Influence of Boundary Conditions on Particle Density

Abstract

The objective of this paper is an analysis of the influence of drying air temperature and the drying air velocity on the particle density. The particle density of apple, banana, potato and carrot slices during convective drying was experimentally determined. Drying experiments were conducted in a laboratory air-dryer, repeated at different air temperatures and air velocities. The drying air temperatures considered were 40, 50, 60 and 70 °C with drying air velocities of 1, 2 and 3 m/s. Two simple mathematical models for correlating the dimensionless particle density with the dimensionless material moisture content, drying air temperature and drying air velocity are proposed. The models were fitted to experimental data and the correlation coefficients and residual sum of squares were estimated. Nomenclature A, B, C -parameter MRD -mean relative deviation m (kg) -mass n -degrees of freedom R -dimensionless particle density RSS -residual sum of squares r -correlation coefficient V (m 3) -volume SE -standard error of the estimate X -dimensionless moisture content x (kg/kg d.b.) -moisture content Y cal -estimated value Y exp -experimental value Greek symbols ρ (kg/m 3) -density Subscripts 0 -initial a -air ap -apple ba -banana ca -carrot p -particle po -potato s -solid

Full text

Journal of Agricultural Science and Technology B 2 (2012)
Earlier title: Journal of Agricultural Science and Technology, ISSN 1939-1250
Influence of Boundary Conditions on Particle Density
Vangelce Mitrevski1, Vladimir Mijakovski1, Filip Popovski2, Dusan Popovski1
1. Faculty of Technical Sciences, University St. Kliment Ohridski, Bitola 7000, Ivo Lolar Ribar, Macedonia
2. International Balkan University, Skopje 1000, Samoilova 10, Macedonia
Received: August 23, 2011 / Published: February 20, 2012.
Abstract: The objective of this paper is an analysis of the influence of drying air temperature and the drying air velocity on the
particle density. The particle density of apple, banana, potato and carrot slices during convective drying was experimentally
determined. Drying experiments were conducted in a laboratory air-dryer, repeated at different air temperatures and air velocities.
The drying air temperatures considered were 40, 50, 60 and 70 °C with drying air velocities of 1, 2 and 3 m/s. Two simple
mathematical models for correlating the dimensionless particle density with the dimensionless material moisture content, drying air
temperature and drying air velocity are proposed. The models were fitted to experimental data and the correlation coefficients and
residual sum of squares were estimated.
Key words: Particle density, air temperature, air velocity, drying, apple, banana, potato, carrot.
Nomenclature
A, B, C - parameter
MRD - mean relative deviation
m (kg) - mass
n - degrees of freedom
R - dimensionless particle density
RSS - residual sum of squares
r - correlation coefficient
V (m3) - volume
SE - standard error of the estimate
X - dimensionless moisture content
x (kg/kg d.b.) - moisture content
Ycal - estimated value
Yexp - experimental value
Greek symbols
ρ (kg/m3) - density
Subscripts
0 - initial
a - air
ap - apple
ba - banana
ca - carrot
p - particle
po - potato
s - solid
Corresponding author: V. Mitrevski, Ph.D., research fields:
drying, heat and mass transfer, optimization. E-mail:
vangelce.mitrevski@uklo.edu.mk.
s0 - initial solid
t - total
w - water
1. Introduction
Drying is essential process for the preservation of
agricultural and other products, including fruits and
vegetables. The major objective in drying of fruits and
vegetables is the reduction of moisture content to a
certain level, which allows safe storage and
preservation. The present demand of high-quality
products in the food market requires dehydrated foods
that maintain at a very high level the nutritional and
sensorial properties of the initial fresh product [1].
The quality of the dried fruits and vegetables is
characterized by the appearance, color, texture, shape,
sizes, density, shrinkage and porosity. This quality
factors are the factors that determine the worth or
value of a food product to the consumer.
The most important physical properties that
characterise the quality of dried and intermediate
moisture foods, are porosity, bulk density and particle
density [2]. Experimental values of density are
necessary for designing the facility of storage,
D
DAVID PUBLISHING

Influence of Boundary Conditions on Particle Density
2
handling and processing of agricultural materials.
The density can be defined in different ways [3, 4]:
true density, substance density, apparent density, bulk
density and particle density. Bulk and particle
densities are vital parameters in the design, modeling
and optimization of food processing operations
because they have a direct affect on the
thermophysical properties of food materials [3].
The effect of material moisture content and
temperature on true density of foods was studied by
Boukouvalas [5]. While variation of bulk and particle
density of potato starch gel with dimensionless
moisture content at various air temperature was
studied by Muhtaseb [6].
The aim of this paper is to investigate the influence
of boundary conditions on particle density of some
fruits and vegetables during convective drying. Some
simple mathematical models for correlating the
dimensionless particle density with the dimensionless
moisture content drying air temperature and drying air
velocity are proposed.
2. Materials and Methods
Fresh apples, bananas, potatoes and carrots were
used in this study. To prepare samples, apples,
bananas, potatoes and carrots were sliced using
electric slicing machine to give a uniform sample
thickness of 3 mm before being reduced to a cylinder
form with diameter of 40 ± 0.1 mm. Several
measurements were made using a calliper and only
samples with a tolerance of ± 5% were used.
The study of influence on boundary conditions of
particle density for apple, banana, potato and carrot
slices was conducted in a laboratory air-dryer (Fig. 1).
The slices were in contact with drying air from top
and bottom surfaces. In each experiment, the shelf
holding three apple, banana, potato or carrot slices
was inserted into the rectangular experimental channel
with dimensions 25 × 200 × 2,000 mm. The slices
were dried until the equilibrium moisture content was
reached. The samples of apples, bananas,
Fig. 1 Experimental apparatus; 1-material, 2-shelf,
3-electrical heaters, 4-transformers, 5-thermocouples,
6-centrifugal fan, 7-anemometer, 8-panel meter, 9-data
acquisition system, 10-stove, 11-balance, 12-hygrometer.
potatoes or carrots were drawn from the dryer every
10 min and their weights and sample volumes were
measured.
The initial moisture content and the initial slices’
dimensions were measured as well. The experiment
was repeated at different air temperatures and
velocities. During the experiments, air temperature
and drying air velocity were controlled. The drying air
temperatures considered were 40, 50, 60 and 70 °C
with drying air velocities of 1, 2 and 3 m/s.
3. Mathematical Models
In reference literature various simple mathematical
models were used to relate the densities to material
moisture content and drying air temperature [5, 6].
Assuming the moist material of dry solids, water and
air, in literature, the following definitions are used.
True density is a density of a pure substance or a
material calculated from its component’s densities
considering conservation of mass and volume:
p
t
pV
m
=ρ
(1)
where mt is total mass, while Vp = Vs + Vw is the true
volume, which is the total volume of the sample,
excluding air pores.
The enclosed water density ρw can be defined as:
w
w
w
V
m
=ρ
(2)

Influence of Boundary Conditions on Particle Density
3
where mw is mass of water, while Vw, is volume of
water.
The particle density of dry solids ρs is defined as:
s
s
s
V
m
=ρ
(3)
where ms is the mass of dry solids, while Vs, is
volume of dry solids.
Reference literature offers four methods for
determination of sample’s volume (volume of dry solid):
- Direct measurement method [7, 8],
- Method of immersing the samples in n-heptanes
[2, 7],
- Image analysis [8],
- Method of immersing the samples in distilled
water [9].
The comparison between these methods indicates
that the differences among maximum errors are less
than 10% [7, 8]. Therefore, it is the method of direct
measurement with caliper that was used to determine
the volume of the sample.
In this paper, simple mathematical models were
used to determine the dimensionless particle density:
s
0s
Rρ
ρ
=
(4)
as a function of dimensionless material moisture
content (Table 1):
0
0
x
xx
X−
=
(5)
and drying air temperature and drying air velocity.
4. Results and Discussions
On the basis of experimental data for each material
(apple, banana, potato and carrot) and each model
from Table 1, the values of parameters A, B and C,
Table 1 Mathematical models.
Model
Type
ρs0/ρs =
1
Logarithm
1-LOG[1+(A+B*V+C*T)*X]
2
Linear
(A+B*V+C*T)*X + 1
correlation coefficient (r) and residual sum of squares
(RSS) were determined. The following seven methods
were used: Quasi-Newton, Simplex, Composition
Simplex and Quasi-Newton, Hooke-Jeeves Pattern
Moves, Composition Hooke-Jeeves Pattern Moves
and Quasi-Newton, Rosenbrock Pattern Search and
Composition Rosenbrock Pattern Search and
Quasi-Newton. When the results for correlation
coefficient r were different, the highest value was
accepted as relevant. The calculations were made with
software package Statistica [10]. The values of
parameters A, B and C are given in Table 2.
The adequacy of the fitted models can be evaluated
by means of coefficient of correlation (r), residual sum
of squares (RSS), standard error of estimate (SE),
mean relative deviation (MRD), and the plot of
residual.
Correlation coefficient (r) is a dimensionless index
that ranges from 0 to 1 and reflects the extent of linear
relationship between two data sets.
The residual sum of squares (RSS) is defined as:
2
cal
n
1i exp
)YY(RSS
∑
=
−=
(6)
where Yexp is the experimental-measured value, Ycal is
the value estimated through the fitting equation and n
is the number of data points.
The standard error of estimate (SE) is the
conditional standard deviation of the dependant
variable and represents a measure of the predictions
accuracy. The standard error of estimate for large data
set is defined with equation (7):
Table 2 Values of parameters A, B and C.
Material
Model 1
Model 2
A
B
C
A
B
C
Apple
1.395036
- 0.085965
- 0.003821
- 0.935973
0.038427
0.002242
Banana
1.00891
- 0.009713
0.001301
- 0.780456
0.002820
0.000119
Potato
0.75377
- 0.112464
0.007534
- 0.575416
0.057700
- 0.004257
Carrot
1.41398 - 0.61549 - 0.000009 - 0.958091 0.030884 0.000488

Influence of Boundary Conditions on Particle Density
4
n
RSS
SE =
(7)
The mean relative deviation (MRD) is an absolute
value because it gives mean divergence of the
estimated data from the measured data.
∑
−
−
=
n
1i exp
calexp
Y
YY
n
1
MRD
(8)
Plotting of the residuals against the independent
variable is also used as a measure of errors
distribution. If the model is correct, then the residual
should be only random independent error with a zero
mean, constant variance and arranged in a normal
distribution. If the residual plots indicate a clear
pattern, the model should not be accepted.
In general, low values of correlation coefficient,
high values of RSS, SE and MRD, and clear patterns
in the residual plots mean that the model is not able to
explain the variation in the experimental data. It is
also evident that a single statistical parameter cannot
be used to select the best model that must always be
assessed based on multiple criteria [11].
The values of correlation coefficients (r) for apple,
banana, potato and carrot are given in Table 3.
From Table 3, it is obvious that the linear model
gives better results compared to the logarithm model.
In Figs. 2-5, the influence of drying air temperature
T on the dimensionless particle density for apple,
banana, potato, and carrot is shown.
Table 3 Values of correlation coefficients and residual sum of squares.
Model
rap
rba
rpo
rca
RSSap
RSSba
RSSpo
RSSca
1
0.97086
0.95933
0.96888
0.96900
0.14728
0.58100
0.23785
0.42046
2
0.98472
0.97434
0.97212
0.98599
0.27895
0.91363
0.26508
0.92269
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
T = 40
o
C, T = 50
o
C, T = 60
o
C, T = 70
o
C
Fig. 2 Variation of dimensionless particle density of apple
with dimensionless moisture content at various air
temperature and v = 2 m/s.
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
T = 40
o
C, T= 50
o
C, T = 60
o
C, T = 70
o
C
Fig. 3 Variation of dimensionless particle density of
banana with dimensionless moisture content at various air
temperature and v = 2 m/s.
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
t = 40
o
C, T = 50
o
C, T = 60
o
C, T = 70
o
C
Fig. 4 Variation of dimensionless particle density of
potato with dimensionless moisture content at various air
temperature and v = 2 m/s.
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
T = 40
o
C, T= 50
o
C, T = 60
o
C, T = 70
o
C
Fig. 5 Variation of dimensionless particle density of carrot
with dimensionless moisture content at various air
temperature and v = 2 m/s.

Influence of Boundary Conditions on Particle Density
5
Air temperature has different effect on particle
density. For apple, the influence of particle density at
a low temperature (40 °C) is greater than at high
temperature (70 °C), while for potato, air temperature
has an opposite effect on particle density. From Fig. 3
to Fig. 5, it is clear that the variation of temperature
has no influence on a particle density of banana and
carrot.
In Figs. 6-9, the influence of drying air velocity on
the dimensionless particle density for apple, banana,
potato, carrot is presented.
Air velocity, on the other hand, has the biggest
impact on particle density. It can be seen that high
value of air velocity has small effect on dimensionless
particle density, apart from banana, where variation of
air velocity has a negligible effect. That can be
explained with the process of caramelization. The
influence of air velocity on particle density can be
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
v = 1 m/s, v = 2 m/s, v = 3 m/s
Fig. 6 Variation of dimensionless particle density of apple
with dimensionless moisture content at various air velocity
and T = 60 °C.
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
v = 1 m/s, v = 2 m/s, v = 3 m/s
Fig. 7 Variation of dimensionless particle density of
banana with dimensionless moisture content at various air
velocity and T = 60 °C.
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
v = 1 m/s, v = 2 m/s, v = 3 m/s
Fig. 8 Variation of dimensionless particle density of
potato with dimensionless moisture content at various air
velocity and T = 60 °C.
X
R
0,0
0,2
0,4
0,6
0,8
1,0
0,0 0,2 0,4 0,6 0,8 1,0
v = 1m/s, v = 2 m/s, v = 3 m/s
Fig. 9 Variation of dimensionless particle density of carrot
with dimensionless moisture content at various air velocity
and T = 60 °C.
explained on the basis of effect of variables on the
mass transfer. At low air velocities, surface resistance
prevails, moisture profiles in the sample are relatively
flat and internal stresses are at minimum [12]. If air
velocities are very high, drying makes the surface of
the samples stiff, limiting the particle density even in
the earliest stages. At low air velocities the surface of
dry sample does not stiffen until the water content
reaches very low values.
5. Conclusions
The influence of drying air temperature and drying
air velocity on the particle density of apple, banana,
potato and carrot slices during convective drying was
studied in this particle. For this purpose, some
experiments were conducted in a laboratory air-dryer.
Two simple mathematical models for correlating the

Influence of Boundary Conditions on Particle Density
6
dimensionless particle density with the dimensionless
material moisture content, drying air temperature and
drying air velocity are proposed. Air temperature has
different effect on particle density. For apple, the
influence of particle density at low temperature is
greater than at high temperature, while for potato, air
temperature has an opposite effect. It was concluded
that the variation of temperature has no influence on a
particle density of banana and carrot. Air velocity has
the biggest effect on particle density. High values of
air velocity have small effect on dimensionless
particle density, for apple, potato and carrot, while for
banana, the variation of air velocity has a negligible
effect.
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