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Heat Transportation

Storyboard

Heat transport through a system composed of multiple media can be estimated by analyzing how heat is conducted within each medium and transferred at each interface. The calculation is performed using the specific parameters of each medium and interface, as well as the temperatures at both ends of the system, thereby providing the temperatures at each interface.

>Model

ID:(1483, 0)



Mechanisms

Iframe

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Code
Concept

Mechanisms

ID:(15277, 0)



Heat transport

Concept

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The basic system includes a transfer generated by the temperature difference (\Delta T), which consists of the temperature difference at internal interface (\Delta T_i), the temperature difference in the conductor (\Delta T_0), and the temperature difference at external interface (\Delta T_e). Therefore:

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



With the heat flow rate (q) being responsible for the transfer between the interior and the conductor, using the internal transmission coefficient (\alpha_i):

q = \alpha_i \Delta T_i



Conduction involves the thermal conductivity (\lambda) and the conductor length (L):

q = \displaystyle\frac{ \lambda }{ L } \Delta T_0



And the transfer from the conductor to the exterior, with the external transmission coefficient (\alpha_e), is represented by:

q = \alpha_e \Delta T_e



All this is graphically represented by:

ID:(7723, 0)



Heat transport between two systems via a third medium

Concept

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The heat flow rate (q) is calculated from the total transport coefficient (multiple medium, two interfaces) (k) and the temperature difference (\Delta T) using the following equation:

q = k \Delta T



where the total transport coefficient (multiple medium, two interfaces) (k) is derived from the external transmission coefficient (\alpha_e), the internal transmission coefficient (\alpha_i), the thermal conductivity (\lambda), and the conductor length (L) through this equation:

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



This is represented in the image below:

ID:(1675, 0)



Temperature Profile

Concept

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Typically, the temperature variation within a conductor follows a linear pattern. However, in the case of gaseous and/or liquid media in contact with the conductor, there is a gradual temperature variation from the center of the medium to the surface, as depicted in the following image:



the outer surface temperature (T_{es}) depends on the outdoor Temperature (T_e), the coefficient of total transportation (k), the external transmission coefficient (\alpha_e), and the temperature difference (\Delta T):

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



the inner surface temperature (T_{is}) is a function of the indoor temperature (T_i) and the internal transmission coefficient (\alpha_i):

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



and the temperature difference (\Delta T):

\Delta T = T_i - T_e

ID:(7722, 0)



Total heat flow transportation

Concept

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When the material includes multiple conductors connected in series, the total transport coefficient (multiple medium, two interfaces) (k) is calculated from the external transmission coefficient (\alpha_e), the internal transmission coefficient (\alpha_i), the thermal conductivity element i (\lambda_i), and the element length i (L_i) using the equation:

\displaystyle\frac{1}{ k }=\displaystyle\frac{1}{ \alpha_i }+\displaystyle\frac{1}{ \alpha_e }+\sum_i\displaystyle\frac{ L_i }{ \lambda_i }



This process is illustrated in the following diagram:

ID:(7721, 0)



Model

Top

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Parameters

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
k
k
Coefficient of total transportation
W/m K
L
L
Conductor length
m
\alpha_e
alpha_e
External transmission coefficient
W/m^2K
\alpha_i
alpha_i
Internal transmission coefficient
W/m^2K
\lambda
lambda
Thermal conductivity
W/m K

Variables

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
q
q
Heat flow rate
W/m^2
dQ
dQ
Heat transported
J
T_i
T_i
Indoor temperature
K
T_{is}
T_is
Inner surface temperature
K
T_e
T_e
Outdoor temperature
K
T_{es}
T_es
Outer surface temperature
K
S
S
Section
m^2
\Delta T
DT
Temperature difference
K
\Delta T_e
DT_e
Temperature difference at external interface
K
\Delta T_i
DT_i
Temperature difference at internal interface
K
\Delta T_0
DT_0
Temperature difference in the conductor
K
dt
dt
Time variation
s

Calculations


First, select the equation: to , then, select the variable: to
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_0 = T_is - T_es DT_e = T_es - T_e DT_i = T_i - T_is q = alpha_e * DT_e q = alpha_i * DT_i q = dQ /( S * dt ) q = k * DT q = lambda * DT_0 / L T_es = T_e + k * DT / alpha_e T_is = T_i - k * DT / alpha_i kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

Calculations

Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

Variable Given Calculate Target : Equation To be used
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_0 = T_is - T_es DT_e = T_es - T_e DT_i = T_i - T_is q = alpha_e * DT_e q = alpha_i * DT_i q = dQ /( S * dt ) q = k * DT q = lambda * DT_0 / L T_es = T_e + k * DT / alpha_e T_is = T_i - k * DT / alpha_i kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt




Equations

#
Equation

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

1/ k =1/ alpha_i + 1/ alpha_e + L / lambda


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

DT = DT_i + DT_0 + DT_e


\Delta T = T_i - T_e

DT = T_i - T_e


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

DT_0 = T_is - T_es


\Delta T_e = T_{es} - T_e

DT_e = T_es - T_e


\Delta T_i = T_i - T_{is}

DT_i = T_i - T_is


q = \alpha_e \Delta T_e

q = alpha_e * DT_e


q = \alpha_i \Delta T_i

q = alpha_i * DT_i


q \equiv \displaystyle\frac{1}{ S }\displaystyle\frac{ dQ }{ dt }

q = dQ /( S * dt )


q = k \Delta T

q = k * DT


q = \displaystyle\frac{ \lambda }{ L } \Delta T_0

q = lambda * DT_0 / L


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

T_es = T_e + k * DT / alpha_e


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

T_is = T_i - k * DT / alpha_i

ID:(15336, 0)



Temperature difference

Equation

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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

T_i
Indoor temperature
K
5208
T_e
Outdoor temperature
K
5207
\Delta T
Temperature difference
K
10161
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15116, 0)



Temperature difference conductor to medium

Equation

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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

T_e
Outdoor temperature
K
5207
T_{es}
Outer surface temperature
K
5214
\Delta T_e
Temperature difference at external interface
K
10167
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15118, 0)



Medium to conductor temperature difference

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}

T_i
Indoor temperature
K
5208
T_{is}
Inner surface temperature
K
5212
\Delta T_i
Temperature difference at internal interface
K
10166
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15117, 0)



Surface temperature difference

Equation

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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 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}

T_{is}
Inner surface temperature
K
5212
T_{es}
Outer surface temperature
K
5214
\Delta T_0
Temperature difference in the conductor
K
10165
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15120, 0)



Total temperature variation

Equation

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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

\Delta T
Temperature difference
K
10161
\Delta T_e
Temperature difference at external interface
K
10167
\Delta T_i
Temperature difference at internal interface
K
10166
\Delta T_0
Temperature difference in the conductor
K
10165
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15115, 0)



Calculation of heat conduction

Equation

>Top, >Model


The heat flux (q) is a function of the thermal conductivity (\lambda), the conductor length (L) and the temperature difference in the conductor (\Delta T_0):

q = \displaystyle\frac{ \lambda }{ L } \Delta T_0

L
Conductor length
m
5206
q
Heat flow rate
W/m^2
10178
\Delta T_0
Temperature difference in the conductor
K
10165
\lambda
Thermal conductivity
J/m s K
5204
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(7712, 0)



Calculation of the total heat transport by a conductor

Equation

>Top, >Model


In this way, we establish a relationship that allows us to calculate the heat flow rate (q) as a function of the total transport coefficient (multiple medium, two interfaces) (k), and the temperature difference (\Delta T):

q = k \Delta T

k
Coefficient of total transportation
W/m^2K
5174
q
Heat flow rate
W/m^2
10178
\Delta T
Temperature difference
K
10161
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

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 heat transported (dQ), the time variation (dt), the section (S)

q = \alpha_i \Delta T_i



q = \alpha_e \Delta T_e



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

q = \displaystyle\frac{ \lambda }{ L } \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

q = k \Delta T

ID:(7716, 0)



Calculation of heat transmission to the conductor

Equation

>Top, >Model


In this way, we establish a relationship that allows us to calculate the heat flow rate (q) based on the temperature difference at internal interface (\Delta T_i), and the internal transmission coefficient (\alpha_i):

q = \alpha_i \Delta T_i

q
Heat flow rate
W/m^2
10178
\alpha_i
Internal transmission coefficient
W/m^2K
10163
\Delta T_i
Temperature difference at internal interface
K
10166
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15113, 0)



Calculation of heat transfer from the conductor

Equation

>Top, >Model


In this manner, we establish a relationship that enables us to calculate the heat flow rate (q) based on the temperature difference at external interface (\Delta T_e), and the external transmission coefficient (\alpha_e):

q = \alpha_e \Delta T_e

\alpha_e
External transmission coefficient
W/m^2K
10162
q
Heat flow rate
W/m^2
10178
\Delta T_e
Temperature difference at external interface
K
10167
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15114, 0)



Temperature on the external surface of the conductor

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

k
Coefficient of total transportation
W/m^2K
5174
\alpha_e
External transmission coefficient
W/m^2K
10162
T_e
Outdoor temperature
K
5207
T_{es}
Outer surface temperature
K
5214
\Delta T
Temperature difference
K
10161
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

With the heat transported (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

q = k \Delta T



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

q = \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)



Temperature on the inner surface of the conductor

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

k
Coefficient of total transportation
W/m^2K
5174
T_i
Indoor temperature
K
5208
T_{is}
Inner surface temperature
K
5212
\alpha_i
Internal transmission coefficient
W/m^2K
10163
\Delta T
Temperature difference
K
10161
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

With the heat transported (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

q = k \Delta T



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

q = \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)



Total transportation coefficient (one medium, two interfaces)

Equation

>Top, >Model


The value of the coefficient of total transportation (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 }

k
Coefficient of total transportation
W/m^2K
5174
L
Conductor length
m
5206
\alpha_e
External transmission coefficient
W/m^2K
10162
\alpha_i
Internal transmission coefficient
W/m^2K
10163
\lambda
Thermal conductivity
J/m s K
5204
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

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 heat transported (dQ), the time variation (dt), the section (S)

q = \alpha_i \Delta T_i



q = \alpha_e \Delta T_e



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

q = \displaystyle\frac{ \lambda }{ L } \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)



Heat flux density

Equation

>Top, >Model


The heat flow rate (q) is defined in terms of the heat transported (dQ), the time variation (dt), and the section (S) as follows:

q \equiv \displaystyle\frac{1}{ S }\displaystyle\frac{ dQ }{ dt }

q
Heat flow rate
W/m^2
10178
dQ
Heat transported
J
10159
S
Section
m^2
5205
dt
Time variation
s
10160
1/ k =1/ alpha_i + 1/ alpha_e + L / lambda q = lambda * DT_0 / L q = k * DT q = alpha_i * DT_i q = alpha_e * DT_e DT = DT_i + DT_0 + DT_e DT = T_i - T_e DT_i = T_i - T_is DT_e = T_es - T_e DT_0 = T_is - T_es T_is = T_i - k * DT / alpha_i T_es = T_e + k * DT / alpha_e q = dQ /( S * dt ) kLalpha_eqdQT_iT_isalpha_iT_eT_esSDTDT_eDT_iDT_0lambdadt

ID:(15133, 0)