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Ind. Eng. Chem. Res. 2009, 48, 7694–7704
Separation of Gas Mixtures with FricDiff: A Comparison between Experimental Data and Simulations Bianca Breure,*,† Lodewijk Schoon,‡ Elias A. J. F. Peters,† and Piet J. A. M. Kerkhof† Separation Processes and Transport Phenomena Group, Faculty of Chemical Engineering and Chemistry, EindhoVen UniVersity of Technology, P.O. Box 513, 5600 MB EindhoVen, The Netherlands, and Shell Global Solutions International B.V., GSICP, Badhuisweg 3, 1031 CM Amsterdam, The Netherlands
This work describes the experimental procedure and presents experimental results on the separation of helium-argon mixtures in a FricDiff module. A FricDiff module consists of two compartments separated by a porous screen. The mixture to be separated flows through one compartment, the separating agent through the other. Experiments were performed at room temperature and a pressure of approximately 2 bar with nitrogen as the separating agent. In the experiments the pressure of the inlet feed mixture was changed such that convective flow through the porous barrier took place either from the feed mixture compartment to the separating agent compartment or vice versa. The experimental data were compared with a numerical model showing good agreement between mole fractions and molar flow rates. Deviations were observed when the inlet pressures of the feed mixture were compared. This may be attributed to an inaccurate pressure measurement in the experimental setup. Introduction
1/2
A mixture of gases can be separated when it is forced to diffuse through an auxiliary gaseous or vaporous component. By doing this, one exploits the differences in relative rates of diffusion of the gas components through the auxiliary species to achieve the separation. In literature several technologies have been developed that achieve a separation of gas or vapor mixtures based on this principle. Examples of these technologies are mass diffusion,1,2 sweep diffusion,3,4 and FricDiff.5-8 In this paper the latter technology is applied for the separation of mixtures of helium and argon, using nitrogen as the auxiliary component (further referred to as separating agent or sweep gas). In the FricDiff process the gas mixture and the separating agent flow at opposite sides of a porous screen. Through the porous screen the material is exchanged between the two compartments, which mainly occurs by diffusion. When the components of the gas mixtures have different diffusive velocities through the separating agent, this exchange will take place selectively. The separating agent will be enriched in the faster diffusing component (usually the component with the smaller molecular weight), whereas the feed mixture will be enriched in the slower diffusing species. In this paper, experimental results are presented for the separation of a helium-argon mixture with nitrogen as the separating agent. The experimental results obtained with this test system are compared with data generated with a numerical model developed in previous papers5,6 and will mainly be used to validate this model. Theory The separation in the FricDiff process is based on a difference in diffusion coefficients. Chapman and Enskog derived a mathematical expression for the diffusion coefficient of a component i in a binary gas mixture consisting of components i and j. This expression can be written as9 * To whom correspondence should be addressed. Tel.: +31 40 247 3683. Fax: +31 40 244 6653. E-mail:
[email protected]. † Eindhoven University of Technology. ‡ Shell Global Solutions International B.V.
Dij )
3 (4πkT/Mw,ij) fD 16 nπσ 2Ω ij
(1)
D
where Mw,i is the molecular weight of component i, Mw,ij ) 2(1/Mw,i + 1/Mw,j)-1, k is the Boltzmann constant, T is the absolute temperature, and n is the number density of the molecules in the mixture. Furthermore, ΩD is the collision integral, which is a function of temperature and fD is a correction factor, which is in the order of unity. A simpler approximation for Dij derived for molecules that behave as rigid spheres is10
Dij )
2 3
kTπ 12 ( M1
+
w,i
1 nπ (di + dj) 2
(
1 Mw,j
)
(2)
2
)
To determine if a mixture can be separated efficiently by the FricDiff principle, it is important to look at the ratio of the binary diffusion coefficients of the components comprising the gas or vapor mixture in the separating agent. Using eq 2, it is easily shown that this ratio is a function of the molecular weights of the components (Mw,i) and their collision diameter (di), according to
(
Dik dj + dk ) Djk di + dk
) (( 2
) )
Mw,i + Mw,k Mw,j Mw,j + Mw,k Mw,i
1/2
(3)
For the separation of a helium(1)-argon(2) mixture with nitrogen(3) as the separating agent, the ratio D13/D23 calculated with the Chapman and Enskog expression is approximately 3.6. This implies that the molar velocity of helium through the porous barrier will approximately be 3.6 times higher than the molar velocity of argon. In a completely isobaric system the selectivity of the separation is to a good approximation given by the ratio of the diffusion coefficients. However, in the case where absolute pressure gradients are present between the two compartments of the FricDiff unit, the selectivity of the separation will be different. This was already shown by Maier11 and Schwertz12 and by Keyes and Pigford.13 Pressure gradients between the
10.1021/ie801738v CCC: $40.75 2009 American Chemical Society Published on Web 07/15/2009
Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009 Table 1. Specifications of Porous Screens Used in the Experiments abbreviation average pore size (µm) porosity (-)a tortuosity (-)a inner radius (mm) outer radius (mm) length (cm)
dp ε τ r1 r2 L
Sika-R 0.5 IS
Sika-R 1 IS
2 0.17 3 5 36
5 0.20 3 5 36
a A value for the factor ε/τ2 for the screens is later found by fitting experimental data.
compartments induce convective mass transport through the barrier, which influences the separation process. The selectivity will be reduced when the gas mixture is forced to flow through the barrier. However, by increasing the pressure of the separating agent, the selectivity of the separation can be improved. If, as a result of convective flow of the separating agent, the transport of the slower diffusing component through the barrier is completely eliminated, selectivities up to 100% can be attained. It should however be noted that these high selectivities are accompanied by significant decreases in the magnitude of the diffusive fluxes through the barrier. In this work a comparison is made between experimental data and data obtained from simulations. The numerical model that is used for this comparison is discussed in the papers by Geboers et al.5 and Selvi et al.6 In this model the flow of the gas mixture and the separating agent through the flow channel is described as a plug flow, assuming that the mixtures in the flow channels are perfectly mixed in the radial direction. Pressure drops along the length of the flow channel are described with the Hagen-Poiseuille equation, whereas mole balances are solved to describe the change in molar flow rates for the different species. The transport of material through the porous screen is described by combining a model for transport through porous media, the Binary Friction model, with an equation of continuity for each component. The model takes into account the effect of pressure differences over the porous barrier on the separation process. In a later paper8 the influence of concentration boundary layers in the flow channels on the separation process is studied. These concentration boundary layers give rise to additional resistances to mass transfer and hence can influence the transport of species between the compartments. Sh-correlations are derived to describe the transport through these boundary layers. The paper also shows that when the porous barrier is of sufficient thickness the resistances to mass transfer in the flow channels are negligible compared to the resistance in the barrier. Later in this work it will be shown that this is also the case for our experimental setup, that is, the dominant resistance to mass transfer is located in the porous barrier. This justifies the use of the numerical model described by Geboers et al.5 and Selvi et al.6 for the comparison between simulations and experimental data. Experimental Section Materials. Experiments were performed with two heliumargon gas mixtures consisting of approximately 50.4 mol % helium and 49.6 mol % argon or of 74.8 mol % helium and 25.2 mol % argon. Nitrogen was used as separating agent and had a purity of 100%. Two cylindrical stainless steel porous screens were used in the experiments (Sika-R 0.5 IS and Sika-R 1 IS), consisting of sintered metal. These porous screens were obtained from Bekaert14 and their specifications are given in Table 1. No values for the tortuosity of the screens were available from the supplier. Values for the porosity ε and tortuosity τ are required in the numerical model to calculate
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the magnitude of the fluxes through the porous screen. A value for the factor ε/τ2 is finally found by fitting of experimental data as is explained in the paragraph “Determining the Factor ε/τ2”. Experimental Procedure. The experimental setup used in this work is shown in Figure 1. The central part of the setup is the FricDiff unit, which consists of two concentric tubes. The porous screen forms the inner tube and has an internal radius of 3 mm and an external radius of 5 mm. The outer tube consists of a stainless steel impermeable tube with an inside radius of 9 mm. This radius is further referred to as r3. The length of the FricDiff unit is 36 cm. The helium-argon gas mixtures flow at the inside of the porous screen and the separating agent flows through the annular space between the porous screen and the impermeable tube. The FricDiff unit was operated both co- and countercurrently. Experiments were performed at room temperature and at a pressure of approximately 2 bar. The flow rates of the two incoming and the two outgoing flows were measured with mass flow meters. Furthermore, the composition of the outgoing streams was analyzed with a mass spectrometer. At the start of the experiment a set point of about 2 bar was given to PCV1, which regulated the pressure of the separating agent at about 2 bar. The flow rates of the outgoing streams were approximately fixed to constant values (during the experiments the values fluctuated around the desired value with (10 g/h). The pressure of the gas mixture, which was controlled with PCV2, was set slightly higher than the pressure of nitrogen. This resulted in a small convective flow of the gas mixture through the porous screen (diffusion was still the dominant transport mechanism). After steady state was reached, the flow rates of the incoming and outgoing flows and the composition of the outgoing flows was measured. Next the set point of PCV2 was decreased with approximately 1 mbar and the procedure was repeated. Experiments were repeated until the regime was reached where convective flow of nitrogen through the porous screen takes place. Going from one convective flow regime to the other convective flow regime, a point of zero net molar flow is passed. The accuracy of the mass flow meters in the setup is a function of the mass flow according to (0.5% ( 0.001/ measured flow (kg/h) × 100%. This implies that lower flow rates result in higher errors. Considerable effort had been made to improve the accuracy of the mass spectrometer, mainly because helium is a difficult gas to measure quantitatively. The relative errors of the mass spectrometer for helium, argon and nitrogen, after calibration, were about 1%. With the quantity and composition of the flows entering and leaving the system known, the selectivity of the separation and the recovery of a component can be determined. Since helium is the faster diffusing component, it will be enriched in the flow channel for the separating agent. The selectivity and recovery (%) of helium are therefore defined by selectivity )
Gmol,He,out 100% Gmol,He,out + Gmol,Ar,out
recovery )
Gmol,He,out 100% Fmol,He,in
(4) (5)
where Gmol,i is the molar flow rate of component i in the compartment for the separating agent and Fmol,i the molar flow rate of component i in the feed mixture compartment. Results and Discussion Experimental Data. In the experiments the pressure at the location where the separating agent is entering the FricDiff unit
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Figure 1. Experimental FricDiff setup for the separation of a helium-argon mixture with nitrogen as the separating agent (PCV ) pressure control valve, MFM ) mass flow meter).
Figure 2. Experimental results of a countercurrently operated FricDiff module equipped with a Sika-R 0.5 IS porous screen. A feed mixture containing 74.8 mol % helium and 25.2 mol % argon was separated with pure nitrogen. In the experiments Fmass,out varied between 37.2 and 51.3 g/h and Gmass,out between 78.1 g/h and 85.4 g/h. Molefractions of helium, argon, and nitrogen at the outlet of the sweepside are plotted as a function of the set point pressure of PCV2.
is kept constant. Also the flow rates of the outgoing streams are fixed at approximately constant values. The pressure at the location where the feed mixture is entering the unit is varied. For each set point of PCV2, flow rates and compositions of the streams entering or leaving the unit are known or measured. The results of a typical experimental run in which the FricDiff unit is operated countercurrently are shown in Figure 2. Using these data, selectivities and recoveries can be determined. In Figure 3 the selectivity for helium and its recovery are plotted as a function of the pressure at the feed mixture inlet. This figure indicates that the selectivity increases with decreasing pressure at the feed inlet. Hence, convective flow of the separating agent through the barrier has a positive effect on the separation of helium from argon. Since the transport of helium through the porous barrier is hampered by the convective flow of nitrogen, recoveries decrease with decreasing pressure at the feedside. Determining the Factor ε/τ2. The data that were obtained with the experimental setup are compared with the data obtained with the numerical model. According to the supplier of the porous screens, the Sika-R 0.5 IS and the Sika-R 1 IS screens have a porosity of ε ) 0.17 and ε ) 0.2, respectively. No information was available on their tortuosity and as an initial guess a value of τ ) 2 was used. Using these values for the porosity and the tortuosity for the factor ε/τ2 in the numerical model resulted in calculated fluxes that were much higher than
Figure 3. Selectivities and recoveries of helium calculated with eqs 4 and 5 using data from Figure 2 as a function of the set point pressure of PCV2.
the values that were obtained experimentally. It was therefore decided to determine the factor ε/τ2 from the experimental data. To determine this factor for the Sika-R 0.5 IS screen, experimental data points for the countercurrent experiments were collected that were obtained under conditions of almost equimolar mass transport through the porous screen. The mole fractions at the inlet and outlet of the unit and the average molar flow rates of the feed mixture and separating agent were determined for these data points. These data were used as input values for the equations mentioned in a paper by Peters et al.7 In that paper the following equations were derived to calculate the mole fractions of the feed mixture components in the outgoing streams of a countercurrently operated FricDiff unit when equimolar mass transport through the porous screen takes place:
(
)
Fmol exp[-Ri] (1 - exp[-Ri]) Gmol yF,i,out ) yF,i,in + yG,i,in Fmol Fmol 1exp[-Ri] 1exp[-Ri] Gmol Gmol Fmol Fmol (1 - exp[-Ri]) 1Gmol Gmol yG,i,out ) yF,i,in + yG,i,in Fmol Fmol 1exp[-Ri] 1exp[-Ri] Gmol Gmol 1-
(6)
where
Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009
(
Ri ) 1 -
)
Fmol NTUi Gmol
(7)
of the FricDiff module were derived in a similar way as was done by Peters et al.,7 resulting in
yF,i,out
Fmol + exp[-Ri] Gmol 1 - exp[-Ri] ) yF,i,in + yG,i,in Fmol Fmol 1+ 1+ Gmol Gmol
yG,i,out
Fmol Fmol 1+ (1 - exp[-Ri]) exp[-Ri] Gmol Gmol ) yF,i,in + yG,i,in Fmol Fmol 1+ 1+ Gmol Gmol (12)
and NTUi )
2πrlmLki,ovc ε Fmol τ2
(8)
In the expression for the number of transfer units for component i, NTUi, the total concentration is indicated by c and rlm is the log mean radius, which is defined by rlm )
(r2 - r1) r2 ln r1
()
(9)
1 1 1 1 ) + + rlmki,ov r1kF,i rlmkbar,i r3kG,i
(10)
where kF,i, kbar,i and kG,i are the resistances to mass transfer of component i in the feed mixture compartment, the porous barrier and the separating agent compartment, respectively. These mass transfer resistances are in turn defined by ShF,iDi3 , r1
kbar,i )
ε Di3 , τ2 rlm
(
)
(
(
)
)
(
(
Fmol NTUi Gmol
)
where
Furthermore, ki,ov is an overall mass transfer coefficient which can be written as
kF,i )
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kG,i )
ShG,iDi3 (r3 - r2) (11)
where Sh are Sherwood numbers and Di3 are binary diffusivities of the mixture components He or Ar in the separating agent which is indicated here by the index 3. The model developed by Peters et al.7 is based on the assumption that the interaction between the feed mixture components is negligible compared to the interactions with the separating agent. The mixture components move in the same direction and therefore the drag forces between these species are much smaller than the drag forces between the individual mixture species and the separating agent that moves in opposite direction. The mass transfer coefficients in eq 11 are therefore only based on binary diffusion coefficients of the mixture components in the separating agent. Furthermore in the barrier it is assumed that mixture componentseparating agent interactions are dominant over species-pore wall interactions. Equation 6 shows that the mole fractions at the outlet of the FricDiff unit are a function of the mole fractions at the inlet and the (average) molar flow rates Fmol and Gmol of the gas mixtures in the compartments. These equations are used to calculate the mole fractions of helium and argon at both outlets. The mole fraction of nitrogen at these outlets follows from the requirement that the mole fractions should sum to one. The only unknown in these equations is the factor ε/τ2 in the definition of NTUi. This factor can be determined by matching the calculated values for yF,i,out and yG,i,out with the experimental values. In Figure 4 results of this matching are shown for yF,He,out, yF,Ar,out and yF,N2,out for the porous screen Sika-R 0.5 IS. The value of the factor ε/τ2 that was found for this screen is 0.0175. For the Sika-R 1 IS screen a similar procedure was followed to determine the value of ε/τ2 as for the Sika-R 0.5 IS screen. However, in this case data points for cocurrent operation of the FricDiff module were used. The reason is that for this mode of operation more data points were available that satisfied the condition of equimolar flow. Equations for cocurrent operation
Ri ) 1 +
)
(13)
and NTUi is still defined by eq 8. Using eqs 12 and 13 and the experimental data points for equimolar flow conditions, a value of ε/τ2 ) 0.036 was found for the Sika-R 1 IS screen. It should be noted that ki,ov in eq 8 was set equal to the resistance to mass transfer present in the porous screen. Table 2 shows the contribution of the different mass transfer resistances to the overall resistance to mass transfer for different values of the factor ε/τ2, which so far is the only unknown. The different mass transfer coefficients are calculated with eqs 10 and 11 using values for the Sherwood number for the fully developed concentration boundary layer regime as discussed by Peters et al.7 The table clearly shows that for all values of ε/τ2 the resistance to mass transfer is dominated by the porous barrier. Comparison with Numerical Data. With the values of ε/τ2 determined, a comparison can be made between experimental and numerical data. In the experiments the composition of the ingoing streams was fixed, as well as the flow rates of the outgoing streams. Also the pressure is fixed at the inlet for the separating agent (it was set to approximately 2 bar). The numerical model requires an additional variable to be specified at the inlets or outlets and it was decided to fix the flow rate of the gas mixture that we want to separate. In Figure 5 an overview is given of the variables at the inlets or outlets that are fixed or calculated in the numerical model. The calculated values are compared to the experimental data. In Figures 6 and 7 a comparison is made between experimental and calculated mole fractions of helium, argon, and nitrogen at the sweep outlet. The experimental data were obtained with the Sika-R 0.5 IS screen. Also error bars are shown in the figures, indicating the uncertainties in the experimental and calculated values. The calculation of these uncertainties is explained in Appendix 1. In Figure 6 results are shown for a FricDiff module that is operated countercurrently. The data points for helium and nitrogen are located very near or on the 45° line, which implies that the agreement between experiments and model is very good. Looking at the results for argon, one can notice that the experimental and calculated argon mole fractions correspond less well. These deviations cannot be explained with the uncertainty analysis (i.e., the error bars do not cross the 45°-line) and can be the result of experimental errors, an error in the numerical model (i.e., phenomena that are not or not well described by the numerical model) and/or an error in the fitted value for the factor ε/τ2. Since the experimental and numerical results correspond well for helium
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Figure 4. Matching of experimental and calculated values for yF,He,out (left, top), yF,Ar,out (right, top) and yF,N2,out (bottom) in order to determine the value of ε/τ2 for the Sika-R 0.5 IS porous screen. Table 2. Contribution of Mass Transfer Resistances of Helium and Argon in the Compartments and Porous Barrier to the Overall Resistance to Mass Transfer for Various Values of the Factor ε/τ2 ε/τ2
kF,He (m/s)
kbar,He (m/s)
kG,He (m/s)
kov,He (m/s)
0.1 0.08 0.04 0.02
0.0251 0.0251 0.0251 0.025
0.000880 0.000704 0.000352 0.000176
0.0446 0.0446 0.0446 0.0446
0.000834 0.000675 0.000344 0.000174
ε/τ2
kF,Ar (m/s)
kbar,Ar (m/s)
0.1 0.08 0.04 0.02
0.00684 0.00684 0.00684 0.00684
0.000240 0.000192 9.608 × 10-5 4.804 × 10-5
kG,Ar (m/s) 0.0122 0.0122 0.0122 0.0122
kov,Ar (m/s) 0.000228 0.000184 9.401 × 10-5 4.752 × 10-5
and argon and since the argon mole fractions are very small, it is very likely that there is an error in the measurement of the argon concentration. This reasoning is confirmed by the observation that the absolute deviations between experimental and calculated mole fractions are of the same order of magnitude or smaller for argon compared to helium and nitrogen. Figure 7 shows results for a FricDiff module that is operated cocurrently. For nitrogen the agreement between experimental data and model results is vey good when we take the error bars into account. Deviations from the 45° line are observed for the data points of helium and argon, which cannot be explained by the error analysis. The agreement between model and experiments is still fairly good.
In Figure 8 the measured values of molar flow rates of the gas mixture at the inlet of the sweep gas flow channel are plotted versus the values obtained with the numerical model. In the numerical model mole balances are closed, but this is not the case in the experiments. Hence, small deviations can be observed between numerical and experimental data as a result of experimental errors. When we look at the error bars we can see that in the majority of the cases these deviations are within the expected range. In Figures 9-11, a comparison is made between experimental and model results for a FricDiff module equipped with a Sika-R 1 IS porous screen. Again the deviations for argon are higher in Figure 9, which shows results for a countercurrently operated module, whereas the majority of the points for helium and nitrogen show good agreement. The argon data points start to deviate from the 45° line for mole fractions g 0.01 as was also observed in Figure 6. The data points for cocurrent operation, which are shown in Figure 10, show a good agreement for all species for the majority of the points. The comparison between experimental and calculated inlet molar flow rates for the sweep gas is shown in Figure 11. With the error bars taken into account, the experimentally determined molar flow rates are in perfect agreement with the flow rates calculated with the model. A comparison is also made between the set point pressures of PCV2 and the pressures calculated with the numerical model. The results are shown in Figure 12. Here we see that
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Figure 5. Overview of inlet and outlet variables that are fixed or calculated in the numerical model in the case of countercurrent (top figure) or cocurrent (bottom figure) operation.
Figure 6. Comparison between numerical and experimental data points for helium (left, top), argon (right, top), and nitrogen (bottom) including error bars. The results are obtained for a FricDiff module with Sika-R 0.5 IS screen that is operated countercurrently.
the experiments and the model give different results, although the trends are similar. The numerical model predicts that lower inlet feed pressures are required to reach a certain mole fraction of helium, argon, or nitrogen in the sweep gas stream at the module outlet. The numerical model also predicts a
larger pressure range than is obtained in the experiments (approximately 0.045 versus 0.015 Pa, respectively) Similar results have been observed for the Sika-R 0.5 IS porous screen and also for countercurrent operation of the FricDiff unit. Since the correspondence between experimental and
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Figure 7. Comparison between numerical and experimental data points for helium (left, top), argon (right, top), and nitrogen (bottom) including error bars. The results are obtained for a FricDiff module with Sika-R 0.5 IS screen that is operated cocurrently.
Figure 8. Comparison between numerical and experimental data points for the sweep gas flow rate (Gmol) indcluding error bars. Results are shown for a countercurrently operated (left panel) and cocurrently (right panel) operated FricDiff module with Sika-R 0.5 IS screen.
numerical mole fractions is fairly good, we do not think these deviations are a result of deficiencies in the numerical model. It is more likely that the observed differences are the result of an inaccurate pressure adjustment in the experiments. In the experimental setup pressures at the inlet of the module are fixed by giving a certain set point to the pressure control valves (valves PCV1 and PCV2). These set point pressures will deviate from the true pressure at these locations. How
big these deviations are is not known. In the experiments the pressure at the separating agent inlet is approximately 2 bar, whereas in the numerical model it is set equal to 2 bar. It is not unlikely that the true experimental separating agent inlet pressure is higher than 2 bar. This would explain why or model predicts lower values for the inlet pressure of the feed mixture, but it does not explain the wider pressure range found by the model (a more or less equal range is expected).
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Figure 9. Comparison between numerical and experimental data points for helium (left, top), argon (right, top) and nitrogen (bottom) including error bars. The results are obtained for a FricDiff module with Sika-R 1 IS screen that is operated countercurrently.
The latter can again be attributed to deviations between the set point pressures of the control valve PVC1 and the true pressure at the feed inlet. At the moment a setup is being built in which pressure differences over the porous barrier can be measured with an accuracy of 10 Pa. With this setup it will be possible to accurately study the influence of pressure differences over the porous barrier on the separation process. Conclusions In this work experimental data were presented on the separation of helium-argon mixtures in a FricDiff module with nitrogen as the separating agent. Experiments were performed with two porous barriers Sika-R 0,5 IS and Sika-R 1 IS, differing in pore size, porosity (ε), and tortuosity (τ). The experimental data were compared with data obtained with a numerical model, described in the papers by Geboers et al.5 and Selvi et al.6 Before the comparison between experimental and numerical data could be made, the factor ε/τ2 required in the numerical model had to be determined. This was done by fitting experimental data obtained under approximately equimolar conditions with mathematical expressions derived by Peters et al.,7 resulting in a factor ε/τ2 ) 0.0175 for the Sika-R 0.5 IS screen and a factor ε/τ2 ) 0.036 for the Sika-R 1 IS screen. Using these values for the factor ε/τ2 in the numerical model resulted in a good agreement between calculated and experimentally measured mole fractions and molar flow rates. Rather large deviations were observed when the
pressures of the feed mixture at the inlet of the module were compared. These deviations may be attributed to an inaccurate absolute pressure adjustment. Our new experimental setup with pressure sensors with an accuracy of 10 Pa, will have to give a decisive answer about this. 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, FIB Industrie¨le Bedrijven, MolaTech and TU Delft. Appendix 1. Uncertainty Analysis. In the numerical model, experimentally measured values are used as input values to calculate the values of other variables (e.g., outlet mole fractions) as is shown in Fig. 5. These measured values have an experimental uncertainty and hence our calculated values also contain an uncertainty. The uncertainties in the measured values are known and are tabulated in Table A1. However, due to the complicated nature of the equations in the numerical model (i.e., coupled ODEs), it is rather difficult to calculate the uncertainties in the numerical results. It is expected that by using eqs 6 and 12
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Figure 10. Comparison between numerical and experimental data points for helium (left, top), argon (right, top) and nitrogen (bottom) including error bars. The results are obtained for a FricDiff module with Sika-R 1 IS porous screen that is operated cocurrently.
Figure 11. Comparison between numerical and experimental data points for the sweep gas molar flow rate (Gmol) including error bars. Results are shown for a countercurrently operated (left figure) and cocurrently (right figure) operated FricDiff module with Sika-R 1 IS porous screen.
we can make a good estimate of the uncertainties in the calculated mole fractions as a result of uncertainties in the molar flow rates. An estimation of the uncertainties in the calculated molar flow rates Gmol,out can be obtained from a molar balance over the module and the known uncertainties in the molar flow rates Fmol,in, Fmol,out, and Gmol,in. In the calculations, the absolute pressure level in the module is set by fixing the pressure of the entering stream of separating agent (the pressure is fixed at 2 bar in the calculations). Since
(small) variations in the absolute pressure level do not have a large impact on the separation process, the uncertainty in the absolute pressure is not taken into account. We first discuss the estimation of the uncertainty in the calculated mole fractions yG,i,out. The uncertainties in the mole fractions yF,i,out, can be derived in a similar way. After this we will show how we can make a good estimate for the uncertainty in the molar flow rates. In case the FricDiff module is operated countercurrently, the mole fractions of helium
Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009 Ri )
2πrlmLkiovcε τ
2
(
)
(
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)
2πLDincε 1 1 1 1 ) ) Fmol Gmol Gmol ln(rout /rin)τ2 Fmol 1 1 β (A.2) Fmol Gmol
(
)
and where Fmol and Gmol are molar flow rates averaged over the inlet and outlet values for the feed mixture and separating agent compartment, respectively. A value for yG,N2,out can be obtained from the constraint that the mole fractions of all components should sum to one. Since the sweep gas inlet stream consists of pure nitrogen, yG,i,in ) 0, for helium and argon. Hence, eq A.1 reduces to the following equation: Fmol yG,i,out )
Gmol 1-
(1 - exp[-Ri]) Fmol Gmol
yF,i,in
(A.3)
exp[-Ri]
Equation A.3 is used to make an estimate of the uncertainty in the value of yG,i,out for helium and argon calculated with the numerical model. Since yG,i,out is a function of the average molar flow rates Fmol and Gmol, the uncertainty in yG,i,out is given by the following equation:15 δyG,i,out ) (
((
∂yG,i,out ∂Fmol
) ( 2
δFmol
+
∂yG,i,out ∂Gmol
)) 2
δGmol
(A.4)
In this case it is assumed that the uncertainties in Fmol and Gmol, given by δFmol and δGmol, are independent. The uncertainty in yG,N2,out follows from the requirement that the mole fractions should sum to zero and is therefore equal to Figure 12. Comparison between inlet feed pressure and mole fractions obtained in the experiments (top) and with the numerical model (bottom). Results are shown for a FricDiff unit with Sika-R 0.5 IS porous screen that is operated cocurrently. Table A1. Uncertainties in Experimentally Measured Values measured value
unit
uncertainty
Fmass,inlet/Fmass,outlet
kg/h
Gmass,inlet/Gmass,outlet
kg/h
δFmass ) ((0.005 ( 0.001/ Fmass (measured))Fmass (measured) δGmass ) ((0.005 ((0.001/ Gmass (measured))Gmass(measured) δyF,i,out ) (0.01yF,i,out (measured) δyG,i,out ) ( 0.01yG,i,out (measured)
yF,i,out yG,i,out
and argon in the sweep gas outlet stream can be approximated by using eq 6, which is written as
δyG,N2,out ) ( √(δyG,He,out)2 + (δyG,Ar,out)2
In eq A.4, average molar flow rates are used (averaged over the inlet and outlet values) for which the uncertainty is given by 2 2 δFmol ) ( √(1/2δFmol,in) + (1/2δFmol,out)
Fmol yG,i,out(Fmol, Gmol) )
1-
Gmol
yF,i,in + exp[-Ri] 1-
1-
where
Fmol Gmol
Fmol Gmol
(A.7)
n yiMw,i. There is an uncertainty in the where Mw,mix ) ∑i)1 measured mass flow rates and in the measured outlet mole fractions. Furthermore, it is assumed that the feed mixture and sweep gas that enter the module have an exact composition without uncertainties (i.e., δyG,i,in ) 0 and δyF,i,in ) 0). The uncertainty in the molar flow rate can therefore be expressed as
(1 - exp[-Ri]) Fmol
(A.6)
with the expression for δGmol derived in a similar way. In the experiments, mass flow rates are measured and not molar flow rates. By using eq A.7, molar flow rates can be calculated when mass flow rates are known. Fmol ) Fmass /Mw,mix
Gmol
(A.5)
yG,i,in
(A.1) δFmol )( Fmol
exp[-Ri]
where
((
δMw,mix Mw,mix
) ( )) 2
+
δFmass Fmass
2
(A.8)
7704
Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009
δMw,mix )
n
∑ (Mw,iδyi)2 i)1
To calculate δyG,i,out for helium and argon with eq A.4, the derivatives ∂yG,i,out/∂Fmol and ∂yG,i,out/∂Gmol have to be determined. These derivatives can be calculated from eq A.3, resulting in ∂yG,i,out
)
(FmolGmol - (FmolGmol + β(Gmol - Fmol)) exp[-Ri]) Fmol(Gmol - Fmol exp[-Ri])2
yF,i,in
Fmol((Gmol2 + β(Gmol - Fmol)) exp[-Ri] - Gmol2) Gmol2(Gmol - Fmol exp[-Ri])2
yF,i,in (A.9)
Values for ∂yF,i,out/∂Fmol and ∂yF,i,out/∂Gmol can be derived in a similar way. Equation 12 gives a good approximation of the separation in a FricDiff module that is operated cocurrently. The uncertainty in yG,i,out for helium and argon again follows from eq A.4, but in this case the derivatives ∂yG,i,out/∂Fmol and ∂yG,i,out/∂Gmol are given by )
∂Fmol (FmolGmol - (FmolGmol + β(Fmol + Gmol)) exp[-Ri]) Fmol(Fmol + Gmol)2 ∂yG,i,out
yF,i,in
)
∂Gmol Fmol((Gmol2 - β(Fmol + Gmol)) exp[-Ri] - Gmol2) Gmol2(Fmol + Gmol)2
yF,i,in (A.10)
In deriving eq A.10, yG,i,in was again set to 0 for helium and argon, and the parameter β is given by Ri )
(A.12)
δGmol,out ) √(δFmol,in)2 + (δGmol,in)2 + (δFmol,out)2 (A.13) Literature Cited
)
∂Gmol
∂yG,i,out
Gmol,out ) Fmol,in + Gmol,in - Fmol,out
Since the uncertainties in Fmol,in, Gmol,in and Fmol,out are known and assumed independent, the uncertainty in Gmol,out can be calculated as
∂Fmol
∂yG,i,out
Again values for ∂yF,i,out/∂Fmol and ∂yF,i,out/∂Gmol for co-current operation of the module can be found using a similar approach. The uncertainty in the calculated molar flow rate is estimated from a molar balance over the module, resulting in
2πLDincε
(
) (
)
1 1 1 1 + )β + 2 F G F G ln(rout /rin)τ mol mol mol mol (A.11)
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ReceiVed for reView November 14, 2008 ReVised manuscript receiVed February 18, 2009 Accepted June 23, 2009 IE801738V
