Storyboard

The moisture content in the soil is dependent on its temperature, making it essential to calculate this distribution. This relationship is determined by the heat flux exchanged with the soil's surface.

>Model

ID:(2054, 0)

Concept

>Top

ID:(15207, 0)

Concept

>Top

In the case of a solid, and similarly for a liquid, we can describe the system as a structure of atoms held together by something that behaves like a spring. When both ends have temperatures of ($$), with the inner surface temperature ($T_{is}$) and the outer surface temperature ($T_{es}$):

The temperature difference implies that the atoms at the ends oscillate differently; atoms in the high-temperature zone will have a greater amplitude in their oscillations compared to the atoms in the low-temperature zone.

However, this difference will gradually lead to the entire chain oscillating 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 zone.

In this way, the temperature difference in the conductor ($\Delta T_0$) leads to ($$) in ($$).

ID:(15234, 0)

Concept

>Top

One of the key factors determining how much heat can be conducted through a solid or liquid is its cross-sectional area, which refers to the quantity of atom chains available. The more of these chains we have, the greater our heat transport capacity.

However, the length of these chains can be counterproductive. As the spring chain becomes longer, our ability to transmit heat decreases, as more atoms must adjust their oscillation amplitudes.

If we represent this with the section ($S$) and the conductor length ($L$), the diagram takes the following form:

Finally, the ability of the medium and material to transport heat, described by coefficients the internal transmission coefficient ($\alpha_i$) and the external transmission coefficient ($\alpha_e$), and the thermal conductivity ($\lambda$), explains how heat moves in response to the temperature difference ($\Delta T$) created by the difference between the indoor temperature ($T_i$) and the outdoor temperature ($T_e$):

This is calculated as:

ID:(15235, 0)

Concept

>Top

In this manner, a relationship is established that allows us to calculate the variation of heat ($dQ$) based on the time variation ($dt$), the section ($S$), the conductor length ($L$), the temperature difference in the conductor ($\Delta T_0$), and the thermal conductivity ($\lambda$):

Mathematically, this can be expressed as follows:

ID:(15236, 0)

Concept

>Top

The primary driver of heat transfer from a medium to a conductor is the difference in temperature. In the medium the indoor temperature ($T_i$), particles have more energy, and when they collide with those in the conductor at ($$), they tend to increase the energy of the latter. This interaction can be represented as follows:

Beyond temperature itself, the heat flow depends on the temperature difference at internal interface ($\Delta T_i$):

Another key factor is the number of atoms whose oscillation amplitude can be increased, which depends on the section ($S$). Lastly, we must consider surface properties, described by the internal transmission coefficient ($\alpha_i$), which corresponds to the relationship between transmitted heat, surface area, temperature difference, and elapsed time:

ID:(15237, 0)

Concept

>Top

In this way, we establish a relationship that allows us to calculate the variation of heat ($dQ$) based on the time variation ($dt$), the section ($S$), the temperature difference at internal interface ($\Delta T_i$), and the internal transmission coefficient ($\alpha_i$):

This can be expressed mathematically as follows:

ID:(15238, 0)

Concept

>Top

The primary driver of heat transfer from a conductor to a medium is the temperature difference. When the outer surface temperature ($T_{es}$), particles have more energy and oscillate with a greater amplitude when interacting with the atoms and molecules of the medium at ($$). This tends to increase the energy of the latter. This interaction can be represented as follows:

Beyond temperature, the heat flow depends on the temperature difference at external interface ($\Delta T_e$).

Another key factor is the number of atoms that can have their oscillation amplitude increased, which depends on the section ($S$). Finally, we must consider surface properties, represented by the external transmission coefficient ($\alpha_e$), which corresponds to the relationship between transmitted heat, surface area, temperature difference, and elapsed time:

ID:(15239, 0)

Concept

>Top

In this manner, we establish a relationship that enables us to calculate the variation of heat ($dQ$) based on the time variation ($dt$), the section ($S$), the temperature difference at external interface ($\Delta T_e$), and the external transmission coefficient ($\alpha_e$):

This can be expressed mathematically as follows:

ID:(15240, 0)

Concept

>Top

Heat transfer and conduction models suggest that it is possible to develop a relationship that incorporates all three mechanisms together. This equation should take into account the variation of heat ($dQ$), the time variation ($dt$), the temperature difference ($\Delta T$), the section ($S$), and the total transport coefficient (multiple medium, two interfaces) ($k$):

Mathematically, this can be expressed as follows:

ID:(15241, 0)

Concept

>Top

One of the effects of heat transfer from a conductor to an external medium is that the area near the interface can start to heat up and create an interference zone in the transmission. This reduces the efficiency of the transfer and tends to create an insulating layer that reduces the energy flow.

However, this effect can change in the presence of wind, as the wind can remove that layer of high-temperature atoms and molecules, making heat transmission more efficient. This means that the transmission coefficient ($\alpha$) depends on the mean speed ($v_m$):

In this case, we model the relationship based on the transmission coefficient without speed ($\alpha_0$) and a reference factor of the media reference speed ($v_0$).

The mathematical relationship describing this is for a gas with the transmission coefficient in gases, dependent on speed ($\alpha_{gv}$), the mean speed ($v_m$), the transmission coefficient in gases, independent of speed ($\alpha_{g0}$), and the transmission coefficient gas velocity factor ($v_{g0}$):

and for a liquid with the transmission coefficient dependent on the speed ($\alpha_{wv}$), the mean speed ($v_m$), the transmission coefficient Independent of the speed ($\alpha_{w0}$), and the factor velocidad del coefiente de transmisión ($v_{w0}$):

This explains how wind can influence the efficiency of heat transfer between a conductor and an external medium.

ID:(3620, 0)

Concept

>Top

Selected parameter

Calculations

ID:(15229, 0)

Equation

>Top, >Model

In the case of a solid, and similarly for a liquid, we can describe the system as a structure of atoms held together by something that behaves like a spring. When both ends have temperatures of ($$), with the inner surface temperature ($T_{is}$) and the outer surface temperature ($T_{es}$):

$ \Delta T_0 = T_{is} - T_{es} $

Conditions

Variables

Relation

ID:(15120, 0)

Equation

>Top, >Model

The temperature difference ($\Delta T$) is calculated by subtracting the outdoor temperature ($T_e$) and the indoor temperature ($T_i$), which is expressed as:

$ \Delta T = T_i - T_e $

Conditions

Variables

Relation

ID:(15116, 0)

Equation

>Top, >Model

In this manner, a relationship is established that allows us to calculate the variation of heat ($dQ$) based on the time variation ($dt$), the section ($S$), the conductor length ($L$), the temperature difference in the conductor ($\Delta T_0$), and the thermal conductivity ($\lambda$):

$\displaystyle\frac{ dQ }{ dt } = \displaystyle\frac{ S }{ L } \lambda \Delta T_0 $

Conditions

Variables

Relation

ID:(7712, 0)

Equation

>Top, >Model

The temperature difference at internal interface ($\Delta T_i$) is calculated by subtracting the inner surface temperature ($T_{is}$) from the indoor temperature ($T_i$):

$ \Delta T_i = T_i - T_{is} $

Conditions

Variables

Relation

ID:(15117, 0)

Concept

>Top

In this way, we establish a relationship that allows us to calculate the variation of heat ($dQ$) based on the time variation ($dt$), the section ($S$), the temperature difference at internal interface ($\Delta T_i$), and the internal transmission coefficient ($\alpha_i$):

$\displaystyle\frac{ dQ }{ dt } = S \alpha_i \Delta T_i $

Conditions

Variables

Relation

ID:(15113, 0)

Equation

>Top, >Model

The temperature difference at external interface ($\Delta T_e$) is calculated by subtracting the outer surface temperature ($T_{es}$) from the outdoor temperature ($T_e$):

$ \Delta T_e = T_{es} - T_e $

Conditions

Variables

Relation

ID:(15118, 0)

Equation

>Top, >Model

In this manner, we establish a relationship that enables us to calculate the variation of heat ($dQ$) based on the time variation ($dt$), the section ($S$), the temperature difference at external interface ($\Delta T_e$), and the external transmission coefficient ($\alpha_e$):

$\displaystyle\frac{ dQ }{ dt } = S \alpha_e \Delta T_e $

Conditions

Variables

Relation

ID:(15114, 0)

Equation

>Top, >Model

In this way, we establish a relationship that allows us to calculate the variation of heat ($dQ$) as a function of the time variation ($dt$), the section ($S$), the total transport coefficient (multiple medium, two interfaces) ($k$), and the temperature difference ($\Delta T$):

$\displaystyle\frac{ dQ }{ dt } = k S \Delta T $

Conditions

Variables

$S$

Section

$m^2$

$\Delta T$

Temperature difference

$K$

$dt$

Time variation

$s$

$k$

Total transport coefficient (multiple medium, two interfaces)

$J/m s K$

$dQ$

Variation of heat

$J$

Relation

Development

With the temperature difference at internal interface ($\Delta T_i$), the temperature difference in the conductor ($\Delta T_0$), the temperature difference at external interface ($\Delta T_e$), and the temperature difference ($\Delta T$), we obtain

$ \Delta T = \Delta T_i + \Delta T_0 + \Delta T_e $

which can be rewritten with the variation of heat ($dQ$), the time variation ($dt$), the section ($S$)

$\displaystyle\frac{ dQ }{ dt } = S \alpha_i \Delta T_i $

$\displaystyle\frac{ dQ }{ dt } = S \alpha_e \Delta T_e $

and with the thermal conductivity ($\lambda$) and the conductor length ($L$)

$\displaystyle\frac{ dQ }{ dt } = \displaystyle\frac{ S }{ L } \lambda \Delta T_0 $

and

$\displaystyle\frac{1}{ k }=\displaystyle\frac{1}{ \alpha_i }+\displaystyle\frac{1}{ \alpha_e }+\displaystyle\frac{ L }{ \lambda }$

as

$\Delta T = \Delta T_i + \Delta T_0 + \Delta T_e = \displaystyle\frac{1}{S} \frac{dQ}{dt} \left(\displaystyle\frac{1}{\alpha_i} + \displaystyle\frac{1}{\alpha_e} + \displaystyle\frac{L}{\lambda}\right) = \displaystyle\frac{1}{Sk} \displaystyle\frac{dQ}{dt}$

resulting in

$\displaystyle\frac{ dQ }{ dt } = k S \Delta T $

ID:(7716, 0)

Equation

>Top, >Model

In the process of heat transfer, the temperature gradually decreases from the system with the highest temperature (internal) to the one with the lowest temperature (external). In this process, it first decreases from the internal average temperature to the temperature difference at internal interface ($\Delta T_i$), then to the temperature difference in the conductor ($\Delta T_0$), and finally to the temperature difference at external interface ($\Delta T_e$). The sum of these three variations equals the total drop, that is, the temperature difference ($\Delta T$), as shown below:

$ \Delta T = \Delta T_i + \Delta T_0 + \Delta T_e $

Conditions

Variables

Relation

ID:(15115, 0)

Equation

>Top, >Model

The value of the coefficient of total transportation (half, two interfaces) ($k$) in the transport equation is determined using the external transmission coefficient ($\alpha_e$), the internal transmission coefficient ($\alpha_i$), the thermal conductivity ($\lambda$), and the conductor length ($L$) as follows:

$\displaystyle\frac{1}{ k }=\displaystyle\frac{1}{ \alpha_i }+\displaystyle\frac{1}{ \alpha_e }+\displaystyle\frac{ L }{ \lambda }$

Conditions

Variables

$k$

Coefficient of total transportation (half, two interfaces)

$J/m s K$

$L$

Conductor length

$m$

$\alpha_e$

External transmission coefficient

$W/m^2K$

$\alpha_i$

Internal transmission coefficient

$W/m^2K$

$\lambda$

Thermal conductivity

$J/m s K$

Relation

Development

With the temperature difference at internal interface ($\Delta T_i$), the temperature difference in the conductor ($\Delta T_0$), the temperature difference at external interface ($\Delta T_e$), and the temperature difference ($\Delta T$), we obtain

$ \Delta T = \Delta T_i + \Delta T_0 + \Delta T_e $

which can be rewritten with the variation of heat ($dQ$), the time variation ($dt$), the section ($S$)

$\displaystyle\frac{ dQ }{ dt } = S \alpha_i \Delta T_i $

$\displaystyle\frac{ dQ }{ dt } = S \alpha_e \Delta T_e $

and with the thermal conductivity ($\lambda$) and the conductor length ($L$)

$\displaystyle\frac{ dQ }{ dt } = \displaystyle\frac{ S }{ L } \lambda \Delta T_0 $

as

$\Delta T_i + \Delta T_0 + \Delta T_e = \displaystyle\frac{1}{S} \displaystyle\frac{dQ}{dt} \left(\displaystyle\frac{1}{\alpha_i} + \displaystyle\frac{1}{\alpha_e} + \displaystyle\frac{L}{\lambda}\right)$

so we can define a combined coefficient as

$\displaystyle\frac{1}{ k }=\displaystyle\frac{1}{ \alpha_i }+\displaystyle\frac{1}{ \alpha_e }+\displaystyle\frac{ L }{ \lambda }$

ID:(3486, 0)

Equation

>Top, >Model

The inner surface temperature ($T_{is}$) is not equal to the temperature of the medium itself, which is the indoor temperature ($T_i$). This temperature can be calculated from the temperature difference ($\Delta T$), the total transport coefficient (multiple medium, two interfaces) ($k$), and the internal transmission coefficient ($\alpha_i$) using the following formula:

$ T_{is} = T_i - \displaystyle\frac{ k }{ \alpha_i } \Delta T $

Conditions

Variables

Relation

Development

With the variation of heat ($dQ$), the time variation ($dt$), the section ($S$), the temperature difference ($\Delta T$), and the total transport coefficient (multiple medium, two interfaces) ($k$), we have

$\displaystyle\frac{ dQ }{ dt } = k S \Delta T $

which, with the internal transmission coefficient ($\alpha_i$) and the temperature difference at internal interface ($\Delta T_i$)

$\displaystyle\frac{ dQ }{ dt } = S \alpha_i \Delta T_i $

results in

$k\Delta T = \alpha_i \Delta T_i$

and with the indoor temperature ($T_i$) and the inner surface temperature ($T_{is}$) and

$ \Delta T_i = T_i - T_{is} $

results in

$ T_{is} = T_i - \displaystyle\frac{ k }{ \alpha_i } \Delta T $

ID:(15121, 0)

Equation

>Top, >Model

The outer surface temperature ($T_{es}$) is not equal to the temperature of the medium, which is the outdoor temperature ($T_e$). This temperature can be calculated from the temperature difference ($\Delta T$), the total transport coefficient (multiple medium, two interfaces) ($k$), and the external transmission coefficient ($\alpha_e$) using the following formula:

$ T_{es} = T_e + \displaystyle\frac{ k }{ \alpha_e } \Delta T $

Conditions

Variables

Relation

Development

With the variation of heat ($dQ$), the time variation ($dt$), the section ($S$), the temperature difference ($\Delta T$), and the total transport coefficient (multiple medium, two interfaces) ($k$), we obtain

$\displaystyle\frac{ dQ }{ dt } = k S \Delta T $

which, with the external transmission coefficient ($\alpha_e$) and the temperature difference at external interface ($\Delta T_e$)

$\displaystyle\frac{ dQ }{ dt } = S \alpha_e \Delta T_e $

results in

$k\Delta T = \alpha_e \Delta T_e$

and with the outdoor temperature ($T_e$) and the outer surface temperature ($T_{es}$) and

$ \Delta T_e = T_{es} - T_e $

results in

$ T_{es} = T_e + \displaystyle\frac{ k }{ \alpha_e } \Delta T $

ID:(15122, 0)

Equation

>Top, >Model

The constant $k$ in the transport equation is determined as follows:

$ \displaystyle\frac{1}{ k }=\displaystyle\frac{1}{ \alpha }+\displaystyle\frac{ L }{ \lambda }$

Conditions

Variables

$k$

Coefficient of total transportation (half a interfaces)

$J/m s K$

$L$

Conductor length

$m$

$\lambda$

Thermal conductivity

$J/m s K$

$\alpha$

Transmission coefficient

$W/m^2K$

Relation

ID:(3619, 0)

Equation

>Top, >Model

The transport constant $k$ through media with thermal conductivity $\lambda_i$ and thickness $L_i$, respectively, and with transfer coefficients at both boundaries $\alpha_1$ and $\alpha_2$, is determined as follows:

$\displaystyle\frac{1}{k}=\displaystyle\frac{1}{\alpha_1}+\displaystyle\frac{1}{\alpha_2}+\sum_i\displaystyle\frac{L_i}{\lambda_i}$

Conditions

Variables

$L_i$

Element length i

$m$

$\lambda_i$

Thermal conductivity element i

$J/m s K$

$k$

Total transport coefficient (multiple medium, two interfaces)

$J/m s K$

$\alpha_1$

Transmission coefficient Interface 1

$W/m^2K$

$\alpha_2$

Transmission coefficient Interface 2

$W/m^2K$

Relation

ID:(7730, 0)

Equation

>Top, >Model

In the event that a medium moves with a constant value of ($$) and the transmission coefficient in gases, dependent on speed ($\alpha_{gv}$) is equal to

$ \alpha_{gv} = \alpha_{g0} \left(1+\displaystyle\frac{ v_m }{ v_{g0} }\right)$

Conditions

Variables

$v_m$

Mean speed

$m/s$

$v_{g0}$

Transmission coefficient gas velocity factor

$m/s$

$\alpha_g$

Transmission coefficient in gases, dependent on speed

$W/m^2K$

$\alpha_{g0}$

Transmission coefficient in gases, independent of speed

$W/m^2K$

Relation

where the transmission coefficient in gases, independent of speed ($\alpha_{g0}$) represents the scenario where the medium does not move, and the transmission coefficient gas velocity factor ($v_{g0}$) is the reference velocity.

The thermal transfer constant for the material in the case of a stationary gas is equal to $5.6 J/m^2sK$, while the reference velocity is $1.41 m/s$.

ID:(7715, 0)

Equation

>Top, >Model

If a medium is moving with a constant of the transmission coefficient dependent on the speed ($\alpha_{wv}$), and the mean speed ($v_m$) is equal to

$ \alpha_{wv} = \alpha_{w0} \left(1+\sqrt{\displaystyle\frac{ v_m }{ v_{w0} }}\right)$

Conditions

Variables

$v_{l0}$

Factor velocidad del coefiente de transmisión

0.0278

$m/s$

$v_m$

Mean speed

$m/s$

$\alpha_l$

Transmission coefficient dependent on the speed

$W/m^2K$

$\alpha_{l0}$

Transmission coefficient Independent of the speed

$W/m^2K$

Relation

where the transmission coefficient Independent of the speed ($\alpha_{w0}$) represents the case where the medium is not moving, and the factor velocidad del coefiente de transmisión ($v_{w0}$) is the reference velocity.

The thermal transfer constant of the material for the case of a stationary liquid is equal to $340 J/m^2sK$, while the reference velocity is $0.0278 m/s$.

ID:(7714, 0)

Equation

>Top, >Model

Modeling thermal conductivity in a porous medium like soil is a challenging task. In this study, a wide range of samples was analyzed, and a numerical model was developed to predict the thermal conductivity in dry soil ($\lambda_b$) based on soil textures [1].

The relationship of the thermal conductivity in dry soil ($\lambda_b$) was determined in terms of the thermal conductivity in sand ($\lambda_a$), the thermal conductivity in silt ($\lambda_i$), the thermal conductivity in clay ($\lambda_c$), and the mass fraction of sand in the sample ($g_a$), the mass fraction of silt in the sample ($g_i$), the mass fraction of clay in the sample ($g_c$), using the following formula:

$ \lambda_b = \lambda_a g_a + \lambda_i g_i + \lambda_c g_c $

Conditions

Variables

Relation

[1] "Physical principles and calculation methods of moisture and heat transfer in cable trenches," Brakelmann, H., etz-Report 19, 93p. (1984), Berlin; Offenbach.

ID:(15130, 0)

Equation

>Top, >Model

Modeling thermal conductivity in a porous medium like soil is a challenging task. In this study, a wide range of samples was analyzed, and a numerical model was developed to predict the thermal conductivity in dry soil ($\lambda_b$) based on soil textures [1].

The relationship of the thermal conductivity in dry soil ($\lambda_b$) was determined in terms of the thermal conductivity in sand ($\lambda_a$), the thermal conductivity in silt ($\lambda_i$), the thermal conductivity in clay ($\lambda_c$), and the mass fraction of sand in the sample ($g_a$), the mass fraction of silt in the sample ($g_i$), the mass fraction of clay in the sample ($g_c$), using the following formula:

$ \lambda = \lambda_w^{\Phi} \lambda_b^{1-\Phi} e^{-\beta \Phi (1-\theta_S)^2}$

Conditions

Variables

Relation

[1] "Physical principles and calculation methods of moisture and heat transfer in cable trenches," Brakelmann, H., etz-Report 19, 93p. (1984), Berlin; Offenbach.

ID:(15131, 0)