Black body radiation problem 
Three concentric thin spherical shells are given their radius are $R$, $2R$ and $3R$. Outer most sphere can't radiate in outer space. The inner most and the outer most shells are maintained at $T_{1} K$ & $T_{2} K$ respectively. Assume the three shells behave as black body. The steady state temperature of the middle shell is $$\left(\frac{T_{1}^{4}}{x}+\frac{T_{2}^{4}}{y}\right)^{1 / 4}$$ Value of $x+y$ is


My first doubt is : can the outermost shell radiate inwards
Second doubt is : where the energy flow (i.e its direction)?

I attempted it as the middle shell temperature is constant therefore power absorbed = power radiated
$$
\sigma 4 \pi R^{2}\left(T_{1}\right)^{4}+\sigma 4 \pi(3 R)^{2}\left(T_{2}\right)^{4}=\sigma 4 \pi(2 R)^{2}(T)^{4}
$$
 A: Yes, outer shell radiates heat inwards & receives the same amount to maintain dynamic thermal equilibrium between any two shells.
Let $T_i$ be the steady state temperature of intermediate shell & $T_1>T_2$
In steady-state condition, the net radiation heat transfer between inner and middle shells i.e. $\sigma A_1(T_1^4-T_i^4)$ will be equal to the net radiation heat transfer between intermediate and outer shells i.e. $\sigma A_i(T_i^4-T_2^4)$ i.e.
$$\sigma 4\pi R^2(T_1^4-T_i^4)=\sigma 4\pi(2R)^2(T_i^4-T_2^4)$$
$$T_i^4=\frac{T_1^4+4T_2^4}{5}$$
$$\implies T_i=\left(\frac{T_1^4}{5}+\frac{T_2^4}{5/4}\right)^{1/4}\equiv \left(\frac{T_1^4}{x}+\frac{T_2^4}{y}\right)^{1/4}$$
By comparison, we get $$x+y=5+\frac54=\color{blue}{\frac{25}{4}}$$
A: Firstly, You haven't mentioned their radii. But from your attempt, it appears that they are R,2R and 3R respectively.
You have made a mistake in the RHS. It should be $2 \sigma 4 \pi (2R)^2 T_2^4$, as the body can radiate through both inner and outer surfaces
