Heat transport in laminar flow of erythrocyte suspensions.

JOURNALOP

APPLIED

Vol. 39, No. 1, July

PHYSIOLOGY Printtd

1975.

in U.S.A.

Heat transport of erythrocyte

in laminar

flow

suspensions

AVTAR SINGH AHUJA Rust College, Holly Springs, Mississ$~i

38635

AHUJA, AVTAR SINGH. Heat transport in laminarj!ow of erythrocvte Physiol. 39( 1) : 86-92. 1975.-Measuresuspensions. J. Appl. ments of thermal conductivity were made in laminar flow of dog and turkey erythrocyte suspensions in a stainless steel tube of about 1 mm ID. These measurements were independent of the shear rate, showing that the red cell motion relative to plasma in flowing blood had no effect on the heat transfer. Measurements of thermal conductivity were further made in suspensions of polystyrene spheres of 100 pm and were found to be dependent upon the shear rate. The Graetz solution corresponding to uniform wall temperature was used for determining the value of thermal conductivity in an apparatus calibrated with tap water. The overall accuracy of the results is within 10%. A model based on the particle rotation with the entrained fluid is proposed. It is pointed out that the diffusion of platelets, red cells, and possibly plasma proteins (such as fibrinogen) will be augmented if they happen to be in the hydrodynamic

field of rotating

It is shown in this paper that there is no augmentation of heat transfer in flowing blood. Measurements were further made in suspensions of polystyrene spheres of size much larger than that of the erythrocyte and an augmentation in heat transfer was detected. This study presents a method of determining the thermal conductivity of blood other than the steady-state (23) and unsteady-state (1) methods. On the basis of particle rotation in the shear field, an augmentation model is proposed which explains the negative results in blood as well as positive results in polystyrene suspensions. From this model augmentation of diffusion of such species as plasma proteins, platelets, and red cells due to red cell rotation is also deduced. The following valid assumptions are made in regard to the laminar flow of blood. I> Whole blood flowing in a tube of inside diameter of about 1 mm, where tube diameter to red cell diameter (10 pm) is more than 100, is homogeneous. 2) Blood in laminar flow with shear rates more than 100 s-l behaves like a Newtonian liquid (24). 3) The hematocrit of whole blood flowing in a l-mm tube is the same as of stationary blood and the average velocity of erythrocytes is very much the same as that of plasma (8). The blood viscosity can therefore be assumed to be uniform across the cross section of the tube and flow to be Poiseuillian.

erythrocytes.

thermal conductivity values of stationary blood suffice for blood in laminar motion; thermal conductivity of polystyrene suspensions dependent on shear rate; model of augmentation of heat transport; augmentation of diffusion of platelets and possibly of plasma proteins

IT IS WELL KNOWN that the transport properties (viscosity, thermal conductivity, and diffusion coefficient) of singlephase fluids (such as water or air) in laminar flow are independent of the state of the motion. However, the transport properties of turbulently flowing single-phase fluids are dependent on the shear rate, in accordance with the Prandtl Mixing Length Hypothesis (19). As blood flows in the microvasculature (in vivo) or in small tubes (in vitro), the erythrocytes rotate, deform, tumble, and move across the flow stream lines in an irregular fashion toward the axis of the blood vessel (3, 4). In analogy with the turbulent flow of fluids, if it could be proved that the irregular motion of the erythrocytes in flowing blood augmented the thermal conductivity of blood, it would indicate that the gas diffusion coefficient of blood is also augmented (because heat and gas transfers in fluids are analogous phenomena). This effect would increase the efficiency of the artificial lung. Further, it is important to know the heat transfer characteristics of flowing blood for the design of a heat exchanger for hypothermia in extracorporal circulation. Augmentation of heat transfer in flowing blood would help increase the efficiency of the heat exchanger.

METHODS

A schematic diagram of the apparatus used for the measurements is shown in Fig. 1. Blood or other test liquid is heated by circulating water in a countercurrent heat exchanger as it moves down the stainless steel tube. Heat gained by blood is p,c(7rD2/4)u(tbe - tbi) and the total heat crossing the tube wall is hrd,L(tw - tB). Equating and, for countercurrent flow, taking (6) tJ$r the average

overall

h = (m c/rd,l)[(tbe (For meanings also given by

tB =

Ati - At, ln (Ati/At,>

heat transfer -

tbJ/(Ati

of the symbols,

coefficient -

is

At,)] ln (Ati/Ate)

see LIST OF SYMBOLS.)

l/h = l/h + b/k,, -I- l/h2

(2) It

is

(3)

where b = (d,/Z) In (Do/D). F or a given tube size and material, b/k,, can be calculated. Assuming that the temperature of the tube wall jumps 86

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HEAT

TRANSPORT

IN

FLOWING

87

BLOOD

suddenly at x = 0 from that of the entering liquid to some higher temperature which remains uniform downstream from x = 0 and blood enters the tube section x = 0 with a fully developed velocity profile, the Graetz solution for uniform wall temperature is (6)

0.0668 Re Pr D/L 1 + O.O4[Re Pr D/L12j3

Nu = 3.65 + Or, simplifying

for computational

hlD

(4)

purposes

0.2672rnc/~L 1 + 0.04[4mc/?rLkb]2’3

= 3*65kb +

(5)

If the value of ha is known, then from Eqs. 2, 39 and 5 the thermal conductivity kb of the test liquid (blood) can be found. For the experiments reported herein, water was selected as a heating fluid, because steam, if used for heating blood, will be severely traumatic to red cells. But the temperature (NOT

To

COMP. AIR-

SCALE)

m INLET

VAC.

FLASK

SECTION = 2.0 cm

EXIT

I

022 HEATED

TUBE:

H 1 CIR. WATER OUT 1,

VAC FLASK

1. Schematic of flowing liquids TC = thermocouple. FIG.

by

diagram Graetz

for measuring solution with

BOUNDARY

the thermal uniform wall

4037.39

conductivity temperature.

COND I T IONS WATER

/___

BATHr=MP-

_----Me------

30. P

.

g

THERMOCOUPLE

h

WALL

TEMP

I

20.

W + I o-

0

b

I

IOI

I 1 ++.HEATED ENTRANCE

201

301

TUBE

LENGTH-

501

LENGTHJM

6.5

2. Longitudinal temperature measured by four thermocouples 1 mm; Re = 169. FIG.

40 I

LENGTH,L

curve

obtained

by

I

0.62 - OC running

1

tap

I

0.78

0.70 water

0.86

through

A4

TUBE

3. Calibration of 1 mm ID.

I

0.54 coI/sec

of the tube wall, when blood flowed, varied,l as shown in Fig. 2 for one flow rate. This means that the inverse of the average heat transfer coefficient of the circulating water, zero and for different flow rates l/h2 > does not approach of the test liquid, the value of l/h2 was different. Since the thermal conductivity of water is known to be independent of the flow rate in the laminar regime, the apparatus was calibrated with (tap) water; i.e., the value of l/h2 was determined for different flow rates of the tap water. Since the bulk temperature of the circulating water, tW , was measured to be constant (i.e., t.,i = tW, = tw), Eq. 2 can be written as

Ln = 1.066Bmm

kJ

IN

I

0.46 fix,

FIG.

OUTLET

I

0.38

LENGTH = 55 0 cm

tube

CIR WTER

430

-

CM

variation embedded in

5s CM

60 1

70 1

1 ’

4J-----II

EXIT

of the the wall.

LEN&H

= 2.0

tube wall Tube ID

CM

as =

=

(&c/r&

L)

In

[(L

-

tbi)/(tw

-

tbJ]

(61

Calculating h from Eq. 6 and hr from Eq. 5 in which thermal conductivity k of water (independent of flow rate) has been taken to be 1.43 X 10m3 Cal/s 0cm. “C and substituting in Eq. 3 yields the values of l/h2 for several flow rates. Assuming that the heat transfer coefficient ha outside the tube depends on the product mc, i.e., the heat extracted by the flowing test liquid for one degree of temperature rise, a plot of l/h2 versus ri?~ is shown in Fig. 3 from which l/h2 can be determined for a known value mc for any test liquid. hl can then be calculated from Eq. 3 and 6 and kb of the unknown liquid from Eq. 5. l Uniform wall temperature was not attained by passing methylene chloride (boiling point = 40.67”C at a pressure of 760 mmHg and heat of vaporization = 78.74 Cal/g) vapors around the finned tube in the heat exchanger. Quite surprisingly, uniform wall temperature was also not achieved with tap water running in the finned tube and steam around it at 100°C at atmospheric pressure.


88

A. S. AHUJA

Tube specijcations. A l-mm-ID tube was used to ensure high shear rates (e.g., of the order of 10,000 s-l) in the laminar regime. Stainless steel hypodermic needle tubing was selected due to the fact that of all metals stainless steel is one of the least traumatic to blood. It has a fairly reasonable thermal conducitivity value of about 0.04 cal/s.cm= “C (6). Exact inside diameter of the tube was 1.07 mm and wall thickness 0.02 mm. Experimental setup. Four 36-gauge copper-constantan calibrated thermocouples were tied with thread and glued to the tube wall at distances of 8, 18, 35, and 58 cm from the inlet end of the tube. The tube was insulated from the copper jacket with rubber stoppers (see Fig. 1). Water, at 37.4”C, from a constant-temperature bath of accuracy O.O6’C, was turbulently circulated in a 1.27-cm jacket around the test section by a centrifugal pump at the rate of about 8 gal/min (ensuring Reynolds number of more than 70,000). The tube was connected to two vacuum flasks-with inlet capacity of 1,000 ml and outlet capacity of 650 ml-through Tygon and polyethylene flexible tubing. A glass tube bent in zigzag form for adiabatically mixing the blood was suspended in the outlet flask through its rubber stopper. Gravity and compressed air forced the test liquid through the tube. An ice bath served as reference junction for the thermocouples. The accuracy of temperature measurement (of Leeds and Northrup K-3 Universal Potentiometer) was o.o13?Z. Procedure. Test liquids were 1) tap water; 2) blood plasma centrifuged from 1,000 ml of dog blood containing about 150 ml of anticoagulant acid-citrate-dextrose (ACD) solution; 3) erythrocyte suspension of hematocrit 8.6% prepared in plasma; 4) 1,000 ml of whole dog blood of hematocrit 38 % and containing 150 ml of ACD solution; 5) 500 ml of turkey blood of hematocrit 25 %, containing 1% heparin and about 200 ml of isotonic saline; and S) polystyrene spheres, as shown in Fig. 4, in various fractions of diameters 88-105 pm, suspended in 5.0 wt % aqueous sodium chloride solution: volume fraction of particles 8.2 %.

TABLE

1. Properties of blood components and polystyrene Density,

RBC Plasma Polystyrene * FromMendlowitz nology (7).

g/cm=

Specific

t From

cd/g

“C

0.77* 0.94* 0.32t

1.09 1.03 1.04-1.065t (14).

Heat,

Encyclopedia

of Chemical

Tech-

The experimental procedure for every test run was essentially the same. It is described below for whole blood. Freshly drawn dog blood (about 400 ml of blood were drawn from the femoral artery into a blood bag containing 75 ml of anticoagulant acid-citrate-dextrose) was cooled to a temperature well below the body temperature (temperatures in the range 2-9°C) and was well mixed. Its bulk temperature was noted by a thermocouple. The hematocrit was determined to be 38 % by spinning the blood in Wintrobe tubes in a centrifuge at 2,009 rpm for 30 min. The outlet flask with stopper and the adiabatic mixer was weighed by means of a balance accurate to 0.1 g. The compressed air valve was opened and its pressure was regulated corresponding approximately to the desired flow. Blood was allowed to run into a beaker until the air pressure became steady and all the air bubbles in the flow appearing at either or both the junctions of the tube and the flexible tubing were expelled by pressing the tubing with fingers at the bubble sites. The polyethylene flexible tubing exiting heated blood was inserted into the glass tube projecting out of the stopper of the outlet flask and the stopwatch was started. At the conclusion of a run, the stopwatch was stopped, temperature of the hot blood (in the range 27-34’C) noted, and the flask with blood was weighed to determine the blood flow rate. The outlet flask was not calibrated with respect to some known temperature, e.g., ice point. It was therefore considered imperative to maintain the flask at outlet bulk temperature of the blood of the preceding run to account for the heat capacity of the flask and the mixer in the succeeding run; the very first run of every test liquid was repeated. This was done by preventing the setting up of free convection in the outlet flask by keeping it covered all the time except when emptying the blood out. The polystyrene particles of density 1.05 g/cm3 and diameters between 88 and 105 pm were suspended in 5.0 g/100 ml NaCl solution so as to match densities and retard particle settling. The inlet flask was shaken to ensure homogeneous suspension before switching on the air. RESULTS

FIG. 4. Photomicrograph 105 pm. Root-mean-square

x78.)

of polystyrene diameter =

spheres of diameters 8894.6 pm. (Magnification

The properties of erythrocytes, plasma, and polystyrene used in calculations are given in Table 1. The densities and specific heats of blood and suspension have been calculated by averaging over the volumes. A sample calculation is given in Singh (21). It should be remarked however that in the calculations heat absorbed by the flask itself was taken into consideration. The flask was taken to be of diameter 7 cm, height 20 cm, and with glass thickness of 0.1 cm.


IN FLOWING

BLOOD

HEAT

TRANSPORT

TABLE

2. Summary of all results including Liquid

Hematocrit,

Tap water Dog blood plasma Dog erythrocyte suspension Dog

whole

Approx

8.6

blood

25.0

Whitmore

(24).

those given in Fig. 5 RBC Ctm

Size,

8 (Biconcave disklike) 8 (Biconcave disklike) 16 (Oval shaped)

38.0

Turkey blood (diluted with isotonic saline) * From

y0

89

t Numbers

in parentheses

Wall

Shear

Rate Range

PROPOSED

AND

Viscosity cm2/s

Reynold’s

Mean Thermal Conductivity, Cal/s. cm. “C

No. Range

2,450-6,200 3,100-6,500 2,900-5,850

0.01 0.015* 0.018”

349-88 1 292-614 228-46 1

1.43 1.35 1.33

2,250-4,150

0.03”

106-197

1.23

w

2,250-4,900

0.025”

126-279

1.27

(5) t

indicate

DISCUSSION

Discussion. It is seen from Fig. 5 and Table 2 that the thermal conductivity of erythrocyte suspensions is inde-

X 1O-3 (6)t (6) t (7) t

no. of observations.

The arithmetic mean of inlet and outlet temperatures for plasma and erythrocyte suspensions varied from 15 to 20°C. Their densities and specific heats have very slight variations over this temperature range and values given in Table 1 were used in the calculations. However, the kinematic viscosity of liquids varies appreciably over the range 1520°C. Since the knowledge of kinematic viscosity for various samples is required only for the calculation of Reynolds number (Table 2 and Fig 5) and not for the calculation of thermal conductivity (see Eq. 5), the knowledge of kinematic viscosity corresponding to mean temperature for each observation was not considered important. The values of kinematic viscosity at 20°C given in Table 2 were used for Reynolds number calculations for erythrocyte suspensions. Using Mooney’s formula (16), the kinematic viscosity of 8.2 % polystyrene suspension in 5 g/ 100 ml aqueous NaCl has been calculated to be 0.012 cm2/s. (Since the mean temperature for polystyrene suspension was from 18-2O”C, the kinematic viscosity of 5 % aqueous NaCl has been taken to be 0.01 cm2/s.) Final results for whole blood (hematocrit 38 %), and for turkey erythrocyte suspension (hematocrit 25 70) are plotted in Fig. 5, and the results for polystyrene suspension are plotted in Fig. 6. The scatter of data in Figs. 5 and 6 is within 10 70. All results except those of polystyrene suspension are summarized in Table 2. The thermal conductivity values for tap water were recalculated from the calibration curve in Fig. 3. The thermal conductivities of water, blood plasma, 8.6 % erythrocyte suspension, whole dog blood, and turkey blood are found to be independent of the shear rate. However, the thermal conductivity of the polystyrene particle suspension increases with the shear rate, as shown in Fig. 6. By extrapolation to zero shear, the mean values of the thermal conductivity in Table 2 and Fig. 5 are the same (within th e experimental accuracy) as determined by the unsteady state method (1). The experiment with 8.6 % erythrocyte suspension was conducted with a view to eliminate or reduce the effect on heat transfer of red cell crowding (hindering each other’s rotations) which may be present in a concentrated suspension or whole blood. The results are accurate to within 10 70. MODEL

Kinematic

s-1

0”

A

1.35

E

wfwEY

U

z

1.27

BLOOD

(HEMATOCRIT

A

25 %)

1ll1lllHlHliIl~l~l~~~l

uB 1.25

A

> a U 3

A

0

x

1.15

z 2

$d:‘~:*>~.&$3~>\ i&,*$2::q$???..%%p.&v~~?~ 1, fluid particles flow outward from the equator due to centrifugal forces and flow inward toward the poles. This secondary stream superposed on the rotational motion of the fluid causes the dependence of heat energy transport on flow rate in laminar motion of a solid-particle suspension. The augmentation in heat (or gas) transport resides in the inertia of the fluid entrained with the rotating spherical particle. One parameter through which this inertia is manifest is the Reynolds number w0r02/v and the other parameter is the Peclet number w0r02/cy, where a! is the thermal diffusivity of the suspending fluid. It has been shown experimentally (5, 22) that augmentation effect is determined

by the entity or the material which is being convected. For example, augmentation in heat transport is less than that in gas transport and, furthermore, augmentation in oxygen transport is different from that in helium transport. Intuitively, the origin of the Peclet number in this connection can be seen as follows. The time period of rotation of a sphere is 7 - l/w, (seconds). The time heat takes in diffusing a distance of one particle diameter Zr, (taken as characteristic length), in a fluid of thermal diffusivity cu(cm2/s), is t - (2r,)2/a (seconds). The augmentation is expected if the time t is of the same order of magnitude as 7 or larger. That is, if 4r02/cr - l/w, or 4 w.r,2/a - 1. In the companion case of gas diffusion in flowing suspensions, cy is replaced by the diffusion coefficient D in these arguments. For a wall shear rate of 8,000 s-l, the average shear rate is 4,000 s-l and the average angular velocity is ~3~ = 2,000 rad/s. For a polystyrene sphere of radius r0 = 50 pm suspended in saline solution of kinematic viscosity v = 0.01 cm2/s and thermal diffusivity a! = 1.43 X 10H3 cm2/s, the Reynolds number o,r02/v = 5 and the Peclet number w,ro2/(x = 35 and according to the proposed model the augmentation in heat (or gas) transport is expected. Now for a dog red cell of radius of the equivalent sphere of the same volume [about 65 pm3 (18)]r, = 2.5 pm suspended in plasma of kinematic viscosity v = 0.015 cm2/s, the Reynolds number w,r,2/v = 0.006 which is very small compared to 1. Similarly, for the thermal diffusivity of plasma of 1.48 X 10B3cm2/s (I), Peclet number w,r,2,/a = 0.05 which is again small. Hence, according to the proposed model, no augmentation in heat transport in flowing blood is expected. It is suggested here that the geometry of the particle may be playing a significant role in heat and mass transfer. The dog or human erythrocyte, which is a biconcave disk of diameter 7-8 pm and of peripheral thickness of 2 pm, is not likely to cause as large an “eddy” as a sphere of the same size would. Moreover all the red cells may not be spinning as they should were they solids; red cells can be easily deformed and some of them may actually be assuming various shapes as they flow without rotating. Augmentation of dijusion of platelets and erythrocytes in the hydrodynamic jield of other erythrocytes. It has been proposed above that the inertia of the fluid rotating with the particle, which manifests itself in w0r,2/v, is responsible for the augmentation of heat or mass transport. Further, it has been proposed that the augmentation depends upon the type of material or entity being transported. This manifests itself through w0r02/a for heat transport or W,ro2/D for mass transport. Consider the case of a sphere (equivalent in volume to red cell) of radius r0 = 2.5 pm rotating with an angular velocity ~3~ = 1,000 rad/s (equivalent to a shear rate of 4,000 s-l in tube flow) and suspended in plasma of kinematic viscosity v = 0.015 cm2/s. An extensive experimentation with flowing particle suspensions has shown that, given other parameters such as tube length and size and particle concentration, the augmentation in heat or mass transport in laminar flow of blood (of hematocrit 40 70) can be detectable (20 %) if ( w,r,2/v) (w,r,2/a! or D) z 3 (22) or if Q! or D is the order of 10V7 cm2/s or less. Under this circumstance, heat or oxygen transport is not augmented because for heat cy - 10m3 and for oxygen D - 10s5. It appears then that the


HEAT

TRANSPORT

IN

FLOWING

91

BLOOD

augmentation of diffusion of platelets for which D - 10-g cm2/s (10) is expected. This indeed is the case as was observed by Petschek and Madras (17) and Friedman and Leonard (9). Since the diffusion coefficient of plasma protein varies from about 2 X lo-’ to 6 X lo-’ cm2/s (Z), on the basis of the order of magnitude estimate the case of augmentation of diffusion of proteins appears to be the borderline case. Again, since the Brownian diffusion coefficient of erythrocytes in plasma is of the order of lO+O cm2/s (10) it appears that the red cell diffusion coefficient will be augmented in the hydrodynamic field of other red cells. CONCLUDING

REMARKS

It has been shown herein that there is no detectable augmentation in heat transport in the Poiseuille flow of blood, although there is a twofold augmentation in laminar flow of polystyrene suspensions. Flowing blood can be treated as a single-phase fluid insofar as heat transfer is concerned and the thermal conductivity values for blood in laminar motion can be taken to be the same as that of stationary blood (1). However, it has been reported in the literature (11, 12) that augmentation in oxygen transport takes place in the Couette flow of blood. A model based on the rotating particle with the entrained fluid has been proposed, which explains the positive results in polystyrene suspensions as well as the negative results in blood. According to this model there should be no appreciable augmentation in oxygen transport in Couette flow of blood. It has been further shown from this model that the diffusion of platelets, red cells, and possibly plasma proteins will be enhanced in the event they happen to be in the hydrodynamic field of rotating erythrocytes. LIST OF SYMBOLS specific heat of the test liquid, inside diameter of the tube, suspending liquid, cm2/s

b

Cal/g cm;

“C diffusion

coefficient

in

the

dm D* h h h2 k k I? .

logarithmic mean diameter between the outside and the inside diameters of the tube = (Do - D)/ln (Do/D), cm outside diameter of the tube, cm average overall heat-transfer coefficient, Cal/s. cm2. “C average heat-transfer coefficient of the test liquid, cals/cm2. “C average heat-transfer coefficient of the circulating water, Cal/s. cm2. “C thermal conductivity, Cal/s. cm. “C thermal conductivity of stainless steel, Cal/s cm. “C heated length of the tube, cm mass flow rate of the test liquid, g/s average Nusselt number = hlD/kb Prandtl number = vb c/kb radial coordinate, cm radius of the spherical particle, cm Reynolds number = u D/vb temperature, “C; time - 4r02/cu, s temperature difference = twi - the, “C temperature difference = tw, - tbi, “C mean velocity of the test liquid, cm/s thermal diffusivity = k/pc, cm2/s dynamic viscosity, dyn s/cm2 kinematic viscosity = v/p, cm2/s density, g/cm3 time period of rotation of the spherical particle -lICJO, s angular velocity of the particle, rad/s l

ZU Pr r r. Re t Ate Ati U a rl Y P 7 WO

Subscripts b blood B bulk e exit i inlet W water W wall

or suspension

The author thanks Dr. P. L. Blackshear, Jr., for suggesting this problem and for providing assistance and support in the completion of it. Assistance provided by Messrs. John H. Traver and Frank D. Dorman, and by Dr. R. E. Collingham is gratefully acknowledged. Free samples of polystyrene spheres provided by Dow Chemical Co., Midland, Mich., are also gratefully acknowledged. This paper is part of the author’s MS thesis done under the name Avtar Singh at the Dept. of Mechanical Engineering, University of Minnesota, Minneapolis. Address reprint requests to A. S. Ahuja, School of Information and Computer Science, Georgia Institute of Technology, Atlanta, Ga. 30332. Received

for

publication

4 November

1974.

REFERENCES 1. AHUJA, A. S. Measurement of thermal conductivity of stationary blood by unsteady-state method. J. A/$. Physiol. 37: 765-770, 1974. 2. ALTMAN, P. L., AND D. S. DITTMER (Editors). Blood and Other Body Fluids. Bethesda, Md. : Fed. Am. Sot. Exptl. Biol., 197 1. 3. BLOCH, E. H. A quantitative study of the hemodynamics in the living microvasculature system. Am. J. Anat. 110: 125-153, 1962. 4. BUGLIARELLO, G., AND J. W. HAYDEN. High-speed microcinematographic studies of blood flow in vitro. Science 138 : 981-983, 1962. 5. COLLINGHAM, R. E. Mass Transfer in Flowing Suspensions (PhD thesis). hlinneapolis, Rlinn. : Univ. of Minnesota, 1968. 6. ECKERT, E. R. G., AND R. M. DRAKE, JR. Heat and Mass Transfer. New York: McGraw, 1959. 7. Encyclopedia of Chemical Technology. New York : Interscience, 1954, vol. 13, p. 158. 8. F~HRAEUS, R., AND T. LINDQVIST. The viscosity of the blood in narrow capillary tubes. Am. J. Physiol. 96 : 562-568, 9. FRIEDMAN, L. I., AND E. F. LEONARD. Platelet adhesion surfaces : consequences of flow, exposure time, blood 10.

and surface nature. Federation Proc. GOLDSMITH, H. L. Red cell motions flow. Federation Proc. 30 : 1578-1588,

193 1. to artificial condition,

30: 1641-1646, 1971. and wall interactions 1971.

in tube

19.

KELLER, K. H. Development of a Couette oxygenator. In: ArtiJicial Heart P ro g ram Conference, edited by R. J. Hegyeli. Washington, D.C: US. Govt. Printing Office, 1969, chapt. 34. KELLER, K. H. Effect of fluid shear on mass transport in flowing blood. Federation Proc. 30: 1591-1599, 1971. LAMB, H. Hydrodynamics. New York: Dover, 1945. MENDLOWITZ, M. The specific heat of human blood. Science 107 : 97-98, 1948. MITVALSKY, V. Heat transfer in the laminar flow of human blood through tube and annulus. Nature 206: 307, 1965. MOONEY, M. The viscosity of a concentrated suspension of spherical particles. J. Colloid Sci. 6 : 162-l 70, 195 1. PETSCHEK, H. E., AND P. N. MADRAS. Thrombus formation on artificial surfaces. In: Artijcial Heart Program Conference, edited by R. J. Hegyeli. Washington, D.C.: US Govt. Printing Office, 1969, chapt. 24. PONDER, E. Hemolysis and Related Phenomena. New York: Grune & Stratton, 197 1. SCHLICHTING, H. Boundary Layer Theory. New York: McGraw,

20.

1960. SIEDER,

11.

12. 13. 14. 15. 16. 17.

18.

liquids

E. N., in

tubes.

AND G. E. TATE. Ind.

Eng.

Chem.

Heat 28:

transfer 1429-1435,

and

pressure

drop

of

1936.


92 21.

A. A. Experimental Determination Stationary and Flowing Blood (RIS Univ. of Minnesota, 1966.

SINGH,~

of thesis).

Thermal Conductivity Minneapolis, Minn.

of

22. : 23.

are by them.

the

present

author,

who

changed

his

name

24.

The Thermal Conductivity (PhD thesis). Minneapolis,

SPELLS, K. E. The Phys.

2 These works after completing

A.

sINGH,2

Suspensions 1968. Med.

WHITMORE,

thermal Biol. 5: 139-153, R. L. Rheology

of Minn.

conductivities 1960/61. of the Circulation.

Stationary : Univ. of some New

S. AHUJA

and Flowing of Minnesota, biological York:

fluids. Pergamon,

1968.


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