Experimental Measurement of Membrane Residue Curve Maps

Jul 23, 2013 - ... Private Bag 3, WITS 2050, Johannesburg, South Africa ... Membrane residue curve maps (M-RCMs) are a useful graphical tool developed...
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Experimental Measurement of Membrane Residue Curve Maps Naadhira Seedat, Pretesh Parag, Dirren Govender, Mark Peters,* Diane Hildebrandt, and David Glasser Centre of Material and Process Synthesis, School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Private Bag 3, WITS 2050, Johannesburg, South Africa S Supporting Information *

ABSTRACT: Membrane residue curve maps (M-RCMs) are a useful graphical tool developed to better understand and design membrane separations [Peters, M.; Kauchali, S.; Hildebrandt, D.; Glasser, D. Ind. Eng. Chem. Res. 2006, 45, 9080]. The maps, initially developed theoretically, track the compositional change of the retentate with time during batch permeation. This article looks at the design and construction of an experimental apparatus for the measurement and existence of M-RCMs. For demonstration purposes, gas separation was chosen for the basis of the membrane separation system. The gas mixture, which consisted of syngas with carbon dioxide as an impurity, was passed through a polyethylene membrane. The theoretical transport of gases moving through the membrane was developed by a solution diffusion model that fitted the experimental system adequately. The experimental setup was successfully designed and operated; the results obtained correlate to the theoretical models chosen. In addition, the experimental apparatus can be used for the validation of M-RCMs for any type of membrane and mixture of gases.

1. INTRODUCTION Separation systems are of great importance in industrial chemical processes. In recent times, alternative methods of separation have been developed and implemented, with membranes being one of the alternatives. Membrane separation processes have been proven to be a more economical and efficient alternative to distillation in certain applications, such as gas separations. Residue curve maps are a graphical tool, originally used in distillation design, and have produced numerous forms of shorthand design methods in the understanding and designing of complex distillation systems. 1,2 Although membrane separations are physically different to operate and design than distillation systems, residue curve maps for membrane systems were developed by Peters et al.3 to better understand and design membrane systems. Although Peters et al.3 developed the theoretical groundwork for membrane residue curve maps (M-RCMs) and later showed their applications,4 the main aim of this paper is to prepare, measure, and evaluate an experimental technique for generating M-RCMs. Membrane separation technology is a rapidly growing development. Membrane systems have proved to reduce costs in the sense that they require low energy and save on capital costs.5 The advantage of implementation into industry is that it they are easily installed, operated, and require limited space. Membrane separation is also an equilibrium-independent process that can reach higher yields than conventional separations processes.3 Membrane gas separation, for example, is gradually becoming a more important separation technology in areas such as air separation, hydrogen recovery, and natural gas purification.6 In this regard, membrane separation is a conceptually simple process. However, there are many design decisions with respect to selection of operating conditions, module configurations, and suitable membrane materials that need to be made.6 M-RCMs not only introduce a novel graphical method of designing membrane systems but also provide insight into © 2013 American Chemical Society

optimization and attainability. The graphical means are derived on the basis of mass balance constraints (see section 2). Initially the use of residue curve maps was constrained by the understanding of equilibrium-based processes, but the construction of the maps is based on material balances that enable the understanding of equilibrium- and nonequilibrium-based systems that can be extended to understanding kinetic systems.3 This work is focused on the validity of residue curve maps for membrane separations. For reasons of simplification, the validation of M-RCMs is done using a nonreactive experimental system. Membrane residue curves will be constructed on the basis of experimental data, and their validity to membrane separations will be analyzed. The validation of membrane residue curves is conducted on a gas separation system, but this confirmation will be implied to other forms of membrane systems. The gas separation involves the diffusion of a gaseous batch feed (retentate) at a high pressure through a membrane into a low pressure permeate side.

2. THEORY The generation of membrane residue curve maps can adequately be understood by considering a batch system of gases enclosed by an appropriate membrane, as shown in Figure 1. It is assumed that a constant pressure difference exists across each side of the membrane. The pressure difference is constant for a specific experimental run but differs from one run to the next (see sections 3 and 4). A higher pressure occurs on the feed side (retentate, R), causing the gases to permeate through the membrane and move into the permeate side (P) of the system. The permeate is removed from the system as it is formed, allowing Received: Revised: Accepted: Published: 11142

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Such a derivation was first analyzed by Peters et al.3 It is important to note that eq 4 is applicable to a gas system, but it renders exactly the same result as the derivation of a liquid system. 2.2. Permeation Model. In membrane separations, the relationship between retentate and permeate compositions is usually described by a nonequilibrated permeation model. Many experimental models exist in describing permeation, but for purposes of simplicity, the relationship of retentate and permeate compositions are assumed to be described by a basic fundamental permeation model, described by eq 5 and adapted from Wijmans and Baker.7 Figure 1. Batch membrane gas system with the retentate, R, at a higher pressure than the permeate, P.

Ji =

dR = Ṗ (1) dt where R refers to the amount of retentate [mol], Ṗ refers to the amount permeated through the membrane per unit time [mol/s], and t refers to time [s]. Performing a component balance on the system, with respect to component i, further relates the permeate flow to the change in retentate. −

d(Rxi) dt

dx dR Pẏ i + xi +R i =0 dt dt

(5)

l

where Ji refers to the molar transport of component i through the membrane [mol/(s m2)], P′i is the permeation coefficient of component i [(mol m)/(s m2 Pa)], l refers to the membrane thickness [m], and PR and PP are the total pressures [Pa] of the retentate and permeate, respectively. According to Wijmans and Baker,7 eq 5 is specific to gas separation and has been derived from first principals. The experiment mentioned in this review is assumed to be conducted under vacuum conditions, resulting in a permeate pressure of zero. This assumption simplifies eq 5 accordingly:

separation to continue until all the high pressure fluid has permeated through the membrane. 2.1. Mass Balance. The system in Figure 1 is based on the assumptions that the gas components are reactive neither with each other nor with the material of the membrane and system. Performing an instantaneous overall mass balance in Figure 1 produces

Pẏ i = −

P′i (PR xi − PPyi )

Ji =

P′i PR xi l

(6)

In essence, the transport of a gas through any membrane media is the amount of component i permeating through a specific membrane area per unit of time. This is mathematically described by Ji =

(2)

Pẏ i (7)

A

where A refers to the effective membrane surface area [m]. By equating eqs 6 and 7 with respect to components i and j and then dividing the expressions by each other, we produce a relationship between the retentate and permeate compositions: yi x = αijM i yj xj (8)

(3)

Equation 2 refers to a component balance over the system where x and y refer to the compositions of the retentate and permeate, respectively, with respect to component i. The algebraic simplification of eq 2 is represented by eq 3. The combination of eqs 1 and 3 results in the isolation of the retentate and is represented by the ordinary differential equations

where αM ij is the permeability coefficient of component i with respect to the permeability coefficient of component j

dxi = (xi − yi ) i = 1, 2, ..., n − 1 (4) dτ where n refers to the number of components within the system and τ is a dimensionless variable with respect to time and is equated to −ln (R0/Rt) where R0 and Rt refers to the retentate initially and at time t respectively. Equation 4 tracks the retentate composition with respect to time for a given set of boundary conditions (feed compositions, x0). A full derivation of eq 4 can be found in Appendix A. By definition, a membrane residue curve map (M-RCM) is a plot of the change in composition of the retentate in a batch still over time.3 This definition is mathematically described by the derived equations 4. It should also be noted that eq 4 is independent of the permeate removal rate, Ṗ , and is only dependent on the change in composition. Therefore, eq 4 can be used to produce a membrane residue curve for a given feed point. A series of curves for different feed points will result in a membrane residue curve map (M-RCM).

αijM =

P′i P′ j

Equation 8 is sufficient for the solution of eq 4, which is then used for the graphical generation of M-RCMs. 2.3. Permeability Coefficients. The permeability of a gas moving through a membrane media is usually a series of correlations that account for system properties and material. A constant relative permeability for each component is assumed for the system in this experiment. This ideal assumption is made for the simplicity of describing the system. The gas system used in this experiment is composed of three gases: hydrogen (A), carbon monoxide (B), and carbon dioxide (C). The constant relative permeabilities at 25 °C of these components relative to the carbon monoxide flux rate are represented as M αAB = 31

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and

M αBC = 60

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M Figure 2. Ideal M-RCM of hydrogen (A), carbon monoxide (B), and carbon dioxide (C) system with αM AB = 31 and αBC = 60.

carbon dioxide is known as the stable node because all curvature moves toward this point. Carbon dioxide has the lowest permeation rate, thus moving through the membrane the slowest; hence, the retentate becomes richer in carbon dioxide. The point of pure carbon monoxide is referred to as a saddle point because trends move toward and away from this point (curvature occurs) but does not reach pure carbon monoxide. It is important to note that the magnitude of the intermediate permeability directly affects the level of curvature in the vicinity of the pure carbon monoxide. Larger permeabilities will produce larger curvature because the trend will move closer to this point before moving toward the slowest permeating component. This curvature is indicative to harder separations due to the intermediate permeation rate of this component.

These relative permeabilities are based on mass transfer through a natural rubber membrane and are adapted from Geankoplis8 and Michaels and Bixer.9 Component A has a high permeability with respect to the other components and will move the fastest through a natural rubber membrane. Component B will behave as an intermediate permeater, and component C will move very slowly through the membrane. Permeability has a large dependence on the partial pressure of the system. The driving force of the system is purely based on the partial pressure of the system; this relation is represented by eq 5. Partial pressure is directly proportional to the concentration of the system. The concentration of the system continuously changes with time, and therefore, the partial pressure of the system will continuously change. 2.4. Theoretical Curves. Figure 2 represents a theoretical M-RCM of a hydrogen, carbon monoxide, and carbon dioxide system. The construction of the map was conducted by solving eqs 4 and 8 simultaneously and using the constant relative permeabilities, with respect to natural rubber, as mentioned before. The axes form the boundaries of the mass balance triangle (MBT) which is the region of all physically obtainable compositions in a ternary system. The curves plotted in Figure 2 within the MBT track the change in retentate (nonpermeate) composition with time. The trend in these curves promotes rapid movement from the fastest permeater (hydrogen) toward the slowest permeating component (carbon dioxide). As the curves move closer to pure carbon monoxide, the intermediate component, the trend becomes more curved in nature. 2.5. Stationary Points. The pure component points are known as stationary points. At these points the composition of a component is equal on both sides of the membrane; that is, xi = yi for component i. There exist three stationary points in Figure 2: pure hydrogen (labeled A), pure carbon monoxide (B), and pure carbon dioxide (C). The stationary point of pure hydrogen is known as an unstable node. This name is given because all curves move away from this point, as hydrogen has the highest permeation rate and therefore moves through the membrane faster than the other components. Thus, the retentate is rapidly depleted of hydrogen. The point of pure

3. EXPERIMENTAL SECTION An apparatus was designed to measure various profiles on a membrane residue curve map (M-RCM). This apparatus operates under batch conditions, and this particular setup will allow one to analyze the change in retentate composition of a gas mixture. This will in turn allow one to generate profiles on a M-RCM for different membrane types and different gas mixtures under a host of conditions; this forms a part of our future research. The current research is the development of the method and apparatus to generate M-RCMs. The experiments run isothermally at room temperature of 25 °C. 3.1. Apparatus and Materials. The apparatus consists mainly of two plastic containers, a vacuum pump, a variable voltage controller, a PVC membrane support covered with a polyethylene membrane, a water manometer, a gas syringe, a flow bubble meter, silicone piping, two computer fans, shut-off valves, and control valves. It is noted that for this particular experiment only a polyethylene membrane material was utilized; however, the apparatus is not selective and any other membrane type could be used and tested. The plastic containers used are sold with the intention of saving space by storing clothing items in the bag and evacuating the air by means of a vacuum cleaner. To serve this purpose, the bag has to be airtight (having a very low or rather negligible 11144

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Figure 3. Plastic container used to hold the gas mixture.

permeation rate) and be well sealed. These properties were taken advantage of; for the purpose of the experiment, one container is to be filled with a given volume of gas and thereafter be evacuated through a membrane by means of a vacuum pump to the next container. These containers were essential to ensure the system is free from leaks of potentially harmful gases to the environment. Furthermore, it is essential to the quality of the results that a negligible amount of gas moves through the walls of the bag, thus ensuring that all the gas is moving through the membrane. This assumption was easily validated, prior to experimentation, by filling each bag with hydrogen and tracking the pressure change over a period of 72 h. The volume change was also measured. It was found that, after 72 h, the pressure dropped by 10 kPa, and there had been no significant change in the volume of the bags. This is sufficient indication that a negligible amount of gas was lost through the walls of the bag, especially when one considers the fact that the duration of an experimental run is no more than a few (1−2) hours. The containers had to be further tailored for the purposes of the experiment. These alterations include the addition of three ports on each container. For each container, two of the ports were for the purposes of piping, and the third was for the electrical wiring required for the computer fans, which were required for agitation of the gas mixture. The agitation was required to obtain a good representative sample, as there are large differences in the densities of the gases (hydrogen rises, whereas carbon dioxide sinks). There also exists a port on each container that was sealed by means of a rubber septum; this served as the sampling point where a gas syringe could be inserted to obtain samples at a given time. Figure 3 shows a single container with the various ports The two-stage vacuum pump used was controlled by means of a variable voltage controller and a control valve controlling the inlet to the vacuum pump. The variable voltage control allows one to limit the voltage fed to the pump, thus directly affecting the shaft power and hence allowing one to control the pump speed. The control valve serves as a fine controller where one can adjust the pressure across the membrane (observed by means of a water manometer). The upstream pressure of the bag when initially filled with gases was a few inches of water above the atmospheric pressure of Johannesburg (approximately 850 kPa). Though the actual upstream pressure of the system is lowered during a run, knowledge of it is not necessary to the results. What is important is to maintain a stable difference in pressure (between upstream and downstream) so as to ensure reliable permeation. A manometer reading of about 0.5 m of water was maintained. This setup is displayed in Figure 4. The membrane support structure was made from PVC (poly(vinyl chloride)) piping. The reason for this material choice was the strength (will not collapse under vacuum conditions) and also was due to the ease by which various

Figure 4. Vacuum pump and associated control mechanisms.

pieces of piping could be connected and sealed. It is noted that this type of piping is commonly used in plumbing; thus, the material, connectors, and sealing adhesives were easily obtained. Holes (3 mm in diameter) were drilled into the structure to provide sites where the gas could flow through. These holes were covered by the membrane and the ends sealed by means of insulation tape. The total area of membrane available for permeation was found to be approximately 32 cm2. The idea was to connect the vacuum pump to the support structure, which would in turn provide a pressure difference across the membrane. This created the driving force for the gas in the container to permeate through the membrane, into the pipe structure, and through to the next container. Figure 5 depicts the membrane support structure created. The experimental setup as a whole is depicted in Figure 6. Table 1 serves to annotate the various pieces of the apparatus by associating labels to the circled numbers in Figure 6. In order to generate data with a high level of confidence, there were certain aspects of the experimental setup that had to minimized: • There can be no punctures in the membrane. These were checked for by using a simple soap bubble check method. • No gas losses can occur through the bag walls. This was done (as described earlier) by filling the bags with hydrogen gas and monitoring the change in pressure and volume. • The vacuum pump operates as designed. • Sufficient mixing in the bag was needed to avoid stratification of the gases. This was achieved by the inclusion of the computer fans. 3.2. Experimental Procedure and Conditions. Using the above experimental setup (Figure 6), the following procedure was carried out at ambient room conditions: (1) Initially, connect valve V-4 to the vacuum pump. 11145

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Figure 5. Membrane wrapped around PVC piping (membrane support).

Figure 6. Experimental setup (refer to Table 1 for key).

Table 1. Labeling key for Figure 6 1 2 3 4 5 6 7 8 9 10 11 12 13 valves (1, 2, 3, 5, 8, 9, and 10) valves (4, 6, 7, and 11)

(8) All valves (both control and shut-off valves) are fully closed. (9) To obtain a three component system, carbon dioxide, carbon monoxide, and hydrogen are filled from separate gas cylinders (10, 11, and 12) into the bag by introducing the gas by means of a mass flow meter. (10) Once the container is filled with the desired amount of the three components, valves V-8, V-9, and V-10 are closed. (11) Thereafter, valve V-4 is closed. (12) The container and associated valves are double checked for leaks by means of soapy water and soap bubbles. (13) The computer fans (2) are then switched on to promote agitation. (14) After a few minutes, an initial sample is taken for the septum port (4) by means of a gas syringe and analyzed using gas chromatography. It is noted that all samples will be taken from the bag containing the retentate. (15) It takes 13 min for the gas chromatograph to analyze the sample. (16) Eight minutes into the sample analysis, the run is started so that the next sample can be taken in 5 min. (17) To start the run, one must ensure that all valves except valves V-1, V-2, V-3, and V-6 are closed. (18) The pump must then be switched on and the control valve (V-11) opened slightly and then adjusted so as to obtain the appropriate pressure. (19) Once the analysis of the initial sample is complete, another sample of retentate is taken (approximately 5 min into the run) and injected into the gas chromatograph for analysis. (20) Thereafter, careful attention is paid to the pressure across the pump, and samples are taken every 16 min.

inlet ports for piping and electrical wire 12 V (dc brushless fans for agitation) drilled PVC piping structure (membrane support) and membrane septum port for sampling “zip-lock” seal bubble meter 12 V dc electrical source vacuum pump water manometer carbon dioxide gas cylinder carbon monoxide gas cylinder hydrogen gas cylinder gas container (vacuum plastic bag) control valves shut-off valves

(2) Fully open valves V-4 and V-11, switch the vacuum pump on, and evacuate contents of the container to the vent system. This is done to ensure all the container contents are removed. (3) Close valves V-4 and V-11 and then reconnect the vacuum pump to valve V-6. (4) Fully open valves V-6 and V-11, switch the vacuum pump on, and evacuate contents of the container to the vent system. (5) Ensure that the vacuum pump (8) is switched off. (6) Ensure that the “zip-lock” seals (5) are sealed properly. (7) Ensure that valves V-1 and V-2 are fully opened because, once the experiment is started, they cannot be accessed. 11146

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(21) Once the run was complete, the experiment valves V-3, V-6, and V-11 are closed once the pump is switched off. (22) The experiment could be run again in the other direction by simply connecting the pipe from valve V-3 to valve V-5 and the pipe from valve V-6 to V-4. (23) Thereafter, one would adopt steps 12−21. The only difference would be the fact that one will be dealing with valves V-5 and V-4 instead of valves V-6 and V-3. (24) Once the runs are complete, one should evacuate the container by connecting the vacuum pump to either valve V-4 or valve V-6, and the outlet of the pump should be directed to the vent system in the lab. 3.3. Sample Analysis. The samples that were collected were analyzed by means of gas chromatography. In particular, a Hewlett-Packard 6980A (Agilent 6890) gas chromatograph was used. The column was of a packed type (carboxen packing). The following specification applies to the gas chromatograph utilized. Support Q: 80/100 mesh. Length/OD 2 m × 1/8 in. Poropak. For the purposes of this experiment, a thermal conductivity detector (TCD) was used because it had a universal selectivity.10 This implied that it can be used for a broad spectrum of components. The carrier gas used was argon, and samples were analyzed at an oven temperature of 35 °C. Furthermore, the samples were analyzed on the basis of prior calibration done by analyzing a known composition of a gas mixture. The calibration was done to an accuracy of 0.5%. Thereafter, when the data obtained was used, the relevant response factors could be found and later used to determine the concentrations of an unknown gas mixture. This was done by relating the response factors and various peak areas obtained for the various components.

Figure 7. Residue curve map depicting the theoretical data against experimental data for the four regions under analysis. The direction of data progression over a run is from the feed point (for a run) moving toward pure C. (See Supporting Information for experimental data.)

runs were required to reproduce the curve near pure B and the middle residue curve. The reason for a limited range in data obtained per run was the limited volume of the sample that could be held in the plastic bags. (An analogy could be drawn to a batch boiling to reproduce distillation residue curve maps. A limited amount of data could be obtained from the batch still as all the liquid boiled off, and the liquid would need to be replenished in order to extend the data range.) 4.1. Experimental Behavior. In the experimental data plotted in Figure 7, progress from the respective feed points toward pure carbon dioxide in all the regions is analyzed. This behavior is expected because all theoretical data moves toward the slowest moving component (the stable node). The largest deviations of experimental data points from theoretical data in all regions occurr at the last experimental point for each run. These deviations are attributed to the nearvacuum conditions that occur toward the end of the experimental run, when most of the gas material had permeated through the membrane. The near-vacuum condition, formed by the use of the vacuum pump, inhibits a high accuracy result because the sample taken may not be a true reflection of the retentate under this state. It is interesting to note the behavior of the middle residue curve in Figure 7. Closer inspection of this particular curve reveals that the intermediate component, namely carbon monoxide, labeled B, does not exhibit any drastic change in composition during an experimental run. This can be deduced from the linear and parallel nature of the data relative to the intermediate diagonal axis. In other words, if one were to draw a straight line through these data points (on the middle profile) parallel to the diagonal axis of the MBT, it will be noticed that the data points agree quite well with this line; this indicates that there is no change in carbon monoxide composition with time. Thus, although there is a change in retentate composition with time, it is apparent that this change, for the most part, only occurs due to the other two species, namely hydrogen and carbon dioxide. When the permeability factors of each of the components are considered, it is clear that hydrogen is the

4. RESULTS AND DISCUSSION In order to verify the theoretical curves, the above-mentioned procedure was conducted. The theoretical curves were generated by the use of the above-mentioned theory and assumptions. The data obtained was plotted against the theoretical curves corresponding to the same feed point chosen on the theoretical curve such that both sets of data could be compared, as in Figure 7. Four regions of the residue curve map were chosen for analysis, namely in the vicinity of the three stationary points, and the fourth was on a single residue curve in the middle region of the map. The stationary points allowed for the investigation of the behavior of the experimental data near the nodes and whether the results obtained correlated to the behavior predicted by theory. The above regions of the residue map were chosen such that a wide area of the residue curves could be represented, ensuring that the results obtained could be inferred at any point on the map. A typical run took place over about 1−1.5 h, and when the duration between sampling was used as discussed before, about 5−8 samples were taken per run. It should be noted that a single experimental run (from filling the bag with “feed” gas to evacuating that gas through the membrane) only produced data over a small range of the entire residue curve within the map. In other words, to cover a wider range of a residue curve, multiple runs were needed. (All runs moved in the same direction on the map: from the feed point toward pure C.) This can be seen in Figure 7, where multiple 11147

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Figure 8. Zoomed in representation of experimental and theoretical data in the vicinity of the stable node, carbon dioxide. The order in which the samples were taken is indicated numerically.

The experimental data in the region of the stable node, carbon dioxide (point C), is represented in Figure 7, and a zoomed in version is given in Figure 8. It is worth mentioning that in this particular instance (where the feed location was close to pure carbon dioxide), the data points do not move sequentially toward the stable node. This phenomenon has no significance to the experimental procedure or apparatus employed but rather can be attributed by the possible error in the gas chromatography detection. During the membrane separations, the change in the retentate composition is fairly small, as can be seen in Figures 7 and 8. Composition detectors such as gas chromatographs have difficulty in detecting such small changes, hence the cause of the abovementioned phenomena. There could possibly be a form of error in the sampling of the retentate due to the degree of agitation in the retentate, which could also result in this phenomenon. However, the bulk of the experimental data obtained fit the theoretical data with minimal or no deviations. With that said, the data fits the theory with good accuracy. The data is produced close to the stable node because all data will move toward pure carbon dioxide. This is expected because according to theory carbon dioxide permeates the slowest, remaining in the retentate. Figure 7 depicts the experimental data obtained for different feed points on the same theoretical residue curve near the saddle node. The component near the saddle node has an intermediate permeation rate (between the components of the stable and unstable node); thus, separation is difficult, resulting in high curvature. The analysis of a residue curve near the saddle point is important as it has predicted difficult-toreproduce experimental data around the curvature of residue curves with minimal deviations. As seen in the Figure 7, the data around the curvature produces very small ranges, though the data still fits the predicted trend and the points do move toward the stable node, thus reproducing the curvature. The predicted large deviation from theoretical data did not occur, and the data produced

fastest permeating species. It can thus be deduced that because the retentate composition of hydrogen is decreasing, and that of carbon monoxide is constant, the net effect is that the retentate becomes continually richer in carbon dioxide. It is important to note that the magnitude of the carbon monoxide permeability is within a sufficient range from each of the other components and therefore does not change drastically under the nonideal conditions. As stated before, a single feed point produces a small range of data on the residue curve; hence, it was recommended we choose the successive feed point to be the preceding run’s last data point. Due to inaccuracies in the mass flow meter, a feed point close to the chosen point on the same residue could be obtained. Therefore, the data obtained is a representation of the behavior down the profile for feed points on the curve. It can be seen in Figure 7 that the trend of each feed point behaves the same, and that is the experimental data move toward pure carbon dioxide, the stable node, with respect to time. It is also important to note that the range of data decreases down the curve per a constant volume of feed. This result is due to the expected behavior around pure carbon dioxide. This phenomenon is explained in section 4.2. 4.2. Stationary Point Behavior. For the evaluation of experimental stationary points effects on M-RCM behavior, points were taken close to pure hydrogen (the unstable node), pure carbon dioxide (the stable node), and pure carbon monoxide (the saddle point). These feeds and associated experimental data are represented in Figures 7 and 8. The experimental results obtained from the evaluation of the profile in the vicinity of the unstable node (hydrogen, point A) are depicted in Figure 7. It can be seen that the experimental data fits the theoretical data with a high accuracy and minimal deviations. The data produced is over a larger range in comparison to that of the other data because separations are usually easier at points close to the unstable node. This trend is due to the fast permeation of the large quantity of hydrogen close to the unstable node. 11148

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around the curvature fits the trend with small errors. In accordance with Figure 7, the errors obtained in the saddle region are greater than the other regions analyzed due to the increased difficulty of separation in this region. The accuracy of experimental data against theoretical data proves the validation of the concept of M-RCM and its applications thereof. This also implies that the experimental setup and procedure were successfully designed and operated. Furthermore, this accuracy validates the assumption of a basic flux model in the prediction of experimental data and more importantly validates the use of residue curve maps in membrane separations. 4.3. Experimental Error Analysis. Apart from small deviations in the data represented in Figures 7 and 8, the data fits the theory with good accuracy. The data that did result in relatively large deviation are accounted for by the following possible experimental errors: • There may be uncontrollable and undetectable leaks. • There may be leaks within the system through micropunctures in the membrane. • The system was not at a complete vacuum condition. • The sample gas permeated through the apparatus. • The permeability coefficient changed with the change in partial pressure of the retentate. • The errors produced by the gas chromatography affected the results because the change in composition with time was small. • The agitation by the small computer fans was not adequate for mixing purposes; hence, samples taken were not completely true to the retentate. These errors were not quantified because the occurrence of the possible errors could not be controlled. Although these errors could have occurred, they can be considered miniscule because the results produced fitted the theory effectively.

This article provides substantial data that correlate to the theoretical models, and therefore, it can be concluded that MRCMs have been validated, and that the initial proposals put forward by Peters et al.1 regarding the nature and behavior of these maps is verified. In addition, the experimental apparatus designed can be used for the validation of M-RCMs for any type of membrane and mixture of gases. With this initial validation in place, future experimentation can use this to research the behavior of more complicated gas mixtures through various types of membranes.

5. CONCLUSION AND RECOMMENDATIONS An experimental setup was successfully designed and operated for the measurement of M-RCMs. For the purpose of demonstrating the method, a relatively simple separation was employed. This was done by separating a gas mixture consisting of synthesis gas (carbon dioxide and hydrogen) and a carbon dioxide impurity through a polyethylene membrane. Four different regions were chosen for discussion, namely the three stationary points and a single curve in the middle of the residue curve map. All of the data points obtained in each region tended toward the stable node (carbon dioxide) in accordance with theory that indicated carbon dioxide was the slowest permeating species in the gas mixture. The bulk of the experimental data obtained fit the theoretical data with a high level of accuracy, although possible causes of error have been mentioned. The data produced at pure hydrogen, the unstable node, were over a larger range in comparison to that of the other data because separations were usually easier at points close to the unstable node. Data points closer to the stable (carbon dioxide) node correlated to theory as carbon dioxide, which permeates the slowest, remained in the retentate. Data at the saddle point (carbon monoxide) showed that deviations increase from theory as the data points move closer to curvature. This trend is best explained by the difficulty of separation associated with saddle point behavior; therefore, predictions of data around this point are associated with a certain level of error tolerance.

where k is a constant. Hence

■

APPENDIX A: DERIVATION OF THE RESIDUE CURVE EQUATION Refer to Figure 1. Equation 2 can be expanded using the wellknown chain-rule: d d d 0 = Pẏ i + (Rxi) = Pẏ i + R (xi) + xi (R ) dt dt dt Substituting in eq 1 ⎛ d ⎞ d d 0 = yi ⎜ − (R )⎟ + R (xi) + xi (R ) ⎝ dt ⎠ dt dt

Rearranging d

R dt (xi) d

− d t (R )

= (xi − yi )

And canceling −R

d (xi) = (xi − yi ) dR

Now, it can be shown that d ⎛ ⎛ k ⎞⎞ 1 ⎜ln⎜ ⎟⎟ = − d R ⎝ ⎝ R ⎠⎠ R

⎛k⎞ dR d ln⎜ ⎟ = − ⎝R⎠ R

Let

dτ = −

dR R

...( ∗)

Thus

d (xi) = (xi − yi ) dτ This is the residue curve equation, as given in eq 4. Further manipulation is needed to define τ. By rearranging eq 1 and substituting into (*) dτ =

P dt R

or τ=

P t R

Thus, at t = 0, τ = 0, and R = R0, hence

⎛ k ⎞ ln⎜ ⎟ = 0 ⎝ R0 ⎠ 11149

dx.doi.org/10.1021/ie302955c | Ind. Eng. Chem. Res. 2013, 52, 11142−11150

Industrial & Engineering Chemistry Research

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or Thus

⎛R ⎞ dτ = d ln⎜ 0 ⎟ ⎝ Rt ⎠ or

⎛R ⎞ τ = ln⎜ 0 ⎟ ⎝ Rt ⎠ as discussed.

ASSOCIATED CONTENT

S Supporting Information *

Stationary point experimental data, saddle point experimental data, and middle region curve experimental data. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

(1) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001. (2) Fien, G. A. F.; Liu, Y. A. Heuristic Synthesis and Shortcut Design of Separation Processes Using Residue Curve Maps: A Review. Ind. Eng. Chem. Res. 1994, 33, 2505−2522. (3) Peters, M.; Kauchali, S.; Hildebrandt, D.; Glasser, D. Derivation and Properties of Membrane Residue Curve Maps. Ind. Eng. Chem. Res. 2006, 45, 9080−9087. (4) Peters, M.; Kauchali, S.; Hildebrandt, D.; Glasser, D. Application of Membrane Residue Curve Maps to Batch and Continuous Processes. Ind. Eng. Chem. Res. 2008, 47, 2361−2376. (5) Rautenbach, R.; Albrecht, R. Membrane Processes; John Wiley: New York, 1989. (6) Pettersen, T.; Lien, K. M. Design Studies of Membrane Permeator Processes for Gas Separation. Gas. Sep. Purif. 1995, 3, 151−169. (7) Wijmans, J. G.; Baker, R. W. The Solution-Diffusion Model: A Review. J. Membr. Sci. 1995, 107, 1−21. (8) Geankoplis, C. J. Transport Processes and Unit Operations, 3rd ed.; Prentice-Hall: Upper Saddle River, NJ, 1993; p 760. (9) Michaels, A. S.; Bixler, H. J. Flow of Gases Through Polyethylene. J. Polym. Sci. 1961, L, 413−439. (10) Sheffield Hallam University. Gas Chromatography. http:// teaching.shu.ac.uk/hwb/chemistry/tutorials/chrom/gaschrm.htm (accessed May 26, 2011).

k = R0

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE

Symbols

A = effective membrane surface area [m] A = hydrogen B = carbon monoxide C = carbon dioxide Ji = flux of component i through membrane [mol/(s m2)] l = membrane thickness [m] n = number of components Ṗ = permeate rate through membrane [mol/s] P′i = permeability coefficient of component i [(mol m)/ (s m2 Pa)] PP = permeate pressure [Pa] RR = retentate pressure [Pa] R = retentate holdup [mol] R0 = retentate holdup at t = 0 [mol] Rt = retentate holdup at time t [mol] t = time [s] x = retentate molar composition x0 = retentate feed molar composition y = permeate molar composition Greek Letters

αM ij = ratio of membrane permeabilities (selectivity) τ = dimensionless time

Subscripts

i = component i j = component j P = permeate R = retentate Superscripts

M = membrane Abbreviations

MBT = mass balance triangle M-RCM = membrane residue curve map PVC = poly(vinyl chloride) 11150

dx.doi.org/10.1021/ie302955c | Ind. Eng. Chem. Res. 2013, 52, 11142−11150