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Ind. Eng. Chern. Process Des. Dev., Vol. 17, No. 1, 1978
Controlling Factors for Oxygen Transfer through Bubbles Wallace W. Bowley and Graeme L. Hammond' Surgical Cardiovascular Research Laboratory, Yale Medical School, New Haven, Connecticut, and the Department of Mechanical Engineering, University of Connecticut,Storrs, Connecticut
In the standard bubble oxygenator used for cardiac surgery, oxygen transfer efficiency is limited by the difficulty in maintaining a uniformly optimum bubble size. To maintain adequate p(02),high oxygen flows are required. This increases turbulence, causes hemolysis, and decreases arterial p(COp).To avoid these problems, a gas exchange system, which controlled bubble size, was investigated. In this system, the bubble chamber consisted of a vertical cylinder containing tightly packed spherical beads forming a porous media through which blood and oxygen could be passed. The interstices between the beads controlled bubble size and produced an extended and tortuous path for each bubble to negotiate. This, also, caused boundary layer reorientation each time a bubble collided with a solid bead, thereby reducing oxygen transfer resistance. An analysis of the oxygen mass transfer in the bead column is included, together with the results of in vitro tests of the system.
There are three basic types of oxygenators which have been used to take the place of the human lung during open heart surgery. These are filmers, bubblers, and membranes. Each device performs a mass transfer function and derives its name from the predominant mass transfer mechanism taking place within the device. Because of ease of production, handling, sterilization, and dependability, bubblers are the most common type of oxygenator currently in use. Bubble oxygenators lend themselves to design concepts in which reliability and economic considerations are simultaneously paramount and compatible. In the bubble oxygenator, oxygenation occurs in two subsystems. The oxygen is brought into intimate contact with unoxygenated (venous) blood at a bubble plate or diffuser. This is usually a flat plate with a large number of small holes through which the oxygen is passed into the venous blood forming a large number of small bubbles. In the second subsystem the bubbles, mixed with blood, are allowed to rise in a bubble chamber. The bubble chamber may be a straight or folded section of open pipe in which the bubbles coalesce, forming larger and larger bubbles. Sufficient time is allowed for diffusion of the oxygen out of the bubble and into the blood while carbon dioxide diffused out of the blood and into the bubble. Oxygen transfer efficiency, in bubble oxygenators, is related to bubble size, boundary layer adherence of blood to bubble, and length of bubble path. In the design of disposable bubble oxygenators, these concepts can be used to increase the mass transfer of oxygen. If oxygen enters a column of blood in an uncontrolled fashion and forms bubbles of varying size which then coalesce and pass through the column a t random, low oxygen transfer efficiency results. To compensate, oxygenblood flow ratios considerably in excess of 1:l are often required. This creates turbulence hemolysis (Anderson and Kuchiba, 1969) and the following chain of events: excessive removal of carbon dioxide, rising blood pH, compensatory cellular cationic exchange of potassium ions for hydrogen ions, and falling blood potassium concentration. This contributes to unstable cardiac rhythm and ventricular fibrillation. T o help understand the special features of bubbles, we designed a system which allowed for control of bubble parameters and therefore provided a model for studying the ability of the bubbles to transfer oxygen in a controllable fashion. 1 Address correspondence to Graeme L. Hammond, M.D., Yale University School of Medicine, Department of Surgery, 333 Cedar Street, New Haven, Conn. 06510.
Experimental Section Materials. The system consisted of two concentric cylinders (Figure 1).The inner cylinder (Figure 2), which measured 36 cm in length and 7 cm in internal diameter (Ld.), served as the oxygenating chamber. This chamber was divided into an upper part measuring 27 cm in length and a lower part measuring 9 cm in length. The lower chamber was designed as a venturi tube and contained oxygen and blood inlet ports. The blood inlet port was flush with the bottom of the venturi tube and entered at a 45' angle. The oxygen port extended 4.5 cm into the lower chamber and terminated just above the narrowest diameter of the venturi tube. The oxygen port measured 0.5 cm i.d. a t its inlet and flared to an outlet i.d. of 2.5 cm. The outlet was covered with a plastic diffuser plate consisting of multiple 0.34-mm perforations. Separating the upper and lower chambers was a plastic mesh. The upper chamber contained 16 500 tightly packed 4-mm diameter plastic beads or 5000 6-mm diameter beads. Figure 3 illustrates the closest packing arrangement. As illustrated in Figure 4, this arrangement creates interstices with two configurations. Accordingly, the formation of two different-sized bubbles is theoretically allowed. A bubble can be formed in the recess existing anterior or posterior to the ogive of three adjacent beads while the base of the bubble is formed by a fourth bead, or a larger bubble can be formed by the ogive of four adjoining beads while the base and apex are formed by a fifth and sixth bead. Each bead has a possible contact site for 14 bubbles, six of which can be large and eight small. The diameter of bubbles released from the diffuser, as calculated from (Bird et al. (1962) and Vander et al. (1970)) Dbu =
[ K ~ uK2 log p -I-A](do)'"
(1)
is 1.997 mm. Bubble diameter in the oxygenating column, calculated from the formulas tan 30" = 0.5R R cos 45O = R+r
(3)
is 1.16 and 1.64 mm for the 4-mm beads and 1.74 and 2.46 mm for the 6-mm beads (Figure 5). Surrounding the outside of the bead chamber was a silicone-coated, meshed Dacron defoamer. The outside cylinder served as the reservoir. A heat exchanger, consisting of tightly wound plastic tubing, was placed in the bottom of the reservoir. The oxygen, which entered just distal to the narrowest
0019-7882/78/1117-0002$01.00/0 0 1978 American Chemical Society
Ind. Eng. Chem.
ProcessDes. Dev., VoI. 17,No. 1, 1978 3
Figure 3. Diagram shows closest bead packing arrangement.
1 BEAD CHAMBER LOIFFUSER 3 M N T U R I TUBE INLET PORT 5BLoOD INLET PORT
Figure 4. Diagram shows how bead packing arrangement permits two types of interstices. This theoretically allows the creation of two sizes of bubbles. The smaller interstice on the right is created by four adjoining beads and the larger interstice on the left by six adjoining beads. There are six large and eight small bubble contact sites on each
bead.
Figure 2. Diagram of oxygenating column showing venturi tube in lower chamber and beads in upper chamber.
point of the venturi tube, increased the negative pressure at, that point. This acted as a weak suction pump which facilitated blood flow into the oxygenator. Because blood entered
the venturi tube at an angle, a vortex was created which helped to mix the blood and oxygen before they entered the bead column. Oxygenation took place as blood and oxygen mixed in the upper portion of the venturi tube and percolated through the head chamber. Assuming oxygen to he incompressible at the pressures involved, for the 4-mm beads, there were 243 cm3 of oxygen, 243 cm3 of blood, and a theoretical bubble surface area of 1.10 m2 in the head chamber a t any one time. For the 6-mm heads, there were 237 cm3 of oxygen, 237 cm3 of blood, and a theoretical bubble surface area of 0.72 m*. For both sized heads, at a blood flow of 3 Llmin, blood velocity through the oxygenating column was 7.3 cmls and the oxygen-blood contact time was 6.5 s. At 4 Llmin, blood velocity was 9.7 cmls and contact time 4.8 s. At 5 Llmin, blood velocity was 12.1 cm/s and contact time 3.9 s. Oxygenated blood exited through the top of the upper chamber. It was defoamed and collected in the reservoir.
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Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
Both loops were connected through T connectors, and blood could be shunted from one loop to the other by switching clamps. Blood temperature was maintained at 37 "C in both loops by heat exchangers either in the oxygenators or in series with the Tygon tubing. Fresh, heparinized canine blood was first drained into the right loop. Here it was cycled repeatedly while a deoxygenating gas mixture of 95% nitrogen and 5% carbon dioxide was being passed through the oxygenator. Blood samples were evaluated for ~ ( 0 ~ p (COz), 1 , p(H), oxygen saturation, and temperature until a steady situation was obtained that approximated venous blood with a p ( 0 2 ) of 35-40 mmHg and a p(C0z) of 45 mmHg. The deoxygenator gas supply was then terminated and the left loop opened. Blood flow was set by the roller pump and oxygen flow by an Ohio oxygen flowmeter. The blood was pumped once through the test oxygenator and then analyzed for ~ ( O Zp(COz), ), p(H), and oxygen saturation. This process was repeated as many times as was deemed necessary to confirm reproducibility. The oxygen transfer rate was then calculated according to a modification of the Galletti (1972) formula Vo, = 10 &l(Sat(Oz)o - Sat(O2)i)HCT X 0.308 X 1.34
+ (1- HCT) X 0.9471
(4)
Figure 5. Method for calculating bubble size on known bead diameter and known shape of interstices. A, In the three-bead configuration, the chord of the bead is equal to the radius, and the joining of three chords forms an equilateral triangle with each apex being 60'. B, In the four-bead configuration, joining of radii of diagonally opposite beads with radii of adjacent beads creates a 45' angle.
The first term of the equation represents oxygen found in hemoglobin while the second term is the oxygen in physical solution; 0.308 is an empirical regression coefficient relating blood hemoglobin concentration to hematocrit; 1.34 is the mass of oxygen (in cubic centimeters) able to combine with 1g of hemoglobin; 0.617 is the water content of red blood cells; and 0.947 is the water content of plasma. Oxygen transfer capabilities were determined in the 4 and 6-mm bead column systems and in the more common bubble oxygenators in clinical use, i.e., Harvey H200, Bentley Q l O O , and Trevenol6LF.
Method. Oxygen transfer efficiency data were obtained from the test apparatus shown in Figure 6. This apparatus consisted of two connecting loops, a deoxygenator (right) loop and an oxygenator (left) loop. The deoxygenator loop consisted of a Bentley QlOO oxygenator (because of its availability) and a calibrated Sarns roller pump. Tygon tubing completed the circuit between blood inlet and outlet ports. The oxygenator loop was identical with the deoxygenator loop except that different oxygenators were used in the circuit.
Results Each point on the curves in the figures represents the average of at least four data points obtained from the experiments. The oxygen transfer in milliliters per minute in relation to different oxygen-blood flow ratios is plotted separately for each oxygenator in Figure 7. As the increase in oxygen content is not linearly related to the partial pressure of gaseous oxygen, and variations in the oxygen content of blood at the inlet will partially determine the volume of oxygen able to enter the blood, the changes in ~ ( 0 2 and ) oxygen saturation are not
B
91% N2 5 % cop-
HEAT EXCHANQER
-
BENTLEY #OXYOENATOR
HEAT EXCHMR
Figure 6. Schematic diagram of test loop used for determining oxygen mass transfer. The deoxygenating loop is on the right and the oxygenating loop on the left.
Ind. Eng. Chem. Process Des. D e v . , Vol. 17, No. 1. 1978
5
OXYGEN CONTENT CURVES AT 3 L I T E R S / M I N BLOOD F L O W
BEADS
- 4mm
0
k E A O S - 6 m m I
0
-
0
-
0
4.0
I
1
P
1.0
2.0
3.0
02/BLOOD F L O W RATIO
0 3
b,,
4 (I/min)
5
Figure 8. P,erformance curves relating mass transfer to ~ oxygen saturation a t a blood flow of 3 L/min.
Figure 7. Mass transfer curves showing transfer of oxygen into blood in mL/min a t varying blood flows and oxygen-blood flow ratios for the separate oxygenators.
OXYGEN CONTENT CURVES AT 4 L I T E R S / M I N BLOOD F L O W
r
expressed as absolute values. Rather, the ratio between change in oxygen content and the oxygen content at the inlet is plotted for 3 , 4 , and 5 Llmin blood flows for each oxygenator at 1:1,2:1, and 3:l oxygen-blood flow ratios. These ratios appear in Figures 8,9, and 10.
Discussion These experiments were conducted under highly controlled laboratory conditions in which a living animal was not used as the reciprocal portion of the oxygenator. They are far removed from the operating room situation, and no attempt is implied to show the superiority of one oxygenator over another. The system discussed in this report was not tested for other important properties such as microbubble production, platelet survival, protein denaturization, etc., which would be necessary before clinical application was contemplated. The purpose of this report, therefore, is only to demonstrate a mechanism for controlling bubble parameters and the effect this mechanism can have on the mass transfer of oxygen. The area across which oxygen may be transferred is the total surface area of all bubbles which may be in a column of blood at any one time. While the volume of a bubble is a function of the radius cubed, its surface area is a function of the radius squared. Consequently, the production of large bubbles creates an imbalance between oxygen availability and surface area across which it can be transferred. Therefore, it would appear advantageous to divide the oxygen flow into a large number of very small bubbles so that each bubble's volume approaches the surface area across which its oxygen is to be released. Unfortunately, however, the gradient of chemical potential for oxygen, which is at a partial pressure of 760 mmHg in the bubble, is approximately 17 times the gradient
( 0 2 and )
12.0 I4'O
I
0--4 BENTLEY
&--+
TRAVENOL
F L - ~ HARVEY
0
LB E A O S , 6 m m
6 0 1
4.0
c
0'
0-.4 B E A D S , 4 m m
I
I
G40t
0
1.0
2.0
3.0
Figure 9. Performance curves relating mass transfer to ~ ( 0 2 and ) oxygen saturation a t a blood flow of 4 L/min.
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Ind. Eng. Chern. Process Des. Dev., Vol. 17, No. 1, 1978
OXYGEN CONTENT CURVES AT 5 L l T E R S i M I N BLOOD FLOW
&--Q
TRAVENOL
(3 8.0
0,
8 fL a
6.0
-LO 4.0
0‘
Figure 11. Schematic diagram for analytical analysis.
1
I
thus the bubble at each Z position is pure oxygen. This implies that there is no diffusion resistance within the bubble itself. (5) The diffusion resistance of the interface between the pure oxygen gas bubble and blood is negligible. Thus, the resistance to diffusion lies solely within the blood. Then for the elemental volume shown in Figure 11,a molar balance yields
I
a
-No2S 1.0
2.0
+ (No2 +
%
dZ)S - NO2’//AbudZ = 0
3.0
O,/BLOOD FLOW RATIO Figure 10. Performance curves relating mass transfer to ~ oxygen saturation at a blood flow of 5 L/min.
and
( 0 2 )
of carbon dioxide, which is a t a partial pressure of 45 mmHg in the blood. Therefore, in a very small bubble, the oxygen would diffuse into the blood, and the bubble would disappear before an appreciable amount of carbon dioxide could diffuse into the bubble. However, if a bubble were just large enough to avoid disappearance, it would still be small enough to pass through the defoamer mesh without contacting a siliconized surface. The bubble, not having collapsed, would pass into the reservoir where its buoyance, being extremely low due to its size, would prevent it from rising to the surface before being pumped out the arterial line. Accordingly, it is very important to have bubbles with the appropriate volume-surface area ratios so that safety is maintained and both adequate oxygenation and carbon dioxide removal are assured. The following analysis is an attempt to evaluate the oxygen concentration in the blood at the discharge of the bead bed or column. This is made in an effort to delineate the significant parameters affecting mass transfer within the device. Knowledge of these parameters is useful in obtaining a better understanding of the mass transfer mechanism itself, but more importantly it identifies key design areas. The analysis is compared with appropriate experimental situations to determine its predictive quality. For the small elemental control volume shown in Figure 11, an oxygen balance will be made with the following assumptions. (1) Steady-state conditions prevail. (2) All molar fluxes are with respect to a stationary observer. (3) The oxygen disappears in the control volume through the bubble surface into the blood. Actually, the oxygen diffuses through the bubble surface into the plasma, through the red blood cell membrane and hemoglobin, where it subsequently reacts chemically with the hemoglobin. This assumption then ignores the fact that blood is not a homogeneous fluid undergoing chemical reactions. (4) The carbon dioxide balance may be neglected, and
The molar fluxes can be written as
An approximation can be made to the mass transfer through the bubble surface (Noz”), if one makes use of the analogy between heat and mass transfer a*.
No2 = k(X02s - x02)
(8)
In eq 8, X Ois the ~ ~oxygen mole fraction at the surface of the bubble, and this is assumed to be the mole fraction of saturated blood a t the temperature and pressure of the system. The X02 is the missing cup mole fraction a t any elevation 2 in the bed. The mass transfer coefficient k is obtained from the heat transfer analogy with a sphere
Rearranging eq 7 , one obtains
Combining eq 10, 8, and 5, one obtains
(11)
Performing the indicated operations, there results an ordinary differential equation describing the molar transfer of oxygen
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
under the assumptions previously stated as
-c
The bubble surface area per unit length (Abu) can be found by equating the oxygen mass flow to a function of the bubble formation rate MOZ
= P02fibu* [(4/3)srbu3]
(13)
Therefore the number of bubbles formed per unit time N b u * is
The bubble transit time through the bead bed is equal to the bubble path length divided by the interstitial velocity. But the interstitial velocity is equal to the superficial velocity divided by the void fraction t ) as
Thus, the bubble transit time is tt =
Lbu y
From these equations, the bubble area per unit length is
If one now makes the additional assumption that the mass transfer by diffusion in the 2 direction is small compared to the mass transfer through the bubble surface within the control volume, then eq 1 2 simplifies to
The boundary conditions for the bead column are
z= 0; xoz = xo2i z= L ; xo2 = Xozf
(18)
Integrating eq 17 with these boundary conditions, one obtains as a final equation 1
(1 - Xozi)
(1 - X02f)
Equation 19 can be solved in an iterative manner to predict the final oxygen concentration exiting from the bead column under the assumptions previously stated. In an oxygenator, bubbles are created by passage of oxygen through a diffuser. In the bead column, Bentley and Harvey oxygenators, the diffuser is a perforated plastic plate, whereas in the Travenol oxygenator it is a core of porus silicate (pumice). Because of limitations of space, coalescence of bubbles, and variations in oxygen and blood flows, it is presently impossible to design a diffuser which will create bubbles of uniform size in a consistent and continuous manner and have them maintain their size over any appreciable distance. The diffuser in the present system has special significance. Since the bubble chamber is filled with beads, a resistance to flow will be encountered which is not present in production
7
bubble oxygenators. To help nullify the resistance, it is critical that thorough mixing of oxygen and blood occur so that a uniform mixture with low density is presented to the bead column. This is expressed by the Fanning friction formula (Bird et al., 1962) as v2 p = fL D p2g Therefore, for resistance to be kept low, the column ideally should be wide and short, the velocity of the fluid slow, and the density of the fluid low. Obviously, many factors help to determine the size of the column, and blood flow requirements determine the velocity. Therefore, it is in the density of the blood-oxygen mixture that an advantage can be obtained by careful diffuser design, the objective being to have the blood foamed as much as possible before entering the bead column. If a bubble oxygenator could be designed to be as efficient as the lung, only 200 mL/min of oxygen would need to be delivered to the blood (Vander, 1970). Therefore, a t a 5 L/min blood flow, an oxygen-blood flow ratio of only 0.04:l would be required. However, since the bubbles play a vital role in removing carbon dioxide, it is not practical to reduce the number and size of bubbles below that which is necessary to maintain normal p(CO2).In these experiments, 6-mm beads decreased the p(CO2) between 5 and 10 mmHg a t a 1:l flow ratio. The 4-mm beads produced better oxygen transfer efficiencies but retained carbon dioxide. Since an important parameter in determining bubble size is carbon dioxide removal, it becomes necessary, once bubble size has been established, to concentrate on other aspects of the bubble to increase oxygen transfer efficiency. One of the most important aspects is boundary layer adherence. The significance of this, although not the rationale, was recognized in 1950 [see Stokes and Flick (1974)]. When a bubble is released into a column of desaturated blood, the blood which forms the surface of the bubble becomes quickly oxygenated. As this blood becomes saturated, the gradient for transfer of oxygen out of the bubble disappears. The situation will then stagnate unless the bubble's boundary layer of oxygenated blood is removed and a new layer of desaturated blood is applied. In effect, each bubble acts as its own filmer. This problem is compounded in all present day bubble oxygenators by the fact that oxygen and blood both move in the same direction. As oxygen-blood flow ratios approach equality, the shearing effect between bubble and blood decreases and the depth of the boundary layer increases. T o a degree, this problem is offset by boundary layer reorientation resulting from turbulence. However, when special attention is paid to this detail, more efficient oxygen transfer occurs. For example, the filming action between bubbles and columns probably accounts for the enhanced performances of the Harvey oxygenator in these experiments as well as those reported by Page and Haller (1974). In the bead chamber, as the bubble encounters each new bead, the boundary layer is reoriented and renewed, thereby continuously applying desaturated blood to the bubble surface and decreasing the diffusion resistance for oxygen transfer. The significance of boundary layer adherence can be appreciated when one recognizes that as the flow of blood through an oxygenator increases, the blood-bubble contact time decreases. Yet, despite this, our data demonstrate that, generally, the mass transfer of oxygen increased as the blood flow increased. This is explained by the greater turbulence at higher flows and, consequently, the greater frequency with which reorientation of the boundary layer occurs. Finally, various maneuvers can be employed to increase the length of bubble-blood contact. Theoretically, it is possible
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Ind. Eng. Chem. Process Des. Dev., Vol 17, No. 1, 1978
500
L
& .+--*---
&-- - 4-- - -o
0 - 6 0 * / 6 b , = 1.0
a - bo2/6bl’2.0
&e = diameter of bead, mm Dbu = diameter of bubble, mm do = diameter of holes in diffuser, mm 0 0 2 - ? , = diffusion coefficient for oxygen into blood, cm2
S-
= volume void fraction, dimensionless f = friction factor g = gravity HCT = hematocrit, % It = mass transfer coefficient, mol cm-2 s-1 K1 = experimental constant in eq 1when p , u, do are in the units given below, 0.012 K2 = experimental constant in eq 1when p, u, do are in the units given below, 0.07 L = length & b u = bubble path length, mm = mass flow rate of oxygen, g s-1 Nbl = molar flux of blood with respect to a stationary observer, mol cm-2 s-l $bu* = number of bubbledunit time, no. s-1 No2 = molar flux of oxygen with respect to a stationary observer, mol cm-2 s-1 N02”’ = volume rate of disappearance of oxygen, mol cm-3 6
0 - 602/Qbl =3.0
-EXPERIMENTAL --- THEORETICAL
Figure 12. Comparison of experimental and theoretical results for 6-mm bead oxygenator. Actual and predicted mass transfer curves show transfer of oxygen into blood in mL/min at varying blood flows and oxygen-blood flow ratios.
to administer a bubble of oxygen at the lower end of a column of desaturated blood and retrieve the same bubble composed almost entirely of carbon dioxide, a t the upper end. However, constrictions of time and space make this ideal circumstance impossible. Nevertheless, variations in local geometry can extend blood-bubble contact while retaining the dimensions of a small bubble chamber. For example, folding of the bubble chamber in the Bentley oxygenator triples the length of the bubble column while still maintaining compactness. In the bead column, the bubble is made to traverse a tortuous path through the beads. Because the bubble size is small compared to the bead, a theoretical path length can be calculated based on the assumption that each bubble will traverse one half the circumference of each encountered bead until it exits a t the top of the column. Accordingly, the bead chamber which is 27 cm in length provides an effective bubble travel path of 47 cm. The theoretical accuracy might be evaluated for a typical situation such as that illustrated in Figure 12. In this particular test situation, the predicted oxygen mole fraction at the exit is of the order of 20% lower than that value obtained from the experiments. The figure shows the corresponding mass transfer rate error which is obtained for various blood flows and oxygen to blood flow ratios. It should be noted that the shape of the curves at each oxygen to blood flow is the same. The theoretical error may be explained by the fact that the experimental data were taken from the oxygenator outlet port, rather than at the exit of the bead column. Accordingly, blood reaching the deformer material has a further opportunity to obtain atmospheric oxygen as the bubbles collapse and run down the defoamer fibers. In addition, the mass transfer coefficient, when evaluated from eq 9, can be in error by
*lo%.
In the future, bubble oxygenators may be replaced by membrane oxygenators. For the present, however, it is interesting to comtemplate the physiology of gas transport in a bubble and the various methods by which the performance of bubblers might be improved. Nomenclature A = experimental constant in eq 1when p , u, do are in the units given below, 1.90 Abu = surface area of bubblehnit of length, cm c = total concentration, mol D = diameter
S-1
P = change in pressure p = density Po2 = partial pressure of oxygen, mmHg
pH = acid-base index, dimensionless bl = volume flow rate of blood, cm3 s-l 02 = volume flow rate of oxygen, cm3 s-1 R = radiusofbead r = radiusof bubble S = effective cross-sectional area for blood flow in bead volume, cm2 Sat02 = % saturation of oxygen in blood tt, = time for bubble to traverse full bead column length, s V = volume, cm3 V2 = velocity squared us = superficial velocity, cm s-1 u.t = interstitial velocity, cm s-1 VOZ= mL/min of oxygen transferred to the blood X O Z = mole fraction of oxygen in blood, dimensionless 2 = distance measured from base of bead column, cm p = viscosity in 10-3 g/cm s u = surface tension, dyn/cm
8
Subscripts be = bead bl = blood bu = bubble f = properties evaluated at film concentration i = in max = maximum possible hemoglobin bound oxygen 0 = out 0 2 = oxygen s = surface L i t e r a t u r e Cited Anderson, M. N., Kuchiba, K., J. Thorac. Cardiovasc. Surg., 57, 238 (1969). Bird, R. E.. et al., “Transport Phenomena”, Wiley, New York, N.Y., 1962. Galletti, P. M.. et al., Trans. Am. SOC.Artif. Intern. Organs, 18, 359 (1972). Page, P. A., Haller, A. J., J. Thorac. Cardiovasc. Surg., 82, 213 (1974). Stokes, T. L., Flick, J. B., J. Thorac. Cardiovasc. Surg., 62, 213 (1974). Vander, A. J., et al., “Human Physiology”, McGraw-Hill, New York, N.Y., 1970.
Received for review January 13, 1975 Resubmitted July 23, 1977 Accepted August 1,1977 Figures 1to 10 are reproduced, with permission, from G. L. Hammond and W. W. Bowlcy, J.Thorac. Cardiovasc. Surg.,71,422-428 (1976). Copyright 1976 C. V. Mosby Co.
