Clear["Global`*"]

a=1 ;
κ=1;
Ta=200;
Ti=-20;

ImaxDefault=30;
sn=((Table[BesselJZero[0,i]//N,{i,1,200}])/a);

T[r_,t_,imax_]:=Ta-2(Ta-Ti)Sum[((BesselJ[0,sn[[j]] r])/(sn[[j]] a    BesselJ[1,sn[[j]] a])) Exp[-κ ((sn[[j]])^2)t],{j,1,imax}];
BesFejk[k_,r_]=(Sqrt[(2/(Pi r))])Cos[r-((k/2)+(1/4))Pi];
Tfejk[r_,t_,imax_]:=Ta-2(Ta-Ti)Sum[((BesFejk[0,sn[[j]] r])/(sn[[j]] a    BesFejk[1,sn[[j]] a])) Exp[-κ ((sn[[j]])^2)t],{j,1,imax}];

Plot[{
T[r,0.002 ,ImaxDefault],
T[r,0.01,ImaxDefault],
T[r,0.05,ImaxDefault],
T[r,0.1,ImaxDefault],
T[r,0.2,ImaxDefault],
T[r,0.9,ImaxDefault]},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Temperatura pri različnih časih",PlotLegends->"Expressions"]

Plot[{
T[r,0.001,5],
T[r,0.001,10],
T[r,0.001,20]
},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Konvergenčna analiza",PlotLegends->"Expressions",PlotRange->All]

Plot[{
T[r,0.001,ImaxDefault],
Tfejk[r,0.001,ImaxDefault],


T[r,0.01,ImaxDefault],
Tfejk[r,0.01,ImaxDefault],

T[r,0.05,ImaxDefault],
Tfejk[r,0.05,ImaxDefault]
},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Primerjava s poenostavljeno",PlotLegends->"Expressions"]

Numerični primer: Baker

Clear["Global`*"]
k=286 (*(W/(m K))*);
cp=380 (*(J/(kg K ))*);
ρ=8900(*(kg/(m^3))*);
a=50(*mm*);
κ= (k/(ρ cp))//N;
Ta=100;
Ti=0;
ImaxDefault=30;
sn=((Table[BesselJZero[0,i]//N,{i,1,200}])/a);
T[r_,t_,imax_]:=Ta-2(Ta-Ti)Sum[((BesselJ[0,sn[[j]] r])/(sn[[j]] a    BesselJ[1,sn[[j]] a])) Exp[-κ ((sn[[j]])^2)t],{j,1,imax}];
Plot[{ (*Tukaj sta za skaliranje časa nujna κ in tudi a, radij*)
T[r,0.01 (a^2)/κ ,ImaxDefault],
T[r,0.05(a^2)/κ ,ImaxDefault],
T[r,0.2(a^2)/κ ,ImaxDefault],
T[r,0.9 (a^2)/κ ,ImaxDefault]},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Temperatura pri različnih časih",PlotLegends->"Expressions",PlotRange->All]

(Subscript[t, segr])=0.9 ((((a/1000)))^2)/κ  (*s*)

Numerični primer: Jeklo

Clear["Global`*"]
k=80 (*(W/(m K))*);
cp=450 (*(J/(kg K ))*);
ρ=7950(*(kg/(m^3))*);
a=50(*mm*);
κ= (k/(ρ cp))//N;
Ta=100;
Ti=0;
ImaxDefault=30;
sn=((Table[BesselJZero[0,i]//N,{i,1,200}])/a);
T[r_,t_,imax_]:=Ta-2(Ta-Ti)Sum[((BesselJ[0,sn[[j]] r])/(sn[[j]] a    BesselJ[1,sn[[j]] a])) Exp[-κ ((sn[[j]])^2)t],{j,1,imax}];
Plot[{ (*Tukaj sta za skaliranje časa nujna κ in tudi a, radij*)
T[r,0.01 (a^2)/κ ,ImaxDefault],
T[r,0.05(a^2)/κ ,ImaxDefault],
T[r,0.2(a^2)/κ ,ImaxDefault],
T[r,0.9 (a^2)/κ ,ImaxDefault]},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Temperatura pri različnih časih",PlotLegends->"Expressions",PlotRange->All]

(Subscript[t, segr])=0.9 ((((a/1000)))^2)/κ  (*s*)

Numerični primer: Guma

Clear["Global`*"]
k=1(*(W/(m K))*);
cp=1880 (*(J/(kg K ))*);
ρ=7950(*(kg/(m^3))*);
a=50(*mm*);
κ= (k/(ρ cp))//N;
Ta=100;
Ti=0;
ImaxDefault=30;
sn=((Table[BesselJZero[0,i]//N,{i,1,200}])/a);
T[r_,t_,imax_]:=Ta-2(Ta-Ti)Sum[((BesselJ[0,sn[[j]] r])/(sn[[j]] a    BesselJ[1,sn[[j]] a])) Exp[-κ ((sn[[j]])^2)t],{j,1,imax}];
Plot[{ (*Tukaj sta za skaliranje časa nujna κ in tudi a, radij*)
T[r,0.01 (a^2)/κ ,ImaxDefault],
T[r,0.05(a^2)/κ ,ImaxDefault],
T[r,0.2(a^2)/κ ,ImaxDefault],
T[r,0.9 (a^2)/κ ,ImaxDefault]},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Temperatura pri različnih časih",PlotLegends->"Expressions",PlotRange->All]

(Subscript[t, segr])=0.9 ((((a/1000)))^2)/κ  *(1/3600)(*ure*)

Numerični primer: Govedina

Clear["Global`*"]

a=25(*mm*);
κ=(4.5 (10^(-4))  *((1000^2)/3600))(*(mm^2)/s*);
κ=(3.8 (10^(-4))  *((1000^2)/3600))(*(mm^2)/s*);
Ta=200;
Ti=-20;
ImaxDefault=30;
sn=((Table[BesselJZero[0,i]//N,{i,1,200}])/a);
T[r_,t_,imax_]:=Ta-2(Ta-Ti)Sum[((BesselJ[0,sn[[j]] r])/(sn[[j]] a    BesselJ[1,sn[[j]] a])) Exp[-κ ((sn[[j]])^2)t],{j,1,imax}];
Plot[{ (*Tukaj sta za skaliranje časa nujna κ in tudi a, radij*)
T[r,0.01 (a^2)/κ ,ImaxDefault],
T[r,0.05(a^2)/κ ,ImaxDefault],
T[r,0.22(a^2)/κ ,ImaxDefault],
T[r,0.8 (a^2)/κ ,ImaxDefault]},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Temperatura pri različnih časih",PlotLegends->"Expressions",PlotRange->All]

(Subscript[t, segr])=0.22 (((a))^2)/κ*(1/60)  (*ure*)

Numerični primer: Govedina

Clear["Global`*"]

a=100(*mm*);
κ=(4.5 (10^(-4))  *((1000^2)/3600))(*(mm^2)/s*);
(*κ=(3.8 (10^(-4))  *((1000^2)/3600))(*(mm^2)/s*);*)
Ta=200;
Ti=20;
ImaxDefault=30;

T[r_,t_,Ta_,imax_]:=Ta-2(Ta-Ti)Sum[(((-1))^(j+1))*((Sin[(Pi j)(r/a)])/((Pi j)(r/a)))Exp[((-κ)/(a^2))(((Pi j))^2) t],{j,1,imax}];
Plot[{ (*Tukaj sta za skaliranje časa nujna κ in tudi a, radij*)
T[r,0.02 (a^2)/κ ,Ta,ImaxDefault],
T[r,0.08(a^2)/κ ,Ta,ImaxDefault],
T[r,0.13(a^2)/κ ,Ta,ImaxDefault],
T[r,0.4 (a^2)/κ ,Ta,ImaxDefault]},{r,0,a},AxesLabel->{"r/a","Temperatura"},PlotLabel->"Temperatura pri različnih časih",PlotLegends->"Expressions",PlotRange->All]

(Subscript[t, segr])=0.13 (((a))^2)/κ*(1/60) *(1/60) (*ure*)

Plot[{ T[0.00001,t (a^2)/κ ,180,ImaxDefault],
 T[a*0.8,t (a^2)/κ ,180,ImaxDefault],
T[0.00001,t (a^2)/κ ,250,ImaxDefault],
 T[a*0.9,t (a^2)/κ,250,ImaxDefault]},{t,0,0.2},AxesLabel->{"t","Temperatura"},PlotLabel->"Temperatura v r=0",PlotLegends->"Expressions",PlotRange->{0,250}]