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CHE 499 (Spring 04)
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Problem set #5
1. (p. 7.41) The blood flow through human tissues is by very small arteries called the arterioles,
which have diameters in the range of 5 to 50 µm and a length of a few centimeters. These are
fed by small arteries, which in turn are fed by the aorta. Each arteriole empties into 10 to 100
capillaries, which have porous walls and are the sites of the exchange between the blood and
interstitial tissue fluid. These are shown in Figure 1. There are about 1010 capillaries in
peripheral tissue. This cascading of blood vessels results in a large increase in the total flow
cross section Au, as listed in Table 1 along with the cross-sectional area and the time-area
averaged blood velocity <uf>.
Table 1 Cross-sectional area (total) and time-area averaged blood velocity.
Au, cm2
<uf>, cm/s
aorta
2.5
33
small arteries
20
4.1
arterioles
40
2.1
capillaries
2,500
0.033
venules
250
0.33
small veins
80
1.0
venue cavao
8
10
As the total flow cross-sectional area Au increases, <uf> decreases (because the mass flow
rate M is conserved). For a very large specific surface area of blood vessels, i.e., Aku/V ,
we have NTU   and the body can use these blood streams to control the local tissue
temperature. For the conditions given below and in Table 1, determine NTU =
( Aku )( Nu)( k f )
for
( M )( C )( D )
p, f
(i) an arteriole, and (ii) a capillary. Note: if L/D > 0.6ReDPr, Nu = 3.66.
La = 2 mm, Da = 50 µm, Lc = 30 µm, Dc = 3 µm, f = 1,000 kg/m3, Cp,f = 3,000 J/kgK,
µf = 10-3 Pas, kf = 0.6 W/mK.
1
Kaviany, Principles of Heat Transfer, Wiley, 2002, p. 350
Figure 1 Blood supply to tissue by arterioles feeding the capillaries.
2. (p.
Consider a spherical organism of radius R within which respiration occurs at a
uniform volumetric rate of rA =  ko. That is, oxygen (species A) consumption is governed by
a zero-order, homogeneous chemical reaction.
(a) If a molar concentration of CA(R) = CA,0 is maintained at the surface of the organism, obtain
an expression for the radial distribution of oxygen, CA(r), within the organism. What is the
minimum value of CA,0 so that the solution will be applicable?
(b) Obtain an expression for the rate of oxygen consumption within the organism.
(c) Consider an organism of radius R = 0.10 mm and a diffusion coefficient for oxygen transfer
of DAB = 10-8 m2/s. If CA,0 = 510-5 kmol/m3 and ko = 1.210-4 kmol/sm3, what is the molar
concentration of O2 at the center of the organism?
14.342)
3. (p. 14.352) Consider a spherical organism of radius R within which respiration occurs at a
uniform volumetric rate of rA =  k1CA. That is, oxygen (species A) consumption is governed
by a first-order, homogeneous chemical reaction.
(a) If a molar concentration of CA(R) = CA,0 is maintained at the surface of the organism, obtain
an expression for the radial distribution of oxygen, CA(r), within the organism.
(b) Obtain an expression for the rate of oxygen consumption within the organism.
(c) Consider an organism of radius R = 0.10 mm and a diffusion coefficient for oxygen transfer
of DAB = 10-8 m2/s. If CA,0 = 510-5 kmol/m3 and k1 = 20 s-1, what is the molar concentration
of O2 at the center of the organism? What is the rate of oxygen consumption by the
organism?
4. (p. 14.372) Consider the problem of oxygen transfer from the interior lung cavity, across the
lung tissue, to the network of blood vessels on the opposite side. The lung tissue (species B)
may be approximated as a plane wall of thickness L. The inhalation process may be assumed
to maintain a constant molar concentration CA,0 of oxygen (species A) in the tissue at its inner
surface (x = 0), and assimilation of oxygen by the blood may be assumed to maintain a
constant molar concentration CA,L of oxygen (species A) in the tissue at its outer surface (x =
L). There is oxygen consumption in the tissue due to metabolic processes, and the reaction is
zero order, with rA =  ko. Obtain expressions for the distribution of the oxygen concentration
2
Incropera, Fundamentals of Heat and Mass Transfer, Wiley, 2002
in the tissue and for the rate of assimilation of oxygen by the blood per unit tissue surface
area.
5. (p. 27.143) A “drug patch” is designed to slowly deliver a drug (species A) through the body
tissue to an infected zone of tissue beneath the skin. The drug patch consists of a sealed
reservoir containing the drug encapsulated within a porous polymer matrix. The patch is
implanted just below the skin. A diffusion barrier attached to the bottom surface of the patch
sets the surface concentration of the drug in the body tissue at 2 mol/m3, which is below the
solubility limit. The mean distance from the drug patch to the infected area of tissue is 5 mm.
To be effective, the drug concentration must be at least 0.2 mol/m3 at the top edge of the
infected zone. Determine the time (in hours) it will take for the drug to begin to be effective
for treatment. The effective molecular diffusion coefficient of the drug through the body
tissue is 110-6 cm2/s.
3
Welty, J. R., Fundamentals of Momentum, Heat, and Mass Transfer, Wiley, 2001, p. 548.