MadSci Network: Physics
Query:

Re: Three cups of tea, same heat. 1 blown on, 1 stirred, 1 left alone.

Date: Tue Jan 22 10:28:34 2002
Posted By: Tom Cull, Staff Scientist MR Division, Philips Medical Systems
Area of science: Physics
ID: 1009481232.Ph
Message:

This is good question for dealing with differential equations and heat transfer

This is good question for dealing with differential equations and heat transfer.  I had a similar question on an exam involving adding creamer early or late to a cup of coffee. Once upon a time, I was a teaching assistant for a physics lab in which the students dropped an unknown metal into a cup of water.  They measured the change in temperature and then tried to figure out which metal they had used from the value of heat capacity that they had calculated.

 

A few terms and assumptions to agree on first will help the discussion. 

 

The physics term thermal bath refers to the environment that is big enough that adding or losing heat does not change its temperature at all.  In this case, the thermal bath is the air in the room.  The heat lost by the tea is not going to raise the temperature of the room.

 

Heat or energy is transferred in three ways: radiation, convection, and conduction. 

 

Radiation is the transfer of energy by photons.  It can happen in a vacuum.  No physical contact on any sort is necessary.  Radiation always happens and cannot really be stopped.

 

Convection involves some medium to move excited particles from one body to another.  For example, air currents come in contact with the surface of the tea and are swept away.  The air currents are now warm or contain more thermal energy that was received from the tea.  Convection is the method used for air conditioning or heating.   Air is warmed or cooled, is forced or blown into a room.  The forced air mixes with the air in the room.  Somewhere there is a vent for the mixed air to leave the room.   This process is continuous as the forced air is blowing.   The temperature will reduce or rise over time as more cool or warm air mixes with the air in the room.

 

You can argue that convection really cannot happen alone.  Radiation or conduction is required to transfer heat from one body to the other and then convection is the currents of the gas or liquid that moves the heat away.

 

Conduction is heat transfer by direct contact.  The molecules of a solid are in direct contact to each other through chemical bonds.  And whenever the temperature is above absolute zero, there is a whole lot of shaking going on.  Physicists quantify these vibrations as bundles or quantum of energy or heat called phonons.

 

In general, radiation is the poorest method for heat transfer, so most insulation design involves reducing convection and conduction.  Conduction in general is the best heat transfer method, so insolating bodies is a good design strategy.  And, convection depends on all sorts of parameters, but can be reduced by “dead spaces.”  For example, a thermos bottle keeps liquids inside hot or cold by using a vacuum.   Inside a thermos is empty space that has been vacuum pumped to reduce convection between the inside and outside walls.  And since the inside and outside walls do not touch in many places there is very little conduction.

 

Let’s look at the three parts of the tea problem. 

 

The tea left alone: 

 

It will cool down following Newton’s Law of Cooling.   Newton’s Law of Cooling is valid when the temperature difference between the object and the thermal bath is not ridiculous.   Newton’s Law of Cooling says that the change in the difference in temperature is dependent on the difference in temperature.  So as the object cools down closer to the temperature of the thermal bath the change in temperature slows down.

 

Newton’s Law of Cooling can be expressed as:

 

DT = DT0 * exp (-k * time),

 

where DT is the temperature difference between the tea and the room, DT0 is the initial temperature difference at time = 0, k is the rate of transfer in (1 /seconds), and time is the elapsed time.  The rate constant k depends on the material and surface area exposed.  Typically k is an empirically measured value for the situation at hand.   When I ran the physics lab the students measured this property as well.  In general it took several minutes for hot water to drop a couple degrees centigrade.   The data fit very well to a graph of this equation.   The tea in the cup will cool down most at the top and probably at the edges of cup.  The center and lower tea will only cool down as its heat is conducted to the surfaces.

 

The tea stirred:

 

The tea stirred will also work similar to the tea left alone.  However, it should cool down faster because the effective surface area of contact will be larger.   By stirring the drink, hot tea below the surface is brought to the surfaces.   The stirring adds convection to the problem making cooling more effective.

 

The tea blown upon:

 

This is similar to the case of stirred tea.  Blowing on the tea also induces convection currents in the tea.   And depending on the speed and heat of the breath this can be more or less efficient that stirring.  If the breath is fast it will probably induce more convection and a bigger temperature difference between the surface of the tea and the blown air.  However, if the breath is warm because the air was held in the mouth for a while it may be less efficient.

 

So what is the answer?

 

Well certainly the tea left alone will take the longest to cool down.   But with the stirred tea and the blown on tea, it depends on the efficacy of stirring and blowing to induce convection.

 

 

Sincerely,

 

Tom “Reggie Jackson” Cull

 

When Reggie Jackson joined the New York Yankees in 1977, he was quoted as, “I am the straw that stirs the drink.”

 


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