Re: Looking for notes: Heat Interchanger (simplified model)

[Brian Whatcott <betwys1@SBCGLOBAL.NET>]

My references are still all packed away - so I am considering Chuck's
 suggestions to Roberto from scratch:

I  suppose that the counter flow heat exchanger he suggested
has its concrete embodiment in a long expanded polystyrene channel
with a central divider along its length made of thin copper foil,
and that hot water  (M1) at temperature T1 enters at the near left end
and exits at Temperature T2 at the  near right end (the near channel),
while cold water (M2)  at temperature T3 enters the channel at the
 far right end and exits at the far left end at temperature T4
(the far channel)

One could model various conditions:
one would expect the ratio of the temperature changes to be
 inversely proportional to the ratio of the mass flow rates,
so (T1-T2)/(T4-T3) = dM2/dt  / dM1/dt


With fast flowing hot water, say 100 times the cold water flow rate,
one would expect an exponential temperature profile
with distance along the far channel, of the general form
T = T3 + (T1-T3)(1 - exp(-x/k)) where x is distance along the
 far channel from the right end and k is a characteristic distance
for the cold water temperature to change through 63% of its
 available range,  perhaps a fifth or less of the total distance
along the channel.

Similarly, if the cold water flow were one hundred times greater
one would expect an exponential temperature distribution with
distance in the near channel of the form
T = T1  - (T1-T3)(1 - exp(-x/k))  where x and k are now referred to
the left end.

On the face of it, when flows are equal, one might expect a linear
temperature change along the channel, with a constant temperature
 difference at any distance x along the channel, between the near
and far channel.   So unlike the temperature resulting from perfect
mixing of two equal flow rates of hot and cold water where the
resultant temperature would be  (T1 + T3)/2 one might expect the
respective temperatures to change from T1 to  T2 = T1 + m . (T1 - T3)
and from T3 to   T4 = T3 + m . ( T1 - T3 ) the m factor expressing
a heat exchange efficiency parameter and the cold water efflux
approaches the high temperature, while the hot water efflux
 approaches the cold water temperature.

To reflect outputs of this kind,  the model would use a function
of two flow rates and a distance from a reference point, ignoring
transient change.

Remembering that one hundred three years ago,
Professor Reynolds was also experimenting with water flow in
a long channel of this kind, I suppose I should expect to find an
exchange efficiency parameter changing as the flow rates progressed
from a Reynolds number of 2 to 3 thousand  which as he found,
marks the onset of turbulent flow, (with reduced temperature
difference across the septum here).

Perhaps a first model might miss some of the behavior I described above,
 for simplicity's sake.  I wonder what Fourier had to say on the subject?

Brian Whatcott    Altus OK    Eureka!

At  05:40 PM 9/7/2003 -0400,  Chuck, you wrote:
> A 'model' is a (usually mathematical) approximation of the real world.
> A GOOD model is one that can be worked 'easily' and gives 'pretty
> good' results.
>
> e.g. the Copernican Model of the solar system (not as exact as the
> reigning Ptolmeic model but far superior in enough ways for it to
> become the winner.
>
> Use a counterflow model and make gross assumptions such as:
>
> Assume a uniform thermal impedance separating the two flows at each point.
> Approximate the thermal profile for given temp parameters.
> Assume given the (2) input and (2) output T's
>
> Reality check - make sure that the heat lost by one side equals the
> heat gained by the other. (sorry Jim)
> Maybe you could (should?) assume only the two input temperatures and
> a constant thermal impedance / unit length.
> Sounds like some good exercise.
> --
> Chuck Britton                                   Education is what is left when
> britton@ncssm.edu                       you have forgotten everything
> North Carolina School of Science & Math         you learned in school.
> (919) 416-2762                                  Albert Einstein,        1936



> OK, ya'll, at 12:33 PM -0300 9/7/03, Roberto Carabajal wrote:
> >Distinguished Teachers:
> >
> >Please, I am looking notes for a very simplest model of heat interchanger
> >between two liquid fluids ( in example two tubes in contact) without fase
> >changes.  The central idea of the class  are the differential equations
> >describing the model,   but I am in trouble to find  the appropriate
> >simplified case  just not too far  from reality.