From: xie@phoenix.Princeton.EDU (Yan-bo Xie)
Subject: Re: any equations that can help determine ice melting rate?
Originator: news@nimaster
Organization: Princeton University
Date: Thu, 20 Aug 1992 20:54:38 GMT

From article <1992Aug20.102335.20303@hemlock.cray.com>, by horan@hemlock05.cray.com (Rick Horan):
> 
> 
> hi,
> 
> i am looking for some type of equation that help me
> determine the rate of ice melting. for example,
> 
> how long would it take 100 lbs of ice to melt if it's size is
> square and at room temp of 78 degrees F. 
> 
> i am not a physicist,
> i am merely a computer geek. a friend of mine called and asked
> if i could find this out for a radio contest to give away a car (or something,
> i don't remember what the prize is) to whoever can guess how long
> it would take to melt a 300 lb ice cube.
>  
> 
> thanx for any info you can give.
> -- 
> ==============================================================================
> Rick Horan
> horan@hemlock.cray.com
> 612-683-5357

It may be not easy to guess.  first of all, if there is wind, the
melting rate may be greater than that when windless.

But when the condition is specified, one can still try hard to
obtain a number from the microscopic theory.

The physics assumptions I am going to make are the following:

1) For given Temperature, the air obeys the Boltzmann distribution.

2) When a particular air molecular hit the surface of the ice,
its emitting distributistribution (constrained by the boundary condition)
is given by the Boltzmann distribution function at T=0C.

3) Therefore one can calculate how much energy does that air
molecular give away to the ice.

4) Know the latent heat for the ice-water-(water air).  In
any general physics books.

5) Now the most difficult part of the problem is to determine
what the effective temperature T is for the air moleculars
which hit on the ice surface.  This is very sensitively dependent of
the condition of the ice.  If there is a strong wind, approximately,
you can just take T=78F.  But if there is no wind as in the case of
a big, but pretty closed room, then T is much below 78F.
Suppose we know the temperature dependence of the air as function
of the distance how far away from the ice, then the effective
T is given by T(d) with d is the mean free path of the air moleculars.

6) If the air (in macroscopic point of view) is really at rest,
the dependence of T(d) needs the detail information how the
room temperature is kept, ...  But in general if there is turbulance
in the room, T(d) would be still much greater than 0C.  However,
since the answer is very sensitivie to the value of T(d)-0C,
the difference between T(d) and the melting temperature.
For example, a small disturblance may yield T(d)=1C, or a large
one may give T(d)=10C.  Again it should be reminded, the condition
is very important here.  To give an example, supposed there is no
wind, and suppose the temperature is a linear function of the height
between the ice surface and the ceiling, then T(d) is very small indeed
(take d=1000A) T(d)=10**(-6)C.  This just gives you some idea about
how this melting rate does depend on the condition.  But if anyway
we assume T(d)=1C, we can still go further.

7).After find T(d), use the Boltzmann function to calculate how
many moleculars hit on the ice surface per second and thus one
can obtain the energy obtained by ice per second.  Divide this number
the latent heat, then one can calcualte how much ice is melted per
second.  It may be important to point out that T(d) may depend
on the shape of the ice as well as its size.

Although this may be not very helpful to your prize, it should
be helpful in understanding the problem, in particular the 
crucial point, namely T(d).

Yan-Bo Xie

Xref: sparky sci.physics:12742 sci.astro:8870 sci.philosophy.tech:2957