I think the error Error, (in PDEtools:-Library:-NormalizeBoundaryConditions) is a side error, due to using real numbers with exact solver. Maple can't solve this PDE.
Rule of thumb. Do not use real numbers with exact solver. Rationalized all values. After doing this, the error goes away, and pdsolve returns just () with no solution. Another thing, I do not like using eval to setup BC. Better using D directly.
Also, for the Laplacian in cylinderical coordinates, you can use VectorCalculus:-Laplacian to build it, instead of typing it yourself. Less change of error.
Also you had h:=(x,t)->0. But there is no x in this problem. I just set h:=0.
a := 35.5*0.0254: a:=convert(a,rational);
b := 36.5*0.0254: b:=convert(b,rational);
L := 0.0254*30*12: L:=convert(L,rational);
alpha := 0.000026972: alpha :=convert(alpha,rational);
Psi_s := 0.0440: Psi_s := convert(Psi_s,rational);
flux_b := 5.1/12: flux_b := convert(flux_b,rational);
h := 0:
the_laplacian_in_cylinderical :=VectorCalculus:-Laplacian( u(r,z,t), 'cylindrical'[r,phi,z] );
pde := diff(u(r, z, t), t) = alpha*the_laplacian_in_cylinderical + h;
ic:= u(r, z, 0) = z*Psi_s/L;
sol := pdsolve([pde,bc,ic],u(r, z, t)) ;
You might want to make sure your IC and BC are consistent. You IC says u=z*PSI/L. This means u_z = PSI/L, a constant. This is at t=0, but for all z and all r. But one of your BC says -apha*u_z = 0 (at z=L). Since alpha is not zero, this means u_z = 0 (at z=L). Which is conflicting with the initial condition. I do not know now if this is why the PDE was not solved.
Either way, Maple simply can't solve this analytically. No error generated.
Maple 2020.2 on windows