Section 8.4 Circulation of the Blood

105

FIGURE 8.3 Turbulent ﬂuid ﬂow.

8.4 Circulation of the Blood

Blood is not a simple ﬂuid; it contains cells that complicate the ﬂow, especiallywhen the passages become narrow. Furthermore, the veins and arteries arenot rigid pipes but are elastic and alter their shape in response to the forcesapplied by the ﬂuid. Still, it is possible to analyze the circulatory system withreasonable accuracy using the concepts developed for simple ﬂuids ﬂowing inrigid pipes.

Figure 8.4 is a drawing of the human circulatory system. The blood in the circulatory system brings oxygen, nutrients, and various other vital substancesto the cells and removes the metabolic waste products from the cells. Theblood is pumped through the circulatory system by the heart, and it leaves theheart through vessels called arteries and returns to it through veins.

The mammalian heart consists of two independent pumps, each made of two chambers called the atrium and the ventricle. The entrances to and exitsfrom these chambers are controlled by valves that are arranged to maintain theﬂow of blood in the proper direction. Blood from all parts of the body exceptthe lungs enters the right atrium, which contracts and forces the blood into theright ventricle. The ventricle then contracts and drives the blood through thepulmonary artery into the lungs. In its passage through the lungs, the bloodreleases carbon dioxide and absorbs oxygen. The blood then ﬂows into theleft atrium via the pulmonary vein. The contraction of the left atrium forcesthe blood into the left ventricle, which on contraction drives the oxygen-richblood through the aorta into the arteries that lead to all parts of the body exceptthe lungs. Thus, the right side of the heart pumps the blood through the lungs,and the left side pumps it through the rest of the body.

106

Chapter 8 The Motion of Fluids FIGURE 8.4 Schematic diagram showing various routes of the circulation.

The large artery, called the aorta, which carries the oxygenated blood away from the left chamber of the heart, branches into smaller arteries, which leadto the various parts of the body. These in turn branch into still smaller arteries,the smallest of which are called arterioles. As we will explain later, the arterioles play an important role in regulating the blood ﬂow to speciﬁc regions in

107

the body. The arterioles branch further into narrow capillaries that are oftenbarely wide enough to allow the passage of single blood cells.

The capillaries are so profusely spread through the tissue that nearly all the cells in the body are close to a capillary. The exchange of gases, nutrients,and waste products between the blood and the surrounding tissue occurs bydiﬀusion through the thin capillary walls (see Chapter 9). The capillaries joininto tiny veins called venules, which in turn merge into larger and larger veinsthat lead the oxygen-depleted blood back to the right atrium of the heart.

8.5 Blood Pressure

The contraction of the heart chambers is triggered by electrical pulses thatare applied simultaneously both to the left and to the right halves of the heart.

First the atria contract, forcing the blood into the ventricles; then the ventriclescontract, forcing the blood out of the heart. Because of the pumping action ofthe heart, blood enters the arteries in spurts or pulses. The maximum pressuredriving the blood at the peak of the pulse is called the systolic pressure. Thelowest blood pressure between the pulses is called the diastolic pressure. In ayoung healthy individual the systolic pressure is about 120 torr (mm Hg) andthe diastolic pressure is about 80 torr. Therefore the average pressure of thepulsating blood at heart level is 100 torr.

As the blood ﬂows through the circulatory system, its initial energy, pro vided by the pumping action of the heart, is dissipated by two loss mechanisms: losses associated with the expansion and contraction of the arterialwalls and viscous friction associated with the blood ﬂow. Due to these energylosses, the initial pressure ﬂuctuations are smoothed out as the blood ﬂowsaway from the heart, and the average pressure drops. By the time the bloodreaches the capillaries, the ﬂow is smooth and the blood pressure is only about30 torr. The pressure drops still lower in the veins and is close to zero justbefore returning to the heart. In this ﬁnal stage of the ﬂow, the movement ofblood through the veins is aided by the contraction of muscles that squeezethe blood toward the heart. One-way ﬂow is assured by unidirectional valvesin the veins.

The main arteries in the body have a relatively large radius. The radius of the aorta, for example, is about 1 cm; therefore, the pressure drop alongthe arteries is small. We can estimate this pressure drop using Poiseuille’slaw (Eq. 8.7). However, to solve the equation, we must know the rate ofblood ﬂow. The rate of blood ﬂow Q through the body depends on the levelof physical activity. At rest, the total ﬂow rate is about 5 liter/min. Duringintense activity the ﬂow rate may rise to about 25 liter/min. Exercise 8-1shows that at peak ﬂow the pressure drop per centimeter length of the aorta

108

Chapter 8 The Motion of Fluids FIGURE 8.5  Blood pressure in a reclining and in an erect person.

is only 42.5 dyn/cm2 (3.19 × 10−2 torr), which is negligible compared to thetotal blood pressure.

Of course, as the aorta branches, the size of the arteries decreases, result ing in an increased resistance to ﬂow. Although the blood ﬂow in the narrower arteries is also reduced, the pressure drop is no longer negligible (seeExercise 8-2). The average pressure at the entrance to the arterioles is about90 torr. Still, this is only a 10% drop from the average pressure at the heart.

The ﬂow through the arterioles is accompanied by a much larger pressure drop,about 60 torr. As a result, the pressure at the capillaries is only about 30 torr.

Since the pressure drop in the main arteries is small, when the body is horizontal, the average arterial pressure is approximately constant throughoutthe body. The arterial blood pressure, which is on the average 100 torr, cansupport a column of blood 129 cm high (see Eq. 7.1 and Exercise 8-3). Thismeans that if a small tube were introduced into the artery, the blood in it wouldrise to a height of 129 cm (see Fig. 8.5).

If a person is standing erect, the blood pressure in the arteries is not uni form in the various parts of the body. The weight of the blood must be takeninto account in calculating the pressure at various locations. For example, theaverage pressure in the artery located in the head, 50 cm above the heart (seeExercise 8-4a) is P head  P heart − ρgh  61 torr. In the feet, 130 cm belowthe heart, the arterial pressure is 200 torr (see Exercise 8-4b).

The cardiovascular system has various ﬂow-control mechanisms that can compensate for the large arterial pressure changes that accompany shifts inthe position of the body. Still, it may take a few seconds for the system tocompensate. Thus, a person may feel momentarily dizzy as he/she jumps upfrom a prone position. This is due to the sudden decrease in the blood pressureof the brain arteries, which results in a temporary decrease of blood ﬂow tothe brain.

The same hydrostatic factors operate also in the veins, and here their eﬀect may be more severe than in the arteries. The blood pressure in the veinsis lower than in the arteries. When a person stands motionless, the bloodpressure is barely adequate to force the blood from the feet back to the heart.

Thus when a person sits or stands without muscular movement, blood gathersin the veins of the legs. This increases the pressure in the capillaries and maycause temporary swelling of the legs.

8.6

Control of Blood Flow

The pumping action of the heart (that is, blood pressure, ﬂow volume and rateof heart beat) is regulated by a variety of hormones. Hormones are molecules,often proteins, that are produced by organs and tissues in diﬀerent parts ofthe body. They are secreted into the blood stream and carry messages fromone part of the body to another. Hormones aﬀecting the heart are produced inresponse to stimuli such as need for more oxygen, changes in body temperature, and various types of emotional stress.

The ﬂow of blood to speciﬁc parts of the body is controlled by the arteri oles. These small vessels that receive blood from the arteries have an averagediameter of about 0.1 mm. The walls of the arterioles contain smooth muscleﬁbers that contract when stimulated by nerve impulses and hormones. The contraction of the arterioles in one part of the body reduces the blood ﬂow to thatregion and diverts it to another. Since the radius of the arterioles is small, constriction is an eﬀective method for controlling blood ﬂow. Poiseuille’s equationshows that if the pressure drop remains constant, a 20% decrease in the radiusreduces the blood ﬂow by more than a factor of 2 (see Exercise 8-5).

A stress-induced heart condition called stress cardiomyopathy (broken heart syndrome) has only recently been clearly identiﬁed by Western medicine.

The syndrome occurs most frequently after a sudden intense emotional traumasuch as death in the family, an experience of violence, or extreme anger. Thesymptoms are similar to an acute heart attack, but the coronary arteries arefound to be normal and the heart tissue is not damaged. It has suggested thatthe condition is triggered by an excessive release of stress-related hormonescalled chatecholamines.

Chapter 8 The Motion of Fluids

8.7

Energetics of Blood Flow

It moves in spurts. During the period of ﬂow, the velocity of the blood is aboutthree times as high as the overall average value calculated in Exercise 8-6.

Therefore, the kinetic energy per cubic centimeter of ﬂowing blood is KE  1 ρv2  1 (1.05) × (79.5)2  3330 erg/cm3 2

2

We mentioned earlier that energy density (energy per unit volume) and pressure are measured by the same unit (i.e., 1 erg/cm3  1 dyn/cm2); therefore, they can be compared to each other. The kinetic energy of 3330 erg/cm3is equivalent to 2.50 torr pressure; this is small compared to the blood pressure in the aorta (which is on the average 100 torr). The kinetic energy in thesmaller arteries is even less because, as the arteries branch, the overall areaincreases and, therefore, the ﬂow velocity decreases. For example, when thetotal ﬂow rate is 5 liter/min, the blood velocity in the capillaries is only about0.33 mm/sec.

The kinetic energy of the blood becomes more signiﬁcant as the rate of blood ﬂow increases. For example, if during physical activity the ﬂow rateincreases to 25 liter/min, the kinetic energy of the blood is 83,300 erg/cm3,which is equivalent to a pressure of 62.5 torr. This energy is no longer negligible compared to the blood pressure measured at rest. In healthy arteries,the increased velocity of blood ﬂow during physical activity does not presenta problem. During intense activity, the blood pressure rises to compensate forthe pressure drop.

8.8 Turbulence in the Blood

η Vc .04  38 cm/sec

ρD

1.05 × 2

For the body at rest, the ﬂow velocity in the aorta is below this value. But asthe level of physical activity increases, the ﬂow in the aorta may exceed thecritical rate and become turbulent. In the other parts of the body, however, theﬂow remains laminar unless the passages are abnormally constricted.

Laminar ﬂow is quiet, but turbulent ﬂow produces noises due to vibrations of the various surrounding tissues, which indicate abnormalities in the circulatory system. These noises, called bruit, can be detected by a stethoscope andcan help in the diagnosis of circulatory disorders.

8.9 Arteriosclerosis and Blood Flow

Arteriosclerosis is the most common of cardiovascular diseases. In the UnitedStates, an estimated 200,000 people die annually as a consequence of thisdisease. In arteriosclerosis, the arterial wall becomes thickened, and the arteryis narrowed by deposits called plaque. This condition may seriously impairthe functioning of the circulatory system. A 50% narrowing (stenosis) of thearterial area is considered moderate. Sixty to seventy percent is consideredsevere, and a narrowing above 80% is deemed critical. One problem causedby stenosis is made clear by Bernoulli’s equation. The blood ﬂow through theregion of constriction is speeded up. If, for example, the radius of the arteryis narrowed by a factor of 3, the cross-sectional area decreases by a factorof 9, which results in a nine-fold increase in velocity. In the constriction, thekinetic energy increases by 92, or 81. The increased kinetic energy is at theexpense of the blood pressure; that is, in order to maintain the ﬂow rate atthe higher velocity, the potential energy due to pressure is converted to kineticenergy. As a result, the blood pressure in the constricted region drops. Forexample, if in the unobstructed artery the ﬂow velocity is 50 cm/sec, then inthe constricted region, where the area is reduced by a factor of 9, the velocityis 450 cm/sec. Correspondingly, the pressure is decreased by about 80 torr(see Exercise 8-8). Because of the low pressure inside the artery, the externalpressure may actually close oﬀ the artery and block the ﬂow of blood. Whensuch a blockage occurs in the coronary artery, which supplies blood to theheart muscle, the heart stops functioning.

Stenosis above 80% is considered critical because at this point the blood ﬂow usually becomes turbulent with inherently larger energy dissipation thanis associated with laminar ﬂow. As a result, the pressure drop in the situation presented earlier is even larger than calculated using Bernoulli’s equation.

Further, turbulent ﬂow can damage the circulatory system because parts of theﬂow are directed toward the artery wall rather than parallel to it, as in laminar Chapter 8 The Motion of Fluids ischemic stroke.

There is another problem associated with arterial plaque deposit. The artery has a speciﬁc elasticity; therefore, it exhibits certain springlike properties. Speciﬁcally, in analogy with a spring, the artery has a natural frequency at which it can be readily set into vibrational motion. (See Chapter 5,Eq. 5.6.) The natural frequency of a healthy artery is in the range 1 to 2 kilohertz. Deposits of plaque cause an increase in the mass of the arterial wall anda decrease in its elasticity. As a result, the natural frequency of the artery issigniﬁcantly decreased, often down to a few hundred hertz. Pulsating bloodﬂow contains frequency components in the range of 450 hertz. The plaquecoated artery with its lowered natural frequency may now be set into resonantvibrational motion, which may dislodge plaque deposits or cause further damage to the arterial wall.

8.10

Power Produced by the Heart

The power PH produced by the heart is the product of the ﬂow rate Q and the energy E per unit volume of the blood; that is,

cm3

erg PH  Q E Q × E erg/sec

(8.9)

sec cm2 At rest, when the blood ﬂow rate is 5 liter/min, or 83.4 cm3/sec, the kinetic energy of the blood ﬂowing through the aorta is 3.33 × 103 erg/cm3. (See previous section.) The energy corresponding to the systolic pressure of 120 torris 160 × 103 erg/cm3. The total energy is 1.63 × 105 erg/cm3—the sum of thekinetic energy and the energy due to the ﬂuid pressure. Therefore, the powerP produced by the left ventricle of the heart is

P  83.4 × 1.63 × 105  1.35 × 107 erg/sec  1.35 W Exercise 8-9 shows that during intense physical activity when the ﬂow rateincreases to 25 liters/min, the peak power output of the left ventricle increasesto 10.1 W.

113

The ﬂow rate through the right ventricle, which pumps the blood through the lungs, is the same as the ﬂow through the left ventricle. Here, however, theblood pressure is only one sixth the pressure in the aorta. Therefore, as shownin Exercise 8-10, the power output of the right ventricle is 0.25 W at restand 4.5 W during intense physical activity. Thus, the total peak power outputof the heart is between 1.9 and 14.6 W, depending on the intensity of thephysical activity. While in fact the systolic blood pressure rises with increased blood ﬂow, in these calculations we have assumed that it remainsat 120 torr.

8.11

Measurement of Blood Pressure

The arterial blood pressure is an important indicator of the health of an individual. Both abnormally high and abnormally low blood pressures indicatesome disorders in the body that require medical attention. High blood pressure, which may be caused by constrictions in the circulatory system, certainlyimplies that the heart is working harder than usual and that it may be endangered by the excess load. Blood pressure can be measured most directly byinserting a vertical glass tube into an artery and observing the height to whichthe blood rises (see Fig. 8.5). This was, in fact, the way blood pressure wasﬁrst measured in 1733 by Reverend Stephen Hales, who connected a long vertical glass tube to an artery of a horse. Although sophisticated modiﬁcationsof this technique are still used in special cases, this method is obviously notsatisfactory for routine clinical examinations. Routine measurements of bloodpressure are now most commonly performed by the cut-oﬀ method. Althoughthis method is not as accurate as direct measurements, it is simple and in mostcases adequate. In this technique, a cuﬀ containing an inﬂatable balloon isplaced tightly around the upper arm. The balloon is inﬂated with a bulb, andthe pressure in the balloon is monitored by a pressure gauge. The initial pressure in the balloon is greater than the systolic pressure, and the ﬂow of bloodthrough the artery is therefore cut oﬀ. The observer then allows the pressure inthe balloon to fall slowly by releasing some of the air. As the pressure drops,she listens with a stethoscope placed over the artery downstream from the cuﬀ.

No sound is heard until the pressure in the balloon decreases to the systolicpressure. Just below this point the blood begins to ﬂow through the artery;

however, since the artery is still partially constricted, the ﬂow is turbulent andis accompanied by a characteristic sound. The pressure recorded at the onsetof sound is the systolic blood pressure. As the pressure in the balloon dropsfurther, the artery expands to its normal size, the ﬂow becomes laminar, andthe noise disappears. The pressure at which the sound begins to fade is takenas the diastolic pressure.

Chapter 8 The Motion of Fluids

EXERCISES

8-1. Calculate the pressure drop per centimeter length of the aorta when the blood ﬂow rate is 25 liter/min. The radius of the aorta is about 1 cm,and the coeﬃcient of viscosity of blood is 4 × 10−2 poise.

8-2. Compute the drop in blood pressure along a 30-cm length of artery of radius 0.5 cm. Assume that the artery carries blood at a rate of8 liter/min.

8-3. How high a column of blood can an arterial pressure of 100 torr support? (The density of blood is 1.05 g/cm3.)

8-4. (a) Calculate the arterial blood pressure in the head of an erect person.

Assume that the head is 50 cm above the heart. (The density of bloodis 1.05 g/cm3.) (b) Compute the average arterial pressure in the legs ofan erect person, 130 cm below the heart.

8-5. (a) Show that if the pressure drop remains constant, reduction of the radius of the arteriole from 0.1 to 0.08 mm decreases the blood ﬂowby more than a factor of 2. (b) Calculate the decrease in the radiusrequired to reduce the blood ﬂow by 90%.

8-6. Compute the average velocity of the blood in the aorta of radius 1 cm if the ﬂow rate is 5 liter/min.

8-7. When the rate of blood ﬂow in the aorta is 5 liter/min, the velocity of the blood in the capillaries is about 0.33 mm/sec. If the averagediameter of a capillary is 0.008 mm, calculate the number of capillariesin the circulatory system.

8-8. Compute the decrease in the blood pressure of the blood ﬂowing through an artery the radius of which is constricted by a factor of 3.

Assume that the average ﬂow velocity in the unconstricted region is50 cm/sec.

8-9. Using information provided in the text, calculate the power generated by the left ventricle during intense physical activity when the ﬂow rateis 25 liter/min.

8-10. Using information provided in the text, calculate the power generated by the right ventricle during (a) restful state; blood ﬂow 5 liter/min,and (b) intense activity; blood ﬂow 25 liter/min.

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8-11. During each heartbeat, the blood from the heart is ejected into the aorta and the pulmonary artery. Since the blood is accelerated during thispart of the heartbeat, a force in the opposite direction is exerted onthe rest of the body. If a person is placed on a sensitive scale (orother force-measuring device), this reaction force can be measured.

An instrument based on this principle is called the ballistocardiograph.

Discuss the type of information that might be obtained from measurements with a ballistocardiograph, and estimate the magnitude of theforces measured by this instrument.

Heat and Kinetic Theory

9.1

Heat and Hotness heat. Heat

may be transformed into work, and therefore it is a form of energy. Heatedwater, for example, can turn into steam, which can push a piston. In fact, heatcan be deﬁned as energy being transferred from a hotter body to a colder body.

In this chapter, we will discuss various properties associated with heat.

We will describe the motion of atoms and molecules due to thermal energyand then discuss diﬀusion in connection with the functioning of cells and therespiratory system.

9.2 Kinetic Theory of Matter

116

117

atoms are bound together, the random motion is more restricted. The atomsare free only to vibrate and do so, again randomly, about some average position to which they are locked. The situation with regard to liquids is betweenthese two extremes. Here the molecules can vibrate, but they also have somefreedom to move and to rotate.

Because of their motion, the moving particles in a material possess kinetic energy. This energy of motion inside materials is called internal energy, andthe motion itself is called thermal motion. What we have so far qualitativelycalled the hotness of a body is a measure of the internal energy; that is, in hotter bodies, the random motion of atoms and molecules is faster than in colderbodies. Therefore, the hotter an object, the greater is its internal energy. Thephysical sensation of hotness is the eﬀect of this random atomic and molecularmotion on the sensory mechanism. Temperature is a quantitative measure ofhotness. The internal energy of matter is proportional to its temperature.

Using these concepts, it is possible to derive the equations that describe the behavior of matter as a function of temperature. Gases are the simplestto analyze. The theory considers a gas made of small particles (atoms ormolecules) which are in continuous random motion. Each particle travels ina straight line until it collides with another particle or with the walls of thecontainer. After a collision, the direction and speed of the particle is changedrandomly. In this way kinetic energy is exchanged among the particles.

The colliding particles exchange energy not only among themselves but also with the wall of the container (Fig. 9.1). For example, if initially thewalls of the container are hotter than the gas, the particles colliding with thewall on the average pick up energy from the vibrating molecules in the wall.

As a result of the wall collisions, the gas is heated until it is as hot as the walls.

After that, there is no net exchange of energy between the walls and the gas.

This is an equilibrium situation in which, on the average, as much energy isdelivered to the wall by the gas particles as is picked up from it.

The speed and corresponding kinetic energy of the individual particles in a gas vary over a wide range. Still it is possible to compute an average kineticenergy for the particles by adding the kinetic energy of all the individual particles in the container and dividing by the total number of particles (for details,see [11-7]). Many of the properties of a gas can be simply derived by assuming that each particle has this same average energy.

The internal energy in an ideal gas is in the form of kinetic energy,1 and therefore the average kinetic energy 1 mv2 is proportional to the tem 2

av

perature. The proportionality can be changed to an equality by multiplyingthe temperature T by a suitable constant which relates the temperature to theinternal energy. The constant is designated by the symbol k, which is called 1The simple theory neglects the vibrational and rotational energy of the molecules.

118

Chapter 9 Heat and Kinetic Theory FIGURE 9.1  Collisions in a gas.

Boltzmann constant. For historical reasons, Boltzmann constant has been sodeﬁned that it has to be multiplied by a factor of 3 to relate temperature to the

2

average kinetic energy of a molecule; thus,

1 mv2

3kT

(9.1)

2

av

2

The temperature in this equation is measured on the absolute temperature scalein degrees Kelvin. The size of the degree division on the absolute scale is equalto the Celsius, or centigrade, degree, but the absolute scale is transposed so that0◦C  273.15 K. Since our calculations are carried only to three signiﬁcant ﬁgures, we will use simply 0◦C  273◦K. The value of Boltzmann constant is k  1.38 × 10−23 J/molecule K The velocity deﬁned by Eq. 9.1 is called thermal velocity.

Each time a molecule collides with the wall, momentum is transferred to the wall. The change in momentum per unit time is a force. The pressureexerted by a gas on the walls of its container is due to the numerous collisionsof the gas molecules with the container. The following relationship betweenpressure P, volume V, and temperature is derived in most basic physics texts: PV  NkT

(9.2)

Here N is the total number of gas molecules in the container of volume V, andthe temperature is again measured on the absolute scale.

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TABLE 9.1  Speciﬁc Heat for Some Substances

Substance

Speciﬁc heat(cal/g◦C)

Water

1

Ice

0.480

Average for human body

0.83 Soil 0.2 to 0.8, depending on water

content

Aluminum

0.214

Protein 0.4 In a closed container, the total number of particles N is ﬁxed; therefore, if the temperature is kept unchanged, the product of pressure and volume is aconstant. This is known as Boyle’s law. (See Exercises 9-1 and 9-2.)

9.3

Deﬁnitions 9.3.1 Unit of Heat calories. One calorie (cal) isthe amount of heat required to raise the temperature of 1 g of water by 1 C◦.2Actually, because this value depends somewhat on the initial temperature ofthe water, the calorie is deﬁned as the heat required to raise the temperatureof 1 g of water from 14.5◦C to 15.5◦C. One calorie is equal to 4.184 J. Inthe life sciences, heat is commonly measured in kilocalorie units, abbreviatedCal; 1 Cal is equal to 1000 cal.

9.3.2 Speciﬁc Heat

Speciﬁc heat is the quantity of heat required to raise the temperature of 1 g ofa substance by 1 degree. The speciﬁc heats of some substances are shown inTable 9.1.

The human body is composed of water, proteins, fat, and minerals. Its speciﬁc heat reﬂects this composition. With 75% water and 25% protein, thespeciﬁc heat of the body would be Speciﬁc heat  0.75 × 1 + 0.25 × 0.4  0.85 The speciﬁc heat of the average human body is closer to 0.83 due to its fat andmineral content, which we have not included in the calculation.

2For the symbol ◦C read degree Celsius. For the symbol C◦, read Celsius degree.

120

Chapter 9 Heat and Kinetic Theory FIGURE 9.2 Heat is transferred from one region to another by (a) conduction, by (b) convection, and by (c) radiation.

9.3.3 Latent Heats latent heat. The latent heat of fusion is the amount of energy requiredto change 1 g of solid matter to liquid. The latent heat of vaporization is theamount of heat required to change 1 g of liquid to gas.

9.4 Transfer of Heat

9.4.1 Conduction