SIMULATION OF A VIBRATED FLUIDISED BED DRYER FOR SOLIDS … · 2015-07-29 · SIMULATION OF A...

XXII IACChE (CIIQ) 2006 / V CAIQ

AAIQ Asociación Argentina de Ingenieros Químicos IACCHE - Interamerican Confederation of Chemical Engineering

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



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;



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.



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:



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)



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)



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:



∑≠=

+=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.



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



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.



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



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.


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.


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.


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



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



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)



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)



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:



{ }∫∑∞

=

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.

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