**Analogy Between
Thermal and Electrical Conduction**

**_____________________________________________________________________________**

**The
Analogy **

** **In the development of his law for electrical circuits, Georg Ohm performed experiments that modeled Fourier’s law
of heat conduction. Consequently, an analogy between heat and electrical
conduction can be observed. Symbolically Ohm’s law can be expressed as

where *I*
is the current flowing through an element, *ΔV*
is the voltage across the element, and *R _{e}*
is the electrical resistance across the element. Similarly Fourier’s law
can be written as

where*
q* is the rate of heat conduction, *ΔT*
is the temperature difference between the surfaces of a slab, and *R _{t}* is
the thermal resistance between the surfaces defined as

where *k*
is the thermal conductivity and *A* is
the effective area of the resistance. By
using the convective heat transfer coefficient *h*, the convective thermal resistance from a surface to a fluid can
also be calculated by the equation

Schematic
Representation of Thermal-Electrical Analogy

Since heat is understood to transfer
through lattice vibrations between adjacent atoms, and electricity is conducted
by free valence electrons of atoms, it may seem intuitive that heat and
electrical conduction are analogous. Although this is most often true,
some non metallic materials like diamond exhibit dissimilar heat and electrical
conductivities.

**Uses
of the Thermal-Electrical Analogy**

As found in the
referenced literature, the analogy is useful in the analysis of several steady
heat transfer problems from property measurement to modeling. In
modeling, a complicated heat transfer analysis can be made much simpler by
creating an “electric circuit” like model of the problem. This can be
seen in the illustrative examples below. In the first example, a simple comparison
between single and triple paned windows is modeled. In the second example an entire house is
modeled using a thermal network. Also
evident from the illustrative examples is the fact that one should take
additional care in determining if system components are acting in series or
parallel.

**Limits of the Thermal-Electrical Analogy**

Although generally extremely useful, the analogy is not without
its own limitations. These include the
non-linearities that occur between voltage and
current at extremely high and low values. The literature survey would
also lead to special cases where the analogy doesn’t quite fit the actual
physics of a problem. Recall the constitutive models of heat transfer are
based on the continuum concept. These
special cases are at extremely large or small values of current and voltage as
well as heat transfer problems at the nano-scale
level. Despite this,
modifications can be made to still find the relationship helpful in the
analysis.

Consider the
steady heat loss out of a residential glass window. The first case is a single pane window, and
the second is a triple paned window.

Given Parameters:

Area of windows: 1
m^{2}

Thickness of single
paned window L: 1
cm

Thickness of
panes in triple paned window l: 0.3
cm

Thickness of air
gap between panes: 0.5
cm

Temperature in
house T_{∞ in }: 20°C

Temperature
outside T_{∞ out }: -10°C

Constant thermal
conductivity of glass k_{1}: 0.78
W/(m*°C)

Constant thermal
conductivity of air k_{2}: 0.026
W/(m*°C)

Convective heat
transfer coefficient outside h_{out}: 40 W/(m*°C)

Convective heat
transfer coefficient inside h_{in}: 10 W/(m*°C)

Radiative heat transfer resistance R_{rad}: 0.03
°C/W

Instead of doing
a heat conduction analysis of each layer starting from first principles, a
simplified thermal resistance network can be used.

**Case 1:**

Calculate thermal resistances:

R_{rad} = given

R_{conv1}=1/(h_{out}×A) =
**0.025 °C/W**

R_{conv2} = 1/(h_{in}×A) = **0.1 °C/W**

R_{glass}=L/(k_{1}×A) = **0.0128 °C/W**

** **

** **

** **

**Case 2:**

Calculate thermal resistances:

R_{rad} = given = **0.03 °C/W**

R_{conv1}=1/(h_{out}×A) = **0.025 °C/W**

R_{conv2} = 1/(h_{in}×A)
= **0.1 °C/W**

R_{glass} = l/(k_{1}×A)
= **0.00385 °C/W**

R_{air}=(L/2)/(k_{2}×A) = **0.1923 °C/W**

** **

** **

** **

Comparison: The rate of heat loss in Case 2 with triple
paned windows was much less than Case 1 with a single pane. Note the high resistance of the air
gaps. Temperatures on the inside and
outside surfaces as well as in the gaps can now be calculated as well.

**Background:**

Energy prices are
increasing. This is likely due to
depleting resources, environmental concerns about its production, and
diplomatic conflicts in traditional energy rich nations. Consequently, the need for energy
conservation is becoming an important economic, political, and environmental
issue. A major consumer of energy is
heating systems in residential homes.
Heating and cooling systems frequently use over 60% of the energy
consumed in residential buildings and even more in office and educational
buildings [13]. Therefore, to conserve
energy, the heating system in residential buildings is a logical place to
start. In addition to making heating
furnaces more efficient, it is essential to retain the heat supplied by the
furnace. This is done by insulating a
home during its construction. Insulating
materials in residential home construction are often rated using *R-values*. This is similar to the thermal resistance
value discussed in the preceding sections, except that thermal resistances are
the R-values divided by the effective area.
To further describe the thermal electrical analogy, a simple thermal
resistance example is discussed using a typical residential home in the Midwest
United States.

**Building used in Example:**

To illustrate the
thermal resistance concept, a house constructed in 1985 in the southwest

**Calculating the Overall Thermal Resistance**

As opposed to
performing a one-dimensional heat transfer analysis on each contributing layer
of construction, the thermal-electrical analogy was used and an overall thermal
resistance network or circuit was used.
This was done by taking into account all the building materials between
the inside and outside environment acting in either series or parallel. A schematic illustration of this circuit can
be seen in figure 3.

**Figure 3. Thermal Resistance Network of House**

Next, the individual thermal resistance of
each contributing material was determined.
This was done by using the dimensional specifications of the building in
combination with either published thermal conductivities or R values of the
various building materials in the house.
These values are tabulated in Table 2.
Using the thermal circuit model, a total thermal resistance for the
house can be calculated by the equation

(8)

where

(9)

and

(10)

where

(11)

and

(12)

By substitution the equation for the total
thermal resistance* R _{total} *of the
house can now be written as

(13)

and by substituting the values in table 1,
the total thermal resistance is

(14)

From the initial
temperature a heat loss rate could be calculated using the total thermal
resistance of the house calculated above using equation (5) and listed in table
1 resulting in **591.41 W**. Using
$1.87 per gallon as published on the DOE website [15], this converts to about
4.1 cents per hour.

**Table 1. Calculated and Measured
Parameters**

**Table 2. Component Resistance Values**

**Confirmation of Calculated Thermal
Resistance**

To check the
calculated thermal resistance, a simple experiment was conducted by taking
temperatures with the furnace turned off for four hours during the middle of
the day. During this period there was
negligible wind and temperature variation outside of the house. Temperatures were taken before shut off and
after 4 hours. One way heat loss can be
measured is by using the equation

(15)

where *ρ _{air}*
is the mass density of the indoor air,

** Results**

From table 1 we
can see a substantial discrepancy in the rate of heat loss calculated using the
thermal resistance and the rate of heat loss using the psycrometric
chart. Before immediately abandoning the
analysis of thermal resistances, several factors contributing to heat gain,
other than the furnace, need to be considered.
For one, light. The house is lit
with incandescent light bulbs which give off considerable amounts of heat due
to their inefficiencies. For instance,
assume during the 4 hour shut-off period there were 5- 60 W incandescent light
bulbs on and running at a typical 10% efficiency. That would contribute 270 W to the
house. In addition to light, consider
there was one adult man sitting in the house during the shut-off. The average adult man gives off body heat at
a rate of 108 W when seated in a room [5].
These factors alone when subtracted from the heat losses calculated
using the thermal resistances result in a much smaller discrepancy of
7.1%.

**References**

1. Kakaç S., Yener, Y.(1993) Heat Conduction 3^{rd} ed.

2.
K. C. Cheng (1992) Heat Transfer Engineering,
Volume 13, Issue 3 1992

3. Rizzoni, Giorgio Principles and Applications of Electrical Engineering
3^{rd} ed. 2000 McGraw-Hill

4. Srivastava G. P (1990),
"The Physics of Phonons." Adam Hilger, IOP Publishing Ltd, Bristol.
URL
http://books.google.com/books?id=OE-bHd2gzVgC&dq=physics+of+phonons&printsec=frontcover&source=web&ots=5U8r_P6sh0&sig=mzLYEu_l1Y5JsPIiUVDfyq2adZI#PPP1,M1

5.
Çengel, Yunus A. (2003), Heat Transfer
An Engineering Approach 2^{nd} ed. McGraw-Hill

6.
Lozano, Francesc (2005), Thermal Conductivity and Specific Heat Measurements for
Power Electronics Pakaging Materials. Thesis from University of Barcelona,
Spain.

7.
Lang, Walter (1990) IEEE Transactions on Electron Devices. Vol 37. No 4.

8.
Wang et al. (2005) IEEE Transactions on Energy Conversion. Vol 20. No. 2.

9.
Tang et al. (2005) IEEE Transactions on Energy Conversion. Vol 20. No. 1.

10.
Kays, W.M., (1980) Crawford, M.E. Convective Heat and Mass Transfer 2^{nd}
ed. McGraw-Hill New York, NY.

11.
Chen, Gang (2003) Rohsenow Symposium on Future Trends of Heat Transfer, MIT
Cambrige, MA

12.
Bennett, Andrew John (1999) Coupled Nano-structure Lattices. Thesis from
Universisty of Toronto. Ontario, Canada.

13. Mull Thomas E. (1998) HVAC Principles and Applications.

14. Moran, Michael J., Shapiro, Howard N. (2004) Fundamentals of
Engineering Thermodynamics. John Wiley & Sons, Inc.

15. DOE Federal
Registrar Vol. 72, No. 54

*Special thanks to the Lindberg household
for letting me freeze them for 2 days.*