Cooling a cup of coffee 1

[Restrell]

Hi,

Id like to answer how much time does it take to the coffee to cool to the surrounding air?

I know the surface will cool from convection, but for example, what happens in the middle of the cup?

If someone could advice in how to make a mathematical model for this Id appreciate it.

Thanks

 

[chicopee]

This is what I visualize:
1)Assuming the top to be insulated and the heat transfer rate thru the bottom is much less than along the wall of the cup, then, the flow will ascend in the middlle of the cup and descend along the wall of the cup, basically forming the shape of a toroid.
2)Assuming the heat transfer rate thru the top is equal to or greater than thru the wall, I believe that there will still be ascent thru the center until it reaches a strata below the fluid level where the flow divides mostly radially toward the wall and vertically to the surface where it will flow radially and downward along that level. That flow at the surface will reach the wall but most of the flow will be directed downward in that strata until it joins the radial flow at the bottom of that strata. At the wall the flow will descend to the bottom of the cup.

 

[btrueblood]

What is the conductivity of the water relative to that of air? At any 2-d slice through the cup, what is the relative heat transfer rate (W/m^2) relative to the convective heat transfer rate at the surface? If h > 10*k_coffee then use a lumped-mass approximation for the liquid.

 

[btrueblood]

and, after searching the terms, if h < 10*k_Coffee or thereabouts, then you need to study

http://en.wikipedia.org/wiki/Marangoni_effect

and/or

http://en.wikipedia.org/wiki/Rayleigh–Bénard_convection

 

[racookpe1978]

Theoretically, it will never "equal" the room (air) temperature, but will only approach it. (log function needed there of the difference between cup internal to cup internal wall to cup outside wall to cup-to-room film temp to room temp.)

Further, you put the hot coffee in the room, the room heats up and the cup cools, therefore it will meet somewhere in the middle. But what is a few 0.001 matter?

Best model needs convection from the top as a hot surface below a cylinder wall, evaporation out the top, radiation out on all sides, convection up the walls as a cylinder, and conduction from the bottom to the 'floor" (table ?)

 

[monkeydog]

Hmmm,
About 1 to 2 months into each new semester, the same physics I question comes up - how strange.

 

[corus]

For a 'model' you have to make assumptions. How many assumptions you make will determine how accurate you are. If you ignore 2D effects, convection of the coffee within the cup, any forced convection on the outer surfaces from someone leaving the door open, and a variety of other things, then the simplest model would be a lumped mass model that assumes the temperature within the cup is constant throughout and that the thickness of the cup itself is negligible. Take a guess at the heat transfer coefficient and you'll have a simple model that gives an exponential decay in temperature.

 

[racookpe1978]

Put the coffee cup (with coffee) back in the microwave. 12.5 seconds in an 1100 watt microwave for each 3/32 inch of coffee contraction from the original fill point, if original coffee level within 3/4 inch from the top of a nominal coffee cup of nominal diameter.