<< Algebra`InequalitySolve`
<< Calculus`VectorAnalysis`

λ = 80;
cp = 449;
ρ = 7874;
T1 = 20;
T2 = 800;
Tpec = 900;
d = 0.2;
σ = 5.667*(10^(-8));
ϵ = 0.9;
InequalitySolve[((α*d)/λ) < 0.25, α]

alfaer = ϵ*σ (((((Tpec + 273))^4) - (((T2 + 273))^4))/(Tpec - T2))

V = (4/3)*Pi*((((d/2)))^3);
m = ρ*V;
S = 4*Pi*((((d/2)))^2);
rce = 
        m*cp*T'[t] == ϵ*
            S*σ*((((Tpec + 273))^4) - (((T[t] + 273))^4));
poc = T[0] == 20;
tmax = 1420;
res = NDSolve[{rce, poc}, 
            T[t], {t, 0, tmax}][[1]];
Plot[{T[t] /. res, 800}, {t, 0, tmax}, 
        PlotRange -> All];

Plot[{(t/(t + 1)), 0}, {t, 0, 400}]

SetCoordinates[
        Spherical[r, theta, fi]];
lapl = 
        Div[Grad[T[r, t]]] // 
          Simplify;
rce = ρ*cp*
            D[T[r, t], t] == λ*
            lapl;
poc = 
        T[r, 0] == T1;
okr1=-λ*Derivative[1, 0][T][r, t]==(ϵ*σ*((((T[r, t] + 273))^4) - (((Tpec + 273))^4)))*(t/(t + 1))/.r -> (d/2);
rmin = (d/200);
okr2= Derivative[1, 0][T][r, t]==0/.r -> rmin;
resND = NDSolve[{rce, poc, okr1, 
              okr2}, T[r, t], {r, rmin, (d/2)}, {t, 0, 
              tmax}][[1]];
Plot3D[
      T[r, t] /. resND, {r, rmin, (d/2)}, {t, 0, 
        tmax}]
tsred = T[r, t] /. resND /. 
          r -> rmin;
tpovrch = T[r, t] /. 
            resND /. r -> (d/2);

Plot[{tsred, tpovrch}, {t, 0, tmax/10}]

Plot[{tsred}, {t, 0, tmax/10000}, PlotRange -> All]