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Heat transport
Storyboard 
In the design of a house, managing heat is crucial, so it's essential to understand the key concepts of temperature, its relationship with heat, and how it is transported through different mediums and interfaces.
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Heat transport mechanism
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In the case of a solid, and similarly for a liquid, we can describe the system as a structure of atoms bound together by something that behaves like a spring. When both ends have different temperatures, with the higher temperature ($T_2$, represented by a red LEGO®) and the lower temperature ($T_1$, represented by a blue one).
The difference in temperatures implies that the atoms at the ends oscillate differently; those in the high-temperature zone will oscillate with greater amplitude compared to those in the low-temperature zone.
However, this difference will gradually cause the entire chain to oscillate in such a way that, in the end, the amplitude will vary along the path, from the highest values where the temperature is also higher, to the lowest values in the lower-temperature region.
In this way, the temperature difference ($\Delta T$, represented by a light green LEGO®) drives a quantity of heat ($\Delta Q$, represented by a rust-colored LEGO®) over time ($\Delta t$, represented by a white LEGO®).
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Dependence of heat conduction on geometry and material
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One of the key factors determining how much heat can be conducted through a solid or liquid is its cross-sectional area, which essentially represents how many chains of atoms are available. The more of these chains we have, the more heat we can transport.
On the other hand, the length of these chains can be counterproductive. The longer the chain of springs, the less heat we can transmit, as many more atoms will need to adjust their oscillation amplitude.
If we represent the cross-sectional area ($S$, represented by a brown LEGO®) and the length ($L$, represented by a tawny LEGO®), the diagram would take this form:
Finally, a material's ability to transport heat, known as thermal conductivity ($\lambda$, represented by a salmon-colored LEGO®), describes how the material responds to the temperature difference ($\Delta T$, represented by a white LEGO®):
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Calculation of heat conduction
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In this way, a relationship is established that allows us to calculate the conducted heat ($\Delta Q$, represented by a LEGO® in rust color) depending on the cross-sectional area ($S$, represented by a LEGO® in brownish-red), the length ($L$, represented by a LEGO® in tawny), the temperature difference between both ends ($\Delta T$, represented by a LEGO® in light green), the elapsed time ($\Delta t$, represented by a LEGO® in white), and the thermal conductivity constant specific to each material ($\lambda$, represented by a LEGO® in salmon):
This can be expressed mathematically as:
 $\displaystyle\frac{ \Delta Q }{ \Delta t } = \displaystyle\frac{ S }{ L } \lambda \Delta T $ |
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Heat transfer to the conductor
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When you have a gaseous or liquid medium at a high temperature, it can transfer heat to a solid or liquid conductor. You can imagine it as if balls were being thrown, representing the molecules of the gas or liquid impacting the atoms of the conductor and causing them to oscillate. In fact, the atoms in the medium are transferring heat to the conductor:
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Dependence of heat transfer on geometry to the conductor
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The primary driver of heat transfer from a medium to a conductor is the temperature difference. If the temperature in the medium is higher ($T_2$, represented by a red LEGO®), the particles have more energy, and when they collide with particles in the conductor at a lower temperature ($T_1$, represented by a blue LEGO®), they tend to increase the energy of the latter. This interaction can be represented as follows:
More than just temperature itself, the heat flow depends on the temperature difference ($\Delta T$, represented by a light green LEGO®). Another key factor is the number of atoms that can have their oscillation amplitude increased, which depends on the surface area of the interface ($S$, represented by a brown LEGO®). Finally, we must consider the surface properties, which are described by a transmission coefficient ($\alpha$, represented by a magenta LEGO®), corresponding to the relationship between the transmitted heat, surface area, temperature difference, and elapsed time:
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Calculation of heat transmission to the conductor
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In this manner, we establish a relationship that allows us to calculate the transmitted heat ($\Delta Q$, represented by a LEGO® brick in rust color) based on the cross-sectional area ($S$, represented by a brown LEGO® brick), the temperature difference between both ends ($\Delta T$, represented by a light green LEGO® brick), the elapsed time ($\Delta t$, represented by a white LEGO® brick), and the thermal transmission constant specific to each interface ($\alpha$, represented by a magenta LEGO® brick):
This can be expressed mathematically as:
| $\displaystyle\frac{ \Delta Q }{ \Delta t } = S \alpha_i \Delta T $ |
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Heat transfer from the conductor
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When you have a solid or liquid conductor at a high temperature, it can transfer heat to a gaseous or liquid medium at a lower temperature. This can be imagined as if the conductor had a ping pong paddle and was striking each atom or molecule in the medium composed of gas or liquid that approaches the interface. In fact, the atoms of the conductor are transferring heat to the medium:
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Dependence of heat transfer on geometry from the conductor
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The primary driver of heat transfer from a conductor to a medium is the temperature difference. When the temperature in the conductor is higher ($T_2$, represented by a red LEGO®), the particles have more energy and oscillate with a greater amplitude when interacting with the atoms and molecules of the medium at a lower temperature ($T_1$, represented by a blue LEGO®). This tends to increase the energy of the latter. This interaction can be depicted as follows:
More than just temperature, the heat flow depends on the temperature difference ($\Delta T$, represented by a light green LEGO®). Another key factor is the number of atoms that can have their oscillation amplitude increased, which depends on the surface area of the interface ($S$, represented by a brown LEGO®). Finally, we must consider surface properties, represented by a transmission coefficient ($\alpha$, represented by a magenta LEGO®), which corresponds to the relationship between the transmitted heat, the surface area, the temperature difference, and the elapsed time:
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Calculation of heat transfer from the conductor
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In this manner, we establish a relationship that allows us to calculate the transmitted heat ($\Delta Q$, represented by a LEGO® brick in rust color) based on the cross-sectional area ($S$, represented by a brown LEGO® brick), the temperature difference between both ends ($\Delta T$, represented by a light green LEGO® brick), the elapsed time ($\Delta t$, represented by a white LEGO® brick), and the thermal transmission constant specific to each interface ($\alpha$, represented by a magenta LEGO® brick):
This can be expressed mathematically as:
| $\displaystyle\frac{ \Delta Q }{ \Delta t } = S \alpha_e \Delta T $ |
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Heat exchange efficiency
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One of the effects of heat transfer from a conductor to an external medium is that the region near the interface can start to heat up, creating an interference zone in the transmission. This reduces the efficiency of heat transfer and tends to create insulation that hinders the energy flow.
However, this effect can change in the presence of wind, as the wind can remove the layer of atoms and molecules at a high temperature, making heat transfer more efficient. This means that the heat transfer coefficient ($\alpha$, represented by a magenta LEGO® block) depends on the wind speed ($v_w$, represented by a lime-colored block):
In this case, we model the relationship based on the heat transfer coefficient when there is no wind ($\alpha_0$, represented by a purple LEGO® block) and a reference velocity factor ($v_{w0}$, represented by a light green LEGO® block).
The mathematical relationship describing this is:
| $\alpha = \alpha_0 \left(1 + \displaystyle\frac{ v }{ v_0 }\right)$ |
This explains how wind can influence the efficiency of heat transfer between a conductor and an external medium.
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Transportation Coefficient Calculation
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If we consider the temperature difference between the interior ($T_2$, represented by a blue LEGO®) and the exterior ($T_1$, represented by a red LEGO®), we can show that their difference ($\Delta T$, represented by a light green LEGO®) is equal to the sum of three temperature differences (exterior-outer wall, outer wall-inner wall, inner wall-interior). If we replace these temperature differences with the equations for external transfer, conduction, and internal transfer, we can derive the general transport equation and an expression for the transport coefficient ($k$, represented by a light pink LEGO®):
The transport coefficient depends on the exterior transport coefficient ($\alpha_e$, represented by a violet LEGO®), interior transport coefficient ($\alpha_i$, represented by a magenta LEGO®), the thermal conductivity coefficient ($\lambda$, represented by a salmon-colored LEGO®), and the length ($L$, represented by a tan LEGO®):
This can be expressed mathematically as follows:
| $\displaystyle\frac{ 1 }{ k } = \displaystyle\frac{ 1 }{ \alpha_e }+\displaystyle\frac{ 1 }{ \alpha_i }+\displaystyle\frac{ L }{ \lambda }$ |
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Total heat transport through the wall
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The heat transfer and conduction models suggest that we can develop a relationship that incorporates all three mechanisms together. This equation should take into account the amount of heat transferred ($\Delta Q$, represented by a rust-colored LEGO®), the elapsed time ($\Delta t$, represented by a white LEGO®), the temperature difference ($\Delta T$, represented by a light green LEGO®), the cross-sectional area ($S$, represented by a brown LEGO®), and the transport constant ($k$, represented by a light pink LEGO®):
This can be expressed mathematically as follows:
| $\displaystyle\frac{ \Delta Q }{ \Delta t } = k S \Delta T $ |
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Heat flow model in housing
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The different parts of the discussed model can be integrated into a model that can be represented as a set of instructions similar to LEGO® instructions for building a model:
This model includes the calculation of:
1. The transport coefficient based on the wind speed at the surface.
2. The total transport coefficient based on the wall's geometry and materials, as well as the conditions of the internal and external surfaces.
3. The heat flow over a specific period of time.
4. The temperature on the external surface of the wall.
5. The temperature on the internal surface of the wall.
To achieve this, it is necessary to define the internal temperature and the external temperature as additional parameters.
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