The basic equation is:

where **σ** = Stefan-Boltzmann
constant; the **ε**'s
are the emissivities of the fire and wall; the
T's are the localized temperatures Kelvin of the fire and the
wall; the A's are the areas used; and the
**θ**'s are the space angles between the
surface's normal and the line joining the fire point and the wall point
(for each end, of course); (r_{1} - r_{2})
is the distance between the fire source point and the
wall point. Subscript 1 refers to the fire (source); subscript
2 refers to the heat exchanger surface (destination).

This can be handled as a quintuple integral,

but solving this equation gets fairly messy with any complicated structure like the JUCA heat exchanger, and a numerical integration has generally been used.

The problem would actually be much more than a quintuple integral, except we made some assumptions to simplify it to the above equation. We assumed that the flames did not vary with time, and that they were uniform over the entire (flat) top surface area of the fire. We assumed that the fire was entirely flat and had no vertical side walls to radiate from. We made certain assumptions about the consistency of convective air flows within the firebox, as would occur after several hours of constant burning.

Using the production of hot smoke as a starting point, it is possible to estimate local smoke temperatures and velocity vectors near all wall segments. The excess air amount, wood consumption rate, flame temperature and distribution, and effectiveness of draft air distribution all represent serious effects, and a smoke production equation is used to provide estimates for these results, assuming the instantaneous smoke temperature begins at the fire temperature and that it immediately mixes following ideal gas statistical behavior.

Now, if we temporarily assume a wall temperature, we can calculate the amount of energy available to the wall (a total of the convective and radiative heat energy) by solving the two component problems using that destination temperature. Next, we must determine the air flow characteristics that will affect the thermal transfer to the wall. A number of surface effects come into play which affect (generally reduce) the amount of heat actually transferred to the metal of the wall.

Since this is a natural convection situation, we can use the Nusselt number, the Grashof number, and the Prandtl number, to get:

where *h*= the desired heat-transfer coefficient; L relates
to the geometry of the structure; **κ**
is the conductivity of the fluid; **γ**
is the fluid density; **Φ**
is the thermal expansion coefficient; c_{p}
is the specific heat; **θ**
is the temp differential; **μ**
is the fluid viscosity.

Using a variety of assumptions, this can often be simplified to:

where

This equation for free convection must be used in the firebox as we use a natural draft fire.

On the other side of the exchanger surface a different set of equations must be used, since it is a forced air convection situation there. The Reiher equation (simplified) becomes:

where *h* is again the desired heat-transfer coefficient;
d is a geometrical factor; G(max) is the mass flow rate of
fluid; is the absolute
viscosity of the fluid;
is the film thermal conductivity. Notice the similarity
of the right hand side to the Reynold's number. It is very
important to correct all the values to the correct temperature
conditions that exist at the wall/air interface as many of them
vary rapidly with temperature.

Now, use the radiation equation above again to calculate the outgoing radiation from that square inch of wall. Then solve:

for equilibrium conditions to estimate a correction to the initial temperature estimate for that square inch, then do it all again until incoming energy balances outgoing energy for every square inch of the structure. Of course, it is necessary to pay attention to the interaction of the separate square inches. Early ones in the exchanger will change the temperatures of both the air being heated and the smoke being tapped, so conditions in later portions and will materially change, generally tending to reduce the apparent effectiveness of the exchanger.

The balance equation above requires use of the *h*
(heat-transfer coefficient) to determine actual heat transfer.
It is also necessary to include a factor relating to the thermal
resistance of the wall structure itself (as an additional R
factor) although, with normal wall materials (like steel), the
bulk of the result will be due to the surface effects described
here. The advantage of the better intrinsic conductivity of an
aluminum exchanger would be mostly lost among these much more
dominant surface effects.

A representative simulation of a computer numerical integration of these equations is included in our web-site. It is at: CAD Design.

In the many thousands of such simulations that have been done, the advantages of larger blowers in the room air side of the heat exchangers is evident.

The numerous simulations also showed the significant advantage of a tapering firebox shape. As heat is removed from the smoke, it becomes cooler and needs to take up less volume (at constant pressure). We call this concept "isobaric equilibrium" and it is the reason why free-standing JUCAs all have a tapering shape. The very popular B-3B and B-3A units have sides that slope in at a 17 degree angle. The larger B-3D and B-3C have an optimal angle of 16 degrees. Discussion of that subject is at Why JUCA sides slope at 17 degrees.

The JUCA Home Page is at: juca