Editing Lumped-element model (section)

==== Thermal purely resistive circuits ====

A useful concept used in heat transfer applications once the condition of steady state heat conduction has been reached, is the representation of thermal transfer by what is known as thermal circuits. A thermal circuit is the representation of the resistance to heat flow in each element of a circuit, as though it were an [[electrical resistor]]. The heat transferred is analogous to the [[electric current]] and the thermal resistance is analogous to the electrical resistor. The values of the thermal resistance for the different modes of heat transfer are then calculated as the denominators of the developed equations. The thermal resistances of the different modes of heat transfer are used in analyzing combined modes of heat transfer. The lack of "capacitative" elements in the following purely resistive example, means that no section of the circuit is absorbing energy or changing in distribution of temperature. This is equivalent to demanding that a state of steady state heat conduction (or transfer, as in radiation) has already been established.

The equations describing the three heat transfer modes and their thermal resistances in steady state conditions, as discussed previously, are summarized in the table below:

{| class="wikitable" style="margin:1em auto; text-align:center;"
|+Equations for different heat transfer modes and their thermal resistances.
|-
!Transfer Mode
!Rate of Heat Transfer
!Thermal Resistance
|-
|Conduction
|<math>\dot{Q}=\frac{T_1-T_2}{\left ( \frac{L}{kA} \right )}</math>
|<math>\frac{L}{kA}</math>
|-
|Convection
|<math>\dot{Q}=\frac{T_{\rm surf}-T_{\rm envr}}{\left ( \frac{1}{h_{\rm conv}A_{\rm surf}} \right )}</math>
|<math>\frac{1}{h_{\rm conv}A_{\rm surf}}</math>
|-
|Radiation
|<math>\dot{Q}=\frac{T_{\rm surf}-T_{\rm surr}}{\left ( \frac{1}{h_rA_{\rm surf}} \right )}</math>
|<math>\frac{1}{h_rA}</math>, where<br /><math>h_r= \epsilon \sigma (T_{\rm surf}^{2}+T_{\rm surr}^{2})(T_{\rm surf}+T_{\rm surr})</math>
|}

In cases where there is heat transfer through different media (for example, through a [[composite material]]), the equivalent resistance is the sum of the resistances of the components that make up the composite.  Likely, in cases where there are different heat transfer modes, the total resistance is the sum of the resistances of the different modes. Using the thermal circuit concept, the amount of heat transferred through any medium is the quotient of the temperature change and the total thermal resistance of the medium.

As an example, consider a composite wall of cross-sectional area <math>A</math>. The composite is made of an <math>L_1</math> long cement plaster with a thermal coefficient <math>k_1</math> and <math>L_2</math> long paper faced fiber glass, with thermal coefficient <math>k_2</math>. The left surface of the wall is at <math>T_i</math> and exposed to air with a convective coefficient of <math>h_i</math>. The right surface of the wall is at <math>T_o</math> and exposed to air with convective coefficient <math>h_o</math>.

Using the thermal resistance concept, heat flow through the composite is as follows:
<math display="block">\dot{Q}=\frac{T_i-T_o}{R_i+R_1+R_2+R_o}=\frac{T_i-T_1}{R_i}=\frac{T_i-T_2}{R_i+R_1}=\frac{T_i-T_3}{R_i+R_1+R_2}=\frac{T_1-T_2}{R_1}=\frac{T_3-T_o}{R_0}</math>
where <math>R_i=\frac{1}{h_iA}</math>, <math>R_o=\frac{1}{h_oA}</math>, <math>R_1=\frac{L_1}{k_1A}</math>, and <math>R_2=\frac{L_2}{k_2A}</math>