Gas−Liquid Desorption through Blind-Ended Microporous Hollow

Gas-Liquid Desorption through Blind-Ended Microporous Hollow. Fibers for an Intravascular Artificial Lung. Harihara Baskaran, Vladislav Nodelman, and ...
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Ind. Eng. Chem. Res. 1998, 37, 4142-4151

Gas-Liquid Desorption through Blind-Ended Microporous Hollow Fibers for an Intravascular Artificial Lung Harihara Baskaran, Vladislav Nodelman, and James S. Ultman* Biomolecular Transport Dynamics Laboratory, Department of Chemical Engineering, The Pennsylvania State University, 106 Fenske Laboratory, University Park, Pennsylvania 16802

The Penn State Intravascular Lung (PENSIL) consists of thousands of microporous, hollow, polypropylene fibers, each with one end sealed and the other end attached to an oxygen (O2) supply tube which is implanted in a major vein. Since diffusion resistance along the inside of the blind-ended fibers is significant, a hypobaric O2 supply might enhance the desorption of carbon dioxide (CO2) from blood. Therefore, CO2 desorption through small-scale PENSIL modules containing 14 polypropylene fibers, 380 µm in diameter and from 0.5 to 5.0 cm in length, was measured as the O2 supply pressure was varied from 300 to 900 Torr. The fiber modules were mounted in a 1.75-in.-diameter transfer chamber in which water saturated with 760 Torr of CO2 flowed around the fibers at 1.2 or 3.4 L/min. At atmospheric pressure, the CO2 desorption rate leveled off with increasing fiber length. At hypobaric conditions, however, the rate of CO2 desorption increased linearly with fiber length, and was as much as eight times greater than at atmospheric pressure. These data were analyzed with a one-dimensional binary diffusion model that utilized an overall mass transfer coefficient as a free parameter. The best estimates of the coefficient (20-40 µm/s) were independent of gas pressure but varied with fiber length. Introduction Hollow fiber mass exchange devices are a favorable alternative to many industrial mass transfer operations because the hollow fiber devices do not have problems with flooding or loading (Sidhoum et al.,1989; Semmens et al., 1990; Karoor and Sirkar, 1993; Kreulen et al., 1993). The possibility of bubbleless mass transfer has also attracted the attention of biomedical researchers interested in developing an artificial lung that can be implanted in a major vein. The earliest of such intravascular artificial lungs consisted of several hundred microporous hollow fibers, approximately 200 µm in diameter and 60 cm in length, placed parallel to the direction of blood flow within the inferior vena cava (Mortensen, 1987). The two open ends of each fiber were manifolded to the inlet and outlet of a gas supply tube that pumped pure oxygen (O2) through the fiber. Since gas transfer was limited by diffusion through the liquid boundary layer on the external surfaces of these fibers, a later improvement in the design was to loop the fibers (Vaslef et al., 1989). Relative to a parallel fiber arrangement, looping the fibers increased the angle of incidence between the fiber axis and the direction of blood flow which is known to increase mass transfer to the fiber surface (Yang and Cussler, 1986). The newest concept for an implantable artificial lung is the Penn State Intravascular Lung (PENSIL). This device will consist of thousands of microporous hollow fibers, 200-400 µm in diameter, 5-10 cm in length, and sealed at one end (Snider et al., 1994). The fibers will be distributed along the length of a central gas supply tube, with their open ends glued to the tube at an angle of 140° relative to the direction of blood flow. The purpose of using fibers that float freely in flowing blood * Corresponding author. Phone: (814) 863-4802. Fax: (814) 865-7846. E-mail: [email protected].

is to improve mechanical flexibility so that the PENSIL can be inserted into the right ventricle and pulmonary artery as well as in the inferior vena cava. Angling the fibers on the gas supply tube is intended to improve mass transfer through the blood. There have been a few studies of the absorption of O2 from such blind-ended fibers into flowing water (Cote et al., 1989, Ahmed and Semmens, 1992a; Johnson et al., 1997). Since the intended application of the fiber modules tested in these experiments was the bubbleless aeration of water, a pressurized supply of pure O2 was manifolded directly to the open ends of the fibers. In this configuration, the transfer of O2 across the fiber surface induces convection from the open end to the sealed end of a fiber, and this maintains a relatively pure O2 environment in the fiber lumen. As both Cote and associates (1989) and Ahmed and Semmens (1992b) pointed out, however, there still exists a counterdiffusion of impurities such as nitrogen and water vapor that diminish the efficiency of oxygenation. Ahmed and Semmens (1992a) have demonstrated that oxygenation can be made more efficient by increasing the O2 supply pressure, thereby increasing O2 convection and diminishing the buildup of impurities by counterdiffusion. Since there is no provision to remove gas, this type of oxygenation system would be incapable of desorbing carbon dioxide (CO2) from blood, as is required in an intravascular artificial lung. In the PENSIL, however, the gas supply provides a constant flow of O2 across each fiber mouth (Figure 1). Carbon dioxide desorption then takes place in two coupled steps: CO2 dissolved in the blood is transferred across a liquid boundary layer into the fiber lumen, and gaseous CO2 diffuses along the fiber lumen toward the open end of the fiber where it is swept away by the gas supply flow. Absorption of O2 from the gas supply tube to the blood takes place in the opposite manner. With this design, it is possible to reach the normal physiological condition where O2 absorption and CO2

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Figure 1. Schematic of a single fiber in PENSIL. Solid arrows indicate O2 being absorbed from the gas supply into flowing liquid, and dashed arrows show CO2 being desorbed from flowing liquid into the gas supply.

desorption occur at similar rates. Diffusion-induced convection in the fiber lumen would then be small, and mass transfer would be analogous to the classical problem of energy transfer from a finned heat exchanger (e.g. Kreith, 1973). This analogy predicts that as fiber length increases, O2 and CO2 transfer rates should reach a limiting value because the diffusion resistance along the fiber lumen will eventually be much larger than the diffusion resistance across the liquid boundary layer on the fiber surface. Both the magnitude of the limiting transfer rate and the characteristic fiber length at which this limit is reached depend directly on the mutual diffusion coefficient of the O2-CO2 gas pair. Since the diffusion coefficient is inversely related to total pressure, the use of a hypobaric O2 gas supply should increase the portion of a fiber surface that participates in gas transfer, allowing fewer but longer fibers to be incorporated in the final PENSIL design. The main purpose of this study was test the hypotheses that there is a limiting gas transfer rate when blind-ended fibers are subjected to a sweep flow and that this limit can be relaxed by the use of a hypobaric gas supply. Modules consisting of 14 380-µm diameter, microporous hollow fibers with lengths of 0.5-5.0 cm were studied at gas supply pressures between 300 and 900 Torr in a 1.75-in.-diameter transfer chamber filled with water flowing at either 1.2 and 3.4 L/min (lpm). In addition to measurement of CO2 desorption from water by a pure O2 gas supply, the desorption of O2 from water to a pure CO2 supply was observed in some experiments. Separate mass transfer coefficients for CO2 and O2 were estimated by simulating these data with an extension of a mathematical diffusion model previously described by Ahmed and Semmens (1992b). Mathematical Modeling. The exchange of a desorbed gas (denoted by a subscript A) with an absorbed gas (denoted by a subscript B) in a blind-ended hollow fiber was modeled by applying the diffusion equation to a differential control volume in the gas-filled lumen of the fiber. Since the length-to-diameter ratio of the cylindrical fibers ranged between 20 and 200, integral averaging over the inner cross-sectional area of the fiber could be used to reduce the three-dimensional diffusion equation to an equation where diffusion only occurred in the axial direction along a fiber (Baskaran, 1997). Other assumptions used in the model were the following: Fiber geometry and permeability properties

are uniform with fiber length. Gas convection inside the fiber is sufficiently small that pressure gradients are negligible. Heat transfer from the surrounding thermostated liquid is sufficiently rapid that temperature variations in the fiber are negligible. Gas follows the ideal gas law within the fiber as well as Henry’s law of solubility in the liquid. Gas transport processes are at steady state. Permeation of water through the wall of the hollow fiber is so rapid that the partial pressure of water throughout the fiber lumen is equal to the vapor pressure of water. Interactions of water vapor with CO2 and O2 are negligible so that Fick’s law of binary diffusion applies to CO2 and O2. Transfer of O2 and CO2 across the fiber wall can each be formulated in terms of constant mass transfer coefficients. The concentrations of O2 and CO2 outside the fiber are constant. Given these assumptions, three dependent variables are necessary to specify the transport of gas A within the fiber lumen: NA (mol/(cm2‚s)), the axial flux of gas A relative to stationary coordinates; vg (cm/s), the average axial velocity of gases A and B; pA (Torr), the partial pressure of desorbed gas A. The distribution of these variables in the axial direction, z (cm), is governed by a material balance on gas A,

4KA dNA + (p - pAL) ) 0 dz dRgT A

(1)

an overall material balance on gases A and B,

dvg 4 + [K (p - pAL) + KB(P - pA - pBL)] ) 0 (2) dz dP A A and Fick’s law of diffusion for gas A,

dpA pAvg RgTNA + )0 dz DAB DAB

(3)

Three boundary conditions are required to solve eqs 1-3. At the open end of the fiber, the partial pressure of gas A can be specified,

pA (z ) 0) ) pA0

(4a)

and at the sealed end of the fiber, both the flux of gas A and the average velocity of gases A and B must be zero.

NA (z ) 0) ) vg (z ) 0) ) 0

(4b)

The parameters associated with these equations are the following: KA and KB (cm/s), overall mass transfer coefficients between the external fluid and the fiber lumen; Rg ((Torr cm3)/(mol‚K)), the universal gas constant; T (K), the absolute temperature; P (Torr), the dry gas pressure; DAB (cm2/s), the mutual diffusion coefficient of gases A and B; L (cm), the length of a fiber; d (cm), the inner diameter of a fiber; pAL (Torr) and pBL (Torr), the partial pressures of gases A and B in equilibrium with those dissolved in the fluid external to the fibers. In eqs 1-4, which will collectively be designated as the “convection diffusion model”, the radial flux of the desorbed gas A between the external fluid phase and the fiber lumen is formulated as

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Figure 2. Fiber manifolds and the transfer chamber. Left: Modules of 14 blind-ended fibers with alternative lengths of 0.5, 1.0, and 1.5 cm. Right: 3-cm fiber module mounted in the transfer chamber between stainless support tubes. Water flow is upward, and the free ends of the fibers are oriented at 140° relative to the flow direction.

KA(pAL - pA) (mol/(cm2‚s)) and the radial flux of absorbed gas B is formulated in an analogous fashion as KB[(P pA) - pBL]. The convective diffusion model can be simplified to the point that it can be solved analytically. In particular, if the mass transfer coefficients of the desorbed gas, KA, and the absorbed gas, KB, were equal to a common value KAB, and the dry gas pressure in the fiber lumen, P, was equal to the partial pressure sum in the external liquid phase, pAL + pBL, then there would be equimolar counterdiffusion of gases A and B such that the molar velocity, vg, was zero. Given these assumptions, the solution to eqs 1, 3, and 4 for the desorption rate MA (mol/s) through a fiber of length L is given by

MA ) MA,max tanh(L/L0)

(5)

where the characteristic length L0 (cm) is defined as

L0 ) (dDAB/4KAB)1/2

(6)

and the maximum transfer rate MA,max (mol/s) is given by

MA,max ) (πd2DAB/4RgTL0)(pA0 - pAL)

(7)

This “equimolar counterdiffusion model” is analogous to heat conduction through an uninsulated solid fin of uniform cross-sectional area that is maintained at a constant temperature at one end and is insulated at the other end (Kreith, 1973). For relatively short fibers whose dimensionless length L/L0 is 0.4 or less, the dimensionless transfer rate MA/ MA,max is approximately equal to L/L0, and gas transfer through a blind-ended fiber approaches the transfer rate for a flow-through fiber that is filled with fresh gas supply.

MA ) (πdLKA/RgT)(pA0 - pAL)

(8)

For longer fibers, however, the value of MA/MA,max begins to level off, reaching a value of 0.96 when L/L0 is equal to 2. In other words, 2L0 represents the “active length” in the sense that making fibers longer than 2L0 improves the transfer rate by less than 4%. This gas transfer limitation occurs because desorbed gas must overcome an axial diffusion resistance in the fiber lumen, resulting in a buildup of desorbed gas toward the sealed end of the fiber. The buildup of desorbed gas in the fiber lumen reduces the driving force for transfer across the fiber wall, an effect that becomes more pronounced for longer fibers. Experimental Methods The microporous polypropylene fibers used in this research had an outer diameter of 380 µm, a wall thickness of 50 µm, a wall porosity of 30%, and a pore size of 0.02 µm (Oxyphan 80, Enka, Wuppertal, Germany). Fiber modules were fabricated by gluing individual microporous polypropylene fibers of equal length into a circumferential row of 14 holes drilled through the wall of a polycarbonate tube segment that was 14 mm in length, 5.8 mm in outer diameter, and 5 mm in inner diameter. The holes were angled so the anchored end of each fiber was tilted at 40° with respect to the polycarbonate tube axis. The free end of each fiber was sealed shut with epoxy. Modules with fiber lengths of 0.5, 1.0, 1.5, 2.5, 3.0, and 5.0 cm were used in this study (Figure 2). Two types of experiments were carried out to determine the desorption rates of O2 and CO2 from the outside to the inside of the fiber modules. These were gas-gas experiments and gas-liquid experiments. In gas-gas experiments designed to measure CO2 desorption, a gas mixture of O2 and CO2 flowed external to a fiber module while a gas supply consisting of pure O2

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Figure 3. Complete apparatus configured for the gas-liquid experiments. Gas component A is desorbed from a circulating water stream (dashed lines) into a gas supply consisting of component B (solid lines).

flowed inside the modules. Alternative compositions of the external gas mixture were chosen so that CO2 desorption could be observed at partial pressure differences of 150, 400, 600, or 760 Torr between the external gas and the gas supply. To minimize the effects of forced convection through the porous fiber wall, both the gas supply and the external gases were maintained at a pressure of 760 Torr. In an analogous set of gasgas experiments designed to measure O2 desorption, the gas supply consisted of pure CO2 and external gas mixtures of CO2 and O2 were used. In gas-liquid experiments carried out to measure CO2 desorption through each fiber module, deionized distilled water was saturated with pure CO2 and then circulated outside a fiber module while a gas supply consisting of pure O2 flowed inside the module. Rates of CO2 desorption were measured at various gas supply pressures between 300 and 900 Torr. These experiments were repeated at water flows of 1.2 and 3.4 L/min (lpm) corresponding to tube Reynolds numbers of 600 and 1600 which bracket the range that is found in those blood vessels where the PENSIL will be placed. In an analogous set of gas-liquid experiments carried out to measure O2 desorption on the 5 cm fiber module alone, water saturated with pure O2 was circulated outside a fiber module while a gas supply consisting of pure CO2 flowed inside the module. Gas-liquid exchange took place in a transfer chamber which consisted of a 50 cm long, 1.75 in. inner diameter, Plexiglas tube that was capped on each end with a polycarbonate section (Figures 2 and 3). Water entered and exited the transfer chamber through these polycarbonate sections at a 45° angle relative to the flow direction in order to minimize the pressure loss. The entrance of the transfer chamber was packed with 5 cm long, 5 mm inner diameter, 0.2 mm wall thickness, stainless steel tubes that facilitated the development

of the velocity field in the liquid. A fiber module was held at the center of the transfer chamber by two 40 cm long, 1/16 in. inner diameter, stainless steel support tubes with two silicon rubber O-rings providing a gastight seal. Most experiments were conducted with the free ends of the fibers pointing at an angle of 140° relative to the direction of water flow, but some experiments were repeated after flipping the fiber module to decrease the orientation angle to 40°. The support tubes also served as the inlet and outlet of the gas supply. A mass flow controller (902C, Sierra Instruments) ensured a constant 10 mL (STP)/min flow of the supply gas B in the inlet support tube, and an electromechanical pressure transducer (DP45, Validyne Engineering) monitored the gas supply pressure. The outlet support tube was connected to a vacuum pump (UN026, KNFNeuberger) through a double-pattern metering valve (Series S, Swagelok) that was used to regulate pressure. A 24 µL (STP)/min sample of the gas supply was continuously withdrawn from the outlet of the apparatus by a mass spectrometer (Extrel, Questor II). The composition of this gas stream was recorded by the mass spectrometer at 1-s intervals and became constant 1-4 h after an experiment began. At that time, the steadystate transfer rate of the desorbed gas species, either O2 or CO2, was calculated from a simplified material balance equation given by

MA ) yAM

(9)

where yA is the mole fraction of absorbed gas A recorded by the mass spectrometer and M (mol/s) is the molar flow rate at which the gas supply is delivered to the fiber module through the mass flow controller. It was assumed in deriving eq 9 that the gas supplied at the inlet of the fiber module does not contain the desorbed gas A and also that the flow rate of gas supply is virtually the

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same at the inlet and outlet of the fiber module. The former assumption was satisfied because a pure O2 gas supply was always used when CO2 desorption was measured, and a pure CO2 gas supply was used when O2 desorption was measured. The latter assumption was reasonable because the net transfer of gases across a fiber module was small compared to the rate at which the gas supply flowed into the module. Before reaching the transfer chamber, water was bubbled with pure gas A using a fritted disk dispersion tube (ACE 7200) inside a polycarbonate carboy (10 L capacity) that was vented to the atmosphere. A cylindrical rod heater (Chromolax, Omega) and a T-type shielded thermocouple in combination with a temperature controller (model CN76130, Omega) maintained a constant temperature of 37 °C inside the carboy. Water was circulated between the carboy and the transfer chamber using a gear pump (model TCM 1079 MCK, Tuthill). A bubble trap was located between the gear pump and the transfer chamber. A rotameter (model F-1660, Gilmont Instruments) and an electromechanical transducer (model DP15-40, Validyne Engineering) were used to measure the water flow rate and water pressure near the fiber module. The experimental apparatus for the gas-gas experiments was different from that of the gas-liquid experiments in two respects. First, the circulating water was replaced by the steady flow of a gas mixture of O2 and CO2. Second, an electromechanical transducer (DP45, Validyne Engineering) was used to monitor the pressure difference between the inside and outside of the fiber module. As in the gas-liquid experiments, the flow of the gas supply into the fiber modules was maintained at 10 mL (STP)/min. The compressed gas sources of O2 and CO2 used in the both the gas-gas and gas-liquid experiments were specified by the manufacturer to be at least 99.99% pure, and the compressed gas mixtures used in the gas-gas experiments and the mass spectrometer calibrations were purchased as certified standards (Scott Specialty Gases, Inc.). Analysis of Data. Equations 1-4 constitute a nonlinear boundary value problem that was solved numerically for NA, vg, and pA as a function of z by using the method of spline collocation at Gaussian points (COLSYS Fortran Subroutine, Netlib Repository). The measured desorption rates could then be compared to the values predicted by the convection diffusion model according to the equation:

MA ) -(πd2/4)(NA)z)0

(10)

The input parameters required by these mathematical simulations were L, d, P, T, DAB, pA0, pAL, pBL, KA, and KB. Whereas the values of L, d, P, T, DAB, pA0, pAL, and pBL were known for each fiber module at each experimental condition, KA and KB were estimated by matching the simulated MA values to the corresponding measurements. When simulating gas-liquid experiments on the 5-cm fiber module for which O2 as well as CO2 desorption were measured, both KA and KB were treated as unknown parameters. Beginning with a set of trial values for KA and KB, a separate solution to eqs 1-4 was found for CO2 desorption into O2 and for O2 desorption into CO2 at each experimental gas supply pressure. This process was repeated, employing a Levenberg-Marquadt algorithm (DRNLIN, IMSLSTAT Library) to update the values of KA and of KB, until the

sum of squared error between the entire set of desorption data and the model predictions was minimized. In the simulation of gas-liquid CO2 desorption data from the other fiber modules for which only CO2 desorption data were available, KB (B ≡ O2) was fixed at the optimal value obtained from the 5-cm module and KA (A ≡ CO2) was treated as the only unknown input parameter to be estimated. The reciprocal of KA is an overall resistance to radial mass transfer that can be expressed as the sum of the individual resistances in the fiber lumen, in the micropores of the fiber wall, and in the fluid flowing external to the fiber. Assuming that the diffusion resistance within the fiber lumen is relatively small, 1/KA can be formulated as

1/KA ) 1/kAM + (HA/RgTCW)(d/do)(1/kAL)

(11)

where kAM (cm/s) is the permeability of the fiber wall to gas A, kAL (cm/s) is the individual mass transfer coefficient of gas A in the external fluid, HA (Torr) is the Henry’s law constant of gas A in the fluid, and CW (mol/ cm3) is the molar density of the fluid. The ratio of the inner to outer diameter of the hollow fibers, d/do, is included in the diffusion resistance of the fluid because the mass balances used to deduce KA were based on a control volume bounded by the internal fiber surface, whereas kAL should be defined with respect to the external fiber surface. When the fiber modules were immersed in a gaseous mixture of A and B, the second term on the right-hand side of eq 11 should be relatively small, so that kAM could be approximated by the value of KA obtained in the gasgas experiments. Once kAM was known, values of kAL were computed by applying eq 11 to the KA values obtained in the gas-liquid experiments. To facilitate comparison with previous measurements of liquid-phase mass transfer coefficients, kAL was also computed in dimensionless form as the Sherwood number on the basis of the outer diameter of the fiber.

Shm ≡ kALdo/DAB

(12)

Results The effect of fiber length on the desorption measured in the gas-gas experiments is shown in Figure 4. Increasing the fiber length by an order of magnitude had no appreciable effect on the desorption rates of O2 and CO2. Halving the driving force by decreasing the partial pressure of the desorbed gas outside the fiber prototype decreased the transfer rate by almost exactly 50%. Desorption rates of O2 were consistently higher than CO2 desorption rates by about 10%. The curves in Figure 4 were obtained by simultaneously regressing the convection diffusion model to all the O2 and CO2 desorption data obtained from all the fiber modules. The estimated mass transfer coefficients were KO2 ) kO2,M ) 251 µm/s and KCO2 ) kCO2,M ) 216 µm/s. The average error between the experimental data and the predicted desorption rate was less than 5%. The effect of partial pressure driving force on the desorption rates of both CO2 and O2 in the gas-gas experiments is shown in Figure 5. The desorption rates increased linearly with driving force in a range of 150760 Torr for CO2 and 380-760 Torr for O2. The lines in Figure 5 represent simulations of the convection diffusion model that incorporated the estimates of kO2,M

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4147

Figure 6. Gas-liquid experiments: effect of fiber length. Filled and unfilled points represent the measured CO2 desorption rates at fiber orientations of 140 and 40°, respectively. The water flow was 3.4 lpm, and the gas supply pressure was 300, 600, or 760 Torr. In most cases, the unfilled points are completely obscured by the filled points, indicating that there was no difference between the 140 and 40° orientations.

Figure 4. Gas-gas experiments: effect of fiber length. Points represent the measured desorption rates of CO2 (upper panel) and O2 (lower panel) at a gas driving force of 380 or 760 Torr. Curves represent regressions by the convection diffusion model employing parameter values of KCO2 ) kCO2,M ) 216 µm/s and KO2 ) kO2,M ) 251 µm/s.

Figure 5. Gas-gas experiments: effect of partial pressure driving force. Points represent the measured desorption rates of CO2 or O2 for the 5-cm fiber module. The lines represent regressions by the convection diffusion model employing parameter values of KCO2 ) kCO2,M ) 216 µm/s and KO2 ) kO2,M ) 251 µm/s.

and kCO2,M. The model predicted these data with an average error of less than 7%. The results from the gas-liquid experiments are summarized in Figures 6-9. The data points in Figure 6 are the CO2 desorption rates measured at a constant water flow of 3.4 lpm at fiber orientations of 40° (unfilled points) and 140° (filled points). The standard deviation of the data was less than 2%. At a gas supply pressure of 760 Torr, the CO2 desorption rate initially increased

Figure 7. Gas-liquid experiments: sensitivity of convection diffusion model to KO2. Points represent the measured desorption rates of CO2 (upper panel) and of O2 (lower panel) for the 5-cm fiber module at an orientation of 140° and a water flow of 3.4 lpm. The three curves represent regressions by the convection diffusion model employing parameter values of KO2 ) 1.4 µm/s (dotted curve), 2.8 µm/s (solid curve), or 4.2 µm/s (dashed curve) and KCO2 ) 40 µm/s (all curves). The curves in the upper panel superimpose, indicating that CO2 transfer was insensitive to the value of KO2.

with fiber length and then leveled off at a relatively small value. At a gas supply pressure of 300 Torr, the desorption rate increased almost linearly with fiber length. At a gas supply pressure of 900 Torr, desorption rate increased to what appeared to be a mild maximum

4148 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 Table 1. CO2 and O2 Desorption from Water at 37 °C Flowing over 5 cm Long Fibers at an Orientation of 140° a gas A

Q

Re

kAL

Shm

ShY-C

ShCote

CO2 CO2 O2 O2

1.2 3.4 1.2 3.4

4.2 11.8 4.2 11.8

36 57 42 74

6.7 10.6 6.1 10.8

17 26 16 24

7.2 10.5 6.7 9.7

a Q (lpm) ≡ volumetric rate of water flow; Re ≡ Reynolds number based on the component of water velocity normal to the fiber averaged over the entire cross-sectional area of the transfer chamber; kAL (µm/s) ≡ liquid-phase mass transfer coefficient of gas A determined from gas-gas and gas-liquid measurements according to eq 11; Shm ≡ Sherwood number based on the measured value of kAL; ShY-C ) 1.38Re0.40Sc0.33, Sherwood number correlation of Yang and Cussler (1986); ShCote ) 0.61Re0.363Sc0.33, Sherwood number correlation of Cote and associates (1989).

Figure 8. Gas-liquid experiments: effect of gas supply pressure. Points represent the measured CO2 desorption rates for a 0.5, 1.5, or 5.0 cm fiber module at an orientation of 140° and a water flow of 1.2 lpm. Curves represent regressions by the convection diffusion model employing parameter values of KCO2 ) 24 µm/s (0.5 cm fibers), 28 µm/s (1.5 cm fibers), or 27 µm/s (5.0 cm fibers) µm/s and KO2 ) 1.6 µm/s (all fiber lengths).

Table 2. CO2 Desorption from Water at 37 °C Flowing over 0.5-3.0 cm Long Fibers at an Orientation of 140° a Q ) 1.2

Q ) 3.4

L

Re′

kCO2,L

Shm

ShCote

Re′

kCO2,L

Shm

ShCote

0.5 1.0 1.5 2.5 3.0

3.4 4.0 4.3 4.3 4.2

31 37 37 44 42

5.9 7.0 7.0 8.1 7.8

5.9 7.0 7.4 7.5 7.0

9.8 11.6 12.4 12.6 11.8

49 57 59 72 70

9.1 10.6 11.0 13.4 13.0

8.7 10.4 11.0 11.2 10.6

a Q (lpm) ≡ volumetric rate of water flow; Re′ ) Reynolds number based on the component of the water velocity normal to the fiber averaged over the cross-sectional area of the transfer chamber located between the gas supply tube and the free end of the fiber; L (cm) ) length of fibers in a module; kCO2,L (µm/s) ≡ liquid-phase mass transfer coefficient of CO2 determined from gas-gas and gas-liquid measurements according to eq 11; Shm ≡ Sherwood number based on the measured value of kCO2,L; ShCote ) 0.61(Re′)0.363Sc0.33, Sherwood number correlation of Cote and associates (1989).

Figure 9. Gas-liquid experiments: effect of water flow. Points represent the measured CO2 desorption rates for the 5-cm module at an orientation of 140° and a water flow of 1.2 or 3.4 lpm. Curves represent regressions of the data by the convection diffusion model employing parameter values of KCO2 ) 27 µm/s (1.2 lpm water flow) or 39 µm/s (3.4 lpm water flow) and KO2 ) 1.6 µm/s (both water flows).

value after which the transfer rate decreased slightly. Changing the orientation angle from 40 to 140° did not affect the transfer rate. Both CO2 and O2 desorption rate data for the 5-cm fiber module are shown in Figure 7 as a function of gas supply pressure at a water flow of 3.4 lpm. For both gases, the desorption rate increased with a decrease in gas supply pressure by as much as a factor of 8. Desorption rates of CO2 were greater than those of O2 by as much as a factor of 12. The solid curves represent the simultaneous regression of CO2 and O2 desorption versus pressure data to the convection diffusion model. Overall mass transfer coefficients obtained for the 5-cm fiber module by this regression were KCO2 ) 40 µm/s and KO2 ) 2.8 µm/s. A similar regression for the same fiber module at a flow of 1.2 lpm yielded KCO2 ) 27 µm/s and KO2 ) 1.6 µm/s. The average errors between the predicted and experimental transfer rates were 12%. Figure 7 also includes simulations that were carried out with alternative KO2 values of 4.2 µm/s (dashed line) and 1.4 µm/s (dotted line) while KCO2 was held constant at its optimal estimated value of 40 µm/s. Although the

O2 transfer rates were roughly proportional to the KO2 value, the predicted CO2 desorption rate was virtually unchanged for this (50% change in KO2 from its optimal estimate of 2.8 µm/s. The lack of sensitivity of CO2 excretion to KO2 occurs because the value of KCO2 is about 15 times greater than the value of KO2. The desorption rate of CO2 for three different fiber modules is shown in Figure 8 as a function of the gas supply pressure at a water flow of 1.2 lpm. This figure demonstrates that transfer rates were increasingly sensitive to gas pressure as fiber length increased from 0.5 to 5.0 cm. The solid lines in Figure 8 represent three independent regressions of the transfer rate versus gas pressure data, one simulation carried out for each fiber module. In all these simulations, KO2 was fixed at the optimal value found for the 5.0 cm fiber prototype at a 1.2 lpm flow and KCO2 was treated as a free parameter. The model results fit these data with an average error of 10%. The effect of water flow on CO2 desorption is illustrated in Figure 9 for 5-cm fiber modules. The desorption rates increased only slightly with increasing flow at the higher gas supply pressures but showed a substantial flow dependence at the lower gas supply pressures. For example, the excretion rate increased by only 10% at a gas supply pressure of 900 Torr but by more than 50% at 300 Torr. Table 1 contains the values of kCO2,L and kO2,L for the 5-cm fiber module that were calculated from KCO2 and KO2 by using eq 11 with the values of kCO2,M and kO2,M obtained from the gas-gas experiments. Table 2 contains the kCO2,L obtained in the same manner for all the

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4149

fiber modules. The kCO2,L varied in a systematic manner with fiber length, appearing to reach a maximum value for the 2.5-cm fiber module. Discussion The convection diffusion model, as defined by eqs 1-4, is virtually equivalent to the model proposed by Ahmed and Semmens (1992b) to describe the aeration of water. The previous model differed in only one respect: the liquid-phase mass transfer coefficients for the absorbing and desorbing gases were assumed to be equal. Whereas this may be appropriate for the aeration of water in which the solubility of the absorbing O2 gas and the desorbed nitrogen gas are similar, it is not valid for an artificial lung in which the solubility of the absorbing O2 is only 5% of the solubility of the desorbing CO2. Both the model of Ahmed and Semmens (1992b) and the present convection diffusion model required that the gas composition at the open end of the fibers be known (eq 4a). When implementing numerical simulations, it was assumed that this composition was equal to that of the gas supply. This approximation is justified in the present experiments because a relatively large sweep flow removed most of the desorbed gas from the open end of the fiber. In the water aeration experiments, however, desorbed gas could readily counterdiffuse into the tube connecting the compressed O2 source to the fibers. By regression of the gas-gas desorption data to the convection diffusion model, the permeability of the polypropylene fibers to O2 was found to be 1.16 times their permeability to CO2. This factor is very close to the square root of the inverse ratio of molecular weights, (44/32)1/2, implying that gas transfer through the micropores of the fiber walls occurred by Knudsen diffusion (Callahan, 1988). Because the values of CO2 and O2 permeabilities are similar, the predictions of the equimolar counterdiffusion model should approximate the gas-gas desorption data. In particular, the active fiber length predicted by the model is about 0.4 cm (eq 6). This is less than the shortest fiber module of 0.5 cm used in the experiments. Thus, the gas-gas desorption rates in Figure 4 appear to be insensitive to fiber length because they are always at their limiting values. Although it would have been interesting to observe gasgas desorption rates below these limiting values, such measurements were precluded because of the difficulty in constructing modules with fibers shorter than 0.5 cm. The equimolar counterdiffusion model also indicates that the desorption rate data should be proportional to the partial pressure driving force between the gas supply and the water stream. This prediction is supported by the desorption data in Figure 5. Even though the mass transfer coefficients for CO2 and O2 are far from being equal during gas-liquid desorption, it is instructive to borrow the concept of active length from the equimolar counterdiffusion model to provide a qualitative interpretation of the data. In particular, the diffusion coefficient in the fiber lumen is inversely related to pressure (Bird et al., 1960), so that active length should increase as pressure is reduced from atmospheric to hypobaric conditions (eq 6). The CO2 desorption data in Figure 6 support this prediction: the active fiber length is about 1 cm at gas supply pressures of 760 and 900 Torr, whereas the active length appears to be greater than 5 cm at a pressure of 300 Torr. The fact that the 0.5 cm fiber module is always

well below its active length explains why desorption for this module is insensitive to the gas supply pressure. On the other hand, the 5 cm fiber module is well above its active length at a pressures of 760 and 900 Torr but is below its active length at a pressure of 300 Torr. This explains why desorption for the 5-cm fiber module is sensitive to pressure. If active length is inversely related to pressure, then gas desorption should be given by eq 8 when the gas supply pressure is sufficiently low. To test this prediction, the data in Figure 8 were extrapolated to find the zero-pressure intercept of desorption rate at every fiber length. The value of KCO2 computed from each intercept by using eq 8 was found to be within 5% of the corresponding KCO2 value obtained by regressing the entire data set from each fiber module to the convection diffusion model. Therefore, at very low gas pressures, the efficiency of desorption in blind-ended fibers approached that in open-ended fibers that are flushed clean with the gas supply. For the gas-liquid data in Figure 8 obtained at 900 Torr, there was a slight decline of CO2 desorption through the 5 cm module relative to desorption occurring in shorter fiber modules (Figure 6). To some extent, this can be attributed to bending of the free ends of the 5 cm fibers near the wall of the transfer chamber. The convection diffusion model does, however, predict a reduced desorption at hyperbaric conditions without the need to account for fiber bending. This prediction may be explained as follows: The 5 cm fibers accumulate CO2 near their sealed end due to an axial diffusion resistance, and since the total gas pressure inside the fiber lumen is higher than the sum of the partial pressures of the gas species in the liquid, some diffusion of CO2 back into the liquid phase could occur near the sealed end of the fiber. A low fiber bubble point of about 1000 Torr prevented measurement of desorption at higher pressures to further examine this behavior. Gas-liquid desorption in a blind-ended fiber must overcome an axial diffusion resistance in the fiber lumen as well as a radial diffusion resistance through the boundary layer of the flowing liquid. The axial diffusion resistance through the fiber lumen is inversely related to the diffusion coefficient and, consequently, is directly related to the gas supply pressure. The diffusion resistance through the boundary layer is inversely related to the liquid flow rate. The data in Figure 9 indicate that water flow had little effect on CO2 desorption at 900 Torr pressure, probably because the axial diffusion resistance was the bottleneck in the overall desorption process. When the gas supply was introduced at a hypobaric pressure, however, a flow effect became evident. This was most likely due to a substantial reduction of the axial diffusion resistance that revealed the effect of flow on the boundary layer resistance. The fiber wall permeabilities estimated in gas-gas experiments were greater than the mass transfer coefficients obtained in gas-liquid experiments by a factor of about 7 for CO2 and 120 for O2. This indicates that the fiber wall offered a negligible resistance to the radial transfer of CO2 and O2 in gas-liquid experiments. On the other hand, the fiber wall was expected to offer the dominant resistance to the radial diffusion of water vapor in gas-liquid experiments, because there is virtually no resistance to the diffusion of water in an

4150 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

aqueous solution. Therefore, the overall mass transfer coefficient for water vapor in gas-liquid desorption is on the order of 10 times the overall mass transfer coefficient for CO2 and about 100 times that for O2, implying that the active length with respect to water vapor is 1/3 and 1/10 of the active lengths of CO2 and O2, respectively. This implies that 70 to 90% of the sealed end of the fiber is saturated with water vapor. In the frontal portion of the fiber that is not saturated, water vapor is diffusing toward the open end of the fiber. This effect, which is ignored in the convection diffusion model, actually adds to the value of the diffusioninduced velocity, vg, thereby enhancing the desorption of CO2 or O2. Therefore, the mass transfer coefficients reported in this study may be somewhat overestimated. To determine whether the numerical values of the mass transfer coefficients estimated by the diffusion convection model were reasonable, the corresponding Sherwood numbers, Shm, were compared to values predicted from existing mass transfer correlations. For a uniform flow of a liquid perpendicular to an array of fibers at a low packing density, Johnson and co-workers (1997) listed two correlations that are valid for the Reynolds number range used in the current study. From the work of Yang and Cussler (1986)

ShY-C ) 1.38Re0.40Sc0.33

(13)

and from the research of Cote and associates (1989)

ShCote ) 0.61Re0.363Sc0.33

(14)

where Sh ) kALdo/DAB is the Sherwood number, Re ) vdo/ν is the Reynolds number, Sc ) ν/DAB is the Schmidt number, v (cm/s) is the liquid velocity impinging on the fiber, and ν (cm2/s) is the kinematic viscosity. Even when the liquid flow does not impinge on a fiber at exactly 90°, these correlations can still be used by computing Re from the velocity component normal to the fiber (Smith, et al., 1972; Willins and Griskey, 1975). For the current study, the means that Re ) (vt sin 140°)do/ν, where vt is the axial velocity of water in the transfer chamber. Since vt sin 140° is identical to vt sin 40°, the gas transfer predicted at a fiber orientations of 40 and 140° should be the same, which is consistent with the data in Figure 6. Table 1 contains the numerical predictions of ShY-C and ShCote for the 5-cm fiber modules. These fibers were sufficiently long that their free ends touched the wall of the transfer chamber. Therefore, the vt used in computing Re was the axial velocity of water averaged over the entire cross-section of the transfer chamber. This is equivalent to the ratio of the volumetric flow rate of water to the chamber area. The resulting ShY-C values are more than twice as large as the measured Shm values, but there is good agreement between ShCote and Shm. Table 2 contains the numerical predictions of ShCote for fiber modules of 3.0 cm or less. The 3.0 cm fibers barely fit into the transfer chamber without touching the chamber wall so that shorter fibers were not exposed to the entire velocity profile of water in the chamber. In this case, the value of vt used for computing Re was the axial velocity of water averaged over the chamber area located between the gas supply tube and the free end of the fibers. If a theoretical formulation for the laminar velocity distribution in the annulus between the

gas supply tube and the chamber wall (Bird et al. 1960) is employed, the average vt and its corresponding Re was found to increase with fiber length. The resulting ShCote predictions were surprisingly similar to the Shm measurements. Both the predicted and measured Sherwood numbers increased with fiber length, reaching an apparent maximum for the 2.5 cm long fibers. Concluding Remarks The objective of this research was to determine the effects of gas supply pressure and fiber length on gasliquid desorption through blind-ended fibers for which composition at the fiber mouth was maintained by a sweep flow of fresh gas. Desorption rates of CO2 and O2 from water were measured for simple 14-fiber arrays. A convection diffusion model was used to simulate these data and also to estimate the corresponding mass transfer coefficients. In future research, this model could serve as a starting point for predicting the fiber length and number of fibers required for satifying the gas transfer requirements in a full-scale intravascular lung. The conclusions reached from the current study were as follows: (1) When a gas supply at atmospheric pressure is employed, there is an active fiber length beyond which an increase in gas desorption is prevented by the axial diffusion resistance in the fiber lumen. (2) The use of a gas supply at hypobaric pressure reduces this diffusion resistance, thereby increasing the active fiber length and enhancing desorption. (3) At sufficiently low gas supply pressures, the fiber lumen is well-mixed by diffusion and the desorption rate is directly proportional to fiber length. Acknowledgment This research was supported in part by a National Institutes of Health Contract NO1-HR-16053. The transfer chamber and fiber modules were fabricated by Mr. Georg Panol. The authors appreciate the technical suggestions made by Dr. Michael Snider and the cooperation of the Bioprocessing Research Center at the Pennsylvania State University in the shared use of their mass spectrometer. Literature Cited (1) Ahmed, T.; Semmens, M. J. The Use of Independently Sealed Microporous Hollow Fiber Membranes for Oxygenation of Water; Experimental Studies. J. Membr. Sci. 1992a, 69, 1. (2) Ahmed, T.; Semmens, M. J. The Use of Independently Sealed Microporous Hollow Fiber Membranes for Oxygenation of Water; Model Development. J. Membr. Sci. 1992b, 69, 11. (3) Baskaran, H. Mass Transfer in Blind-Ended Hollow Fiber Prototypes of the Penn State Intravascular Lung. Ph.D. Dissertation, The Pennsylvania State University, University Park, PA, 1997; pp 127-130. (4) Baskaran, H.; Nodelman, V.; Ultman, J. S.; et al. Small Intrapulmonary Artery Lung Prototypes; Mathematical Modelling of Gas Transfer. Am. Soc. Artif. Int. Org. J. 1996, 42, M597. (5) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (6) Callahan, R. W. Novel Uses of Microporous Membranes: a Case Study. Am. Inst. Chem. Eng. Symp. Ser. 1988, 84 (No. 261), 54. (7) Cote, P.; Bersillon, J.; Huyard, A. Bubble-Free Aeration Using Membranes: Mass Transfer Analysis. J. Membr. Sci. 1989, 47, 91. (8) Johnson, D. W.; Semmens, M. J.; Gulliver, J. S. Diffusive Transport Across Unconfined Hollow Fiber Membranes. J. Membr. Sci. 1997, 128, 67.

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4151 (9) Karoor, S.; Sirkar, K. K. Gas Absorption Studies in Microporous Hollow Fiber Membrane Modules. Ind. Eng. Chem. 1993, 32, 674. (10) Kreith, F. Principles of Heat Transfer; Intext Educational Publishers: New York, 1973; p 60. (11) Kreulen, H.; et al. Determination of mass transfer rates in wetted and nonwetted microporous membranes. Chem. Eng. Sci. 1993, 48, 2093. (12) Mortensen, J. D. An Intravenacaval Blood Gas Exchange (IVCBGE) Device: A Preliminary Report. Trans. Am. Soc. Artif. Int. Org. 1987, 33, 570. (13) Semmens, M. J.; Foster, D. M.; Cussler, E. L. Ammonia Removal from Water Using Microporous Hollow Fibers. J. Membr. Sci. 1990, 51, 127. (14) Sidhoum, M.; Majumdar, S.; Sirkar, K. K. An Internally Staged Hollow-Fiber Permeator for Gas Separation. AIChE J. 1989, 35, 764. (15) Smith, R. A.; Moon, W. T.; Kao, T. W. Experiments on Flow about a Yawed Circular Cylinder. J. Basic Eng. 1972, 94, 771.

(16) Snider, M. T.; et al. Small Intrapulmonary Artery Lung Prototypes: Design, Construction, and InVitro Water Testing. Am. Soc. Artif. Int. Org. J. 1994, 40, M533. (17) Vaslef, S. N.; Mockros, L. F.; Anderson, R. W. Development of an Intravascular Lung Assist Device. Trans. Am. Soc. Artif. Int. Org. 1989, 35, 660. (18) Willins, R. E.; Griskey, R. G. Mass Transfer from Cylinders at Various Orientations to Flowing Gas Streams. Can. J. Chem. Eng. 1975, 53, 500. (19) Yang, M.; Cussler, E. L. Designing Hollow-Fiber Contactors. AIChE J. 1986, 32, 1910.

Received for review September 22, 1997 Revised manuscript received July 22, 1998 Accepted July 27, 1998 IE9706875