CHE 499 (Spring 04) ______
LAST NAME, FIRST
Problem set #6
1. (p. 3.16[1]) During hibernation of warm-blooded animals (homoisotherms), the heart beat and the body temperature are lowered and in some animals the body waste is recycled to reduce energy consumption. Up to 40% of the total weight may be lost during the hibernation period. The nesting chamber of the hibernating animals is at some distance from the ground surface, as shown in Figure 1(i). The heat transfer from the body is reduced by the reduction in the body temperature T1 and by the insulating effects of the body fur and the surrounding air (assumed stagnant). A simple thermal model for the steady-state, spherical, one-dimensional heat transfer is given in Figure 1(ii). The thermal resistances of air and soil can be determined from
Rk,air = , Rk,soil =
An average temperature T2 is used for the ground surrounding the nest. The air gap size RaRf is an average taken around the animal body.
Determine the heat loss from the body for (i) L = 2.5Ra, and (ii) L = 10Ra.
R1= 10 cm, Rf = 11 cm, Ra= 11.5 cm, T1= 20oC, T2= 0oC.
For air ka = 0.0267 W/mK, for fur kf = 0.036 W/mK, for soilks = 0.52 W/mK.
Figure 1. Conduction heat transfer from a warm-blooded animal during hibernation.
2.(p. 3.56[2]) A fermentation broth consists of an aqueous solution of nutrients and cells. As the cells grow, they cluster into spherical pellets of radius R(t). On average, the cell density inside a pellet is 0.02 mg of cell mass per cubic millimeter of pellet volume. The dissolved oxygen concentration in the broth is 5 g/cm3. The cells utilize oxygen at a rate of 1.2 mmol of oxygen per hour per gram of cell mass, via a zero-order reaction. Assume that the diffusion coefficient DAB of oxygen within the pellet is 1.810-5 cm2/s. How large can R(t) become before the oxygen concentration becomes zero at the center of a pellet? Assume that the broth external to the pellets is well mixed. The cells and broth have densities close to that of water.
3.Repeat problem (2), but allow for the presence of a finite convective resistance to mass transfer to the surface of the pellet, such that the flux to the surface is given by
molar flux = kc(CPCB)
where CB is the broth oxygen concentration and CP is the (unknown) concentration of oxygen at the pellet surface. Calculate the critical value of R(t) (at which the oxygen concentration becomes zero at the center of a pellet) for two cases: 2kcR/DAB = 2, and 2kcR/DAB = 20.
4. (p. 26.16[3]) The “drug patch” shown in the figure below releases a water-soluble epidermal growth factor (species A) to repair a specific region of wounded tissue on the human body. A slow release of the drug is critical for regulating the rate of tissue repair. The drug layer (pure solute A) rests on top of a diffusion barrier. The diffusion barrier is essentially a micro-porous polymer material consisting of tiny parallel pores filled with liquid water (species B). The diffusion barrier controls the rate of drug release. The thickness (L), pore size (dpore), and porosity of the diffusion layer determine the dosage rate of the drug to the tissue directly beneath it. The maximum solubility of the drug in water is 1 mole/m3 at 25oC. The pore diameter (dpore) is 110-8m, and the equivalent molecular diameter of the drug is 0.2510-8 m. The total surface area of the patch is 4 cm2, but the cross-sectional area of the pores constitute only 25% of the surface area for flux. The diffusion coefficient of the drug in water at infinite dilution is 110-10 m2/s at 25oC.
(a)Determine the effective diffusion coefficient of the drug in the diffusion barrier using the Renkin equation.
Dm/D = (1 a/r)2
(b)Estimate the thickness of the diffusion barrier (L) necessary to achieve a maximum possible dosage rate of 0.05 mole per day, assuming that the drug is instantaneously consumed once it exits the diffusion barrier and enter the body tissue.
(c)The above calculations assume a temperature of 25oC. However, the human body is actually at 37oC, and it is likely that the diffusion barrier equilibrates to that temperature. If all other process parameters are constant, what is the percentage change in dosage rate if the temperature is increased to 37oC? You may assume that maximum solubility of the drug in water is unchanged at 1 mole/m3 in the range 25oC to 37oC.
2-9, 2-11, 2-12, 2-13 (Text)
[1] Kaviany, Principles of Heat Transfer, Wiley, 2002, p. 350
[2] Middleman, An Introduction to Mass and Heat Transfer, Wiley, 1998, p.107
[3] Welty, J. R., Fundamentals of Momentum, Heat, and Mass Transfer, Wiley, 2001, p. 548.