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Unformatted text preview: Chapter 6 Diffusion:
Random Walksin
Air and Water This chapter deals with the process of random motion and the way in which it can act
as a mechanism of transport. The focus here is on the process of molecular diffusion
and its biological consequences. For instance, we will see how the diffusion of
gases can limit the size of plants and animals and explain why an ostrich egg is
more porous than that of a hummingbird. We will explore the role of diffusive
transport in foraging strategy: Is it better to go looking for food or to wait for the
food to come to you? And we will see how the mean free path of molecules in air
sets the minimal size of insect tracheoles. The principles outlined in this chapter are quite general, however, and we will
make considerable use of them in later chapters. 6.1 The Physics 6.1.1 Molecular Velocity In chapter 3 we deﬁned temperature in terms of the average kinetic energy of
molecules using the relationship = , (6.1) where m is the mass of the molecule, u its speed, and k is Boltzmann’s constant,
1.38 x 10"23 J K‘ 1. The brackets, 0, denote that velocity is averaged over time for
a single molecule, or averaged over many molecules at any one time. This simple
expression has some intriguing consequences. Consider, for instance, a typical molecule of air (nitrogen) with a weight per
molecule of about 4.7 x 10“26 kg. At room temperature (290 K), we have seen
(chapter 3) that the average speed of a nitrogen molecule is . @2) = ﬂ .~. scams—1. (6.2) m This is a sizable speed.1 For example, an unimpeded nitrogen molecule would
ﬁnish the hundredyard dash in about a ﬁfth of a second. And this speed is not
peculiar just to nitrogen. At 290 K oxygen moves at an average speed of 475 m 5“,
carbon dioxide at 405 m s“, and water vapor at a brisk 634 m s‘ 1. The high velocity of molecules at room temperature raises visions of extremely
effective transport systems. If you want to move oxygen from point Ato point B, you simply take the gas one molecule at a time, point it in the right direction,2 and '0f the total kinetic energy of a molecule, 1/3 (on average) is associated with motion along each
particular axis. Thus, at 290 K the average velocity of a nitrogen molecule along the 9: axis is kT
1/<uZ) = g z 292ms". (63) The overall speed of 508 ms‘ is (uZ) + (v2) + (wz). let it go. Thermal kinetic energy does the rest! But these visions seem at odds with
everyday experience. If you put a drop of dye in a glass of still water, it may take
hours to spread throughout the container. Similarly, it can take several minutes
for the scent from a bottle of perfume opened in one corner of a room of still air
to reach the opposite corner. How can this be if molecules are moving at several
hundred m s”1 ? ' Even worse, these everyday experiences are at odds with careful experiments.
In an ordinary glass of water, convection currents created as the water evaporates
account for most of the movement of dye. If one makes sure that no convection is
present, a drop of dye may take months (rather than hours) to spread throughout
the container. Similarly, convection currents in air account for most of the transport
of perfume. Clear evidence of these currents are the motes of dust that dance in a
sunbeam. In the absence of convection, scent takes hours (rather than minutes) to
spread across a room. Our task in this chapter is to reconcile these experimental results regarding
the macroscopic transport of molecules in ﬂuids with the microscopic behavior
of individual molecules. This reconciliation is found inthe process of random
motion. Each molecule moves at considerable speed, but it cannot move very far
in a straight line without colliding with another ﬂuid molecule. Like a billiard ball
on a crowded table, at each collision the molecule ﬂies off in a new direction only
to collide again. The direction taken by a molecule after a collision is purely a
matter of chance. If it hits another molecule squarely, it may rebound back from
where it came. A glancing blow results in a less drastic change in direction. As a
result of its collisions, each molecule performs a random walk through space. It is
to the characteristics of such walks that we now turn our attention. 6.1.2 Random Walks The concept of a random walk is best understood through the use of a simple
example. Consider motion along the x axis. We start by placing a hypothetical
particle at the origin, a: = 0, and every T seconds we allow the particle to step
a distance 6 along the axis. Thus the average speed of the particle is 6/7'. The
direction of each step, however, is purely a matter of chance. Half the time the
particle steps to the right, half the time to the left. In practice, we can simulate this
process by tossing a coin before'each step. If the coin comes up heads, the particle
moves to the right; if tails, it moves to the left. After each step in this random walk we note the position of the particle. In this
manner we can track the particle along the axis as a function of the number of steps
taken, or, equivalently, as a function of time. Because each step is chosen at random,
the precise path taken by a particle is impossible to predict (ﬁg. 6.1). However,
if we repeat the experiment a number of times, we can perceive a pattern in the
average manner in which particles move. It is this predictability of the average
that renders the concept of a random walk so useful. To explore the properties of the average, we examine the statistics of parti
cle motion. To do so, we track the random walks of N particles. Let rim) be the 2You may object to the idea of “pointing” a thermally agitated molecule. Because thermal motion
is disordered, the act of aiming a molecule in essence cools it. It would be possible in theory, however,
to let each molecule rattle around in a box until by chance it is moving in the proper direction, and then
suddenly release it. 85 Diﬂusion:
Random Walks in
Air and Water 86 Chapter 6 Fig. 6.1 Six particles, all starting at the same
spot, take quite different paths in a random walk.
On average, however, the particles go nowhere at all. Distance From Origin 0 20 40 60 80 100 120 Steps location on the 3: axis of the ith particle after the particle has taken n steps. We
know from our assumptions that ' trim) = 33,(n — 1) :i: 6. (6.4) In other words, the position of the particle after n steps is either one step length to
the right or one step length to left of the position of the particle after 71. — 1 steps.
After we have repeated our experiment N times, we can compute the average position of particles after n steps: ($01)): 51:1(n) + $202]. . . + zN(n) 1 N
= ~ Zoe(n), (66)
N i=1 (6.5) where the symbol )3 means that we add the position after 11 steps of all particles numbered 1 to N. I 
We can expand this expression by replacing rim) With 55,(n — 1) :1: 6, which we know to be equivalent (Eq. 6.4). Thus, N ($01)) = %Ztcc.<n — 1) i 61 (6.7)
i=1
1 N 1 N
= NZxZ(n— 1)+N216. (6.8) But, since half the time 6 is to the right and half the time to the left, the average of
i6 must be 0. As a consequence, ($01)) = (x01 — 1)). (6.9) In other words, the average position of a particle after n steps is the same as the
position after n — 1 steps. Taken to its logical conclusion, this means that, on
average, particles starting at the origin remain at the origin. This is just what you might expect given equal probability of stepping right or left. The fact that particles go nowhere on average does not imply that all particles
remain at the origin, however. All that the average tells us is that for every particle
that ends up at x = +3, for instance, there is likely to be a particle that ends up at
a: = —3. As long as an equal number of particles have net movements left and right,
the average can still be 0, although there is considerable spread of particles.3 To
see how far away from the origin particles are likely to travel, we need to calculate
our average in a different way. It would be possible to avoid the problem of averaging positive with negative
values by taking the absolute value of each particle’s location after n steps and
averaging these values as before. Instead, it has become traditional to circumvent
the problem by taking the square of the location on the :3 axis. Because the squares
of both positive and negative numbers are always positive, we can, by averaging
the squares of locations, gain information as to the average distance traveled from
the origin. Again we begin by expressing the location after n steps in terms of the
location after 71 —_ 1 steps. Expanding the expression, we see that4 em) = [scan — 1) i 612 (6.10)
=m§(n—1)i26m,(n— 1)+62. (6.11) When we average this value for N particles, we see that N
(35201)) = % 2 [953(1) » 1) i 26mm — 1) + 62] (6.12)
2 l l 1:] 1 N 1 N
= — _1 __ 2 , ~ __ 2 .
N 23:47: )+ N I; d: 613,02 1)+ N g 6 (613) l N 1 N
= NZ}:£3(""1)+N;52 (6.14)
= ($201  1)) +52 (6.15) In this computation we have again made use of the fact that the average of i6 = O. This result tells us that the average (or mean) square of position after 71 steps is
greater by 62 than the average square of position after n — 1 steps. Now, we know
from our initial assumptions that the mean square position after 0 steps is O [i.e.,
(m(0)) = 0], so (222(1)) = 62. Aftertwo steps, the average square of position is 262.
and so forth. Thus, (93201)) = “52 (6.16) In other words, the square of the displacement from the starting point increases
directly with the number of steps taken. We can convert from the square of dis
placement to displacement itself simply by taking the square root of eq. 6.16: V = 5rrms = (6.17) 3The distinction between average and individual movement reminds me of the story of two statis
ticians who went duck hunting. A bird ﬂew past. and both hunters ﬁred. One shot passed 2 m in front
of the duck, the other 2 m behind. The statisticians proceeded to congratulate themselves because on
average they had hit the bird dead center. 4Recall that :rZ(n — l) is the z position of the ith particle after (71 ~ 1) steps. The square of this
value is $301 — 1), not x (n — 1)2. 87 Diffusion:
Random Walks in
Air and Water 88 . Chapter 6 The value \/ (x2(n)) is called the root mean square or rms displacement, and it
is a measure of how far, on average, particles have moved from their starting point after n steps. Readers familiar with statistics may note that the mean square
displacement is the same as the variance of displacement, and the root mean square
displacement is the standard deviation. We now return to our original assumptions regar
that the particle takes a step every 7' seconds. Thus, the number ‘ .
equal to 15/7, where t is the time since the particle started its walk. Inserting this expression for n in eq. 6.16, we see that ding the random walk and recall
of steps taken is 2
(32%)) = 6715. (6.18) particle (in other words, the probable ’" The mean square distance traveled by a '
time at a rate governed by the ratio of square of distance) increases directly with
62 to T. 6.2 The Diffusion Coefﬁcient We now are in a position to relate this theoretical consideration of a random
walk to the process of diffusion. To do so, we deﬁne a diﬂusion coeﬁ‘iczent, ’D: 'D (6.19) ill 62
5;. The reason for the factor of l / 2 will become clear later in this chapter; it eventually . . . . 2 _
makes the math simpler. The diffusmn coefﬁCient has the units m s .
Using this deﬁnition, we can restate our conclusion regarding the random walk: ($20)) = 21% (6.20)
wrms = ($26)) = V (6.21) Eq. 6.21 is of special importance, and is therefore worth dwelling upon for a
moment. It says that the distance a particle is likely to travel from its point of
origin while performing a random walk (the rms distance) increases not With time, as one might guess, but rather with the square root of time. In other words, if a particle travels an average of 1cm in 1 s, it takes an average of 100s to travel 10 cm, and 10, 000 s to travel 100 cm. . ' I ”
As a result, if we try to express the rate of travel as a “diffusmn velocrty we come to a curious conclusion. Setting velocity equal to rms distance per time, /2D
“diffusion velocity” = (6.22) The shorter the period over which we measure velocity, the greater the velocity
we measure. Conversely, the longer the period of measurement, the slower the
velocity measured. The fact that the rate of transport in diffusion is a function of
time is one of the principal characteristics of diffusive processes. ‘
Note that eq. 6.22 can be applied only when t > T. It was calculated assuming
that the particle changes its direction at random. For periods shorter than 7", we have
assumed that the particle is in the midst of a step and not subject to directional change. Therefore, at t < T, our assumption is violated and eq. 6.22 leads to
spurious results. We now examine the diffusion coefﬁcients for three molecules with biological
importance: oxygen, carbon dioxide, and water vapor. 6.2.1 Diffusion Coeﬂicients in Air The diffusion coefﬁcients for oxygen and water vapor in air can be described by the equation, D(T,p> = Mk! ’2. (6.23) P where ’D(T, p) is the diffusion coefﬁcient at absolute temperature T and pressure
p and p0 is one normal atmosphere. The coefﬁcients k; and k2 vary from one type
of molecule to the next. For oxygen, [01 = 1.13 x 10‘9 m2 s“1 and k2 = 1.724
(Marrero and Mason 1972). For water vapor, k1 = 0.187 x 10’9 m2 s‘1 and
k2 = 2.072. The form of eq. 6.23 tells us that the diffusion coefﬁcient rises with
any increase in temperature, but decreases with any increase in pressure. The diffusion coefﬁcient for carbon dioxide in air is described by a slightly
more complex equation (Marrero and Mason 1972), D(T, p) = lek1 exp (~k3/T) 1%, (6.24) where k3 is an additional parameter. In this case k1 = 2.70 x 10‘9m2 5“, k2 =
1.590, and k3 = 102.1 K. . These results are summarized in table 6.1 and ﬁgure 6.2. The diffusivity of
water vapor is about 50% greater, and the diffusivity of 02 about 30% greater,
than that of carbon dioxide. For all three gases, the diffusion coefﬁcient is about
30% higher at 40° C than at 0° C. At the top of Mount Everest, where the pressure
is only a third of that at sea level, the diffusion coefﬁcients of these gases would
be three times those shown here. The distance (arms) that an oxygen molecule is expected to travel in air is shown
as a function of time in ﬁgure 6.3. In one second a molecule will, on average, move
about a centimeter from its starting point. Several hours are required, however,
before the molecule reliably travels one meter. 6.2.2 Diffusion Caeﬁicients in Water The diffusion coefﬁcients of oxygen and carbon dioxide are about 10,000 times
smaller in water than they are in air (table 6.2). This tremendous difference in
diffusivity will dominate our discussion of the biological consequences of diffu
sion, but there are two further contrasts to note. First, the variation in diffusion
coefﬁcient with temperature is much greater in water than in air. For example, at
40° C the diffusion coefﬁcient of 02 in water is 3.2 times that at 0° C, whereas in
air it is increased by a factor of only about 1.3. The variation is somewhat less for
C02; 7) in water is about 2.1 times higher at 40° C than at 0° C. Second, the dependence on temperature affects the rank order of the diffusivities
in water (table 6.2; ﬁg. 6.4). At low temperature, the diffusion coefﬁcient of C02
is larger than that of 02, but above about 6° C, the diffusivity of oxygen is greater
than that of carbon dioxide. ‘, The distance (arrms) that an oxygen molecule is expected to travel in water is
shown as a function of time in ﬁgure 6.3. These distances are surprisingly small. 89 M Diﬂ’usion:
Random Walks in
Air and Water Table 6.1 The diffusion coefﬁcients of various
gases in air at one atmosphere. Diffusion Coefﬁcient, TC) C) Dm (m2 s" x 10*) 02 C02 H20 0 17.9 13.9 20.9
5 18.5 14.4 21.7
10 19.1 14.9 22.5
15 19.7 15.4 23.3
20 20.3 16.0 24.2
25 20.9 16.5 25.1
30 21.5 17.0 26.0
35 22.1 17.6 26.8
40 22.7 18.1 27] Source: Calculated from data presented by
Marrero and Mason (1972). Notes: Note that these values represent the best
ﬁt to empirical measurements, but that actual
measured values may vary by d: 5% from
those shown here. The symbol D771 is used
here to distinguish molecular diffusion from
other types of diffusion (e.g., the diffusion of
heat). 05.: L,” ‘E . i E 90
Chapter 6 For example, relying on diffusion alone, a molecule moves only about 5 cm in a
million seconds.
30 10a
10"
9A 20 A 10'1
3 5
’< c
Tm E 10 ’
E 7.
"E 1 '5.
Q 10 ’ 10.
L 104
La,“
0 .1. r ._l  4—1» 4—; I i. .I—L A _I. L _L.._— 100 
0 10 20 40 10"10'510“10"10"10" 10° 10' 102 10’ 10‘ 10’ 10° 101
Time (5) Temperature (“C) Fig. 6.2 The diffusion coefﬁcients of gases in air
increase slightly with an increase in temperature.
(Data from table 6.1) Fig. 6.3 The rootmean—square distance traveled
by an oxygen molecule is much greater in air
than in water. Note, however, that in either
medium it takes a very long time to travel a
meter by diffusion alone. Table 6.2 The diffusivity of 02 and C02 in
water at one atmosphere (from Armstrong 1979). Diffusion Coefﬁcient, T(° C) Dm (m2 s”l X 10‘9) 02 C02 Id—
0 0.99 1.15
5 1.27 1.30
10 1.54 1.46
15 1.82 1.63
20 2.10 1.77
25 2.38 1.92
30 2.67 2.08 _______.__._.___,_—.—’— Note: The symbol ’Dm is used here to distinguish
molecular diffusion from other types of
diffusion (e.g., the diffusion of heat). 6.2.3 Mean Free Path
Let us now examine the concept of a diffusion coefﬁcient in greater detail. We ﬁrst note that because the coefﬁcient has dimensions of length squared per time, it
can be treated as the product of velocity and a distance. Thus, the rate at which a
particle is diffusively transported depends both on how fast it moves while in free
ﬂight between collisions (u = 6/7) and on the average distance it moves before
again colliding (6 / 2), a distance called the mean free path, 1. We have already
calculated how fast ﬂuid molecules move at room temperature (6/7 x 500 m s” 1 [eq. 62]), and are now in a position to estimate their mean free path.
To do so, we rely on the diffusion coefﬁcients as presented in tables 6.1 and 6.2. We will discuss later in this chapter how these empirical measurements are
made, but for now accept them as given. The diffusion coefﬁcient for an oxygen
molecule in air is about 2 x 10"5 m2 s” 1. Noting that 6 = 219/ u, we calculate that
the mean free path in air is 8 x 10‘8 m. In other words, every 0.08 pm an oxygen
molecule collides with another air molecule and careens off in a new direction.
Traveling at its velocity of 500ms", the 02 molecule travels this distance in
about 0.00016 its. To put it another way, an air molecule experiences about 6.25
billion collisions every second, each one capable of changing the direction of its
ﬂight. With so many collisions going on, it is no wonder that air molecules behave
differently in bulk than might be expected from the behavior of a single molecule
on its own. In water, the diffusion coefﬁcient of an oxygen molecule is 10,000 times smaller
than that in air, 2 x 10—9 m2 s". We know, however, that at a given temperature,
an oxygen molecule dissolved in water has the same kinetic energy as one in air,
and therefore must have the same average velocity. As a consequence, the reduced
diffusion coefﬁcient in water must be due to a reduction in the mean free path.
Again equating 6 and 2D/u, we calculate that the mean free path of an oxygen
molecule in water is about 10“ '1 m, only a twentieth of the diameter of a hydrogen
atom! Because water molecules are packed so tightly together, there is virtually no
“free space” through which an oxygen molecule can ﬂy, and the randomly walking
molecule undergoes about 60,000 billion collisions per second. The net result is a
reduction in the rate at which the molecule is transported from one spot to another. Din (n12 s'I x 10") 0 1° 20 30 40 Temperature ("C) 6.3 The Sherwood Number Before we consider the biological consequences of diffusion and the 10 000
fold difference in diffusion coefﬁcients between air and water, we need to, take
a moment to consider under what circumstances diffusion is the primary means
by which substances are transported through the ﬂuid environment to or from an
organism. In particular, we need to compare transport by diffusion with the rate at
which a substance can be transported by convection, where convection includes
transport by either the bulk movement of a ﬂuid relative to an organism} or by
movement of an organism relative to the ﬂuid. The ratio of transport by convection Ethat by diffusion is quantiﬁed by a dimensionless quantity, the Sherwood number .% Sh ,
1) (6.25)
where u is the relative velocity between the organism and the bulk of the ﬂuid
6?, IS a characteristic length of the organism (here taken to be the length along the
direction of motion), and D is the diffusion coefﬁcient. We derive the Sherwood
number later in this chapter. When Sh is large, convective transport to an organism is much greater than
that by diffusion. But ’D is so small in either air or water (10‘9 to 10‘5 In2 s“)
that the Sherwood number is bound to be large unless the organism in question is
exceedingly small, the relative velocity is exceedingly slow, or both. To give this concept some tangibility, we calculate the Sherwood number for
representative organisms. Consider ﬁrst a sessile plant or animal. If a stationary
terrestrial organism is 1cm long, the Sherwood number is 500 when the wind
blows at just 1 m s“. The breeze would have to fall below 2mm 5‘1 before Sh
would be less than 1, at which point diffusion would begin to outweigh convective
transport. A velocity this slow is unusual. Even in the complete absence of wind
the convective currents set up by hot or cold objects are virtually guaranteed to
exceed 2mm 5—], and, as a result, a terrestrial organism 1 cm in length almost
always operates at a Sherwood number greater than 1. In water, the convective ﬂow must be slower still before diffusive transport outwelghs convective transport. For example, a stationary organism 1cm long would be serviced primarily by convection unless the
0.2 mm 5—1 ! current speed dropped below 91 Diﬁusion:
Random Walks in
Air and Water Fig. 6.4 The diffusion coefﬁcients of both
oxygen and carbon dioxide gases in water
increase with an increase in temperature. but the increase in D", is greater for oxygen. (Data
from table 6.2) 92 _______________________________—.———_———————————————————— Chapter 6 Mobile organisms can effect the Sherwood number through the increase in
relative velocity due to their locomotion. How small must a mobile organism be
before diffusive transport outweighs convective transport? We may make a rough
calculation by noting that the maximal speed of many organisms is about ten of
their own body lengths per second (see ﬁg. 6.5). Thus, we can estimate the velocity
term in the Sherwood number as at most 10 EC per second and 1023,
Sh~ D. (6.26) This relationship is shown in ﬁgure 6.6. The Sherwood number is less than 1
(indicating that diffusion is the dominant form of transport) when 66 is less than
about 1.4 mm in air. Thus aerial organisms up to the size of gnats and no—seeums
might live in a diffusioncontrolled world. In this chapter we will generally limit
ourselves to the discussion of terrestrial organisms smaller than this size. 10‘ 10’ 10' 10" 10’
102 10' :5
V)
T: 10" Lengths Per Second ’ é
a‘ n
E 100\,/ = 10
1, 101 z
3 " Swimming 1% 10“
6? 10" a ,
. g 10‘
10" i
10"
10 5 104
10,, .. L. malt mL. M 10:
107 10° 10’ 10‘ 10] 102 10'I 10n 10 101 10'° 10'5 10“ 10" 10'2
Length (In) Body Length (tn) Fig. 6.5 The speed of locomotion increases in an orderly fashion as the size of an animal
increases. Swimming data (solid dots) span sizes
from bacteria to whales (Okubo 1987). The best
ﬁt line for these data is speed (m s‘ l) = 1.4 x
Z086. Flying data (open dots) span sizes from
fruit ﬂies to ducks (insect data from Denny 1976,
duck datum from McFarlan 1990). The bestﬁt
line for these data is speed (ms") = 13.8 X6030,
but given the small number of points, this line
should not be taken too seriously. The point for
running (cross) is for a human being (2 m tall)
running at 10 m s‘1 (fast sprint pace). (Adapted
and expanded from a ﬁgure in Mann and Lazier
1991 by permission of Blackwell Scientiﬁc Publications, Inc.) Fig. 6.6 The Sherwood number (Sh) for
organisms moving at a speed equal to ten times
their body length per second is much larger in
water than in air. In water, (3c must be less than about 14 um before diffusion is the dominant form
of transport. As a consequence, we will limit our discussion of aquatic organisms
in this chapter to small phytoplankton and bacteria. In summary, the message of the Sherwood number is this: for most large or
ganisms and most realworld ﬂows, transport by diffusion is small compared to transport by convection.
I do not mean to imply that diffusion is unimportant for other organisms. We will explore several examples in chapter 7 in which diffusive transport forms an
integral and important part of the system by which gases and nutrients are delivered.
However, in these later examples, it is primarily convection that controls the rate at
which diffusion transports material (by a mechanism we have not yet discussed),
and it is for this reason that these examples are best left to a later chapter. Here we
focus on examples where diffusion alone governs the rate of transport. 6.4 Fick’s Equation Before we begin this exploration, we need one further mathematical tool. To
this point we have treated diffusion from a microscopic perspective, following
individual molecules as they undergo random walks. For many practical prob
lems, it is easier to calculate transport when diffusion is examined from a more macroscopic point of view. 10“ Consider the situation shown in ﬁgure 6.7. Again we deal with random motion
in a single direction, but do so in a threedimensional container. For example, at
time t we may assume that N (:13) particles are located at at, a distance 6 / 2 to the
left of the plane shown in the ﬁgure. Similarly, N (x + 6) particles are located at
z + 6, a distance 6 / 2 to the right of the plane. All of these particles undergo the
same type of motion as prescribed in the previous example, which is to say that
at time t + 7' half the particles at a: have by chance moved to the right through the
plane and end up at a: + 6. Similarly, half the particles at a: + 6 move to the left. The
net number of particles that cross the plane in the positive a: direction in time 7' is net number crossing = w — 2 2 (6.27) If we assume that A is the area of the plane through which particles pass, we can
expressthis process as aﬁux density, J3, i.e., the net number of particles crossing
our plane per time per area. Dividing eq. 6.27 by A and rearranging, we see that =aNm+o—Non JI 2A7 (6.28)
We now perform a bit of mathematical legerdemain. Multiplying the righthand
side of this equation by 62/62 and again rearranging, we conclude that n=—fl[ N(.'r+6) N(x)
2T6 — A6 —A6_ (629) This expression can be simpliﬁed. First, we recall that by our deﬁnition 62/(27')
is the diffusion coefﬁcient, D. The factor of 1/2 that appears in this equation as
a result of the averaging process is the reason we included a factor of 1/2 in our
deﬁnition of D (eq. 6.19). Next we note that the product A6 is a volume, so that
the terms in brackets each represent the number of particles per volume occurring at a given position. That is, these terms are measures of concentration, 0. Thus, _,DC(.7: +6) — C(33). Jar: 6 (6.30)
The fraction in this equation is expressed in the form used to deﬁne a derivative; in
other words, it represents the change in concentration per change in distance along the :1; axis. As 6 —> 0, we may express this concentration gradient5 as BC/Bx, with the ﬁnal result, L72: = 81: v (6.31) The ﬂux of particles in the a: direction is proportional to the gradient in concen
tration in the a: direction and to the diffusion coefﬁcient. The negative sign tells
us that transport proceeds from a position of higher concentration to one of lower
concentration. This differential equation, known as F ick’s ﬁrst equation of diﬁfusion, is the
basis for much of the analysis of diffusive transport that follows. Fick’s equation as expressed here refers to the ﬂux through a plane. At times it
will be more convenient to deal with the ﬂux through a spherical surface. In this 5Because. the concentration may also vary along the y and z axes, we express this gradient as a
partial derivative. In the rare circumstances where C varies only with cc, we could use dC/dar. 93 Diffusion:
Random Walks in
Air and Water Fig. 6.7 In a derivation of Fick’s law, we keep
track of the number of particles (N) at two '
points on the maxis. Particles taking a step to the
left from (a: + 6) cross through the plane shown,
as do particles taking a step to the right from w. 1________________________________—————————————— case, the net movement of molecules'is directed radially, either into or out of the sphere, and the ﬂux density, J,, is 60
j, = —’D——, I (6.32) where r is radial distance. I
Note that concentration, as used in these equations, can be ex I d
of ways. If one is concerned with the transport of mass, concentration expresse 1
as mass per volume is appropriate, and ﬂux pressed in a variety density has the units of kg in"2 s‘ . Alternatively, if one is concerned primarily with how many particles are transported
regardless of their mass, one should express concentration as moles per volume.
For ease of computation, it is convenient to express concentration as moltzs p3er
cubic meter rather than the more traditional moles per liter (1 liter 1t) ml ).
In this case, J has units of mo] III—2 s“. In this chapter we deal primarily With
the diffusive ﬂux of metabolic gases into and out of organisms, and therefore care more about the number of particles than their mass. 6.5 Deriving the Sherwood Number Before leaving Fick’s equation, we use it to derive the Sherwood number. Recall
that Sh is the ratio of the rate at which mass is transported by convection to that at
which mass is transported by diffusion. Consider these transports in the following Dismce simple situation (ﬁg. 6.8). A planar surface lies next to a gradient in concentration
such that the concentration rises from 0 at the surface to Coo ‘over a distance
EC perpendicular to the surface. A substance dissolved in the ﬂuid can reach the
surface either by moving diffusiver down the concentration gradient or by movmg convectively toward the surface. I Consider convection ﬁrst. The amount of dissolved substance delivered to the
surface per time is equal to the product of moles per volume in the ﬂuid (concentra
tion) and the volume per time delivered to the surface. blow, volume is equal to the
product of area and length, so volume per time is equivalent to area times length
per time, or area times velocity. Thus the rate at which a substance is delivered by Concentration convection is mom = CooAu, (6.33)
time
where u is the convective velocity of the ﬂuid toward the surface. The rate of Fig. 6.8 The situation shown here is used in an delivery per area (a ﬂux density) is thus
informal derivation of the Sherwood number. convective ﬂux density = Coon. (6.34) Now the rate at which moles are delivered by diffusion (per area) is simply the
diffusive ﬂux density as expressed by Fick’s equation. In this particular case, Coo , (6.35)
to diffusive ﬂux density = D where the gradient in concentration has been expressed as the total change in
concentration (Coo) over the entire thickness of the gradient 13C. Dividing convective ﬂux density by that due to diffusion, we see that _ ulc Sh D, (6.36) as proposed in eq. 6.25. Note, however, that EC here is a measure of the thickness of
the concentration gradient, whereas earlier we deﬁned it as a characteristic length
of an object. As we will see in chapter 7, the thickness of a concentration gradient
is often a function of the size of an organism, and therefore this subtle switch in
nomenclature does not present a problem. It does, however, mean that unless 6c
is chosen speciﬁcally to be the thickness of the concentration gradient (i.e., the
boundary layer as discussed in chapter 7), Sh is not equal to the ratio of transport
by convection to that by diffusion, but rather is proportional to the ratio. As a
result, the Sherwood number is best thought of as an index (rather than a direct
measure) of the ratio of convective to diffusive transport. 6.6 Other Forms of Diffusion In this chapter we deal with the transport of molecules as they move diffusively
due to thermal agitation. However, there is nothing in our analysis so far that is
intrinsically tied to molecules. Anything that is transported as particles undergo
a random walk can be described by the concepts presented here and modeled
as a diffusive process. For example, in chapter 7 we use an analogue of Fick’s
equation to describe the way in which momentum is transported, and in chapter 8
we use the concept of diffusion to quantify the transport of heat. The notion that
“things” are carried down a gradient through the action of random motion is one
of the grand, unifying concepts of physics, and there are a number of different
diffusion coefﬁcients, each tailored to what it is that is diffusing. To avoid confusion
when dealing with the diffusion of molecules we refer to the molecular diffusion
coefﬁcient, 17",. 6.7 Diffusion Velocity vs. the Speed of Locomotion We are now in a position to explore the biological consequences of diffusion,
and we begin by comparing the time it takes a mobile organism to move a cer—
tain distance with the time required for a molecule to diffusiver move that same
distance. Consider ﬁrst an organism that releases a chemical signal into its envi—
ronment. For instance, the carbon dioxide excreted by an aerobic bacterium might
act as a “come hither” signal to nearby bacteriaeating ﬂagellates. Which is likely
to arrive at the predator ﬁrst, the chemical signal or the bacterium? Let us sup
pose that the bacterium is 10 ,um away from the ﬂagellate and swims toward it. A
bacterium with a body length of 2 pm moves at a speed of about 20 ,um s‘l (Berg
1983). Thus it takes the bacterium half a second to swim to its captor. In contrast,
C02 molecules travel 10 am in only about 0.03 s. In other words, at the small scale
of this predatorprey interaction, the chemical signal announcing the presence of
prey is diffusively transported more than ten times faster than the prey itself. The same logic applies to the delivery of food or oxygen. On the large scale
at which terrestrial mammals live, one is often conscious of depleting the local
resources and havingto move elsewhere. For instance, a cow must move to ﬁnd 95 Diﬁusion:
Random Walks in
Air and Water 97 food as it consumes what is available locally. At the small size of a bacterium, tables 6.1 and 6.2. Specifying the concentrati . . _
however, diffusion replaces the local supply faster than the organism can move, but We can employ one of the ClaSSical methadsnfgradient IS less stralghtforward, Diffusion:
as if the grass grew faster than the cow could eat it. As Berg (1983) points out, it up. Books such as Berg (1983) Crank (19750 solvmg such problems: we look Random Walks in
the only reason for a bacterium to move is if it ﬁnds itself in a particularly poor are ﬁlled with the analytical solutions to . )’ and _Caf§1aw and Jaeger (1959) Air and Water
pasture. this is one. a variety of diffusron problems, of which
Consider an example. The “diffusion velocity” (eq. 6.22) of an oxygen molecule Under these circumstances the c . ,
in water is equal to the swimming speed of abacterium (20 pm s‘l)when the period of the sphere is (Berg 1983) , oncemrauon at a dIStance TI from the center
under consideration is about 10 s. At shorter times, diffusive transport is faster than
swimming; at longer times swimming is faster than diffusion. In 10 s a bacterium T
can swim 200 ,um. Thus, if the local concentration of oxygen is favorable at a C(T) = C°° (1 _ P), T, > 7“ (6.37)
distance greater than 200 um away, it is advantageous for the bacterium to swim Taking the derivativ f th_ ‘ I
there rather than wait for delivery by diffusion. On the other hand, if the area of centraﬁon gradient is 0 ‘3 expresswn Wlth FCSPBCI to r’, we ﬁnd that the con
favorable oxygen concentration is only 100 am away, the bacterium is better off S d0
staying put. F = C00 42., (6.38)
In air, these arguments apply over larger scales. For example, in the last chapter T
we proposed that bacteria might be capable of swimming in air. The argument just leading to the condusmn that the ﬂux density 0f gas at the sphere’s surf ‘
made, however, makes it unlikely that locomotion would ever be of any substantial = T ) l3 ace (mtde
advantage in terms of aquiring food or oxygen because the swimming speed of an J (T) = _ DmCoo
aerial bacterium (which we calculated to be about 1 mm s“) exceeds the diffusion T ' (639)
velocrty only if the time consrdered is greater than about 40 5. Thus, only if the NOW, this ﬂux density is the number of moles of gas passin throu h .
bacterium needed to move to a greener pasture 4 cm away (a huge distance relative the surface per second. To calculate the total rate at W}. ich g l g “m area 0f .
to the size of the organism) would it be better to swim to the pasture rather than to we need to multiply by the where’s surface are 4 3 m0 CS enter the Sphere a
wait for the pasture to come to it. Oxygen would be rapidly delivered and carbon gas, J , is 8’ mt ' Thus’ the Inward ﬂux 0f u
dioxide effectively removed without the bacterium ever moving a ﬂagellum. J = —47rDmCoo1~ (6 40) ’ .
i
6.8 Diffusion and Metabolism thfighggig‘alvl: dpmﬁ: that gas .moves into the Sphere. m
Having seen that diffusion can effectively deliver oxygen to small organisms, of 471'7‘3 / 3. As a result?ng maxitrlritttlitnortrizzbcdfliihjatsghictze’itzhlff? has la mlume t“
we now explore the limits of the process: at what rate can an organism metabolize Supplied by diffusion per body volume per time is t , I it e m0 es 0f gas
if its only means of receiving gaseous fuel is through diffusion? For instance, how ’
fast can an animal or plant respire if it must rely on diffusion for the delivery of M _ 3DmCDo
oxygen? Or, how fast can a plant photosynthesize if carbon dioxide is transported _ _r2—’ (6.41)
by diffusion alone? a t  . I i
We ﬁrst carefully deﬁne the situation. To keep matters simple, we assume wﬁfﬁlgggolilsrga:sttliigiiedhlthvilue ls proportional to the maXimal rate at
that our organism is a sphere of radius r immersed motionless in a stationary ﬂuid rate must be if gas is Provided b difquiZnailger the sphere, the lower its metabolic
where, in the absence of the organism, the gas in question is present at concentration of both oxygen and carbon d‘ y d . _ one Becallse the diffuswn coefﬁcient
Coo (expressed as me] m‘3). We assume that at a great distance from the sphere, this mm c m at are 10’000 times that 1“ water, a Sphere
concentration is maintained even when the organism consumes gas. Furthermore, 10‘
we may assume that any gas molecule that contacts the surface of the sphere is ,8 10’
immediately bound and made available to metabolic processes inside the cell; as in 10‘
a result, the concentration at the surface is maintained at 0. This is a feat beyond E (10’
the capability of any real organism, but it allows us to calculate the maximum rate 5 10’
at which 02 or C02 can be delivered in theory if not in practice. 1°;
Note that the approach outlined here treats the inside of the organism as a g 10"
“black box” that consumes oxygen or carbon dioxide at a rate determined solely : 1:1
by the rate at which gas is delivered to the periphery of the sphere. The speciﬁcs of g 10, E
internal transport and the mechanism of metabolism are left intentionally vague. i 10.. 5
We now proceed to solve Fick’s equation to calculate the ﬂux of gas to the 10., t MA“, Wm“ M I I l Fi 6 9 A h _ I 
surface. To do so we need to know the diffusién coefﬁcient and the concentration 10" 10’ 10“ 10" ‘10} ‘ Amie" tufteages, niestfgoi’ifc:Zﬂhﬁﬁfldﬁams“;
e empirical values cited in , Radius (m) Oxygenis delivered by diffusion aloneéaTsliils
constraint is much more severe in water, gradient. For the diffusion coefﬁcients we rely on th 110 Chapter 7 From Fick’s equation we know that Chapter 6
C — C
J = —'Dm7rr2N 2 e ’ , (6.57) J};
, th t Much of ﬂuid dynamics involves the tugofwar between inertia, with its tendency Density and
and‘ remangmg’ we see a to support continued motion, and viscosity, which tends to bring objects to a
J E (6.58) halt. As we have seen in chapter 5, the relative contributions of these tendencies VISCOSIty Together: D in governing patterns of ﬂow can be described conveniently by the Reynolds number, and in this chapter we eXplore the ways in which Reynolds numbers can
be used to highlight relevant differences between air and water. We will see why the
high terminal velocities of terrestrial organisms make for a sparse aerial plankton
but efﬁcient aerial “ﬁlter feeders.” We will show how turbulence and a rapid
reproductive rate allow diatoms to avoid the abyss. We calculate the maximal speed
at which animals can walk in air and water, see how crickets use boundary layers
as an aid in discriminating the frequency of sounds, and explore the mechanics of
ﬁsh olfaction. m = mama, — 02)' s egg provides the means by which The Many Guises of
Reynolds Number Thus, an apparatus very much like a bird’ diffusion coefﬁcients can be measured. . '
Rahn and Paganelli (1979) actually used another rearrangement of thrs equation to measure the area of pores in birds’ eggs. They knew the diffusion coefﬁcrent
of water from previous empirical measurements and could easrly measure the
thickness of the shell (= 6). By placing each egg in a desiccator and measuring the
rate at which it lost mass, they could then solve for the effective pore area. .
Many other methods have been devised for measuring diffusron coefﬁcrents. To explore these consult Marrero and Mason (1972).
7.1 He Revisited 6.12 Summary . . .
The diffusion coefﬁcients of gases are 10,000 times larger in arr than in water. As a result, terrestrial organisms that rely on the diffusive delivery of oxygen
and carbon dioxide can be larger and can metabolize faster than their aquatic counterparts. . . . _ ‘ i
The effective transport of gases by diffusron' 1n arr makes it possrble for ter
'ratory system, and this ' ’ ' ' lrespr
restrral animals and plants to resprre usrng a trachea A h I
possibility has been realized repeatedly in the course of evolution. The high dif— fusivity of gases in air also makes it possible for birds’ eggs, insects, and leaves
to metabolize even though only a small fraction of their surface area is permeable
to gases. The mean free path of a molecule in air sets a lower lrmrt to the effective size ofpores, and small insect tracheoles approach this limit. Before embarking on this exploration, let us brieﬂy review the concept of a Reynolds number. ‘
Recall that Re is a dimensionless value proportional to the ratio of inertial and viscous forces. Its mathematical expression involves four parameters: the density of the ﬂuid, pf; the ﬂuid’s dynamic viscosity, p; the relative velocity between object and ﬂuid, u; and a characteristic length, KC: 3c
He: pf” . (7.1) p. In this chapter we are primarily concerned with the combined effects of density
and viscosity. It is appropriate, then, to use an alternative expression in which p f
and ,u are combined into the kinematic viscosity, V = p. / p f: E
R8 = u— (7.2) 6.13 . . . and 2 Warning
0 nonintuitive results, and a complex geom— V The rocess of diffusion can lead t . t
p have a drastic effect on the rate of drffusrve etry or the presence of convection can
transport. As a consequence, you are cautrone
conclusions reached in this chapter. When fac d against the blind application of the
ed with the complexities ‘of the real The kinematic viscosity of air is about ﬁfteen times that of water, varying
slightly with temperature (table 7.1; ﬁg. 7.1). As a result, the Reynolds number for Table 7.1 The kinematic viscosity of air and d t onsun a Standard text on diffusion (6%., Crank 1975). a given EC and u is about ﬁfteen times larger in water than in air. Therein hangs water
worlda you are urge o b . f discussion exhaust the subject of diffusion and the tale of this chapter. T(°C) Kinematic Viscosity (m2 s“ X 10‘5)
By no me?“ does t Is He 1. “ed ourselves to cases where the Sherwood Now, eq. 7.2 is only one particular example of a class of expressions known Seawater
biology.'In this chapter we have rmIt m and interesting effects of the disparity in generically as Reynolds numbers, all having this same general form. The precise Dry Air Fresh Ware} (5' = 35)
numberis less than 1. But many’mptzi a t are seen at higher Sherwood numbers, manifestation of the Reynolds number applicable to a particular case depends 0 13 3 1 79 1 84
difoS’on coefﬁments. between a“ an Vila er f th t two cha ters ’ typically on the choice of velocity, characteristic length, and density, and these can 10 14:2 131 1:3 5
and these form a 133518 for the exploranons 0 6 ﬂex p ' be chosen in creative fashions that give the Reynolds number a variety of different 20 151 101 106
“looks.” The Re we have dealt with so far involves the length of an organism 30 ’6‘0 0'80 0'85
40 17.0 0.66 0.70 along the direction of ﬂow, the velocity of the organism relative to the bulk of the
surrounding ﬂuid, and the density of the ﬂuid itself. In this chapter we encounter
several alternative guises for Be in which length, velocity, and density are chosen
so as to scale particular phenomena to certain characteristics of the ﬂow. In each ...
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This note was uploaded on 07/23/2008 for the course GEOSC 203 taught by Professor Anandakrishnan during the Fall '07 term at Pennsylvania State University, University Park.
 Fall '07
 ANANDAKRISHNAN

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