan 0i,9 where S is
the transverse distance between gage tube and scale, Fig. 10A.
This discrepancy corresponds to distance 66' or dd' (aa' or cc' in
the opposite condition) shown in Fig. 10B. Representative
values are given in Table 5 of this discussion for S = 1 in. and,
since the discrepancies for upper and lower readings are additive,
the error may easily become appreciable.
The values of Table 5 are primarily of academic interest, as
it is usually difficult to determine the amount of swing \p. The
simplest and best method of correction is to place the scale as
close as possible to the manometer tubes (preferably between
them), and to exercise care in aligning and leveling the camera.
Otherwise two scales should be used, one on either side of the
manometer and with their zeros accurately adjusted to the same
level. Thus, whatever the resultant shape of the original
rectangle, the horizontals ab and cd will be always defined, respectively,
by like graduations on the two scales; this may, of
course, involve some additional labor in reading. A word of
caution should be given regarding linear measurements taken
from photographs, as the linear magnification will be nonuniform
from point to point except under strict conditions of parallelism.
Hence, the necessity, apart from the convenience, of referring the
measurements to an actual scale incorporated with the manometer.
(c) Even if the errors considered in sections (a) and (6) are
• This assumes that the lens axis is normal to the camera back, and
directed centrally toward the subject.
T A B LE 5 E R R O R S IN R E C O R D E D P O S IT IO N O F O N E M E N ISC U S
D U E TO N O N PA R A L L E L IS M O F M A N O M E T E R A N D C A M ERA
BACK (B O T H V E R T IC A L )
(T ransverse distance betw een tu b e and scale 1 in.)
' -------------------- H orizontal swing \p---------- ■— ——•
E levation
of ray fli 5 deg 10 deg 15 deg
deg in. in. in.
5 0.008 0.015 0.023
10 0.015 0.031 0.046
15 0.023 0.046 0.069
T A B L E 6 R E F R A C T IV E C O R R E C T IO N S IN V E N T U R I
M A N O M E T E R
(Scale assum ed to be in plane of tu b e axes; cam era back parallel to plane of
m anom eter)
Angle subtended
---------- by differential--------- . -------------- Corrections--------------- .
di t 0ib T op B ottom
Item
T otal (top), (bottom ),
deg deg deg
m eniscus,
in.
meniscus.
in.
T otal,
in.
1 10 5 5 0.008 0.008 0.016
2 30 15 15 0.024 0.028 0.052
3 60 30 30 0.055 0.071 0.126
3a
3b
60
60
15
45
45
15
0.024
0.115
0.151
0.028
0.175
0.143
4 90 45 45 0.115 0.151 0.266
N o t e : W a ll of tube =» 0 .2 1 in.; bore of tube — 0 . 2 0 in. Refractive
indexes: Air 1 .0 0 ; Pyrex glass 1 .4 8 ; water 1 .3 3 .
eliminated, there still remain the corrections due to refraction.
These have already been discussed by the author, and it will
suffice here to indicate how the values may be obtained analytically.
The following equations assume that the scale is in the
plane of the tube axes, and that the camera back is parallel to this
plane. In computing these corrections, the value of ffi (9i or 8ib)
is obtained as suggested under section (a), and the other angles
follow from Snell’s law of refraction:
where N is the absolute index of refraction and 0 is the angle
between the ray and the normal to the refracting surface; subscripts
refer to the different media. The corrections to the
meniscus readings are always negative and are as follows:
For top meniscus
where W = wall thickness and B = bore of tube. Table 6 of this
discussion shows representative values for assigned values of 6i.
The values given considerably exceed those encountered during
the tests, and are included merely to demonstrate the trend of the
correction. Items 3, 3a, and 36 of Table 6 show the effect of
unequal angles above and below the optical axis, while item 4
shows what might be expected if a “wide-angle” lens were used
on the camera to permit photographs to be taken in cramped
quarters.
Such corrections as those indicated should be quite satisfactory
if actual conditions were the same as those assumed, but
this is seldom likely to be the case, as irregularities in the glass
tubing or slight deviations from the vertical would alter considerably
the optical characteristics. By far the safest procedure is
to keep the angular subtense as small as possible by proper arrangement
of the camera; any correction which may arise will
then be small, and the calculated value not likely to depart much
from the proper value. Although the equations appear rather
formidable, a few values will suffice to plot curves (corrections
against 0i) adequate to cover any given series of tests.
With the mercury head gage, the recording is usually much