A mathematical model for low-pressure superheated steam drying of a biomaterial

A mathematical model for low-pressure superheated
steam drying of a biomaterial
Peamsuk Suvarnakuta a, Sakamon Devahastin b,?, Arun S. Mujumdar c
a Department of Food Science and Technology, Thammasat University, Pathum Thani 12121, Thailand
b Department of Food Engineering, King Mongkut’s University of Technology Thonburi, 126 Pracha u-tid Road, Bangkok 10140, Thailand
c Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore
Received 12 April 2006; received in revised form 4 September 2006; accepted 10 September 2006
Available online 15 September 2006
Abstract
Low-pressure superheated steam drying (LPSSD) has recently received much attention as an alternative drying technique for heat-sensitive
biomaterials. Although there are a number of works that report studies of this drying technique experimentally, there are a very limited number of
works that report attempts to model this drying process. The aim of the present study was therefore to propose the use of a simple three-dimensional
liquid diffusion based model to predict the evolutions of the moisture content and temperature of a product undergoing LPSSD. The effect of the
product shrinkage was also included directly in the model and the effect of this inclusion on the predictability of the model is shown. The model
was found to be able to predict the heat and mass transfer behavior as well as the change of a selected chemical quality, i.e., -carotene, of a model
biomaterial viz., carrot cube reasonably well over some range of moisture content if accurate values of the heat transfer coef?cient were used.
© 2006 Elsevier B.V. All rights reserved.
Keywords:
-Carotene evolution; Carrot; Deformation; Finite element method; Heat and mass transfer; Liquid diffusion model; Shrinkage
1. Introduction
theoretically. A semi-empirical mathematical model was devel-
oped based on a theoretical drying mechanism, which assumes
During the past decade there has been considerable interest in
that the water removal is carried out by evaporation in a moving
applying superheated steam to dry various products with some
boundary allowing the vapor to ?ow through the dry layer built
success [1–3]. Despite the many advantages of near-atmospheric
as drying proceeds. Despite its simplicity, the model was found
pressure superheated steam drying [4], there still exist some
to predict the drying kinetics of the tested materials adequately.
limitations, especially when applying it to drying heat-sensitive
Defo et al. [7] developed a two-dimensional mathematical
materials, e.g., foods and other biomaterials [5]. Since most
model based on the combination of the mass conservation equa-
foods or other heat-sensitive biomaterials are damaged at the
tion and Darcy’s law with negligible temperature gradients to
saturation temperature of superheated steam corresponding to
simulate superheated steam vacuum drying of sugar maple sap-
the atmospheric or higher pressures, one possible way to alle-
wood. The values of the convective mass and heat transfer
viate the above-mentioned problem is to operate the dryer at
coef?cients were obtained by curve ?tting the simulated results
reduced pressure.
to the experimental data. The predicted and experimental drying
The concept of low-pressure (or sub-atmospheric pressure)
curves were in good agreement when transfer coef?cients were
superheated steam drying (LPSSD) has been applied to various
adjusted as a function of wood moisture content.
types of heat-sensitive materials. Elustondo et al. [6] studied
From an experimental side, Devahastin et al. [8] and Suvar-
sub-atmospheric pressure superheated steam drying of shrimp,
nakuta et al. [9] studied experimentally LPSSD of a model
banana, apple, potato and cassava slices both experimentally and
heat-sensitive biomaterial, i.e., carrot. The effects of the oper-
ating pressure and temperature on the heat transfer and drying
characteristics of the product as well as its various physical and
?
chemical properties were evaluated. However, as mentioned ear-
Corresponding author. Tel.: +662 470 9246; fax: +662 470 9240.
E-mail address: sakamon.dev@kmutt.ac.th (S. Devahastin).
lier, no attempts were made to develop a mathematical model to
0255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.cep.2006.09.002

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predict the temperature and moisture content evolutions of the
was reduced considerably. A variable-speed electric fan was
sample during the process.
used to disperse steam throughout the drying chamber. The
In addition to the above-mentioned works, which focus
change of the weight of the sample was detected continuously
directly on modeling of LPSSD, there are also a number of
using a load cell (Minebea, model Ucg-3 kg, Japan) and an
other works that consider modeling of near-atmospheric pres-
indicator and recorder (A&D Co., model AD 4329, Japan).
sure superheated steam drying (SSD). These include the use
The temperatures of the steam and of the drying sample were
of various mathematical models to predict the heat and mass
measured continuously using type K thermocouples, which
transfer of many products undergoing SSD and also to study
were connected to an expansion board (Omega Engineering,
the effects of various operating parameters on the tempera-
model no. EXP-32, USA). Thermocouple signals were then
ture and moisture content pro?les of those products [10,11].
multiplexed to a data acquisition card (Omega Engineering,
A model that considers the effect of initial steam condensa-
model no. CIO-DAS16Jr., USA) installed in a PC. LABTECH
tion, which is inevitable in SSD, is also available in literature
NOTEBOOK software (Version 12.1, Laboratory Technologies
[12].
Corp., USA) was then used to read and record the temperature
Another aspect that is of great importance during drying of
data.
most biomaterials is product deformation. The degree of shrink-
age and its variation with drying conditions as well as product
2.2. Materials and methods
moisture content in?uences the heat and mass transport within
the product. There are a few numerical studies that investigated
Fresh carrot (Daucus carota var. sativa) was obtained from a
the effect of product deformation, which involves modeling of
supermarket and stored at 4 ?C. Prior to the start of each drying
coupled heat and mass transfer and stress, on drying and heat
experiment carrot was peeled and diced (only the cortex part)
transfer [13]. Mihoubi et al. [14] numerically studied and ana-
into 1 cm3 cubes.
lyzed the distribution of temperature, moisture, strain, and stress
To perform an LPSSD experiment approximately 35 cubes
of a shrinkable product during drying. A validation of the model
of carrot were placed as single layer on the sample holder. More
was achieved by the comparison of the numerical and experi-
detailed procedures of an LPSSD experiment could be found in
mental data. The experimental temperature and moisture pro?les
Devahastin et al. [8]. The experiments were performed at the
compare well with the model predictions. They also found that
steam absolute pressure of 7 kPa and the steam temperatures of
the distribution of displacement was not necessarily uniform
60, 70 and 80 ?C. The ?ow rate of steam into the drying chamber
within the material (the stress variation is larger at the surface
was maintained at about 26 kg/h and the speed of the fan was
than in the sample body) and may cause some bending and crack-
?xed at 1100 rpm.
ing within the material.
Since the coupled mechanical and heat and mass transfer
3. Model development
models are rather sophisticated and contain many parameters
that must be known prior to the simulation, the need for a simple
A mathematical model was developed to predict the tem-
model that utilizes semi-empirical or empirical shrinkage infor-
perature and moisture content pro?les of carrot cube of well-
mation still exists. The objective of this work was therefore to
de?ned geometry undergoing LPSSD. This was accomplished
develop a simple three-dimensional liquid diffusion based model
by identifying the simplifying assumptions, de?ning the gov-
to predict the evolutions of moisture and temperature of a model
erning conservation equations for heat and mass transfer along
biomaterial viz., carrot cube during LPSSD. The model consists
with appropriate initial and boundary conditions. Since carrot
of coupled heat conduction and mass diffusion equations along
shrinks signi?cantly during drying, the shrinkage effect was also
with an empirical equation, which describes shrinkage of the
included empirically in the model to investigate the effect of
product during drying. An empirical equation that expresses the
shrinkage on the predictability of the model.
-carotene degradation in carrot was also included in the model
so as to predict the evolution of
-carotene content in carrot
during drying.
3.1. Assumptions
2. Experimental set-up, materials and methods
To formulate a mathematical description of the LPSSD pro-
cess the following assumptions were made:
2.1. Experimental set-up
1. Since only the cortex part of carrot was used in the experi-
A schematic diagram of the low-pressure superheated steam
ments, the sample was assumed to be isotropic and homoge-
dryer and its accessories [8] is shown in Fig. 1. The dryer con-
nous.
sists of a drying chamber, a steam reservoir, which received
2. Because of the pre-warming up of the drying chamber, initial
steam from a boiler and maintained its pressure at around
condensation was neglected. Previous studies [8] also showed
200 kPa (gage); a liquid ring vacuum pump. An electric heater,
this to be a very short period in steam drying.
rated at 1.5 kW, which was controlled by a PID controller,
3. Mass transfer within carrot was controlled only by liquid
was installed in the drying chamber. Due to the use of the
diffusion. It was thus assumed that no vaporization occurred
heater, the initial steam condensation during the start-up period
within the drying material.

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Fig. 1. A schematic diagram of the low-pressure superheated steam dryer and associated units [8]. 1, Boiler; 2, steam valve; 3, steam reservoir; 4, pressure gauge; 5,
steam trap; 6, steam regulator; 7, drying chamber; 8, steam inlet and distributor; 9, electric fan; 10, sample holder; 11, electric heater; 12, on-line temperature sensor
and logger; 13, vacuum break-up valve; 14, insulator; 15, on-line weight indicator and logger; 16, vacuum pump; 17, PC with installed data acquisition card.
4. The density and thermal properties of carrot cube were con-
3.3. Initial and boundary conditions
sidered as a function of the product moisture content. Mass
diffusivity of carrot was considered as functions of both prod-
At the onset of the LPSSD process the temperature and mois-
uct temperature and moisture content.
ture content of carrot cube were uniform:
5. Shrinkage of material was signi?cant that it was accounted
for in all three directions (see Fig. 2). The volumetric shrink-
T = Ti
(3)
age depends on the operating temperature and moisture
content (as experimentally studied by Panyawong and Deva-
X = Xf,i
(4)
hastin [15]).
For carrot cube subjected to convective drying the boundary
condition (Eq. (5)) at the surface was used:
3.2. Governing equations for heat and mass transfer
?k(?T · n) = h(Tsteam ? Ts) ? ?LvDeff(?Xf · n)
(5)
Heat transfer within carrot cube was driven by conduc-
tion as temperature gradients developed in all directions. The
where the term on the left hand side refers to heat conducted
conduction equation to describe energy transfer is written as
from the outer surface to the inside of the cube, the ?rst term
follows:
on the right hand side is heat penetrating from low-pressure
?T
?
?T
?
?T
?
?T
superheated steam to the solid body by means of convection and
?Cp
=
k
+
k
+
k
(1)
?t
?x
x ?x
?y
y ?y
?z
z ?z
the second term on the right hand side denotes latent heat of
vaporization.
where kx = ky = kz = k due to the product isotropy.
Mass transfer at the surface was modeled by assuming that
The moisture transfer within carrot cube was postulated to
there was no mass transfer resistance at the surface of carrot
occur only by liquid diffusion. The three-dimensional form of
cube since water possessed no self-resistance in its own body.
the Fick’s second law was therefore applied to simulate the mois-
Therefore, the mass transfer boundary condition is as follows:
ture transfer. The equation to describe the mass transfer during
LPSSD is then:
Xf = 0
(6)
?Xf
?
?X
?
?X
?
?X
=
D
f
+
D
f
+
D
f
where Xf denotes free moisture content (Xf = X ? Xeq). This con-
?t
?x
eff ?x
?y
eff ?y
?z
eff ?z
dition simply implies that the moisture content at the surface was
(2)
always at its equilibrium value at the corresponding operating
condition.
4. Parameter estimation
4.1. Thermophysical properties of carrot
The physical and thermal properties of carrot used in the
model are shown in Table 1.
Based on the correlation proposed by Mulet [18] the effec-
Fig. 2. Schematic diagram and directions of shrinking carrot cube undergoing
LPSSD.
tive diffusion coef?cient, Deff, of carrot can be estimated by the

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Table 1
Table 2
Properties of carrot used in the model
Empirical constants of Eq. (11) at different temperatures [15]
Property
Source
Temperature
Expression
Drying temperature (?C)
a
b
c
range
60
?0.8288
1.8170
0.0366
Density ?
[15]
25–80 ?C
? = 15.63X + 839
70
?0.9026
1.9663
?0.0433
(kg m?3)
80
?0.9492
1.9938
?0.0421
Speci?c heat
Cp (J kg?1 K?1)
[16]
Above freezing
Cp = 837 + 4186X/1 + X
Thermal
[17]
Above freezing
k = 0.148 + 0.641X/1 + X
Table 3
conductivity k
Heat transfer coef?cients (W m?2 K?1) of LPSSD at different operating
(W m?1 K?1)
pressures
7 kPa
10 kPa
13 kPa
following form:
Mean
S.D.
Mean
S.D.
Mean
S.D.
c
D
2
7.661
0.61
8.353
0.44
9.168
0.33
eff = exp ?c1 ?
+ c
T
3X
(7)
abs
where Tabs is the absolute product temperature (K) and c1, c2
and c
respectively. The values of the empirical constants are listed
3 are the constants. In this study, these constants were
obtained by ?tting the simulated results to the existing exper-
in Table 2. Eq. (11) was implemented directly along with the
imental results [8] at various drying conditions. As a result, D
Arbitrary Lagrange-Eulerian (ALE) formulation of COMSOL
eff
was found (in terms of the absolute temperature and free mois-
MultiphysicsTM 3.2 (see Section 5).
ture content) to be:
4.3. Heat transfer coef?cient
Deff = exp ?3 ? 6420 + 0.0005X
T
f
(8)
abs
Based on the previous experimental data for molecular sieve
For the cases when the shrinking assumption was not used
beads undergoing LPSSD [19] the values of the heat trans-
the values of D
fer coef?cient (h, W m?2 K?1) of LPSSD process could be
eff were obviously different. In such cases, Deff
was found to be correlated by:
estimated from the drying rates (evaporation rates) during the
constant rate period as follows:
Deff = exp ?3 ? 5880 + 0.0005X
T
f
(9)
q
hA T
NLv
abs
N =
=
or
h =
(12)
Lv
Lv
A(Tsteam ? Ts)
A simple empirical model that enables prediction of the
-
where T
carotene degradation as a function of carrot moisture content and
s is the surface temperature of the drying material, which
is boiling point of water at the corresponding operating pressure,
temperature during LPSSD has been proposed in the following
and T
form [9]:
steam is the temperature of the low-pressure superheated
steam. Lv is the latent heat of vaporization at the operating pres-
?
2
t
X
T
X
sure and A is the surface area of all molecular sieve particles
=
p
1.104 + 0.261
? 0.192
? 0.561
?
used in each experiment [19]. The values of the heat transfer
i
Xi
Ti
Xi
coef?cient were then calculated and are listed in Table 3.
T 2
X 3
T 3
?
p
p
6.61 × 10?3
+ 0.390
+ 0.011
Ti
Xi
Ti
5. Model implementation
(10)
The
above
model
was
solved
using
COMSOL
where ?i and ?t are the initial and instantaneous -carotene
MultiphysicsTM 3.2 (Comsol AB, Sweden) with the chemical
contents (mg/100 g solid), respectively. Ti and Tp are the initial
engineering module extension. The package is based on
and instantaneous temperatures of carrot (?C), respectively.
the ?nite element method. The Arbitrary Lagrange-Eulerian
(ALE) formulation was used to solve a problem with moving
4.2. Shrinking correlation
boundaries; this allowed consideration of a drying process of a
shrinking body as is the case of the present study. The direct
The following relationship between the shrinkage and the
(UMFPACK) linear system solver was used in the simulation.
moisture ratio of carrot undergoing LPSSD was reported by
Panyawong and Devahastin [15] and used in this study:
6. Results and discussion
V
X 2
X
= a
+ b
+ c
(11)
V
6.1. Drying kinetics and heat transfer behavior
0
Xi
Xi
where a, b and c are the empirical constants, V0 and V are
Selected experimental LPSSD results [8] were used to ver-
the volumes of carrot cube before and after drying (cm3),
ify the present model. The experimental and simulated drying

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Fig. 3. Comparison between predicted (assuming shrinkage) and experimental moisture content and temperature variation with time of carrot cube at 60 ?C and
7 kPa (Xeq = 0.10 kg/kg (d.b.)). Lines represent predicted data; symbols represent experimental data.
conditions are as follows: the drying temperatures of 60, 70 and
almost all conditions. However, at the highest operating pressure
80 ?C, the operating pressures of 7, 10 and 13 kPa. From the
(13 kPa) the model was not able to predict the experimental data
mesh independence test, it was found that a mesh size of 903
well (see Figs. 8 and 9). This is due to the fact that a larger amount
elements was enough to give mesh-independent results.
of steam condensation occurred during the start-up period at a
Since it was noted by Suvarnakuta et al. [19] that there existed
higher operating pressure (since steam at a lower degree of super-
noticeable variations in the weight measurement during molec-
heat tended to condense more easily) and this understandably
ular sieve beads drying experiments, the estimated values of
led to under-prediction of the moisture ratio since the model, as
the heat transfer coef?cient were set in the range of ±20% to
mentioned earlier, did not take into account the effect of initial
investigate the sensitivity of this parameter on the predictability
condensation. The same reason could also be used to explain a
of the product temperature and moisture content. This ±20%
rather larger discrepancy between simulated and experimental
allowance was also due to the fact that the similarity principle
results at lower drying steam temperature as well.
requires that the same conditions of steam velocity and identical
Representative product center temperature pro?les over time
geometry of tested samples were employed. Since the molecular
are also presented in Figs. 3–9. The center temperature increased
sieve beads and carrot cubes have different geometries, it was
over time until it reached the boiling point of water at the cor-
expected that the values of the heat transfer coef?cient would be
responding operating pressure; after this point, the temperature
different.
slowly increased while latent heating prevented a temperature
Figs. 3–9 present the evolutions of the moisture ratio (MR)
rise. It is seen from these ?gures that the simulated predictions
and the center temperature of carrot cube undergoing LPSSD at
did not quite agree with the experimental data. This is due to the
different operating conditions. The equilibrium moisture con-
fact that carrot undergoing LPSSD in fact shrank non-uniformly;
tents reported were obtained from Panyawong and Devahastin
this characteristic is indeed typical of most foods and biomateri-
[15]. It was found that the trends of the moisture ratio predic-
als [20]. Thus, the assumption of uniform shrinkage used in this
tion are in good agreement with the experimental data under
study was not quite correct. Another factor that contributed to the
Fig. 4. Comparison between predicted (assuming shrinkage) and experimental moisture content and temperature of carrot cube at 70 ?C and 7 kPa (Xeq = 0.06 kg/kg
(d.b.)).

---

Fig. 5. Comparison between predicted (assuming shrinkage) and experimental moisture content and temperature of carrot cube at 80 ?C and 7 kPa (Xeq = 0.03 kg/kg
(d.b.)).
Fig. 6. Comparison between predicted (assuming shrinkage) and experimental moisture content and temperature of carrot cube at 70 ?C and 10 kPa (Xeq = 0.07 kg/kg
(d.b.)).
deviation of the simulated results from the experimental data is
gradients were generated within the product, which could in turn
the fact that once the boiling point was reached there was vapor
drive the liquid-form moisture out of the product faster than
generation, which could give rise to an increase in the internal
what was permissible by liquid diffusion alone. Furthermore,
pressure in pores. Thus, it was possible that hydrostatic pressure
the change in porosity and physical structure of carrot during
Fig. 7. Comparison between predicted (assuming shrinkage) and experimental moisture content and temperature of carrot cube at 80 ?C and 10 kPa (Xeq = 0.06 kg/kg
(d.b.)).

---

Fig. 8. Comparison between predicted (assuming shrinkage) and experimental moisture content and temperature of carrot cube at 70 ?C and 13 kPa (Xeq = 0.10 kg/kg
(d.b.)).
Fig. 9. Comparison between predicted (assuming shrinkage) and experimental moisture content and temperature of carrot cube at 80 ?C and 13 kPa (Xeq = 0.10 kg/kg
(d.b.)).
drying could, in principle, change the diffusivity from what was
6.2. Effect of shrinkage on predictability of the model
predicted by the empirical correlations used. Note that the cor-
relations in the literature are based on original dimensions of
Fig. 10 illustrates the simulated product temperature and
the material, i.e., they do not account for changing geometry of
moisture content evolutions of carrot cube undergoing LPSSD
the product. This effect has been shown by Islam and Mujumdar
without consideration of the product shrinkage. It is seen that
[21] via mathematical modeling and experiments conducted in
the assuming non-shrinking model was not able to predict both
a heat pump dryer using air as the drying medium.
the moisture content and, in particular, temperature of carrot
Fig. 10. Comparison between predicted (assuming non-shrinkage) and experimental moisture content and temperature of carrot cube at 80 ?C and 7 kPa.

---

Fig. 11. Comparison between predicted and experimental -carotene degrada-
Fig. 13. Comparison between predicted and experimental -carotene degrada-
tion of carrot at 60 ?C.
tion of carrot at 80 ?C.
undergoing LPSSD. Consideration of the product shrinkage is
fer coef?cients. The -carotene degradation predictions, which
therefore very important to the success of the model.
followed from the simulated moisture content and temperature
It can be observed from Fig. 10 that when shrinkage effect
based on the assuming shrinking model, show good agree-
was not considered the simulated moisture contents were higher
ment with the experimental data under all conditions. Because
than the experimental data, especially during the later stage of
assuming shrinking model could predict the product tempera-
drying where internal diffusion controlled the drying process.
ture and moisture content better, it is not surprising that it could
This is due to the fact that the assuming non-shrinking model
also predict more precisely the -carotene degradation than the
over predicted the diffusion path of moisture compared with that
assuming non-shrinking model.
in the real situation. Therefore, at the same instant, predicted
moisture content was higher than that of the experimental data
and assuming shrinking predicted data. Also, the poor prediction
7. Conclusions
is possibly due to vapor evolution within the product and possible
generation of pressure gradients not accounted for by the model.
A simple three-dimensional liquid diffusion based model was
A similar reason applies in the case of the product temperature.
proposed to predict the evolutions of moisture and temperature
Non-shrinking assumption implies that the area for heat/mass
of a biomaterial undergoing LPSSD. The model consists of the
transfer was constant, which was obviously not the case.
heat conduction and mass diffusion equations and utilizes an
empirical equation, which describes shrinkage of the product.
6.3. Prediction of ?-carotene degradation
The model was found to predict the moisture content as well as
product temperature reasonably well over some ranges of mois-
Figs. 11–13 show the predicted
-carotene degradation in
ture content. The assumption of product deformation and the val-
carrot during LPSSD using the middle values of heat trans-
ues of the heat transfer coef?cient had effects on the shape of the
moisture content and product temperature curves signi?cantly.
The model was used in this study to show the effect of direct
inclusion of uniform shrinkage of the product in the model on
the shapes of drying and product temperature curves. However,
a more realistic information on the deformation kinetics of the
drying material should be used instead of the uniform shrinkage
assumption employed in this study. The use of more realistic
boundary conditions and the inclusion of terms that can take into
account hydrostatic pressure gradients that may exist within the
product during drying are also suggested.
Acknowledgments
The authors express their sincere appreciation to the Com-
mission on Higher Education, the Thailand Research Fund
(TRF), National Science and Technology Development Agency
(NSTDA) and the International Foundation for Science (Swe-
Fig. 12. Comparison between predicted and experimental -carotene degrada-
tion of carrot at 70 ?C.
den) for supporting this study ?nancially.

---

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unit vector
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absolute temperature (K)
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product temperature (?C)
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volume (cm3)
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Document Outline

• A mathematical model for low-pressure superheated steam drying of a biomaterial
  • Introduction
  • Experimental set-up, materials and methods
    • Experimental set-up
    • Materials and methods
  • Model development
    • Assumptions
    • Governing equations for heat and mass transfer
    • Initial and boundary conditions
  • Parameter estimation
    • Thermophysical properties of carrot
    • Shrinking correlation
    • Heat transfer coefficient
  • Model implementation
  • Results and discussion
    • Drying kinetics and heat transfer behavior
    • Effect of shrinkage on predictability of the model
    • Prediction of beta-carotene degradation
  • Conclusions
  • AcknowledgmentsList of symbolsGreek lettersSubscripts
  • References

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