The continuous inverse model

Let us assume a spherical mineral with radius R which initially contains a gas with concentration C0(r), r being the radial distance from the center. Upon incremental heating, this gas is lost to the extraction line and at the ith heating step when time is tf, the fraction of initial gas remaining is/(tf). Loss takes place by radial diffusion with temperature-dependent, hence time-dependent, coefficient 3 (t). We assume that the total amount of gas held by the mineral at t=0 is equal to one, i.e., that 

We further change to the dimensionless spatial coordinate x=r/R and introduce the dimensionless time t such that 

The radial distribution of concentration is given for the sphere by Carslaw and Jaeger (1959) as 

An analog expression can be found for parallel diffusion in Chapter 8. By integration of this expression over the entire sphere volume the fraction / (r) of initial gas remaining at dimensionless time r becomes 

The kernel function could also be made simpler by using 

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