The Dissolving Sphere Reprise

To illustrate what is meant by scaling, let us return to the problem of the dissolving sphere and try to make sure that all the important dependent variables are in the interval [0, 1]. The independent variable, time, must be allowed to run its course. We have certainly done this with the radius of the sphere, for r can only diminish so, if the initial radius is R,x = r/R is obviously the correct choice. With an eye to extending the model later, we define U as the terminal velocity of a sphere of radius R, and because this decreases with decreasing radius, v = u/U is certainly in [0,1]. The very simple relationship v = x2 holds as long as our assumption of the validity of Stokes law is true. 

Thus far, we are merely confirming what we did before, but when we come to the equation for the flux of solute from the surface, we have probably wrapped too much up in the dimensionless time. The empirical factor 1 + 0.672 Re1/2Sc1/3 in kc accounts for the convective enhancement of the mass transfer and the time the sphere takes to be dissolved in a stagnant fluid is the time for x to decrease from 1 to 0, which is 

This suggests that if we incorporate the solubility constant and the factor 2 into the dimensionless time, 

All times would be of magnitude less than 1. If we now write the convective factor as 

we will leave the solution of this equation until later. 

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