4-5. Transient Diffusion in a Permeable Tube with Open Ends The objective is to examine in more detail the effects of the capillary ends in the permeability experiment described in Example 3.5-2. In that experiment (Fig. 3-16) an open-ended cylindrical tube (isolated kidney capillary) of radius a and length 2L was immersed initially in a fluid with solute concentration C0. At t=0 the concentration in the surrounding fluid was changed rapidly to a new constant value, C10. For the similarity method to work, how must the concentration variable be modified? EXAMPLE 3.5-2 Capillary Permeability Experiment This example illustrates the use of tire seales in reducing the dimensionality of a model. It is based on experiments in Daniels et al. (199) that were designed to determine the permeability of renal glomerular capillaries. -ff the de monly defined as km=J/C, where C is the difference in solute concentration between the elle nal solutions. Thus, km resembles a mass transfer coefficient, but involves a transmentrane concentration difference rather than one within a single fluid phase. The experiments were designef to determine km by optically monitoring the time-dependent concentration of a fluorescent tracer in "windows" placed inside and outside a single capillary. Confocal microscopy was used lo cteain the needed spatial resolution. The capillaries were in glomeruli isolated from rat kidneys and dextran of 70kDa molecular weight. In the particular experiments to be analyzed lere, the cells that form part of the capillary wall were removed by detergent lysis, leaving as the only tarrier the glomerular basement membrane (GBM). The GBM is a hydrogel containing a crosslinked network of collagen and other biopolymers and is about 90% water by volume. An idealized representation of the system is shown in Fig. 3-16. The capillary segmentis modeled as a cylinder of length 2L, radius a, and wall (membrane) thickness h. The observation windows (shown as dashed rectangles) are assumed to be equidistant from the open ends. Initially, the capit lary was equilibrated with a bath containing a concentration C0 of the dextrin. Then, the bath flild was changed rapidly (in about 2 s). lowering its dextran concentration to about 25% of C0. This resulted in outward diffusion both across the membrane and from the ends, as indicated by the arruws. The objective is to show that a lumped model is adequate near the windows and that axial diffosion can be neglected there. Parameter values for this system are summarized in Table 3.2. The intramembrane diffusinity Dm is unknown, but can be related to the aqueous diffusivity Dc and the membrane/aqueous partition coefficient K. The diffusivity ratio shown is based on the reported value of km(=16X 107m/s) and the pseudosteady relationship km=KDm/h that follows from Example 351. The parti- Figure 3-16. Schematic for the capillary permeability experiment dikcuked in Example 3.5-2. The kidoey cap illary segnent (wich cells removed) has an ininer radius a, a wall thickness h, and a kength 21. . In a espenment. a sadden reduction in the external concentration causes outwand diffusion across the wall and through the open ends. The iaternal and exiemal concentrations are monitoned optically in the ares shown by the dapied rectangles. Table 3-2 3.5 Time Scales in Moteling 93 Parameter Values for the Canillam n. tion coefficient, also unknown, is assumed to be in the range 0.01 to 0.1. The transmembrane concenination difference was found to decay exponentially with time, and tp is the time required for it to fall to e1 times its initial value. It follows from the lumped model (discussed below) that tp=a/(2km). The various time scales are examined now. If the bath concentration could be lowered instantaneously, the time required for the change to propagate across the membrane would be on the order of t1=h2/Dm. Based on the assumed range for K, this is between 0.013 and 0.13 s. Relative to ip(=12s) or even the actual time for the bath exchange (=2s), the membrane response is nearly instantaneous. Accordingly, as seen in Example 3.5-1, the membrane concentration profile will become pseudosteady almost immediately and remain so until a final equilibrium is reached. This confirms that km can be viewed as a constant. Once a concentration change has occurred across the membrane, it will propagate radially within the capillary. The characteristic time for this, t2=a2/Da=0.34 s, again is much smaller than the process time. Thus, adjustments within the capillary lumen are rapid, and radial concentration variations quiekly become pseudosteady. Whether they can be neglected entirely is assessed by calculating what will be termed the membrane Biot number," Bim=kma/D. This differs from the usual mass-transfer Biot number in Ex. 3.4-1 in that the "external" resistance is that in the membrane, rather than a surrounding fluid; as with Bi, it is interpretable as a ratio of internal to extermal resistances. (The partition coefficient K has not been ignored in Bim. but rather is embedded within km ) It is found that Bim=0.015, which indicates that the resistance to radial diffusion in the lumen is small relative to that in the membrane. This justifies the use of a partially lumped (onedimensional) model inside the capillary, in which radial concentration variations are neglected and C=C(z,f) only. The remaining issue is whether axial diffusion must be included when calculating kmfromthe measurod concentrations. If the observation windows are centered, as assumed, the time for diffusion from the ends of the capillary to be "felt" will be on the order of ts=L2/Ds=1.2103s=20min. This eeally exceeds not only tp but also the total observation time, which was about 2min. With nothing to ceate axial concentration variations in the vicinity of the windows, it can be assumed that C=C(t) only. Acoordingly, a fully lumped model was used for the data analysis (Daniels et al. 1993). Axial diffusion