Temperature distribution for Pb in electron beam

Since the thermal conductivity of Pb changes dramatically with temperature the simple analytical solution for constant k is not applicable over most of the volume. However, this rapid change occurs outside the beam area so the heat equation becomes homogeneous( the external power density g = P/(pa*a*t) = 0.). We know the power in, the beam power, and we know in equilibrium that the power out at the cooled edges( r=b) must be the same.

Suppose the volume in red is the beam. This goes to a radius a. At radius b the temperature is fixed at Tb. From the BNL tables of the integral of thermal conductivity of lead we can calculate the temperature Ta at radius a. Within the region r<a the thermal conductivity of lead is practically constant at k=0.35 W/(cm.K). For Ta > 300K we can write

If the beam power is uniformly distributed across this region then the equation for the temperature distribution for 0<r<a is

At r = 0 this gives a fixed rise in temperature of  P/(4ptk) = 273K

a=2mm, P=60W, t=0.05cm

effect of different boundary temperatures and central temperature for uniform beam power
 Tb (K) b (mm) Ta (K) at 2mm T at center (K) 4.2 4 221 494 4.2 6 366 639 4.2 7 527 800 4.2 8 600 873 4.2 10 721 994 6 4 300 573 6 6 443 716 6 8 676 949 6 10 797 1070

Thus, the best case is if we could bring the cooling to within 4 mm of the center (494K). The more realistic condition of 6K cooling with a 4mm radius for the cooling line is too close to the melting point for Pb to be practical (573K).