Laplace length scale

Phase separation in macroscale equipment either uses density differences between the two fluids to drive the separation, as in settlers, or these differences play an important role in the technical layout of the separator, e.g. in distillation towers. In macroscopic two-phase flow, length scales vary between the size of the apparatus and the interface-dictated Laplace length scale /o/(g AQ)) of entrained bubbles or drops. The former is often on the order of meters, whereas the latter is on the order of millimeters. This significant disparity in length scales makes it virtually impossible to separate macroscopic two-phase flows in a single step. 

A simple example for which problems (9-16) and (9 17) can be solved easily is the case of a heated sphere. In this case, we may choose the sphere radius as the characteristic length scale for nondimensionalization, and the problem is to solve (9 16) subject to the boundary condition 0 = 1 at r = 1. This can be done easily. We may first note that a general solution of Laplace s equation in spherical coordinates is... 

If we go further and assume an extreme form of the mean-field approximation, namely that the liquid is homogeneous even on the scale of length of u(r), then we can put g(r) = 1 in (4.112). An integration by parts then returns us to Laplace s equation for [Pg.92]

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