I need to find a formula or group of formulas that will tell me the
amount of energy (heat) transfer due to the convective flow of a
viscous liquid in a closed loop pipe.
Specifically, I have a tank T1 (surface area A1 at an ambient temp of
TF1 of volume V1) connected by an insulated pipe from the top of T1 to
the top of a second tank T2 (surface area A2 at an ambient temp of TF2
of volume V2).  T2 is above T1 and TF1 > TF2. A second pipe connects
the bottom of T2 to the bottom of T1. The entire system is filled with
oil with no air.   Total oil volume is TV1.  Assume no energy loses in
pipe.

I need a set of formulas to let me compute the amount of heat energy
that will transfer from T1 to T2 due to only the convective flow due
to the heat differential. A2 is much bigger than A1 and the
temperature difference can range from 5 to 40 degrees between TF1 and
TF2.  Vertical separation of T1 and T2 is about 20 to 30 feet.  Pipes
are 1" diameter.  T1 and T2 are thin metal.  You can make reasonable
assumptions for parameters I have not listed and use known constants
and physical attributes of components.
It would be most helpful if you would do an example but not necessary.

When you say heat energy do you mean to say: enthalpy?

Also if there is no heat transfer from the fluid to the ambient air
outside the system then you can calculate the amount of enthalpy
exchanged - the gradient of the temperature in your system will of
course be zero after a good amount of time.  If this is not the case
then of course this becomes quite a more complicated problem.  It
might be helpful for the researchers to recieve the conduction and
convection coefficient for the pipe and ambient system/air
respectively.  With these the problem would be alot easier.

The way I understand it you have no fluid flow, except for the natural
convection of the oil and the natural convection of the ambient air
(which aides in heat transfer from the oil, through the pipe wall, and
into the ambient surroundings).  You also did not mention the ambient
temperature.

I am interested in only the heat transfer of the oil.  That is, in
calories, BTUs, or joules (over time).  I expect this to be a moderate
heat transfer.

Assume that the ambient tempertures around T1 and T2 will never change
- that is, the heat tranfer of the oil will have no effect on the
temperatures around the two tanks.

The ambient temp for T1 is 5 to 40 degrees F above that of T2.  The
only transfer of heat is from the convective flow of the oil from T1
to T2 and the ambient air around T2.  T1 is actually under water.  To
simplify, the pipe can be considered to be a perfect insulator with no
loses.  I am only concerened with the two tanks.  The surface area of
T2 is perhaps 5 times that of T1.

I have thought about this problem a little bit more.  Basically what I
am understanding is that the pipes are perfect insulators, and they
are lossless which really simplifies this problem.  Also the second
tank is above the first and is in a colder ambient temperature than
the first.  I am now going to make the assumption that this whole
system runs in steady state.  that is the time derivatives are all
zero.  So this meens that:

1:  Heat must enter the system from the ambient temperature through
the surface of T1.
2:  Heat must leave the system through the surface to the ambient in T2.
3:  The pipes are perfectly insulated.

Therefore I propose that this problem is reduced to:
heat transfer into T1 = heat transfer out of T2. (steady state assumption)
But you want to find the heat transfer from T1 to T2.  So we now know that:

The rate of heat transfer through your pipes = heat rate entering T1
OR
the rate of heat transfer through your pipes = heat rate leaving T2

So all you need to do is find one of these quantities and you will
know the others because its a simply energy conservation problem!

Your analysis in reducing the problem to its essence is correct if I
was measuring an existing system to determine heat flow.  I am,
however, trying to create a math model so I can play with various
parameters as a function of initial desing.  The solution I seek is
the AMOUNT of heat that will circulate as a result of the convective
flow of the liquid.  If I have a formula that allows me to vary
surface area, heat coefficients/capacity, volume and viscosity, then I
can explore the effect of using different liquids, changing the size
and shape of T1 and T2 and the pipe between them.

My objective is to find an ideal solution that will give the maximum
degree of heat transfer given a specific set of temperatures for T1
and T2.

For instance, In general, viscosity goes up as heat capacity goes up
(thick oil holds more heat than water).  However, will the convective
flow of the fluid slow down as viscosity increases, negating the
increase in heat capacity?  Would a larger pipe reduce the convective
flow because it has a finite lifing power to its flow for a given heat
differential OR will a large pipe increase convective efficiency by
allowing more volume to move from T1 to T2?   If the differential
temperature is high, a viscous liquid should work well, however, if
the differential temperature is low, should I move to a less viscous
liquid that has less heat capacity but may move faster?  If the
temperature difference is very (50+ degrees) would I transfer more
heat if I were able to add additional connecting pipes using valves?

If I can find the math for all this, I can create a computer model
that will explore all these and other aspects of this design.

Thanks

Ok the most basic of equations are:

for the heat transfer between the ambient and the T1 and T2 ->
Q (heat rate) = -k*A*(dt/dx)
where Q is the rate of heat conduction.
k is the thermal conductivity of the metal between the two fluids.
A is the cross sectional area
dt = temperature difference
dx = metal thickness

For each tank and pipe you might consider using:
Qin + m*(h+v^2/2+g*z) = Qout + m*(h+v^2/2 + g*z)
Qin = heat rate into the volume
Qout = heat rate leaving
m = mass flow rate
h = enthalpy in (h = Cp*T) on left side of equation
h = enthalpy out on right side of equation
v = velocity in (on left side of equation)
v = velocity out (on right side of the equation)
g = gravity (9.81 m/s^2)
z = height above some plane below the system (in your case)

For mass conservation you might consider using something like:
rho*v*A = rho *v*A (out).
rho = density of fluid
v = velocity
A = area of pipe inlet or outlet.
the right side of the equation consider the terms to be the outgoing terms
the left side should be considered incoming terms.

Ok have fun! :)

saem_aero-ga,

Thank you so much for your kind patients and effort to help me resolve
this.  I am sure I can make your latest formulas work but I will have
to study up on it a little and work out the proper program - perhaps I
can use Excel.

You have been most helpful and certainly deserve more than my humble
offering but I hope you will take it with my heartfelt gratitude.  At
my end, I have no way to indicate to pay you.  We have exchanged notes
in the "comment" box.  You have to enter something in the "answer" box
so that I can accept it.  Once I do, the service will pay you.  Thank
you again and if you ever need any business management and business
development advise (my field of expertise), let me know.

Tom

Tom,
  I am not a google answer researcher - I was just having fun. :)  If
you need more details I suggest you look at the book...

Thermodynamics: An Engineering Approach by Cengel and Boles from mcgraw hill press.

This is a introductory text which has alot of great stuff which one
might find useful.

Best,
Steve.