1V[ed. & Biol. Eng. & Comput., 1977, 15, 106-117
Blood oxygenation in coiled silicone-rubber tubes of complex geometry U. B a u r m e i s t e r
D.F. James
W . Zingg
Institute of Biomedical Engineering, University of Toronto, Toronto, Canada
A b s t r a c t - - O x y g e n transfer to blood and water was investigated experimentally for steady flow in a "complex coil, a silicone-rubber tube formed into a sequence of helical coil sections of alternating orientation. The series of reorientations caused the secondary flow induced by the helical motion to be continually re-established, so that mixing of the liquid was enhanced. Complex coils of various configurations were tested in an oxygen atmosphere and a significant improvement of the oxygen-transfer to the liquids was measured while the pressure drop increased moderately. The case of ideal oxygen transfer, i.e. a perfectly mixed fluid, was treated theoretically and a comparison of these results with the experimental data demonstrated how much the fluid-side resistance was reduced by complex coiling. K e y w o r d s - - B l o o d oxygenation, Augmentation of gas transfer, Secondary flow, Helical coil, Mixing
Symbols a = internal tube radius, cm al, a2, a3 = coefficients of the modified Hill equation (eqn. 5) c = concentration, m o l e / c m 3 c~ = amount of oxygen per unit blood v o l u m e b o u n d to haemoglobin at full saturation (s = 1), cm 3 (s.t.p.)/100 ml blood or m o l e / c m 3 d = internal tube diameter, cm dn = hydraulic d i a m e t e r = (4-times area)/ circumference), c m D = coil diameter, cm De =- Rea/(d/D), D e a n n u m b e r D~, = oxygen diffusivity in the membrane, cm2/s
f =- Ap(d/L)/(v 2 g/2), friction factor f~ = 64~Re, friction factor for laminar flow in straight tubes G =- 4(~D)m/(~D)wb, dimensionless membrane permeability Hb = total haemoglobin concentration of the blood = (g haemoglobin)/(100 ml blood), g ~ L = tube length, c m P = ( P - Po)/(P~- Po), dimensionless partial pressure of oxygen P : = dimensionless partial pressure P in the ideally mixed fluid p = partial pressure (of oxygen), m m H g pco~ = partial pressure of carbon dioxide mmHg po2 = partial pressure of oxygen, m m H g pg = gas-side partial pressure of oxygen, mmHg First received 2Oth November 1975 and in final form 7th July 1976
106
inlet oxygen partial pressure of the fluid, m m H g Pa = outlet oxygen partial pressure of the fluid, m m H g Ap = static pressure drop along the tube axis, m m H g p H = standard measure for the concentration of free hydrogen ions Q = flow rate, m l / m i n R -= r/a, dimensionless radial co-ordinate r = radial co-ordinate, cm Re =- vd/v = 2a v/v, Reynolds n u m b e r s = degree of oxygen saturation of blood haemoglobin (0 ~< s ~< 1) Sc = (v/Do2)wb, Schmidt number for oxygen diffusion in whole blood ( = wb) T = temperature, deg C v = m e a n velocity, cm/s W =- w/a, dimensionless wall thickness w = wall thickness, cm Z ~- (z/a) G/(Re Sc), dimensionless axial coordinate z = axial co-ordinate, cm (eD)m = m e m b r a n e oxygen-permeability, m o l e / (s c m m m H g ) ~t: = effective oxygen-solubility o f the fluid, for blood, ~t: is defined by eqn. 4, mole/(cm a mmHg) ~t:* = r dimensionless effective oxygen solubility for whole blood Ctm = m e m b r a n e oxygen solubility, m o l e / (cm 3 m m H g ) ~twb = oxygen solubility of the whole blood, including plasma and red cells, m o l e / (cm 3 m m H g ) V = kinematic viscosity, cm2/s Po =
Medical & Biological Engineering & Computing
March 1977
1 Introduction
IN THE artificial lungs currently in use for open-heart surgery, oxygen is transferred to the blood in an extracorporeal circuit by bringing pure oxygen into direct contact with the liquid. This gas-blood interface promotes a rapid mass transfer, but it also causes serious blood damage (LEE et aL, 1961), limiting the use of these devices to a few hours. Since much longer periods are required for some clinical treatments, e.g. reversible respiratory insufficiencies as caused by pneumonia, fat embolism or burns (PEIRCE, 1972), there is considerable interest in the development of 'membrane oxygenators' (DRINKER, 1972). In these devices, the blood and oxygen are separated by a thin gas-permeable membrane, which significantly reduces blood damage but also inhibits gas transfer. With a membrane, there is a b l o o d membrane interface instead of a blood-gas interface, and although the blood damage is much less in the former case, it is still this factor which limits the duration of application. Much of the current research work therefore centres on developing membrane oxygenators which are efficient and cause a minimum amount of blood damage. The analysis of membrane oxygenators has shown (MocKROS et al., 1971) that the resistance to gas transfer across silicone-rubber tubes, or membranes founded into channels, can be made small compared with the resistance across the laminar film of blood. Accordingly, gas transfer can be improved only by reducing the fluid-side resistance, e.g. by mixing the blood, and the demand for such improvements will likely increase as more permeable membranes become available. Improving mass transfer is a common problem in industry but, when blood is the working fluid, some special restrictions rule out the usual engineering solutions. These restrictions essentially reduce to the requirement that the shear stress at the wall must be within a certain range; stresses that are too low promote thrombus formation (MADRASe t al., 1971), while ones that are too high cause haemolysis (BLACKSHEAR, 1972). As a consequence it is widely accepted that the flow should not be turbulent, which would otherwise be an excellent mechanism for mixing, and the boundary layer should not separate. Of the various techniques developed to increase the convective mass transfer, the use of laminar secondary flow appears attractive because the motion can be confined to meet these conditions and still achieve efficient mixing. A good example of secondary-fluid motion occurs in flow through a curved tube (DEAN, 1928). The primary flow is along streamlines parallel to the tube axis, but centrifugal forces due to the tube curvature set up a secondary motion perpendicular to the main flow, i.e. a fluid particle has a velocity component in planes normal to the tube axis. Typical fluid trajectories in these planes are shown in Fig. 1 for an Medical & Biological Engineering & Computing
established steady flow through a helically coiled tube. The strength of the secondary flow is a function of the Reynolds number Re and the curvature ratio D / d , where D and d are the coil diameter and internal tube diameter, respectively. For large values of D/d, dynamic similarity is governed by a single dimensionless group termed the Dean number, De -- Re ~/(d/D) (WHITE, 1929). Some studies in heat transfer (KALB and SAEDER, 1974) have shown that the transfer rate is increased by the secondary motion, with only a moderate increase in pressure loss. The first application of this technique to blood oxygenation was made by WEISSMANand MOCKROS (1969). Although their predictions of improved oxygen transfer proved too optimistic, it was shown both experimentally and theoretically by DORSON et al. (1971) that the membrane area could be halved by coiling the tubes. With this improvement, but with a considerable blood-side resistance remaining, they developed tube oxygenators to the stage of experimental clinical application. Numerous other means of inducing secondary flow in the blood have been considered, and these have been discussed in a review by RICHARDSON (1971). One of the most worthwhile of these is the method of 'actively' inducing secondary flow by oscillating a curved channel about its centre (DRINKER, 1972). By adjusting the parameters of the oscillation, the convective transport can be increased to the point where the gas-exchange process is practically membrane-limited. This and other active methods have the disadvantage that additional mechanical equipment is needed for the oxygenator, ~)
~'
ECTIONA
CROSSSECTION A~ . . ~ ~
Fig. I Secondary flow streamlines Established laminar flow in helical coils at low Dean numbers and large values of (DEAN, 1928)
D/d
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107
but these methods have demonstrated that secondary flows can greatly increase the mass transfer. There is no indication that secondary flows have any additional detrimental effects on the blood. Passively induced secondary flows, on the other hand, are not as strong as those actively induced, but oxygenators employing this principle for clinical use have the advantage that no moving parts are required. This simplifying feature leads to the possibility of developing a mass-produced disposable oxygenator, but the outstanding problem with this technique is the relatively low efficiency. Our overall objective, accordingly, was to improve the efficiency of the passive method, and the work reported herein is part of that programme. The base for our work is secondary flow in a helical coil. In the steady state, a stable secondary flow is established, like that shown in Fig. 1, and when the oxygen concentration gradient across the tube wall is nonzero, a stable concentration profile likewise is developed. The lines of constant oxygen concentration in blood in the steady state have been calculated by WEISSMANand MOCKROS (1969) and are remarkably similar to the streamlines depicted in Fig. 1. Our approach is to improve fluid mixing by continually disturbing this stable secondary flow---essentially by periodically reorienting the curvature of the helix so that the direction of the secondary motion is altered. This technique will be referred to as 'complex coiling' and some examples of coils in this configuration are shown in Fig. 2. Each complex coil consists TYPE /dlSO"
TYPE 2190"
of a series of coil sections, each with the same number of turns, arranged such that the axes of adjacent coil sections form a given angle, generally 90 or 180 ~ By this configuration, secondary flow must be re-established in each coil section, and the direction of the secondary motion will then be 90 or 180 ~ to the original direction. F o r example, the direction of the centripetal force in Fig. 1 is right to left; if the coil axis is changed by 90 ~ the direction will then be vertically upwards or downwards, and after a 180 ~ change, the force will be right to left. This continual reorientation should improve mixing inside the tube, and the problem then is to find the magnitude of the increase in convective mass transfer. It is difficult to estimate the increase by theoretical means, not only because of the complex geometry and the nonlinear properties of the blood, but also because of the transitory state of the fluid motion throughout. Accordingly, an experimental programme was undertaken and this proceeded in three steps: (a) the secondary motion in a complex coil was investigated qualitatively by a dye injection study, (b) measurements of oxygen transfer and pressure drop were made in straight and complex coils with water, (c) similar measurements were made with blood. A basic problem in a study of this sort is to determine how close any improved mixing is to ideal radial mixing. The experiments may show that oxygen transfer in a complex coil is greater, say, than in a straight coil, but it is also important to indicate how much of a gap still remains for ideal radial mixing, i.e. perfect mixing of the fluid so that gas transfer is limited only by the membrane. Oxygen transfer with ideal mixing can be found only from theoretical work, of course, and consequently this 'membrane-limited' case was investigated for flow in circular tubes. This analysis is given in the next Section, prior to the description of the experiments, and the results will not only provide a reference curve to contrast with the experimental data, but will also be used to estimate how much the variation of blood properties, like pH, p C O : and temperature, can influence the oxygen uptake. 2 M'~mbrane-limited case
"--4d Fig. 2 Segments of three complex coils tested in the experimental work The number in the type code represents the number of turns in one coil section, and the second number indicates the angle between the axes of two adjacent coil sections. A full coil has 56 turns
108
The situation analysed in this Section is oxygen transfer to blood or to water flowing in a gaspermeable helically coiled tube surrounded by oxygen at a constant partial pressure pg. The essential idealisation is that radial mixing within the tube is perfect. Hence, the analysis basically reduces to finding the gas concentration, and therefore the gas transfer, in the tube wall. The situation is assumed to be axisymmetric so that the problem is described by the circular cylindrical co-ordinates (r, z); this presupposes that the wall thickness w is much less than the coil diameter D, i.e. w/D .~ 1,
Medical & Biological Engineering & Computing
March 1977
and the coil radius is much larger than the tube radius. The concentration c, or equivalently the partial pressure p, of the oxygen in the liquid is constant over the cross-section due to perfect radial mixing, i.e. the partial pressure p(r, z) equals p(a, z) for r ~< a, where a is the internal tube radius. Therefore the problem is to find p(a, z), the change in p in the downstream direction. The inlet partial pressure is Po. Several authors have analysed similar situations for a fiat membrane or a straight tube (WEmsMAN 1969, BISCHOFF and REOAN, 1971), both neglected axial diffusion within the membrane, and did not state any limits for the validity of this simplification. Other authors (DoRsoy et al., 1971) have considered a more simplified situation in which the partial pressure p(a, z) did not vary along the tube axis, i.e. the partial pressure difference (pg-p(a, z)) was kept constant at the inlet value (pg-po). This simplification will not be made here since the change of the partial pressure p(a, z) with z was considerable for some experiments. In addition, the axial diffusion will be retained in the first part of the analysis. This allows us to state a general condition for which the axial diffusion can be neglected. Within the tube wall (a ~< r ~< a + w), the governing equation for oxygen transfer is ~c/t3t = D,, A2 c = (aD)m A2 p, where Dm is the 02-diffusivity and (aD)m the O2-permeability of the tubular membrane. Therefore, in a steady state the governing equation is r
~r
r
+ 7
= 0
. . . .
(1)
The boundary conditions are specified at the inlet (z = 0), at the outer surface (a+ w, z) and at the inner surface (a, z)
p(r, O) = po . . . . . . . . .
(2a)
p(a + w, z) -- pg . . . . . . . .
(2b)
tgp (a, z) = v t3p (a, z) aaf c~-r~ 2~. D~
(2c)
When these substitutions are made, the result is eqn. 2c. The substitution t3c = aft?p can be applied straightaway for water, for which ~y can be taken constant at a given temperature. But in blood when the oxygen becomes bound to the haemoglobin the equivalent of af is a function of p and other parameters. In this case, the differential increase in oxygen concentration is given by d c = =wb d p + c~ ds. The first term (awbdp) is the direct result of Henry's law and represents the increase of oxygen in physical solution, hence =,~b is the 'physical solubility' for whole blood. The second term (c~ ds) represents the increase in oxygen which is bound to the haemoglobin, and is given in terms of the increase in saturation s. The saturation depends primarily on the oxygen partial pressure p in the blood plasma, and is defined as the ratio of the number of molecules bound by haemoglobin to the maximum number that can be bound. Full saturation, s = 1, is reached at partial pressures around 1 5 0 m m H g when each haemoglobin molecule has four oxygen molecules bound to it. The resulting oxygen concentration in the blood when s = 1 is the saturation concentration c~, and this quantity must be proportional to the amount of haemoglobin present, specifically, Cs = 1-36Hb where Hb is the haemoglobin concentration in grams per cent and c, has units of cm 3 (s.t.p.)/100 ml blood). The oxyhaemoglobin-dissociationcurve is the plot of s as a function of p, with T (temperature) and pH as parameters, and these curves are known for various species (ALTMAN, 1971). At constant T and pH, ds = (ds/~3P)pH. r d p , SO that the preceding expression for dc may read ~s
This has the form of Henry's law for a liquid, i.e. dc = af dp, where the af for blood is understood to be af = awb+
The last condition is not obvious for it is a statement for the mass balance of the gas in the liquid, derived by considering a disc-shaped element of liquid of radius a and thickness dz. The amount of gas transport into the element in the radial direction is Dm t3c/~r (a, z) 2~radz, which equals the increase of convective gas transport in the axial direction, v c3c/~z (a, z) ~ra2 dz (diffusion in this direction is negligible by comparison), v is the mean velocity of the liquid in the tube. Since ~c/~r (a, z) is the radial concentration gradient within the membrane at r = a, t3c may be replaced by a,,~3p, and since 8c/#z (a, z) is the axial gradient in the liquid, the correct replacement for #c in this quantity is t3c = otfc3p, where ~f is the 02-solubility of the fluid.
Medical & Biological Engineering & Computing
-Z--
\ op ]pH, r
e.
.
.
.
.
.
(4)
Since aw~ is known and cs = 1.36Hb, finding this 'effective solubility' =s means finding values of (t~s/~P)pn, r- These were estimated from the modified Hill equation (RossING and CAIN, 1966) log s/ (1 -- s) = ax log p [mm Hg] + a2 + a3 (pH, T)
(5)
where the constants were chosen (al = 2.9 and a2 = - 3 . 0 5 ) to fit data for pig blood (0.2 ~ s ~< 0-98), since this type was used in the oxygenation experiments. It was presumed that the function a3 (pH, T ) = 1.18 ( p H - 7 ) - 0 . 0 4 7 T[~ was the same for pigs as for dogs, since this form of
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a3 (pH, T) was found by RossI~G and CAIN (1966) for dog blood. Hence, the dependence of as on p was established with pH, T and Hb as parameters. We return our attention to the governing equations and, as the first step, in seeking solutions, make the problem dimensionless. Let the dimensionless co-ordinates be R =- r]a and Z =- (z/a) G/(Re So) and let the dependent variable be P - (p-po)/(p,-po). Here G ~ 4(aD)~/(aD),~b, which is four times the ratio of the membrane permeability to the blood permeability; Re = v2a/Vwb is the Reynolds number of the blood flow, where vwb is the kinematic viscosity, and Se =- (v/D)~b is the Schmidt number for the oxygenation of whole blood. In this way, eqns. 1 and 2 become
1
(~ [R gP'~
R ~g L
(
G
)2(~2P
"~R-) + R-eSc-
t?Z2 = 0
(6)
P(R, 0) = 0 . . . . . . . . .
(7a)
P(1 + W, Z) = 1 .
.
(7b)
~RC~P (1, Z) = as*{e(1 , Z) } -c~P ~ - (1, Z)
(7c)
.
.
.
.
.
.
where W =- w/a and as* = as/aw~, i.e. the ratio of the 'effective solubility' a s in eqn. 4 to the 'physical solubility' awb. For water or plasma (Hb = cs = 0), as* equals unity, but for blood, as* is a nonlinear function of P (and pH, T, pg, Po and Hb) where P refers to the blood, so that in eqn. 7c P is evaluated at the membrane-blood interface (1, Z): Since G/(ReSc) was typically 0(10 -2) for the oxygenation experiments, the axial diffusion term in eqn. 6 can in fact be neglected and the solution of the remaining equation is P(R, Z) = fl(Z) In R +f2(Z) where f l and f2 are undetermined functions found from the boundary conditions at R = t and R = 1 + W. At the inner wall, P(1, Z) = f2(Z) is just the dimensionless partial pressure of the fluid, henceforth denoted by Ps(Z). Since Ps is the nondimensional form of p(a, z), it is the objective of this analysis and so the solution for P(R, Z) will be expressed in terms of this function. When the other boundary condition, eqn. 7b, is substituted in the preceding equation for P, then for 1 ~ R
o\ .7
2
/,-
. _,,,o. ,, o,,,o-.
J"
O HELICAL COIL
,,
z w
z~ STRAIGHT
TUBE
o 10-I
,
z,
,
q
6
i
R
8
I
[
L
100
2
I
..
J
4
i
,
6
,
I
S
J
I
101
Z / In(l+W) Fig. 9 Coiling coefficient for hefical and complex coils The hydraufic diameter d l was used instead o f d to compute Reand De. The continuous curves were fitted to the data, and curve w is an empirical equation for hefical coils (D/d >~ 15.5) by WHITE (1929)
116
Medical & Biological Engineering & Computing
March 1977
accordingly a smaller priming volume. This would reduce the detrimental effects due to blood-surface interaction and allow more compact designs. Acknowledgment--The principal author (U.B.) was supported by a scholarship of the Canada Council. The assistance of the staff of the Hospital for Sick Children, Toronto, and of the Department of Mechanical Engineering at the University of Toronto, is gratefully acknowledged. References
ALTMAN, D. L. (1971) Respiration and circulation. Biological Handbook, Federation of American Societies for Experimental Biology BAURMEISTER,U. (1974) Oxygen transport to blood flowing in coiled semipermeable tubes. M.Sc. thesis, Univ. of Toronto BXSCHOFF,K. B. and T. M. REGAN (1971) Comments on diffusion in membrane-limited blood oxygenators. Amer. Inst. Chem. Eng. J. 17, 225 BLACKSHEAR,P. L. Jun. (1972) Mechanical haemolysis in flowing blood. In: Biomechanics, its foundations and objectives. Eds. Y. C. FUNG, N. PERRONE and M. ANLI~:ER, 501. Prentice Hall, New Jersey BUCKLES,R. G., E. W. MERRILL and E. R. G1LLILAND (1967) An analysis of oxygen absorption in a tubular membrane oxygenator. Amer Inst. Chem. Eng. J. 14, 703 DEAN, W. R. (1928) The streamline flow of fluid in a curved pipe. Phil. Mag. 4, 673 DORSON, W. J. Jun., LARSEN,K. G., ELGAS,R. J. and VOORHEES, M. E. (1971) Oxygen transfer to blood: data and theory. Trans. Am. Soc. Artif. Intern. Organs 17, 309 DRAVID, A. N., SMITH, K. A., MERRILL, E. W. and BRIAN, P. L. T. (1971)Effect of secondary fluid motion
on laminar flow heat transfer in helically-coiled tubes. tlmer. Inst. Chem. Eng. J. 17, 1114 DRINKER, P. A. (1972) Progress in membrane oxygenator design. Anaesthesiology 37, 242 KALR, C. E. and SAEDER, I. D. (1974) Fully developed viscous-flow heat transfer in circular tubes with uniform wall temperature. Amer. Inst. Chem. Eng. J. 20, 340 LEE, W. n., KRUMMHAAR,D., DERRY, G., SACHS, D., LAWRENCE, S. H., CLOWES, G. H. A. and MALONEY, J. V. (1961) Comparison of the effects of membrane and non-membrane oxygenators on the biochemical and biophysical characteristics of blood. Surg. Forum 12, 200 MADRAS, P. A., MORTON, W. A. and PETSCHECI(,H. E. (1971) Dynamics of thrombus formation. Fed. Proc. 30, 1665 MOCKROS, L. F. and. WEISSMAN, M. H. (1971) The artificial lung. In: Biomedical engineering. Eds. L. H. U. BROWN, J. E. JACO8S and L. STARK, 325. F. A. Davis Co., Philadelphia PEIRCE, E. C. II (1972) The role of the artificial lung in the treatment of respiratory insufficiency: A perspective. Chest 62, 107 S RICHARDSON, P. D. (1971) Effects of secondary ftow in augmenting gas transfer in blood. In: Advances in cardiology vol. 6. Eds. R, H. BARTLETT,P. A. DRINKER and P. M. GALETTI,2, S. Karger, Basel ROSSING, R, G. and CAIN, S. M. (1966) Dog oxyhaemoglobin dissociation curve. J. Appl. PhysioL 21,198 WElSSMAN, M. H. (1969) Diffusion in membrane-limited blood oxygenators. Amer. Inst. Chem. Eng. J. 15, 627 WEISSMAN, M. H. and MOCKROS, L. F. (1969) Oxygen and carbon dioxide transfer in membrane oxygenators. Med. & Biol. Eng. 7, 169 WHITE, C. M. (1929) Streamline flow through curved pipes. Proc. Roy. Soc. 123A, 645
Oxygenation du sang dans des tubes spirales en caoutchouc de silicone la g6om6trie comp|exe Sommaire---Le passage de l'oxyg~ne dans le sang et dans l'eau fut 6tudi6 de fa~on exp6rimentale pour un d6bit constant dans une 'spirale complexe', tube en caoutchouc de silicone constitu6 d'une s6rie de sections h61icoidales b. orientation alternge. La s6rie de r6orientations provoquait le r6tablissement continu du flux secondaire induit par le mouvement hdicoidal, de mani~re ~t accro~tre le m61ange du liquide. Des spirales complexes de diverses configurations furent soumises /~ des essais dans une atmosphere d'oxyg~ne et une am61ioration significative du passage de l'oxyg~ne dans les liquides fur mesur6e alors que la chute de pression s'intensifiait mod6r6ment. Le cas d'un transfert id6al d'oxygSne--par example dans un liquide parfaitement m61ang6--fut trait6 de fa~on th~orique et une comparaison de ces r~sultats avec les donndes exp6rimentales a demontr6 h quel point la rdsistance lat6rale du liquide 6tait r6duite par le syst~me de spirales complexes.
Sauerstoffanreicherung des Blutes in gewundenen Silikonkautschukrohren komplexer Geometrie Zusammenfassung--Sauerstoffiibertragung an Blut und Wasser wurde experimentell f'tir stetige Str~Smung in einer 'komplexen Spule' untersucht, einem Silikonkautschukrohr, das in eine Folge yon Wendelspulen alternativer Orientierung geformt wurde. Die Reihe yon Wieder-Orientierungen verursachte die fortgesetzte Wiederherstellung des v o n d e r Spiralbewegung induzierten Sekund~rstromes, so dab die Misehung der Fliissigkeit gef~Srdert wurde. Komplexe Spulen verschiedener Konfiguration wurden in einer Sauerstoffatmophiire geprfift und es wurde eine betr~ichtliche Verbesserung dee Sauerstofftibertragung an die Fltissigkeiten gemessen, wiihrend der Druckanfall ein wenig anstieg. DeE Fall der idealen Sauerstofftibertragung-d.h. an eine vollkommen gemischte Fltissigkeit- wurde theoretisch behandelt und ein Vergleich dieser Resultate mit den experimentellen Daten zeigte, um wieviel der Widerstand yon Seiten der Fliissigkeit durch komplexe Spulung verringert wurde.
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Blood oxygenation in coiled silicone-rubber tubes of complex geometry U. B a u r m e i s t e r
D.F. James
W . Zingg
Institute of Biomedical Engineering, University of Toronto, Toronto, Canada
A b s t r a c t - - O x y g e n transfer to blood and water was investigated experimentally for steady flow in a "complex coil, a silicone-rubber tube formed into a sequence of helical coil sections of alternating orientation. The series of reorientations caused the secondary flow induced by the helical motion to be continually re-established, so that mixing of the liquid was enhanced. Complex coils of various configurations were tested in an oxygen atmosphere and a significant improvement of the oxygen-transfer to the liquids was measured while the pressure drop increased moderately. The case of ideal oxygen transfer, i.e. a perfectly mixed fluid, was treated theoretically and a comparison of these results with the experimental data demonstrated how much the fluid-side resistance was reduced by complex coiling. K e y w o r d s - - B l o o d oxygenation, Augmentation of gas transfer, Secondary flow, Helical coil, Mixing
Symbols a = internal tube radius, cm al, a2, a3 = coefficients of the modified Hill equation (eqn. 5) c = concentration, m o l e / c m 3 c~ = amount of oxygen per unit blood v o l u m e b o u n d to haemoglobin at full saturation (s = 1), cm 3 (s.t.p.)/100 ml blood or m o l e / c m 3 d = internal tube diameter, cm dn = hydraulic d i a m e t e r = (4-times area)/ circumference), c m D = coil diameter, cm De =- Rea/(d/D), D e a n n u m b e r D~, = oxygen diffusivity in the membrane, cm2/s
f =- Ap(d/L)/(v 2 g/2), friction factor f~ = 64~Re, friction factor for laminar flow in straight tubes G =- 4(~D)m/(~D)wb, dimensionless membrane permeability Hb = total haemoglobin concentration of the blood = (g haemoglobin)/(100 ml blood), g ~ L = tube length, c m P = ( P - Po)/(P~- Po), dimensionless partial pressure of oxygen P : = dimensionless partial pressure P in the ideally mixed fluid p = partial pressure (of oxygen), m m H g pco~ = partial pressure of carbon dioxide mmHg po2 = partial pressure of oxygen, m m H g pg = gas-side partial pressure of oxygen, mmHg First received 2Oth November 1975 and in final form 7th July 1976
106
inlet oxygen partial pressure of the fluid, m m H g Pa = outlet oxygen partial pressure of the fluid, m m H g Ap = static pressure drop along the tube axis, m m H g p H = standard measure for the concentration of free hydrogen ions Q = flow rate, m l / m i n R -= r/a, dimensionless radial co-ordinate r = radial co-ordinate, cm Re =- vd/v = 2a v/v, Reynolds n u m b e r s = degree of oxygen saturation of blood haemoglobin (0 ~< s ~< 1) Sc = (v/Do2)wb, Schmidt number for oxygen diffusion in whole blood ( = wb) T = temperature, deg C v = m e a n velocity, cm/s W =- w/a, dimensionless wall thickness w = wall thickness, cm Z ~- (z/a) G/(Re Sc), dimensionless axial coordinate z = axial co-ordinate, cm (eD)m = m e m b r a n e oxygen-permeability, m o l e / (s c m m m H g ) ~t: = effective oxygen-solubility o f the fluid, for blood, ~t: is defined by eqn. 4, mole/(cm a mmHg) ~t:* = r dimensionless effective oxygen solubility for whole blood Ctm = m e m b r a n e oxygen solubility, m o l e / (cm 3 m m H g ) ~twb = oxygen solubility of the whole blood, including plasma and red cells, m o l e / (cm 3 m m H g ) V = kinematic viscosity, cm2/s Po =
Medical & Biological Engineering & Computing
March 1977
1 Introduction
IN THE artificial lungs currently in use for open-heart surgery, oxygen is transferred to the blood in an extracorporeal circuit by bringing pure oxygen into direct contact with the liquid. This gas-blood interface promotes a rapid mass transfer, but it also causes serious blood damage (LEE et aL, 1961), limiting the use of these devices to a few hours. Since much longer periods are required for some clinical treatments, e.g. reversible respiratory insufficiencies as caused by pneumonia, fat embolism or burns (PEIRCE, 1972), there is considerable interest in the development of 'membrane oxygenators' (DRINKER, 1972). In these devices, the blood and oxygen are separated by a thin gas-permeable membrane, which significantly reduces blood damage but also inhibits gas transfer. With a membrane, there is a b l o o d membrane interface instead of a blood-gas interface, and although the blood damage is much less in the former case, it is still this factor which limits the duration of application. Much of the current research work therefore centres on developing membrane oxygenators which are efficient and cause a minimum amount of blood damage. The analysis of membrane oxygenators has shown (MocKROS et al., 1971) that the resistance to gas transfer across silicone-rubber tubes, or membranes founded into channels, can be made small compared with the resistance across the laminar film of blood. Accordingly, gas transfer can be improved only by reducing the fluid-side resistance, e.g. by mixing the blood, and the demand for such improvements will likely increase as more permeable membranes become available. Improving mass transfer is a common problem in industry but, when blood is the working fluid, some special restrictions rule out the usual engineering solutions. These restrictions essentially reduce to the requirement that the shear stress at the wall must be within a certain range; stresses that are too low promote thrombus formation (MADRASe t al., 1971), while ones that are too high cause haemolysis (BLACKSHEAR, 1972). As a consequence it is widely accepted that the flow should not be turbulent, which would otherwise be an excellent mechanism for mixing, and the boundary layer should not separate. Of the various techniques developed to increase the convective mass transfer, the use of laminar secondary flow appears attractive because the motion can be confined to meet these conditions and still achieve efficient mixing. A good example of secondary-fluid motion occurs in flow through a curved tube (DEAN, 1928). The primary flow is along streamlines parallel to the tube axis, but centrifugal forces due to the tube curvature set up a secondary motion perpendicular to the main flow, i.e. a fluid particle has a velocity component in planes normal to the tube axis. Typical fluid trajectories in these planes are shown in Fig. 1 for an Medical & Biological Engineering & Computing
established steady flow through a helically coiled tube. The strength of the secondary flow is a function of the Reynolds number Re and the curvature ratio D / d , where D and d are the coil diameter and internal tube diameter, respectively. For large values of D/d, dynamic similarity is governed by a single dimensionless group termed the Dean number, De -- Re ~/(d/D) (WHITE, 1929). Some studies in heat transfer (KALB and SAEDER, 1974) have shown that the transfer rate is increased by the secondary motion, with only a moderate increase in pressure loss. The first application of this technique to blood oxygenation was made by WEISSMANand MOCKROS (1969). Although their predictions of improved oxygen transfer proved too optimistic, it was shown both experimentally and theoretically by DORSON et al. (1971) that the membrane area could be halved by coiling the tubes. With this improvement, but with a considerable blood-side resistance remaining, they developed tube oxygenators to the stage of experimental clinical application. Numerous other means of inducing secondary flow in the blood have been considered, and these have been discussed in a review by RICHARDSON (1971). One of the most worthwhile of these is the method of 'actively' inducing secondary flow by oscillating a curved channel about its centre (DRINKER, 1972). By adjusting the parameters of the oscillation, the convective transport can be increased to the point where the gas-exchange process is practically membrane-limited. This and other active methods have the disadvantage that additional mechanical equipment is needed for the oxygenator, ~)
~'
ECTIONA
CROSSSECTION A~ . . ~ ~
Fig. I Secondary flow streamlines Established laminar flow in helical coils at low Dean numbers and large values of (DEAN, 1928)
D/d
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107
but these methods have demonstrated that secondary flows can greatly increase the mass transfer. There is no indication that secondary flows have any additional detrimental effects on the blood. Passively induced secondary flows, on the other hand, are not as strong as those actively induced, but oxygenators employing this principle for clinical use have the advantage that no moving parts are required. This simplifying feature leads to the possibility of developing a mass-produced disposable oxygenator, but the outstanding problem with this technique is the relatively low efficiency. Our overall objective, accordingly, was to improve the efficiency of the passive method, and the work reported herein is part of that programme. The base for our work is secondary flow in a helical coil. In the steady state, a stable secondary flow is established, like that shown in Fig. 1, and when the oxygen concentration gradient across the tube wall is nonzero, a stable concentration profile likewise is developed. The lines of constant oxygen concentration in blood in the steady state have been calculated by WEISSMANand MOCKROS (1969) and are remarkably similar to the streamlines depicted in Fig. 1. Our approach is to improve fluid mixing by continually disturbing this stable secondary flow---essentially by periodically reorienting the curvature of the helix so that the direction of the secondary motion is altered. This technique will be referred to as 'complex coiling' and some examples of coils in this configuration are shown in Fig. 2. Each complex coil consists TYPE /dlSO"
TYPE 2190"
of a series of coil sections, each with the same number of turns, arranged such that the axes of adjacent coil sections form a given angle, generally 90 or 180 ~ By this configuration, secondary flow must be re-established in each coil section, and the direction of the secondary motion will then be 90 or 180 ~ to the original direction. F o r example, the direction of the centripetal force in Fig. 1 is right to left; if the coil axis is changed by 90 ~ the direction will then be vertically upwards or downwards, and after a 180 ~ change, the force will be right to left. This continual reorientation should improve mixing inside the tube, and the problem then is to find the magnitude of the increase in convective mass transfer. It is difficult to estimate the increase by theoretical means, not only because of the complex geometry and the nonlinear properties of the blood, but also because of the transitory state of the fluid motion throughout. Accordingly, an experimental programme was undertaken and this proceeded in three steps: (a) the secondary motion in a complex coil was investigated qualitatively by a dye injection study, (b) measurements of oxygen transfer and pressure drop were made in straight and complex coils with water, (c) similar measurements were made with blood. A basic problem in a study of this sort is to determine how close any improved mixing is to ideal radial mixing. The experiments may show that oxygen transfer in a complex coil is greater, say, than in a straight coil, but it is also important to indicate how much of a gap still remains for ideal radial mixing, i.e. perfect mixing of the fluid so that gas transfer is limited only by the membrane. Oxygen transfer with ideal mixing can be found only from theoretical work, of course, and consequently this 'membrane-limited' case was investigated for flow in circular tubes. This analysis is given in the next Section, prior to the description of the experiments, and the results will not only provide a reference curve to contrast with the experimental data, but will also be used to estimate how much the variation of blood properties, like pH, p C O : and temperature, can influence the oxygen uptake. 2 M'~mbrane-limited case
"--4d Fig. 2 Segments of three complex coils tested in the experimental work The number in the type code represents the number of turns in one coil section, and the second number indicates the angle between the axes of two adjacent coil sections. A full coil has 56 turns
108
The situation analysed in this Section is oxygen transfer to blood or to water flowing in a gaspermeable helically coiled tube surrounded by oxygen at a constant partial pressure pg. The essential idealisation is that radial mixing within the tube is perfect. Hence, the analysis basically reduces to finding the gas concentration, and therefore the gas transfer, in the tube wall. The situation is assumed to be axisymmetric so that the problem is described by the circular cylindrical co-ordinates (r, z); this presupposes that the wall thickness w is much less than the coil diameter D, i.e. w/D .~ 1,
Medical & Biological Engineering & Computing
March 1977
and the coil radius is much larger than the tube radius. The concentration c, or equivalently the partial pressure p, of the oxygen in the liquid is constant over the cross-section due to perfect radial mixing, i.e. the partial pressure p(r, z) equals p(a, z) for r ~< a, where a is the internal tube radius. Therefore the problem is to find p(a, z), the change in p in the downstream direction. The inlet partial pressure is Po. Several authors have analysed similar situations for a fiat membrane or a straight tube (WEmsMAN 1969, BISCHOFF and REOAN, 1971), both neglected axial diffusion within the membrane, and did not state any limits for the validity of this simplification. Other authors (DoRsoy et al., 1971) have considered a more simplified situation in which the partial pressure p(a, z) did not vary along the tube axis, i.e. the partial pressure difference (pg-p(a, z)) was kept constant at the inlet value (pg-po). This simplification will not be made here since the change of the partial pressure p(a, z) with z was considerable for some experiments. In addition, the axial diffusion will be retained in the first part of the analysis. This allows us to state a general condition for which the axial diffusion can be neglected. Within the tube wall (a ~< r ~< a + w), the governing equation for oxygen transfer is ~c/t3t = D,, A2 c = (aD)m A2 p, where Dm is the 02-diffusivity and (aD)m the O2-permeability of the tubular membrane. Therefore, in a steady state the governing equation is r
~r
r
+ 7
= 0
. . . .
(1)
The boundary conditions are specified at the inlet (z = 0), at the outer surface (a+ w, z) and at the inner surface (a, z)
p(r, O) = po . . . . . . . . .
(2a)
p(a + w, z) -- pg . . . . . . . .
(2b)
tgp (a, z) = v t3p (a, z) aaf c~-r~ 2~. D~
(2c)
When these substitutions are made, the result is eqn. 2c. The substitution t3c = aft?p can be applied straightaway for water, for which ~y can be taken constant at a given temperature. But in blood when the oxygen becomes bound to the haemoglobin the equivalent of af is a function of p and other parameters. In this case, the differential increase in oxygen concentration is given by d c = =wb d p + c~ ds. The first term (awbdp) is the direct result of Henry's law and represents the increase of oxygen in physical solution, hence =,~b is the 'physical solubility' for whole blood. The second term (c~ ds) represents the increase in oxygen which is bound to the haemoglobin, and is given in terms of the increase in saturation s. The saturation depends primarily on the oxygen partial pressure p in the blood plasma, and is defined as the ratio of the number of molecules bound by haemoglobin to the maximum number that can be bound. Full saturation, s = 1, is reached at partial pressures around 1 5 0 m m H g when each haemoglobin molecule has four oxygen molecules bound to it. The resulting oxygen concentration in the blood when s = 1 is the saturation concentration c~, and this quantity must be proportional to the amount of haemoglobin present, specifically, Cs = 1-36Hb where Hb is the haemoglobin concentration in grams per cent and c, has units of cm 3 (s.t.p.)/100 ml blood). The oxyhaemoglobin-dissociationcurve is the plot of s as a function of p, with T (temperature) and pH as parameters, and these curves are known for various species (ALTMAN, 1971). At constant T and pH, ds = (ds/~3P)pH. r d p , SO that the preceding expression for dc may read ~s
This has the form of Henry's law for a liquid, i.e. dc = af dp, where the af for blood is understood to be af = awb+
The last condition is not obvious for it is a statement for the mass balance of the gas in the liquid, derived by considering a disc-shaped element of liquid of radius a and thickness dz. The amount of gas transport into the element in the radial direction is Dm t3c/~r (a, z) 2~radz, which equals the increase of convective gas transport in the axial direction, v c3c/~z (a, z) ~ra2 dz (diffusion in this direction is negligible by comparison), v is the mean velocity of the liquid in the tube. Since ~c/~r (a, z) is the radial concentration gradient within the membrane at r = a, t3c may be replaced by a,,~3p, and since 8c/#z (a, z) is the axial gradient in the liquid, the correct replacement for #c in this quantity is t3c = otfc3p, where ~f is the 02-solubility of the fluid.
Medical & Biological Engineering & Computing
-Z--
\ op ]pH, r
e.
.
.
.
.
.
(4)
Since aw~ is known and cs = 1.36Hb, finding this 'effective solubility' =s means finding values of (t~s/~P)pn, r- These were estimated from the modified Hill equation (RossING and CAIN, 1966) log s/ (1 -- s) = ax log p [mm Hg] + a2 + a3 (pH, T)
(5)
where the constants were chosen (al = 2.9 and a2 = - 3 . 0 5 ) to fit data for pig blood (0.2 ~ s ~< 0-98), since this type was used in the oxygenation experiments. It was presumed that the function a3 (pH, T ) = 1.18 ( p H - 7 ) - 0 . 0 4 7 T[~ was the same for pigs as for dogs, since this form of
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a3 (pH, T) was found by RossI~G and CAIN (1966) for dog blood. Hence, the dependence of as on p was established with pH, T and Hb as parameters. We return our attention to the governing equations and, as the first step, in seeking solutions, make the problem dimensionless. Let the dimensionless co-ordinates be R =- r]a and Z =- (z/a) G/(Re So) and let the dependent variable be P - (p-po)/(p,-po). Here G ~ 4(aD)~/(aD),~b, which is four times the ratio of the membrane permeability to the blood permeability; Re = v2a/Vwb is the Reynolds number of the blood flow, where vwb is the kinematic viscosity, and Se =- (v/D)~b is the Schmidt number for the oxygenation of whole blood. In this way, eqns. 1 and 2 become
1
(~ [R gP'~
R ~g L
(
G
)2(~2P
"~R-) + R-eSc-
t?Z2 = 0
(6)
P(R, 0) = 0 . . . . . . . . .
(7a)
P(1 + W, Z) = 1 .
.
(7b)
~RC~P (1, Z) = as*{e(1 , Z) } -c~P ~ - (1, Z)
(7c)
.
.
.
.
.
.
where W =- w/a and as* = as/aw~, i.e. the ratio of the 'effective solubility' a s in eqn. 4 to the 'physical solubility' awb. For water or plasma (Hb = cs = 0), as* equals unity, but for blood, as* is a nonlinear function of P (and pH, T, pg, Po and Hb) where P refers to the blood, so that in eqn. 7c P is evaluated at the membrane-blood interface (1, Z): Since G/(ReSc) was typically 0(10 -2) for the oxygenation experiments, the axial diffusion term in eqn. 6 can in fact be neglected and the solution of the remaining equation is P(R, Z) = fl(Z) In R +f2(Z) where f l and f2 are undetermined functions found from the boundary conditions at R = t and R = 1 + W. At the inner wall, P(1, Z) = f2(Z) is just the dimensionless partial pressure of the fluid, henceforth denoted by Ps(Z). Since Ps is the nondimensional form of p(a, z), it is the objective of this analysis and so the solution for P(R, Z) will be expressed in terms of this function. When the other boundary condition, eqn. 7b, is substituted in the preceding equation for P, then for 1 ~ R
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2
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Z / In(l+W) Fig. 9 Coiling coefficient for hefical and complex coils The hydraufic diameter d l was used instead o f d to compute Reand De. The continuous curves were fitted to the data, and curve w is an empirical equation for hefical coils (D/d >~ 15.5) by WHITE (1929)
116
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March 1977
accordingly a smaller priming volume. This would reduce the detrimental effects due to blood-surface interaction and allow more compact designs. Acknowledgment--The principal author (U.B.) was supported by a scholarship of the Canada Council. The assistance of the staff of the Hospital for Sick Children, Toronto, and of the Department of Mechanical Engineering at the University of Toronto, is gratefully acknowledged. References
ALTMAN, D. L. (1971) Respiration and circulation. Biological Handbook, Federation of American Societies for Experimental Biology BAURMEISTER,U. (1974) Oxygen transport to blood flowing in coiled semipermeable tubes. M.Sc. thesis, Univ. of Toronto BXSCHOFF,K. B. and T. M. REGAN (1971) Comments on diffusion in membrane-limited blood oxygenators. Amer. Inst. Chem. Eng. J. 17, 225 BLACKSHEAR,P. L. Jun. (1972) Mechanical haemolysis in flowing blood. In: Biomechanics, its foundations and objectives. Eds. Y. C. FUNG, N. PERRONE and M. ANLI~:ER, 501. Prentice Hall, New Jersey BUCKLES,R. G., E. W. MERRILL and E. R. G1LLILAND (1967) An analysis of oxygen absorption in a tubular membrane oxygenator. Amer Inst. Chem. Eng. J. 14, 703 DEAN, W. R. (1928) The streamline flow of fluid in a curved pipe. Phil. Mag. 4, 673 DORSON, W. J. Jun., LARSEN,K. G., ELGAS,R. J. and VOORHEES, M. E. (1971) Oxygen transfer to blood: data and theory. Trans. Am. Soc. Artif. Intern. Organs 17, 309 DRAVID, A. N., SMITH, K. A., MERRILL, E. W. and BRIAN, P. L. T. (1971)Effect of secondary fluid motion
on laminar flow heat transfer in helically-coiled tubes. tlmer. Inst. Chem. Eng. J. 17, 1114 DRINKER, P. A. (1972) Progress in membrane oxygenator design. Anaesthesiology 37, 242 KALR, C. E. and SAEDER, I. D. (1974) Fully developed viscous-flow heat transfer in circular tubes with uniform wall temperature. Amer. Inst. Chem. Eng. J. 20, 340 LEE, W. n., KRUMMHAAR,D., DERRY, G., SACHS, D., LAWRENCE, S. H., CLOWES, G. H. A. and MALONEY, J. V. (1961) Comparison of the effects of membrane and non-membrane oxygenators on the biochemical and biophysical characteristics of blood. Surg. Forum 12, 200 MADRAS, P. A., MORTON, W. A. and PETSCHECI(,H. E. (1971) Dynamics of thrombus formation. Fed. Proc. 30, 1665 MOCKROS, L. F. and. WEISSMAN, M. H. (1971) The artificial lung. In: Biomedical engineering. Eds. L. H. U. BROWN, J. E. JACO8S and L. STARK, 325. F. A. Davis Co., Philadelphia PEIRCE, E. C. II (1972) The role of the artificial lung in the treatment of respiratory insufficiency: A perspective. Chest 62, 107 S RICHARDSON, P. D. (1971) Effects of secondary ftow in augmenting gas transfer in blood. In: Advances in cardiology vol. 6. Eds. R, H. BARTLETT,P. A. DRINKER and P. M. GALETTI,2, S. Karger, Basel ROSSING, R, G. and CAIN, S. M. (1966) Dog oxyhaemoglobin dissociation curve. J. Appl. PhysioL 21,198 WElSSMAN, M. H. (1969) Diffusion in membrane-limited blood oxygenators. Amer. Inst. Chem. Eng. J. 15, 627 WEISSMAN, M. H. and MOCKROS, L. F. (1969) Oxygen and carbon dioxide transfer in membrane oxygenators. Med. & Biol. Eng. 7, 169 WHITE, C. M. (1929) Streamline flow through curved pipes. Proc. Roy. Soc. 123A, 645
Oxygenation du sang dans des tubes spirales en caoutchouc de silicone la g6om6trie comp|exe Sommaire---Le passage de l'oxyg~ne dans le sang et dans l'eau fut 6tudi6 de fa~on exp6rimentale pour un d6bit constant dans une 'spirale complexe', tube en caoutchouc de silicone constitu6 d'une s6rie de sections h61icoidales b. orientation alternge. La s6rie de r6orientations provoquait le r6tablissement continu du flux secondaire induit par le mouvement hdicoidal, de mani~re ~t accro~tre le m61ange du liquide. Des spirales complexes de diverses configurations furent soumises /~ des essais dans une atmosphere d'oxyg~ne et une am61ioration significative du passage de l'oxyg~ne dans les liquides fur mesur6e alors que la chute de pression s'intensifiait mod6r6ment. Le cas d'un transfert id6al d'oxygSne--par example dans un liquide parfaitement m61ang6--fut trait6 de fa~on th~orique et une comparaison de ces r~sultats avec les donndes exp6rimentales a demontr6 h quel point la rdsistance lat6rale du liquide 6tait r6duite par le syst~me de spirales complexes.
Sauerstoffanreicherung des Blutes in gewundenen Silikonkautschukrohren komplexer Geometrie Zusammenfassung--Sauerstoffiibertragung an Blut und Wasser wurde experimentell f'tir stetige Str~Smung in einer 'komplexen Spule' untersucht, einem Silikonkautschukrohr, das in eine Folge yon Wendelspulen alternativer Orientierung geformt wurde. Die Reihe yon Wieder-Orientierungen verursachte die fortgesetzte Wiederherstellung des v o n d e r Spiralbewegung induzierten Sekund~rstromes, so dab die Misehung der Fliissigkeit gef~Srdert wurde. Komplexe Spulen verschiedener Konfiguration wurden in einer Sauerstoffatmophiire geprfift und es wurde eine betr~ichtliche Verbesserung dee Sauerstofftibertragung an die Fltissigkeiten gemessen, wiihrend der Druckanfall ein wenig anstieg. DeE Fall der idealen Sauerstofftibertragung-d.h. an eine vollkommen gemischte Fltissigkeit- wurde theoretisch behandelt und ein Vergleich dieser Resultate mit den experimentellen Daten zeigte, um wieviel der Widerstand yon Seiten der Fliissigkeit durch komplexe Spulung verringert wurde.
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March 1977
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