 Alright, the last problem that we're going to look at using the Excel spreadsheet for solving the heat diffusion equation using the finite difference technique is going to be a problem that involves radiative heat transfer. And what we're going to work on here is we're going to work on a problem involving a bar of copper that is 10 mm by 10 mm and it is sitting on an insulated surface and so there is insulation on the bottom of the bar and then the 10 mm by 10 mm bar will be, there is internal generation in the bar at a rate of 5 x 10 to the 5 watts per meter cubed. The copper bar thermal conductivity is 401 and there is radiative and convective heat transfer from the three upper exposed surfaces of the square bar. The environmental conditions that this bar is sitting in, the free stream ambient temperature 25 degrees C as is the surroundings and so converting that to kelvin that is 298 kelvin. Convective heat transfer coefficient, we have a mix between force to natural about 20 watts per square meter kelvin. Emissivity, we'll assume this copper bar has an emissivity of 0.75 so maybe it is a little tarnished. And the grid spacing that we're going to use is 1 mm for both delta x and delta y and with that the 10 mm by 10 mm bar so in the horizontal direction 10 mm divided by the 1 mm grid spacing gives us 10 cells plus 1 so 11 cells in the horizontal and for the vertical again 10 mm divided by the 1 mm grid spacing plus 1 gives us 11 cells in the vertical direction. So let's go ahead and set that up and then we'll see how it operates when we use Excel. So I'm going to begin by clicking in a cell in a corner here and I'm going to color it just like we've done with all the others and we said that there were 11 in the horizontal so there we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. I'll color code those and you know what I'm going to do? Let me set the width of all of them to the same so we'll do something like that. There we go and then 11 in the vertical and I will color code that and then we fill it in here and again enter in the values starting initial condition for our grid just like we did with the other examples. Alright and if you want to put your boundary conditions on here I'm just going to put the lower one that was an insulated boundary so we'll make that a gray. Now let's do pink. Pink for fiberglass pink. Okay so there we have our bar. It doesn't look square but it is because we're going to set delta x and delta y equal to 1 millimeter and let's go in and start entering in the values. We said that the internal generation was 5 times 10 to the 5 so 5e05, k was 401 and delta x and delta y was 1 millimeter so 0.001. Now one thing that we should do let's go under options here and depending on the version of Excel that you're using if you go under formulas sometimes it's under calculations. There's so many different versions of Excel but enable iterative calculation you want to have that checked and here you can specify the number of iterations. I'm going to specify it to be 5000 or the maximum change to be 0.0001 so that's a very very small change. You can play with this and that will set when you push F9 it will either get to 5000 iterations or this being the maximum change. So we'll set that for this particular calculation. Other boundary conditions that we have let's go find the insulated and then the radiative and convective. So scrolling down that's a convective boundary insulated there we go. Okay so we need to set 401 delta x 0.00 what did I say 1 millimeter so that is 0.001 and q dot was 5e05. Okay so we've done the insulated boundary now we have to find the radiative heat transfer with convection, constant heat flux, composite solids, another composite solid, radiation and convection. Okay this is the one that we want. Emissivity we said was 0.75 so we enter that. Surrounding temperature be careful here I mix units I have degrees Celsius and Kelvin so this one is 298k. Stefan Boltzmann constant oh boy I deleted that let me pull it from down below it's right here I deleted that earlier that should be in your spreadsheet there we go. H was the convective heat transfer coefficient 20 watts per meter squared Kelvin thermal conductivity we said copper pure copper 401.001 for the delta x delta y 1 millimeter and q dot was 5 to the 0.5 and heat convection now this is in degrees C it was 25 degrees C. Now one thing when when you're dealing with the excel spreadsheet and radiation you'll notice these are still showing divide by zero and what we need to do is find the boundary conditions that we're going to use so we're going to use this one because if I go to our object that would be the boundaries over here so we're certainly using a right boundary with convection and radiation so I'll scroll back down to that okay here we are so what you need to do once you've entered these values you need to go into the cell go up into the formula bar click there and push enter and then it converts it to a number if you try copying and pasting this into your spreadsheet into your grid it's going to mess it up I don't know why excel does that don't ask me that's just the way that it works and probably some of you out there know why send me an email if you know why and I'll fix it in the spreadsheet and now top boundary condition there we go bottom we don't have a bell item we do have an upper right hand corner so I'm going to click there go up into the formula bar hit enter how we do have an upper left so I'll click there up in the formula bar and the right we don't have a lower right we don't have a lower left so I think I've done all of the ones that we need so what I'm now going to do I'm going to copy and paste in the boundary conditions into our grid and we'll begin with the insulated boundary and so what I will do we had a lower surface so these are the middle ones so I'll copy going up to our grid there we go so I'm going to paste into those and then what I'm going to do for the corners let's go back to the insulated boundary insulation I'm going to pick this corner and this corner so this will be the lower the bottom right so I copy I apologize for the bell but that's the quickest way for me to get around excel then the other one down here I'll copy and then I'm going to paste into that one and if you want to see what cells are being used just like I said before go up in the formula bar and you can see that it's using those cells there these ones in the middle they're just using the ones around there okay so now what we need to do we need to handle the radiative boundaries I'm going to begin with the corners so let's start with the upper left and we'll scroll all the way down to where we had the radiation where was radiation oops I'm gone too far down radiation is here so let's get the upper left so I'm going to click there control C I'll go up so that's the upper left that we're dealing with and it's going to complain here and the reason is because it's scientific so change that to number everything is good and then the other one let's get the upper right where's upper right there it is back up to our grid here we go again it's playing the game and by the fact that it's giving us a scientific I guess I must have set that cell scientific when I set up the excel spreadsheet anyways you can change it now what we need to do we need to do the left the right and the upper so let's go get those cells let's start with the left finding our radiative condition with convection there's the left click there control C go back up this is kind of like playing Minecraft okay let's see I don't play Minecraft I watch people play Minecraft and I'm puzzled by it but anyways okay the right surface control C I apologize if you're a gamer and you feel offended by that comment I used to spend a lot of time playing video games when I was young I guess I got it out of my system okay there we go that's the last one radiation and convective boundary upper we click there control C going back up and now what I'm going to do is I'm going to copy and paste control V and if you're wondering again are those cells correct are they pulling in the right conditions yeah that looks good clicking there yeah it's pulling in from the inside that one looks good these ones here yeah they're pulling in from there that's pulling from the inside everything looks good so the last thing that we need to do here let's check the corners I think I've already done these but I'll do them again corners look good okay the last thing we need to do we need to copy the interior nodes so we've done the boundary conditions the interior node was with generation we have it right here so I do control C I do control V paste it in there we go let excel run and what you're noticing down in the bottom here you see the number of iterations it's kind of slow and the reason is because we're watching it if you scroll down and don't watch it and we'll do that in the next run you'll notice that it moves much much quicker so let this one do its thing it's trying to converge and and so what it's doing is it's going through a calculation after calculation applying the finite difference technique and iterative convergence process is what we're essentially doing here so I'm going to hide it I'm going to push f9 now watch down here the number of iterations much much quicker so that's the way to move the simulation a lot quicker than watching the numbers and it's just because it takes time to generate those numbers and display them and if you're not displaying them the computer is much quicker and so what we do we keep pushing this let's take a peek oh look at that we're already at 67 degrees so it's getting warmer that's good I know what the answer is but I know where we should be going average oh that's kind of neat it's showing us the average there so we know what the temperature is average in our grid as we can see it's still changing and you'll notice that it's starting to converge when when you push f9 and it no longer does the iterations so we're still going and now you can see it's changing very little each time I push f9 so we've exceeded the thresholds that we set so that means we're pretty close to having a converged solution let's go take a peek there we go that's what it looks like so this is a copper bar 10 millimeters by 10 millimeters with internal generation radiative heat transfer around the perimeter insulated bottom so let's select it coming up insert we're going to do a contour plot there we go let's position the contour plot down below the color yeah that's probably as good as we're going to get we can try other colors if we want yeah that one's not bad we can see the difference now the thing about excel I talked about this earlier with the contour plot it does funny things so go under layout axis depth axis and show reverse and that is what we're looking at that's the actual result of our simulation so that is an example of using the excel spreadsheet for solving a problem with radiative heat transfer the most important thing to remember when you enter the radiation value go into the cell and click and enter up in the formula bar if you recall what that was where were we right here so for example if we wanted to use this bottom you'd have to go into this cell here go up into the formula bar click enter and it populates it if you try copying and pasting this into your spreadsheet it's going to mess you up and it won't work so that is the excel spreadsheet and that is the finite difference method it's kind of a neat little tool for quick and dirty calculations not always the most convenient but it works and it's quick so that gives you an introduction to finite difference and heat transfer calculations