<< Calculus`VectorAnalysis`

SetCoordinates[Spherical[r, φ, ϕ]];

CoordinatesToCartesian[{r, φ, ϕ}]

Div[λ*Grad[T[r]]] // Simplify

qvect = -λ*Grad[T[r, t]]
n0 = {1, 0, 0}
Dot[qvect, n0]

<< Miscellaneous`ChemicalElements`

λ = ThermalConductivity[Aluminum][[1]];

FullForm[ThermalConductivity[Aluminum]]

Density[Aluminum]

ρ = Density[
            Aluminum][[1]];

    cp = 1000  SpecificHeat[Aluminum][[1]];

mm = 0.001;
rmin = 0.1  mm;
R = 25  mm;
Tpoc = 20;
α = 5;
rce = ρ*cp*
          D[T[r, t], 
            t] == λ (((2 D[T[r, t], r])/r) + 
              D[T[r, t], {r, 2}])

okrIn = D[T[r, t], r] == 0 /. r -> rmin

Plot[Tanh[0.1  t], {t, 0, 100}, PlotRange -> All]

To = 200;
okrOut = -λ*D[T[r, t], r] == 
          Tanh[0.1  t]*α*(T[r, t] - To) /. r -> R

poc = T[r, 0] == Tpoc;

tmax = 10000;
teplota = 
        T[r, t] /. NDSolve[{rce, poc, okrIn, okrOut}, 
              T[r, t], {r, rmin, R}, {t, 0, tmax}][[1]];

Plot3D[teplota, {r, rmin, R}, {t, 0, tmax}, 
        AxesLabel -> {"r[m]", "t[x]", "T[r,tmax]"}];

rezR = teplota /. t -> tmax;
Plot[rezR, {r, rmin, R}, 
        AxesLabel -> {"r[m]", "T[r,tmax]"}, 
        PlotStyle -> {Hue[0.6], 
            Thickness[0.008]}];