I am a pro at become distracted by side projects. In any case it is very similar to the coffee cup model, expect that the temperatures will switch. Each simulation just brings more questions, namely:
What happens if their is condensation of the outside? How much energy does it take to for the condensation? Is the convection more efficient for a film of water compared to silica glass?
Can I add ice cubes?
How does this compare to experimental data?
What are the effects of different geometries? Would the cup that we (Zilla, Nate, plus other) actually be more effective?
Check out the results for a rather simple cup:
Linear Heat Loss - Convection is proportional to the temperature difference
All of the heat flux is through the top of the glass
Turns out through a bit of research COMSOL already published a simulation similar to mine - a cold glass cooling down, complete with a nice animation. I can't quite figure out how they did theirs, but the model I have currently cannot be correct - it simply takes way to long to converge. Below is the initial profile and the profile at 65s; the well developed temperature contours present in the model without fluid dynamics are gone and we have the cooler liquid sinking to the bottom.
What is the effect of having the little cardboard thing around a coffee cup? Do they serve as an applicable amount of insulation, or just keep your hand from being too hot? What about the effects of a cap?
First Model
Heat Transfer in Solid's physics - the water (coffee) was assumed to be a solid; it couldn't move due to density differences. I assumed the cup was antisymmetric, so you only get a slice.
Natural Convection from air on the boundaries. The top boundary was from modeled as a plate above, the bottom of the cup was modeled as a horizontal plate below (these are actually quite different - when the heat comes from the top it can raise away, but on the bottom it cannot), and from a vertical plate on the side.
The fluid was assumed to be water with an initial temperature of 212 F, and the cup was assumed to be paper with an initial temperature of 50F. The ambient temperature was also assumed to be 50 F.
Density, heat capacity, and thermal conductivity values from library materials or the internet.
(Pro tip - creating both sides of the geometry with antisymmetric settings causes terrible convergence to non-physical values.)
Initial Temperature Profile
Temperature Profile of a point Halfway up the r=0 axis of the cup. x-axis is time in seconds, y-axis is temperature in F.
What we observe is that the temperature at the center of the coffee changes 14 F in 30 min, while huge temperature gradients appear in the fluid. Temperature gradients would then effect the density, so some sort of circulation would result.
What's the effect of a 3% change in density? Source Data
Anywhos, using coupled fluid dynamics with heat transfer has already taken 2 hours to simulate 28.9s; so I think I just should have borrowed a thermal camera from Dr. Hiato and done the experiment. To do the effects of the cardboard holder (which this entire shinding is about) I will need to find a better method.
The solver crashed sometime during the night; something about an illegal mathematical expression, so I don't have the correct physics / model because it I get an illegal mathematical expression that probably means I am not converging (I could simply be taking too large of a step size initially). I did get the velocity profile of the first two time steps, 0 and 10s. Initially it looks like that I am getting small currents (10 microns/s) near the top of the cup (these have to circulate, I think that is why the velocity is highest in the middle), and at 10s they are much larger.
The interesting thing from these results is why the velocity current of the top extend all the way to the symmetry axis?
This was from a while ago, but I forgot to post it. It shows the actually image and the processed image, along with the plot of the largest white space area. From here I gathered that my simple threshold is not very accurate.
I think that it is cool though.
Hopefully the next few days will be a little less busy, and I can improve the algorithm.
I got a little ahead of myself this morning, so I thought best to take a moment and explain what I am doing.
For a while now (about a year or so) I have been interested in how milk swirls in coffee. When I was at CMU I used to go to Kiva Han and get a $1.00 refill (if you bring your own cup) just to watch it swirl. When I went home from the summer I played a little bit with using a clear cup so I could better see what goes on.
I originally thought that the process was mostly diffusion and temperature gradient limited since the milk never seemed to come all the way to the top of the mug (milk is more dense than water), but watching it in a clear glass I noticed that most of the milk goes straight to the bottom. (This should have been evident due to the milk being in a stream and initially not pooling).
About a week ago I video taped a series of experiments in which I poured various amounts of food coloring into different containers of water in order to observe the transport profiles. I decided to use a food color model to eliminate the density effects and temperature effects (I assumed that the food coloring has a density similar to water, and if it doesn't the amount I add is too small to have a noticable effect), and to provide a clear view of what is going on.
Trial 1
The first trial I tried was designed to observe the transport profile. I imediatly noticed that the equalibrium was reached too rapidly, and that I was correct that the liquid went straight to the bottom.
Trial 2
In this trial I tried to limit the effects of advection by using less liquid (I estimated that I added less than 10mL of dye) and by pouring off a spoon to limit the momentum of the dye. I was stupid to dip the spoon in (I wanted to make sure I got all the dye off); this set up advection. I tried to limit the distance the dye would fall, but there is still a very noticeable movement of dye toward the bottom after the dye has been added. This could be due to a density difference (currently testing that by mixing dye and water and letting it sit - they should separate out if there is a significant density difference), or simply an unavoidable consequence of dumping liquids together.
The movie is ~10 minutes long, but equilibrium is reached after about 3 minutes.
Trial 4 and 5
I am convinced that in the dye model most of the transport is due to the conservation of momentum from dumping the liquid in. (As an example, consider dropping a bouncy ball. Conservation of momentum makes it bounce back up; but a fluid is much more elastic.) In these trials I wanted to investigate the effects of different boundary conditions (the fluid being constrained in different geometries) and the effect of height. I haven't really thought about what these trials tell me yet expect for 'Damn! My hand really shakes!'.
Coffee Trial
I actually only did one trial with coffee (I only had the camera for a day, and a man can only drink so much coffee). I noticed a curious phenomenological effect - the milk looks like it goes to the top, and then bounces back down. The water currents seem to be more visible in coffee than the water, leading me to doubt the validity of my model.
Further Work
I am working on developing the Matlab code to track the position of the centriod (I will post the profiles and movies of the gray scale sometime tomorrow once I can use the lab computer to run it).
I am using the area of the red dye as an indication of the amount of diffusion. I am thinking about how to normalize the output to concentration - perhaps by normalizing to the initial area of red in the image, or maybe to the area under the curve so I could compare between trials, or both.
Working on developing an analytical model of diffusion, heat transfer, and buoyancy. I want to find the time scale of these processes.
Working on developing an analytical model that solves the Navier-Stokes. I think in order to do that I have to use tensors, and quite honestly I didn't eat enough veggies in Vogel's class and learned nothing from Math Meth / Phys Anal.
The frame is broken up into the red, green, and blue components
r = frame(:,:,1);
g = frame(:,:,2);
b = frame(:,:,3);
% A net frame is computed, which only has the red pixels
net = r - g - b;
% Thresholding the image
bw = net > binaryThresh;
% Creating a component map baseed on area
cc = bwconncomp(bw,4);
stats = regionprops(cc,'Area');
if isempty(stats) || isempty(fieldnames(stats))
fprintf(1,'Empty Stats structure at frame %d.\n',i);
break;
else% Creating an index of all areas greater than the threshold
idx = find([stats.Area]>areaThresh);
bw2 = ismember(labelmatrix(cc),idx);
maxArea(i)= max([stats.Area]);
end
Empty Stats structure at frame 476.
end
plot(maxArea);
xlabel('Frame (t)');
ylabel('Area (conenctration)');
title('Concentration Evolution');
