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Chemical Engineering and Processing 47 (2008) 548–556

Gas–liquid mass transfer in bubble column reactor: Analytical

solution and experimental confirmation
H. Dhaouadi a,∗ , S. Poncin b , J.M. Hornut b,c , N. Midoux b
a Faculté des Sciences, Département de Chimie, Laboratoire de Chimie Appliquée et Environnement, Bvd de l’Environnement, 5000 Monastir, Tunisia
b Laboratoire des Sciences du Génie Chimique, CNRS, ENSIC Nancy, France
c IUT/UHP le Montet Nancy Brabois, France

Received 23 January 2006; received in revised form 8 November 2006; accepted 20 November 2006
Available online 24 November 2006

Abstract
Gas–liquid mass transfer behavior in a bubble column is investigated. The used reactor is of 6 m height and has an internal diameter of 0.15 m.
The classical ‘gassing out’ dynamic method is used to obtain the dissolved oxygen (DO) concentration profiles. An analytical description of the
dissolved oxygen concentration evolution with time following the switch in the inlet gas is presented and allows the evaluation of the gas–liquid
volumetric mass transfer coefficient for the different superficial gas velocities used. The comparison between the kL a values determined from this
exact solution and those obtained in the same reactor from a more complete and sophisticated model, numerically solved, shows a good agreement.
© 2006 Elsevier B.V. All rights reserved.

Keywords: Bubble column; Mass transfer; Analytical solution

1. Introduction progress in this field during the past 30 years has been reviewed
with a focus on the past 10 years. He concluded that the main
Bubble column reactors are frequently used in biological, developments are made on three fronts: (i) formulation of inter-
chemical and petrochemical industry. Compared to other multi- facial forces (ii) closure problem for the eddy viscosity and (iii)
phase reactors, such as packed towers and trickle bed reactors, modeling of the correlations arising out of Reynolds averaging
bubble columns offer some distinct advantages, among which its procedure. He also underlined that for mass transfer coeffi-
simple construction and excellent mass and heat transfer char- cient estimation, the prevailing procedures are largely empirical.
acteristics can be mentioned. However, bubble column reactors Arsam et al. [2] worked on a large scale slurry bubble reac-
have an inherent limitation of having less degree of freedom tor. They found that kL a values intimately follow the behavior
available to a designer to tailor their performance characteris- of bubble size distribution. They have tested several correla-
tics. In a bubble column reactor, local flow, turbulence and gas tion found in the literature and concluded that these correlations
hold-up distribution are interrelated in a complex way with the are not capable of predicting the experimental kL a values with
operating and design variables. A change in any one of these vari- acceptable accuracy since these particular correlations do not
ables will affect all the other reactor parameters. Therefore, an account for the effect of pressure. They therefore correlate kL a
accurate mathematical model to predict these complex interac- values in terms of the physical properties of the gas–liquid sys-
tions and their variation with the design and operating variables tem and the operating variables used. Krishna et al. [3] in their
is essential to compensate the reduced degrees of freedom. paper have compared simulations with measured experimental
Many papers and works are published in the bubble reactor data and have concluded that mass transfer from the large bubble
field. Joshi [1] published a paper dealing with computational population is significantly enhanced due to frequent coalescence
flow modeling and design of bubble column reactors. The and break-up into smaller bubbles. They also outlined that CFD
simulations marked the strong influence of column diameter on
hydrodynamics and mass transfer. For mass transfer, a simple
∗ Corresponding author. Tel.: +216 73500276; fax: +216 73500278. model is used and do not take into account neither time delay
E-mail address: hatem.dhaouadi@fsm.rnu.tn (H. Dhaouadi). of the used probe nor the pressure effect on gas solubility. The

0255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.cep.2006.11.009
 H. Dhaouadi et al. / Chemical Engineering and Processing 47 (2008) 548–556 549

used model is: penetration theory for kL calculation as:

   
CL kL a 2 √ ωL ρL
= 1 − exp − t (1) kL = √ D (4)
CL∗ 1 − εG π μL
The interfacial area is directly obtained from the predicted
In another paper, Vandu and Krishna [4] studied the influence bubble size distribution as a = k 6ε k
dk . In a review paper
of scale on the volumetric mass transfer coefficient in bubble recently published, Kantarci et al. [10], concluded that volumet-
columns. The time delay of the used probe is taken into account ric mass transfer coefficient kL a increases with gas velocity, gas
(ksensor = 1/τ p ) and the used model is: density and pressure whereas decreases with increasing solid
concentration and liquid viscosity. For kL a estimation several
CL 1 empirical correlations has been reported.
∗ = 1−
CL (1/τp ) − (kL a/1 − εG ) In this paper an analytical solution for the mass transfer in
     bubble column reactors is presented and experimentally vali-
1 kL a kL a t
× exp − t − exp − (2) dated. The analytical solution is based on some assumptions;
τp 1−εG 1 − εG τp the main one is non-dispersive plug flow of the liquid in the
column. The time constant of the oxygen probes is taken into
They conclude that for the estimation of the mass transfer account by assuming first order behavior. The pressure effect on
coefficient for bubble column reactor of large diameter and gas solubility along the reactor is taken into account. This yields
operating at high gas velocities (more than 8 cm s−1 ) and ele- a very satisfactory agreement between the analytical solution
vated pressure (kL a/εG ) is practically independent of column and experimental data.
diameter and superficial gas velocity, and its value is around In a second time, the analytical solution is compared to the
0.48 s−1 for their work and 0.45 s−1 for Jordan and Shumpe [5] one given by the numerical resolution of the model equation in
re-analyzed data. Those values are to be compared with the sen- the real time domain using commercial software (MODEST)® .
sor time constant (ksensor = 0.45 s−1 ). In a third paper, Vandu et
al. [6] extended the previous model to the slurry bubble case 2. Mathematical modeling of mass transfer
and they conclude that for superficial gas velocity more than
10 cm s−1 , (kL a/εG ) is practically independent of column diam- In the analytical solution presented here, the liquid flow in
eter and superficial gas velocity, and its value is in the range the bubble column reactor is considered to be plug flow. The
of 0.36–0.55 s−1 . More recently, Vandu et al. [7], studied the influence of the mixed zone at the bottom part of the reactor,
hydrodynamics and mass transfer in bubble column reactor. The where gas is sparged is neglected. The oxygen mole fraction
volumetric mass transfer coefficient kL a is determined by fitting variation of the gas flowing through the reactor is considered
the experimental data according to a model based on oxygen bal- to be negligible. The static pressure effect on gas concentration
ance on liquid and gas phase and taking into account time delay (and therefore on the oxygen solubility) and the Clark probes
of oxygen probes. Pressure effect on gas solubility is not taken time constant are however taken into consideration.
into account. The obtained model is numerically solved. Linek The oxygen balance on liquid phase can be written as follow
et al. [8, parts I and II] studied the mechanism of mass transfer ∂cL ∂(v̇L cL ) c

G
from bubbles to dispersion in bubble column reactor. According (1 − εG ) + = kL a − cL (5)
∂t ∂z m
to the authors, different, often contradictory influence of oper-
ating conditions on mass transfer coefficient kL is made, which where m = He/RT
can be partially explained by errors made in kL a values used in The oxygen balance on gas phase, supposed to be plug flow,
its evaluation. They conclude that the main parameter to corre- can be written as follow
late the kL is the power dissipated in the liquid phase rather than ∂cG ∂(v̇G cG ) c

G
the bubble diameter and the slip velocity. They also underlined εG + = −kL a − cL (6)
∂t ∂z m
that air absorption did not yield correct kL data because of the
use of an improper gas phase mixing model for absorption data Time delay of DO probe response is also taken into account
evaluation. The used model for kL determination is: through Eq. (7):

 0.25  0.50 dcm 1

eν D (1 − εG ) = (cL − cm ) (7)
kL = c1 (3) dt τp
ρ ν
The following Eq. (8) is valid just after the switch from
CFD has also been recently employed by Dhanasekaran et nitrogen to air.
al. [9] for mass transfer estimation in a generalized approach to  
v̇G
model oxygen transfer in bioreactors using population balances. cG (z, t) = cG0 (z)H t−z (8)
εG
According to the authors, and contrarily to Linek et al., a key
factor that influences oxygen transfer is bubble size distribution. The bottom column pressure is taken as the reference one.
For the volumetric mass transfer estimation they use the Higbie’s When the variation of the oxygen mole fraction in the gas phase
550 H. Dhaouadi et al. / Chemical Engineering and Processing 47 (2008) 548–556

(air) is neglected, the gas phase oxygen concentration becomes The model is, in a second time, written for one velocity direc-
proportional to the static pressure: tion only ignoring radial effects and using the generic transport
z equation. The basic equation in phase k is
p(z) = p(0) − [p(0) − p(Z)] (9)  
Z ∂(ρεΦ)k ∂ ∂ μeff ∂Φ
= − (ρεv̇Φ)k + ε + SΦ (16)
   ∂t ∂z ∂z σ ∂z k
p(z) p(Z) z
cG (z) = cG0 = cG0 1− 1− where SΦ is the source term.
p(0) p(0) Z

For the conservation of mass, variable Φ = 1 and σ = ∞, the
z
= cG0 1 − a (10) source term is
Z  
∂ ∂ε
SΦ = μeff ± Mk (17)
The boundary conditions are: cG (0, t) = cG H(t) and ∂z ∂z k
cL (0, t) = 0. where Mk describes the interfacial mass transfer. Thus
Solving the obtained equation system yield the measured DO
∂(ρε)k ∂
concentration as time function. The obtained solution is: = − (ρεv̇)k ± Mk (18)
∂t ∂z
H(␪ − NmrεG ζ)F1 − H(θ − (1 − εG )Nζ)F2
Cmes = (11) Next, in order to solve the continuity equation, the volume or
BT
height has to be divided into sections where no spatial variations
with: for εk exist. So dεk /dz = 0 and constant density ρk we find
 
BT 2 mrεG Nζ − θ ∂εk ∂v̇k Mk
F1 = BT + exp = −εk ± (19)
B−T T ∂t ∂z ρk
 
B2 T mrεG Nζ − θ
− exp (12) 3. Experimental investigation and results
B−T B
Experiments were carried out in a bubble column reactor with
  
BT 2 (1 − εG ) Nζ − θ a height of 6 m and internal diameter of 0.15 m (Fig. 1). The liq-
F2 = exp(−Nζ) BT + exp uid is water and the gas feed can be switched from air to nitrogen
B−T T
  or vice versa. Gas flow is controlled and regulated using a digi-
B2 T (1 − εG ) Nζ − θ tal control mass flowmeter TYLAN® RO28 FC262 100 SPLM.
− exp (13)
B−T B Pressure, for gas hold-up estimation, is measured using a dig-
ital TEFLINOX® ED 510/324 pressure probes mounted flush
and
with the column wall. Pressure probes are connected to a data
B = 1 − (1 + mr)εG (14) acquisition system (RTI 815) allowing the immediate reading
and recording of pressure value. The gas hold-up εG in the reac-
In the particular case where the reactor is closed to the liquid tor is measured using pressure values. The pressure difference
phase, the analytical solution becomes: P between two levels of the column, measured for aerated
 (P) and unaerated liquid (P0 ), yields the mean gas hold-up
H(θ − N  ζ) (1 − εG )T 2
Cmes (ξ, θ) = (1 − εG )T + between those levels.
(1 − εG )T (1 − εG ) − T  
   εG = 1 −
P
N ζ−θ (1 − εG )2 T (20)
× exp − P0
T (1 − εG ) − T
   The oxygen concentrations are measured at six different lev-
N ζ−θ els by “Clark” probes (PONSELLE MESURE® ) also connected
× exp (15)
(1 − εG ) to a data-acquisition system. The gas sparger is a perforated
tubes type (64 holes of 1 mm diameter each one). At the column
The volumetric mass transfer coefficient is determined by height used here, its not any longer possible to ignore the col-
adjusting the experimental probe response to the analytical solu- umn height effect on the gas solubility: static pressure causes
tion by taking kL a as an optimizing parameter using the Excel® a decrease by a factor of 1.6 of the solubility between the bot-
Solver. DO concentration are calculated according to the exact tom and the top of the column. This leads to an increase of the
solution (15), we then calculated the R2 values which corre- dissolved oxygen concentration in the lower part of the column,
spond to the squared difference between the experimental and and a decrease in the upper part. The unsteady-state gassing-in
analytical values of the DO at several time steps. The kL a value method is used for kL a determination (Fig. 2); it is technically
corresponds to the one minimizing the objective function value very easy and gives, if well analyzed, reliable results. Omitting
(the sum of the R2 values for all the time data). the considerations mentioned above (especially about the static
Details of the obtained Eq. (15) are given in appendix. The height effect) would have resulted in major errors in the deter-
time constant τ p of the oxygen probe is estimated by a classical mination of kL a. Typical DO curve used for kL a determination
concentration switch method. during the gassing-in period are given in Fig. 3.
 H. Dhaouadi et al. / Chemical Engineering and Processing 47 (2008) 548–556 551

Fig. 1. Experimental setup.

As it clearly appear in Fig. 4, the exact solution used here, with

only one single adjustable parameter describes the experimental
result in a very satisfactory manner.
The kL a values from the exact solution are now compared
with those obtained using more sophisticated model derived
from principal Navier–Stokes equations, following the princi-
ples given, e.g. by Hillmer et al. [11] for slurry bubble columns.
The same assumption as in the exact solution are made but
are numerically solved using a commercial software, MODEST
(MODel-ESTimation; Haarrio [12]).
The present model leads to a set of ordinary (ODE) and partial
second order differential equations (PDE). In solving the PDE’s,
the bubble column reactor was first discretised into 40 sections
and by employing the method of lines [13] the created ODEs
Fig. 2. ‘Gassing out’ dynamic method raw data.

Fig. 4. Experimental to analytical solution adjustment for two given axial posi-
Fig. 3. Typical DO profile during the “gassing on” period. tion.
552 H. Dhaouadi et al. / Chemical Engineering and Processing 47 (2008) 548–556

Fig. 5. Runs of the numerical resolution made with the MODEST software.
 H. Dhaouadi et al. / Chemical Engineering and Processing 47 (2008) 548–556 553

Table 1
Model parameters value used for the numerical runs with the MODEST software
Figure 5 Patm (mmHg) T (◦ C) kL a (s−1 ) ug (cm s−1 ) Z (m) Time run (s)

a 730 15 0.01 8 7 400

b 800 30 0.01 15 4 100
c 730 15 0.08 8 7 100
d 800 30 0.15 15 4 100
e 730 15 0.15 8 7 100
f 760 25 0.07 1–20 7 90
g 760 25 0.01–0.20 5 7 150
h 760–2660 25 0.08 8 4 80

were integrated in time. The differential equations were solved

with LSODE [14]. In parameter estimations, the objective func-
tion was minimized by the simplex method. The closeness of
the data and the values predicted by the model can be measured,
in principle, with several criteria. The most common objective
function according to which the parameters are estimated is the
sum of residual squares. The goodness of fit is the coefficient of
determination, the R2 -value given by the expression
 
||y − y p || 2
R2 = 100 1 − (21)
||y − ȳ||2

So R2 ≤ 100—the closer the value is to the number 100, the

more perfect is the fit. Fig. 6. kL a values obtained with the analytical and numerical solution and
The parameters to be estimated in this study were the kL a and comparison with Deckwer and Takavoli empirical correlations.
the oxygen equilibrium concentration. The gas hold-ups were
taken from separate measurements described in the previous clude, after a sensitivity analysis of the model parameters, that
section. the kL a is independent from the liquid velocity and propose the
Several runs of the numerical resolution with the MOD- following empirical correlation: kL a = 0.467u0.82
G . Takavoli et
EST software have been made and the results are reported in al. [16], uses a PFM (Plug Flow) model which clearly over-
Fig. 5a–g. The effect of the atmospheric pressure, temperature, estimates the kL a values. The authors propose the same kind
depth, superficial gas velocity and column height have been of empirical correlation than Deckwer with an additional term
numerically evaluated. Values of the different parameters are related to the system viscosity. The proposed correlation is:
kL a = 0.014u0.66 −0.5 . For k a determination, each experi-
reported in the table below Table 1. The column has been numer- G μ L
ically divided into 40 cells. At each cell level, we can get the DO mental data point in Fig. 6 is issued from a mean value of six
profile for a set of different parameters. We have choose to put ones corresponding to the fitting result of the six oxygen probe
DO observer each meter of the column height at levels not far records for a given superficial gas velocity. Perfect agreement
from the experimental ones. From Fig. 5a, depth effect on gas between analytical, numerical and Deckwer correlation is found,
solubility is clearly shown. Levels would be mixed at an average but all, compared to Takavoli one underestimate the kL a value
value if liquid velocity was not nil. Because of the absence of for all the superficial gas velocity studied domain. In order to
liquid feed, gas saturation level at different depth is separated. highlight the importance of the pressure effect on gas solubility
Fig. 5g shows kL a effect on DO oxygen profile. Greater is the for the kL a determination, several runs have been undertaken.
kL a value more rapidly liquid gas saturation is reached, not in The experimental data points have been fitted once again with
a linear way. Fig. 5f shows that superficial gas velocity has a the used solution but neglecting the pressure effect. The results
moderate effect on DO profile, less than pressure one (Fig. 5h). show that the kL a values are overestimated (more than 45% rel-
The following table summaries the parameter values used for ative error for the high gas velocity to more than 120% for the
the presented numerical resolution runs. low ones).
Once the analytical solution experimentally validated and
compared to the numerical solution, the obtained results have 4. Conclusion
been compared to those available in the literature. We have
chosen to compare our experimental result of both analytical The principal object of this paper is to present an analytical
resolution and numerical one to Deckwer et al. [15] correla- solution for the estimation of volumetric mass transfer coeffi-
tion and the one published by Takavoli et al. [16]. Deckwer cient in bubble column reactors. Pressure effect on gas solubility
et al. [15] uses the ADM (axial dispersion) model. They con- is taken into account and this is of an importance especially in
554 H. Dhaouadi et al. / Chemical Engineering and Processing 47 (2008) 548–556

the case of tall reactors. The main advantage of the analytical Subscripts
solution proposed here is the facility with which the volumetric c column
mass transfer coefficient can be correctly evaluated for a given D delay
experimental data points. The drawback is that the analytical eff effective
expressions obtained are quite cumbersome. G gas
The analytical solution has been compared to the numerical k phase k
solution made with commercial software and also shows a good L liquid
agreement. m measured
p probe
0 level of the gas sparger
Appendix A. Nomenclature
Appendix B. Analytical solution of the equations system

a pressure coefficient defined by Eq. (6) B.1. Dimensionless variables

b parameter
c concentration (mol m−3 ) Concentrations : CG =
cG
and CL =
mcL
(B.1)
c1 constant c G0 cG 0
C dimensionless concentration
d bubble diameter (m) Abscissa and number of transfer units :
D gas diffusivity in the liquid (m2 s−1 ) z kL a kL aεG
e total specific power dissipated in the liquid volume ζ= , N= Z, and N  = Z (B.2)
Z v̇L v̇G
(W m−3 )
F intermediate variable
H Heavyside function Time : θ = kL at and T = kL aτp (B.3)
He Henry coefficient (Pa m3 mol−1 ) v̇L
kL a volumetric mass transfer coefficient (s−1 ) Linear velocity ratio : r= (B.4)
mv̇G
L Laplace transform
m dimensionless Henry coefficient B.2. Dimensionless equations
M mass transfer term (kg m−3 s−1 )
n exponent Taking into account Eqs. (B.1)–(B.4) into Eqs. (1), (2), (4)
N, N number of transfer units and (6) gives:
p pressure (Pa)
Q volume flow rate (m3 s−1 ) Eq. (5) ⇒ (1 − εG )
∂CL
+
1 ∂CL
= (CG − CL ) (B.5)
s Laplace variable (s−1 ) ∂θ N ∂ζ
S source term ∂CG 1 ∂CG
t time (s) Eq. (6) ⇒ mεG + = −(CG − CL ) (B.6)
∂θ Nr ∂ζ
T dimensionless time
u superficial velocity (m s−1 ) Eqs. (8) + (10) ⇒ CG (ζ, θ) = H(θ − NmrεG ζ) (B.7)
v̇ interstitial velocity (m s−1 )
V velocity ratio B.3. Deviation variables
y data point value
ȳ average value of all data points
yp prediction given by the model L[CG ] = ĈG , L[CL ] = ĈL and L[Cm ] = Ĉm ,
z distance (m) ĈG (ζ, θ) = CG (ζ, 0) − CG (ζ, θ),
Z column height (m)
ĈL (ζ, θ) = CL (ζ, 0) − CL (ζ, θ)

Greek letters
ε gas hold-up ∂ĈL ∂ĈL (ζ, θ) ∂CL (ζ, 0)
ζ dimensionless abscissa Eq. (B.5) ⇒ + =
∂θ ∂ζ ∂ζ
θ dimensionless time
μ viscosity (Pa s) − CG (ζ, 0) + CL (ζ, 0) + [ĈG (ζ, θ) − ĈL (ζ, θ)]
ν kinematics viscosity of the liquid phase (m2 s−1 )
ρ density (kg m−3 ) thus:
σ Schmidt number (=μeff /ρDeff ) ∂ĈL ∂ĈL (ζ, θ)
τp time constant of the oxygen probe (s) + = [ĈG (ζ, θ) − ĈL (ζ, θ)] (B.5 )
∂θ ∂ζ
Φ variable of Eq. (12)
ω turbulent dissipation rate (m2 /s3 ) Eq. (B.6) ⇒ ĈG (ζ, θ) = CG (ζ, 0)H(θ − mrζ) (B.6 )
 H. Dhaouadi et al. / Chemical Engineering and Processing 47 (2008) 548–556 555

θ=0 ĈG (ζ, 0) = 0; ĈL (ζ, 0) = 0 (B.7 ) So we finally get

  
BT 2 ␪ − mrεG Nζ
B.4. Laplace transform BTCmes = H(␪−NmrεG ζ) BT + exp −
B−T T
 
B2 T ␪ − mrεG Nζ
1 dC̄G − exp −
Eqs. (B.5 ) + (B.7 ) ⇒ + (1 + msεG )C̄G = C̄L B−T B
Nr dζ
 
(B.8) BT 2
−H(␪ − (1 − εG )Nζ) exp(−Nζ) BT +
B−T
1 dC̄L  
Eqs. (B.6 ) + (B.7 ) ⇒ + [(1 − εG )s + 1]C̄L = C̄G ␪ − (1 − εG )Nζ 2
B T
N dζ × exp − −
(B.9) T B−T
 
  ␪ − (1 − εG )Nζ
1 1 × exp − (B.16)
Eq. (7) ⇒ s+ Cmes = CL (B.10) B
T T
1
Eq. (B.7) ⇒ CG = exp(−NmrεG ζs) (B.11) B.6. Expression of the measured concentration (Cm )
s
probe transfer function: H(␪ − NmrεG ζ)F1 − H(␪ − (1 − εG )Nζ)F2
Cmes =
BT
C̄m 1/θp
= (B.12) with
C̄L s + 1/θp  
BT 2 θ − mrεG Nζ
F1 = BT + exp −
B.5. Solution in the Laplace domain B−T T
 
B2 T θ − mrεG Nζ
The Laplace transform of the concentration is obtained by − exp −
solving Eqs. (B.8)–(B.12). The general solution of the differen- B−T B
tial Eq. (B.8) is given by the relation (B.13): and
1   
C̄L = {− exp{[(1 − εG )s + 1]Nζ} BT 2 θ − (1 − εG ) Nζ
s[(1 − εG mr − εG )s + 1] F2 = exp(−Nζ) BT + exp −
B−T T
+ exp(−mrεG Nζs)} (B.13)  
B2 T θ − (1 − εG ) Nζ
− exp −
Taking into account Eq. (B.12) we get: B−T B
1/T
C̄mes = {exp(−mrεG Nζs) In the particular case, where the reactor is closed to the liquid
s(s + (1/T ))[(1 − εG mr − εG )s + 1]
phase (no new water is fed continuously), v̇L → 0 and N → ∞.
− exp{−[(1 − εG )s + 1]Nζ}} (B.14) So H(θ − (1 − εG )Nξ) = 0, ∀, θ. We also have B = 1 − εG . We get:
Cmes = H(θ−Nmrε G ζ)F1
. Replacing with F1 and N we finally get:
Supposing B = 1 − (1 + mr)εG we get: BT

1 H(θ − N  ζ) (1 − εG )T 2
BT Cmes = {exp(−mrεG Nζs) Cmes (ξ, θ) = (1 − εG )T +
s(s + (1/T ))(s + (1/B)) (1 − εG )T (1 − εG ) − T
  
− exp(−Nζ) exp(−(1 − εG )Nζs)} (B.15) N ζ−θ (1 − εG )2 T
× exp −
T (1 − εG ) − T
The inverse Laplace function gives:   
  N ζ−θ
exp(−mrεG Nζs) × exp .
BTCmes (θ, ζ) = L−1 1 − εG
s(s + 1/T )(s + 1/B)
 
−1 exp (− (1 − εG ) Nζs)
− exp (−Nζ) L
s(s + 1/T )(s + 1/B) References
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