I’m probably going to mess up the math but here is a shot… A calorie is defined as the amount of energy it takes to raise one gram of water by one degree Celsius. A food calorie is actually a kilo-calorie, or 1000 “small” calories. There is just under a 1000 grams of water in 1 liter (at everyday temps at 1 ATM) , so a food calorie raises a liter of water by 1 degree Celsius. The average human core temp is 37 degrees Celsius. So suppose your fridge is 3 degrees Celsius. So if you drink a liter of refrigerated water, you have to spend 34 food calories to heat it up to body temp.
So if you eat a 700 calorie Big Mac, just wash it down with 20 or so liters of cold water and you’re good
you need a serious reference if you want to run an experiment like this. First of all if you want to measure changes in energy expenditure that are not due to physical activity (i.e. movement), you won’t go far with activity trackers, that most likely use your anthropometric characteristics to get a resting metabolic rate, and that’s it (for sure they don’t measure the diet induced thermogenesis - DIT -, and no, the fact that bodymedia includes heat flux, temperature, gsr, does not mean that you can measure such differences, they are most likely merely using these signals to detect different levels of physical exertion, allowing them to distinguish for example sedentary activities from intense activities where no arm movement is involved (e.g. biking), which is something other physical activity trackers used by the medical community can’t capture (e.g. the actigraph)).
This being said, other methods are not really accessible and quite expensive (indirect calorimetry or DLW). I know this doesn’t really help, but with your current reference it’s going to be a waste of time. One last point, beware that even other references have limitations in accuracy, and might not be good enough for the order of magnitude of what you would like to detect (e.g. 5% error typically reported for indirect calorimeters such as the Cosmed K4b2, around 10% error for DLW - the latter being a less interesting reference, since it averages everything over days).
Yes, if you will drink the colder water then it will burn the calories. For example you drink a glass of cold water at body temperature. Then after sometime go for urine. If the urine is warm then your calories has burned. Means calories has burned to bring the fallen temperature of water similar to body temperature until urine would be cold.
But you should depend upon ice water to replace exercise or warm water for burning the water.
Not to forget the amount of water.
If you dring more liters your body has also more work to do to get it out of the body again. So your kidney also burn calories!
I think you must also measure the amount of water. Or you have to take the same amount all the time doing your measurements.