Reaction Engineering Approach (REA) To Modeling Drying Problems: Ideology Versus Reality

13 Pages·0.284 MB·English

by  Chen X.D., Putranto A.

19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

REACTION ENGINEERING APPROACH (REA) TO MODELING DRYING
PROBLEMS: IDEOLOGY VERSUS REALITY

X.D. Chen1,2*, A. Putranto2,3

1School of Chemical and Environmental Engineering, College of Chemistry, Chemical Engineering and Material
Science, Soochow University, Suzhou, Jiangsu Province, PR China

2Department of Chemical Engineering, Monash University, Clayton, Victoria, Australia

3Department of Chemical Engineering, Parahyangan Catholic University, Bandung,
Indonesia

*Corresponding author Tel.:[phone], Email: [email protected]

Abstract: Among the drying models available in literature, REA model is semi-empirical, which was first proposed
in 1996. It was described based upon a basic physical chemistry principle. The ‘extraction of water from moist
material’ is signified by applying the activation energy concept. The single expression of the extraction rate
represents the competition between evaporation and condensation. It also encompasses the internal specific surface
area and mass transfer coefficient thus is linked to material characteristics. What have emerged is that the REA can
be classified in two categories: Lumped (L) -REA and Spatial (S) -REA which can be used to deal with drying of a
material as a whole or to deal with the process by considering the local phenomena within the material, respectively.
Both models have been proven to serve the intended purposes very well. The REA is very effective to generate the
parameters since only one accurate drying run is required to establish the relative activation energy function. The
paper covers fundamentals of the REA, which has not been fully explained as well as the most recent development
and applications. The future of the REA including the role to assist in process design and product quality of several
heat and mass transfer processes is addressed thoroughly.

Keywords: reaction engineering approach (REA), drying, relative activation energy, mass transfer, heat transfer

Arrhenius equation [1]:
FUNDAMENTALS OF THE REACTION dlnk E
- A = A (2)
ENGINEERING APPROACH (REA) dT RT2
where E is the activation energy of the reaction (J.mol-
A
To begin the discussion on the REA concepts, we may 1). This means that the value of lnk changes against
A
need to outline the physical chemistry principles of
temperature is proportional with the value of E . If the
A
chemical reaction with little originality on our part to
activation energy E is larger, the more sensitive of the
A
form the basis of the REA idea. The most prominent
reaction rate towards temperature change. E can be
A
idea of reaction engineering is the expression of the
variable against temperature, when multiple reactions
chemical reaction rate. A chemical reaction rate of
are occurring simultaneously.
species A, involved in the reaction of two species (A

and B) to yield a product is commonly expressed as:
When the range of temperature is not large, E may be
A
dc considered as a constant. In this case, which is more
- A =k cnAcnB (1)
dt A A B commonly adopted in real life, the rate constant is
expressed as:

where n and n are the orders of reactions associated k =k e-EA RT (3)
A B A Ao
with species A and B respectively, kA is a rate constant where kAo is a constant.
and CA and CB are the concentrations of species A and
B, respectively. This rate constant increases with Evaporation-condensation is one of the fundamental
temperature T, approximately increasing by 2-4 times processes in many fields of science and engineering.
with a temperature increment of 10 K. The relationship For decades, various experiments have been done to
between a reaction rate constant, k, and temperature, T, measure the absolute values of the evaporation or
has been generically described using the famous condensation rate, but there still remain many problems
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

due to the lack of knowledge of the underlying
molecular mechanisms. There is still a large room In the same paper, Gray wrote down the following
available to explore the fundamental aspects of equations to describe a wet-combustion system where
evaporation-condensation. This phenomena is further the temperature of the (combustible) solid material is
complicated by the presence of other species than assumed to be uniform so is the water content within
water during drying process. One could, however, the solid matrix:
understand intuitively that removal of water (in vapor) dx
is an energetic process involving latent heat. One has to =  1- x- xe-u - xe-w u(4)
d c e w
put in energy to ‘activate’ water molecules that could
become free from the material being dried. When the where x, u and, are the dimensionless liquid water
water molecules are not associated with the solid or concentration, temperature of the (combustible)
solute molecules and also they stay in bulk liquid material and time respectively. x is defined as the
domain, their interactions between one another would current water content (g) divided by the initial water
only be the kind as if there is no solid. Evaporating content (g) that is available in liquid form (in fact it is
them into gaseous form would mainly be done by the total water content in the system boundary), i.e.
providing energy sufficient to overcome the latent heat mw/mw,o.  represents the model constants in the above
of water evaporation. Of course, another condition is equation, this system is also, in the theory of thermal
that the vapor can be transported into the gas medium ignition and combustion, called as the “Semenov
or into a vacuum as such. It is interesting to note that Approach” signifying the uniformity of the variables
since condensation process is spontaneous which does throughout the material of concern [3]. This is a useful
not need to overcome a kind of activation energy (an assumption, which paves the way for a large number of
energy barrier) meaning the activation energy for mathematical analyses that have considerable physical
condensation is zero, the activation energy of the meanings already relevant to practical conditions.
evaporation process for pure water would be equal to
the heat of reaction (the latent heat of water In equation (4) the first term on the RHS represents the
vaporization), i.e. the forward reaction activation condensation process, denoted by subscript “c”, the
energy less the reverse reaction activation energy second term evaporation, denoted by “e”. The last term
which is zero for condensation. The above was a basic on the RHS represents the consumption of water due to
motive and a basic idea in formulating the rate of the water induced or involved chemical (exothermic)
drying as a competition between evaporation and reaction, wet-oxidation, denoted by “w”. When we
condensation to be described later. remove this wet-oxidation term, an inert system, we
have:
Evaporation-condensation is a sound concept dx
= 1-x- xe-u (5)
describing the influence of phase change upon not only d c e
heat but also mass transfer of a system. One example

that has attracted some attention in the scientific
The above equation represents the water exchange
community was the work on "wet combustion" by
(condensation less evaporation) between the moist
Professor Brian F Gray. In 1990, a mathematical
material and the environment/surrounding. (1-x)
model for wet-combustion system, the exothermically
signifies a conservation of the ‘total water content in
reactive (porous solid) system which is also influenced
the domain of interest’. The evaporation term is
by the presence of water, was published by Professor
considered to be of the first order as far as water
Brian Gray who was a Senior Professor of
‘reactant’ is concerned. The most important description
Mathematics at School of Chemistry in Macquarie
of the evaporation term is the employment of the
University, Sydney, Australia. Though before that
Arrhenius dependence function (a well-known function
‘landmark’, the role of water on the exothermic (solid)
systems, such as spontaneous heating or spontaneous in physical chemistry), i.e. . ‘’ in the context
combustion, several models have been proposed in a of Gray’s analysis denotes the dimensionless latent heat
more engineering manner [2]. However, the role of of water vaporization. This might not be completely
water as a direct participant in chemical reactions (in new even at that time but certainly a great approach to
oxidation in particular) had not been considered describe the physical picture, as we understood about
quantitatively. In the paper by Gray, he added a term in the moisture movement in and out of a porous solid
the mass balance and in the energy balance matrix. Gray [4] and Gray and Wake [5] took fully the
respectively, which accounts for the direct participation advantages of this simple and effective formulation to
of water in chemical reaction (exothermic one), in explore fruitfully the wealth of the behaviors of the
conjunction with the water effect through evaporation wet-combustion system. In any case, a steady state can
(liquid to vapor) and condensation (vapor to liquid). be attained for the system described by equation (5) as:
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

 1-x= xe-u (6) concentration (ρv,s) can then be scaled against saturated
c e vapor concentration (ρ ) using the following equation
v,sat
Re-ranging the above, giving: [6-7]:
1-x = e e-u (7)  =expæç-DEvö÷ (T) (9)
x  v,s è RT ø v,sat s
c s
which essentially suggests that as the material gets where ΔE represents the additional difficulty to
v
hotter, the liquid water content in the system reduces remove moisture from the material beyond the free
more. The activation energy of the evaporation water effect. This ΔE is the average moisture content
v
‘reaction’ was assumed to be the latent heat of water (X) dependent. T is the surface temperature of the
s
vaporization. This has good intuitive ground (Chen, material being dried (K) and ρ for water vapor can
v,sat
1998). As the dimensionless temperature ‘u’ gets still be estimated by:
infinitely large, x becomes zero. This system, however,
does not seem to capture the physics that when the  =4.84410-9(T -273)4-1.480710-7(T -273)3
water content in the environment becomes zero, even v,sat s s (10)
2.657210-5(T -273)2-4.861310-5(T -273)
when temperature is moderate, the liquid water content s s
inside the material can become zero. In other words, 8.34210-3
the above system might have neglected another based on the data summarized by Keey [10].
dimension of the drying system.
When material is ‘thermally’ thin, the surface
THE FORMULATION OF REACTION temperature is considered to be the same as the sample
ENGINEERING APPROACH (REA) AND temperature [11-12], i.e. Ts ≈ T.
DETERMINATION OF THE PARAMETERS The mass balance (8) can then be neatly
expressed as:
In REA, evaporation is modeled as zero-order kinetics dX  -DE 
m =-h A exp( v) (T)- (11)
with activation energy while condensation is described s dt m  RT v,sat v,b
as first order wetting reaction with respect to drying air
From equation (11), it can be observed that the REA is
solvent vapor concentration without activation energy
expressed in the first order ordinary differential
[6]. The REA approach offers the advantage of being
equation with respect to time. It can be seen that
expressed in terms of simple ordinary differential
evaporation rate is expressed as zero order with
equation with respect to time. This negates the
activation energy while condensation is represented as
complications arising from use of partial differential
first order with respect to drying air concentration
equation [7]. This approach has been firstly employed
without activation energy [6-7]. Equation (11) is the
to express the overall drying rate for the entire object
core of reaction engineering approach i.e. the L-REA
being dried – a lumped approach. A summary of the
(lumped reaction engineering approach). It must be
developments of the lumped approach of REA was
noted that, though the average moisture content is used,
given by Chen [6].
the L-REA does not assume uniform moisture content.

The L-REA may also be applied to cases where the
Generally, with no assumption, the drying rate of a
temperature within the material is not uniform as long
material can be expressed as:
as the surface temperature can be determined or
dX
m =-h A( - ) (8) predicted accurately. In order to yield the moisture
s dt m v,s v,b content and temperature profiles during drying,
where ms is the dried mass of thin layer material (kg), X equation (11) yields as a mass balance and need to be
is the average moisture content on dry basis (kg.kg-1), t coupled with the appropriate heat balance.
is time (s), ρv,s is the water vapor concentration at
interface (kg.m-3), ρv,b is the water vapor concentration The activation energy (ΔEv) (which is the characteristic
in the drying medium (kg.m-3), hm is the mass transfer of the material being dried; it is material dependent)
coefficient (m.s-1) and A is surface area of the material needs to be determined experimentally. Upon the
(m2). attainment of the drying data, notably the surface
temperature of the material (or the sample temperature
Equation (8) is a basic mass transfer equation. The in the case of thermally thin situation) and moisture
convective mass transfer coefficient (hm) is determined loss against time as well as the information about the
based on the established Sherwood number correlations external mass transfer coefficient, one can obtain the
for the geometry and flow condition of concern or can activation energy. This can be done by re-arranging
be established experimentally for the specific drying equation (11) into:
conditions involved [8-9]. The surface vapor
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

 dX 1  difference of moisture content during drying (X-Xb).
-m 
 s dt h A v,b The relative activation energy is zero near the start and
DE =-RT ln m (12) this increases as drying progresses. When the
v s   T   equilibrium moisture content is achieved, the relative
v,sat s
  activation energy is 1. This indicates that the difficulty
The rate of moisture loss dX /dt is experimentally to remove the moisture from the materials being dried
increases as the drying continues. The relative
determined. The surface area A should also be recorded
activation energy (ΔE / ΔE ) is a material
in drying experiments in the case of shrinkable v v,b
characteristic but it is independent on drying air
material. The dependence of activation energy on
conditions since different conditions would result in the
average moisture content on a dry basis (X) can be
similar profiles of activation energy [6-7]
normalized as:

DE
v = fX - X  (13) THE SPATIAL REACTION ENGINEERING
DE b APPROACH (REA)
v,b
where f is a function of water content difference, ∆E
v,b
is the ‘equilibrium’ activation energy representing the The diffusion-based models commonly use effective
maximum ΔE determined by the relative humidity and diffusivity in order to lump the whole phenomenon of
v
temperature of the drying air: drying. The effective diffusivity is actually a complex
DE =-RT lnRH  (14) interaction of variables involved during drying
v,b b b including temperature, moisture content and pressure.
RHb is the relative humidity of drying air, Xb is the The use of liquid diffusion only may not be sufficient
equilibrium moisture content under the condition of the to represent the drying process since unreasonable
drying air (kg.kg-1) and Tb is the drying air temperature profiles are resulted [13]. For better understanding of
(K). transport phenomena of drying process, a multiphase
model can be implemented. The equilibrium and non-
The REA model parameters i.e. the relative activation equilibrium multiphase model can be applied but the
energy can be generated from one set of accurate latter is suggested since it is more general and can be
drying run which consists of measurement of sample used to assess the applicability of the equilibrium one
mass, sample surface temperature and sample surface [14]. However, the non-equilibrium one requires an
area during drying. So far in the successful applications accurate formulation of the local
of the REA concept, the drying experiments are usually evaporation/condensation rate inside the materials
performed with final moisture content being very low. being dried as affected by local structure and
composition.
From the experiments mentioned above, the
experimental data of sample mass, surface area, Due to accuracy of the REA to model the global drying
volume and temperature can be obtained. The rate of rate, the REA is implemented to model the local
moisture content change (dX/dt) can be obtained from evaporation/condensation rate in a multiphase drying
the experiment of the weight loss curve. The heat and model. The combination of the REA with a system of
mass transfer coefficient is determined based on equations of conservation of heat and mass transfer is
established Sherwood and Nusselt number correlations called the S-REA (spatial reaction engineering
or established experimentally for the specific drying approach). The S-REA consists of a mass balance of
conditions involved. The ambient water vapor liquid water, mass balance of water vapor and heat
concentration (ρv,b) is determined based on the balance. For uniform convective drying of cubic object
humidity and temperature of the drying air employed in in a heated environment, three-dimensional modeling
the laboratory. By using the experimental data, can be established. The mass balance of water in the
calculated mass transfer coefficients, ρv,sat and ρv,b, the liquid phase (liquid water) is written [13-16]:
activation energy (ΔE) can be calculated using
v (C X)  (CX)  (CX)
equation (12). In addition, the equilibrium activation s = (D s ) (D s )
t x w x y w y (15)
energy (ΔE ) can be evaluated using equation (14).
v,b
The ρ can be related to the equilibrium moisture  (C X) .
v,b  (D s )-I
content (Xb) through the desorption isotherm such as z w z
GAB isotherm. The activation energy (ΔE) can be where X is the concentration of liquid water (kg
v
scaled (from 0 to 1) by dividing it with the equilibrium H O.kg dry solids-1 C is the solids concentration (kg
2 ), s
activation energy (ΔE ) to yield the relative activation dry solids.m-3) which can change if the structure is
v,b
energy (ΔEv / ΔEv,b) as indicated in equation (13). The shrinking, Dw is liquid diffusivity (m2.s-1), I. is the
relative activation energy can be related to the evaporation or condensation rate (kg H O.m-3.s-1) and
2
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

. dT
I is usually defined as positive when evaporation =0,
occurs locally. The liquid diffusivity represents the dx
movement of liquid water inside the pore structure of (24)
the materials due to capillary action as a result of water x=L,
concentration gradient. In practice, the liquid dX C
-C D =h  ( v,s - ) (25)
diffusivity needs to be extracted from the available s w dx m w  v,b
effective diffusivity data [17].
dC C
-D v =h  ( v,s - ) (26)
v dx m v  v,b
The mass balance of water vapor is expressed as [13-
dT
16]: k =h(T -T) (27)
C  C  C dx b
v = (D v) (D v) where ε and ε are fraction of surface area covered by
t x v x y v y w v
(16) liquid water and water vapor respectively.
 C . .
 (D v)I The internal evaporation rate (I) can be described as:
z v z .
where Cv is the concentration of water vapor (kg.m-3) I =hminAin(Cv,s -Cv) (28)
and D is the effective water vapor diffusivity in pore where A is the internal surface area per unit volume
v in
channels (m2.s-1). available for phase change (m2.m-3), h is the internal
m,in
surface mass transfer coefficient (m.s-1).
The heat balance is represented by the By implementing the REA, internal-surface water
following equation [13-16]: vapor concentration can be written as [16]:
T  T  T -DE
C = (k ) (k ) C =exp( v)C (29)
p t x x y y (17) v,s RT v,sat
where C is the internal-solid surface water vapor
 T . v,s
 (k )-IDH concentration (kg.m-3), C is the internal saturated
z z v v,sat
water vapor concentration (kg.m-3) and ΔE is the
where T is the sample temperature (K), ΔH is the v
V activation energy (J.mol-1) similar to the one described
vaporization heat of water (J.kg-1), k is the sample
in equation (14).
thermal conductivity (W.m-2.K-1) and ρ is the sample
Therefore, the internal evaporation rate can be
density (kg.m-3), k and ρ may be functions of
expressed as [16]:
temperature and moisture content.
. -DE
I =h A (exp( v)C -C ) (30)
For cubic samples dried uniformly from all directions min in RT v,sat v
(x, y and z directions), the mass balance of water in In equation (30), the REA is used to describe the local
liquid phase can be simplified into [20-21]: evaporation rate as affected by pore structure (porosity,
shrinkage, local moisture content and local
(C X)  (C X) .
s =3 (D s )-I (18) temperature). These microstructural effects would be
t x w x
‘encapsulated’ in the term h A .
m,in in
while the mass balance of water in vapor phase can be

expressed as:
The effective vapor diffusivity is deduced from [22]:
Cv =3  (D Cv)I. (19) D =D  (31)
t x v x v vo
In addition, the heat balance can be represented as: while D is the water vapor diffusivity (m2.s-1) which
vo
T  T . is dependent on temperature. For food and biological
C =3 (k )-IDH (20)
p t x x v materials, Dv can be expressed as [16]:

The initial and boundary conditions for equations (18)
to (20) may be written as: D =2.0910-5 2.13710-7(T -273.15) (32)
vo
t=0, X=Xo, Cv=Cvo, T=To
(21) Until now, there is no method to measure the ‘effective
x=0, dX , liquid diffusivity’ [13]. Many drying papers estimated
=0
dx the effective liquid diffusivity based on drying kinetics
(22) data. Several sets of drying followed by complex
dCv =0, optimization procedures are used to generate the
dx effective diffusivity function [18, 23-26]. The literature
(23) on the effective diffusivity may be used as a basis to
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

 
determine the effective liquid diffusivity to be used in d mC T dX
p avg hAT -T m DH (34)
S-REA. A little adjustment on the effective liquid dt b s s dt V
diffusivity to generate the effective liquid diffusivity
where m is the sample mass (kg), C is the specific heat
p
function since the effective liquid diffusivity in these of the sample (J.kg-1.K-1), T is the average
avg
existing literatures is used to represent the whole temperature (K), h is the heat transfer coefficient (W.m-
phenomenon in drying (liquid diffusion, vapor 2.K-1), T is the drying air temperature (K), T is the
b s
diffusion, Darcy flow, evaporation/condensation).
surface temperature (K), m is the dried sample mass,
s
∆H is the vaporization heat of water (J.kg-1).
v
APPLICATION OF THE LUMPED REACTION

ENGINEERING APPROACH (L-REA) FOR
For modeling the intermittent drying, the heat balance
MODELING INTERMITTENT DRYING
is employed according to the drying air temperature in

each section. In addition, the equilibrium activation
In this study, the experimental data are derived from
energy shown in equation (14) is evaluated according
the work of Vaquiro [18]. For better understanding of
to the corresponding drying air temperature and
the modeling, the experimental details are reviewed
humidity in each drying period.
briefly here. Mango tissues used for drying

experiments were formed into cubes with side-length
Figures 1 to 2 show the results of modeling of
of 2.5 cm with initial moisture content of 9.3 kg.kg-1
intermittent drying of mango tissues using the L-REA.
and initial temperature of 10.8 ºC. Drying was
For intermittent drying at drying air temperature of 45
conducted in a laboratory drier described in detail by
ºC, the L-REA describes both moisture content and
Sanjuan et al [19]. The drying air temperature and air
temperature profile very well. A very good agreement
velocity were controlled at preset values by PID
is observed between experimental and predicted data.
control algorithms while air humidity was maintained
Similar results are also revealed for intermittent drying
constant during drying. The weight of the sample was
at drying air temperature of 55 ºC and 65 ºC. The
measured periodically to obtain weight loss as well as
predicted moisture content and temperature match well
the centre temperatures every 2 minutes.
with the experimental data. The good predictions of

moisture content and temperature profile are revealed
The intermittency is created by heating and resting by R2 and RMSE shown in Table 2.
period listed in Table 1. During resting period, the

samples stay at environment with ambient temperature
Benchmark towards modeling proposed by Vaquiro et
of 27±1.6 ºC and relative humidity of 60%.
al [18] employing diffusion model was conducted and

it is revealed that the L-REA gives comparable or even
Table 1. Schemes of intermittent drying of mango
better results. Modeling proposed by Vaquiro et al [18]
tissues [18]
showed a kink of temperature profile at the beginning
Drying air Period of Period of Period of
of drying; which was not observed by modeling using
temperature first resting second
the L-REA. In addition, the underestimation of
(ºC) heating (at 27 ºC heating
moisture content profile at last period of drying at
(s) ± 1.6) (s) (s) drying condition of 65 ºC is not revealed by the L-
45 16200 10800 36360 REA; as shown by modeling by Vaquiro et al [18].
55 9480 10800 33720
65 7800 10800 16200

The L-REA shown in equation (11) is used for
the modeling. The relative activation energy is
generated from convective drying run at drying air
temperature of 55 ⁰C. Using the procedures explained
in Section 2, the relative activation energy is written as:

DE
v =-9.9210-4(X -X )39.7410-3(X -X )2(33)
DE b b
v,b
-0.101(X -X )1.053
b

The heat balance for intermittent drying of mango Figure 1. Moisture content profile of mango tissues
tissues can be expressed as: during intermittent drying at drying air
temperature of 45 ºC and resting at 27 ºC
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

The accuracy of the S-REA to model the intermittent
drying is validated towards the experimental data of
intermittent drying of mango tissues [18] whose
experimental setup and settings are explained
previously,

The mass balance of water in liquid phase (liquid
water), the mass balance of water in the vapor phase
(water vapor) and the heat balance are shown in
equations (18), (19) and (20) respectively while the
initial and boundary conditions for equations are shown
in equations (21) to (27). For modeling the intermittent
drying using the spatial reaction engineering approach
(S-REA), the relative activation energy used in the
lumped reaction engineering approach (L-REA) and

shown in equation (33) is used to describe the local
Figure 2. Temperature profile of mango tissues
drying/condensation rate. The equilibrium activation
during intermittent drying at drying air
(ΔE ) energy shown in equation (14) is evaluated
temperature of 45 ºC and resting at 27 ºC v,b
according to the corresponding drying air temperature

and humidity in each drying period. It is also combined

Table 2. R2 and RMSE of modeling of intermittent with the relative activation energy shown in equation
(12) to yield the local drying/condensation rate. In
drying of mango tissues using the L-REA (lumped
addition, the heat balance implements the
reaction engineering approach)
corresponding drying air temperature in each drying

period by using the corresponding drying air
Drying air R2 X R2 T RMSE RMSE
temperature in the boundary conditions indicated in
temperature X T
equation (27).
(ºC)

45 0.998 0.996 0.083 0.483
The profiles of moisture content during intermittent
55 0.998 0.997 0.087 0.554
drying are shown in Figure 3. A good agreement
65 0.998 0.997 0.082 0.686
between the predicted and experimental data is

observed and confirmed by R2 and RMSE listed in
It can be said that the L-REA is accurate to model
Table 3. The results of modeling match well with the
intermittent drying of mango tissues although the REA
experimental data of moisture content. The S-REA
is represented in a lumped model. This is because the
models the average moisture content during the
relative activation energy (ΔE/ΔE ) implemented
v v,b intermittent drying at drying air temperature of 45, 55
allows the natural transition along drying time. The
and 65 °C very well. Benchmarks against modeling
relative activation energy keeps increasing during
implemented by Vaquiro et al [18] revealed that the
drying indicating the increase of difficulty to remove
REA yields better results as Vaquiro et al [18] showed
water from materials. This increases significantly
a slight underestimation of drying rate of intermittent
during heating period while this only increases slightly
drying at drying air temperature at 45 °C during drying
during resting period. This natural transition during
time between 20000 and 50000s. Similarly, Vaquiro et
drying is not observed by empirical models and CDRC
al [18] also revealed a slight overestimation of drying
(characteristics drying rate curve) [27]. It was revealed
rate of intermittent drying at drying air temperature of
that the empirical models could not model the
65 °C after drying time of 20000s. The slight
intermittent drying of banana tissues well. It was also
underestimation and overestimation of the drying rate
analyzed that CDRC might not be able to handle this
are not shown by the modeling using the S-REA. This
since drying rate of intermittent drying could not be
indicates that the S-REA can be used to describe the
represented simply as linear and exponential
moisture content of intermittent drying of mango
decreasing drying rate [27].
tissues well and the REA can be applied to model the

local evaporation and condensation rate well.
APPLICATION OF THE SPATIAL
REACTION ENGINEERING APPROACH (S-REA)
FOR MODELING INTERMITTENT DRYING
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

observed. This is in agreement with the relatively
uniform moisture content at the end of drying as
explained above.

Figure 3. Average moisture content profiles of
mango tissues during intermittent drying at
different drying air temperatures

The spatial profiles of moisture content during the
Figure 4. Spatial moisture content profiles of mango
intermittent drying at drying air temperature of 55 °C
tissues during intermittent drying at drying air
are indicated in Figure 4. The moisture content of the
temperature of 55 °C
outer part of the samples is lower than that in the inner

part of the samples which indicates that moisture
migrates outward during intermittent drying. Initially,
the gradient of moisture content inside the samples is
relatively high but during drying time between 9480
and 20280s the gradient is relatively low since the
samples are under resting period. This period seems to
allow the moisture to redistribute inside the samples
and low gradient of moisture content inside the
samples are generated. Towards the end of drying,
although the samples are under heating period again,
the relatively uniform moisture content is observed
which indicates that the equilibrium condition is almost
approached.

Figure 5 presents the spatial profiles of water vapor Figure 5. Spatial water vapor concentration profiles
concentration during the intermittent drying at drying of mango tissues during intermittent drying at
air temperature of 55 °C. The profiles of water vapor drying air temperature of 55 °C
concentration are significantly affected by the local
composition and structure of the samples being dried. The profiles of temperature during the intermittent
The water vapor concentration achieves a maximum at drying are indicated in Figures 6 and7. Figure 6 shows
particular position inside the samples. At core of the the profiles of centre temperature during the
samples, the moisture content is relatively high which intermittent drying of mango tissues at drying air
may result in lower porosity and retard the local temperature of 45, 55 and 65 °C. A good agreement
evaporation rate. At outer part of the samples, the local between the predicted and experimental data is
evaporation rate may be enhanced as a result of the observed which is also supported by R2 of higher than
higher porosity but this seems to be balanced by higher 0.992 and RMSE lower than 0.828. The results of
diffusive flux of water vapor due to the higher porosity modeling match well with the experimental data.
and temperature. During first heating period, the Benchmarks against modeling implemented by
gradient of water vapor concentration is relatively high Vaquiro et al [18] show that the S-REA yields better
but this decreases as drying progresses. The relatively results since the modeling by Vaquiro et al [18]
uniform concentration of water vapor is shown during indicated kinks and underestimation of temperature
resting period which could be due to relatively low profiles in the beginning of first heating period.
temperature. During the second heating period, the However, these are not observed by the modeling using
relatively uniform water vapor concentration is the S-REA. This indicates that the S-REA is indeed
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

accurate to model the temperature profiles of The S-REA yields advantages of resulting in the spatial
intermittent drying of mango tissues. profiles of local evaporation rate during drying as the
REA is used to predict the local evaporation rate. The
profiles of local evaporation rate during intermittent
drying of scheme 1 at drying air temperature of 55 °C
are shown in Figure 8. The local evaporation rate at
core of samples is lower than that of the outer part of
samples. This could be because of lower porosity at
core of the samples as a result of higher moisture
content. During resting period, the spatial profiles of
local evaporation rate are more uniform than those
during first heating period which may be due to lower
temperature inside the samples. Towards the end of
drying, the gradient of local evaporation rate decreases.
This could be because the moisture content decreases
as drying progresses which increases the porosity.
From the beginning of drying to drying time around
Figure 6. Centre temperature profiles of mango 5000s, the local evaporation rate increases which could
tissues during intermittent drying at different be because of the increase of temperature. After this
drying air temperatures period, the local evaporation rate tends to decrease
which could be because of the decrease of moisture
content. During resting period, the local evaporation
Figure 7 represents the spatial profiles of temperature rate only changes slightly which may be because of the
during the intermittent drying at drying air temperature relatively low temperature. During the second heating
of 55 °C. During the first heating period, the period, the local evaporation decreases. This may be
temperature in the outer part of the samples is higher because the moisture content inside the sample is
than that of the inner part because the samples receive depleted. At the end of drying, essentially there is no
heat by convection from the drying air used for water much difference in evaporation rate inside the samples
evaporation and penetrated inwards by conduction. because the moisture content has nearly achieved
However, the gradient of temperature inside the equilibrium, under the drying conditions.
samples is not high.. During the first heating period,
the temperature of samples increases but the
temperature decreases between drying time of 9480
and 20280s as a result of resting period. This is
followed by a further increase of temperature during
the second heating period. At the end of the
intermittent drying, the temperature approaches the
drying air temperature.

Figure 8. Profiles of evaporation rate inside mango
tissues during intermittent drying at drying air
temperature of 55 °C

It has been demonstrated that the S-REA is accurate to
model the intermittent drying of mango tissues very
well. The S-REA can also project the concentration of
water vapor and local evaporation rate during

Figure 7. Spatial temperature profiles of mango intermittent drying so that better understanding of
tissues during intermittent drying at drying air transport phenomena of drying processes can be
temperature of 55 °C gained. It is argued here that the S-REA is not only
19th International Drying Symposium (IDS 2014)
Lyon, France, August 24-27, 2014

robust to model the convective drying (Putranto and T  T
C = (k ) (37)
Chen, 2013) but also the intermittent drying. p t x x

QUESTIONS ON LIQUID DIFFUSIVITY
where C is the solids concentration (kg solids.m-3), X
s
is the moisture content (kg water.kg dry solids-1), C is
Modeling of drying process is relatively complex and v
the concentration of water vapor (kg.m-3), T is
there have been several mechanistic models proposed
temperature (K), D is the capillary diffusivity (m2.s-1),
including liquid diffusion, capillary flow, evaporation w
D is the effective vapor diffusivity (m2.s-1), t is time
condensation, ,Luikov and Whittaker approach [28-32]. v
(s), x is the axial dimension (m), ρ is the sample density
Luikov’s approach [31] assumes the thermal and
(kg.m-3) and C is the sample specific heat (J.kg-1.K-1),
moisture potential gradient within a porous body cause p
k is the thermal conductivity (W.m-2.K-1).
the vapor and liquid water transfer so that the flux of

liquid water and water vapor is proportional to the
The initial and conditions of equations (35) to (37) are
thermal gradient and moisture potential gradient. As
[13, 15]:
mentioned earlier, the coefficients of effective water

liquid diffusivity, effective water vapor diffusivity and
t=0, X=X , C=C T=T (38)
thermal diffusivity are implemented to link the fluxes o v vo, o
x=0,
and the gradients.
X
=0 (39)
There is little question that the effective liquid x
diffusion model is the simplest model proposed [13]. C (40)
v =0
The effective diffusivity should be influenced by the x
composition of the materials. The multi-component T
=0 (41)
diffusion model is often implemented to couple this x
effect [33-34]. Therefore, the effective diffusivity is x=L,
essentially a lumped parameter whose variability is
X
dependent on the drying condition, material structure -C D =0 (42)
s w x
and composition, and sometimes sample sizes.The last
C
aspect rules out the upmost fundamental nature of the -D v =h ( - )
effective liquid diffusivity. For moderate drying v x m v,s v,b (43)
condition, it would be better to see the effective T
diffusivity as a liquid depletion coefficient in order to kx =h(Tb-T)-DHVhm(v,s-v,b) (44)
avoid confusion with the original meaning of

diffusivity [13].
where L is the sample half thickness, T is drying air
b
temperature (K), h is the mass transfer coefficient
For better understanding of transport phenomena m
(m.s-1), h is the heat transfer coefficient (W.m-1.K-1),
during drying, the use of multiphase drying model is
ΔH is the vaporization heat of water (J.kg-1), ρ is the
suggested. It consists of the mass balance of water in V v,s
surface water vapor concentration (kg.m-3) and ρ is
liquid and vapor phases as well as the heat balance. By v,b
the concentration of water vapor at drying medium
this approach, the spatial profiles of moisture content,
(kg.m-3).
concentration of water vapor and temperature can be

generated. For the multiphase approach, which does
From equations (35) to (37), it can be observed that
not use the source and depletion term, the model for
there is no interaction among the liquid water and
symmetrical convective drying of a slab can be written
water vapor apart from the effective diffusivity of
as follows [13,15].
water vapor, which should be a function of porosity,

dependent on the moisture content. In addition, the
The mass balance of liquid water:
boundary conditions indicate that at the interface, the
(C X)  (C X)
s = (D s ) (35) vapor diffusive transport inside the samples is balanced
t x w x by the convective water vapor. Therefore, the
The mass balance of water vapor: equilibrium relationship between the moisture content
C  C and concentration of water vapor has to be
v = (D v) (36)
t x v x implemented at the boundary [13].

The heat balance:
However, if equations (35) to (37) are solved
simultaneously with the initial and boundary conditions