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Research Paper

Mathematical model of mechanical behaviour ofJatropha curcas L. seeds under compression loading

David Herak a,*, Abraham Kabutey a, Monika Divisova a, Satya Simanjuntak b

aDepartment of Mechanical Engineering, Faculty of Engineering, Czech University of Life Sciences Prague, Kamycka 129, Prague,

Czech Republicb Faculty of Economy, University of Tapanuli, Silangit, North Tapanuli, Indonesia

a r t i c l e i n f o

Article history:

Received 16 October 2012

Received in revised form

10 December 2012

Accepted 18 December 2012

Published online 1 February 2013

* Corresponding author.E-mail address: herak@tf.czu.cz (D. Hera

1537-5110/$ e see front matter ª 2012 IAgrEhttp://dx.doi.org/10.1016/j.biosystemseng.20

The tangent curve function was used to develop mathematical models to describe the

mechanical behaviour of Jatropha curcas L. seeds at different pressing vessel diameters and

seed pressing heights under compression loading. Three different pressing vessel diame-

ters; 60, 80 and 100 mm in relation to seed pressing heights; 30, 40, 50, 60, 70 and 80 mm

were used for the compression tests. Based on the statistical analysis results, the tangent

curve function was fitted by determining the force coefficient of mechanical behaviour and

coefficient of mechanical deformation behaviour described the deformation behaviour of

the seed pressing heights of jatropha with respect to the pressing vessel diameter. In

addition, the average stress coefficient of mechanical behaviour, average force coefficient

of mechanical behaviour and average compression coefficient with respect to the pressing

vessel diameter were all fitted by the general tangent curve equation.

ª 2012 IAgrE. Published by Elsevier Ltd. All rights reserved.

1. Introduction In the literature, the physical properties and the mechan-

To design suitable pressing equipment for processing Jatropha

curcas L. seeds with minimum energy performance, it is

important to fully understand the mechanical behaviour of

bulk material under compression loading. Essentially, this

means that the relationship between the compressive force

and deformation of the bulk seeds during linear pressing

needs to be determined, analysed and then to transformed to

the deformation characteristics used in nonlinear environ-

ments involving the use of screw extruders or presses.

However, the amount of the pressing force can influence

the output oil. Therefore, determining the correct pressing

force could help achieve greater amounts of oil with

minimum energy input (Herak, Gurdil, Sedlacek, Dajbych, &

Simanjuntak, 2010; Kabutey, Herak, & Hanus, 2010).

k).. Published by Elsevier Lt12.12.007

ical behaviour of jatropha seeds have been described

(Garnayak, Pradhan, Naik, & Bhatnagar, 2008; Karaj & Muller,

2010; Pradhan, Naik, Bhatnagar & Vijah, 2009; Sirisomboon &

Kitchaiya, 2009; Sirisomboon, Kitchaiya, Pholpho, &

Mahuttanyavanitch, 2007). Nevertheless, studies on the

mechanical behaviour of jatropha seeds, have only described

the amount of the rupture force and other mechanical prop-

erties without giving detailed description of the complete

course of deformation characteristics. This is very relevant for

energy requirement analysis for obtaining the oil (Karaj &

Muller, 2011; Pradhan et al., 2011). Recently, some authors

have also reported these deformation characteristics on

jatropha seeds using the tangent curve equation (Herak et al.,

2010). The tangent curve equation was later modified to

describe the mechanical behaviour of bulk seeds of jatropha,

d. All rights reserved.

Nomenclature

AD Force coefficient of mechanical behaviour, (kN)

ADA Average force coefficient of mechanical

behaviour, (kN)

BD Coefficient of mechanical deformation behaviour,

(mm�1)

C Stress coefficient of mechanical behaviour,

(N mm�2)

CA Average stress coefficient of mechanical

behaviour, (N mm�2)

CV Coefficient of variation, (%)

D Inner diameter of pressing vessel, (mm)

F General compressive force, (N)

Fcrit Critical value that compares a pair of models, (e)

FD Compressive force, (N)

Frat Value of the F test, (e)

G Compression coefficient, (e)

GA Average compression coefficients, (e)

H Initial height of bulk seeds, (mm)

I, II Subscripts indicate different diameter of pressing

vessels, (e)

Mc Moisture content, (% d.b.)

ms Mass of the mixture, (g)

Pf Porosity, (%)

Pvalue Significance level at which it the hypothesis of the

equality of models can be rejected, (e)

R2 Coefficient of determination, (e)

SD Pressing plunger area, (mm2)

V Initial volume of pressing vessel, (m3)

x Deformation of bulk seeds, (mm)

rb Bulk density, (kg m�3)

rt True density, (kg m�3)

s Compressive stress inside bulk seeds, (MPa)

Table 1 eDetermined physical properties of bulk seeds ofjatropha; data in the table are means ± SD.

H (mm) V (mm3) ms (g) Mc (% d.b.) Pf (%)

D [ 60 mm

30 84,834 � 5650 30.04 � 0.90 8.5 � 0.2 63.49 � 3.22

40 113,112 � 4600 45.20 � 0.29 8.5 � 0.2 58.80 � 2.86

50 141,390 � 3800 54.83 � 0.74 8.5 � 0.2 60.02 � 2.99

60 169,668 � 6200 66.56 � 1.03 8.5 � 0.2 59.56 � 3.42

70 197,946 � 4800 75.16 � 1.62 8.5 � 0.2 60.86 � 3.12

80 226,224 � 6340 88.04 � 1.55 8.5 � 0.2 59.88 � 3.61

D [ 80 mm

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sunflower, rapeseeds and other materials such as spruce

wood chips and paper chips (Herak, Kabutey, & Sedlacek,

2011; Herak, Kabutey, Sedlacek, & Gurdil, 2011). The descrip-

tion of the mechanical behaviour and deformation charac-

teristics using the tangent curve function has been verified by

the finite element method (Petru et al., 2012). The modified

tangent curve equation can be described by:

FDðxÞ ¼ AD½tanðBDxÞ�2 (1)

where FD(x) is the compression force (N) for deformation of

bulk seeds, x (mm), AD is force coefficient of mechanical

behaviour (N) with respect to the pressing vessel, D (mm) and

BD is coefficient ofmechanical deformation behaviour (mm) in

relation to the diameter of the pressing vessel. The deforma-

tion characteristic (Eq. (1)) shows that the force coefficient of

the mechanical behaviour AD influences the slope of the

deformation characteristic whereas deformation coefficient

of the mechanical behaviour BD also influences the range of

deformation. The product of these two coefficients AD$BD (N

mm) is essentially the initial rigidity of the system.

The aim of the study was to verify the general tangent

curve equation suitable for the description of mechanical

behaviour of J. curcas L. seeds under compression loading for

different pressing diameters and seed pressing heights.

30 150,816 � 12,000 56.86 � 0.53 8.5 � 0.2 61.13 � 2.95

40 201,088 � 12,680 77.13 � 1.69 8.5 � 0.2 60.46 � 3.21

50 251,360 � 13,520 98.35 � 1.24 8.5 � 0.2 59.66 � 3.45

60 301,632 � 11,950 116.44 � 0.36 8.5 � 0.2 60.20 � 3.12

70 351,904 � 12,650 141.32 � 0.51 8.5 � 0.2 58.60 � 3.45

80 402,176 � 12,980 157.49 � 1.21 8.5 � 0.2 59.63 � 2.65

D [ 100 mm

30 235,650 � 14,320 87.76 � 3.38 8.5 � 0.2 61.61 � 3.12

40 314,200 � 15,630 119.42 � 2.23 8.5 � 0.2 60.82 � 3.45

50 392,750 � 14,980 156.34 � 2.29 8.5 � 0.2 58.96 � 3.22

60 471,300 � 15,690 188.33 � 2.11 8.5 � 0.2 58.80 � 3.45

70 549,850 � 16,380 218.78 � 5.22 8.5 � 0.2 58.98 � 2.93

80 628,400 � 16,900 255.13 � 6.08 8.5 � 0.2 58.14 � 2.69

H e different pressing seed heights, V e initial volume of bulk

seeds,msemass of bulk seeds,Mcemoisture content of bulk seeds

in dry basis, Pf e porosity of bulk seeds, D e inner diameter of

pressing vessel.

2. Materials and methods

2.1. Sample

J. curcas L. seed of variety IPB2 obtained from North Sumatra,

Indonesia was used in this experiment. The physical prop-

erties are presented in Table 1. The moisture content Mc

(% d.b.) of the samples was determined using moisture

equipment (Farm Pro, model G, Czech Republic). Samples of

weight 100 g of a batch of jatropha seeds were randomly

selected for the moisture content determination. The mass of

each sample ms (g) was determined using an electronic

balance (Kern 440-35, Kern & Sohn GmbH, Balingen,

Germany) and the porosity Pf (%) was calculated from bulk

and true density using the relationship given by porosity

formula (Eq. (2)) (Blahovec, 2008).

Pf ¼�1� rb

rt

�100 (2)

where: Pf (%) is the porosity, rb (kg m�3) is the bulk density

and it was determined as the weight of the sample divided

by initial volume of pressing vessel V (m3) and

rt ¼ (980 � 12) kg m�3 is the true density which was deter-

mined by the hydrostatics method (Blahovec, 2008). Three

samples were tested and the results averaged.

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2.2. Compression test

To determine the relationship between the pressing force and

deformation, the compressive device (ZDM, model 50,

Germany) was used to record the course of deformation

function. Three pressing devices and plungers of diameters

60, 80 and 100 mm (Fig. 1) were used. Six initial heights 30, 40,

50, 60, 70 and 80 mm of the bulk seeds were measured and

pressed at the rate of 1mms�1 under the temperature of 20 �C.The measuring range of force was between 0 kN and 100 kN.

The experiment was repeated three times for each initial

height. Individual measurements were digitally recorded and

analysed with the addition of a deformation of 0.5 mm.

2.3. Determination of general curve

The measured amounts of compressive force and deforma-

tion of jatropha seeds for different pressing heights and

diameters of pressing vessels were analysed with computer

program Mathcad 14 (MathCAD 14, PTC Software, Needham,

MA, USA), (Pritchard, 1998) uses LevenbergeMarquardt algo-

rithm for data fitting (Marquardt, 1963) which is optimal for

tangent curve approximation. Tangent curve equation (Herak

et al., 2010), which was originally used to describe the

mechanical behaviour of the jatropha seed under compres-

sion loading, wasmodified as a general curvewhich is suitable

to describe the deformation characteristics (Herak, Kabutey,

Sedlacek, & Gurdil, 2011) and was verified by the finite

element method (Petru et al., 2012). This curve can be

described by Eq. (3) which is derived by modifying the basic

Fig. 1 e Scheme of pressing vessel.

equation of tangent curve (Eq. (1)) in which element in the

brackets is multiplied by ratio of initial pressing heights, H.

FDðxÞ ¼ AD

htan

�BDH

xH

�i2(3)

Product of coefficients of mechanical deformation behav-

iour BD and initial heights H can be called G (e), the

compression coefficient, and it is described by Eq. (4).

G ¼ BDH (4)

Substituting Eq. (4) into Eq. (3), the following can be obtained

considering the initial height of pressing:

FDðxÞ ¼ AD

htan

�GxH

�i2(5)

The compression stress sD(x) (MPa) inside the pressing

vessel with the bulk seeds (Eq. (6)) can be determined by

dividing equation (Eq. (5)) by pressing plunger area SD (mm2)

(Eq. (7)), where D (mm) is inside diameter of pressing vessel.

sDðxÞ ¼ FDðxÞSD

(6)

SD ¼ p D2=4 (7)

Considering the assumption which is very well known from

the mechanics of heterogeneous materials (Budynas &

Nisbett, 2008), that the compression stress inside the bulk

seeds is not dependent on diameter of pressing vessel and is

based on the assumption that the porosity of the material is

constant. From these assumptions, it follows that the pressing

process of the various pressing vessel diameters could be

similar. This hypothesis of the properties of the bulk seeds can

be described by Eq. (8), where subscripts I and II indicate two

different diameters of pressing vessels.

sðxÞ ¼ sDIðxÞ ¼ FDI ðxÞ

SDI

¼ sDIIðxÞ ¼ FDII ðxÞ

SDII

(8)

Dependency between individual courses of compression

force (Eq. (9)) can be given by adjusting the previous equation.

FDI ðxÞ ¼ FDII ðxÞSDI

SDII

(9)

Substituting Eq. (9) into Eq. (5) an equation describing how

the compressive force also depends on the diameter of the

pressing vessel can be derived.

FDI ðxÞ ¼ ADII

htan

�GxH

�i2D2I

D2II

(10)

The ratio of the force coefficient of mechanical behaviour

and the corresponding square of pressing vessel diameter can

be denoted as C (N mm�2) this being the stress coefficient of

mechanical behaviour (Eq. (11)).

C ¼ ADII

D2II

(11)

By simply adjusting the compressive force equation (Eq.

(10)) and using Eq. (11), the general tangential curve which

describes the dependency between compressive force and

deformation of bulk seeds, the inner diameter of pressing

vessel and initial height of pressing can be derived.

Fig. 2 e Measured amounts of mechanical characteristic of different pressing seed heights for diameter of pressing vessel,

D [ 60 mm with displayed amounts of coefficients of variation and their fitted functions by general equation Eq. (12). The

numbers in the chart indicate initial pressing height.

Fig. 3 e Measured amounts of mechanical characteristic of different pressing seed heights for diameter of pressing vessel,

D [ 80 mm with displayed amounts of coefficients of variation and their fitted functions by general equation Eq. (12). The

numbers in the chart indicate initial pressing height.

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Fig. 4 e Measured amounts of mechanical characteristic of different pressing seed heights for diameter of pressing vessel,

D [ 100 mm with displayed amounts of coefficients of variation and their fitted functions by general equation Eq. (12). The

numbers in the chart indicate initial pressing height.

Table 2eDetermined coefficients of deformation characteristics for different initial pressing seed heights and diameters ofpressing vessel for bulk seeds of jatropha and their statistical analysis.

H (mm) AD (kN) BD (mm�1) Frat (e) Fcrit (e) Pvalue (e) R2 (e) CV (%)

D [ 60 mm

30 4.284 0.064 2.345 � 10�3 3.934 0.954 0.993 5.0

40 4.142 0.053 3.460 � 10�4 3.936 0.961 0.994 4.5

50 4.527 0.041 9.798 � 10�6 3.924 0.985 0.996 5.2

60 4.114 0.033 5.739 � 10�6 3.915 0.998 0.995 5.3

70 3.504 0.029 1.126 � 10�3 3.919 0.998 0.996 5.0

80 2.320 0.026 5.404 � 10�3 3.918 0.942 0.995 5.0

D [ 80 mm

30 5.245 0.076 5.393 � 10�3 3.957 0.942 0.996 5.9

40 6.948 0.050 6.210 � 10�4 3.945 0.98 0.998 5.3

50 5.057 0.041 2.125 � 10�3 3.949 0.963 0.999 6.1

60 5.142 0.035 3.758 � 10�3 3.949 0.951 0.999 5.2

70 5.367 0.029 9.348 � 10�3 3.936 0.923 0.998 5.3

80 4.778 0.026 9.298 � 10�3 3.915 0.923 0.999 6.0

D [ 100 mm

30 11.251 0.066 1.756 � 10�3 3.957 0.967 0.999 5.3

40 12.090 0.047 3.046 � 10�3 3.947 0.956 0.999 6.2

50 11.331 0.040 0.015 3.940 0.904 0.998 5.0

60 13.352 0.032 9.167 � 10�3 3.938 0.924 0.997 4.9

70 10.500 0.028 0.015 3.934 0.902 0.999 5.4

80 11.251 0.025 0.016 3.936 0.899 0.998 5.1

He different pressing seed heights, ADe force coefficient of mechanical behaviour, BDe deformation coefficient of mechanical behaviour, Frat e

value of the F test, Fcrit e critical value that compares a pair of models, Pvalue e the significance level at which it can be rejected the hypothesis of

equality of models, R2 e coefficient of determination, D e inner diameter of pressing vessel, CV e coefficient of variation.

b i o s y s t em s e ng i n e e r i n g 1 1 4 ( 2 0 1 3 ) 2 7 9e2 8 8 283

Fig. 5 e Dependency between deformation coefficients of mechanical behaviour and initial pressing seed heights (mm). The

legend in the chart indicates diameter of pressing vessel (mm).

b i o s y s t em s e n g i n e e r i n g 1 1 4 ( 2 0 1 3 ) 2 7 9e2 8 8284

Fðx;D;HÞ ¼ CD2$ tan GxH

(12)

Table 3 e Determined average amount of stresscoefficients of mechanical behaviour, force coefficients ofmechanical behaviour and compression coefficients, datain the table are means ± SD.

D (mm) CA (N mm2) ADA (kN) GA (e)

60 1.06 � 0.21 3.82 � 0.74 2.03 � 0.07

80 0.85 � 0.11 5.42 � 0.71 2.09 � 0.09

100 1.16 � 0.09 11.63 � 0.90 1.96 � 0.04

D e inner diameter of pressing vessel, CA e average stress coeffi-

cient of mechanical behaviour, ADA e average force coefficient of

mechanical behaviour, GA e average compression coefficients.

h � �i2

Based on Eq. (12) can be seen that the dependency between

force coefficient of mechanical behaviour AD and diameter of

pressing vessel D can be described by:

ADðDÞ ¼ CD2 (13)

3. Results and discussion

3.1. Compression test

Measured data of individual pressing curve for different

diameter of pressing vessel are shown as follows Fig. 2 (60mm

Ø), Fig. 3 (80 mm Ø), Fig. 4 (100 mm Ø). The compression test

was repeated three times for each initial pressing height

where the coefficient of variation of compression force was

determined (Table 2). Individual dependencies between

compressive force and deformation were fitted by tangent

curve equations (Eq. (1)) and the coefficients are presented in

Table 2. The LevenbergeMarquardt approximation process

(Lourakis, 2005), (Marquardt, 1963) with the aid of genfit

function (Pritchard, 1998), which is included in the MathCAD

software was used to determine of the coefficients. The

analysis of the experimental results showed that the fitted

curves can be used to describe accurately the measured

amounts over the whole range of deformation based on their

coefficients of determination (R2) values (Table 2). An ANOVA

was carried out using the MathCAD 14 software for a level of

significance 0.05. It was shown (Figs. 2e4) that the measured

dependencies can be described by tangent curve (Eq. (1)). The

values of Fcrit (critical value comparing a pair of models) were

higher than the Frat values (value of the F-test) for all the

measured bulk seeds and values of Pvalue (significance level at

which it can be rejected the hypothesis of equality of models)

(Table 2) were higher than 0.05. This shows that measured

values and tangent curve values are statistically significant.

The validity of these equations is limited to the region where

the deformation of seeds varies from zero to the maximum.

The physical properties of the bulk seeds of jatropha, pre-

sented in (Table 1) shows that moisture content and porosity

being constant can be substituted as means of Mc ¼ (8.5 � 0.2)

% in dry basis (d.b.) and porosity, Pf ¼ (59.98 � 1.26) %

respectively. Although the bulk seeds may vary biologically,

the moisture content of the bulk seeds remained constant

Fig. 6 e Dependency between force coefficients of mechanical behaviour (kN) and initial pressing seed heights (mm). The

numbers in the chart represent diameter of pressing vessel (mm) and lines represent the mean of force coefficients of

mechanical behaviour.

0

2

4

6

8

10

12

14

0 2000 4000 6000 8000 10000 12000

Ave

ra

ge

fo

rc

e c

oe

ffic

ie

nt o

f m

ec

ha

nic

al b

eh

av

io

ur

(k

N)

Square of inner diameter of pressing vessel (mm 2)

Fig. 7 e Dependency between average force coefficient of mechanical behaviour (kN) and squared diameter of pressing

vessel (mm2). Fitted line in the chart is given by Eq. (13).

b i o s y s t em s e ng i n e e r i n g 1 1 4 ( 2 0 1 3 ) 2 7 9e2 8 8 285

Table 4 e Statistical analysis of general deformationcurve for different initial pressing seed heights anddiameters of pressing vessel.

H (mm) Frat (e) Fcrit (e) Pvalue (e) R2 (e)

D [ 60 mm

30 1.110 2.755 0.294 0.981

40 0.063 2.756 0.800 0.981

50 0.389 2.750 0.534 0.979

60 1.101 2.745 0.295 0.982

70 0.008 2.747 0.926 0.983

80 1.541 2.746 0.216 0.980

D [ 80 mm

30 0.334 2.767 0.564 0.984

40 0.144 2.760 0.704 0.978

50 0.334 2.763 0.564 0.982

60 0.027 2.763 0.868 0.983

70 0.004 2.756 0.947 0.979

80 0.014 2.745 0.904 0.978

D [ 100 mm

30 0.059 2.767 0.808 0.979

40 0.042 2.762 0.837 0.975

50 1.470$10�05 2.758 0.996 0.982

60 0.053 2.752 0.817 0.983

70 0.083 2.755 0.773 0.981

80 0.072 2.756 0.789 0.983

H e different initial pressing seed heights, Frat e value of the F test,

Fcrit e critical value that compares a pair of models, Pvalue e the

significance level at which it can be rejected the hypothesis of

equality of models, R2 e coefficient of determination, D e inner

diameter of pressing vessel.

b i o s y s t em s e n g i n e e r i n g 1 1 4 ( 2 0 1 3 ) 2 7 9e2 8 8286

since a single batch of jatropha seeds under the same

temperature and storage conditions was used.

3.2. General curve

The deformation coefficients of mechanical behaviour, BD and

the dependency on the initial pressing seed heights in relation

to the diameter of the pressing vessel are shown in Fig. 5.

These coefficients were used to determine the compressive

coefficients, G using Eq. (4) for the different pressing vessels

(Table 3). The dependency between deformation coefficients

of mechanical behaviour and seed pressing heights (Fig. 5)

showed that compressive coefficient is precisely the coeffi-

cient of hyperbolic function as described in Eq. (4). The results

of previous studies (Herak, Kabutey, & Sedlacek, 2011; Herak,

Kabutey, Sedlacek, & Gurdil, 2011) showed that the compres-

sive coefficient which was constant had no influence on the

diameter of pressing vessel and initial pressing seed heights.

Based on this assumption, the average compressive coeffi-

cient, G ¼ 2.02 � 0.07 was determined as mean of individual

compressive coefficient for each diameter of pressing vessel

(Table 3). For each diameter of the pressing vessel, the rela-

tionship between the force coefficient of the mechanical

behaviour, AD and initial pressing seed heights (Fig. 6) was

determined. These coefficients were substituted by their

average amounts (Table 3). This hypothesis was based on the

elementary mathematics about the similarity of the gonio-

metric functions (Rektorys, 1981) and it has been verified by

few studies (Herak, Kabutey, & Sedlacek, 2011; Herak,

Kabutey, Sedlacek, & Gurdil, 2011).

The relationship between average force coefficients of

mechanical behaviour and the squared diameter of pressing

vessel are shown in Fig. 7. This relationship can be fitted by

a line which intersects the origin of coordinate system, that is,

zero diameter of pressing vesselmust be equal to zero amount

of stress coefficient as explained by Eqs. (11) and (13). There-

fore a fitted line can be described accurately by Eq. (13) which

describes the stress coefficient of mechanical behaviour,

C ¼ (1.02 � 0.16) N mm�2 with very high coefficient of deter-

mination R2 ¼ 0.98. Average stress coefficient of mechanical

behaviour is not dependent on the diameter of pressing vessel

(Table 3). However, the different values obtained for the

various pressing diameters could be due to the biological

variability of the bulk seeds. Equation (13) confirms the

assumption of the squared dependency between average force

coefficient of mechanical behaviour and diameter of pressing

vessel. Using Eq. (13) and the amount of average compressive

coefficient, a general equation which can be used to describe

the mechanical behaviour of J. curcas L. seeds under

compression loading according to deformation of bulk seeds

can be derived. The general equation (Eq. (12)) can be used to fit

data from the compressive tests (Figs. 2e4) for each diameter

of pressing vessel with respect to initial pressing seed heights.

Clearly, Eq. (12) accurately described the measured

amounts of compressive force over the whole range of

deformation with very high coefficients of determination R2

(Table 4). Statistical analyses by ANOVA showed that values of

Fcrit were higher than Frat values for all the measured data.

Amounts of Pvalue were also had higher significance than 0.05

(Table 4) and the measured values (Figs. 2e4) and general

predicted values (Eq. (12)) were statistically significant.

Clearly, the fitted curve does not exactly describe the start of

the relationship between compressive force and deformation

characteristics of the pressing vessel diameter 100 mm

compared to themeasured data. This could be due to themore

free spaces occurring between the seeds within the measured

pressing area. Also the size of the seeds and the air gaps inside

the bulk seeds could contribute to the deviation. Comparing

the results of this study to the results previous studies which

was focused on rapeseeds (Herak, Kabutey, & Sedlacek, 2011;

Herak, Kabutey, Sedlacek, & Gurdil, 2011), it is clear that the

shape, dimensions and dimensional similarity of the seeds

tends to influence the start of the deformation curve.

However, with mechanical properties such as deformation

energy and compressive force, the effect is not so important.

The validity of Eq. (12) is limited to the region from zero to the

maximum deformation of bulk seeds. It is evident that results

of the present study are similar to the results of the previous

studies focused on the mechanical behaviour of rapeseeds

under compression loading at uniform diameter of pressing

vessel (Herak, Kabutey, & Sedlacek, 2011; Herak, Kabutey,

Sedlacek, & Gurdil, 2011).

However for description of mechanical behaviour of bulk

seeds, different solution methods generally based on the

Darcy’s Law (Fasino & Ajibola, 1990; Fomin, 1978) and fluid

flow through porous media (Mrema & Mc Nulty, 1985; Singh &

Kulshreshtha, 1996) have been used. Darcy’s law and rheo-

logical properties of deformable solid matrix of bulk seeds are

fundamental for developing mathematical models to

b i o s y s t em s e ng i n e e r i n g 1 1 4 ( 2 0 1 3 ) 2 7 9e2 8 8 287

mechanical behaviour of oilseeds (Ocenasek & Voldrich, 2009;

Omobuwajo, Ige & Ajaji, 1998; Petru et al., 2012; Raji & Favier,

2004). The mechanical behaviour of bulk oilseeds can be

described also by methods based on the Terzagi’s model

(Shirato, Murase, Iwata, & Nakatsuka, 1986; Willems, Kuipers

& De Hann, 2008) or the energetic balance model (Zheng,

Wiesenborn, Tostensona, 2005). However, these models

cannot be used to resolve individual particles and their

properties, as well as relationships between particles, unlike

the tangent curve equation which considers the bulk seeds as

a unit, the constrains between the pressing vessel and the

bulk seeds, and also the pressing process. Obviously, the

tangent curve method can be used for the description of

mechanical behaviour of different types of bulk oilseeds and

other materials such as wood or paper chips under compres-

sion loading where the rigidity of pressing vessel is much

greater than the rigidity of the bulk samples (Herak, Kabutey,

& Sedlacek, 2011; Herak, Kabutey, Sedlacek, & Gurdil, 2011;

Lehtikangas, 2001; Plistil, Brozek, Malatak, Roy, & Hutla, 2005;

Wilaipon, 2009). Notwithstanding this fact, the development

of mathematical models based on tangent curve equations for

the description of deformation characteristics and mechan-

ical behaviour of oilseeds could also be influenced bymoisture

content, temperature and pressing velocity (Garnayak et al.,

2008; Herak et al., 2010; Kabutey, Herak, & Sedlacek, 2011;

Pradhan et al., 2009). Thus, mathematical models could

predict the change of the tangent curve coefficients based on

the compression factors.

4. Conclusion

A general equation describing the mechanical behaviour of J.

curcas L. seeds under compression loading was determined by

tangent curve equation (Eq. (12)) with stress constant of

mechanical behaviour C ¼ (1.02 � 0.16) N mm�2 and

compression coefficient G ¼ 2.02 � 0.07. The general equation

was statistically analysed and the results showed that

measured amounts of deformation characteristic were

statistically significant similar to the amounts determined

from this general equation (Eq. (12)) with its validity limited

from zero to maximum deformation of the bulk seeds. The

moisture content and porosity of the bulk seeds of jatropha

were found to be constant but their changes could also

influence the mechanical behaviour of oilseeds such as

jatropha under compression loading. The mathematical

models described in this study should provide the basis for the

development of further models which will describe the non-

linear mechanical behaviour of oilseeds involving the use of

screw extruders or presses.

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