How Does Temperature Change Over Time in a Coke Bottle?

[phystudent515]

Homework Statement

This is not really a homework assignment but I think it sounds like it is, so I just put this here regardless.

I want to know the temperature in a coke bottle as a function of time. I can measure the initial temperature of the coke bottle, I know the dimensions of the bottle, the material and the ambient temperature of the room. If there is anything else trivial I could measure or determine that would be relevant I could probably obtain that too. What does the equation look like that will determine this temperature as a function of time?

Homework Equations

Newton's law of cooling?

The Attempt at a Solution

Sorry I don't know.

 

[Simon Bridge]

Temperature of what in a coke bottle? Air? Coke? What? Is the bottle open or closed? How accurate do you need to be?

You'd need to know how convection gets set up in the room around the bottle. In Newtonian cooling you have forced convection: so you'd be blowing air over the bottle.

You'd need to know the thermal conductivity through the walls of the bottle - is it glass (what kind) or PET?

Is the room heated? Is there light falling on the bottle (and room) that could have an effect on the experiment?

This is the sort of thing you normally do empirically.

However, a really simple model would just treat the (closed) bottle like a room - and you work out the heat transfer due to conduction through the sides assuming the outside of the bottle is maintained at a constant temperature. You've probably seen these calculations. The instantanious heat flow would look like Ohm's law, use the specific heat at constant volume to convert heat flow to temperature ... though you can do it in terms of temperature difference as well... the actual change in temperature will depend on the instantanious temperature so you'll end up with an exponential much like Newtons cooling.

 

[Chestermiller]

Homework Statement

This is not really a homework assignment but I think it sounds like it is, so I just put this here regardless.

I want to know the temperature in a coke bottle as a function of time. I can measure the initial temperature of the coke bottle, I know the dimensions of the bottle, the material and the ambient temperature of the room. If there is anything else trivial I could measure or determine that would be relevant I could probably obtain that too. What does the equation look like that will determine this temperature as a function of time?

Homework Equations

Newton's law of cooling?

The Attempt at a Solution

Sorry I don't know.

Simon Bridge gave a pretty good assessment of this problem. I have a few additional comments. Let's assume that there is liquid present within the bottle. This is a transient problem, and there may be temperature variations and convection within the liquid as it heats up. If you assume that there is a stirrer present within the liquid, then at high stirring rates, the liquid can be assumed to be well-mixed, and at a uniform temperature. Neglecting viscous heating from the stirring, this situation would provide an upper bound to the rate at which the liquid heats up (since there would then be zero thermal resistance within the liquid). If the bottle is thin plastic like present day coke bottles, then the main resistance to heat transfer will be on the outside air side, resulting (as Simon Bridge has indicated) from forced and natural convection in the room. Ordinarily, for this case, the heat transfer coefficient on the outside air side will be on the order of 1 to 10 BTU/hr-ft²-deg-F. For old time coke bottles, made of thick glass, the resistance to heat transfer through the glass should be considered. The equivalent heat transfer coefficient would be the glass thermal conductivity divided by the glass thickness. This would have to be compared with the heat transfer coefficient on the outside air side to see which, if either, thermal resistance dominates. In any case, you could evaluate the overall heat transfer coefficient. You would multiply this by the contact area of liquid within the bottle to get the UA factor for Newton's cooling law. Incidentally, in this case you would neglect the thermal inertia of the glass.

The limiting case for slowest cooling would be if the liquid within the bottle experiences no stirring, or even natural convection. Therefore, it could be treated as a rigid solid. The heat transfer within the liquid would then be strictly conduction. You could solve this problem approximately by treating the bottle as a cylinder, and neglecting heat transfer at the ends. Transient solutions for heating and cooling of rigid cylinders with convective heat transfer at the surface can be found in the literature (say Carslaw and Jaeger). You might also want to include the thermal inertial of the glass for this case.

In any case, the two solutions you get by the above simplified calculations would be provide upper- and lower bounds to the actual heating rate. These might be sufficient for your purposes. To be more precise, you would have to solve the problem for the true geometry, including natural convection in the liquid. This would require numerical solution of the differential transient heat balance equation.