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51[B R A R Y

Mlchigan StatUniversity

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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

REFERENCES

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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

.010

.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

("H

O

m

m

D

*1

92

2.7

“‘1

‘O

(\J

if

r—i

"Y‘

C1

(*3

oy

=1.780

.9651

.107

.p

15

1.“7

1.32

7y

=$449

.9142)“

.061

.-

;-

1.21

0.70

5.00

.".“2

2.“

“J

N‘.

D

I

I?)

\J

(‘1

C

6

:\

7 J

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l

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(“J

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D

II

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11

y=

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1.31!

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2.76

3.0

9.08

2.6

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r '1

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3

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Krrffirgrt

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

.0“

on 3.3“

2.62

2.90

2.8

2.6

2.7

2.9

“9

XJPW'

6E89'

CI

9016'

100'

(x

LWI'I

C

go

PC

08'

EO'E

7L

(J

XQRE'

19‘

6F

ZEE6'

790

1\

JC

KO

8

"71¢.

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1.9

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CC

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7,14

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rv

1.1T1

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Add

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(:‘TFTAF‘

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rx

r—<

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v—.~

ILu).

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c-.-

Avy'n'r..

..v

_31:181.-

.-

-v'rlrfir

-;._.L

0

RESULT“

FROM

LINEAR

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

I”):

H

.98“2

.057

.2

01.97

1.27

2.95

3.16

3.9“

3.0

.986“

'.0“7

.“

01.75

1.06

2.92

3.02

3.50

3.0

Li'\

CO

IAN")

”J

(or

1\I

N

ON

CA

CO

O

m

:1”

II

>3

(“’7

1.33

0.67

2.9“

2.98

2.66

3.1

(1::

”O

L" I

C)

L.“

m

L! .

UK

M

:T

on

LG

II

>3

2'

10

1.29

0.73

3.06

3.15

2.58

3.0

(\J

[\,

L‘\

C)

V J

’3“

1:.

ON

(\1

H

[\

H

II

>3

Ll’\

10

1.59

1.31

3.02

3.09

3.18

3.0

O

(\1

m

\o

C)

CD

H

O\

O\

:1-

N

GO

r-I

ll

>3

O

1.71

1.37

3.1“

3.20

3.“2

2.9

0‘.

CO

(I)

H

II

>3

[\.

.992“

.063

.2

51.66

1.35

3.10

2.98

3.32

3.0

O\

[\

N

(I)

H

I |

>3

1‘0

960

.0“2

.

O

(U

1.75

1.37

3.1“

3.10

3.50

3.0

(\J

m

:r

H

II

>3

ON

.9902

.063

.“

51.57

1.20

3.20

3.13

3.1“

3.1

10

.9919

0.30

.8

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

‘—

.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

.99““

.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

>>

O'\

10

y

53

"71111111111111.1111ITS