Problem abstract:
You are an avid coffee drinker, you always order coffee when you go out to dinner and you drink your coffee with milk. If you order coffee at the beginning of a meal but want to drink it at the end and still have the coffee be as hot as possible:
Do you
a. add the milk to your coffee as soon as the waiter pours it?
Or
b. add the milk to your coffee only seconds before your first sip at the end of your meal?
Thus your challenge in this exam is:
1. Theoretically Analyze the cool down of a standard ceramic coffee cup left out on an open desk with and without milk.
a. Write out and numerically model the theoretical cooldown rate of a full coffee cup answering the following questions:
b. What are the fundamental modes of heat transfer relevant to this problem - using characteristic values from your text ((you must document and show all calculations), Create a table of heat loss by heat transfer mode and make assumptions about what modes of heat transfer can and should be neglected in your energy model, and what modes are dominant.
c. numerically integrate the characteristic equation after (appropriately linearizing any non-linear terms). Then calculate the individual heat rates of each of following modes(showing all your work):
a. Natural convection off the sides and top of the liquid surface
b. Radiation heat transfer from the sides and top of the cup
c. Conduction through the liquid to the air surrounding the cup
d. Conduction heat rates through the cup into the surface below
e. Evaluate the evaporation cooling rate assuming a stagnant mass of extremely dry air above the cup, be prepared to answer and model how this calculation varies with some relative humidity present in the air above the cup?
2. create a table of order of magnitude heat transfer rates for each of the 4 modes above
3. how do you model the evaporative heat transfer process stating all assumptions?
4. what is the final equation describing the heat transfer rates and the change of temperature of the coffee in the cup?
5. Analytically graph the T(t) for the problem based on your model. Make sure your model includes the capability to add a given amount milk at any time t and then calculate the cooldown that follows. T(t)
6. Now conduct a simple experiment measuring the coffee temperature as a function of time and superimpose it on your theoretical calculations.
7. Redo the above experiment removing the dominant heat transfer mode and document the temperature history change. How do your results prive that this was indeed the dominant heat transfer mode?.
There should be a curve with only one modality active at a time, that is do 1 experiment where conduction and convection are minimized or eliminated and evaporation clearly dominates. Then do 1 cool down where evaporation is minimized and the other modes are present so that you can compare the relative effects of each mode on the overall cool down rate.
Show the experimental cool down curves along with experimental cool down rates clearly labeling the curves.
Project deliverables:
1. Full report with theory,
2. Analysis with numerical solution of differential equations and Comparison to experimental results
4. Conclusions, optimal strategy to have coffee be as warm as possible when you are about to drink it.