rce=500==25*(10^(-6))*(15*(T-20)+0.6*5.6*(10^(-8))*((((T+273))^4)-(293^4)));
res=Solve[rce,T]

Timing[Solve[rce,T]]
Select[T/.res,#>0&][[1]]//Quiet

T/.res

Cases[res,{T->_?((Im[#]==0&&Re[#]>0)&)},{0,∞}]

((Im[#]==0&&Re[#]>0)&)[5]

list=Range[10]
Cases[list,_]
Cases[list,_Integer]
Cases[list,_?(#>5&)]
fce[a_]:=a>5;
Cases[list,_?fce]

rce=500==25*(10^(-6))*(15*(T-20)+0.6*5.6*(10^(-8))*((((T+273))^4)-(293^4)))

mm=0.001
eqna=500==25(mm^2)*(15*(T-20)+0.6*5.6*(10^(-8))*(Ψ-(293^4)));
ClearAll[Tnext]
Tnext[Tprevious_]:=((Ψ/.Solve[eqna,Ψ][[1]]/.T->Tprevious)/.Ψ->(((Tprevious+273))^4))^(1/4)-273;
NestList[Tnext,1000,5]
Nest[Tnext,-20000,5]

Tnext[1000]

((Ψ/.Solve[eqna,Ψ][[1]]/.T->Tprevious))
((Ψ/.Solve[eqna,Ψ][[1]]/.T->Tprevious)/.Ψ->(((Tprevious+273))^4))^(1/4)-273

NestList[Sin[x],x,5]

ClearAll[Tnext]
Tnext[Tprevious_]:=((T/.Solve[eqna,T][[1]]/.Ψ->Tprevious)/.T->(((Tprevious+273))^4))^(1/4)-273;
NestList[Tnext,1000,5]
Nest[Tnext,1000,5]

pol=Plus@@Table[(Subscript[a, i])*(x^i),{i,0,5}]

Solve[pol==0,x]

rce=500==25*(10^(-6))*(15*(T-20)+0.6*5.6*(10^(-8))*((((T+273))^4)-(293^4)));

lhs=rce[[1]];
rhs=rce[[2]];
Tnext[Tprevious_]:=Module[{taylor},
taylor=Series[rhs,{T,Tprevious,1}]//Normal;
T/.Solve[taylor==lhs][[1]]
]
NestList[Tnext,1000,15]

rhs=rce[[2]];
lhs=rce[[1]];
Tnext[Tprevious_]:=Module[{ser},
ser=Series[rhs,{T,Tprevious,1}]//Normal;
T/.Solve[ser==lhs][[1]]
];