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Correlation Between Fluctuation

and Dissipation in the Case of

the Adsorption of Acetone and

Ethanol by the Activated CarbonF. Mhiria and A. JemnibaInstitut Preparatoire aux Etudes dIngenieurs de Monastir, Rue Inb El Jazzar, 5019 Monastir,

Tunisie; [email protected] (for correspondence)b Ecole Nationale dIngenieurs de Monastir, Rue Inb El Jazzar, 5019 Monastir, Tunisie

Published online 23 July 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/ep.10475

The previous studies have shown that the gasadsorption process in a microporous solid is due tothree phenomena: the direct adsorption (transporta-

tion phenomenon), the diffusion, and the fluctua-tions of the energy and the adsorbed matter quantity.These studies assume that the links between the differ-ent fluctuations are constant. In this article, we pro-

pose a new model of theses links based on the proba-bilistic-statistical properties of correlations and on themicropore filling theory. Currently, there is no resultleading to the establishment of the general form of the correlation which is based essentially on the exist-ing general theory of the actual statistical probability,the experiment and the scientific logic. A good agree-ment between the theoretical results and the measure-ments is obtained, which confirms the efficiency of the proposed model. 2010 American Institute of Chemi-

cal Engineers Environ Prog, 30: 294302, 2011Keywords: kinetics, adsorption, microporous solid,fluctuation, Fiks

INTRODUCTION

The development of the industrial applications ofthe adsorption phenomenon, requires a good knowl-edge of the transfers of heat and mass in the adsorb-ent mediums [1]. Consequently, the determination ofthe parameters which control these transfers and

especially the adsorption kinetics have a great interest[2] in several fields: Chemistry, Biology, Pharmaco-chemistry, applications in the environment and the

separation of the molecules (molecular sieves).Generally, the adsorption speed is controlled bythe adsorbed mass and the resistance to the heattransfer instead of the adsorption intrinsic kinetics [3].

A direct measurement on an adsorbent sample ena-bles to plot some characteristics which provide aneasy method for the adsorption kinetic study. Never-theless, the interpretation of many data shows someunavoidable difficulties. In fact, there are many differ-ent resistances to the transfer of mass and heat,

which limits the adsorption speed. The influence ofthe thermal resistance on the kinetic has been studiedby Valentina et al. [4]. Since 1950 many theories ofthe adsorption kinetics have been developed. Some

lead to equations barely-applicable as they integratemany factors which depend on the physico-chemicalproperties of the adsorbents and adsorbats [3]. Othersare applied only by concentration intervals [5]. In lit-erature, there are some empirical models the parame-ters of which are not explicitly defined from a physi-cal point of view [5, 6]. Moreover, in these proposedmodels, the number of constants experimentallydetermined is generally greater than three [711].

The irreversibility of the adsorption and the de-sorption phenomena have been proved by manyauthors [12]. It has been shown that the irreversibilitydepends on the temperatures and to reach the ther- 2010 American Institute of Chemical Engineers

294 October 2011 Environmental Progress & Sustainable Energy (Vol.30, No.3) DOI 10.1002/ep

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mal equilibrium 1.5 h is required for each measure-ment [1317]. The needed time to reach this equilib-rium ensures the existence of fluctuations during theadsorption. Besides, in the same study the variationsof the differential heat with respect to the adsorbedmatter quantity illustrate the fluctuations of this heat.The fluctuation phenomena have been observed byother authors [18] who introduce the correction func-

tion in the equation of Dubinin-Astakhov and on theadsorption and desorption energies. These fluctua-tions or the adsorption oscillations are explained andconfirmed by the theory of the thermodynamics ofsystems out of equilibrium [1921] by incorporatingthe Onsager theorem [22] or the fluctuation dissipa-tion theorem [23, 24]. These theorems have beenused by Mhiri et al. [25] and Mhiri and Jemni [26].

Some researchers have tried to linearize the plotsof the adsorption kinetics [27]. However, this assump-tion, deduced from the chemical kinetics [28], yields agood approximation only in reduced intervals of theplots. From a microscopic point of view, the experi-ment shows that the adsorption of a gas in a micro-

porous solid is due to two phenomena: the transportand the diffusion [15, 16, 29] which introducesanother complexity in the study of the adsorptionkinetics. In fact, from the study of Browniens move-ment (Langevin equation) we show that both phe-nomena dont have the same speed.

The adsorption kinetic approach from Ficks law[30] needs the modification of this law and the con-sideration of the fluctuation. The authors using Fickslaw without considering the fluctuation and the dif-ference between the transport and the diffusion haveto define up to five diffusivities [11]. A recent studybased only on mathematical results [31] proposes amodification of the Ficks law. This study doesnt pro-

pose any experimental approach and doesnt con-sider the heat and mass fluctuations either.

THEORETICAL STUDY

General Theoretical StudyThe adsorption of a gas in a microporous solid is a

matter transfer with a movement quantity from oneenvironment to another. This property is the featureof a transportation phenomenon [32]. In the idealcase, the main physical aspects of these phenomenaare described by the Ficks law. Generally, these phe-nomena obey some equations more complicated thanFicks law which can be considered only a first

approximation. For example, the diffusion with achemical reaction (production and dissipation of mol-ecules) or the balance in a nuclear thermal reactor,

yield equations more intricate than those of Ficks[32].

The transportation phenomenon which occursbetween the ambient gas around the microporoussolid and the solid itself is followed by a diffusioninside the solid in the direction of the decreasingconcentrations and tends to uniform the moleculardistribution of the diffusing substance in all theallowed space. Thus, there is a tendency that makesthe diffusion take place. This tendency has to be con-

sidered in a statistical or microscopic way as it may

have some fluctuations which are created for shortperiods of time due to the molecular flow reverse insome places [32]. Moreover, the adsorption of a gasin a microporous solid is an exothermic phenomenonas the overheating of a solid region may cause thetransfer of a matter quantity to the less hot region.Both remarks are responsible for the fluctuation ofthe adsorbed quantity in the elementary volume ofthe microporous solid.

In conclusion, in a microscopic way, the adsorp-tion process in elementary volume results from thesuccession of three phenomena which occur duringthe time: the direct adsorption of the exterior to theadsorbant (transportation phenomenon), the matter

diffusion inside the adsorbent, and the exchange ofthe adsorbed matter due to fluctuation (see Figure 1).The differential equation which describes the

adsorbed quantity during the time results from a mat-ter balance in an adsorbent elementary volume [25,26].

The adorbat accumulation is equal to the gain perdiffusion inside the adsorbent plus the quantitydirectly adsorbed from out side plus the matterexchange due to fluctuations:

@nr; t

@t DDnr; t Ar; t Br; t (1)

In a spherical symmetry case and from [25], thisequation is written as:

@nr; t

@t DDnr; t b nr; t ZtRr (2)

where A(r,t) 5 b n(r,t) with b a constant in the isobarcase as it depends only on the pressure and on theadsorbents active surface [25]. Equation 2 has beensolved using the variable separation method, themodel chosen for B(r,t) is Z(t) R (r) where Z(t) isthe function which describes the deep link between

Figure 1. Physical model 1: direct adsorption (trans-portation phenomenon); 2: diffusion; 3: fluctuation.

Environmental Progress & Sustainable Energy (Vol.30, No.3) DOI 10.1002/ep October 2011 295

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Establishment of the general form of the soughtfunctional relation;

Computing of parameter estimates of the equationwhich fit as most as possible the experience data.Its to be noted that the first step is less familiar

than the second as no resolution methods are avail-able in the literature [34].

The most used and studied dependences are the

linear regression and parabolic regression. In ourcase, these regressions dont provide the wished pre-cision degree. So, we have to try other dependenceforms such as hyperbolic, exponential, or of power.In the case of the study of the adsorption kineticsand for the dependences between the fluctuation andthe dissipation, these dependences can be formulatedas linear ones via the variable change. The depend-ence which describes our experiments is the expo-nential one.

Study of the Function Z(t)This function expresses the deep link between the

fluctuation and the energy and matter dissipation, itisnt measurable because of its random behavior.Moreover, there is no mathematical study which ena-bles to determine its expression. This function ismathematically-inaccessible. Only some propertiescan be described from the experiment. In the iso-therm case, adsorption Z(t) satisfies the followingproperties: It depends on the adsorbed molecule number

n(rc,t), on the volume available at time instant tand on the temperature.

It is finite for any quantity of the adsorbed matter. Its first derivative is finite for any adsorbed matter

quantity.

When the system goes to saturation, this functiontends to be a constant. Theres no volume avail-able for the matter fluctuation and the heat diffu-sion permanent regime is established.

Initially, when the microporous solid is empty ofadsorbat, this function must be null. If theres noadsorbat in the adsorbent, theres no fluctuation.

The function Z 5 Z(n) and n 5 n(rc,t) are func-tions of time, we can express Z as a function of time

Z 5 Z(t). Moreover, our study handles the kineticsand it is irrelevant to express all functions withrespect to the same variables. Mathematically, in theisotherm case these properties are: The limit of Z(t) exists for any value of t.

The limit of the first derivative exists for any valueof t. It goes to a constant value once t goes to infinity

and its first derivative goes to zero. Initially (t 5 0) this function has to be null and its

first derivative should be finite.From these mathematical properties, the form Z 5

Z(t) in the isotherm case is given by Figure 2.The most effective interaction on the adsorption

speed are: the interaction between the energy dissi-pation and the matter fluctuation and the interactionbetween the matter fluctuation and the energy fluctu-ation. In such a case, the function Z(t) can be written

as: Z(t) 5 h(t) 1 g(t) where h(t) is the function thatcharacterizes the link between the energy dissipationand the matter fluctuation and g(t) characterizes thelink between the matter fluctuation and energy fluc-tuation.

Study of h(t)At the start of the adsorption (t small), the avail-

able volume for the adsorbed molecules is big, which

enables an easy movement of molecules: the matterfluctuation is important. However, at the end ofadsorption (t big), the adsorbent solid is close to sat-uration and the volume available for the free movingof molecules is so small that the matter fluctuation isnear zero, which enables us to write h(t) as:ht E1 e

ts where 1 e

ts characterizes the

mean values of matter fluctuation with respect totime and E is a constant (in isotherm case) whichcharacterizes the mean dissipation of energy.

Study of g(t)

In the beginning of the adsorption, the energy

fluctuation has to be important because of the highnumber of hits between the molecules of the adsor-bat and the adsorbed molecules and the microporoussolid edges since the available volume is relativelybig and the molecules have a high degree of move-ment freedom. Once the solid moicropores are filled,the available volume for the movement of moleculesdecreases as well as their free movement degree,

which leads to a reduction of the energy fluctuation.Moreover, the heat conduction by the microporoussolid tends to the steady state, which enables to char-acterize the energy fluctuation by the function e

ts. In

the isotherm case, the mean value of the fluctuation

Figure 2. Theoretical evolution of Z correlation withtimes in isotherm case.


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of the matter due to the energy fluctuation isF(12e2x

2

t) (see Eq. 4) where (12e2x2

t) characterizesthe adsorbed quantity during the time and F the per-centage of molecules whose positions are fluctuatedbecause of the energy fluctuation. The function Z(t)is written as:

Zt Fets1 ex

2t E1 ets (9)

By hypothesis, the interactions between the differ-ent fluctuations are linear and from Eq. 8, the func-

tion Y(t) is written as the product of two functions as:

Yt C1 ets1 ex

2t (10)

The term (12e2x2

t) describing the links betweenthe matter and energy dissipations and fluctuationand from previous reasoning has an effect only onsmall concentrations. This result can be proved theo-retically (Annex I). We prove that this term is differ-ent from unity if

nt;Tn1T

h12. The product of the two ex-ponential terms in the expression of Y(t) is familiar inadsorption kinetic theories [5, 35].

EXPERIMENTAL STUDY

The measurements are collected at high National

School of industrial technique and mines of Nantes(France) within the Department of energetic andenvironment Engineering.

Experimental Condition

The measurements of the adsorbed mass are real-ized in a fluidized bed in isotherm conditions (seeFigure 3). The used adsorbent is the activated carbonPIC AN C60, the physicochemical characteristics arepresented in Table 1. The grain radius ranges from0.3 to 0.6 mm; the bed mass is 0.3 kg; the bed heightis 0.12 m and the reactor diameter is 100 mm. All the

Figure 3. Representative diagram of the installation utilized for the experimental tracing of isotherms. 1: filter; 2:Mass volumetric regulator; 3: Wash-bottle; 4: fluidized bed reactor; 5: Conical float rotameter; 6: Exchanger withcool water; 7: Detector with ionization of flame; 8: Thermocouple K.

Table 1. Characterization of activated carbon NC 60.

Origin Coconut Oxidation temperature 2038C

Activation CO21H2O (a 9008C) Volumic mass 1.4 g cm23

Specific surface 1240 m2 g21 Specific heat 0.80.9 J g21 K21

Porous volume 0.55 cm g21

Table 2. Values of speed fluidization according to thetemperature.

T5 308C T5 508C T5 758C T5 1008C

0.23 0.210 0.195 0.18


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tests are carried out with a solvent concentration atthe input of 30 gm23. All tests performed at a U0speed 5 2 Umf. The speed minimal of fluidizationmeasured according to the temperature Umf is givenin Table 2.

The activated carbon particles are washed withdistilled water then placed in a drying oven at 1058Cduring 24 h before fluidizing them to wash them andremove the fine impurities.

The reactor has been dimensioned by taking intoaccount the physical properties and the minimalspeed of the particle fluidization of the activated car-bon. The tubular oven with 0.81-W power allows toheat the gas at the input for an isotherm functioning.

An air temperature control is achieved in a wind boxof the column by a PID generator (Eurotherm 2132)acting on the uphill power. The cold water exchangerenables to evacuate the yielded energy during theadsorption. Its composed of 18 copper spirals and aPID regulator which allows to regulate the water flowdepending on the temperature in the activated car-bon bed. The steam saturated air flow is generated

by the pinching of a dry air flux in a liquid solvent.The pinching system is thermostat to avoid the con-centration fluctuation of the saturation of the ambienttemperature. This flux saturated in the solvent isdiluted in a wind box by a pure air current.

By considering the variation of the partial molarflow of the adsorbate between the column input andoutput equivalent to the adsorbed adsorbate quantityper unit of time, we can write the following relation:

Input 2 output 5 accumulation:

G0X0X Mbeddn

dt

With X x

1 xand x

C0

M

RT

P

We deduce n

n X0G0

Mbed

Zttf

t0

1 X

X0dt

Table 3. Experimental values of constants from Equations (5) and (10).

T(8C)

C s(S)x2 (s21) Mhiri

F. Jemni A (2010)a (m21) Mhiri F.Jemni A (2010)

Ethanol Acetone Ethanol Acetone Ethanol Acetone Ethanol Acetone

30 10.418 9.721 80.645 93.750 9.059 3 1024 1.195 3 1023 0.192 0.19150 12.000 9.550 52.631 78.947 1.499 3 1023 1.556 3 1023 0.247 0.218

75 10.279 9.454 39.443 41.666 2.7003

10

23

2.1173

10

23

0.330 0.255100 10.540 9.000 39.473 37.500 4.666 3 1023 2.819 3 1023 0.435 0.295

Figure 4. Evolution of relative quantity of ethanoladsorbed in activated-carbon () theoretical curves(Eq. 7), () experimental curves, 1 experimentalpoints; 1 308C, ^: 50, O: 75 and D at 100.

Figure 5. Evolution of relative quantity of acetoneadsorbed in activated-carbon () theoretical curves(Eq. 7), () experimental curves, 1 experimentalpoints; 1 308C, ^: 50, O: 75 and D at 100.


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By integrating the previous equation we obtain thevalue of n.

APPLICATION OF THE THEORETICAL MODEL

To experimentally illustrate the proposed theoreticalmodel we have exploited the real part of Eq. 5 whereY(t) is replaced by its expression given in Eq. 10.

On the basis of the conjugate gradient identifica-

tion technique, we have estimated the constants Cand s of expression Eq. 10 for each isotherm. Thecomputing results of constants C and s are summar-ized in Table 3 and the evolution of the experimentaland theoretical adsorbed quantities are plotted in Fig-ures 4 and 5.

CONCLUSION

In this article, we have proposed a theoreticalapproach of the correlation between matter andenergy fluctuations and dissipations. This approachas well as its theoretical limit (annex I) are confirmedexperimentally.

The proposed approach is a continuation of the

previous works where an adsorption kinetic theoryhas been developed. The main advantage of thistheory is that it takes into account the nonhomogene-ity of the systems and the fluctuation observed exper-imentally.

This theory takes into account all the internal sys-tem values because of the use of the statistical theoryand the random function while solving the estab-lished differential equation.

From the previous studies and the present one, wecan affirm that the adsorption of a gas in a micropo-rous solid is the result of three phenomena: a trans-portation phenomenon followed by a diffusion phe-nomenon and both phenomena are accompanied by

a fluctuation.In spite of the assumption that the adsorption is iso-

bar, the concordance between the theory and theexperiment is very satisfactory. This is due to the factthat the studied adsorptions are free and not forced; thepressure variations are very small and can be neglected.The relative error between practice and theory is lessthan 10% at any instant over 10 s. This result is expectedas the initial speed is very high and the measurementsduring the time interval [0, 10 s] are less assured.

Annex I

In statistical thermodynamics, the entropy is definedas a value which measures the degree of a system dis-order in a microscopic level. If X is the number ofmicrostates, the cell number of the correspondingspace at a thermodynamical state; the Boltzman postu-late, to which Plank has given its final form, establishesthe link between the entropy and X as S5f(X) where fis a universal function. The unique solution of thisequation is f(X) 5KlnX, so that:

S KlnX

In the case of the adsorption of a gas in a micropo-rous solid, X is the number of possibilities to arrange

the adsorbed molecules over the total number of thereceiver sites n1T:

X n1T!

nt; T!n1T nt; T!

With Stirling formula, we have

S Kn1T lnn1T nt; T lnnt; T

n1T nt; T lnn1T nt; T

The maximum of the entropy is :

@ SK

@n 0 ) lnnt; T lnn1T

nt; T 0 ) nt; T n1T

2

The function S 5 f(n) has a maximum fornt; T n1T2 , the entropy S decreases as we have anegative temperature.

From this study the term 1 ets which links the

fluctuations to dissipations has an effect oncent;Tn1T

h12.If this ratio equals 12, the term 1 e

ts is set to unity.

NOMENCLATURE

A accumulation due to direct adsorption(mol kg21 s21)

1b relaxation time relative to pressure in

isobar condition (s)

C integrate constantC0 mass concentration of the solvent at thecolumn input (g m23)

D diffusion constant of micro pores(m2 s21)

G0 model flow of inert gas (mol s21)

I Complex number i2 5 21K proportionality constantm mass of adsorbed particles (kg mol21)Mbed mass of activated carbon bed (kg)M molar mass of solvent (g mol21)N number of adsorbed molecules by vol-

ume unit (mol m23)n(r

c,t) Number of adsorbed molecules in (mol

kg21) by spherical volume of rc radius attime t

n(r,t) average density of particles in point attime t (mol kg21)

P pressure (Pa), assumed to be constantR perfect gas constantr instantaneous particle position in spheri-

cal coordinates (m)rc radius of spherical grain of adsorbent

(m)t time (s)T temperature (8K)1x

relaxation time relative to temperature(s)

X0 the partial molar fraction of initial sol-vent in the air


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