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51[B R A R Y
Mlchigan StatUniversity
C
ABSTRACT
A STUDY OF THE SIZE, SHAPE AND SURFACE
ROUGHNESS OF APPLE FRUIT
by Eudell G. Vis
A study of the physical properties of apple fruit is
a logical approach in an investigation for developing
principles for mechanical orientation of the fruit in a
specified position. Orientation of the fruit is necessary
for quality determinations, packing and labeling, and in
certain processing operations. The properties investigated
in this study were size, shape, and surface roughness.
Mean radii were determined for several transverse
sections of individual apple by rotating the apple at a con-
stant angular velocity while the radius was measured by an
LVDT. By approximating the apple as a series of short
cylinders, the analog computer was utilized to calculate an
approximate volume and surface area. The approximate volume
was consistently within 10 per cent of the volume as deter—
mined by the conventional water displacement method. The
same was true for the approximate surface area compared
with surface area determined by the paper weighing method.
A linear regression analysis was performed to correlate
the measured mean radii to an elliptical model of an apple.
,For this model a longitudinal section consisted of a portion
of two identical ellipses translated and rotated from the
Eudell G. Vis
longitudinal axis. Variables describing the elliptical
model were tabulated and compared to the measured diameter
and length of the fruit. Longitudinal sections of apples
constructed using values obtained from the regression
analysis were more accurate in the cheek area than on the
stem and calyx ends. The characteristic shape associated
with a particular variety was not evident from the analysis.
The depth of the stem and calyx cavities were measured
to determine if differences in size could be used as criteria
to distinguish between the cavities. For McIntosh the stem
cavity depth was three times greater than the calyx cavity
depth. For Jonathan and Delicious the stem cavity depth
was nearly twice the calyx cavity depth. The results indi-
cate that the development of a mechanical sensor may be
feasible.
MicroscOpic surface features were studied by using a
Microcorder to record pen—drawn surface roughness profiles.
The number of roughness irregularities with respect to
location on the surface was investigated. The roughness
irregularities that had a width less than 0.02 inch and
height from 100 to 1000 microinches did not vary in number
: , 1Approved I A A441“
Major Professor
Approved éfi'k- M
Department Chairman
with respect to location.
A STUDY OF THE SIZE, SHAPE, AND SURFACE
ROUGHNESS OF APPLE FRUIT
By
Eudell G. Vis
A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Department of Agricultural Engineering
1967
ACKNOWLEDGMENTS
I would like to extend my thanks to Dr. B. A. Stout
(Agricultural Engineering) who served as my major
professor. His helpful suggestions and encouragement are
greatly appreciated.
Also, a special thanks to Dr. J. V. Beck (Mechanical
Engineering) who served as my minor professor.
Suggestions by Dr. S. Persson (Agricultural Engineer—
ing) on the use of the analog computer and the help Dr.
H. P. Rasmussen (Horticulture) provided in the surface
roughness study are sincerely appreciated.
The work reported herein was conducted under contract
No. l2-lA-lOO-8902(51) between Michigan State University
and the Market Quality Research Division, Agricultural
Research Service, U. S. Department of Agriculture.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . .
LIST OF TABLES . . . . . . . . . . .
LIST OF FIGURES . . . . . . . . . .
Chapter
1. INTRODUCTION . . . . . . . . . . .
2. LITERATURE REVIEW . . . . . .
2.1 Size and Shape of Fruit
2.2 Surface Roughness
2.3 Survey of Existing Fruit Handling
Equipment
2.“ Patent Review
3. OBJECTIVES . . . . . . . . . .
A. EXPERIMENTAL PROCEDURE AND EQUIPMENT
4.1 Instrumentation and Theoretical Discus-
sion for Size and Shape Measurements
“.2 Procedure to Measure Size and Shape of
Stem and Calyx Cavities
A.3 Description oflflicrocorder and EXperi—
mental Procedure
5. RESULTS AND DISCUSSION . . . . . . . .
5.1 Size and Shape
5.11 Volume and Surface Area Results
5.12 Linear Regression Analysis of Mean
Radii to the Equation of an
Ellipse
5.2 Discussion of Application of Measurements
from Size and Shape of Stem and Calyx
Cavities Experiment
5.3 Discussion of Surface Roughness
iii
Page
11
vi
13
1A
25
Chapter
6. SUMMARY .
7. CONCLUSIONS
REFERENCES
APPENDIX
iv
Page
38
A0
41
“5
LIST OF TABLES
Table Page
1. Comparison of volume measurements . . . . 26
2. Comparison of surface area measurements . . 28
3. Results from modified linear regression
analysis computer program . . . . . . 33
A. Mean values for height and width of stem and
calyx cavities . . . . . . . . 3A
5. Number of roughness irregularities per
.location . . . . . . . . . . . . 35
6a-6h. Results from linear regression analysis,
correlating mean radii to the elliptical
model for an apple fruit 46—53
V"
Figure
l.
10.
ll.
12.
LIST OF FIGURES
Approximation of an apple by a series of short
cylinders . . . . . . . . . . .
Nomenclature describing the parts of an apple
fruit (Mohsenin, 1965) . . . .
Apparatus and instrumentation used for volume,
surface area, and mean radius measurements
LVDT probe with rotating ball tip measuring
the fruit radius . . . . .
The integral of the differential element
multiplied by L is the expression programmed
on the analog Computer to obtain the volume
of an apple fruit . . . . . . . . .
Flow diagram illustrating instrumentation and
analog computer components necessary to com—
pute mean radii, surface area, and volume of
an apple fruit . . . . . . . . .
Top shows where measurements were taken on
typical Michigan Apples: Left-—McIntosh,
Right-—Jonathan, and Bottom-—Red Delicious
Microcorder components——pilotor, tracer with
skids resting on fruit, amplifier, and
recorder . . . . . . . . . . .
Apples used in surface roughness experiment.
Fruit grown in Michigan. Variety: A-—
Jonathan, B--McIntosh, and C—-Red
Delicious . . . . . . . . . .
Elliptical model of an apple fruit used in
linear regression analysis . . . .
MicrOSCOpic view (125X) of a cross—section of
the skin and cuticle of a Michigan Jonathan
Apple
Surface roughness profile as recorded by the
Microcorder
vi
Page
15
l5
l7
l7
l9
19
21
23
23
31
LA)
‘4
1. INTRODUCTION
Investigating engineering prOperties of apple fruits
reflects a recent philOSOphy of engineers working with
biological products that such properties should be quantita—
tively defined. Basic properties of biological materials
which are properly evaluated could be utilized extensively
in theoretical modeling, develOpment of processes and
machines, and in design work. Such is the case of metals
for which many engineering properties are well known and
useful mathematical descriptions can be readily employed.
For this study, physical properties will be considered
a sub—class of engineering properties relative to physical
characteristics of an object or substance such as weight,
size, shape, volume, area, and surface characteristics.
Why is the study of physical properties of apple fruit
important? Several reasons are, mechanization in handling
fruit, development of products from apple fruit which require
new automated processes, heating and cooling studies, and
studies in the areas of physiology and pathology. Machines
become economically feasible as seasonal labor becomes more
scarce and labor cost increases. Larger fruit farms with
centralized storage and packing facilities are factors pro—
moting efficient utilization of machines. Packing fruit in
smaller sized units for retail sales requires more labor or
mechanization of operations. The consumer is willing to pay
higher prices for uniformly sized and colored fruit that is
packaged in an attractive manner. The quality of processed
fruit is important to minimize labor required to trim bruised
portions. Exact peeling and coring operations by machine
eliminate unnecessary waste. Packing line Operations that
are or could be mechanized are bagging the fruit, tray pack—
ing fruit, labeling of individual fruit, and quality
evaluations by optical instruments.
Surface area, shape, and volume of fruit are important
in heat transfer studies. Surface area estimates are utilized
in the fields of pathology and physiology. Volume measure-
ments enter into space requirement calculations for storage
and shipping.
Recent development of Optical instruments for nonde-
structive methods of internal quality evaluation of agricul—
tural products such as fruit and eggs for commercial
application necessitates development of fairly sophisticated
handling procedures. Optical measurements vary with size,
shape and orientation of the fruit in relation to line of
travel of the light beam. Consequently, fruit uniform in
size and shape oriented in a specified position can be
evaluated more precisely.
2. LITERATURE REVIEW
2.1 Size and Shape of Fruit
Downing (1865) described apple fruit as globular,
oblate, oval, ovate, and conic for various varieties.
Mohsenin (1965) proposed a standard chart for describing
shape of apple fruit with similar terms such as round,
oblong, oblate, conic, unequal, oblique and truncate
approximating the longitudinal section and regular, irregu-
lar, and ribbed to describe the cross or transverse section.
Apple varieties were identified by considering the variable
characteristics: shape, length of stem, stem basin, eye
basin (calyx indentation), calyx, and skin was reported by
Williams and Child (1965). The least variable character—
istics for a particular variety were inspected in unknown
fruits to identify varieties. For example, length of stem
is normally constant within a variety. Variety shape was
classified by terms similar to those reported by Downing
and Mohsenin. Size of stem and calyx indentation, skin
color, amount of wax present on the surface, and conspicuous—
ness of the lenticels were observed.
The equation of an ellipsoid was utilized by Baten and
Marshall (1943) for surface area calculations which was a
lengthy process, however, today with computers the problem
could be programmed. A mathematical model of an apple fruit
developed by Moustafa (1967) approximates the fruit surface
with a portion of an ellipse displaced and at an angle from
a reference axis, which in effect is the stem—calyx axis,
hereafter, referred to as the longitudinal axis. The
ellipse is rotated about the longitudinal axis to generate
the complete surface. Volume and surface area can be cal-
culated from this model. Groves (1935) used the equation of
a cardioid, p = a (1 — cos 0) to describe the apple fruit
surface for surface area calculations.
Baten and Marshall (19A3) also reported a statistical
analysis relating the surface area of apple fruits to the
transverse cross sectional area, diametrical measurements,
and weight of the apple. In a study of heat transfer
involving Massachusetts McIntosh apples, Frechette and
Zahradnek (1965) tested Baten's results for McIntosh apples
grown in Massachusetts and reported a similar linear relation—
ship between surface area and fruit weight.
The % ratio was used as an index of fruit shape by
Westwood and Blaney (1965). This index did not indicate
subtle differences in shape, but was adequate to distinguish
between flat and elongated fruits.
Houston (1957) defined a new criterion of size as the
average of the projected areas taken along three mutually
perpendicular axes. Experimental measurements were obtained
to determine the relationship between the criterion area and
the volumes of lemons, potatoes, and carrots. To evaluate
the validity of this criterion of size, the experimental
measurements included: (a) variation of criterion area with
orientation, (b) variation of criterion area with shape, and
(c) variation of criterion area through the size range for
each product tested. AC = KV2/3 was the resulting mathe-
matical relationship, where AC is the criterion of size, K
a dimensionless constant, and V the volume.
Sphericity defined by Curray (1951) related the shape
of a solid to a sphere and roundness as a measure of the
sharpness of the corners of a solid.
Object area
Circle area
Roundness =
Object area is largest projected area measured in a
horizontal plane when object assumes a natural rest position
and the circle area is the area of the smallest circum-
scribing circle.
U
Sphericity = fig
s
where De is the diameter of a sphere of the same volume as
the product and DS the diameter of the smallest circum—
scribing sphere, or usually the longest diameter of the
product. Curray also discussed a shape factor which is a
dimensionless ratio describing the relationship between
characteristic dimensions of a body. For a spheroid,
characteristic dimensions are the semi-major axis (a) and
the major axis (b). Shape factor for
spheroid = b/a
sphere = b/a = 1
line of length 2a = b/a = 0.
Griffiths and Smith (1964) represented the volume of
an irregular object as V = ap bq or ... hw, V is the measured
volume of the body and abc...h are axis or diameters within
the body and each may be considered a measure of size.
2.2 Surface Roughness
A study of the surface characteristics of apple fruit
was initiated to determine if the characteristics varied
with respect to location on the fruit surface. If a measur-
able difference existed this fact could conceivably be
employed as an index for orientation.
Instruments are available that can produce pendrawn
profiles primarily of machined and finished surfaces. The
Microcorderl provides roughness profiles of surfaces that
are used extensively in research, inspection, and production
departments. This instrument has a tracer point (stylus)
with a small radius of curvature that follows the profile of
the surface. Vertical deflection of the stylus is converted
into an electrical signal, which is amplified to control the
movement of the pen that traces the profile on strip chart.
1Manufactured by Micrometrical Division, Bendix
Corporation, Ann Arbor, Michigan.
Movement of the tracer point is referenced to the
skids, therefore, large wavy undulations are not recorded.
However, microscopic surface irregularities are sensed by
the stylus and appeal:suitablywnagnified on the trace of the
surface profile on shart paper. On apple fruit the micro—
corder would record the minute surface irregularities, but
not the macroscopic features of the fruit such as the
general shape of the fruit.
The outer surface of apple fruits according to Smock
and Neubert (1950) is a layer of waxy and oil substance
called the cuticle. Directly below the cuticle are the
layers of epidermal cells followed by the layers of hypo-
dermal cells. On a mature apple the cuticle completely
covers the epidermal cells. A mature McIntosh apple has an
average cuticle thickness of 23 nanometers (10-9 meters).
Fisher (1965) describes the phenolic materials in the apple
cuticle in relation to the protection of the fruit on the
tree and in storage. Martin (1958) reported cuticle weight
in micrograms per square centimeter on apple fruit. Baker
(1962) investigated the growth of the cuticle during growth
and storage for three varieties, Cox, Worcester, and Bramley.
Skene (1963) reported using an electron microscope to
observe the wax platelets on certain apple varieties. For
Cox's Orange Pippin variety, the surface appeared to be
overlapping platelets ranging from 0.5 to A microns across.
Platelets for the Crawley Beauty variety were larger and had
a wrinkled, corrugated appearance. Skene (196A) reported the
variation in growth rate may account for the time that
minute cracks appear in the skin cuticle, but not the
distribution of the cracks. Further research (Skene, 1966)
indicated that the growth rate at the apple surface was
greater in the cheek area than on the stem and calyx ends.
A recorded trace of the surface profile at different
locations on the apple could be analyzed to determine if
surface characteristics differ with respect to location on
the apple fruit. Evaluation of surface profiles are briefly
described in mechanical engineering handbooks, however,
Brosheer (1947) gives a more detailed description of surface
roughness nomenclature as approved by American Standards
Association B 46.1 - 1962 published standards.
Important characteristics of total profiles are rough—
ness, waviness, size and shape of flaws, lay, nominal sur-
face, and mean surface. Roughness are the finely spaced
irregularities, often superimposed on a wavy surface. The
wavy surface is composed of surface irregularities which are
of greater spacing than the roughness irregularities. Flaws
are irregularities in the profile that occur at random
intervals. Metals have surface flaws such as a scratch, a
ridge, or a crack which can be attributed to imperfections
in structure or a machining error. Lay is defined as the
direction of the predominant surface pattern. A nominal
surface is the result if all peaks were leveled off into the
valleys and is the term commonly used to describe a geometri-
cally perfect two dimensional surface. A mean surface is
obtained by drawing a line which follows the wavy contour
of the profile. Height of the roughness irregularities are
measured from the mean surface.
2.3 Survey of Equipment in the
Fruit Handling Industries
Birth (1963) reported instruments can be used to
detect levels Of anthrocyanin in cherries, chlorOphy in
peaches, oranges, and apples, blood in eggs, water core in
apples, and brown substances in potatoes. Romani et. a1
(1963) used a spectrophotometer to determine light trans-
mitting characteristics of apricots. Interesting facts
reported in the use of these instruments by Birth and Romani
is that the orientation of the fruit in the optical instru-
ment affects the light transmitted through the fruit. Birth
(1964) reported obtaining best results in detecting water
core when the longitudinal axis of the fruit coincided with
the Optical axis in the sample compartment. Detecting water
core in Delicious apples is important since moderately water—
cored apples are acceptable for fresh market at harvest time.
In many instances, this condition spreads throughout the core
and surrounding tissue during storage, but does not affect
the outward appearance of the apple. Consequently, a signifi—
cant number of Delicious apples are marketed with serious
internal defects.
Procedure for fruit evaluation by Optical instruments is
to initially analyze the light transmitting properties of the
10
sample with a single beam spectrophotometer. Light Of dif—
ferent wavelengths are passed through the sample to deter—
mine which wavelengths will give optimum readings with a
two wavelength differential spectrophotometer. This
instrument records the difference between light transmitted
at two different wavelengths (A. O. D.). Light of wave-
lengths 670 and 730 nanometers have been found to give good
A. O. D. readings of apple fruits.
Deweyl reported that researchers interested in fruit
handling at Washington State University had the following
experiences. Measuring internal quality Of apple fruit by
light transmittance was feasible if the fruit was oriented
with stem up. They also noted brush roll sorting and sizing
equipment naturally orient Delicious apples with longitudinal
axis parallel to the rollers. One sizing unit had spring
loaded fingers to orient Delicious apples, however, this
concept was not effective on Winesap apples. F M C has a
machine to orient eggs accurately, which indicates Objects
with consistent shapes can be effectively oriented. Manu-
facturers are also interested in developing equipment for
automatic filling Of trays for apples and labeling of
individual fruits.
Continental Can Company2 reported average mechanical
peeling and coring losses for apple size 2% to 2% inches at
1D. H. Dewey, Personal correspondence, 1966.
9
2Canning Memorandum, "The canning of apples, sliced,‘
Continental Can Company, Inc., Chicago, Illinois.
ll
33 per cent and for larger apples 3 inches and over at 21
per cent. Equipment developed for orienting the apples
prior to these Operations, position the fruit either stem
up or calyx up. More exact positioning, with stem always
up, might result in reduced losses.
Lorenzen and Lamouria (1964) developed a system for:
aligning, metering, orienting, cutting, pitting, realigning,
reorienting, and spreading of apricots in preparation for
drying. Each fruit was oriented by existing drag forces
generated by the flow of water. The drag forces develop a
moment that rotates the fruit into a stable position.
2.4 Patent Review
An invention by Hait (1960) for orienting fruit with
one indentation and a suture line was designed for such
fruit as peaches and apricots. A mechanism similar to a
cup rotates the fruit to align the suture line and simul-
taneously a small wheel locates the indentation. Keesling's
invention (1965) has two feelers that locate the indenta—
tions and thus orients the longitudinal axis of the fruit.
This method is intended for orienting apple fruits particu-
larly for the peeling, coring, and slicing Operations.
Several devices for orienting pears based on the fact
that pears have an uneven distribution Of mass along its
longitudinal axis. The center of gravity located near the
bulb end stabilizes that end while the slender stem is
rotated in the desired position. Gardiner (1964) invented
12
an apparatus that uses a shuffling action to orient pears.
Anderson (1966) invented a mechanism to align fruit that
have an uneven distribution of weight about the plane pass-
ing through the largest transverse diameter. Thompson
(1947) utilized convoluted threads on tubes to transfer and
align the pear. Chamberlin (1965) developed a mechanism
to orient pears in a cup with cylindrical rollers in the
bottom of the cup to rotate the pear about several axis.
Another machine designed by Chamberlin (1965) orients pears
by pushing them through a trough with converging walls at
the bottom. As the pear passes between the walls, the small
end projects downward and is thereby oriented. Coons (1950)
develOped a mechanism that rolls pears down a sloping trough.
When the stem end of the pear is directed toward the low
end of the trough, the pear stops rolling and remains in
this position.
3. OBJECTIVES
Review of literature indicated Opportunity and need
for the development of new and improved handling equipment,
especially in the area of orientation of apple fruits. To
date, considerable time has been devoted to developments
in this area, however, no great success has been achieved.
The approach in this study was to investigate basic
prOperties Of apple fruit that may be beneficial in
developing principles for mechanical orientation of fruit
in a specified position. Orientation is defines as having
the longitudinal axis within 10 degrees Of a specified
position. For practical commercial adaptation the rate of
orientation should be at least 90 to 100 bushels per hour.
The objectives of this investigation were limited to:
l. A study Of size and shape of mature apple fruit.
2. A study of surface characteristics of mature
apple fruit.
13
4. EXPERIMENTAL PROCEDURE AND EQUIPMENT
4.1 Instrumentation for Size
and Shape Measurements
An apparatus was constructed to rotate an apple about
the longitudinal axis at a constant angular velocity. A
linear variable differential transformer (LVDT) with a
displacement of plus or minus one inch was used to sense
the radius of the apple as it was rotated. The output
voltage was amplified, fed into the analog computer, and a
mean radius was calculated for that particular transverse
section Of the apple. Approximating the apple by a series
of short cylinders % inch in height (Figure 1) the volume
and the value of frde were computed simultaneously on the
analog computer. The first measurement was taken approxi—
mately % inch from the stem end or in other words on the
cavity shoulder (Figure 2). Then measurements were taken
at 8 inch intervals on the cheek and the last measurement
taken at the basin shoulder. Thus, by measuring the mean
radius at % inch intervals the volume and surface area for
an apple were calculated.
Components of the apparatus for rotating the apple
were an electric motor, speed reducer, electric clutch and
brake, cam and micro—switch, and two prongs between which
the apple was located (Figure 3). Connected to the end Of
the LVDT probe was a rotating steel ball 0.375 inch in
14
15
....11 , e H .
Figure l.--Approximation of an apple by a series of
short cylinders.
AbscUnion
Cavity zone Stern
Show”: Sum End
\\\ Chvfly
/
F Con Lam
-———-Funh
\ \Chat \1 ‘ r s."
L. 3 r (RN.
0 Bundle:
’ Bosh
CoUmImes ‘qfiflflflg§;;'
Bosin Calyx End 0: Apu
Figure 2.-—Nomenclature describing the parts of an
apple fruit (Mohsenin, 1965).
16
diameter (Figure 4). The LVDT was centered directly above
the longitudinal axis and adjusted to have zero output at
the axis. LVDT output voltage was amplified by the
Daytronic and Dana amplifiers to have one inch LVDT dis—
placement equal to 2 volts input to the analog computer
(Figure 3).
The analog computer circuit for computing volume was
based on the integration of the expression % rzde, where
r = f (r) and is continuous in the interval aiefs (Figure 5).
The integral gives the total area bounded by rl at a, r2at B,
and the surface of the apple.
The cross sectional area of an apple can be Obtained by
fgflkrzde
since r = f (6). Volume would be the area multiplied by the
length, AL.
The mean radius is
- fgrde
However, dt can be substituted for de since the angular
velocity was constant giving E =
ti
ftordt
tifto dt
l7
\
1
I
i‘-
. . v -'
‘ O s
o ‘t -L» ,.,
[153 In
-" In’
674837
Figure 3.--Apparatus and instrumentation used for
volume, surface area, and mean radius measurements.
: 2 .662971—1
Figure 4.--LVDT probe with rotating ball tip
measuring the fruit radius.
18
This circuit was unstable on the analog computer since the
numerator was a finite value and the denominator was zero
at the beginning of the time interval. The expression
[Eidt is a constant, hence, a value for [Eirdt was computed
on the analog computer and a value for E was calculated
later.
The surface area for each short cylinder is 2nrdt.
Total surface area for one apple was calculated by adding
the surface area for each short cylinder and taking the
first (stem) radius and the last (calyx) radius to calculate
the area of circles with those radii to approximate the area
of the stem and calyx ends.
The flow diagram (Figure 6) illustrates the arrangement
Of components on the analog computer for computation of
ۤfr2dt and frdt. Computer output, Y1 and Y were recorded1
on an x-y recorder.
4.2 Procedure to Measure Size and
Shape Of Stem and Calyx Cavities
Commercial machines capable of orienting apple fruit
either stem or calyx up are available. This orientation is
necessary prior to the mechanical peeling and coring Opera-
tion on processed fruit. Considering this as an initial
step for exact orientation subsequent steps could theoreti-
cally be employed in a more exact orientation process. TO
help predict possible solutions, size and shape Of the stem
and calyx cavities of three Michigan apple varieties,
Jonathan, McIntosh, and Red Delicious, were investigated.
l9
2
I . z
dA= I/2 rzde
_4_
Figure 5.-—The integral of the differential element
multiplied byAIJis the expression programmed on the analog
computer to obtain the volume of an apple fruit.
_“
1
Transducer Transducer A . .
(LVDT) Amplifier Amplifier
Figure 6.--Flow diagram illustrating instrumentation
and analog computer components necessary to compute mean
radii, surface area, and volume Of an apple fruit.
20
A mathematical model of an apple developed by Moustafa
(1967) was used as a basic model. The model consists of a
portion of an ellipse rotated about the longitudinal axis
of an apple, generating the outer surface of the apple. In
two dimensions, a longitudinal section of the apple consists
of portions of two identical ellipses joined at the stem—
calyx or longitudinal axis.
Forty apples from each variety were cut through the
longitudinal axis and surface outline traced on paper. The
major axes were drawn through the ellipses, the longitudinal
axis were constructed at the low and high points of the stem
and calyx cavities respectively. Figure 7 shows where the
stem and calyx cavity depth and width were measured. The
maximum length was measured as the length of the major axis
of the ellipse and the maximum diameter measured perpendicu—
lar to the longitudinal axis of the traced outline. Apples
in the test were packaged as 2% inches and larger.
4.3 Description of Microcorder
and Experimental Procedure
Instrument specifications indicate the tracer point
has a maximum vertical displacement of 0.016 inch with respect
to the tracer skids. The skids, located on a skidmount, are
approximately 0.2 inches in length as measured in the same
direction as the line Of travel of the skids. Tracing speed
is 0.005 inch per second. Vertical sensitivity recorded on
the chart was 0.001 inch per % inch division on the chart or
21
c c
We
57 .. i so __.,,
24
1 3° 2:1 r
i .l
l2 H g f
45 9 9
43
I
1 'f
" «S
C
Figure 7.--Top shows where measurements were taken on
typical Michigan Apples: Left--McIntosh, Right—-Jonathan,
and Bottom--Red Delicious.
22
a magnification of 250 X. Horizontal sensitivity was 0.020
inches per 8 inch division on the chart or a 25 X magnifica-
tion.
Initially the same apparatus constructed for the size
and shape experiment was used to rotate the apple at a slow
rate to approximate the tracing speed in a stationary
position. In this case the apple was assumed to be
cylindrical in the cheek area where the measurements were
taken. One problem developed was that vibrations from the
motor and drive train were being picked up by stylus through
the apple. Therefore, it was decided to hold the apple
stationary by placing them in modeling clay while a surface
roughness trace was recorded by the Microcorder (Figure 8).
The stylus was moved approximately 0.2 inche over the surface
at each setting. Three profiles were recorded on each of
three general areas--the stem, cheek, and calyx of the fruit.
Thus, nine profiles were recorded for each fruit. A con-
siderable amount of time was necessary to record each trace,
therefore, the experiment was limited to three Michigan
varieties, Jonathan, McIntosh, and Red Delicious with traces
recorded from three apples from each variety (Figure 9).
Direction of tracer movement on the cheek was a line
parallel to the equator of the apple. On the stem and calyx
ends a uniform surface area was selected that could be
oriented in a horizontal plane with tracer movement in the
same direction as on the cheek. Location and the color Of
the apple at that location were recorded for each trace.
-m
‘-'
* -: "1.14,.
67483-6
Figure 8.—-Microcorder components--piloter, tracer
with skids resting on fruit, amplifier, and recorder.
67483-9
Figure 9.--Apples used in surface roughness experiment.
Fruit grown in Michigan. Variety: A-Jonathan, B—Mclntosh,
and C-Red Delicious.
24
To analyze a profile, a mean surface was drawn through
the profile with a french curve. This mean surface is not
the true mean surface excluding roughness irregularities,
but is the wavy features of the surface as sensed by the
stylus with reference to the skids. Values for waviness
heights and width can be measured from the mean surface.
However, the width of the waves when present, were nearly
equal or greater than the length of trace, hence, meaningful
values could not be Obtained. Another parameter describing
the nature of the surface is roughness irregularities.
Roughness may be visualized as irregularities superimposed
on a wavy surface. For this experiment, the roughness ir—
regularities having a width less than 0.02 inch and a height
in the range of 100 to 1000 micro-inches that appeared in a
0.2 inch trace were counted and recorded as the parameter
describing the surface. The irregularities may be either
above or below the mean surface. Table 9 contains the
results of the experiment.
5. RESULTS AND DISCUSSION
5.1 Size and Shape
5.11 Volume and Surface Area Results.--The volume
quantities as computed by the analog computer were compared
with volume measurements obtained by the conventional water
displacement method. A linear regression analysis revealed
the linear equation relating the two methods, a correlation
coefficient, and a standard error of estimate.
The following symbols represent:
y — volume by LVDT-—analog computer method.
x - volume by water displacement or surface area
by paper weighing method.
r — correlation coefficient
se- standard error of estimate.
The three Michigan varieties were run as one test.
The experimental (LVDT) values were reasonably close tO the
volume obtained by the water displacement method (Table l).
The slope of the regression equations for the McIntosh and
Jonathan were 1.016 and 0.972 respectively, which indicates
the two measuring methods were consistent for these varieties.
For Red Delicious the slope had a greater deviation from one,
however, the standard error of estimate was the lowest. The
correlation coefficient was highest for the Jonathan variety
which also had the largest size range within the sample Of
10 apples.
arm1]v L.) A
26
TABLE 1
ME MEASUREMENTS,
Michigan AUUIGS..v‘p \ 'H“ r). ‘7
"Q ‘“tJ ””n‘d Apple fimcer
l 4 S 6 10
ucIntosh Water 015-
placement 9.80 8.85 10.55 9.64 8 85 9 85 11.40 9.80 11.60 10.80
A”;T 1- 80 9 55 11.00 13.25 8 30 9.85 11.10 10.35 12.40 10.80
y = 11511 +1 "116 (X)
r = d‘fia
S_= .5‘)4
e
Janathan Water 015-
placement 15.90 13.90 13.25 8.70 7 15 8.10 11.45 9.7 7.35 11.00
LVDT 16.80 14.20 12.90 9.43 6.94 9.05 11.10 10.25 8.30 11.10
y = .6533 + 972 (x)
r = .9~38
5e: .329
Red Water Dis-
Dclicisus placement 13.13 12.00 14.25 13.75 13.40 14.70 14.20 11.80 11.80 12.05
LVDT 13.20 11.25 14.40 14.20 13.75 14.55 13.15 11.38 11.20 11.33
y = -2.694 + 1.185 (x)
r = .9420
s = 512
e
New Ycrk Appges
Red Rome Water Dis-
placement 18.59 18 19 15.11 15.63 18.83 1’ 85 15.28 16.50 17.02 17.37
LVDT 20.99 19.80 16.46 7 79 20.77 19.03 16 68 18.61 18.61 19.98
y = 3689 + 1 142 (x)
1‘ = 1:95.31";
sfi= 44;
McIntosh Water fis-
placement 1: an 14.1% 12.38 19.61 15.46 13 37 12 09 14.18 12.55 12.90
LVDT 1t 78 14.93 13.80 14.73 10 57 14 *4 13.54 16.55 13.83 14 83
y = 5.807 + 0.686 (x1
(,9: Bij‘wi
5’: 1.078
Ulsnington Apples
Winesap Water Dis-
placement 13.19 12.84 12.26 13.48 13.95 12.32 12.84 12.90 13.77 13.13
LVDT 14.81 14.49 14.79 15.44 15.21 14.05 14.71 14.36 16.06 14.86
y = 4.396 + .802 (x)
r = .7795
= {12se .38
Red Rome Water 015- _
placement 17.18 16.59 16 94 17.23 16.13 18.81 15.43 18.22 14.38 16.48
LVDT 18.62 16.96 18.92 19.31 17.28 20.14 16.64 18.34 15.18 18.49
y = .9765 + 1.016 (x)
r = .8872
se= .716
Red Water 015— - , ' , ’
Delicious placement 16.53 16.48 16.42 16.77 15.66 16.30 15.66 _lo.24 17.29 15.25
LVDT 16.65 16.58 16.09 17.38 16.31 17.78 17.36 17.87 18.70 17.35
y - 1.133 +.365 (x)
r = .2878
s = .769e
27
Actual surface area values were obtained by peeling
the apples, tracing around the peeling on graph paper,
cutting out the traced portions and weighing the paper.
Weight for one square inch of graph paper was found and
from this the actual surface area was found from the weight
Of the cut portions of graph paper. Standard error of
estimate ranged from 1.106 to 1.983 (Table 2). Correla—
tion coefficients were lower which indicates the model was
less accurate for surface area calculations.
The New York and Washington varieties were run in the
second test. Only volume comparisons were made for this
test. The LVDT values were consistently greater than the
water displacement values by approximately 10 per cent.
Calibration of instruments for radial measurement accuracy
was checked periodically by rotating a wooden cylinder with
known diameter, therefore, the most likely source of error
was in the approximating model. The apples in this test were
generally larger which indicates this approximation is less
accurate for larger fruit. Also, the correlation coefficients
are lower. The range of size of apples within each variety
used in the second test was generally very small. Conse—
quently, the constants in the linear regression equation had
a greater deviation from a perfect linear regression equation
where the y intercept in the equation would be zero and the
slope equal to 1.
TABLE
2
SURFACE
AREA
MEASUREMENTS,
SQUARE
INCHES
Variety
Method
Michigan
Apples
Apple
Number
McIntosh
Jonathan
Red
Delicious
Paper
Weighing
LVDT
y S
e
Paper
Weighing
LVDT
y r S
e
Paper
Weighing
LVDT
y r s+
e
3-099
r=
.7446
1.983
31.80
1.859
.9323
1.373
.2583
.8631
1.106
23.84
21.55
26.66
24.37
+1.195
(x)
28.05
29.80
30.09
+.964
(x)
28.37
26.89
29.79
26.82
+1.007
(x)
24.32
26.47
28.11
27.48
30.90
31.22
23.07
25.92
21.97
24.14
30.17
31.34
5
20.99
22.39
19.60
19.98
29.55
30.25
6
22.43
23.88
22.32
23.96
30.34
32.01
25.87
26.37
27.38
26.93
30.63
30.30
21.92
18.72
24.18
24.88
26.48
27.76
25.45
28.73
21.22
21.86
26.26
26.20
10
25.31
26.09
25.25
26.37
29.33
27.59
28
29
5.12 Linear Regression Analysis Of Mean Radii to the
quation of an Ellipse.--A linear regression analysis was
conducted in an attempt to fit mean radii as measured in the
preceding section to an equation Of an ellipse. Mean radii
r1, r2, r3 ... r1 had been measured as described previously.
Using a mathematical model similar to one developed by
Moustafa (1967) the measured mean radii values could be
modified and a best fit curve Of an ellipse fitted to the
data.
The equation for an ellipse is
X1 2 Y1 2
a2 b2
and solving for Yi we Obtain
Yl=b "gz' (X1).
A linear regression equation is of the form y = a' + b'x.
From an analogy of the two equations
2
y_Yl
a'= b2
2
bl:—
b 2a
x = Xi
The constants a and b can be calculated once a linear
regression equation is calculated.
30
A computer program was written to vary the position
Of the line B-B (Figure 10). Values for C varied from 0.2
to 0.8 in increments of 0.2 inches and the value of Z varied
from 0 to 15 degrees in 5 degree increments. The Y term1
used was
Y = r — C i AADJ F1
where AADJ DISCC (tan E), and DISCC = distance from origin I
l
Of coordinate system to cross section where ri was measured. j
Since tan Z E sin 8 for small angles, AADJ = DISCC (sin 2) j
was the expression used in the computer program. Values for
Y1 and S1 (X1 = DISCC) were squared and a linear regression
analysis performed on values of Y12 vs X12. Sixteen dif-
ferent equations for each apple were Obtained when an analysis
for each position of line B-B was performed. The equation
with the highest correlation coefficient would be the best
fit line to the modified data. Constants E 830 b were then
calculated from the best fit linear regression equation and
value Of C and Z were recorded. Standard error of estimate
was also recorded.
The correlation coefficients were of the order of 0.95
or higher for nearly all apples regardless Of variety.
Standard error Of estimate varied from a high of 0.293 to a
low of .005. Therefore the fit of the linear regression to
the modified data was very good. Model diameter (c + b) was
_fi_c_, Figure 10.——Elliptical model of an apple fruit used
in linear regression analysis.
P.
1
mean radii Of fruit as measured in LVDT
experiment.
translation Of axis from longitudinal axis of
fruit. Range from 0.2 to 0.8 inches.
rotation of axis with respect to line B-B which
is parallel to the longitudinal axis of the fruit.
origin of x' - y' coordinate system.
approximated by X , distance from O to the point
where r intersects longitudinal axis of fruit.
approximated by Y = r — C - X (sin Z) where Z
remains small, 0 - 15 degrees.
32
close to the actual maximum diameter as measured with a
calipers (Table 6). However, 2a was not a good approximation
for the length Of the apple fruit.
A model apple constructed using values of C, E, a,
and b which had the best correlation did not approximate
very closely the length Of the apple or the stem and calyx
cavities of the actual apple. For all varieties, when C was
equal to 0.2 inches the size Of the stem and calyx cavities
was too small. With C equal to 0.8 the cavities were too 7
large irreSpective of the rotation E. Of the total of 80 E}
apples, 46 had C values of 0.8, 5 of 0.6, 4 of 0.4, and 25
of 0.2 inches. Majority of the fruit had E values ranging
from 0—10 degrees with only two of the 80 apples having a z
value of 15 degrees. The most consistent variety was
McIntosh where 19 of 20 apples had E values of 0 or 5 degrees
and 75 per cent had C values of 0.8 inches.
It is evident from this analysis that the ellipse that
fit the data in the cheek area did not approximate a complete
longitudinal section of an apple.
The computer program was modified so that when six
mean radii were calculated, the length of the apple was
assumed to be 3 inches and Y1 was set equal to zero at the
stem and calyx ends or in other words the Yl values at the
ends of the major axis was read into the program to be zero.
The results shown in Table 3 indicates that this approach
gives more realistic values of C and E and a good comparison
between the model and measured values of the diameter and
33
TABLE 3
RESULTS FROM LINEAR REGRESSION ANALYSIS, CORRELATING MEAN
RADII TO ELLIPTICAL MODEL FOR APPLE FRUIT
(with yi = 0 at xi = L/2)
Sample NO. 1* Sample No. 2**
Regression Equation y = .8707 y =1.212
-.38 (x) -.535 (x)
Correlation Coefficient .9984 .9940
Standard Error of
Estimate .021 .058
C (in.) .6 .6
E (deg.) 5 5
a (in.) 1.50 1.50
b (in.) .93 1.1
Model Dia. (in.) 3.09 3.40
Actual Dia. (in.) 3.02 3.40
Model Length (in.) 3.0 3.0
Actual Length (in.) 2.85 2.90
*Washington——Red Delicious Sample NO. 10.
**New York—-Red Rome Sample NO. 5.
34
length. These values are also more consistent with the
measured values reported by Moustafa (1967).
5.2 Discussion of Application of
Measurements from Size and Shape
of Stem and Calyx Cavities
Experiment
The mean values tabulated in Table 10 show that the
size of the stem cavities was significantly larger than the
calyx cavities for all varieties. For individual fruits
the same trend was evident. Development of mechanical
devices to sense the difference in size of the two cavities
would be necessary for the purpose of orientation. The
sensors placed on mechanical arms could fit in the cavities
if the apple was oriented with the longitudinal axis in a
specified direction. The function of the sensing device
would be to determine the orientation Of the stem and calyx
cavities. The sensing device could be designed to measure
the volume of the cavities or possibly just the depth.
TABLE 4
MEAN VALUES FOR DEPTH AND WIDTH OF CAVITIES OF APPLES
(MEASUREMENTS ON 1/50 INCH SCALE)
Stem Calyx
Variety Height Width Height Width
McIntosh 19 57 6 43
Jonathan 22 60 12 49
Delicious 20 60 ll 49
35
5.3 Discussion of Surface Roughness
An analysis of variance for each variety indicated no
significant difference between locations at the 95% level.
Thus, the results of this experiment Show that the roughness
characteristic did not vary in a predictable manner over
the general surface area locations for the small number of
apples tested.
TABLE 5
NUMBER OF ROUGHNESS IRREGULARITIES PER LOCATION*
Variety Sample No. Stem Cheek Calyx
McIntosh l 14 3 ll
2 5 4 10
3 3 4 15
Delicious 1 18 28 7
2 8 23 30
3 23 12 10
Jonathan 1 27 42 43
2 24 34 36
3 34 63 61
*Values are total number Of irregularities for three
traces in each location with each trace 0.2 inch in length.
Figure 11 is a cryostat section 8 microns thick at a
magnification of 125X. The cuticle is clearly visible in
the photograph and the surface outline is irregular as
reported by Skene (1963). Skene (1964) also reported that
a greater number of cracks appeared in the stem and calyx
cavities than on the cheeks Of the fruit. The size and shape
36
of the cracks were not reported, therefore, it is impossible
to determine if the cracks would appear on the Microcorder
trace. Several profiles of the McIntosh variety exhibited
non—typical irregularities which may have been cracks in the
cuticle.
The Jonathan variety had a greater number of surface
irregularities than the McIntosh and Delicious varieties.
Figure 11 shows a microscopic view of a cross section Of a
Jonathan apple skin and cuticle. Figure 12 is a Microcorder
trace taken from the same apple.
The force of the stylus on a surface according to the
manufacturer is a maximum of 2% grams on 0.0005 inch radius
tip. As expected no damage to the cuticle was visible under
the microscope. The Microcorder might be superior to the
micrOSCOpe in one aspect that it traces out the profile
from which quantitative measurements can be taken.
Results of the sample of apples tested showed that
the Surface roughness parameter would be of questionable
value as an index for orientation.
Figure 11. -—Microscopic View (125X) of a cross-
section of the skin and cuticle of a Michigan Jonathan
Apple.
_ -- . 1996--.--7.7[ 7
Val-my -J a .5 .
bra-n - 7 I .
...‘..........1 ' = I , : ' i _ '
= 1006‘,. Menu g . . :1 ‘
' L10208 '1 ' .1
.. ‘1 ht...” '-_ . \‘ _ ' :1 5
Figure l2.--Surface roughness profile as recorded
by the Microcorder.
6. SUMMARY
Study of the physical properties of apple fruit,
such as size, shape, and surface roughness would be helpful
in the development of new mechanical handling equipment.
An acute shortage of labor, centralization of storage and
packing Operations, and the continuing demand for better
quality fruit are factors that influence the demand for
efficient mechanical handling equipment. Newly developed
optical instruments that detect water core in apple fruit
are effective when the apple is oriented in a specified
position with respect to the light source and the light
detecting component. Mechanical orientation of individual
fruit is necessary for commercial adaptation of these
instruments.
Review of literature indicated most information on
size and shape are general descriptions of fruit. 'Data on
size is usually reported as the maximum characteristic
dimension usually diameter of length. Since apple fruit are
irregular in shape it seemed advantageous to obtain average
measurements and apply them to geometrical models.
An apparatus and instruments were setup to rotate the
apple one revolution per 12.56 seconds about its longitudinal
axis while a linear variable differential transformer (LVDT)
measured the instantaneous radius of the apple. The
38
39
electrical signal from the LVDT was fed into an analog
computer to calculate the volume of an apple by approximating
the apple by a series of short cylinders. Also, a value
for frde was obtained from the computer for transverse sec-
tions of each apple at % inch intervals from which the
surface area and mean radii were determined. Comparison
of the approximated volume to volume as measured by water
displacement and approximated surface area to surface area
as obtained by the paper weighing method were reported.
The mean radii measurements were utilized in a linear
regression analysis in an attempt to fit a mathematical
model of an apple similar to the one developed by Moustafa
(1967) to the data.
The size of the stem and calyx cavities was measured
for the purpose of determining if some type of sensors could
be developed to distinguish the size difference between the
two cavities. These sensors might be useful when utilized
in conjunction with existing apple orienting equipment.
Microscopic surface roughness of apple fruit was
explored with the use of the Microcorder. If surface
roughness varied in a predictable manner with reSpect to
location on the fruit surface a roughness parameter might
be helpful for orientation. In the recorded roughness pro-
files a mean surface was drawn through the profiles. Rough—
ness irregularities that had a width less than 0.02 inch
and a height between 100 and 1000 microinches were counted
and recorded as the parameter describing the surface roughness.
7. CONCLUSIONS
The model approximating the apple by a series of short
(% inch) cylinders can be used to calculate volume and
surface area to within 10 per cent of the actual values.
Correlation coefficients in the linear regression
analysis were high indicating that a portion of an
ellipse does follow the surface contour of an apple on
the cheek area. For the elliptical model, the parameters
C and Z varied within the samples for each variety.
Whether the model can be used to describe varietal shape
differences has not been determined.
For all apples tested the stem cavity was significantly
larger than the oalyx cavity indicating the development
of a sensing device to distinguish between the two
cavities would be feasible.
The number of roughness irregularities did not vary in a
predictable manner with respect to location on the frUit
surface. Therefore, a surface roughness parameter would
be of questionable value to be used as an index for
orientation of the fruit.
HO
REFERENCES
11.1
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M2
”3
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APPENDIX
”5
TABLE
6a
RESULTS
FROM
LINEAR
REGRESSION
ANALYSIS,
CORRELATING
MEAN
RADII
TO
ELLIPTICAL
MODEL
FOR
APPLE
FRUIT
FRUIT
GROWN
IN—-MICHIGAN:
VARIETY-MCINTOSH
Sample
Regression
Correlation
Standard
cz
ab
Model
Actual
Model
Actual
No.
Equation
Coefficient
Error
of
(in.)
(deg.)
(in.)
(in.)
Dia.
Dia.
Length
Length
Estimate
(in.)
(in.)
(in.)
(in.)
\0
CD
.:r
:1-
II
>:
H
.9927
.025
.8
01.09
0.67
2.9“
2.78
2.18
2.2
\D
\O
:r
M
II
>3
N
.9972
.013
.8
51.0“
0.59
2.78
2.65
2.08
2.2
3y
=.“857
.9928
.028
.8
01.06
0.70
3.00
2.90
2.12
2.1
“y
=1.391
.9883
.058
~.2
51.“1
1.18
2.76
2.80
2.82
2.1
GO
ON
C
m
0
||
>3
Ln
.9“72
.0“9
.5
00.88
0.56
2.72
2.70
1.76
2.0
(I)
r—{
...—4
M
II
>5
\0
.993“
.018
.*
51.0“
0.56
2.72
2.80
2.08
2.1
L0
r—1
KO
,_4
II
>3
t‘
.9957
.0“8
.‘
V1.31
1.27
2.9“
2.82
2.5“
2.2
.6“98
.109
.‘
‘1.30
0.67
2.9“
2.76
2.00
2.2
:r
(\J
:1-
:1-
ll
>3
CI)
88
.9970
.033
.4
51.28
1.1“
2.37
2.85
2.28
2.3
“
359
.989“
.031
.f
51.06
0.66
2.92
2.80
2.12
2.2
“6
TABLE
6b
RESULTS
FROM
LINEAR
REGRESSION
ANALYSIS,
CORRELATING
MEAN
RADII
TO
ELIPTICAL
MODEL
FOR
APPLE
FRUIT
FRUIT
GROWN
IN——MICHIGAN:
VARIETY-RED
DELICIOUS
Sample
Regression
Correlation
Standard
0z
ab
Model
Actual
Model
Actual
No.
Equation
Coefficient
Error
of
(in.)
(deg.)
(in.)
(in.)
Dia.
Dia.
Length
Length
Estimate
(in.)
(in.)
(in.)
(in.)
.8226
.9972
.020
.“
51.29
0.91
—.“9“x
(\1
.67
3.0
2.58
2.6
II
>5
...: 2y
=.6898
.9993
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.6
10
1.20
0.83
2.86
2.9
2.“0
2.6
-.“8“x
3y
=.“681
.996“
.019
.8
10
1.28
0.69
2.98
3.1
2.56
2.7
—.285x
“y
=.“560
.9875
.032
.8
10
1.32
0.68
2.96
3.0
—.26“x
ay
=1.710
.992“
.077
-‘
10
l-“6
1-31
—.812x
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RESULTS
FROM
LINEAR
REGRESSION
ANALYSIS,
FRUIT
GROWN
TABLE
6d
IN——NEW
YORK:
CORRELATING
MEAN
RADII
TO
VARIETY——RED
ELLIPTICAL
MODEL
FOR
ROME
APPLE
FRUIT
Sample
No.
Regression
Equation
Correlation
Coefficient
Standard
Error
of
Estimate
N (I)
310
(in.)
Actual
Dia.
(in.)
10
ll
>3
|l
>3
||
>3
ll
>3
1|
>3
2.3“5
-.876x
2.229
—.817x
.OMOO
.38“x
.63“2
.391x
.812“
.“u3x
2.125
-.883x
2.092
—.910x
2.02“
-.7“2x
.7905
-.“66x
.72“2
—.3“8x
.9978
.9756
.9963
.99u8
.9896
.9916
.9u09
.991“
.9903
.9700
.0“5
.1“0
.018
.031
.0“9
.088
(\J
10
1.29
1.28
1.35
1.56
1.67
1.31
1.“5
0.90
1.“6
0.89
0.85
3.20
3.20
3.“O
3.32
3.30
3.36
3.28
3.18
3.13
3.“0
3.28
3.18
3.17
3.26
3.29
2.58
2.56
2.70
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on 3.3“
2.62
2.90
2.8
2.6
2.7
2.9
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FROM
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REGRESS
ON
ANALYSIS,
CORRELATING
MEAN
RADII
TO
ELLIPTICAL
MODEL
FOR
APPLE
FRUIT
FRUIT
GROWN
IH—-NA3LIHGPON:
VARIETY——RED
DELICIOUS
Sample
Regression
No.
Equation
0
L) I.)
$.40)
H-H
Standard
0z
a6
Model
Actual
Model
Actual
Error
of
(in.)
(deg.)
(in.)
(in.)
Dia.
Dia.
Length
Length
Estimate
(in.)
(in.)
(in.)
(in.)
C24.)
0
H <1)
12H
(0 o
m CH
I\
0
Kg
r—‘I
II
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.057
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01.97
1.27
2.95
3.16
3.9“
3.0
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01.75
1.06
2.92
3.02
3.50
3.0
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2.98
2.66
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LG
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0.73
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3.15
2.58
3.0
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3.18
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3.20
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2.9
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II
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51.66
1.35
3.10
2.98
3.32
3.0
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N
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H
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1‘0
960
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1.75
1.37
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3.10
3.50
3.0
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m
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H
II
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.9902
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51.57
1.20
3.20
3.13
3.1“
3.1
10
.9919
0.30
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51.33
0.7“
3.08
3.02
2.66
2.9
Ln
CO
m
Ln
I I
>3
51
TABLE
6g
RESULTS
FROM
LINEAR
REGRESSION
ANALYSIS,
CORRELATING
MEAN
RADII
TO
ELLIPTICAL
MODEL
FOR
APPLE
FRUIT
FRUIT
GROWN
IN-—WASHINGTON:
VARIETY—-RED
ROME
Sample
Regression
Correlation
Standard
0z
ab
Model
Actual
Model
Actual
No.
Equation
Coefficient
Error
of
(in.)
(deg.)
(in.)
(in.)
Dia.
Dia.
Length
Length
Estimate
(in.)
(in.)
(in.)
(in.)
II
>3
v—I
2.1“2
‘.9933
.087
.2
51.“8
1.“6
3.32
3.“0
2.96
2.7
-.979x
II
>3
N
.6819
.9“60
'.081
.8
01.25
0.83
3.25
3.28
2.50
2.7
—.“39x
3y
-2.290
.9959
.07“
.2
5l.“6
1.51
3.“2
3.33
2.92
2.7
-l.070x
uy-=
2.256
.9938
.090
.2
10
1.u6
1.50
3.“0
3.32
2.92
2.7
-1.056x
II
>3
Ln
.7“75
.9811
.05“
.8
01.22
0.87
3.3“
3.38
2.““
2.6
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.50“x
-
6y
=.793“
.9918
.0“2
.8
01.37
0.90
3.“0
3.“0
2.7“
2.8
.“23x
7y
=.7089
..9835
.0“9
.8
01.20
0.8“
3.28
3.25
2.“0
2.6
-.“90x
8y
=.767“
.99“2
.036
.8
51.32
0.88
3.36
3.30
2.6“
2.8
—.““1x
9y
=.58““
.9792
.0“7
.8
01.18
0.77
3.1“
3.10
2.36
2.5
-.“19x
10
y=
.67u5
.9871
.ou8
.8
51.32
0.82
3.2“
3.28
2.6“
'2.7
-.385x
52
TABLE
6h
RESULTS
FROM
LINEAR
REGRESSION
ANALYSIS,
CORRELATING
MEAN
RADII
TO
ELLIPTICAL
MODEL
FOR
APPLE
FRUIT
FRUIT
GROWN
IN-—WASHINGTON:
VARIETY-—WINESAP
Sample
Regression
Correlation
Standard
cz
a0
Model
Actual
Model
Actual
No.
Equation
Coefficient
Error
of
(in.)
(deg.)
(in.)
(in.)
Dia.
Dia.
Length
Length
Estimate
(in.)
(in.)
(in.)
(in.)
.“86“
.9881
.029
.8
01.20
0.70
3.00
2.96
2.“0
2.6
-.3“1x
ll
>3
r-i
1.699
.99u7
’.062
.2
51.u5
1.29
2.98
2.95
2.90
2.6
.790x
ll
>3
N 3y
=.“101
.9976
’.01“
.8
10
1.26
0.6“
2.88
2.91
1.52
2.6
.262x
“ye:
.“036
.5071
.183
.8
0l.“2
0.6“
2.88
3.18
2.8“
2.6
.199x
5y
=.5977
.9300
.091
.8
01.19
0.77
3.16
3.18
2.38
2.6
—.“2“x
6y
=.“872
.9939
.021
.8
01.19
0.70
3.00
2.89
2.38
2.6
.3“5x
7y
=.5697
.9998
.005
.8
51.16
0.76
3.11
3.02
2.32
2.5
—.“2“x
1.690
.9927
.036
.2
51.77
1.30
3.00
2.92
3.5“
2.6
-.5“0x
II
>3
oo
1.801
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.065
.2
51.50
1.3“
3.08
2.9“
3.00
2.7
—.799x
.5318
.9860
.031
.8
01.25
0—73
3.06
3.09
2.50
2.6
—.3“2
ll
>>
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10
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53
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