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Transport phenomena in laminar flow of blood
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1978 Phys. Med. Biol. 23 928 (http://iopscience.iop.org/0031-9155/23/5/009) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.237.165.40 This content was downloaded on 01/10/2015 at 15:54
Please note that terms and conditions apply.
PHYS. MED. BIOL., 1978, Vol. 23, No. 5 , 928-936.
Printed in Great Britain
Transport Phenomena in Laminar Flow of Blood AVTAR S. AHUJA,? PH.D., WILLIAM R. HENDEE, PAUL L. CARSON, PH.D.
PH.D.
and
Department of Radiology, University of Colorado Medical Center, Denver, CO 80262, U.S.A.
Received 14 February 1977, infinal form 2 June 1971 ABSTRACT. Recently it hasbeen shown experimentally that transport of heat and gas (specifically oxygen and helium) areaugmentedinthelaminar flow of aqueous suspensions of polystyrene spheres 50 and 150 pm in diameter. In this report, data on heat and gas transport are correlated. Application of this correlation t o flowing blood leads to the following conclusions. Thereisno significant augmentation of oxygen and heat transport in flowing blood even a t shear rates much higher than physiological shear rates; an observation which is in accord with the experimental results. The augmentation of the diffusion coefficient of plasma proteins in flowing blood, thoughnotvery high, appearsto be measurable. Of thetotal measured augmentation of about 6000-30 000% in platelet diffusivity in flowing blood, quoted from theliterature,about 500% is attributable from this correlation to fluid mechanical forces, and thebalance is hypothetically attributed to other forces (electrical or biochemical) present in blood.
1. Introduction Transport phenomena of fluids embrace three properties. These properties are viscosity (momentumtransport),thermalconductivity(heattransport) and diffusion coefficient (mass transport). The viscosity of particle suspensions is a well understood property (Goldsmith and Mason 1967) and the thermal conductivity and diffusion coefficient of stationary particle suspensions have been explored both theoretically (Fricke1924, Bruggeman 1935, Maxwell 1954) and experimentally (Johnson 1958, Stein, Martin and Keller 1971, Ahuja 1974, 1975a).However,onlyrecently has it been shown experimentally that the thermalconductivity(Ahuja 1975b) and the diffusion coefficient of helium (Collingham 1968) are augmented in the laminar flow of suspensions of 50 and 100 pm polystyrenespheres.This report presentscertain basic ideas of transport phenomena in flowing suspensions and shows a correlation between the heat and gas transport data. Application of this correlation in medicine leads to comments on the transport of heat, oxygen (and other gases), plasma proteins and platelets in flowing blood which from a fluid mechanical viewpoint can be treated as a suspension of red blood cells (RBCS)in plasma. An understanding of the enhanced diffusion of platelets in blood may elucidate the formation of blood clots and thrombi.
t Present address: School of Medicine, Case Western Reserve University, Wearn Research Building, 2065 Adelbert Road, Cleveland, O H 44106, U.S.A. 0031-9155/78/0005-0928 $02.00
@ 1978 The Institute of Physics
Transport Phenomena in Blood
929
It is well known that transport properties of single-phase fluids such as water or air are independentof the stateof motion in thelaminar regime, and that the effects of motion are included in the transfer coefficients (Eckert and Drake 1959). However, for two-phase fluid flow such as the flowof particle suspensions or blood, there are two approaches to the problem of describing transport phenomena. Since heat and mass transport in flowing suspensions or blood are analogous phenomena, these two approaches are discussed with respect to the transport of heat. I n one approach, application of the Graetz solution yields the effective thermal conductivity of flowing suspensions as a function of the shear rate. This approach,which may be termed the conductive approach, is adopted in the literature on the heat (Ahuja 1975b, c, 1976) and mass (Collingham, Blackshear and Eckert 1970) transport of flowing particle suspensions, and on the heat transport (Ahuja 1975d) and mass transport of gases (Collingham 1968), proteins andplatelets(Petschekand Madras 1969, Turitto, Benis and Leonard 1972) in flowing blood. The effect of perfusion of tissues has also been included in the thermal conductivity of tissues (Grayson 1952). The second approach, which may be termed the convectiveapproach, is usually adopted in single-phase fluid flow theory. In this theory, the thermal conductivity offlowing suspensions is assumed to be the same as that of stationary suspensions, and theNusselt number is computed fromthe measured heat transfer coefficient. This approach is adopted in the thermal design of a bloodheat exchanger (AhujaandHendee 1978). I n this paper,the first approach is adopted for investigating the transport phenomena in the laminar flow of blood. Augmentation in thermal conductivity ( k ) of a flowing suspension or blood is defined as k,,,/k,tst - 1, where subscripts ‘mov’ and ‘stat’ stand for moving and stationary suspensions or blood. The augmentation in diffusion coefficient is defined similarly.Augmentation in thermal diffusivity ( a = k / p c ) is also defined similarly because the density p and specific heat c of the suspension or blood are independent of the state of motion,
2. Correlation of heat and mass transportdata
It is well known that transport properties of single-phase fluids (such as water or air) are independent of the state of motion in the laminar regime (EckertandDrake 1959). By employing the Graetzsolution, it has been shown experimentally that the thermal conductivity (Ahuja 1975b) or mass diffusion coefficient (Collingham 1968)of a particlesuspension in laminarmotion is a function of particle radius a and volume fraction +,tube length L and U and transport properties radius R, shear rate or particle angular velocity (kinematic viscosity v, thermal diffusivity a or diffusion coefficient D ) of suspending fluid. By constructing dimensionless groups from these variables, or by following systematic dimensional analysis (Ahuja 1976), augmentation of the thermaldiffusivity or diffusion coefficient of a flowing suspension may be
930
Avtar S. Ahuja et al.
written as amov/astat-
1 or DmovlDstat- 1 = F(+,wa2/vf, wa2/afor wa2/Df,R/a, L/a) (1)
where subscriptfindicates the suspendingfluid.The term wa2/vf is the Reynolds number, and the expressions wa2/af and wa2/Df, respectively, are the Peclet numbers for heat and mass transfer (for meanings of symbols, see the Appendix). For single-phase fluids in laminar flow, the shear rate varies linearly withthe radial distance from the axis of the tube, zero at theaxis and maximum at the wall. Therefore, the average shear rate is half the wall shear rate. The angular velocity of the particle, which is half the shear rate at the point where the centre of the particle happens to be located, varies linearly with the radial distance from the tube axis for the denser particles (such as RBCS) (Jeffrey and Pearson 1965), yielding the average angular velocity of the particle to be half the average shear rate or one quarter of the wall shear rate. When a suspension flows in a tube, particles under theinfluence of the shear field rotate and entrain thesurrounding liquid by friction. If the shear field is strong enough and the particles are large enough so that wa2/vf 2 1, then each particle acts like a centrifugalfan in the sense that thefluid particles are thrown outward from the region near the equator, and return to a region around the of heat or mass transport resides in this poles. The augmentation churning effect superimposed upon the rotational motion of the surrounding fluid. Heat transport data (Ahuja 197513) obtained for a number of values of the variables in eqn (1) have been correlated by eqn (1) of Ahuja (1975~) and the effects of particle interactions on heat transport inflowing suspensions, ignored in eqn ( l ) ,have been postulated (Ahuja 1975~).Particle interactions havebeen classified into particle crowding and formation of doublets or multiplets (Goldsmith and Mason 1967). It is assumed that both typesof interactions will cause a reduction in augmentation. Particle crowding affects the augmentation through the mutualinterference of the hydrodynamic environment of particles whereby particle rotational motion is affected. It is further supposed that a doublet of spheres will cause less churning of the surrounding fluid compared with the churning caused by the two spheres were they free to spin. Collingham’s (1968) gas transfer data, corresponding to several variables as listed above, have remained uncorrelated. I n fig. 1, heat (Ahuja 197513) and helium transport (Collingham 1968) data arecorrelated byplotting the or diffusion coefficient against percentage augmentation in thermal conductivity +(waz/vf)(wa2/af or waZ/D,) x (R/a)z(L/2a)x x doublet collision frequency ratio. The doublet collision frequency ratio indicated in fig. 1, which yields the effect of particle collisions on heat or mass transport, is the cube of the ratio of particlediameter to the 100pm diameter of an arbitrarily chosen reference particle (Ahuja 1975~).Augmentation in the laminar flow of aqueous suspensions of 50 and 100 pm polystyrene spheres is seen to be about 200% for heat transport and 500% for helium transport.
Transport Phenomena in Blood
Y
0 wa2 wa2 4 (T)(F
or
I
I
0.5
1.0
ma2 R L %)(--) (%)
93 1
I
1.5 2.52.0
1
35
3.0
LO
x 10-8 x (doublet collision frequency ratio)
Fig. 1. Plot of augmentation in thermal conductivity kmovlkBtst- 1, or diffusion coefficient 1, against values of the expression given for the abscissa of helium, Dmov/Dstittas shown for a laminar flow of suspensions. For meanings of symbols, see text. In the heat transfer experiment, the suspension was heated by circulating water in a countercurrent heat exchanger as it moved down the stainless steel tube (Ahuja 1975b). The Graetz solution corresponding to a uniform wall temperature was used. 0 , polystyrene spheres of volume fraction 8.8% suspended in 5.2% by weight aqueous NaC1, particle diameter 100 pm, tube diameter d = 1 mm and length L = 55 cm, tube Reynolds number = 576-1240. In thegas transfer experiment, the suspension was saturated with helium. Watersaturated argon flowed in an annulus around a Silastic tube and helium diffused from the flowing suspension into the annulus through the permeable wall of Silastic (Collingham 1968). The Graetz solution corresponding tothe uniform wall concentration was used. Polystyrene spheres were suspended in 7.4% by weight aqueous NaC1, tube diameter d = 1.5 mm and length L =288 cm, the gas was helium. 0,particle diameter 100 pm and volume fraction 7.5%. e, particlediameter 50 pm and volume fraction 7.23%.
2.1. Generality of correlation of heat and mms transfer data Augmentation in heat conductivity is smaller than the augmentation in the diffusioncoefficient of gases, other factors of the experiments being given, ina fluid (for because the Brownian diffusioncoefficient for a gas ( example, water) is about two orders of magnitude smaller thantheheat of water. An abscissa value of 0.5 in fig. 1 is an upper conductivity ( limit for heat transfer data in the laminar flow regime, while an abscissa value of 4 can easily be attained for gas transfer. Correlation of heat and gas (helium) transfer data infig. 1 is for two particle sizes (50 pm and 100 pm), two tubes (one of d = 1 mm and L = 55 cm, and the other of d = 1.5 mm and L = 288 cm) and particle volume fractions of 7.23, 7.5 and 8-8yo.In addition, coordinates of fig. 1 have been seen in detail (Singh 1968) to correlate data for He, 0, and heat, five different tubes (diameters 1-2 mm and lengths 40-288 cm), two suspending media (aqueous sodium N
N
33
932
Avtar S. Ahuja et al.
chloride and aqueous glycerine), two particle diameters (50 pm and 100 pm) and a variety of particle volume fractions (0*3-8-8%). The correlation presented in fig. 1 appears to be representative for applications to blood. 3. Application of correlation to flowing blood 3.1. Necessity of a simulated study as it relates to blood In vivo blood flow is characterised by tapered and radially pulsating blood vessels which may possibly have curvature, pulsatile flow of blood, biconcave disc-like structure of RBCS which are flexible and distort their shape as they move, presence of white blood cells, platelets, proteins, several types of ions and dissolved gases. Complete simulation of the in vivo conditions of blood is not easy and may not even be necessary. Assumptions must be made in the application of a physical theory to so prohibitively complex a situation. (i) Assumption that blood is a Newtonianfluid. It is now well known t h a t a t shear rates greaterthan 50 s-l blood is a Newtonianfluid (Copley 1972). (ii) Assumption of Poiseuille flow of blood. I n this paper, the results insuspension flow are applied to in vitro blood flow. It is an established fact that blood flow in large tubes (of diameter > 300 pm) follows the Poiseuille law (FBhraeus and Lindqvist 1931). (iii) Assumption of a spherical and rigid RBC. To apply the results of the simulatedexperimentswith rigid sphericalparticles to blood, the biconcave disc of the RBC has to be replaced by an equivalent rigid sphere which yields the same results as thedisc. In thecase of blood, we know of no way of establishing this equivalence. To further the argument,we used a sphere of the same volume as the humanRBC. I n section 4 effects of the size and the flexibility of the RBC sphere on the mass transport in blood are discussed. 3.2. Heat and gas transport
in blood
Theradius of asphere of human RBC volume ( = 87 pm3 (Altmanand Dittmer 1971)) is a = 2.75 pm. For a haematocrit C$ = 0.3, angular velocity of the RBC W = 274200) radian s-1 (corresponding to a wall shear rate of 800 s-l), diffusion coefficient of oxygen inplasma D, = 2 x 10-5 cm2s-l, kinematic viscosity of plasma vf = 0-015 cm2s-1, tube radius d = 0.50 mm and tube length L = 300 cm, the abscissa in fig. 1 is 0-0003and the augmentation in diffusion coefficient Dmov/Dstat-1 is negligible. This conclusion, that no significant augmentation of oxygen transport in flowing blood occurs a t even high shear rates, is in accord with direct measurements in blood (Collingham 1968). The same conclusion has been reached from direct measurements for heat transport in laminar flow of blood (Ahuja 1975d). These results suggest that as far as heat or gas transfer is concerned, blood can be treated as a single-phase fluid, That is, RBC tumbling and rotation in flowing blood (even for large turkey RBCS) have no measurable effect on the transfer of heat or gas in blood.
Transport Phenomena
in Blood
933
3.3. Transport of plasma proteins in blood
The diffusion coefficients of plasma proteins in water a t 20 'c vary from 2 x lo-' to 6 x lo-' cm2 s-l (Altman and Dittmer 1971). Consider the case 1971). of fibrinogen for which D, = 2 x 10" cm2 s-1 (Altman and Dittmer Then for the same haematocrit, shearrate and tube dimensions as given above, the value of the abscissa in fig. 1 is 0-03 and the augmentation in diffusion coefficient of fibrinogen in flowing blood is about 50%. The augmentation of the diffusion coefficient of plasma proteins in flowing blood, though not very high, appears to be measurable. 3.4. Transport of platelets in blood
Brownian diffusion coefficient of platelets in blood plasma is given by the R = 1.98 cal Stokes-Einstein law as D, = RT/6,7aN. For the gas constant K-l mol-l, an absolute temperature T = 310 K (37 "c), the dynamic viscosity of plasma 7 = 0.015 Poise, the radius of the equivalent sphere of a platelet a = 1 pm and Avogadro's number N = 6.02 x D, 1.5 x cm2 s-l. Assuming that the diffusion coefficient D, is zero forplateletsin RBC, the Fricke equation (Fricke 1924, Stein et al. 1971)
with the shape factor X for RBCS taken to be 1.1 (Fricke 1924), yields DStat/Df N 0.8 x cm2 5-l asthe diffusion coefficient of plateletsin - 0.55 or D,, stationary blood of 30% haematocrit. For the case of the diffusion of platelets in flowing blood, with a Brownian cm2 and all other diffusion coefficient of plateletsinplasma D, = 1.5 x quantities as given in section 3.2, the abscissa in fig. 1 is 3.6. The augmentation of platelet diffusion is about 55076 which, with a diffusion coefficient of platelets cm2 s-l, yields a diffusion Coefficient in instationary blood Dstat = 0.8 x cm2s-l. Themeasuredvalues of the diffusion moving blood D,,, g 5 x areintherange coefficientof plateletsin flowing blood in vitro, D,,, 5x 2.5 x lo-' cm2 s-l (Turitto et al., 1972) forarange of 40-440 s-1 in the wall shear rate and 3 6 5 0 % for the haematocrit. Augmentation in platelet diffusivity, Dmo,/Dstat- 1 , is 6150-31 150% and the ratioof the measured value of platelet diffusivity to thatdeduced from the simulated experiments is10-50. 4.
Discussion
4.1. Effect of size of spherical RBC
For a physiological averageshear rate of 200 s-l in a given tube as in section 3.2, the abscissa values in fig. 1 are about 1.0, and 5.0 for RBCS of radii 2.75 and 4.1 pm, respectively, yielding D,, of platelets of about 4 x 10-9 and 6 x 10-9 (extrapolated) cm2 s-l. These resultsindicatediscrepancyranges (that is, the ratio of the measured value of the platelet diffusivity to that
Avtar S. Ahuja et al.
934
deduced from the simulatedexperiments) of about 12.5-62.5 and 8-42. Physiological shear rates lower than 200 S-llead to still higher discrepancy values. 4.2. Effect of RBCJlexibility
Because part of the flow energy is spent in distorting the shape of a flexible in reduced angularvelocity W of the RBC, flexible RBC spheres are likely to give less augmentation in platelet diffusivity than rigid RBC spheres. direction. The quantitative argumentsassuming a rigid RBC are in the right RBC, resulting
4.3. Hypothesis regarding the excess platelet diffusivity in $owing blood
The discrepancy between the measured value of the platelet diffusivity and that deduced from the simulated experiments described in sections 3.4 and 4.1 cannot be accounted for on the basis of variations in RBC size and angular velocity. Therefore, there is a strong indication that, of a total augmentation of about 6000-30000% in platelet diffusivityin flowing blood, only about 500% is explained by fluid mechanical forces. Since flowing blood has other (electrical or biochemical) forces present, it is logical, at least a t this stage of the development of thisproblem, to attribute the excess platelet diffusivity to those forces than fluid mechanical ones. 4.4. Comment on the physical basis of the phenomenon of augmentation in
diffusivities of a variety of species To our knowledge, the mechanism of augmentation of heatand mass transport in flowing blood, in all its ramifications, is not yet understood. The physical basis advanced in the literature has been attributed to the Prandtl Mixing Length Hypothesis (from turbulent flow theory) from which a relation for blood in laminar flow has been obtained as (Keller 1971)
D,,,
lOP(average shear rate)n
(3)
where n = 1 or some fraction (Petschek and Madras 1969, Friedman, Liem, Grabowski, Leonard and McCord 1970). A caution has been sounded about its grossness (Keller 197 1 ). Eqn (3) gives only the effect of shear rate on the diffusivity, ignoring the effects of particle size and volume fraction, tubedimensions, diffusion coefficient of thetype of species being transported(heat, gases, platelets,proteins), kinematic viscosity of suspending liquid and particle interactions. Also, the currently available experimental measurements are not sensitive to the effects of haematocrit and shear rate (Turitto et al. 1972). Eqn (1) and fig. 1 give the effects of all these variables quantitatively and so bydoing indicate the physical basis of the mechanism of augmentation of heat and mass transport in flowing blood. 5.
Conclusions
Heat and gas transport data in flowing suspensions of polystyrene particles have been correlated. An application of this correlation to flowing blood yields a possible mechanism of the augmentation of transport of a variety of species.
Transport Phenomenu in Blood
935
The assumptions of the study are valid for in vitro blood flow. This study accounts for all significant variables which enter into in vitro blood flow and quantitates their effects on augmentation of platelet diffusivity. A comparison between the values of platelet diffusivity obtained from the correlation of heat and mass transfer data in flowing suspensions and those measured in flowing blood yields an excess platelet diffusivity unaccounted for by fluid mechanical forces. The excess platelet diffusivity is contributed probably by other forces (electrical or biochemical) present in blood. Another potential application (notdescribed in this paper)of the augmented transport phenomena in flowing suspension is in the area of the energy transport. For a constant energy expended in propelling a fluid in a tube, the heat and gas carrying capacity of the fluid could be increased significantly if the fluid were a suspension ratherthan a single-phase fluid. Such an increase would be advantageous in devices such as heat exchangers.
This investigation was supportedin partbyGrant awarded by the National Cancer Institute, DHEW.
KO. 5T32-CA09073,
APPEXDIX List of symbols a particle or RBC radius (cm) C specific heat (calg-1 'c-1) d tube diameter = 2R D diffusion coefficient (cmzs-l) k thermal conductivity (cal s-l cm-l 'c") L tube length (cm) R tube radius (cm) X particle shape factor in eqn (2) a thermal diffusivity (cm2 s-1) B ratio of species (or platelet) solubility in particles or suspending fluid or plasma V kinematic viscosity (cm2 s-l) density (g cm-3) particle volume fraction or fractional haematocrit W angular velocity (radian s-1) Subscripts f suspending fluid or plasma pparticle or RBC mov moving suspension or blood stat stationary suspension or blood
RBCS
to that in
RI~SUM~ Phenomenes de transport dansdes flux laminaires de sang L'on a montre recemment par experiences que la transport de chaleur et de gaz (surtout l'oxygene et l'helium) est augment6 dans le flux laminaire de suspensions aquemes de spheres de polystyrene d'un diambtre de 50 et de 150 pm. Ce rapport Btablit IS, corr6lation entre les donn6es sur le transport de la chaleur et de gaz. L'application de cette corr6lation iL des flux de sang mene aux conclusions suivantes. I1 ne se produit pas d'augmentation significative du transport d'oxygene et de chaleur dans le sang en 6coulement, m6me avec des taux de cisaillement bien
936
Transport Phenomena in Blood
plus blevbs que lea taux de cisaillement physiologiques; c’est 18 une observation conforme avec les resultats expbrimentaux. L’augmentation du coefficient de diffusion des protbines de plasma dans le sang enbcoulement, sans &re trbs blevbe, semble mesurable. Sur le total d’une augmentation mesurbe d’environ 6000-30 000% quant au coefficient de diffusion des hbmatoblastes dans le flux de sang, comme le montre la littbrature, environ 500% sont attribuables, d’aprbs cette corrblation, 8 des forces mbcaniques fluides et le reste semble provenir d’autres forces (blectriques ou biochimiques) prbsentes dans le sang.
ZUSAMMENFASSUNQ Transportphiinomene bei laminarer Blutzufuhr Kurzlich ist experimentell gezeigt worden, dass der WBrme- und Gastransport (insbesondere Saueratoff und Helium) in dem laminaren Fluss wiisseriger Aufschliimmungen von Polystryrolsphiiren mit einem Durchmesser von 50 und 150 wm gesteigert wird. I n dieser Abhandlung werden die Daten iiber Wiirme- und Gastransport in Korrelation gebracht. Die Anwendung dieser Korrelation auf zirkulierendes Blut ergibt die folgenden Schliisse. I m zirkulierenden Blut findet keine bedeutsameSteigerungdes Sauerstoff- und Wiirmetransports statt, auch wenn die gegebenen Scherungszahlen weit hoher als die physiologischen sind. Diese Beobachtung stimmt mit den Versuchsergebnissen uberein. Die Erhohung des Diffusionskoeffizienten der Plasmaproteine im zirkulierenden Blut ist offensichtlich messbar, auch wenn sie nur geringfiigig ist. Die aufgrund des Schriftmaterials insgesamt gemessene Steigerung des Diffusionsvermogens von Blutpliittchen im zirkulierenden Blut von etwa 6000-30 O O O ~ oist, wie diese Korrelation ergibt zu etwa 500% auf fliessende mechanische Kr(lfte zuriickzufuhren. Man nimmt an, dass der Augsleich durch andere im Blut vorhandene Kriifte (elektrische oder biochemische) gesohaffen wird.
REFERENCES A. S., 1974, J . Appl. Physiol., 37, 765. AHUJA,A. S., 1975a, J . Appl. Phys., 46, 747. AHUJA, A. S., 197513, J . Appl. Phys., 46, 3408. AHUJA, A. S., 1975c, J . Appl. Phys., 46, 3417. AEUJA, A. S., 1975d, J . Appl. Physiol., 39, 86. AHUJA,A. S., 1976, J . Appl. Phys., 47, 775. AHUJA, A. S., and HENDEE, W. R.,1978, Phys. Med. Biol., 23, 937. ALTMAN, P. L., and DITTMER, D. S. (Eds), 1971, Blood and Other Body Fluid8 (Fed. Soc. Exptl. Biol., Bethesda, MD). BRUGGEMAN, D. A. G., 1935, Ann. Phys. (Leipz.), 5, 636. COLLINGHAM, R.E., 1968, Ph.D. Thesis, University of Minnesota, Minneapolis. COLLINGHAM, R. E., BLACPSHEAR, P. L., m d ECPERT,E. R. G., 1970, in Proc. 4th Int. Heat Transfer Conf., Paris. COPLEY, A. L., 1974, Rheol. Acta, 13, 845. ECEERT, E. R. G., and DRAKE,R. M., JR., 1959, Heat and Mass Trartsfer (New York: McGraw-Hill). FAHRAEUS, R., and LINDQVIST, T., 1931, Am. J . Physiol., 96, 562. FRICEE, H., 1924, Phys. Rev., 24, 575. FRIEDMAN, L.I., LIEM,H., GRABOWSEI, E. F., LEONARD, E. F., and MCCORD,C. W., 1970, Trans. Am. Soc. Artif. Int. Organs, 16, 63. GOLDSMITH, H. L., and MASON,S. G., 1967, in Rheology: Theory and Applications, Ed. F. R.Eirich (New York: Academic) Vol. 4, Ch. 2. GRAYSON, J., 1952, J . Physiol., Lond, 118, 54. JEFFREY, R.C., and PEARSON, J. R.A., 1965, J . Fluid Mech., 22, 721. JOHNSON, F. A., 1958, U.K.Atomic Energy Authority, Research Group Report NO. AERE R/R2578 (unpublished). KELLER, K.H., 1971, Federation Proc., 30, 1591. MAXWELL,J. C., 1954, A Treatise o n Electricity and Magnetism (New York: Dover) Vol. 1, p. 440. H. E., and MADRAS,P.N., 1969, in Artijicial Heart Program Conference, Ed. PETSCHEK, R.J. Hegyeli (U. S. Govt. Printing Office, Washington, DC). SINGH,A., 1968, Ph.D.Thesis, University of Minnesota, Minneapolis. (A. Singh is subsequently known as A. S. Ahuja.) STEIN,T. R., MARTIN, J. C., and KELLER,K. H., 1971, J . Appl. Physiol. 31, 397. TURITTO, V. T., BENIS,A. M,, and LEONARD, E. F.,1972, Ind. Eng. Chem. Fundam., 11,216. AHUJA,
Am.
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Transport phenomena in laminar flow of blood
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1978 Phys. Med. Biol. 23 928 (http://iopscience.iop.org/0031-9155/23/5/009) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.237.165.40 This content was downloaded on 01/10/2015 at 15:54
Please note that terms and conditions apply.
PHYS. MED. BIOL., 1978, Vol. 23, No. 5 , 928-936.
Printed in Great Britain
Transport Phenomena in Laminar Flow of Blood AVTAR S. AHUJA,? PH.D., WILLIAM R. HENDEE, PAUL L. CARSON, PH.D.
PH.D.
and
Department of Radiology, University of Colorado Medical Center, Denver, CO 80262, U.S.A.
Received 14 February 1977, infinal form 2 June 1971 ABSTRACT. Recently it hasbeen shown experimentally that transport of heat and gas (specifically oxygen and helium) areaugmentedinthelaminar flow of aqueous suspensions of polystyrene spheres 50 and 150 pm in diameter. In this report, data on heat and gas transport are correlated. Application of this correlation t o flowing blood leads to the following conclusions. Thereisno significant augmentation of oxygen and heat transport in flowing blood even a t shear rates much higher than physiological shear rates; an observation which is in accord with the experimental results. The augmentation of the diffusion coefficient of plasma proteins in flowing blood, thoughnotvery high, appearsto be measurable. Of thetotal measured augmentation of about 6000-30 000% in platelet diffusivity in flowing blood, quoted from theliterature,about 500% is attributable from this correlation to fluid mechanical forces, and thebalance is hypothetically attributed to other forces (electrical or biochemical) present in blood.
1. Introduction Transport phenomena of fluids embrace three properties. These properties are viscosity (momentumtransport),thermalconductivity(heattransport) and diffusion coefficient (mass transport). The viscosity of particle suspensions is a well understood property (Goldsmith and Mason 1967) and the thermal conductivity and diffusion coefficient of stationary particle suspensions have been explored both theoretically (Fricke1924, Bruggeman 1935, Maxwell 1954) and experimentally (Johnson 1958, Stein, Martin and Keller 1971, Ahuja 1974, 1975a).However,onlyrecently has it been shown experimentally that the thermalconductivity(Ahuja 1975b) and the diffusion coefficient of helium (Collingham 1968) are augmented in the laminar flow of suspensions of 50 and 100 pm polystyrenespheres.This report presentscertain basic ideas of transport phenomena in flowing suspensions and shows a correlation between the heat and gas transport data. Application of this correlation in medicine leads to comments on the transport of heat, oxygen (and other gases), plasma proteins and platelets in flowing blood which from a fluid mechanical viewpoint can be treated as a suspension of red blood cells (RBCS)in plasma. An understanding of the enhanced diffusion of platelets in blood may elucidate the formation of blood clots and thrombi.
t Present address: School of Medicine, Case Western Reserve University, Wearn Research Building, 2065 Adelbert Road, Cleveland, O H 44106, U.S.A. 0031-9155/78/0005-0928 $02.00
@ 1978 The Institute of Physics
Transport Phenomena in Blood
929
It is well known that transport properties of single-phase fluids such as water or air are independentof the stateof motion in thelaminar regime, and that the effects of motion are included in the transfer coefficients (Eckert and Drake 1959). However, for two-phase fluid flow such as the flowof particle suspensions or blood, there are two approaches to the problem of describing transport phenomena. Since heat and mass transport in flowing suspensions or blood are analogous phenomena, these two approaches are discussed with respect to the transport of heat. I n one approach, application of the Graetz solution yields the effective thermal conductivity of flowing suspensions as a function of the shear rate. This approach,which may be termed the conductive approach, is adopted in the literature on the heat (Ahuja 1975b, c, 1976) and mass (Collingham, Blackshear and Eckert 1970) transport of flowing particle suspensions, and on the heat transport (Ahuja 1975d) and mass transport of gases (Collingham 1968), proteins andplatelets(Petschekand Madras 1969, Turitto, Benis and Leonard 1972) in flowing blood. The effect of perfusion of tissues has also been included in the thermal conductivity of tissues (Grayson 1952). The second approach, which may be termed the convectiveapproach, is usually adopted in single-phase fluid flow theory. In this theory, the thermal conductivity offlowing suspensions is assumed to be the same as that of stationary suspensions, and theNusselt number is computed fromthe measured heat transfer coefficient. This approach is adopted in the thermal design of a bloodheat exchanger (AhujaandHendee 1978). I n this paper,the first approach is adopted for investigating the transport phenomena in the laminar flow of blood. Augmentation in thermal conductivity ( k ) of a flowing suspension or blood is defined as k,,,/k,tst - 1, where subscripts ‘mov’ and ‘stat’ stand for moving and stationary suspensions or blood. The augmentation in diffusion coefficient is defined similarly.Augmentation in thermal diffusivity ( a = k / p c ) is also defined similarly because the density p and specific heat c of the suspension or blood are independent of the state of motion,
2. Correlation of heat and mass transportdata
It is well known that transport properties of single-phase fluids (such as water or air) are independent of the state of motion in the laminar regime (EckertandDrake 1959). By employing the Graetzsolution, it has been shown experimentally that the thermal conductivity (Ahuja 1975b) or mass diffusion coefficient (Collingham 1968)of a particlesuspension in laminarmotion is a function of particle radius a and volume fraction +,tube length L and U and transport properties radius R, shear rate or particle angular velocity (kinematic viscosity v, thermal diffusivity a or diffusion coefficient D ) of suspending fluid. By constructing dimensionless groups from these variables, or by following systematic dimensional analysis (Ahuja 1976), augmentation of the thermaldiffusivity or diffusion coefficient of a flowing suspension may be
930
Avtar S. Ahuja et al.
written as amov/astat-
1 or DmovlDstat- 1 = F(+,wa2/vf, wa2/afor wa2/Df,R/a, L/a) (1)
where subscriptfindicates the suspendingfluid.The term wa2/vf is the Reynolds number, and the expressions wa2/af and wa2/Df, respectively, are the Peclet numbers for heat and mass transfer (for meanings of symbols, see the Appendix). For single-phase fluids in laminar flow, the shear rate varies linearly withthe radial distance from the axis of the tube, zero at theaxis and maximum at the wall. Therefore, the average shear rate is half the wall shear rate. The angular velocity of the particle, which is half the shear rate at the point where the centre of the particle happens to be located, varies linearly with the radial distance from the tube axis for the denser particles (such as RBCS) (Jeffrey and Pearson 1965), yielding the average angular velocity of the particle to be half the average shear rate or one quarter of the wall shear rate. When a suspension flows in a tube, particles under theinfluence of the shear field rotate and entrain thesurrounding liquid by friction. If the shear field is strong enough and the particles are large enough so that wa2/vf 2 1, then each particle acts like a centrifugalfan in the sense that thefluid particles are thrown outward from the region near the equator, and return to a region around the of heat or mass transport resides in this poles. The augmentation churning effect superimposed upon the rotational motion of the surrounding fluid. Heat transport data (Ahuja 197513) obtained for a number of values of the variables in eqn (1) have been correlated by eqn (1) of Ahuja (1975~) and the effects of particle interactions on heat transport inflowing suspensions, ignored in eqn ( l ) ,have been postulated (Ahuja 1975~).Particle interactions havebeen classified into particle crowding and formation of doublets or multiplets (Goldsmith and Mason 1967). It is assumed that both typesof interactions will cause a reduction in augmentation. Particle crowding affects the augmentation through the mutualinterference of the hydrodynamic environment of particles whereby particle rotational motion is affected. It is further supposed that a doublet of spheres will cause less churning of the surrounding fluid compared with the churning caused by the two spheres were they free to spin. Collingham’s (1968) gas transfer data, corresponding to several variables as listed above, have remained uncorrelated. I n fig. 1, heat (Ahuja 197513) and helium transport (Collingham 1968) data arecorrelated byplotting the or diffusion coefficient against percentage augmentation in thermal conductivity +(waz/vf)(wa2/af or waZ/D,) x (R/a)z(L/2a)x x doublet collision frequency ratio. The doublet collision frequency ratio indicated in fig. 1, which yields the effect of particle collisions on heat or mass transport, is the cube of the ratio of particlediameter to the 100pm diameter of an arbitrarily chosen reference particle (Ahuja 1975~).Augmentation in the laminar flow of aqueous suspensions of 50 and 100 pm polystyrene spheres is seen to be about 200% for heat transport and 500% for helium transport.
Transport Phenomena in Blood
Y
0 wa2 wa2 4 (T)(F
or
I
I
0.5
1.0
ma2 R L %)(--) (%)
93 1
I
1.5 2.52.0
1
35
3.0
LO
x 10-8 x (doublet collision frequency ratio)
Fig. 1. Plot of augmentation in thermal conductivity kmovlkBtst- 1, or diffusion coefficient 1, against values of the expression given for the abscissa of helium, Dmov/Dstittas shown for a laminar flow of suspensions. For meanings of symbols, see text. In the heat transfer experiment, the suspension was heated by circulating water in a countercurrent heat exchanger as it moved down the stainless steel tube (Ahuja 1975b). The Graetz solution corresponding to a uniform wall temperature was used. 0 , polystyrene spheres of volume fraction 8.8% suspended in 5.2% by weight aqueous NaC1, particle diameter 100 pm, tube diameter d = 1 mm and length L = 55 cm, tube Reynolds number = 576-1240. In thegas transfer experiment, the suspension was saturated with helium. Watersaturated argon flowed in an annulus around a Silastic tube and helium diffused from the flowing suspension into the annulus through the permeable wall of Silastic (Collingham 1968). The Graetz solution corresponding tothe uniform wall concentration was used. Polystyrene spheres were suspended in 7.4% by weight aqueous NaC1, tube diameter d = 1.5 mm and length L =288 cm, the gas was helium. 0,particle diameter 100 pm and volume fraction 7.5%. e, particlediameter 50 pm and volume fraction 7.23%.
2.1. Generality of correlation of heat and mms transfer data Augmentation in heat conductivity is smaller than the augmentation in the diffusioncoefficient of gases, other factors of the experiments being given, ina fluid (for because the Brownian diffusioncoefficient for a gas ( example, water) is about two orders of magnitude smaller thantheheat of water. An abscissa value of 0.5 in fig. 1 is an upper conductivity ( limit for heat transfer data in the laminar flow regime, while an abscissa value of 4 can easily be attained for gas transfer. Correlation of heat and gas (helium) transfer data infig. 1 is for two particle sizes (50 pm and 100 pm), two tubes (one of d = 1 mm and L = 55 cm, and the other of d = 1.5 mm and L = 288 cm) and particle volume fractions of 7.23, 7.5 and 8-8yo.In addition, coordinates of fig. 1 have been seen in detail (Singh 1968) to correlate data for He, 0, and heat, five different tubes (diameters 1-2 mm and lengths 40-288 cm), two suspending media (aqueous sodium N
N
33
932
Avtar S. Ahuja et al.
chloride and aqueous glycerine), two particle diameters (50 pm and 100 pm) and a variety of particle volume fractions (0*3-8-8%). The correlation presented in fig. 1 appears to be representative for applications to blood. 3. Application of correlation to flowing blood 3.1. Necessity of a simulated study as it relates to blood In vivo blood flow is characterised by tapered and radially pulsating blood vessels which may possibly have curvature, pulsatile flow of blood, biconcave disc-like structure of RBCS which are flexible and distort their shape as they move, presence of white blood cells, platelets, proteins, several types of ions and dissolved gases. Complete simulation of the in vivo conditions of blood is not easy and may not even be necessary. Assumptions must be made in the application of a physical theory to so prohibitively complex a situation. (i) Assumption that blood is a Newtonianfluid. It is now well known t h a t a t shear rates greaterthan 50 s-l blood is a Newtonianfluid (Copley 1972). (ii) Assumption of Poiseuille flow of blood. I n this paper, the results insuspension flow are applied to in vitro blood flow. It is an established fact that blood flow in large tubes (of diameter > 300 pm) follows the Poiseuille law (FBhraeus and Lindqvist 1931). (iii) Assumption of a spherical and rigid RBC. To apply the results of the simulatedexperimentswith rigid sphericalparticles to blood, the biconcave disc of the RBC has to be replaced by an equivalent rigid sphere which yields the same results as thedisc. In thecase of blood, we know of no way of establishing this equivalence. To further the argument,we used a sphere of the same volume as the humanRBC. I n section 4 effects of the size and the flexibility of the RBC sphere on the mass transport in blood are discussed. 3.2. Heat and gas transport
in blood
Theradius of asphere of human RBC volume ( = 87 pm3 (Altmanand Dittmer 1971)) is a = 2.75 pm. For a haematocrit C$ = 0.3, angular velocity of the RBC W = 274200) radian s-1 (corresponding to a wall shear rate of 800 s-l), diffusion coefficient of oxygen inplasma D, = 2 x 10-5 cm2s-l, kinematic viscosity of plasma vf = 0-015 cm2s-1, tube radius d = 0.50 mm and tube length L = 300 cm, the abscissa in fig. 1 is 0-0003and the augmentation in diffusion coefficient Dmov/Dstat-1 is negligible. This conclusion, that no significant augmentation of oxygen transport in flowing blood occurs a t even high shear rates, is in accord with direct measurements in blood (Collingham 1968). The same conclusion has been reached from direct measurements for heat transport in laminar flow of blood (Ahuja 1975d). These results suggest that as far as heat or gas transfer is concerned, blood can be treated as a single-phase fluid, That is, RBC tumbling and rotation in flowing blood (even for large turkey RBCS) have no measurable effect on the transfer of heat or gas in blood.
Transport Phenomena
in Blood
933
3.3. Transport of plasma proteins in blood
The diffusion coefficients of plasma proteins in water a t 20 'c vary from 2 x lo-' to 6 x lo-' cm2 s-l (Altman and Dittmer 1971). Consider the case 1971). of fibrinogen for which D, = 2 x 10" cm2 s-1 (Altman and Dittmer Then for the same haematocrit, shearrate and tube dimensions as given above, the value of the abscissa in fig. 1 is 0-03 and the augmentation in diffusion coefficient of fibrinogen in flowing blood is about 50%. The augmentation of the diffusion coefficient of plasma proteins in flowing blood, though not very high, appears to be measurable. 3.4. Transport of platelets in blood
Brownian diffusion coefficient of platelets in blood plasma is given by the R = 1.98 cal Stokes-Einstein law as D, = RT/6,7aN. For the gas constant K-l mol-l, an absolute temperature T = 310 K (37 "c), the dynamic viscosity of plasma 7 = 0.015 Poise, the radius of the equivalent sphere of a platelet a = 1 pm and Avogadro's number N = 6.02 x D, 1.5 x cm2 s-l. Assuming that the diffusion coefficient D, is zero forplateletsin RBC, the Fricke equation (Fricke 1924, Stein et al. 1971)
with the shape factor X for RBCS taken to be 1.1 (Fricke 1924), yields DStat/Df N 0.8 x cm2 5-l asthe diffusion coefficient of plateletsin - 0.55 or D,, stationary blood of 30% haematocrit. For the case of the diffusion of platelets in flowing blood, with a Brownian cm2 and all other diffusion coefficient of plateletsinplasma D, = 1.5 x quantities as given in section 3.2, the abscissa in fig. 1 is 3.6. The augmentation of platelet diffusion is about 55076 which, with a diffusion coefficient of platelets cm2 s-l, yields a diffusion Coefficient in instationary blood Dstat = 0.8 x cm2s-l. Themeasuredvalues of the diffusion moving blood D,,, g 5 x areintherange coefficientof plateletsin flowing blood in vitro, D,,, 5x 2.5 x lo-' cm2 s-l (Turitto et al., 1972) forarange of 40-440 s-1 in the wall shear rate and 3 6 5 0 % for the haematocrit. Augmentation in platelet diffusivity, Dmo,/Dstat- 1 , is 6150-31 150% and the ratioof the measured value of platelet diffusivity to thatdeduced from the simulated experiments is10-50. 4.
Discussion
4.1. Effect of size of spherical RBC
For a physiological averageshear rate of 200 s-l in a given tube as in section 3.2, the abscissa values in fig. 1 are about 1.0, and 5.0 for RBCS of radii 2.75 and 4.1 pm, respectively, yielding D,, of platelets of about 4 x 10-9 and 6 x 10-9 (extrapolated) cm2 s-l. These resultsindicatediscrepancyranges (that is, the ratio of the measured value of the platelet diffusivity to that
Avtar S. Ahuja et al.
934
deduced from the simulatedexperiments) of about 12.5-62.5 and 8-42. Physiological shear rates lower than 200 S-llead to still higher discrepancy values. 4.2. Effect of RBCJlexibility
Because part of the flow energy is spent in distorting the shape of a flexible in reduced angularvelocity W of the RBC, flexible RBC spheres are likely to give less augmentation in platelet diffusivity than rigid RBC spheres. direction. The quantitative argumentsassuming a rigid RBC are in the right RBC, resulting
4.3. Hypothesis regarding the excess platelet diffusivity in $owing blood
The discrepancy between the measured value of the platelet diffusivity and that deduced from the simulated experiments described in sections 3.4 and 4.1 cannot be accounted for on the basis of variations in RBC size and angular velocity. Therefore, there is a strong indication that, of a total augmentation of about 6000-30000% in platelet diffusivityin flowing blood, only about 500% is explained by fluid mechanical forces. Since flowing blood has other (electrical or biochemical) forces present, it is logical, at least a t this stage of the development of thisproblem, to attribute the excess platelet diffusivity to those forces than fluid mechanical ones. 4.4. Comment on the physical basis of the phenomenon of augmentation in
diffusivities of a variety of species To our knowledge, the mechanism of augmentation of heatand mass transport in flowing blood, in all its ramifications, is not yet understood. The physical basis advanced in the literature has been attributed to the Prandtl Mixing Length Hypothesis (from turbulent flow theory) from which a relation for blood in laminar flow has been obtained as (Keller 1971)
D,,,
lOP(average shear rate)n
(3)
where n = 1 or some fraction (Petschek and Madras 1969, Friedman, Liem, Grabowski, Leonard and McCord 1970). A caution has been sounded about its grossness (Keller 197 1 ). Eqn (3) gives only the effect of shear rate on the diffusivity, ignoring the effects of particle size and volume fraction, tubedimensions, diffusion coefficient of thetype of species being transported(heat, gases, platelets,proteins), kinematic viscosity of suspending liquid and particle interactions. Also, the currently available experimental measurements are not sensitive to the effects of haematocrit and shear rate (Turitto et al. 1972). Eqn (1) and fig. 1 give the effects of all these variables quantitatively and so bydoing indicate the physical basis of the mechanism of augmentation of heat and mass transport in flowing blood. 5.
Conclusions
Heat and gas transport data in flowing suspensions of polystyrene particles have been correlated. An application of this correlation to flowing blood yields a possible mechanism of the augmentation of transport of a variety of species.
Transport Phenomenu in Blood
935
The assumptions of the study are valid for in vitro blood flow. This study accounts for all significant variables which enter into in vitro blood flow and quantitates their effects on augmentation of platelet diffusivity. A comparison between the values of platelet diffusivity obtained from the correlation of heat and mass transfer data in flowing suspensions and those measured in flowing blood yields an excess platelet diffusivity unaccounted for by fluid mechanical forces. The excess platelet diffusivity is contributed probably by other forces (electrical or biochemical) present in blood. Another potential application (notdescribed in this paper)of the augmented transport phenomena in flowing suspension is in the area of the energy transport. For a constant energy expended in propelling a fluid in a tube, the heat and gas carrying capacity of the fluid could be increased significantly if the fluid were a suspension ratherthan a single-phase fluid. Such an increase would be advantageous in devices such as heat exchangers.
This investigation was supportedin partbyGrant awarded by the National Cancer Institute, DHEW.
KO. 5T32-CA09073,
APPEXDIX List of symbols a particle or RBC radius (cm) C specific heat (calg-1 'c-1) d tube diameter = 2R D diffusion coefficient (cmzs-l) k thermal conductivity (cal s-l cm-l 'c") L tube length (cm) R tube radius (cm) X particle shape factor in eqn (2) a thermal diffusivity (cm2 s-1) B ratio of species (or platelet) solubility in particles or suspending fluid or plasma V kinematic viscosity (cm2 s-l) density (g cm-3) particle volume fraction or fractional haematocrit W angular velocity (radian s-1) Subscripts f suspending fluid or plasma pparticle or RBC mov moving suspension or blood stat stationary suspension or blood
RBCS
to that in
RI~SUM~ Phenomenes de transport dansdes flux laminaires de sang L'on a montre recemment par experiences que la transport de chaleur et de gaz (surtout l'oxygene et l'helium) est augment6 dans le flux laminaire de suspensions aquemes de spheres de polystyrene d'un diambtre de 50 et de 150 pm. Ce rapport Btablit IS, corr6lation entre les donn6es sur le transport de la chaleur et de gaz. L'application de cette corr6lation iL des flux de sang mene aux conclusions suivantes. I1 ne se produit pas d'augmentation significative du transport d'oxygene et de chaleur dans le sang en 6coulement, m6me avec des taux de cisaillement bien
936
Transport Phenomena in Blood
plus blevbs que lea taux de cisaillement physiologiques; c’est 18 une observation conforme avec les resultats expbrimentaux. L’augmentation du coefficient de diffusion des protbines de plasma dans le sang enbcoulement, sans &re trbs blevbe, semble mesurable. Sur le total d’une augmentation mesurbe d’environ 6000-30 000% quant au coefficient de diffusion des hbmatoblastes dans le flux de sang, comme le montre la littbrature, environ 500% sont attribuables, d’aprbs cette corrblation, 8 des forces mbcaniques fluides et le reste semble provenir d’autres forces (blectriques ou biochimiques) prbsentes dans le sang.
ZUSAMMENFASSUNQ Transportphiinomene bei laminarer Blutzufuhr Kurzlich ist experimentell gezeigt worden, dass der WBrme- und Gastransport (insbesondere Saueratoff und Helium) in dem laminaren Fluss wiisseriger Aufschliimmungen von Polystryrolsphiiren mit einem Durchmesser von 50 und 150 wm gesteigert wird. I n dieser Abhandlung werden die Daten iiber Wiirme- und Gastransport in Korrelation gebracht. Die Anwendung dieser Korrelation auf zirkulierendes Blut ergibt die folgenden Schliisse. I m zirkulierenden Blut findet keine bedeutsameSteigerungdes Sauerstoff- und Wiirmetransports statt, auch wenn die gegebenen Scherungszahlen weit hoher als die physiologischen sind. Diese Beobachtung stimmt mit den Versuchsergebnissen uberein. Die Erhohung des Diffusionskoeffizienten der Plasmaproteine im zirkulierenden Blut ist offensichtlich messbar, auch wenn sie nur geringfiigig ist. Die aufgrund des Schriftmaterials insgesamt gemessene Steigerung des Diffusionsvermogens von Blutpliittchen im zirkulierenden Blut von etwa 6000-30 O O O ~ oist, wie diese Korrelation ergibt zu etwa 500% auf fliessende mechanische Kr(lfte zuriickzufuhren. Man nimmt an, dass der Augsleich durch andere im Blut vorhandene Kriifte (elektrische oder biochemische) gesohaffen wird.
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