[CFD] How does the Surface-to-Surface (S2S) Radiation Model Work?

hello everyone this is Aiden for fluid mechanics 101 and today I'm going to be talking to you about how does the surface to surface radiation model work

so just to give you an overview of what I'm going to talk about today I'm gonna start by talking about when do I need to account for radiation in my CFD models

then going to look at what are view factors and how they calculated then I'm gonna move on to look at the surface to surface radiation model specifically and

I'm gonna finish off by talking a little bit about the radiosity vector because that's something you may see mentioned in your CFD manual or in the user guide

so to start off with some motivation for this talk when do I need to account for radiation in my CFD model well the short answer is you need to account for radiation when the temperature

differences are large and what do we mean by large temperature differences well if your simulation is going to be around room temperature for example and

the temperature differences are very small say 5 10 15 degrees then radiation is probably not going to be important as the dominant heat transfer mechanisms

are going to be conduction and convection so you can fairly accurately neglect radiation in those models and what happens if the temperature

differences are much larger so what if we have a temperature difference of 100 degrees 200 degrees or even more well we may remember back from high school that

the radiative heat flux Q is given by equation 1 so Q the radiative heat flux is given by the product of Sigma the stefan-boltzmann constant the area of

the surface and the temperature to the fourth power so what that really tells us is the radiative heat flux scales with the fourth power of temperature or the fourth power of the temperature

difference so as temperature differences get large all of a sudden radiative heat transfer becomes extremely dominant and we do need to consider it

simulations and just a quick point before we move on here the notation I'm going to use I'm going to use capital Q for the radiative heat flux given in

watts and I'm going to use lowercase Q for the radiative heat puffs per unit area so given an equation - and that's what per meter squared so just take a

note and remember that motivation the notation as we carry on through the slides so another point to quickly address before we move on conduction and

convection our transport phenomena that usually occur through the fluid volume whereas radiation is a surface phenomenon that occurs on the surface of

the solid bodies that we're solving for so if we have a complex domain normally with the CFD framework we construct and

mesh throughout the fluid volume and use that to consult to solve for the conductive and convective heat transfer but with radiation radiation is only

occurring off the surface of our bodies as we usually find that the medium when that B water another fluid or a gas doesn't tend to participate in the

radiative phenomena so radiation is only occurring off the surface and in our usual finite volume CFD framework we don't have an easy way to directly

account for radiation because we solve for all of the cell volumes throughout the bottle throughout food volume and for this region we're gonna have to use some type of special treatment to

account for radiation and a special radiation model and in today's talk I'm going to talk to you specifically about the surface to surface radiation model

which is one of the most popular radiation models you can use so just a quick reminder of how we normally solve an account for conduction and convection

in our fluid volume well if we consider a cell that's on the wall of a solid body given in the figure below we know

that our cell centroid is a YP away from the wall and our wall is at some fixed temperature T sub W and we

know the temperature at the cell centroid T P and we calculate that temperature by solving an energy conservation equation and what we find

is that the heat flux off the wall into the fluid volume is given by equation three and that says that the heat flux is given by the product of the density

specific heat capacity and the thermal diffusivity alpha multiplied by the temperature difference T sub W minus the

temperature at the cell centroid T P divided by that distance from the wall so what we're doing there is we're really using a form of furriers law with

a linear temperature difference between that cell centroid and the wall and in a previous video I went through how we

account for the heat flux from the wall if our temperature profile is nonlinear if we perhaps have a turbulent temperature profile between the wall and

the wall adjacent cell centroid and there's a link to that video in the description below this one if you'd like to go back and have a recap for how we account for a nonlinear profile but to

summarize we normally just modify the thermal diffusivity alpha to account for the nonlinear profile and that's how we normally account for the conduction from

the wall into the fluids that's how we normally do it and then we solve our energy conservation equation to get the temperature in the entire fluid domain but what we're looking at today is what

happens if we have a radiation as well and the short story as radiation is an additional heat flux from the surface of the wall into the volume that's to solve there

and we're going to be looking at the surface to surface radiation model which we use to calculate that heat flux Q sub rad and then once we've done that caplet

heat flux put it into a coat of three and then we solve our energy equation for the temperature throughout the entire fluid volume accounting for conduction and convection in the volume

and if you need to remind you or a recap video of how we do that I've also linked the another video for how we solve the energy conservation equation in the link in the description for this video so you

can always go back and have a look at that one if you need to recap so today we're just going to focus on that radiative heat flux there and there again that's just a recap today we're

going to be focusing on our equation for the radiative heat flux Q sub rad and then we put that into the energy equation use that to compute the

temperature field in the volume so to start things off we know that for a blackbody radiator the radius of heat flux is given by equation four and

that's Q is equal to the product of the stefan-boltzmann constant the area of the surface and the temperature to the fourth power and just to remind of you

of course a black body is a perfect emitter of radiation and for our CFD models our surfaces are often not black

bodies they're made up of materials such as perhaps Steel's aluminium or other regular materials that we see maybe

paint and for these surfaces the bodies are called gray bodies they're not perfect emitters of radiation and how we account for gray body radiation is we

multiply that black body radiation by an emissivity given by epsilon and as you can see in equation 5 epsilon is somewhere between 0 and 1 and if epsilon

is 1 then we have a black body and for our grade bodies epsilon is going to be somewhere between 0 one and it's just worth pointing out here that for the surface to surface radiation model we assume that epsilon

doesn't depend on the wavelength we assume it's a constant and it's a property of the surface that we define beforehand so equation 5 is how we can

quite easily account for the radiation given off by our hot surfaces but what we also need to consider is what happens when radiation that's given off by that

surface is there an incident on another surface so quite often in our CFD domain we're going to have a variety of services and they're all going to be emitting radiation and have a radiation

incident upon them so what do we do to account for radiation that's incident on a surface well if we look at the diagram you can see down on the slide we're

going to consider a single cell k and that cell is going to have some radiation incident upon it and we're going to say that incident radiation is

given by one and then of all of that radiation that's incident on the surface some fraction is going to be absorbed by the cell that's alpha K some fraction

will be transmitted and will pass through the cell boundary and then some fraction will be reflected by the cell and we give that the notation Rho sub K

and it's also worth pointing out here that for the surface to surface radiation model we assume that the incident radiation is diffuse and that

means it doesn't matter what angle that incident radiation hits the boundary cell face out we treat the radiation all

the same and equation six there is basically an expression of conservation of energy so of all the radiation that's incident on the cell some fraction is

absorbed some fraction transmits through the cell and some fraction is reflected so if the next assumption that we use in the surface to surface radiation model

is that all of our surfaces in our domain are opaque and that means that they don't allow any radiation to pass through the cell at all so all of the radiation is either observe

or reflectors none of it is allowed to pass through the cell and what does that mean that means that we take tau K to be zero in equation seven the next thing we

do is we use Kirchhoff's law which states that the absorber T alpha is equal to the emissivity and what that really tells us of course is there if

you remember back from from physics a body a black body which is a perfect emitter of radiation is also a perfect for desorber of radiation that's really what kirchoff's law is telling us so

alpha is equal to Epsilon and we substitute that into equation seven and what that gives us is equation eight which you can see there and says that you map the fraction of radiation that's

reflected by the cell it's going to be equal to one minus the amount emitted by the cell and what does that really mean of course if we have our cell that's a

perfect emitter then it's also going to be a perfect absorber so none of the radiation will be reflected at all whereas if we have the other case where our cell doesn't it miss any radiation then it will reflect all of the

radiation that's incident upon it and for all of the cases in between we use equation eight to get how much radiation is reflected by the cell so now we've

put together all of the simple tools that we need to come up with a radiation model for our entire domain we thought about how much radiation is emitted that's given by the blackbody radiation

and we've also thought about how much what happens to radiation when it's incident upon a cell is absorbed or as it reflected so now we can come up with

a radiative energy balance for a boundary cell face that's a cell on the face of a solid boundary in our domain and if we can write a balance for one

cell we can repeat that balance for all of the cells in the mesh and use that to come up with an equation and a model which we can solve for the radiative

energy so now what we're going to do is if we consider our boundary cell phase K we're going to use the subscript K to denote the cell that we're interested in

and if we think about the total radiation that's given out from the cell so equation nine tells us that the total radiative heat

flux per unit area that's given off by the cell is going to be the sum of the radiation emitted by the cell due to its temperature and some fraction

of the radiation that's incident on the cell that is reflected away from it and of course the emission is given by epsilon times the stefan-boltzmann

constant times the fourth power of the temperature of that cell and then the reflected energy is given by the reflectivity multiplied by the radiation

that's incident upon that cell and that's our radiative heat balance for our cell but before we can proceed any

further it's worth noting a few points the first point is that we know what the temperature of that cell is we know what TK is because we've solved that using

the energy equation in a previous iteration of our CFD solution we know what the temperature is but at this point we don't know what the reflectivity of the cell is that's where

okay and we also don't know how much radiation is incident on that cell yet so we need to do a bit of work on equation 9 before we can put it into a form that we can solve for all the cells

in the mesh and that's what we're going to do here the first thing we can do is we can substitute in that relation that we derived for the reflectivity of the

cell so Rho is equal to one minus epsilon and we can substitute that into our radiative heat balance and that's going to give us equation 11 so now we

have the total heat flux per unit area out of the cell is given by the emissive

power plus one minus the emissivity times the radiative heat flux per unit area that's incident on the cell so we're making progress now we're about

halfway through deriving a full radiative heat balance for our cell and the one thing we have left to try and work out is that QK incident on the cell

we need to know how much radiation is incident on that cell and the point to remember here is that the radiation that's incident on our cell K

comes from all of the other cells in the mesh so even though we're considering cell K by itself to get the total radiation incident on the cell we now

need to work out how much radiative heat flux is coming from all of the other cells in the mesh so all of the cells are going to be linked in some way and that's what we're going to look at now

so the way we do this linking the cells together in the mesh is we need to consider a new factor and you'll often see view factors talked about in your

CFD manual and in health guides online so what is a view factor usually we use the notation F with a subscript with two

indices for subscripts so here we'll use F with J and K as subscripts and what the view factor f means is it's the

fraction of radiative energy that's leaving boundary phase J and then arriving at boundary phase k and it's very important to remember the order of

the incident the indices here so f JK means the fraction of radiation leaving cell J that's then arriving at face k so sometimes you may see the form given in

equation 13 where there's an arrow going from J to K to remind you that the radiation here is leaving J and it's arriving at face K and of course this is

a fraction so it's going to be somewhere between 0 and 1 and what I've got for you there in the diagram is to remind you that if we have cell J there at the

top which is another cell in the mesh we haven't considered and then we've got cell K at the bottom which is our cell that we're interested in and the radio the radiation that's given off cell J

some of it will go to cell K and some of it will go to other cells in the mesh and f JK gives us the fraction of that radiation that's given off cell J that's

then incident upon cell k and that's what we want to know here we want to know what the view factor is so a few more points before moving on

with the view factors of course if we think of energy conservation then of all that radiation that's given off cell J some of it may go to sell one to sell

two to sell three to sell k the sum of all that radiation must be conserved and that tells us that the sum of the few factors or the sum of the fractions must

equal one so if we look at equation 14 then if we take F JK that's the view factor of the radiation

leaving cell J incident on cell k if we sum over cell K is that summing over all the cells that that radiation is incident upon then the sum of the

fractions must equal one and that's one of our view factor identities and it's purely a reflection of conservation of energy and that's a useful identity or useful fraction is often used later on

when you're manipulating view factors as we'll see but this is really where it comes from another small point with view

factors to remember is that if a view factor if a cell can't see another cell because radiative rays travel in two straight lines if the cell can't see

another cell then the view factor is zero its view it can't see the other cell and I've given you a small diagram there just to help illustrate this remember that the radiation can't pass

through the cell it doesn't transmit through the cell so I'll sell J there can't see the other two cells that are on the boundary so the view factor for

these cells would be zero and of course remembering that still if we sum up all the view factors for cell J they will still equal one so that radiation that's being emitted off cell J will have to

reach some other cell somewhere but it can't reach our two cells that are on the boundary there so how do we calculate view factors well it turns out

the calculating view factors requires a fairly complicated integral which you can see there in equation 15 and what it basically says is the view factor

f JK again the fraction of radiation leaving cell J that's incident on cell K is the integral over the area of cell J

and the integral over the area of cell K of two cosine Thetas over PI R squared times by the Kronecker Delta there and

at first glance you might think well what does this integral really mean and where does it come from it looks fairly complicated but where does it come from

well the theatres there give the angle of the cell normals to a ray that passes between cell J and cell K so in that

diagram there you can see that there's a ray of radiation leaving cell J that passes to cell K and the unit normals of these cells make some angle theta to

that ray and of course if the angle 0 the radiations passing directly to the cell so the cosine theta 0 whereas if the cells are at 90 degrees to each other that cosine is going to be equal

to 0 and the integral and the PI R squared basically originate from we're assuming that we place a row of point sources all

the way along cell J and each of these point sources moves out in a spherical direction and that's where we get the PI R squared from and we sum up over all

these point sources over both areas and that gives us the D factor now the Kronecker Delta there which you see comes in and that will be given a value

of 1 if the cells can see each other or a value of 0 if the cells can't see each other and of course the cells might not be able to see each other if there's another cell or another object in the

way and still you can see from this view factor integral that it's fairly complicated to compute particularly numerically if you've got a lot of cells in the mesh their orientations are very

different and you have to evaluate can the cells see each other that's going to be a fairly tricky thing to be numerically and it's not something that I'm going to go into detail for here however there are lots of papers and

lots of resources you can look at in the literature if you're interested in how that you factors calculated but for now it's sufficient to say that the CFD solver in the CFD code we usually compute all of

the view factors for you before you start your simulation and of course it's also worth pointing out that we have to evaluate the new factor for every cell

in the mesh incident on every other cell in the mesh so if we have 10,000 boundary cell phases then we have to

compute 10,000 squared view factors which is a fairly intensive computation even for modern day computers so few factors are fairly complicated to

calculate but you should have a fairly good idea now of what they are and what they do so now what we want to do is move back to our radiative energy balance again and look at how we're

going to carry on doing that now that we've got an idea of what the view factors are so if we go back to equation 16 and remind ourselves remember we're

looking at the total rate of heat flux out of the cell and that's given by that's epsilon Sigma T to the 4 which is

the radiative energy given off the cell due to its temperature plus some fraction of the radiation that's incident on the cell that's reflected

off the cell and what radiation is incident upon the cell now if we look at equation 17 we can start off by saying

the total radiative heat flux that's incident on the cell is given by the summation over now all of the other cells in the mesh that's all of the

cells J so remember here we're looking at cell K that's the cell were interested in and all of the other cells in the mesh are going to be given by the subscript J so this time we're summing

over J and what we do is we multiply by the amount of radiation given off those cells that's Q J multiplied by their

view factor f JK so the fraction of radiation from all of the other cells J that's incident on cell K and we sum over all of those cells and that will

give us the total radiation in on our cell K which is what we want what we're going to do now is do a bit of simplification and to do that we're

going to split the radiative heat flux into the product of the cell area as a and its radiative heat flux per unit

area that's lowercase Q and we've done that for both sides of the equation in equation 17 so what do we want to do now we want to cancel out these areas

because if we remember back to our CFD solver we introduced the areas later on so in our radiative balance we don't want to have any areas involved at all

in solving and setting up our radiative equation so if we start with equation 18 on the left hand side again we have the area multiplied by the radiative heat

flux in area and then on the right hand side the summation of the radiation given off all of the other cells J in the mesh and what we're going to do to

do the simplification is use the reciprocity relationship which is given in that boxed equation in equation 19 and if you're interested in deriving

this relationship you can get it from the view factor integral that we looked at earlier by multiplying it by the respective areas and changing the

indices there and you'll arrive at the reciprocity relationship there what it allows us to do is because the view

factor for cell J onto K so f JK multiplied by its area AJ is equivalent to the opposite view factor that's the

view factor from k2 J multiplied by its area we can go to equation 18 there and on the right hand side substitute the

area sub J times by that view factor with the opposite area and view factor given there in equation 19 and when we

do this we see that we have a K on both sides of the equation and they cancel out and that gives us equation 20 there so QK that's the radiation incident on

our cell that we want is equal to the summation over J J being all the other cells in the mesh of the view factor

swapped around from k2 J multiplied by the radiation given off those cells this is fantastic now that we've got equation 20 we can substitute that back into our

radiative balance and make some progress and this is what will happen in equation 21 we take that radiative heat flux per unit area on the right hand side which

has given the under brace there and we substitute in the summation over the view factors and the radiative heat flux out of those cells and that gives us

equation 22 which is our final radiative balance equation that we were looking for and what equation 22 tells us is

that the radiative heat flux given off cell K is going to be equal to the emissive power from that cell so epsilon

Sigma times its temperature to the 4 plus the reflectivity of that cell multiplied by the summation of the

radiative heat flux given off all of the other cells in the mesh so all of the cells are now linked together in one equation and the unknown that we have

here is the radius of heat flux given off all of the cells in the mesh and these are all linked together so if you're a little bit more astute you may notice now that what we've actually got

here is we've got a matrix equation and that's what I'm going to show you on the next slide so in order for you to help see where this matrix equation comes

from if you take the summation that was on the right hand side there and bring it onto the left hand side we can now see that on the left hand side we have for our cell K we've got the heat flux

out of the cell and then we've got minus the reflectivity times by the summation of the heat flux over all the other cells in the mesh and then that gives us on our right-hand

side the temperature of our cell K to the fourth power and in order to see that this is a matrix equation we think that K is going to be the cell we're

interested in so that's the diagonal elements and then the off diagonal elements are given by the radiative heat flux that's emitted by the other cells

that's incident on our cell K and that gives us the matrix equation which you can see there in equation 24 we've got our matrix of coefficients on the left-hand side which are ones on the

diagonals and then the product of the emissivity and the view factors given there on the left-hand side and then our

unknowns our unknown vector is a vector of heat fluxes given off our cells which are unknown and then on the right-hand side we have our unknown emissive power

which is proportional to the temperature of that cell to the power 4 and in simplified matrix form you may see it

written as you can see there in equation 25 K being the matrix of coefficients J being the radiosity vector that's our

unknowns our vector of heat fluxes and the right-hand side being the emissive power E and before carrying on it's worth noting that that matrix of

coefficients K on the left hand side we have all of that information we know the emissivity of all of the surfaces in our mesh we've also calculated the view

factors before starting our CFD simulation so this is why when you load up your CFD code and you turn on a radiation model often the CFD solar will

calculate all of the few factors for you then and there before you start the computation and this is so that you can assemble that matrix of coefficients there on the left hand side and we also

know what the right hand side is because we know what the temperature is of all of the surfaces in the mesh and we got that from solving our energy conservation equation for

duction and convection in our previous iteration so we know what that is and what we do is we solve that matrix equation for our radiosity vector J and

that gives us our radiative heat flux which we can then apply as a boundary condition for our conduction convection CFD equation then we move to the next

iteration we calculate the temperature field and we use that temperature field as a forcing term back on our radiosity equation there in equation 24 and we

update and calculate the new radiative heat flux vectors so what you can probably see here from looking at these equations is that the radiation equation

is solved alongside the conduction convection energy conservation equation that we normally solve in CFD they relieve their coupled and they're

solved during each iteration of our overall CFD solver and just to reminder for you that after you've solved that

radiation matrix equation to get the vector J the radiosity vector then each of those radiative heat flux is given by

the Q's is the radiation that's emitted off the surface of each of our cells in our mesh and we apply that and couple

that with our conduction and convection matrix equation using equation 28 which is the boundary condition that we apply on the surface of all of our cells and

the radiation is an additional heat flux off the surface there it's just a few points before we wrap up and it's worth

noting that that radiosity equation is a dense matrix equation so we have nonzero coefficients for all of the terms in the

matrix and that's because a single cell in the mesh the radiation that's incident on that boundary cell comes from all of the other cells in the mesh and this is different to when we solve

for conduction and convection because the conduction and convection only depends on the cells that neighbor that individual cell so the CFD equations are sparse and easy

to solve whereas the radiosity equation is a dense matrix equation and is a lot more expensive to solve so worth noting as well but the view factor integral is expensive to compute and quite

challenging and for this reason you may see a variety of algorithms used by different CFD solvers to make view factors and the radiosity equation

easier to solve and one such example is a technique called clustering which you can always go ahead and read more about if you're interested to see how CFD

solvers do that so just to wrap up here if you're interested in a bit more further reading a very useful textbook is the fundamentals of heat and mass transfer by n cropper oh and I've put a

link in the description to this video for where you can find an online PDF of that textbook if you don't have it yourself and it's a very detailed chapter on radiation and radiation modeling which you can use and look in a

bit more detail if you're interested and there's also of course the CFD user guide for whichever CFD code you use and of course the answers fluent user manual

is very good and very useful for that and it's section thirteen point three point seven is the section that you want if you want to look at that and again with that one I've linked that in the

description below for this video and that just about wraps up the video if you have any comments questions or suggestions as always I'd love to hear from you in the comments section below

and I'll endeavor to get back to you as quick as I can with those and in the meanwhile if you want to recap on the energy conservation equation the conduction and convection equation I've got a link to the video that I've done

that in the description below and also another video for thermal or temperature wall functions and how we account for nonlinear temperature profiles at the wall and that's also

linked in the description below until next time thank you very much for listening and I look forward to hearing from you