ORIGINAL
Mass transfer during catalytic reaction in electroosmotically driven flow in a channel microreactor
Himanshu Sharma •Nadapana Vasu •
Sirshendu De
Received:6November 2009/Accepted:28November 2010/Published online:14December 2010ÓSpringer-Verlag 2010
Abstract Analytical solution for concentration profile in a microreactor is obtained during heterogeneous catalytic reaction.Reaction occurs in rectangular microchannel with catalyst-coated walls.Flow is induced electroosmotically in the microchannel.A general solution is obtained for first order reaction using a power series solution.Profiles of conversion,cup-mixing concentration of reactant,etc.and variation of Sherwood number is analyzed as function of operating variables.Analytical solution is compared with numerical results.
List of symbols A Parameter defined by Eq.5A m Constant in Eq.18a m mth coefficient of polynomial in Eq.14c Concentration,kg/m 3c 0Inlet concentration of solute,kg/m 3c*Dimensionless concentration c 1
Constant in Eq.10c cm *Dimensionless cup-mixing concentration D Diffusivity of the reactant,m 2/s Da Damkohler number for first order reaction (k 1h/D)
Da 0Damkohler number for nth order reaction (k 1h c n À1
)E x
Electric field in axial direction,V/m f Dimensionless quantity in Eq.12h Channel half height,m I 1,2,3,4Integrals in Eqs.19and 20k 1First order reaction rate constant,m/s k x Axial mass transfer coefficient,m/s L Channel length,m
Sh x Sherwood number
Sh L Length averaged Sherwood number u Velocity component in x direction,m/s u*Dimensionless axial velocity
u HS Helmholtz–Smoulochosky velocity,m/s x Axial flow direction,m
x*Dimensionless axial coordinate X Conversion
X m x-varying part of c*
y Coordinate normal to axial flow direction,m y*Dimensionless normal coordinate
Y m y-varying part of c*(mth eigenfuntion)
Greek symbols e Dielectric constant of fluid,C V -1m -1j *Non-dimensional double layer factor (j h )j -1Debye length,m k 1First eigenvalue k m mth eigenvalue l Viscosity,Pa s f Applied electric potential,V
1Introduction
Microreactors are attractive new tools for chemical engi-neers due to their diverse benefits over the conventional reactors [1–7].The advantages of microreactors include,(1)high surface area to volume ratio resulting to high heat and mass transfer rates;(2)better reaction and flow control;(3)better control of runaway reactions and segregation of affected unit,etc.Liquid phase reactions studied in mic-roreactors include synthesis of diazo dyes,nitration reac-tions,preparation of enamines,photochemical reactions,etc [2,8–12].Heterogeneous catalytic reactions have also
H.Sharma ÁN.Vasu ÁS.De (&)
Department of Chemical Engineering,Indian Institute of Technology,Kharagpur,Kharagpur 721302,India e-mail:sde@in
Heat Mass Transfer (2011)47:541–550DOI 10.1007/s00231-010-0743-y
been conducted in microreactors.Catalyst can be deposited on the inside wall by thinfilm deposition by physical means or chemical vapor deposition,wet preparation techniques,aerosol spraying,ink-jet printing,etc[4,13].In this category,dehydration of hexanol and hexane in pres-ence of Zirconia catalyst[14],synthesis of4-Cyanobi-phenyl using immobilized catalyst[7],three phase hydrogenation reaction[15],etc.,have been studied in microfluidic devices.Electroosmotically drivenflow in a microchannel has unique advantages,like,(1)having an almost slugflow apart from narrow wall region,causing minimal hydrodynamic dispersion;(2)precise electro-kinetic control for better regulation of reaction[16].
Modeling of microreactors mostly involves the detailed flow characterization.Classical models forflow with and without slip velocity at the wall are available[17–20]. Analytical solution for velocityfield in a membrane mic-roreactor is reported[21].Solute electrokinetic transport in nanochannels is analyzed theoretically and species sepa-ration by ion balance(EKSIV method)is demonstrated by Pennathur and Santiago[22].It may be pointed out that this work deals only separation of ions by electrokinetic phe-nomena in presence of a charged wall.It may be empha-sized that this work does not include a reactive system. CFD modeling of mass transfer with non-catalytic chemi-cal reaction in a microreactor
under pressure driven slug flow is available[23].Mass transfer andfluidflow for electrokinetically driven non-catalytic reaction has been studied by Fletcher et al.[16,24].Very few works are available for detailed mass transfer modeling in microre-actors.Mass transport and surface reactions in pressure driven microfluidic systems has been extensively covered by Gervais and Jensen[25].However,in this work[25],flow is induced in the microreactor by applying a pressure gradient and electroviscous effects are neglected.An explicit analytical solution for velocity,temperature and concentration distributions for electroosmoticflow of power lawfluids in microchannels is reported by Das and Chakraborty[26].However,in this analysis,constant axial concentration gradient is assumed.This oversimplifies the problem assuming concentration is a linear function of axial position.Zeng et al.[27],in their analysis of mass transport in a microchannel bioreactor compared the deviation between constant axial concentration gradient (analytical solution)and varying axial concentration gra-dient(numerical solution).It is observed that the error in assuming constant axial concentration gradient is around 25%.This deviation increases significantly in the down-stream of the channel.Thus,a varying axial concentration gradient,which is a more realistic phenomena taking place is to be considered for analysis.It is also to be noted that the analysis of Zeng et al.[27],considered only a mean velocity neglecting its variation in the transverse direction.However,detailed and rigorous mass transfer analysis for electroosmotically driven catalytic microreactors is not available to the knowledge of the authors in literature.This gap is attempted to bridge in the present work.
In this work,a detailed mass transfer modeling of cat-alytic reaction system in a microreactor with catalyst-coated walls is undertaken for electroosmoticflow.It may be emphasized that(1)the present work is not limited by the assumption that axial concentration gradient in the microreactor is constant as reported by Das and Chakraborty[26],(2)appropriate velocity profile for a purely electroosmoticflow is considered in place of aver-age axial velocity as done by Zeng et al.[27]and Lee et al.
reaction mass[28].An analytical solution is obtained by using a power series method.The profiles of cup-mixing concentration, conversion along the microreactor are obtained as a func-tion of operating variables in terms of non-dimensional numbers.The Sherwood number relations for such channel are also developed.Analytical solution is compared with numerical results.The analysis is helpful for efficient design of microreactors.
2Theory
The schematic of the microchannel reactor is shown in Fig.1.The inner walls(top and bottom)of the channel are coated with the catalyst.
The governing equation of concentration inside the microchannel is:
u
o c
o x
¼D
o2c
o y
ð1Þ
For a purely electroosmoticflow inside a microchannel, the fully developed velocity profile,under Debye–Huckel approximation[29]is expressed
as,
Fig.1Schematic diagram of an electroosmotically driven microreactor
u ¼u HS
1À
cosh j y ðÞcosh j h ðÞ
ð2Þwhere,u HS ¼Àef E x
l
is Helmholtz–Smoulochowski velocity.The boundary conditions of Eq.1are presented below.At the channel center,the symmetric condition holds.
at y ¼0;
o c o y
¼0ð3a Þ
At the channel wall,the diffusive flux of species is same as that consumed by the first order reaction in presence of catalyst.at y ¼h ;
D o c
o y
þk 1c ¼0ð3b Þ
In case of an nth order reaction taking place at the channel wall,the boundary condition at channel wall is given as at y ¼h ;
D o c
o y
þk 1c n ¼0ð3c Þ
At the channel entrance,solute concentration is same as that of the feed at x ¼0;c ¼c 0;ð3d Þ
Defining the non-dimensional variables as:x üx =L ;
y üy =h ;
c üc =c o ;
u üu =u HS
the dimensionless form of Eq.1becomes Au Ão c Ã
o x ¼o 2c Ã
o y
Ãð4Þ
where,A ¼
u HS h 2ð5Þ
and
u Ã
¼1À
cosh j Ãy ÃðÞ
ð6Þ
where,j üj h .The dimensionless forms of the boundary conditions in case of a first order reaction taking place at wall of microreactor are,at y Ã
¼0;o c Ã
o y Ã
¼0ð7a Þat y ü1;
o c Ã
o y
ÃþDac ü0ð7b Þ
where,Da ¼k 1h =D .This parameter,Damkohler number,
physically indicates the comparison between reactive and diffusive terms.
For,nth order reaction,the non-dimensional form the above boundary condition becomes,
at y Ã
¼1;
o c Ã
o y
þDa 0c Ãn ¼0ð7c Þ
where,Da 0¼k 1hc n À10
D at x ü0;
c Ã
¼1ð7d Þ
It may be noted here that the solution for the first order reaction only is presented below using series solution.To solve the linear governing equation,Eq.4,separa-tion of variable method is applied
c üX 1m ¼0
X m ðx ÃÞY m ðy ÃÞð8Þ
Substitute c Ãm ¼X m ðx ÃÞY m ðy Ã
Þin Eq.4the governing equation of X m becomes
A X m dX m
dx
üÀk 2m ð9Þ
where,k m is the mth eigenvalue of the system.The solution of above equation becomes,
X m ðx Ã
Þ¼c 1exp
Àk 2m x ÃA
ð10Þ
where,c 1is a constant.The corresponding governing
equation for Y m which becomes an eigenvalue problem,is presented as,d 2Y m dy Ã
þk 2
m fY m ¼0ð11Þ
where,
f ¼1À
cosh j Ãy ÃðÞcosh j ÃðÞ
ð12Þ
The relevant boundary conditions for Eq.11are,at y ü0;
dY m
dy Ã
¼0ð13a Þat y ü1;
dY m
dy Ã
þDaY m ¼0ð13b ÞA power series solution of the form in Eq.14is used to
solve Eq.11.
Y ¼X 1m ¼0
a m y Ãm ð14Þ
The detailed solution is presented in the Appendix A .Using the relation given in Eq.13b ,the eigenvalues are obtained.The governing equation of the eigenvalues is given in Eq.36.For the sake of analytical treatment,the first eigenvalue is derived as,
k 21¼Da 1þDa =2  þ1Àcosh j ÃðÞcosh j ÃðÞ  :Da j Ã2Àsinh j ÃðÞj :cosh j ÃðÞ
ð15Þ
In terms of first eigenvalue,the solution of concentration is given as,
c Ãðx Ã;y ÃÞ¼c 1exp Àk 21x à a 0þa 2y Ã2þa 4y Ã4
þÁÁÁ
ð16Þ
The above equation can be rearranged as,
c Ãðx Ã;y ÃÞ¼A m exp Àk 21x ÃA
1þa 02y Ã2þa 04y Ã4
þÁÁÁ
ð17Þ
Using boundary condition Eq.7d ,the following exp-ression is obtained,1¼A m Y m ðy ÃÞ
ð18Þ
Using orthogonal property of eigenfunction Y m ,and weight function f as presented in Eq.12,the expression of the constant A m ,is obtained.
A m ¼R 10Y 1Àcosh j Ãy ÃðÞcosh j ÃðÞ  dy Ã
R 10
Y 21Àcosh j Ãy ÃðÞcosh j Ã
ðÞ
dy üI 1
I 2ð19Þ
The expression of I 1and I 2are given in the Appendix A .2.1Calculation for cup-mixing concentration
Cup-mixing concentration is defined as the cross-section averaged concentration.
c cm ü2R 1
0u Ãðy ÃÞc Ãðx Ã;y ÃÞdy Ã2R 1
0u Ãðy ÃÞdy üI 3
I 4ð20Þwhere,the expressions of I 3and I 4are given in the
Appendix A .
The conversion can be calculated by the following equation X ¼1Àc cm Ã
ð21Þ
2.2Calculation for Sherwood number
From the definition of Sherwood number of the system [30],the expression of mass transfer coefficient at any x*is given as (based on the first eigenvalue),
k x ¼Àu HS h c Ã
cm L  dc Ãcm
dx Ã
and
Sh x ¼k x h D ¼u HS h 2DL  Àdc Ã
cm dx Ãc Ãcm  !¼A
Àdc Ãcm dx Ã
c Ãcm
!ð22Þ
From Eq.20,the expression of c cm *can be expressed as,
c Ãcm
¼A m exp Àk 21x
ÃA
F ð23Þwhere,F is a constant.Taking derivative of Eq.23with
respect to x*,the following expression is obtained,
dc Ãcm ¼Àk 21
A m  exp Àk 21x Ã
F ð24ÞThus,the expression of Sherwood number becomes,k x h
D
¼Sh x ¼k 21ð25Þ
The averaged Sherwood number becomes,Sh L ¼
Z 1
Sh x dx ük 21
ð26Þ
2.3Numerical solution
Equation 4is a linear parabolic partial differential equation
with Neumann boundary condition at y*=0and mixed boundary conditions at y*=1.The initial-boundary con-dition in x*is presented by Eq.7d .The equations are solved by using PDE solver toolbox of MATLAB.In order to calculate c cm *and Sh L ,a standard (Simpson’s one-third rule)numerical integration technique is used.
For reactions of order other than first order,obtaining analytical solution is not possible using series solution methodology.This case has been solved by obtaining numerical solution using MATLAB.
3Results and discussion
In order to estimate conversion and Sherwood number in
the microchannel,Eq.4along with its boundary conditions is solved both analytically and numerically as presented earlier.The variation of profile of cup-mixing concentra-tion along the microreactor for various values of j h is shown in Fig.2a.Comparison of analytical and numerical solution is also presented in the same figure.It can be observed from the figure that although the first eigenvalue is co
nsidered,the analytical solution follows the numerical one quite closely.This is because the first term of the analytical solution is the dominant one.Two trends can be observed from the figure.First,the cup-mixing concen-tration decreases with the reactor length.This is expected as the reaction proceeds continuously leading to depletion of concentration of reactant in the downstream of the reactor.On the other hand,the cup-mixing concentration decreases more at higher values of j h .Higher values of j h indicate a thin electric double layer and the excess ions are concentrated near the channel wall.Since the reaction is occurring only in the presence of catalyst on the channel wall,the reaction rate is slowed down due to presence of excess ions which obstructs the access of reactants to the
wall.This discussion results in that the reaction conversion increases with lower values of j h.This is presented in Fig.2b.It can be observed from Fig.2b that at Da=5and A=1.5the conversion increases to98%from35%as j h decreases from10to0.5.
Effects of reaction rate parameter Da on the cup-mixing concentration are presented in Fig.3a.It can be observed from thefigure that cup-mixing concentration is less at higher value of Da at a particular location.As mentioned earlier,the parameter Da presents a comparison of deple-tion of reactants by reaction and diffusion.As Da increa-ses,the reaction rate increases and as a result,more reactants are consumed leading to a decrease in the cup-mixing concentration.It is also clear from thefigure that the
analytical solution is close to the numerical one.The variation of analytical solution is attributed to the fact that onlyfirst eigen value is considered.Variation of conversion profile with Da is presented in Fig.3b.The conversion profile increases with Da.As discussed earlier,increase in Da indicates a faster reaction rate,leading to more conversion.For example,at j h=5and A=1.5,the conversion increases from30to80%when Da increases from0.5to10at the end of the channel.
Analytical solutions of eigenvalue for two limiting cases of Da,which are Da?0and Da??,are obtained and presented in Appendix B and C,respectively.Also,in Fig.3b,the conversion profiles obtained using eigenvalues from Eq.50(for Da?0)and Eq.53(for Da??)are compared with conversion profiles from Eq.20.
Effects of the parameter A on the profile of cup-mixing concentration is presented in Fig.4a.It is observed that the profile of cup-mixing concentration decreases faster and

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