The next fireplace was born from a drawing elaborated with the marble worker on paper and then digitized. The design of the flue and various draft adjustment operations follow the design phase of the external frame. Subsequent operations led to the creation of a groove and several air intakes to ensure correct draft. These subsequent operations were necessary since it was carried out without taking into account a section of high resistance flue which led to smoke problems given the size of the mouth.

The initial project starts from the design of the architrave profile in compliance with the constraints imposed by the marble worker and the desired aesthetic standards.

Then follows a phase of careful study with the marble worker for the design of the columns and other aesthetic and support elements.

After the construction of the external frame, the various components were positioned on the wall structure made specifically, leaving the project of the fire chamber and the hood to the construction company that also took care of the assembly.

## Some problems

Although the first ignition tests were completed successfully, several draft problems emerged in the following months.

A first problem is due to the size of the mouth (110×95 cm), which is too large compared to the section of the flue, the environment and the resistance to fumes. A second problem, on the other hand, is linked to the path of the flue which has to cover a rather long horizontal section (in proportion to the total length) as well as several too narrow curves.

Based on the section of the flue and the height, referring to the regulations in force, the section of the mouth had to be smaller or have an equally high flue chamber. The incorrect design and construction by the construction company made further improvements necessary.

## Chimney

The current smoking rod provides a height of 9m for a total distance of 12m with a diameter of 30cm.

It was not possible to change the height nor the section of the flue so we chose to work on the smoke inlet section.

Not being able to create a groove or a smoke chamber with a siphon, an alternative way was sought to increase the draft while remaining in the design of the traditional fireplace.

A first step was to reduce the initial section of the flue to increase the air speed. This step proved to be very effective. Furthermore, by modeling a sheet metal, a narrowing was made to the smoke inlet section. This narrowing widens towards the area of the double elbow bend. The double curve creates a larger area where the smoke “stagnates”. This allows, in the event of temporary draft problems (eg strong winds on the chimney, ignition phase, etc.) the smoke does not fall back into the brazier, giving the flue time to take it all out. This made it possible to avoid the escape of smoke even in the initial phase in which the flue still has to heat up.

*Narrowing of the smoke inlet section*

## Air intake

It was also useful to redo the insulation of the exposed part of the flue and increase the area of the direct air intake bringing it to approximately 22×33.5 cm^{2 }. This is also followed by the project of a grille designed in such a way as to maximize the passage of air.

The air intake allows you to replace an internal 80×30 window that overlooked another room by taking air at room temperature. The air intake takes air from outside the house, so much colder and denser. By carrying out several tests, it was found that the dimensions (even if smaller) are sufficient. In addition, the air intake is directed on the flame and, in addition to being more efficient, it also allows you to avoid annoying drafts inside the room.

## Fireplace draft

It is known as the stairwell of a house, as well as its flue or roof skylights are able to trigger the connective and ascending movements of the air. It is the heated air that expands proportionally to the increase in temperature and, by decreasing its specific weight, tends to rise.

From thermodynamics we know that the fireplace lit in the room is such that the hot gases rise in the flue to exit at a certain altitude h. When the fireplace is lit in winter, the temperature of the combustion products is much higher than the external one and the hot gas rises because it has a lower weight than the external air column since the density is inversely proportional to the temperature. This is natural convection, where the fluid naturally moves due to the difference in density.

We can therefore distinguish the ideal case by considering an isotherm and the real case.

Considering the isotherm , we can write the Bernoulli equation between section 1 of the room and section 2 of the chimney with a difference in height h, suppose that there are no fans we have:

dz_a+\cancel{dz_f}+dz+{dp\over \gamma}+{dw^2\over 2g}=0

Furthermore, if we consider that if the density γ is almost constant we can say that:

\gamma={pg\over RT}={p_{atm}g\over RT_i}=\bold{cost} \\ \text{ con }T_i=\bold{cost}\quad p_{atm}\approx \bold{cost}

We can integrate between 1 and 2 by delimiting the control volume:

z_a+z_2-z_1+{p_2-p_1\over \gamma _i}+{w_2^2+\cancel{w_1^2}\over 2g}=0

If the width of the room is much greater than the section of the chimney, the speed w1 is negligible compared to the square of the exit speed. Furthermore, suppose the friction due to distributed pressure losses.

Then:

z_a=\lambda{w_2^2\over g}{h\over D}+{w_2^2\over 2g}={p_2-p_1\over \gamma_i}+z_1-z_2 \text{ con }p_1-p_2=\rho_egh\\ \Longrightarrow {w_2^2\over 2g}\underbrace{\left(\lambda{h\over D}+1\right)}_\beta={\gamma_e h \over \gamma_i}-h

We can also enter additional terms to account for concentrated losses.

And so we then get:

{w_2^2\over 2g}\beta={h\left({\gamma_e\over \gamma_i}-1\right)}=h\left({T_i\over T_e}-1\right)=h\left(T_i-T_e\over T_e\right)

This expression gives us the physical sense of the problem. In fact, the fluid moves by natural convention when I light the fire, otherwise nothing moves. This simplified expression is good for giving a first physical interpretation. *β* is that which opposes cause and effect.

That is:

\color{red}{w_2^2\over 2g}\beta={h\left(\gamma_e-\gamma_i\over \gamma_i\right)}

The first term represents the draft resistance of the fireplace while the second term represents the cause. If the equation is verified or if the second term is greater than the first, we will have a natural draft. It will therefore work only for the difference in density due to the difference in temperature, otherwise we would have to prepare a fan to overcome friction.

Considering Ohm’s relation Δ*V* = *IR *we could make an analogous comparison (cause and effect). In a verification problem it could be that the cause term is greater than the effect product by resistance and then the chimney pulls. However, it could also be that the cause of the motion is less than the product due to the effect of the resistance, so the chimney does not pull and then it is necessary to increase the right term with a load due for example to a fan (typically used in large chimneys) .

### Non-ideal case

Moving on to the **real case** we have *q *≠ 0.

We then schematize the problem by considering the flue directly and go back to the problem of the methane pipeline where a gas enters at temperature *T*_{0} and runs through the pipe of height *h*). There is a temperature jump and a heat exchange that will take into account how the flue wall is made. Taking then the Bernoulli equation applied to the case of the pipeline which, being vertical, also takes into account the term *dz *. In this case we call it *dx* and it is the fourth term which therefore takes into account the height difference:

p\:dp+BT\:dx-AT{d\gamma\over \gamma}+dx\cdot {gp^2\over RT}=0

Let’s assume again that the temperature follows the exponential trend of the thermostat problem, so we can integrate:

\int_0^up\:dp={p_u^2-p_0^2\over 2}\\

\int_0^uBTdx=BT\int_0^u\left[T_e-\left(T_o-T_e\right)e^{ax}\right]dx

Dependent on γ, it will have to be verified by successive approximations.

{d\gamma \over \gamma}={dp\over p}-{dT\over T}\\ \rightarrow -AT{d\gamma\over \gamma}=-AT{dp\over p}+{}a dT\rightarrow\int _0^uAT{dp\over p}=AT_m\ln \left(p_u\over p_0\right)

Engineering negligible except for large chimneys as h is relatively low.

\int_0^u{dp^2\over RT}dx={g\over R}p_2^2\int_0^u{dx\over T}={gp_o^2\over R}\int_0^u{1\over T_e-\left(T_o-T_e\right)e^{ax}}dx

The reduction in the density of the gas that heats up in the lower part of the duct is also associated with an increase in gas pressure. This pressure variation at the ends of the duct can be obtained from the following formula:

\Delta p=CP_ah\left({1\over T_{ext}}-{1\over T_{int}}\right)

## Design

The design of a cap, added later, also follows, which allows the flue to be closed, avoiding the return of soot and the smell of burning when the fireplace is turned off.

And the project of a customized brazier that adapts perfectly to the room allowing you to contain the necessary wood.