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Scientia Horticulturae 114 (2007) 21–26

Mass modeling of pomegranate (Punica granatum L.) fruit

with some physical characteristics

F. Khoshnam, A. Tabatabaeefar *, M. Ghasemi Varnamkhasti, A. Borghei

Department of Agricultural Machinery, Faculty of Biosystems Engineering, University College of Agricultural and Natural Resource,

University of Tehran, Karaj, Iran

Received 14 March 2007; received in revised form 4 May 2007; accepted 15 May 2007

Abstract

Among physical characteristics, dimensions, mass, volume and projected areas are important parameters in sizing and grading systems. Fruits

with the similar weight and uniform shape are desirable in terms of marketing value. Therefore, grading fruit based on weight reduces packing and

handling costs and also provides suitable packing patterns. The different grading systems require different fruit sizing based on particular

parameters. In this study pomegranate mass was predicted by applying different physical characteristics with linear and nonlinear models as three

different classifications: (1) single or multiple variable regressions of pomegranate dimensional characteristics, (2) single or multiple variable

regression of pomegranate projected areas and (3) estimating pomegranate mass based on its volume. The results showed that mass modeling of

pomegranate based on minor diameter and three projected areas are the most appropriate models in the first and second classifications, respectively.

In third classification, the highest determination coefficient was obtained for mass modeling based on the actual volume as R2 = 0.99 whereas

corresponding values were 0.93 and 0.79 for assumed pomegranate shapes (oblate spheroid and ellipsoid), respectively. In economical and

agronomical point of view, suitable grading system of pomegranate mass was ascertained based on minor diameter as nonlinear relation

M = 0.06c2 � 4.11c + 143.56, R2 = 0.91.

# 2007 Elsevier B.V. All rights reserved.

Keywords: Pomegranate; Mass modeling; Physical characteristics; Grading; Packing; Saveh township

1. Introduction

The pomegranate (Punica granatum L.) fruit, native of Iran,

is extensively cultivated in Spain, Egypt, Russia, France, China,

Japan and USA and in India (Patil and Karade, 1996).

Pomegranate is consumed directly as fresh seeds as well as

fresh juice which can also be used in flavoring and coloring

agents (La Rue, 1969). The edible part of the fruit contains

considerable amounts of acids, sugar, vitamins, polysacchar-

ides, polyphenoles and important mineral. Pomegranate fruits

are most widely grown in Iran. The annual production of

pomegranate is equal to 700,000 tonnes and more than

150,000 tonnes is exported to other countries.

Physical characteristics of agricultural products are the

most important parameters in design of grading, conveying,

processing and packaging systems. Among these physical

* Corresponding author. Tel.: +98 261 2801011; fax: +98 261 2808138.

E-mail address: atabfar@ut.ac.ir (A. Tabatabaeefar).

0304-4238/$ – see front matter # 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.scienta.2007.05.008

characteristics, mass, volume, projected areas and center of

gravity are the most important ones in sizing systems (Malcolm

et al., 1986; Safwat, 1971). Other important parameters are

width, length, and thickness (Mohsenin, 1986). There are some

situations in which it is desirable to determine relationships

among physical characteristics; for example, fruits are often

graded by size, but it may be more economical to develop a

machine which grades by weight. Therefore, the relationship

between weight and the major, minor and intermediate

diameters is needed (Stroshine and Hamann, 1994).

The regression analysis was used by Chuma et al. (1982)

to develop equations for predicting volume and surface area.

They used logarithmic transformation to develop equations for

wheat kernels at 15.7%. They suggested that the volume was

related to the surface area by a linear regression relationship:

V = 1.10S + 17.2. Frequently, the surface area of fruit is

determined on the basis of its diameter or weight. Knowing the

diameter or weight of a fruit, its surface area may be calculated

using empirical equations, or read from an appropriate plot

(Sitkei, 1986; Frechette and Zahradnik, 1968).

Nomenclature

a major diameter (mm)

b intermediate diameter (mm)

c minor diameter (mm)

CPA criteria projected area (mm2)

d average diameter of calyx (mm)

GMD geometric mean diameter (mm)

Ki regression coefficient

M mass (g)

PA1 first projected area (mm2)

PA2 second projected area (mm2)

PA3 third projected area (mm2)

R2 coefficient of determination

V volume (cm3)

Vellip volume of ellipsoid(cm3)

Vm measured volume (cm3)

Vosp volume of oblate spheroid (cm3)

W weight of displaced water (g)

Greek symbol

g weight density of water (g/cm3)

Fig. 1. Components of WinArea-Ut-06 system (Keramat Jahromi et al., 2007).

F. Khoshnam et al. / Scientia Horticulturae 114 (2007) 21–2622

Consumers prefer fruits with equal weight and uniform

shape. Mass grading of fruit can reduce packaging and

transportation costs, and also may provide an optimum

packaging configuration (Peleg, 1985). Sizing by weighing

mechanism is recommended for the irregular shape product

(Stroshine and Hamann, 1994). Since electrical sizing

mechanism is expensive and mechanical sizing mechanism

reacts poorly; therefore, for pomegranate, dimensional method

(of length, area and volume) can be used. Determining

relationships between mass and dimensions and projected areas

may be useful and applicable (Stroshine and Hamann, 1994;

Marvin et al., 1987). In weight sizer machines, individual fruits

are carried by cups or trays that may be linked together in a

conveyor and are individually supported by spring-loaded

mechanism. As the cups travel along the conveyor, the supports

are engaged by triggering mechanisms which allow the tray to

dump if there is sufficient weight. Successive triggering

mechanisms are set to dump the tray at lower weight. If the

density of the fruit is constant, the weight sizer sorts by volume.

The sizing error will depend upon the correlation between

weight and volume (Stroshine and Hamann, 1994).

In the case of mass modeling, Tabatabaeefar et al. (2000)

determined models for predicting mass of Iranian grown orange

from its volumes, dimensions, and projected areas. They

reported that among the systems that sorted oranges based on

one dimension, the system that applies intermediate diameter is

suitable with nonlinear relationship.

The physical properties of different fruits and vegetables

have been determined by other researcher; caper fruit (Sessiz

et al., 2007), potato (Singh et al., 2006; Sadowska et al.,

2004a,b), gumbo fruit (Akar and Aydin, 2005), pear (Wang,

2004), onion (Abhayawick et al., 2002), apple (Woensel et al.,

1987). Also, some physical properties of pomegranate have

been investigated and reported by several researchers (Kingsly

et al., 2006; Fadavi et al., 2005; Kaya and Sozer, 2005; Al-

Maiman and Ahmad, 2002; Safa and Khazaei, 2003) and

others.

No detailed studies concerning mass modeling of pome-

granate have been performed up to now. The objective of this

research was to determine an optimum pomegranate mass

model based on its physical characteristics. This information is

used to design and develop of sizing systems.

2. Materials and methods

This research was conducted on Malase-Torsh Saveh variety

(export variety) obtained from three different regions as Agh-

Dareh, Solghan and Soghanligh located in Saveh township

(latitude: 358, 010E and longitude: 50, 200N). The number of

fruits obtained from the aforementioned regions was 81, 57 and

54, respectively.

The mass of each pomegranate (M) was measured using a

digital balance with accuracy 0.01. Three mutually perpendi-

cular axes (a) major diameter (total height of fruit), (b)

intermediate diameter and (c) minor diameter were measured

by applying WinArea-Ut-06 system (Fig. 1) developed by

Mirasheh (2006). Average diameter of pomegranate calyx (d)

was measured. Then projected areas (PA) in three perpendicular

directions of pomegranate using WinArea-Ut-06 were deter-

mined. Dimensional characteristics obtained from WinArea-

Ut-06 are based on image processing. Captured images from a

camera are transmitted to a computer card which works as an

analog to digital converter. Digital images are then processed in

the software and the desired user needs are determined.

Through three normal images of the fruit, this device was

capable of determining the required diameters as well as

projected areas perpendicular to these dimensions. Total error,

for those objects that was taken at the camera field, was less

than 2%. This method have been used and reported by several

researchers (Rafiee et al., 2006; Keramat Jahromi et al., 2007).

The average projected areas (known as criteria projected areas),

F. Khoshnam et al. / Scientia Hor

geometric mean diameter and sphericity were calculated as

suggested by Mohsenin (1986):

Criteria projected areas ðCPAÞ ¼ PA1 þ PA2 þ PA3

3;

Geometric mean diameterðGMDÞ ¼ffiffiffiffiffiffiffiabc

3p

sphericity ¼ GMD

a

The actual volume of pomegranate was determined by the

water displacement method (Akar and Aydin, 2005; Aydine and

Musa Ozcan, 2007). Randomly selected pomegranate was

placed with a metal sponge sinker into a measuring cylinder

containing known water volume such that the fruit did not float

during immersion in water; weight of water displaced by the

fruit was recorded. The volume of each fruit was calculated by

following equation (Mohsenin, 1986).

Actual volumeðcm3Þ ¼ W

g

where W and g were considered as weight of displaced water

and weight density of water, respectively.

The bulk density was determined using the mass–volume

relationship by filling an empty plastic container of pre-

determined volume and weight; the pomegranates were placed

from a constant height and weight (Fraser et al., 1978).

Spreadsheet software, Microsoft EXCEL 2003, was used to

analyze data and determine regression models between the

studied parameters.

In order to estimate the pomegranate mass from dimensional

characteristics, projected areas and volume, three classifica-

tions of models were considered as follows:

1. S

Ta

So

Pro

Ma

Int

Mi

Av

Ge

Sp

Ma

Vo

Bu

Fir

Se

Th

ingle or multiple variable regressions of pomegranate

dimensional characteristics: major diameter (a), intermedi-

ate diameter (b), minor diameter (c) and average diameter of

calyx (d).

2. S

ingle or multiple variable regressions of pomegranate

projected areas: PA1, PA2 and PA3.

3. S

ingle regression of pomegranate volumes: actual volume,

volume of the fruit assumed as oblate spheroid and ellipsoid

shapes.

ble 1

me physical characteristics of Malase-Torsh Saveh variety

perty Mean M

jor diameter (mm) 116.0 1

ermediate diameter (mm) 95.3 1

nor diameter (mm) 91.1 1

erage diameter of calyx (mm) 17.7

ometric mean diameter (mm) 100.2 1

hericity (%) 89.4

ss (g) 290.9 4

lume (cm3) 296.2 4

lk density (g/cm3) 0.982

st projected area (mm2) 67.48

cond projected area (mm2) 73.19

ird projected area (mm2) 71.06

In the case of first classification, mass modeling was

accomplished with respect to major, intermediate and minor

diameters and the average diameter of calyx. Model obtained

with four variables for predicting of pomegranate mass was:

M ¼ k1aþ k2bþ k3cþ k4d þ k5 (3)

In this classification, the mass can be estimated as a function of

one, two, three and four dimension(s).

In second classification models, mass of pomegranate was

estimated based on mutually perpendicular projected areas as

following:

M ¼ k1PA1 þ k2PA2 þ k3PA3 þ k4 (4)

In this classification, the mass can be estimated as a function of

one, two or three projected area(s).

In the case of third classification, to achieve the models

which can predict the pomegranate mass on the basis of

volume, three volume values were measured or calculated. At

first, actual volume Vm as stated earlier was measured then the

pomegranate shape was assumed as a regularly geometrical

shape, i.e. oblate spheroid (Vosp) and ellipsoid (Vellip) shapes

and, thus, their volumes were calculated as:

Vosp ¼4

3p

�a

2

��b

2

�2

Vellip ¼4

3p

�a

2

��b

2

��c

2

�

In this classification, the mass can be estimated as either a

function of volume of supposed shapes or the measured actual

volume as represented in following expressions:

M ¼ k1Vosp þ k2 (5)

M ¼ k1Vellip þ k2 (6)

M ¼ k1Vm þ k2 (7)

3. Results and discussion

A summary of some selected physical characteristics of the

pomegranate fruit and linear regression models based on the

selected independent variables have been represented in

Tables 1 and 2, respectively.

ticulturae 114 (2007) 21–26 23

aximum Minimum Standard deviation

35.7 69.5 7.68

26.5 79.5 6.94

06.3 76.9 6.82

28.1 12.0 2.71

17.0 84.8 6.57

95.8 76.6 3.2

39.1 187.0 52.3

45.0 191.5 53.9

1.029 0.922 0.014

90.38 49.05 9.35

99.67 53.23 9.21

96.6 51.12 9.34

Table 2

Pomegranate mass models based on selected independent variables

No. Models Parameter Soghan-lighe Solghan Agh-Dareh Total of observations

1 M = k1a + k2 R2 0.68 0.57 0.50 0.57

R.S.E. 32.34 33.85 34.79 34.57

2 M = k1b + k2 R2 0.93 0.93 0.84 0.88

R.S.E. 15.10 13.58 19.48 18.50

3 M = k1c + k2 R2 0.92 0.91 0.92 0.91

R.S.E. 16.61 15.97 13.62 15.73

4 M = k1d + k2 R2 0.28 0.07 0.11 0.15

R.S.E. 48.07 43.93 46.47 48.33

5 M = k1a + k2b + k3c + k4 R2 0.95 0.96 0.95 0.94

R.S.E. 12.19 11.16 11.01 12.76

6 M = k1a + k2b + k3c + k4d + k5 R2 0.96 0.96 0.95 0.94

R.S.E. 11.81 11.26 11.05 12.67

7 M = k1PA1 + k2 R2 0.98 0.96 0.97 0.97

R.S.E. 7.33 10.19 8.64 9.60

8 M = k1PA2 + k2 R2 0.97 0.96 0.96 0.96

R.S.E. 9.68 10.32 9.23 10.30

9 M = k1PA3 + k2 R2 0.97 0.95 0.97 0.96

R.S.E. 9.99 11.85 8.60 10.28

10 M = k1PA1 + k2PA2 + k3PA3 + k4 R2 0.99 0.97 0.99 0.98

R.S.E. 5.24 8.72 4.59 6.74

11 M = k1V + k2 R2 0.99 0.99 0.99 0.99

R.S.E. 3.48 4.71 3.47 4.10

12 M = k1Vosp + k2 R2 0.96 0.93 0.92 0.93

R.S.E. 11.93 13.99 13.88 14.19

13 M = k1Vellip + k2 R2 0.86 0.80 0.77 0.79

R.S.E. 21.07 23.28 23.84 13.76

F. Khoshnam et al. / Scientia Horticulturae 114 (2007) 21–2624

3.1. First classification models, dimensions

Among the first classification models Nos. 1, 2, 3, 4, 5 and

6 shown in Table 2, model 6 where all four dimensions

were considered had the highest R2 value and regression

standard error was also the lowest for all the three regions.

However, all four diameters must be measured for the model

6, which make the sizing mechanism more tedious and

expensive. Among the models 1, 2, 3 and 4, model 3 had

the highest R2 value and the lowest R.S.E. for the entire

regions. Therefore, model 3, among the one-dimensional

models was selected as the best pomegranate mass model

Fig. 2. Pomegranate mass model based on minor diameter.

with minor diameter as shown in Fig. 2. Eleven models for

predicting mass of apples based on geometrical attributes

were recommended by Tabatabaeefar and Rajabipour (2005).

They recommended an equation calculating apple mass on

the basis of minor diameter as M = 0.08c2 – 4.74c + 5.14,

R2 = 0.89. In another study, Lorestani and Tabatabaeefar

(2006) determined models for predicting mass of kiwi

based on physical attributes. They recommended an equation

to calculate kiwi fruit mass based on intermediate diameter

as

M ¼ 293b� 64:15; R2 ¼ 0:78

The mass model of pomegranate (for the entire regions)

based on the model 6 (all the four diameters) is given in

Eq. (8).

M ¼ 1:17aþ 2:44bþ 3:99cþ 0:71d � 453:48;

R2 ¼ 0:94; R:S:E ¼ 12:67 (8)

For the entire regions, the best equation to calculate mass of

pomegranate based on the minor diameter was given in non-

linear form of Eq. (9).

M ¼ 0:06c2 � 4:11cþ 143:56; R2 ¼ 0:91; R:S:E ¼ 15:43

(9)

Fig. 4. Pomegranate mass model based on volume of assumed oblate spheroid

shape.

Fig. 3. Pomegranate mass model based on one projected area.

F. Khoshnam et al. / Scientia Horticulturae 114 (2007) 21–26 25

3.2. Second classification models, projected areas

Among the second classification models, Nos. 7, 8, 9 and 10,

shown in Table 2, the model 10 for the entire regions had

maximum R2 value and minimum R.S.E. The overall mass

model based on three projected areas (model 10) for the entire

regions, was given in Eq. (10) as:

M ¼ 2:72PA1 þ 1:09PA2 þ 1:80PA3 � 101:44

R2 ¼ 0:98; R:S:E: ¼ 6:74(10)

The overall mass model of pomegranate based on the one

projected area as shown in Fig. 3, was given as nonlinear form

in following equation:

M ¼ 1:29ðPA1Þ1:28 ;R2 ¼ 0:96 (11)

The mass model recommended for sizing kiwi fruits based on

any one projected area was reported by Lorestani and Tabata-

baeefar (2006) as:

M ¼ 1:098ðPCÞ1:273; R2 ¼ 0:97

where PC is third projected area. Each one of the three

projected areas can be used to estimate the mass. There is a

need to have three cameras, in order to take all the projected

areas and have one R2 value close to unit or even lower than R2

for just one projected area; therefore, model using only one

projected area, possibly model 7 can be used.

3.3. Third classification models, volume

Among the models in third classification (models 11, 12, 13),

the R2 for model 11 had maximum value and minimum R.S.E.

Among the models 12 and 13, the model 12 for the entire regions

had the highest R2 value and the lowest R.S.E. Therefore, model

12 was recommended for predicting pomegranate mass. The

mass model of overall pomegranates based on measured volume

as shown in Fig. 4, was given as linear form of Eq. (12).

M ¼ 0:96V þ 4:25; R2 ¼ 0:99 (12)

Tabatabaeefar (2002) determined physical properties of com-

mon varieties of Iranian grown potatoes. Relationships among

physical attributes were determined and a high correlation was

found between mass and volume of mixed potatoes with a high

coefficient of determination as:

M ¼ 0:93V � 0:6; R2 ¼ 0:994

Measuring of actual volume is time consuming task, there-

fore, mass modeling based on it is not reasonable; consequently it

seems suitable to mass modeling of pomegranate be accom-

plished based on volume of assumed oblate spheroid shape.

4. Conclusions

1. The recommended equation to calculate pomegranate mass

based on minor diameter (model 3 was the best) was as

nonlinear form:

M ¼ 0:06c2 � 4:11cþ 143:56; R2 ¼ 0:91

2. T

he mass model recommended for sizing pomegranates

based on any one projected area (model 7 is suitable) was as

nonlinear form:

M ¼ 1:29ðPA1Þ1:28; R2 ¼ 0:96

3. T

here was a very good relationship between mass and

measured volume of pomegranates for the entire regions

with R2 as 0.99 (highest R2 value among all the models).

4. T

he model which predicts mass of pomegranates based on

estimated volume, the shape of pomegranates considered as

oblate spheroid was found to be the most appropriate (model

12 is recommended).

5. A

t last, mass model No 3 from economical standpoint is

recommended.

Acknowledgments

The authors express their appreciation to University of Tehran

for full support of the project. It is necessary to thank Reza

Mirasheh, Kamran Kheiralipour, Mahdi Keramat Jahromi, and

Mojtaba Naderi Boldaji for their valuable helps and supports.

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