Electrical Analogy For Thermal Conduction

Edited By Vishal kumar | Updated on Jul 02, 2025 06:37 PM IST

The concept of electrical analogy for thermal conduction bridges two fundamental branches of physics: electricity and heat transfer. Just as electric current flows through a conductor due to a potential difference, heat flows through a material driven by a temperature difference. In this analogy, temperature corresponds to voltage, thermal conductivity to electrical conductivity, and heat flow to electric current. This analogy is not just a theoretical tool but finds real-life applications in understanding and designing efficient thermal systems, such as in the insulation of homes, where materials are chosen based on their ability to resist heat flow, much like choosing a resistor in an electric circuit. This approach helps engineers and scientists analyze complex thermal systems using well-understood electrical principles, making it easier to predict and manage heat transfer in various practical scenarios.

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Electrical Analogy for Thermal Conduction
Solved Examples Based on Electrical Analogy For Thermal Conduction
Summary

Electrical Analogy for Thermal Conduction

Firstly we discuss Electrical vs Thermal Conductivity
Thermal conductivity and electrical conductivity are two extremely important physical properties of matter. The thermal conductivity of the material defines the rate at which a material can conduct thermal energy. The electrical conductivity of the material describes the amount of electrical current that will result due to a given potential difference. These are both fairly well-characterized properties, with huge applications in fields such as power generation and transmission, electrical engineering, electronics, thermodynamics and heat, and many other fields.

Electrical Conductivity

The resistance of a component depends on various parameters. The length of the conductor, the area of the conductor, and the material of the conductor are to name some. The conductivity of a material can be defined as the conductance of a block having unit dimensions made out of the material. The conductivity of a material is the inverse of the resistivity. Conductivity is usually denoted by the Greek letter σ. The SI unit of conductivity is Siemens per meter.

Thermal Conductivity

Thermal conductivity is the ability of a material to conduct thermal energy. The thermal conductivity is a property of the material. The thermal conductance is a property of the object. The most important law behind thermal conductivity is the heat flow equation. In a mathematical form, this can be written as dH/dt = kA(∆T)/l, where k is the thermal conductivity, A is the cross area, ∆T is the temperature difference between the two ends and l is the length of the object. ∆T/l can be termed a temperature gradient. The thermal conductivity is measured in watts per kelvin per meter.

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Electrical Conduction	Thermal conduction
1. Natural flow of electric charge is from higher potential to lower potential	1. Heat flows from higher temperature to lower temperature
2. The rate of flow of charge is defined as electric current. i.e., $I=\frac{d q}{d t}$	2. The rate of flow of heat may be called heat current. i.e., $H=\frac{d Q}{d t}$
3. Ohm's law gives the relation between the electric current and the potential difference $I=\frac{V_1-V_2}{R}$ where R is the electrical resistance of the conductor	3. Similarly, the heat current may be related to the temperature difference as$H=\frac{\theta_1-\theta_2}{R}$ where R, is the thermal resistance of the conductor
4. From the above point the electrical resistance is defined as $ \begin{aligned} R=\frac{\rho l}{A} & =\frac{l}{\sigma A} \\ \rho & =\text { Resistivity } \end{aligned} $ where, $\sigma=$ Electrical conductivity $ \frac{d q}{d t}=I=\frac{V_1-V_2}{R}=\frac{\sigma A}{l}\left(V_1-V_2\right) $	4. Similarly from the above point the thermal resistance may be defined as $R=\frac{l}{K A}$ where K = Thermal conductivity $\frac{d Q}{d t}=H=\frac{\theta_1-\theta_2}{R}=\frac{K A}{l}\left(\theta_1-\theta_2\right)$

Solved Examples Based on Electrical Analogy For Thermal Conduction

Example 1: For a rod of length 'l', area of cross-section 'A, and thermal conductivity K, the thermal resistance R is given as

1) $R=\frac{l}{K A}$
2) $R=\frac{l^2}{K A}$
3) $R=\frac{K}{l A}$
4) $R=\frac{A}{l K}$

Solution:

From the Electrical Analogy for Thermal Conduction

For a rod of length '$l$', area of cross-section 'A, and Thermal conductivity K,

Thermal resistance is defined as $R=\frac{l}{K A}$

Hence, the answer is the option (1).

Example 2: What is the dimension of Heat current (H)?

1) $M L^2 T^{-3}$
2) $M L^2 T^{-2}$
3) $M L^{-3} T^{-3}$
4) $M L^3 T^{-2}$

Solution:

According to Electrical Analogy for Thermal Conduction

The rate of flow of heat is called heat current.

i.e., $H=\frac{d Q}{d t}$

$
\text { So }[H]=\frac{[\text { heat }]}{[T]}=\frac{[\text { work }]}{[T]}=\frac{[F . L]}{[T]}=\frac{\left[M L T^{-2} * L\right]}{[T]}=\left[M L^2 T^{-3}\right]
$

Hence, the answer is the option (1).

Example 3: The temperature $\theta$ at the junction of two insulting sheets, having thermal resistances $R_1$ and $R_2$ as well as top and bottom temperatures $\theta_1$ and $\theta_2$ (as shown in the figure) is given by :

1) $\frac{\theta_1 R_2+\theta_2 R_1}{R_1+R_2}$
2) $\frac{\theta_1 R_2-\theta_2 R_1}{R_1-R_2}$
3) $\frac{\theta_2 R_2-\theta_1 R_1}{R_2-R_1}$
4) $\frac{\theta_1 R_1+\theta_2 R_2}{R_1+R_2}$

Solution:

The heat flow rate will be the same throughout both

$
\begin{aligned}
& \therefore \frac{\theta_1-\theta}{\mathrm{R}_1}=\frac{\theta-\theta_2}{\mathrm{R}_2} \\
& \mathrm{R}_2 \theta_1-\mathrm{R}_2 \theta=\mathrm{R}_1 \theta-\mathrm{R}_1 \theta_2 \\
& \theta=\frac{\mathrm{R}_2 \theta_1+\mathrm{R}_1 \theta_2}{\mathrm{R}_1+\mathrm{R}_2}
\end{aligned}
$

Hence, the answer is the option (1).

Example 4: Two metallic blocks $\mathrm{M}_1$ and $\mathrm{M}_2$ of the same area of the cross-section are connected to each other(as shown in the figure). If the thermal conductivity of $\mathrm{M}_2$ is K then the thermal conductivity of $\mathrm{M}_1$ will be :
[Assume steady state heat conduction]

1) 10 K
2) 8 K
3) 12.5 K
4) 2 K

Solution:

Since the two blocks are in series, the heat current through both of them will be the same as the steady state
$
\begin{aligned}
& \mathrm{H}_1=\mathrm{H}_2 \\
& \frac{\Delta \mathrm{T}_1}{\mathrm{R}_1}=\frac{\Delta \mathrm{T}_2}{\mathrm{R}_2} \\
& \frac{(100-80)}{\left(\frac{\mathrm{l}_1}{(\mathrm{k}) \mathrm{A}_1}\right)}=\frac{(80-0)}{\left(\frac{\mathrm{l}_2}{\mathrm{k}_1 \mathrm{~A}_2}\right)}
\end{aligned}
$
Here, $\mathrm{A}_1=\mathrm{A}_2=$ Area of cross-section (given) $\mathrm{k}_1 \rightarrow$ Thermal conductivity of block $\mathrm{M}_1$ $\mathrm{l}_1=16 \mathrm{~cm}, \mathrm{l}_2=8 \mathrm{~cm}$

$
\begin{aligned}
& \frac{20}{\left(\frac{16}{k_1}\right)}=\frac{80}{\frac{8}{k_0}} \\
& \mathrm{k}_1=8 \mathrm{k}
\end{aligned}
$

Hence, the answer is the option (2).

Example 5: As per the given figure, two plates $A$ and $B$ of thermal conductivity K and 2 K are joined together to form a compound plate. The thickness of the plates are 4.0 cm and 2.5 cm respectively and the area of the cross-section is $120 \mathrm{~cm}^2$ for each plate. The equivalent thermal conductivity of the compound plate is $\left(1+\frac{5}{\alpha}\right) \mathrm{K}$, then the value of $\alpha$ will be_____________.

1) 21

2) 56

3) 47

4) 6

Solution:

$\begin{aligned} & \text { Thermal resistance }=\mathrm{R}=\frac{1}{\mathrm{KA}} \\ & \mathrm{R}_{\mathrm{eff}}=\left(\mathrm{R}_1+\mathrm{R}_2\right) \\ & \frac{\left(\mathrm{L}_1+\mathrm{L}_2\right)}{\mathrm{Keff} \times \mathrm{A}}=\frac{\mathrm{L}_1}{\mathrm{~K}_1 \mathrm{~A}}+\frac{\mathrm{L}_2}{\mathrm{~K}_2 \mathrm{~A}} \\ & \frac{6.5}{\mathrm{Keff}}=\frac{4}{\mathrm{k}}+\frac{2.5}{2 \mathrm{~K}} \\ & \mathrm{Keff}=\frac{13 k}{10.5}=\left[1+\frac{5}{21}\right] \mathrm{k} \\ & \alpha=21\end{aligned}$

Hence, the answer is the option (1).

Summary

The electrical conduction analogy is a flow of heat through a material analogous to the flow of electric current through a conductor. The temperature gradient is similar to the voltage difference; the continuous flow of heat refers to the electric current; and thermal resistance becomes equivalent to electrical resistance. The electricity flows from the region of higher voltage to that of lower voltage. Similarly, heat flows from a higher to a lower temperature. This analogy helps in understanding and calculating the heat transfer more easily.

Frequently Asked Questions (FAQs)

1. What is the electrical analogy for thermal conduction?

The electrical analogy for thermal conduction is a conceptual model that compares the flow of heat through materials to the flow of electric current through conductors. In this analogy, temperature difference is equivalent to voltage, heat flow rate is analogous to electric current, and thermal resistance corresponds to electrical resistance.

2. How does Fourier's law of heat conduction relate to Ohm's law?

Fourier's law of heat conduction is mathematically similar to Ohm's law for electrical circuits. Fourier's law states that heat flow rate is proportional to temperature difference and inversely proportional to thermal resistance, just as Ohm's law states that electric current is proportional to voltage and inversely proportional to electrical resistance.

3. What is thermal resistance in the context of the electrical analogy?

Thermal resistance is the property of a material that resists the flow of heat, analogous to electrical resistance in circuits. It is determined by the material's thickness, cross-sectional area, and thermal conductivity. Higher thermal resistance means the material is a better insulator.

4. How is heat flow rate analogous to electric current?

Heat flow rate, measured in watts (W), represents the amount of thermal energy transferred per unit time, similar to how electric current represents the flow of charge. Both quantities describe the rate at which something (heat or charge) moves through a medium.

5. What does temperature difference correspond to in the electrical analogy?

Temperature difference in thermal conduction corresponds to voltage (potential difference) in electrical circuits. Just as voltage drives the flow of electric current, temperature difference drives the flow of heat from higher to lower temperatures.

6. What is the thermal equivalent of an electrical diode?

While there isn't a perfect thermal equivalent to an electrical diode, some materials and devices exhibit thermal rectification, allowing heat to flow more easily in one direction than the other. This behavior is analogous to how a diode allows current flow primarily in one direction.

7. What is the thermal equivalent of AC in electrical systems?

Oscillating or periodic heat transfer, where temperatures and heat flow rates vary cyclically over time, can be considered analogous to AC (alternating current) in electrical systems. This occurs in situations with periodic heating and cooling cycles.

8. How can the concept of thermal bridges be related to electrical concepts?

Thermal bridges are paths of least resistance for heat flow, often causing unwanted heat transfer. They can be compared to low-resistance paths or short circuits in electrical systems, allowing for increased flow (of heat or current) compared to surrounding areas.

9. What is the equivalent of a series circuit in thermal conduction?

In thermal conduction, layers of different materials stacked one after another represent a series thermal circuit. The total thermal resistance is the sum of individual thermal resistances, similar to how total resistance in a series electrical circuit is the sum of individual resistances.

10. How can the concept of thermal resistance networks be compared to electrical resistance networks?

Thermal resistance networks, like electrical resistance networks, can be analyzed using similar principles. Complex heat transfer problems can be simplified by breaking them down into networks of thermal resistances, which can be solved using techniques similar to those used for electrical circuits.

11. How does the concept of thermal diffusivity relate to the electrical analogy?

Thermal diffusivity, which describes how quickly a material can conduct thermal energy relative to its ability to store it, doesn't have a direct electrical analogue. However, it can be thought of as similar to the speed at which an electrical signal propagates through a material, combining aspects of conductivity and capacitance.

12. How does the concept of thermal diffusion length relate to the electrical analogy?

Thermal diffusion length, which describes how far heat can diffuse in a material during a specific time period, doesn't have a direct electrical analogue. However, it can be loosely compared to the skin depth in AC electrical systems, which describes how deep current penetrates a conductor at a given frequency.

13. How does the concept of thermal radiation relate to the electrical analogy?

Thermal radiation, unlike conduction, doesn't have a direct electrical analogue in the conduction model. However, it can be loosely compared to electromagnetic radiation in electrical systems, as both involve energy transfer without a physical medium.

14. How can thermal conductivity be related to electrical conductivity?

Thermal conductivity is analogous to electrical conductivity. Materials with high thermal conductivity allow heat to flow easily, just as materials with high electrical conductivity allow electric current to flow easily. However, it's important to note that good electrical conductors are not always good thermal conductors.

15. What is the significance of the electrical analogy in understanding heat transfer?

The electrical analogy provides a familiar framework for understanding heat transfer, making it easier to visualize and analyze thermal systems. It allows us to apply concepts and problem-solving techniques from electrical circuits to thermal problems, simplifying complex heat transfer scenarios.

16. How can thermal insulators be compared to electrical insulators?

Thermal insulators, like electrical insulators, have high resistance to flow. In thermal systems, insulators have high thermal resistance and low thermal conductivity, impeding heat flow. Similarly, electrical insulators have high electrical resistance, impeding the flow of electric current.

17. How does a parallel thermal circuit compare to a parallel electrical circuit?

In a parallel thermal circuit, heat flows through multiple paths simultaneously, just as current flows through parallel branches in an electrical circuit. The total thermal resistance of parallel paths is calculated similarly to parallel electrical resistances, resulting in a lower overall resistance than any individual path.

18. What is the thermal equivalent of Kirchhoff's laws in electrical circuits?

While there isn't a direct equivalent, the principle of energy conservation in thermal systems is analogous to Kirchhoff's laws. In steady-state heat transfer, the sum of heat flows into a point equals the sum of heat flows out, similar to Kirchhoff's current law.

19. How can thermal capacitance be related to electrical capacitance?

Thermal capacitance, which represents a material's ability to store thermal energy, is analogous to electrical capacitance. Just as an electrical capacitor stores electric charge, thermal capacitance describes how much thermal energy a material can store for a given temperature change.

20. What is the thermal equivalent of an electrical short circuit?

A thermal short circuit would be a path of extremely low thermal resistance, allowing heat to flow very easily. This could be represented by a highly conductive material creating a direct path between two temperature points, similar to how a wire creates a low-resistance path in an electrical short circuit.

21. How does the concept of thermal time constant relate to electrical time constant?

The thermal time constant, like its electrical counterpart, represents the time it takes for a system to reach approximately 63% of its final steady-state value after a change. In thermal systems, it depends on thermal resistance and capacitance, just as the electrical time constant depends on resistance and capacitance.

22. What is the thermal equivalent of electrical power dissipation?

In thermal systems, power dissipation is represented by the rate of heat transfer. Just as electrical power is the product of voltage and current (P = VI), thermal power can be expressed as the product of temperature difference and heat flow rate, or as heat flow rate squared times thermal resistance.

23. How can thermal impedance be compared to electrical impedance?

Thermal impedance, like electrical impedance, represents the opposition to flow in dynamic systems. In thermal systems, it accounts for both thermal resistance and thermal capacitance effects, especially in transient heat transfer situations, similar to how electrical impedance combines resistance and reactance.

24. What is the significance of thermal contact resistance in the electrical analogy?

Thermal contact resistance is analogous to contact resistance in electrical connections. It represents the additional thermal resistance that occurs at the interface between two materials, impeding heat flow. This concept is important in understanding heat transfer across material boundaries.

25. What is the thermal equivalent of an electrical ground?

In thermal systems, a heat sink or a constant temperature reservoir can be considered analogous to an electrical ground. It acts as a reference point for temperature differences and can absorb or supply heat without significantly changing its own temperature.

26. How does the concept of steady-state in thermal systems relate to DC in electrical systems?

Steady-state heat transfer, where temperatures and heat flow rates remain constant over time, is analogous to DC (direct current) in electrical systems. Both represent situations where the flow (of heat or charge) is constant and unidirectional.

27. How can thermal insulation be compared to electrical insulation in the analogy?

Thermal insulation, like electrical insulation, serves to reduce the flow of energy. Thermal insulators have high thermal resistance and low thermal conductivity, impeding heat flow, just as electrical insulators have high electrical resistance, impeding the flow of electric current.

28. What is the significance of thermal conductance in the electrical analogy?

Thermal conductance is the reciprocal of thermal resistance, analogous to electrical conductance. It represents how easily heat flows through a material or system. Higher thermal conductance means easier heat flow, similar to how higher electrical conductance means easier current flow.

29. What is the thermal equivalent of electrical resistance per unit length?

The thermal equivalent of electrical resistance per unit length is thermal resistance per unit length. It represents how much a material resists heat flow for a given length, analogous to how electrical resistance per unit length describes opposition to current flow along a conductor.

30. How does the concept of thermal equilibrium relate to the electrical analogy?

Thermal equilibrium, where there is no net heat flow between systems, is analogous to electrical equilibrium where there is no potential difference (voltage) between points. In both cases, the absence of a driving force (temperature difference or voltage) results in no flow (of heat or charge).

31. What is the thermal equivalent of an electrical capacitor?

The thermal equivalent of an electrical capacitor is a heat storage element or thermal mass. Just as a capacitor stores electrical energy, a thermal mass can store thermal energy, affecting the dynamic behavior of the system during temperature changes.

32. How can the concept of thermal feedback be related to electrical feedback?

Thermal feedback, where the output of a thermal system affects its input, is analogous to electrical feedback in circuits. For example, the temperature of a device affecting its own heat generation is similar to how the output of an amplifier can be fed back to its input to modify its behavior.

33. What is the significance of thermal time lag in the electrical analogy?

Thermal time lag, representing the delay between a change in input and the corresponding change in output temperature, is analogous to phase lag in AC electrical systems. It's important in understanding the dynamic behavior of thermal systems, especially in periodic heating and cooling scenarios.

34. How can thermal superposition be compared to electrical superposition?

The principle of thermal superposition states that the temperature distribution caused by multiple heat sources can be found by adding the individual effects of each source. This is analogous to the superposition principle in electrical circuits, where the effects of multiple sources can be analyzed independently and then combined.

35. What is the thermal equivalent of electrical noise?

While there isn't a direct equivalent, thermal fluctuations or random heat transfer variations can be considered analogous to electrical noise. These fluctuations can affect the precision of temperature measurements and heat transfer calculations, similar to how electrical noise affects signal quality.

36. How does the concept of thermal runaway relate to electrical concepts?

Thermal runaway, where increasing temperature leads to further temperature increase in a positive feedback loop, is analogous to certain unstable electrical systems. It can be compared to situations in electrical circuits where increasing current leads to further current increase, potentially causing system failure.

37. What is the significance of thermal interface materials in the electrical analogy?

Thermal interface materials, used to improve heat transfer between surfaces, can be compared to techniques for improving electrical connections. They reduce thermal contact resistance, similar to how conductive pastes or soldering improve electrical connections by reducing contact resistance.

38. How can the concept of thermal spreading resistance be related to electrical concepts?

Thermal spreading resistance occurs when heat flow spreads from a small area to a larger area, increasing overall thermal resistance. This concept is analogous to current spreading in electrical systems, where current distribution affects the effective resistance of a conductor.

39. What is the thermal equivalent of electrical inductance?

While there isn't a direct thermal equivalent to electrical inductance, thermal inertia or the tendency of a material to resist temperature changes can be considered somewhat analogous. It represents a delay in temperature response, similar to how inductance causes a delay in current response in electrical circuits.

40. What is the significance of thermal contact conductance in the electrical analogy?

Thermal contact conductance, the inverse of thermal contact resistance, represents how easily heat flows across the interface between two materials. It's analogous to electrical contact conductance, which describes how easily current flows across connections in electrical circuits.

41. How can the concept of thermal effusivity be related to electrical concepts?

Thermal effusivity, which describes a material's ability to exchange thermal energy with its surroundings, doesn't have a direct electrical analogue. However, it combines aspects of thermal conductivity and heat capacity, somewhat similar to how electrical impedance combines resistance and reactance in AC circuits.

42. What is the thermal equivalent of an electrical transformer?

While there isn't a direct thermal equivalent to an electrical transformer, heat exchangers can be considered somewhat analogous. They transfer heat between two fluids without mixing them, similar to how transformers transfer electrical energy between circuits without direct electrical connection.

43. What is the significance of thermal boundary conditions in the electrical analogy?

Thermal boundary conditions, which define the temperature or heat flux at the boundaries of a system, are analogous to voltage or current constraints in electrical circuits. They determine how heat flows within the system, similar to how electrical boundary conditions affect current flow.

44. How can the concept of thermal shock be related to electrical concepts?

Thermal shock, caused by rapid temperature changes, can be loosely compared to voltage spikes or surges in electrical systems. Both can cause stress and potential damage to the system if not properly managed.

45. What is the thermal equivalent of electrical resonance?

While there isn't a direct thermal equivalent to electrical resonance, certain thermal systems can exhibit resonant-like behavior under periodic heating. This can occur when the frequency of temperature oscillations matches the natural frequency of the thermal system, leading to amplified temperature fluctuations.

46. How does the concept of thermal management in electronics relate to the electrical analogy?

Thermal management in electronics involves controlling heat flow to maintain optimal operating temperatures. This can be compared to power management in electrical systems, where current and voltage are controlled to ensure proper operation and prevent damage.

47. What is the significance of thermal anisotropy in the electrical analogy?

Thermal anisotropy, where a material's thermal properties vary with direction, is analogous to anisotropic electrical conductivity in certain materials. Both concepts describe how the flow of energy (heat or charge) depends on the direction within the material.

48. How can the concept of thermal bottlenecks be related to electrical bottlenecks?

Thermal bottlenecks, points in a system where heat transfer is restricted, are analogous to electrical bottlenecks where current flow is limited. Both can lead to localized increases in temperature or voltage, respectively, and can limit the overall performance of the system.

49. What is the thermal equivalent of electrical skin effect?

While there isn't a direct thermal equivalent to the electrical skin effect, the concept of thermal penetration depth in oscillating heat transfer problems bears some similarity. It describes how deep thermal oscillations penetrate into a material, analogous to how AC current is concentrated near the surface of a conductor due to the skin effect.

50. How does the concept of thermal figure of merit relate to electrical efficiency?

The thermal figure of merit, often used in thermoelectric materials, combines thermal and electrical properties to describe the efficiency of energy conversion. While not directly analogous, it serves a similar purpose to measures of electrical efficiency in power systems, indicating how effectively energy is utilized or converted.