Doubt regarding View factor/shape factor of infinite inclined plate

[harsh_23]

Homework Statement

We need to prove that fraction of total radiation from plate i falling on plate j is equal to 1-sin(α/2) where both plates are infinite long, share a common edge , have same width and are inclined at an angle α

Relevant Equations

in general fab represent fraction of total radiation from a falling on b

so sum of all fai =1

The figure is infinite out and in the page.

We need to find $$f_{ij}$$ means if $$I$$ is the total radiation plate i is emitting then $$f_{ij}$$ represent fraction of total radiation Falling on plate $$j$$ from plate $$i$$

$$So in general fab represent fraction of total radiation from a falling on b$$
$$so \ sum \ of \ all\ fai =1$$
So in the above image I want to prove why $$f_{ij} = 1-sin(α/2)$$

First thing I have observed in others proofs that they consider radiation going to the surrounding through the gap between the plates and they consider the gap as an imaginary surface let's call $$k$$ with width l




Now we can see that side l is simply $$2w×sin(α/2)$$

Also $$f_{ki}=f_{kj}=1/2$$
(by symmetry)(means 1/2 of $$k$$ radiation is going to plate $j$ and other to I)

Then by reciprocity theorem

(It's a general theorem in radiation)

(It states that $$A_a×f_{ab}=A_b×f_{ba}$$ (here $$A$$ represent area))

$$A_i×f_{ik}=A_k×f_{ki}$$ (now area of plates is proportional to width)

$$W×f_{ik}=2wsin(α/2)×1/2$$

$$f_{ik}=sin(α/2)$$

And $$f_{ik}+f_{ij}=1$$

So $$f_{ij}=1-sin(α/2)$$

Now my doubt is that radiation shouldn't only be going through the gap but also to the surrounding

So $$f_{ik}+f_{ij}≠1$$

If I call surrounding s then

$$f_{ik}+f_{ij}+f_{is}=1$$

And should be same for all

And even if I don't consider the radiation to surrounding then considering the gap be bit spherical (concave) then some radiation should go to itself so saying

$$f_{ki}=f_{kj}=1/2$$ should be false as some of the total radiation is going to itself.

So someone pls help Clearing My doubt...

 

[haruspex]

Now my doubt is that radiation shouldn't only be going through the gap but also to the surrounding

I think you are supposed to take all the radiation as coming from the side facing the other plate, so it must either go to the other plate or to the gap.

considering the gap be bit spherical (concave) then some radiation should go to itself

But then you cannot apply the reciprocity.

 

[harsh_23]

So is there a way by which radiation outside to surrounding can be blocked for the question to be valid?

 

[haruspex]

So is there a way by which radiation outside to surrounding can be blocked for the question to be valid?

Sorry, I do not understand your question. The question posed as homework in post #1 is valid.