CHE 499 (Spring 04) ______

LAST NAME, FIRST

Problem set #5

1. (p. 7.4[1]) 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 1010capillaries 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 1Cross-sectional area (total) and time-area averaged blood velocity.

Au, cm2 / uf, cm/s
aorta
small arteries
arterioles
capillaries
venules
small veins
venue cavao / 2.5
20
40
2,500
250
80
8 / 33
4.1
2.1
0.033
0.33
1.0
10

As the total flow cross-sectional area Auincreases, <ufdecreases (because the mass flow rate 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 = for

(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= 10-3Pas, kf= 0.6 W/mK.

Figure 1 Blood supply to tissue by arterioles feeding the capillaries.

2.(p. 14.34[2]) 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?

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 in the tissue and for the rate of assimilation of oxygen by the blood per unit tissue surface area.

5. (p. 27.14[3]) 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.

[1] Kaviany, Principles of Heat Transfer, Wiley, 2002, p. 350

[2] Incropera, Fundamentals of Heat and Mass Transfer, Wiley, 2002

[3] Welty, J. R., Fundamentals of Momentum, Heat, and Mass Transfer, Wiley, 2001, p. 548.