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SIMULATION OF A VIBRATED FLUIDISED BED DRYER FOR
SOLIDS CONTAINING A MULTICOMPONENT MOISTURE
A. Picado1,2* and J. Martínez2
1Faculty of Chemical Engineering, National University of Engineering (UNI)
PO Box 5595, Managua, Nicaragua
E-mail: [email protected]
2Dept of Chemical Engineering and Technology, Royal Institute of Technology (KTH)
SE-100 44 Stockholm, Sweden
E-mail: [email protected]
Abstract. The drying of solids in a continuously worked vibrated fluidised
bed dryer is studied by simulations. A model considering the drying of a
thin layer of particles wetted with a multicomponent mixture is developed.
Particles are assumed well mixed in the direction of the airflow and only the
longitudinal changes of liquid content, liquid composition and particle
temperature are considered. Interactive diffusion and heat conduction are
considered the main mechanisms for mass and heat transfer within the
particles. Assuming a constant matrix of effective multicomponent diffusion
coefficients and thermal conductivity of the wet particles analytical
solutions of the diffusion and conduction equations are obtained. The
variation of both the diffusion coefficients and the effective thermal
conductivity of the particles along the dryer is taken into account by a
stepwise application of the analytical solution in space intervals with
averaged coefficients from previous locations in the dryer. The analytical
solution gives a good insight into the selectivity of the drying process and
can be used to estimate aroma retention during drying. The solution is
* To whom all correspondence should be addressed
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computationally fast; therefore, the experimental verification of this
approximate model would introduce an important computational economy
since the rigorous treatment of multicomponent drying involves tedious and
time-consuming calculations.
Keywords: Aroma Retention, Drying Selectivity and Multicomponent Drying.
1. Introduction
Continuously worked vibrated fluidised bed dryers (VFBD) have been used to dry a
variety of particulate solids such as inorganic salts, fertilizers, foodstuffs,
pharmaceuticals, plastics, coated materials, etc. In some industrial processes, the VFBD
is the only drying unit responsible for moisture removal, but it is also used as a second
stage in two stages drying processes. For instance, the first stage is performed in a spray
dryer to concentrate the product and the VFBD is used in a second stage to reduce the
moisture content to the value required by the final product. This second stage saves
energy and assures better control of the product quality (Cruz et al., 2004). Other
advantages of VFBD are: good performance, relative low investment cost, low
maintenance costs, robustness of the equipment and versatility. Many different types of
particulate solids, from chemicals to foodstuffs, usually with large continuous
throughputs are treated in this way. In most of the cases, the moisture to be removed
consists of water but there are important applications such as the drying of
pharmaceuticals, plastics and coated materials where the moisture consists of a
multicomponent mixture. The drying of foodstuffs is a special case of multicomponent
drying since the moisture usually consists of water and a large number of low
concentration volatile compounds (e.g., coffee, cocoa or milk).
Considerable work has been devoted to the study of VFBD concerning particle
behaviour and its interaction with the gas, wall and effects of vibration, as well as mass
and heat transfer during drying (Pan et al., 2000; Pakowski et al., 1984; Hovmand,
1987). Eccles and Mujumdar (1992) carried out an extensive review of work on VFBD.
There are numerous incremental models to simulate the drying process in continuous
fluidised bed dryers (Keey, 1992; Kemp and Oakley, 2002; Izadifar and Mowla, 2003;
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Daud, 2006). Most of the equipment models assume plug flow of the solids but solids
non-ideal flow has been also studied. Gas cross flow is modelled in some extent. On the
other hand, the material model does not include the drying of solids wetted with a
mixture of solvents. These cases are important because of the great influence of the
composition of the remaining mixture on product quality.
A great deal of multicomponent drying research has been performed by Schlünder
and co-workers in Karlsruhe (Schlünder, 1982; Thurner and Schlünder, 1986; Riede and
Schlünder, 1990; Wagner and Schlünder, 1998). The research has been focused on the
behaviour of the evaporating mixtures in a rather simple geometry. Depending on the
prevailing drying conditions, drying of solids containing multicomponent mixtures can
be controlled by transport in the liquid phase, in the gas phase or by equilibrium. Gas-
phase-controlled drying of a multicomponent liquid film in continuous contact with the
gas phase has been studied by Vidaurre and Martínez (1997). Luna and Martínez (1999)
showed that a deep understanding of the process can be obtained by a stability analysis
of the ordinary differential equations that describe the dynamical system. Liquid-phase-
controlled drying of multicomponent mixtures has been analysed by Pakowski (1994).
Gamero et al. (2006a) studied the continuous evaporation of a falling liquid film into an
inert gas numerically. Recently, Gamero et al. (2006b) reported an analytical solution
for batch drying of a multicomponent liquid film in non-isothermal conditions assuming
constant physical properties. The changes of physical properties during the process were
accounted for by a stepwise application of the solution with averaged coefficients from
previous steps.
The purpose of this study is the development of a model to simulate the drying of
particulate solids containing multicomponent liquid mixtures in a vibrated fluidised bed
dryer. The model is developed by incorporating a material model for a single spherical
particle wetted with a liquid mixture in an incremental equipment model assuming plug
flow of the solids. The model would be a useful tool to explore the selectivity of the
drying process and choose appropriate drying conditions to control the composition of
the final moisture.
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2. Theory
A schematic description of the VFBD is shown in Figure 1. In such equipment,
effective mixing of the particles takes place and a homogeneous material at a vertical
cross section of the dryer is usually obtained. The residence time distribution of the
particles measured at the outlet does not differ very much of that calculated for a plug
flow model (Strumiłło and Pakowski, 1980). Vibration allows for lower gas velocities
to achieve a good contact between the gas phase and the wet particles.
Fig. 1. A Plug Flow Vibrated Fluidised Bed Dryer.
2.1. Mass and Energy Balances in the Dryer
In the analysis of the dryer, it is assumed that the bed of particles is moving forward
with a uniform velocity and that the dryer has been operated during sufficient time for
steady state conditions be reached. A moisture balance applied to the volume element
shown in Figure 2 yields:
giii
s GaMdz
dXF −= i = 1, . . . . n (1)
where n is the number of components in the moisture. Since all the evaporated liquid
goes to the gas the changes of air humidity are given by the following balances:
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dzdX
HFdYF ibsig −= i = 1, . . . . n (2)
In Equations (1) and (2) the air humidity, Yi, and the solid liquid content, Xi, are in
dry basis. F is a mass flow of inert per cross section in the direction of the flow. The
subscripts s and g denote solid and gas respectively. M is the molecular weight, a is the
specific evaporation area per bed volume, Gg,i is the molar evaporation flux of
component i, and Hb is the bed height.
Fig. 2. Scheme of a differential dryer element.
If heat losses in the dryer are neglected the energy balance over the volume element
becomes:
dzdI
FF
HdI s
g
sbg −= (3)
where I is the enthalpy of the phases per unit mass of inert. The bed height is calculated
as:
B)1)(1(vSH
bppb ε−ε−ρ
= (4)
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where S is the flow of dry solids, v is forward bed velocity, ρ is the density, ε is the
porosity and B is the dryer wide. The subscripts p and b denotes particle and bed
respectively. To integrated Eq. (1) along the dryer, apart from inlet conditions, the
evaporation fluxes must be provided. These fluxes depend on the temperature and liquid
composition at the surface of the particles. This information can be obtained by
analysing what happens with a single particle moving along the dryer.
2.2. Drying of a Single Particle
The drying of a single particle into an inert gas is schematically described in Figure
3.
Fig. 3. Schematic drying of a single particle into an inert gas.
2.3. Governing Equations
If diffusion inside the particle is the main contribution to mass transfer, the process is
described by the diffusion equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=∂∂
rr2
rzv 2
xxDx 2
(5)
If conduction is the only mechanism for heat transfer within the particle the
corresponding equation to describe changes of temperature is the conduction equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=∂∂
rT
r2
rTD
zTv 2h
2
(6)
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where x is a column vector with the molar fractions of the independent diffusing
component in the liquid, D is the matrix of multicomponent diffusion coefficients and
Dh the heat diffusivity.
Equations (5) and (6) represent a system of partial differential equations. If
evaporation and convection heat occurs only at the surface of the particle and the initial
composition as well as temperature of the particles are given functions of r, the initial
and boundary conditions are:
At z = 0 and 0 ≤ r ≤ δ, { }r0xx= ; { }rTT 0= (7)
At r = 0 and z > 0, 0=∂∂
rx ; 0=
∂∂
rT (8)
At r = δ and z > 0, 1ng −=∂∂
− ,L rC GxD ; gGT
g, )T(ThrTk λ+−=
∂∂
− ∞ (9)
where λ is a column vector of heat of vaporisation. The superscript T denotes
transposition. The subscript n-1 in the column vector of evaporation fluxes in gas phase
indicates that only n-1 of the fluxes are considered to match the dimension of the
independent diffusion fluxes within the particle.
2.4. The Matrix of Multicomponent Diffusion Coefficients
The matrix of multicomponent diffusion coefficients, D, is of order n-1 × n-1. This
expresses the fact that the nth component does not diffuse independently. In non-ideal
mixtures, the matrix of multicomponent diffusion coefficients is defined as:
ΓBD 1ι −= (10)
where ι embodies the constriction and tortuosity factors to take into account that the
liquid is confined in a porous particle. The matrix B, which can be regarded as a kinetic
contribution to the multicomponent diffusion coefficients, has the elements:
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∑≠=
+=n
ik1 ik
k
in
iii D
xDx
Bk
; ⎟⎟⎠
⎞⎜⎜⎝
⎛−−=≠
inijiij D
1D1xj)(iB (11)
where i, j = 1, 2,…n-1 and D ij are the Maxwell-Stefan diffusion coefficients. The
elements of the matrix of thermodynamic factors, Γ, are given by:
j
i
j
iij lnx
lnγxx
δ∂∂
+=Γιj (12)
where γi is the activity coefficient of compound i and δi,j is the Kronecker delta (δi,j = 1
for i = j and δi,j = 0 for i ≠ j). For ideal solutions, the matrix of thermodynamic factors
reduces to the identity matrix.
2.5. Mass and Heat Transfer Rates
If diffusional interactions in gas phase are included evaporation fluxes may be
written as:
}{g ∞−= yyKG δ (13)
Here the matrix K is the matrix product βEk in which β embodies an extra
relationship between the fluxes to calculate molar fluxes from diffusion fluxes, E is a
matrix of correction factors to account for the finite mass transfer rate and k is a matrix
of mass transfer coefficients at zero mass transfer rates. The columns vectors yδ and y∞
are the molar fractions of the vapours at the gas-liquid interface and the bulk of the gas
respectively. For details see Taylor and Krishna (1993). The convective heat flux can be
expressed by:
)T(T h q ,g δ∞ −= (14)
where h is a heat transfer coefficient between the heating medium and the particles.
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2.6. Coupling between Phases
If the gas phase is considered to be in equilibrium with the liquid at the interface,
then at r = δ:
nn xKxγPy γ== 0
tδ P
1 (15)
is obtained, with Pt being the total pressure. P0 and γ are diagonal matrices containing
the saturated vapour pressures of the pure liquids, and activity coefficients respectively.
The subscript n indicates that the vector x contains the molar fractions of the n
components of the liquid mixture.
2.7. Integrating along the Dryer
The solution of Eqs. (5) and (6) subjected to inlet and boundary conditions (7)
through (9) provides the temperature and liquid composition gradients within the
particle. In addition, mass and heat transfer rates at the particle surface are obtained.
The analytical solution assuming constant transport coefficients as well as heat and
mass transfer rates is shown in details in Appendix A. Since these conditions change
along the dryer, the analytical solution is applied to an interval dz, with inlet conditions
and averaged transport coefficients corresponding to the outlet conditions of the
previous step. As the integration of Eq. (1) proceeds the procedure is repeated. The
outlet composition of the gas at each step dz is calculated using Eq. (2). Then, the
energy balance (3) allows for the calculation of the exhaust gas enthalpy using the
particle mean temperature to calculate the outlet enthalpy of the wet solids. Since the
gas enthalpy is a function of gas composition and temperature, the outlet gas
temperature can be calculated from a non-linear equation that relates gas temperature
with enthalpy. Integration proceeds in this way until the exit of the dryer is reached.
3. Results and Discussion
Calculations were performed with particles containing two different liquid mixtures:
ethanol-2-propanol-water, and acetone-chloroform-methanol. The evaporation fluxes
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were calculated according to Eq. (13) using an algorithm reported by Taylor (1982)
with diffusion through stationary gas as bootstrap relationship. The matrix of correction
factors was evaluated using the linearised theory. Mass and heat transfer coefficients at
zero-mass transfer rates were computed by correlations of Kunii and Levenspiel (1969)
with binary diffusion coefficients in gas phase predicted by the method of Fuller et al.
(1966). Physical properties of pure component and mixtures were evaluated using
methods described by Poling et al. (2000). Activity coefficients were calculated
according to the Wilson equation with parameters from Gemhling and Onken (1982).
Antoine method was used for computing the vapour pressure of pure liquids. For
determining the Maxwell-Stefan diffusion coefficients in liquid phase the method of
Brandowski and Kubaczka (1982) with an empirical exponent of 0.5 was used for both
liquid systems. Physical properties of Pyrex were used for the solid.
A typical result for a simulation for a solid containing ethanol-2-propanol-water is
shown in Figure 4.
In this mixture the volatility of water is much less than ethanol and 2-propanol.
According to the theory, to remove water preferentially and keep the volatiles in the
solid, the resistance against mass transfer within the solid must be high. This situation is
favoured by an intensive drying regime. The resistance within the solid increase when
the ratio between constriction and tortuosity has a low value and the diameter of the
particles is large. Drying intensity can be increased by increasing external factors such
as gas velocity and temperature. The Tables below show the influence of these
parameters on the ratio of retention defined as )X/X/()X/X( 0ei,0i,e .
The results revealed that retention of volatile compounds is favoured by the resistance
against mass transfer within the solid. However increasing gas velocity and temperature
has a negative effect. The selectivity of the process is not expected to be affected by the
external conditions but to induce internal resistance. Clearly, in the conditions
examined, the effects of gas velocity and gas temperature on particle temperature and
transport coefficients seem to have an opposite effect.
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Fig. 4. Drying simulations for particles containing ethanol-2-propanol-water. ug0 =
1.5 m/s, Tg0 = 343.15 K, Y0 = [0 0 0 1], S = 7 10-2 kg/s, δ = 3 10-3 m, v = 0.02 m/s.
Table 1. Influence of gas velocity on volatile retention. Tg0 = 343.15 K, S = 7 10-2 kg/s,
δ = 3 10-3 m, v = 0.02 m/s.
Components Retention ratio
ug0 = 1.0 m/s ug0 = 1.5 m/s ug0 = 1.9 m/s
Ethanol 0.9000 0.8583 0.8654
2-propanol 0.9596 0.9314 0.9244
Water 1.0836 1.1267 1.1278
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Table 2. Influence of the particle diameter on volatile retention. ug0 = 1.5 m/s, Tg0 =
343.15 K, S = 7 10-2 kg/s, v = 0.02 m/s.
Components Retention ratio
δ = 0.002 m δ = 0.003 m δ = 0.004 m
Ethanol 0.7484 0.8583 0.9161
2-propanol 0.8821 0.9314 0.9592
Water 1.2222 1.1267 1.0751
Table 3. Influence of the solid structure on volatile retention. ug0 = 1 m/s, Tg0 = 343.15
K, S = 7 10-2 kg/s, δ = 3 10-3 m, v = 0.02 m/s.
Components Retention ratio
ι = 0.35 ι = 0.65 ι = 1.0
Ethanol 0.9483 0.9110 0.9000
2-propanol 0.9771 0.9623 0.9596
Water 1.0447 1.0757 1.0836
Table 4. Influence of the gas temperature on volatile retention. ug0 = 1.5 m/s, S = 7 10-2
kg/s, δ = 3 10-3 m, v = 0.02 m/s.
Components Retention ratio
Tg0 = 60 °C Tg0 = 70 °C Tg0 = 80 °C
Ethanol 0.8648 0.8583 0.8528
2-propanol 0.9388 0.9314 0.9243
Water 1.1179 1.1267 1.1347
Simulation results for the drying of particles wetted with a liquid mixture consisting
of the highly volatile components, acetone-chloroform-methanol are shown in Figure 5
and Table 5. It is clear that drying rates are higher than of the mixture containing water
and particle temperature decreases much more along the dryer. In the presence of such
solvents, the main concern should be to keep the concentration of all or some
components in the product below certain limits. The results of the simulations shown in
Table 5 evidence the particular features of multicomponent drying that can lead to
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unexpected results and the application of unconventional measures to fulfil product
quality requirements. Slight changes of liquid composition in the feed by adding small
amount of the other components to the solid reduce methanol concentration in the
product to less than 25 % of that of the first case. Furthermore, the final total liquid
content is reduced despite the higher total liquid content of the feed.
Fig. 5. Drying simulations for particles containing acetone-chloroform-methanol. ug0
= 1.5 m/s, Tg0 = 343.15 K, Y0 = [0 0 0 1], S = 7 10-2 kg/s, δ = 3 10-3 m, v = 0.02 m/s.
Table 5. Adding solvents to the solid feed. 1) Acetone, 2) Chloroform, 3) Methanol.
x0 (kmol/kmol) X0 (kg/kg) Xe (kg/kg) X3,e, Methanol (mg/kg)
[0.20 0.20] 0.2900 5.516 10-2 2.798
[0.21 0.20] 0.2935 5.779 10-2 2.436
[0.20 0.21] 0.2917 5.367 10-2 0.681
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4. Conclusions
The incremental model to simulate drying of particles containing liquid mixtures in a
vibrated fluidised bed dryer describes qualitatively well the main features of
multicomponent drying established theoretically and experimentally in previous works,
particularly, the effects of the solid resistance against mass transfer on the retention of
volatile components. Factors intrinsically connected to an increase of solid resistance,
such as a more intricate solid structure and larger particle diameters, increase volatile
retention. Remarkably, external factors that make drying more intensive and thereby
more evident the existence of internal resistance, such as gas velocity and temperature,
seems to have an opposite effect on volatile retention. A deeper study using other
conditions is necessary to elucidate this behaviour. Simulations with a mixture
containing highly volatile components showed that the composition of the remaining
liquid in the product can be controlled by adding small amount of the other components
to the solid feed. For instance, the concentration of methanol in the product can be kept
under a certain limit by adding small amount of chloroform to the solid feed. This
unconventional solution in drying practice evidences the complex features of
multicomponent drying and the need for suitable tools to predict the entire trajectory of
a drying process. To make this model such a useful tool for aiding dryer design requires
the experimental verification of the model.
Acknowledgments
The authors gratefully acknowledge the financial support provided by the Swedish
International Development Cooperation Agency (Sida/SAREC) for this work.
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Appendix A: Analytical Solution of the Equations for the Particle
Equations (5) and (6) can be made dimensionless by introducing the following
dimensionless variables:
Lz
=τ ; δ
ζ r= ; ⎟
⎟⎠
⎞⎜⎜⎝
⎛
−
−=
0g
g
TTTT
θ (A1)
The system of partial differential Eqs. (5) and (6) become:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=∂∂
ζζ2
ζτ 2dxxDx 2
; ⎟⎟⎠
⎞⎜⎜⎝
⎛ζ∂
∂+
∂∂
=∂∂ θ
ζ2
ζθκ
τθ
2
2
(A2)
with
2δvL
dDD = ; 2δ
κvLDh= (A3)
The inlet and boundary conditions are:
At 0=τ and 0 ≤ ζ ≤ 1, { }ζ= 0xx ; { }ζθθ 0= (A4)
At 0=ζ and 0>τ , 0=∂∂ζx
; 0=∂∂ζθ
(A5)
At 1=ζ and 0>τ , bζy+xx
φ=∂∂
− ; baθζθ
+=∂∂
− (A6)
with
{ }1n
1d
LCδvL
−γ−= ΚΚφ D ; { } 1n
1d
Lb Cδv
L−∞
−−= yDy Κ (A7)
and
δkha= ;
{ })Th(T
ab
0g
gT
−−=
Gλ (A8)
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AAIQ Asociación Argentina de Ingenieros Químicos IACCHE - Interamerican Confederation of Chemical Engineering
The subscript n-1 indicates that the matrix product consists of the first n-1 column
and rows of the original matrix product. The same applies to the resulting column
vector in Eq. (A7).
Equations (A2) may be now transformed into ones describing linear flow in one
direction by introducing the following new dependent variables
)(ζ byxu += φ ; )bθa(ζ +=Θ (A9)
Equations (A2) become:
2
2
ζ~
τ ∂∂
=∂∂ uDu
; 2
2
ζκ
τ ∂Θ∂
=∂Θ∂
(A10)
with 1
d~ −= φφDD (A11)
The new inlet and boundary conditions are:
At 0=τ and 0 ≤ ζ ≤ 1, { }ζ0u=u ; { }ζ0Θ=Θ (A12)
At 0=ζ and 0>τ , 0=u ; 0=Θ (A13)
At 1=ζ and 0>τ , 0=−+∂∂ uIu )(ζ
φ ; 0=Θ−+∂Θ∂ )1a(ζ
(A14)
where I is a diagonal matrix of ones. The composition in Eq. (A10) can be de-coupled
through the similarity transformation
DPDP ˆ~1 =− (A15)
uPu 1ˆ −= (A16)
The matrix P is the modal matrix whose columns are the eigenvectors of D~ and D a
diagonal matrix of its eigenvalues. The transformation yields:
2
2
ζˆˆ
τˆ
∂∂
=∂∂ uDu (A17)
with initial and boundary conditions:
At 0=τ and 0 ≤ ζ ≤ 1, }{ˆˆ 0 ζuu= (A18)
At 0=ζ and 0>τ , 0ˆ =u (A19)
At 1=ζ and 0>τ , uξuf ˆ
ζˆ −=
∂∂
(A20)
XXII IACChE (CIIQ) 2006 / V CAIQ
AAIQ Asociación Argentina de Ingenieros Químicos IACCHE - Interamerican Confederation of Chemical Engineering
where
( )PIPξ f −= − φ1 (A21)
Since the solution demand ξf to be a diagonal matrix a new diagonal matrix ξ is
defined so that it satisfies:
uξuξ f ˆˆ = (A22)
giving the new boundary conditions:
uξuˆ
ζˆ −=
∂∂
(A23)
Equation (A17) is not explicitly dependent on temperature and can be solved
separately. Under the assumption that the matrix ξ is constant the de-coupled
differential equations can be solved by the method of variable separation. The solution
reported by Carslaw and Jaeger (1959) is:
{ }∫∑∞
=
τ−
⎟⎟⎠
⎞⎜⎜⎝
⎛
+++
=1
0 mm1m
2m
22mˆ ζd)ζsin()ζ(ˆ)ζsin(
)(e2ˆ
2m νuν
Iξξνξν
u νD0 (A24)
To preserve the formalism of matrix product, the integral in Eq. (A24) is a diagonal
matrix that contains the value of the integral. The eigenvalues in Eq. (A24) are defined
implicitly by
m1
mtan νξν −= (A25)
Finally, by using Eq. (A9) and the inverse of Eq. (A16) u is transformed back to
obtain the liquid composition:
⎟⎠
⎞⎜⎝
⎛ −= −byuPx
ζˆ1φ (A26)
At the centre of particle, when ζ = 0, the composition is undetermined and the
expression must evaluated as a limit. The limit of the expression is related to the
derivative of the transformed composition with respect to the dimensionless space
⎟⎠
⎞⎜⎝
⎛=⎟⎠
⎞⎜⎝
⎛→→ ζd
ˆdζˆ
ζζ
uu00
limlim (A27)
By evaluating the derivative of Eq. (A24) at ζ = 0:
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AAIQ Asociación Argentina de Ingenieros Químicos IACCHE - Interamerican Confederation of Chemical Engineering
{ }∫∑∞
=
τ−
→ ⎟⎟⎠
⎞⎜⎜⎝
⎛
+++
=⎟⎠
⎞⎜⎝
⎛ 1
0 mm1m
2m
22mˆ
ζζd)ζsin()ζ(ˆ
)(e2
ζdˆd 2
m νuνIξξν
ξνu νD00
lim (A28)
Equation (A26) provides the mole fractions of n-1 components in the liquid. The
mole fraction of the nth component is calculated taking advantage of:
∑−
=
−=1n
1jjn x1x (A29)
For the temperature:
{ }∫∑ νΘν⎟⎟⎠
⎞⎜⎜⎝
⎛
+ν
+ν=Θ 0
∞
=
τκν− 1
0 m,hm,h1m
2m,h
22m,h ζd)ζsin()ζ()ζsin(
1-a(a1-a(
e22
m,h
))
(A30)
with eigenvalues defined implicitly by
m,h1
m,h )a1tan ν−=ν −( (A31)
Substitution back to temperature:
⎟⎠
⎞⎜⎝
⎛ −Θ−
−= bζa
)TT(TT 0g
g (A32)
The values of the centre are calculated using a similar relation between the limits. In
this case:
⎟⎟⎠
⎞⎜⎜⎝
⎛ Θ=⎟⎟
⎠
⎞⎜⎜⎝
⎛ Θ→→ ζζ ζζ d
d00
limlim (A33)
Applied to Eq. (A30):
{ }∫∑ Θ⎟⎟⎠
⎞⎜⎜⎝
⎛
+
+=⎟⎟
⎠
⎞⎜⎜⎝
⎛ Θ ∞
=
−
→
1
0 ,0,1
2,
22,
0)sin()(
1)-a(1)-a(
2lim2
, ζζνζννν
ζτκν
ζd
ae
dd
mhmhm mh
mhmh (A34)
Even though the solution is only valid for constant physical properties the variation
of coefficients for the whole process can be taken into account by a stepwise application
of the analytical solution along the process trajectory. That is, by performing the
solution in successive steps where the final conditions of the previous step are used to
calculate the coefficients and as initial condition of the next step.