Ind. Eng. Chem. Res. 2008, 47, 3937–3942
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SEPARATIONS Transfer Units Approach to the FricDiff Separation Process E. A. J. F. Peters,* B. Breure, P. van den Heuvel, and P. J. A. M. Kerkhof Separation Processes and Transport Phenomena Group, Department of Chemical Engineering and Chemistry, EindhoVen UniVersity of Technology, P.O. Box 513, 5600 MB EindhoVen, The Netherlands
FricDiff (friction difference) is a recently introduced separation technology. The separation occurs because of differences in interspecies friction in a multicomponent mixture. We present a description of a FricDiff unit. Such a unit consists of two compartments, the feed-side and the sweep-side, with a porous screen in between. The gas mixtures at the feed-side and the sweep gas interdiffuse through the screen. The basic modeling assumption is that the binary interaction of each feed-component with a counterflowing sweep gas is dominant. The interaction between the components diffusing in the same direction is neglected. This assumption leads us to introduce the number of “binary transfer units”. We show that more detailed models introduced earlier are approximated well by this approach. Also, experiments seem to exhibit the same scaling, although the constants needed to fit the experimental results deviate quite a lot from the theoretical predictions. The equations derived can be used straightforwardly for incorporating a FricDiff unit in a process design. 1. Introduction In this paper, we present a simplified model to describe the separation that takes place in a FricDiff separation module. A typical FricDiff module consists of two compartments separated by a porous screen; see Figure 1. Here, we prefer the terminology screen instead of membrane to emphasize that the porous material is nonselective. Through the two compartments gas mixtures flow countercurrently. Usually one incoming stream is the mixture to be separated, say a water-alcohol vapor, the other one is a sweep gas, say CO2. Differences in diffusivity of the components through CO2 give rise to a kinetic separation. When counterdiffusing with CO2 inside the screen, the diffusional velocity of water is much higher than that of alcohol. Therefore the outgoing stream at the feed-side will have an increased alcohol to water ratio compared to the entering stream. The exiting stream at the sweep-side will have a decreased alcohol content relative to water. The idea to separate a binary gas mixture by exploiting the difference in species velocities when counterdiffusing into a third sweep gas is not new. Probably one of the first applications of this principle was for the separation of chloride isotopes by Harkins and collaborators.1 Here, a hydrogen chloride stream inside a clay pipe passes an airstream flowing outside. Both gases are at atmospheric pressure. Interdiffusion of the gases, as opposed to Knudsen diffusion, was identified as the main mechanism causing separation.2 An early application of the principle to separate a gas mixture (helium-neon) by diffusing against a vapor (water) can be found in patent applications of Hertz.3,4 Here, the interdiffusion does not occur at isobaric conditions. The diffusion is opposing a net flow of the sweep vapor. The fluxes of the counterdiffusing gases are small. The components of the gas mixture have a partial pressure gradient that counteracts the friction force with the sweep vapor. The gradient will be steeper for the component that experiences more friction, i.e., has a lower diffusivity. Two enriched gases (one richer in the slowly diffusing component, and the other in * To whom correspondence should be addressed. Tel.: +31 40 247 4922. E-mail:
[email protected].
the faster component) are obtained after condensing the water in the resulting three-component mixtures. An isobaric method for separating hydrogen from nitrogen or carbon monoxide was described by Maier.5 The design of his apparatus is similar to the FricDiff unit, i.e., two compartments with a porous screen in between. The processes discussed by Benedict and Boas6–8 are the following: a unit similar to that of Maier, a cascade of these units, and a column process. The vertical column consists of three concentric tubes. Vapor is introduced through the center porous tube and condenses on the cold surface of the outer tube. In between the inner and the outer tube, a porous tubular screen is placed. There is a net flow of vapor in the outward radial direction. The heavy component is dragged along more than the light one. The mode of operation of the column is quite different from that of the FricDiff or Maier unit and more similar to the mechanism described by Hertz. In the units, the mixture is introduced at one side and the sweep gas at the other. In the column of Benedict and Boas, however, the gas mixture is present at both sides of the screen and the sweep vapor is crossflowing in the radial direction. In the same year, a similar column process was described by Cichelli et al.9,10 In this case, there is no porous screen present. The condensed sweep vapor (plus additional liquid supplied at the top) flows downward inducing a circulating gas flow in the column. This circulation takes care of the counterflow of the light (i.e., faster diffusing) and heavy components. In the processes discussed so far, the sweep gas was a vapor, and the mixture, a noncondensible gas mixture. Clearly the opposite choice, i.e., a noncondensing sweep gas and a vapor mixture is also possible. An obvious application for kinetic separation of a vapor mixture is the breaking of an azeotrope. In an earlier paper,11 we claimed novelty by describing the process for breaking an azeotrope. This claim of originality is not justified. In his papers,12,13 Schwertz described a unit to separate a gas mixture using a vapor. In one of his patent applications,14,15 however, he described a module for separating an azeotropic vapor-mixture using a noncondensible gas. His module operated in a cocurrent manner and not in a counter-
10.1021/ie071395z CCC: $40.75 2008 American Chemical Society Published on Web 05/09/2008
3938 Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008
its solution. The key assumption is that the species of the incoming mixture do not feel each other. Let us consider the Maxwell-Stefan description of multicomponent transport, Figure 1. Schematic representation of a tubular FricDiff module. It consists of two concentric tubes of which the inner one is porous. The depicted mode of operation is countercurrent. The feed, a water-alcohol vapor, enters at the left-hand side in the core region. The sweep gas, carbon-dioxide, enters from the right in the outer region of the module. The gases counterdiffuse through the porous screen, which is nonselective. The main function of the screen is to allow for small pressure differences between the inside and outside region, without giving rise to much convective transport.
current manner. A countercurrent version of the process for separating azeotropes has also been described in literature.16 The column type of process where the vapor mixture is crossflowing through a stagnant sweep gas is known as diffusiondistillation.17,18 Here, liquid is evaporated on an outer tube and condensed on the inner one, where a liquid layer flows downward. The separation is influenced by both the liquidvapor equilibrium and the differences of diffusive transport of the mixture components through the stagnant layer. One can also combine both these mechanisms with a (selective) membrane.19,20 Summarizing, separation by means of counterdiffusion with a sweep gas has been invented and subsequently partially forgotten, several times. There is no doubt that this has to do with the (economic) feasibility of the process. Reviewing the old literature, we still see much potential in what we named “FricDiff”, especially in the light of modern, energy, and environment requirements. The FricDiff module was first described in Geboers et al.11 The model presented was a plug flow model. A similar model is used in a parameter study of the process.21 This means that in the description of the transport in the feed and sweep compartments no account is taken of axial dispersion effects. Moreover the models assumed that the mass-transport resistance in the radial direction resides fully in the screen. More recently a detailed model that solves flow and concentration fields at the feed and sweep side was presented.22 Dispersion effects arise due to the coupling of convection (dominantly in the axial direction, but radial dependent) and diffusion (mainly in the radial direction). In both models,11,22 an advanced multicomponent transport description, namely, the binary friction model,23 was used to describe the mass transport inside the porous screen. The evaluation of the models requires numerical solution of a large set of (partial) differential equations. Here we present a much more simplified model. The goal is to provide a description that can easily be used to model the FricDiff module for, e.g., incorporation into larger scale process simulations. The predictions of the modeling will be compared with predictions of the more detailed models. The model gives predictions for the outgoing streams in terms of the incoming compositions, “binary transfer units” (NTUi), and feed to sweep ratio. Here, binary transfer units are component specific transfer units based on the binary diffusion coefficients of the individual vapor components, i, in a sweep gas background. It will also be shown that these binary transfer units characterize experimental results well. Modeling In this section, we will present the assumptions used in modeling the FricDiff module and give the final equation and
- ∇ yi )
yiyj
∑D j
(Vbi - Vbj)
(1)
ij
From this point of view, components experience drag forces due to velocity differences, Vi - Vj, with other components. The total drag force should be balanced by the thermodynamic driving force (which, for ideal gases at isobaric conditions, is proportional to the gradient in mole fractions, yi). Because the species of the feed mixture are expected to move in the same direction, drag forces between them are small. The only species moving in the opposite direction through the screen is the sweep gas. Therefore, this friction force is dominant and should almost fully balance the thermodynamic driving force. We will denote the n species with an index i. The sweep gas is indicated by the highest index i ) n. The diffusional molar flux through the screen of component i * n is assumed to be dominated by i - n friction. When determining the velocity of species i, we take Vj ≈ Vi (for i, j * n); we further assume equimolar transport, such that (1 - yn)Vi + ynVn ≈ 0. With the assumption, the Maxwell-Stefan equation for species i, eq 1, simplifies to the Fick equation, Ni(r, z) ) -Dinc
∂yi ∂r
(2)
where Ni ) cyiVr, i is the molar flux in the radial direction. The total molar concentration, c, is assumed to be constant throughout the module. For a plate, r is the coordinate in the direction perpendicular to the plane,; for a cylindrical configuration, it is the radial direction. Note that these modeling assumption are made for each species individually. In the end, we will find Vi * Vj. Therefore, the assumptions made are not fully consistent with each other. The result, however, is a simplified set of decoupled equations, where dominant interactions are taken into account, which have a small deviation from the exact solution of eq 1. For a plate geometry, the flux in the screen is independent of r, i.e., Ni(r, z) ) Ni(z). For a cylindrical geometry, the molar flux in the radial direction Ni(r, z) is inversely proportional to r, so Ni(r, z) ) Nav,i(z)Rav/r, where Nav,i is the flux at radius Rav. Integrating eq 2 over the radial direction (or width for a plate) of the screen gives Nav,i(z) ) kav,ic(yF,i(z) - yS,i(z))
(3)
Here kav,i is the overall mass transfer coefficient (evaluated at R ) Rav) for component i, c is the total molar density, yF,i(z) and yS,i(z) is the molar fraction at the feed and sweep side at the streamwise position z. The overall mass transfer coefficient has three contributions, namely, from the screen, the feed side, and the sweep side. We will assume here laminar flow. On the feed and sweep sides, diffusion in the radial direction is assumed to be quick compared to convection. Therefore, we will neglect boundary-layer effects, but concentration-polarization effects are accounted for. The flux at the sweep side is, e.g, N ) kF,ic(yF,i - yFM,i), where yF,i is the cup-averaged molar fraction of component i at the feed side and yFM,i is the molar fraction on the feed-screen interface. The partial mass transfer coefficients for the plate configuration are
Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3939
kF,i )
ShFDin , δF
kM,i )
ε Din , τ2 δM
kS,i )
ShSDin δS
(4)
kav,i-1 ) kF,i-1 + kM,i-1 + kS,i-1
(5)
For the tubular configuration, RF is the radius of the porous annular screen at the feed side and RS is the outer radius at the sweep side. If one uses the logarithmic mean, δM , Rav ) |lnRS/RF|
where δM ) |RS - RF|
(6)
the relations in eq 4 hold also for the cylindrical case, with δF ) RF and δS ) RT - RS. Here, RT is the radius of the impenetrable wall. For the feed side, which is a cylinder with a porous boundary, one has ShF ) 24/11. For the sweep side, the Sherwood number depends on the ratio of inner and outer radius of the annulus, ShS(RS/RT), as derived in the Appendix, eq 23. Using these partial transfer coefficients the fluxes NF, NM, and NS are evaluated on radial position RF, Rav, and RS, respectively. To obtain an overall transfer coefficient for the flux at Rav in the tubular case, one needs to compute the following: kav,i-1 )
( ) RF k Rav F,i
-1
+ kM,i-1 +
( ) RS k Rav S,i
-1
(7)
For a plate geometry, we will use W as the width of the plate. In the cylindrical case, the circumference at Rav is Wav ) 2πRav
(8)
The results given below are valid for both the plate and annular geometry. As further assumptions, we will take equimolar (and isobaric) operation conditions. The sweep gas flux therefore follows from the sum of all other component fluxes, Nn ) -
∑N
i
(9)
i*n
At the feed and sweep side, we will discard axial dispersion. The influence of the shape of the laminar flow profile is not fully neglected, however. As stated before, the mole fractions used for the feed and sweep side should be interpreted as cupaveraged values. As shown in the Appendix, the perturbation of the radial concentration profile, by the flow field gives rise to the Sherwood relations. The molar balances for i * n, for stationary operation, are dyF,i ) -Nav,iWav ) -kav,iWavc(yF,i(z) - yS,i(z)) dz dyS,i -S ) + Nav,iWav ) kav,iWavc(yF,i(z) - yS,i(z)) dz
(1 - exp[-Ri]) yS,i,in F F 1 - exp[-Ri] 1 - exp[-Ri] S S F F (1 - exp[-Ri]) 1S S yF,i,in + yS,i,in yS,i,out ) F F 1 - exp[-Ri] 1 - exp[-Ri] S S i
yF,i,out
with Sherwood numbers ShF ) ShS ) Shplate ) 35/13 (see the Appendix). The porous material is characterized by a porosity ε and a tortuosity τ. Din is the binary diffusion coefficient of component i in a background of component n. In the definition, δF, δM, and δS are the distances between interfaces of the regions considered. For the plate configuration, the overall mass-transfer coefficient is
F
(10)
where F is the total molar flow at the feed side, and S that at the sweep side. Solving these equations for countercurrent flow where yF, i(0) ) yF, i, in and yS, i(L) ) yS, i, in gives, for the outgoing streams,
F 1 - ) exp[-R ] ( S y )
F,i,in +
(11) Besides the incoming molar fraction of a component, the outgoing fractions depend on the ratio F/S and the dimensionless constant Ri. Let us define the number of binary transfer units based on the feed stream as NTUi )
kav,iAavc F
(12)
where Aav ) WavL is the surface area of the porous screen. The number of transfer units is a measure of the flow through the screen compared to the feed flow. Using this definition, one finds that, for the model discussed above, F NTUi (13) S The model discussed so far is highly idealized. In reality, components do feel each other. The transfer coefficients, eq 7, will be composition dependent. Furthermore, the assumptions of isobaric and equimolar transport might be violated. To account for these effects in some effective way, we will therefore pose that
(
Ri ) 1 -
(
Ri ) β 1 -
)
F NTUi S
)
(14)
where NTUi is the ideal number of transfer units that follows from the analysis above. The term β is a constant smaller than 1. It is hoped to incorporate nonidealities not included in the analysis above. Our main assumption is that this constant can be taken as the same for all species and only depends on the module under consideration. This assumption will be referred to below as the binary transfer unit approach. 3. Comparison with Simulation Results In this section, we will compare simulation results obtained with the detailed models of the FricDiff module11,22 with the current approximate treatment using NTUi. It will be shown that, up to good approximation, the binary transfer unit description agrees with these models for β ) 1. The FricDiff module with a tubular geometry is considered. The radii characterizing the porous screen are RF ) 1.50 mm and RS ) 3.00 mm. The radius of the impenetrable tube is RT ) 5.2 mm, and its length is 100 mm. The porous screen is modeled as having a porosity of ε ) 0.20 and a tortuosity of τ ) 1.31. For use of the BFM model, a pore-radius of 2.5 µm is used. The process conditions are an atmospheric pressure of 1.013 bar and a temperature of 383 K. A three component mixture of (1) ethanol or iso-propanol, (2) water and (3) CO2 is considered. In the modeling, and in the analysis, we use the following binary Maxwell-Stefan diffusion coefficients for ethanol: D12 ) 1.77 × 10-5, D13 ) 1.36 × 10-5, and D23 ) 2.48 × 10-5 m2/s. For iso-propanol, the used values are the following: D12 ) 1.51 × 10-5, D13 ) 1.15 × 10-5, and D23 ) 2.48 × 10-5 m2/s. The feed entering the inner cylinder does not contain component 3. Three sets of compositions of the vapor mixture are used: y1 ) 0.25, 0.5, and 0.75 (y2 ) 1 - y1).
3940 Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008
Figure 2. Results of the Navier-Stokes model for alcohol-water separation using the FricDiff module. Several initial compositions and flow rates have been plotted in one graph using the scaling as suggested by the analysis presented in the text. The compositions are scaled using the ingoing feed compositions. Composition is indicated by the symbol shape. Agreement with the binary transfer unit analysis is good.
Figure 4. Experimental results for ethanol-water and iso-propanol-water separation. The data are rather scattered, but the trend seems to agree with the analysis. Nonidealities seem to be larger than in the simulation results.
in Figure 3. These simulation results have much less spread around the binary transfer unit curves. Graham’s law states that for the isobaric case the flow through the screen obeys M1N1 + M2N2 + M3N3 ) 0. One could in principle incorporate Graham-like transport through the screen, but it will make the model much more complicated. Moreover, in practice, one might control flows rather than pressures. If one controls flows, such that in the compartments Fin ) Fout and Sin ) Sout, then the transport is equimolar (but nonisobaric). Comparison with Experimental Results
Figure 3. Results of a plug-flow model where the transport through the porous screen is modeled by means of an equimolar Maxwell-Stefan model. The spread is much less than that in Figure 2.
The resistance to mass transport is dominated by the screen. The Sherwood numbers for the feed and sweep side are ShF ) 24/11 and ShS(RF/RT) ) 2.54. Using these numbers, one finds for ethanol that kF,1 ) 0.0244 m/s, kM,1 ) 0.00106 m/s, and kS,1 ) 0.0157 m/s The logarithmic mean is Rav ) 2.16 mm. The overall mass-transfer coefficient computed using eq 7 is kav,1 ) 9.5 × 10-4 m/s. The reason why the screen dominates is its low porosity and relatively large thickness. For a thinner, more porous screen, the main resistance would have been in the feed and/or sweep side. We performed simulations using this set of parameters for both the plug-flow model and the more detailed Navier-Stokeslike model. In Figure 2, only the results for the Navier-Stokes model are shown. The reason is that, for purpose of comparison with the binary transfer unit model, results do not differ appreciably. Both models are fitted reasonably well with eq 11 (with yS,i,in ) 0). Also, both models show about the same spread for the different compositions sets around the fitted curve. Note that in this case β ) 1 in eq 14. Therefore, we concluded that the cause of the spread is something that both detailed models have in common but the current approach does not contain. A clue for the missing ingredient is shown in Figure 3. From the analysis of the simulations, we find that the transport through the screen is nonequimolar but obeys to a good approximation Graham’s law. Also when checking the simulation results, where the BFM is used for the screen, we see that flows F and S are not constant and can change up to 30% (but usually less). When we change the model for transport through the screen from BFM to a simple isobaric-equimolar Maxwell-Stefan model, we find the data
We have performed some preliminary experiments with ethanol-water and iso-propanol-water mixtures. The experimental procedure is described in ref 11. The dimensions of the module (and porous screen) are those given in the previous section. The porous material used is stainless steel. We have also performed experiments using a ceramic screen of somewhat different dimensions. These data were poorly reproducible, probably due to adsorption effects.11 The ceramic data points (not shown) are scattered around the stainless steel data (so they are at least consistent). The experimental conditions were atmospheric pressure and a temperature of 110 OC. The feed mixture consisted of pure alcohol-water vapor with no CO2. The entering sweep gas consisted of pure CO2. For the ethanol-water case we used an 86 mol % ethanol mixture. The ipa-water mixture contained 69 mol % iso-propanol (ipa). Figure 4 shows a collection of measurements, all for F/S ) 0.25. Even the stainless-steel data are rather scattered. The trend seems to agree well with the analysis, if the fitted “nonideality constant” is taken to be β ) 0.23. This value is rather low, compared to the detailed simulations which fit well with β ) 1. One cause of the spread is the incomplete condensation of the vapor streams in the current setup. The amount alcohol and water in the outgoing flows was determined after condensing the vapor. We did not recover everything, some alcohol and water was lost. In the data presented in Figure 4, this last amount was distributed over the two flows (keeping the ratio of feed over sweep component flows constant). The error bars in Figure 4 show the possible error due to this limitation. This source of error is clearly too small to shift β ) 0.23 to β ) 1. Conclusions and Discussion In this paper, we compared a simple model based on binary transfer units with results obtained from more detailed models and from experiments. The simple analysis seems to agree well
Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3941
Figure 5. Sherwood number in an annulus with a penetrable boundary at RS and an impenetrable boundary at RT as function of the ratio η ) RS/RT.
with the simulation results. This means that for many cases where a high accuracy is not very important, e.g., in preliminary stages of design of equipment, the current approach is very useful. Our different models ranging from very detailed,22 to intermediate,11 to very simple (this paper) produce results that differ little. At least, this is the case for the operation window discussed in the current paper. There are conditions where models in refs 22 and 11 give predictions that differ appreciably from the binary transfer unit model. The plug-flow model11 does not take into account transport resistances in the feed and sweep side. Including a transfer coefficient (or rather a matrix) for the feed and sweep side, for the case of Maxwell-Stefan multicomponent transport, is not trivial.24 If the configuration and operation of the FricDiff module are such that these are important then the current model is expected to give better results than the unaltered plug-flow model. The plug-flow model, on the other hand, describes the multicomponent transport inside the screen, much more accurately using the full BFM model. Also the current model makes an isobaric assumption, and therefore can not predict influences of pressure differences between the feed and sweep side. The most detailed, and computational demanding model,22 is able to take into account effects that are discarded in both the current model and the plug-flow modeling. That model includes details of multicomponent transport at the feed and sweep side, such as the possible influence of (multicomponent) concentration boundary layers. For cases where these effects are expected to be important, the Navier-Stokes-like model of ref 22 is preferred and the current model will give large deviations. For the result presented here, the major part of the difference between the detailed models and the current approach can be explained by nonequimolar flow through the screen due to Graham’s law. The sets of conditions presented in this paper correspond with those where we performed experiments. The experiments can be fitted with the current model (for β ) 0.23 in eq 14) but are not in good agreement with our other modeling results (β ) 1). One possible cause is that relevant phenomena are not incorporated into the model. We have refined our models to a high extent.22 This extra sophistication does not change results with large factors except for special operation conditions. We therefore think extra modeling efforts will not move the simulation results away from β ) 1. We believe that the presented experiments, or their interpretation, are less trustworthy than the modeling results. One speculation is that, for some reason, there is a pressure difference present between feed and sweep side that causes a convective flow through the screen, limiting the possibility of separation
due to counterdiffusion. Currently, we are building a new setup where pressure differences can be measured accurately and flows (and other conditions) be better controlled and measured. A second cause of error in the interpretation of the experiments is the characterization of the porous material. An unknown factor in eq 7 is the tortuosity, τ. We determined this tortuosity by measuring the flux of pure CO2 through the screen when applying a pressure difference. For this computation we assumed a single pore radius of Rp ) 2.5 µm. The molar flow through a pore is proportional to Rp4 (Hagen-Poiseuille). We used this number for a diffusive process where the flow through a pore is proportional to its cross-sectional area (Rp2). In case of a poresize distribution, therefore, large pores are much more biased for convective flow than for diffusive flow. In mathematical terms, when there is a pore-size distribution the average value of 〈Rp4〉 can deviate quite a lot from 〈Rp2〉2. This can explain a missing factor shifting β toward 1. A reviewer suggested the possibility of partial blocking of small pores due to condensation. Acknowledgment The FricDiff project is supported by a grant (ISO44051) of SenterNovem, an agency of the Dutch ministry of Economic Affairs, and is performed in cooperation with Akzo Nobel Chemicals, Purac Biochem, Shell, Bodec Process Technology, FIB Industrie¨le Bedrijven, MolaTech, and TU Delft. Appendix: Sherwood Number for an Annular Region In the annular region between the screen and the outer wall, we will assume that diffusion in the radial direction proceeds on a small time-scale. The concentration does vary only a little bit in the radial direction. Convective transport in the axial direction acts as a perturbation. The molar flux near the screen is N ) -D ∂c/∂r at r ) RS. We want to express the flux as N ) k(c(RS) - jcS), where the transfer coefficient k is related to the diffusion coefficient and the width of the region by means of a Sherwood number k ) ShSD/(RT - RS). Here, c(RS) is the concentration at the screen and jcS is the cup-averaged concentration on the sweep side. Equating both expressions for the flux, one finds that ShS )
|
RT - RS ∂c cS - c(RS) ∂r r)Rs
(15)
In this appendix, we will evaluate this number for an annular region. We will assume that diffusion in the radial direction is fast, so we are not in a boundary layer regime. To emphasize the fact that diffusion is dominant we introduce a small variable ε. The dimensionless, stationary diffusion equation is -
1 ∂ ∂c ∂ r + ε (Vz(r)c) ) 0, r ∂r ∂r ∂z
( )
|
∂c )0 ∂r r)1
(16)
Here, the outer impenetrable wall is located at r ) 1. The inner permeable wall will be located at r ) η. Let us expand the concentration in series of ε, c(r, z) ) c1(r, z) + εc2(r, z) + O(ε2)
(17)
Inserting eq 17 into eq 16 and collecting the zeroth- and firstorder terms gives ε0 : ε1 :
( ) ( )
|
∂c1 1 ∂ ∂c1 )0 r ) 0, r ∂r ∂r ∂r r)1 ∂c2 ∂ 1 ∂ ∂c2 r + (Vz(r)c1) ) 0, )0 r ∂r ∂r ∂z ∂r r)1 -
|
(18)
3942 Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008
For ε f 0, the concentration is constant in the radial direction, c1(r, z) ) c1(z). Because both ∂c1/∂r ) 0 and c1(η, z) - jc1(z) ) 0, the zeroth-order does not contribute to the Sherwood number. Using these properties and inserting eq 17 into 15 gives ShS )
|
1 - η ∂c2 + O(ε) c2 - c2(η) ∂r r)η
Literature Cited (19)
In the limit ε f 0, the Sherwood number is determined by the next order, namely, c2. The formal solution for c2 follows from eq 18, c2(r, z) - c2(η, z) ) -f(r)
∂c1 , ∂z
with
(20) r′′ V (r′′) dr′′ dr′ r′)η r′′)r′ r′ z Inserting the expression into eq 19, the c1 dependence cancels because it is independent of r. Introducing the cup-averaging, we get f(r) )
∫ ∫ r
ShS )
1
f′(η)(1 - η)
∫
1
η
∫
1
η
Vz(r)r dr
value limηf∞ ShS(η) ) Shcyl ) 24/11. The values for the flat plate and the cylinder can also be obtained by explicitly performing the analysis for these simpler geometries.
(21)
f(r)Vz(r)r dr
For laminar flow in an annular region, the velocity profile is given by ln(r) (22) ln(η) The proportionality constant is unimportant since it cancels in eq 21. Computing f(r) using eq 20 and next evaluating eq 21 gives the final expression for the Sherwood number, Vz(r) ∝ (1 - r2) - (1 - η2)
ShS(η) ) [72(η - 1)3(η - 1)2(1 - η2 + (η2 + 1) log(η))2]/[η(9(5η2 - 11)(η2 - 1)3 + ln(η)(3 ln(η)(11η8 - 36η4 - 48η2 + 24 ln(η) + 73) 4(η2 - 1)2(19η4 - 8η2 - 62)))] (23) In Figure 5, we provide a plot of this relation. In the general dimensional case, the parameter η corresponds to the ratio of the inner and outer tube diameter η ) RS/RT. In the limit η f 1 (an infinitely thin gap), one finds the flat-plate value for the Sherwood number Shp ) 35/13. The solution for η > 1 corresponds to an impenetrable boundary on the inner cylinder and a porous boundary at the outer cylinder. The value for the limit η f ∞ corresponds to the case of a cylinder with a porous boundary, since the size of the inner cylinder is negligible. The
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ReceiVed for reView October 16, 2007 ReVised manuscript receiVed March 3, 2008 Accepted March 6, 2008 IE071395Z
