Ind. Eng. Chem. Res. 1992,31,732-735
732 Greek Letters
6 = volumetric coefficient of thermal expansion, 1/K 6 = phase shift, deg 9 = shear rate, l/s r) = viscosity of Newtonian fluid or apparent viscosity of non-Newtonian fluid, N.s/m2=1000 CP w = angular frequency, rad/s p = density of fluid, kg/m3 T = shearing stress, N/m2 Registry No. Carbopol934,9007-16-3; Separan 273,3722428-5.
Literature Cited Cho, Y. I.; Hartnett, J. P. Non-Newtonian Fluids in Circular Pipe Flow. Adu. Heat Transfer 1982,15,59-141. Dodge, D. W.; Metzner, A. B. Turbulent Flow of Non-Newtonian Systems. AZChE J. 1959,5,(2), 189-204. Dow Chemical Company. Let Separan Settle Your Liquid-Solids System; Design Products Department: Midland, MI, 1988. Friend, W. L.; Metzner, A. B. Turbulent Heat Transfer inside Tubes and the Analogy Among Heat, Mass, and Momentum Transfer. AZChE J . 1955,4 (4). B.F. Goodrich Chemical Group. Carbopol Water Soluble Resins; The B.F. Goodrich Company, Speciality Polymer and Chemicals Division: Cleveland, OH, 1989.
Green, A. E.; Rivlin, R. S. Steady Flow of Non-Newtonian Fluids through Tubes. Q. Appl. Math. 1956,15,299-308. Hartnett, J. P.; Xie, C. Influence of Polymer Concentration on Laminar Heat Transfer to Aqueous Carbopol Solutions in a 2:l Rectangular Duct. AZChE Symposium Series; AIChE: Philadelphia, 1989;Vol. 85 (No. 269),pp 454-459. Kostic, M. Heat Transfer and Hydrodynamics of Water and Viscoelastic Fluid Flow in a Rectangular Duct. Ph.D. Thesis, University of Illinois at Chicago, 1984. K o k k i , W.; Chou, C. H.; Tiu, C. Non-Newtonian Flow in Ducts of Arbitrary Cross-sectional Shape. Chem. Eng. Sci. 1966, 21, 665-679. Kwack, E. Y.; Hartnett, J. P.; Cho, Y. I. Chemical Effects in the Flow of Dilute Polymer Solutions. Lett. Heat Mass Transfer 1980,7, 1-6. Shah, R. K.; London, A. L. Laminar Flow Forced Convection in Ducts; Academic Press: New York, 1978. Wheeler, J. A.; Wider, E. H. Steady Flow of Non-Newtonian Fluids in a Square Duct. Trans. SOC.Rheol. 1966,10,353-367. Wibulswas, P. Laminar Flow Heat Transfer in Non-circular Ducts. Ph.D. Dissertation, Department of Mechanical Engineering, University of London, 1966.
Received for review January 11, 1991 Revised manuscript receiued April 10, 1991 Accepted May 3, 1991
Energy Cost of Intracellular Organization Grace H. Okamoto* and Edwin N. Lightfoot Department of Chemical Engineering, University of Wisconsin, 1415 Johnson Dr., Madison, Wisconsin 53706
Intracellular diffusional processes are examined by use of a simple model selected to highlight the role of Brownian motion in determining cellular anatomy. Limiting calculations of energy consumption show that mechanical transport of formed elements costs very little, but it is prohibitively expensive for small metabolites. Biologicai systems have evolved efficient means to maintain spatial organization of intracellular elements in the presence of randomizing diffusive forces.
Introduction The purpose of this paper is to consider the role of intracellular diffusion, in an attempt to assess the impact on intracellular energy consumption. The underlying rationale for this analysis is the belief that the evolution and metabolic behavior of biological structures at the cellular level have been heavily influenced by mass-transport considerations. We shall see that, in the absence of evolutionary adaptation, such transport can be too slow for optimum operation and that the cell has reacted with remarkable effectiveness to overcome this transport constraint. Understanding these constraints and adaptations requires an effective knowledge of pertinent time constants and a careful study of intracellular anatomy and physiology. In fact, the study of intracellular transport is particularly instructive from a modeling standpoint due to the complexity of the cell interior and because intracellular mass transport was only recently recognized as a significant problem. We begin briefly with a bit of historical perspective in the form of Figure 1: a continuous stirred tank reactor (CSTR) model of a typical mammalian cell. A CSTR model assumes that the cell is just an unorganized collection of metabolites and enzymes. Since the cell is seldom more than about 10 pm across, diffusional processes are expected to be very rapid, and the unwary investigator
* Author
to whom correspondence should be addressed.
Table I. Intracellular Diffusion Coefficients Adapted after Mastro et al.. (1984)" radius, i O W , cm2/s a H 2 0 l A compound Mr scab ~ 3 e e l h 170 PCAOL 3.2 64 33 1.9 sorbitol 2.5 1.9 182 94 50 methylene blue 2.6 320 3.7 40 15 sucrose 2.6 324 4.4 52 20 eosin 8 648 6.0 40 5.0 dextran 3600 12.0 18 3.5 5.0 inulin 3.0 5.0 5500 13.0 15 dextran 9.2 2.5 loo00 23.3 3.7 dextran 24000 35.5 6.3 1.5 4.2 actin 5.3 0.03 167.0 43000 23.2 bovine serum albumin 68000 36.0 6.9 0.10 71.0 "Literature references for individual materials will be found in the original table.
wodd be tempted to assume that the cell interior is always well mixed. For a typical cell with a radius of 5 pm (L= 5 pm), the diffusional times can be estimated with the following equation.
Tn = (2.5 X
%
L2/3)
cm2)/31
(1)
When considering a typical globular protein in water, the diffusivity (3))is approximately 7 X cm2/s, and the time to diffuse across the cell (Tn) is less than ' / 2 s.
0888-5885/92/2631-0732$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., VoL 31, No.3,1992 733
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