Underground Storage of Helium Diffusion through a Spherical Surface

Illustration 1.3 Underground Storage of Helium Diffusion through a Spherical Surface 

The previous illustration considered the rate of diffusion through a cylindrical wall and the resulting concentration profile within that wall. A similar approach can be used to calculate these quantities for diffusion through a spherical wall (Eigure 1.3b). This case arises much less frequently, as it requires the steady production of the diffusing species within the spherical cavity, or 

The case to be considered here falls in the latter category and involves the diffusional losses of helium from an underground storage facility. The background to this problem is as follows  

The problem here will be to estimate the losses that occur by diffusion through the surrormding salt and rock, assuming a solid-phase diffusivity Dj of helium of 10 in. /s, i.e., more than three orders of magnitude less than the free-space diffusivity in air. The helium is assmned to be at a pressure of 10 MPa (-lOO atm) and a temperature of 30dC. The cavity is taken to be spherical and of radius 100 m. Applying Pick s law. Equation 1.4a, and converting to pressure we obtain 

We have here an example of some practical importance, which nevertheless yields to a simple application and integration of Pick s law. Two features deserve some mention. The first is the formulation of the upper integration limit in Equation 1.9b. We use the argument that far away from the spherical cavity, i.e., as r °o, the concentration and partial pressure of helium 

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