We consider a one-dimensional heat source with a heat production function w(x). The temperature distribution is given by the heat equation,

w(x)+a d = 0- (1.14)

We consider the simplified situation where the heat source is at x = 0 and w(x) = w0S(x), where S(x) is the usual Dirac distribution and a the thermal conductivity. It follows that

d2T (x) dx2

and by integration the derivative of the temperature follows as

dT (x) w0

=—— 0 H(x)+ constant (1.16)

dx a

where H(x) is the Heaviside step function. A discontinuity of dT(x)/dx is observed at x = 0.

Let us consider a confining layer thickness L = LB + LH where LB;H are the distances of the storage site to the lower (higher) borders of the layer. The temperature decreases from its maximum value Td at the storage site to T0 at

both bounding limits. It is then easy to obtain Td:

w0 Lb lh

TD — T0 + ■

a Lb + Lh

From equation (I.17) one sees that Td is maximum for the symmetric configuration Lb — Lh, which is unfavourable. In contrast, as shown by the quadratic dependence of the diffusion time [equation (I.7)] this con­figuration maximizes the diffusion times, which is optimal in that respect. If the storage temperature is one of the design constraints of the storage and should not exceed Tdesign, equation (1.17) shows that the average heat production density should be proportional to Tdesign — T0, and, con­sequently, that the surface of the storage (and its cost) should be inversely proportional to Tdesign — T0. It might be necessary to find a trade-off between the diffusion time and the temperature of the storage.