### Thermogalvanic Cell

June 25, 2014

Rules of thumb are nice to have when no other

data are available. The first such rule I learned is, "

a pint's a pound the world round." If you insist on using the

English system of units, as we do in the

United States, this means that a

gallon of

water weighs about eight

pounds, as does a gallon of

milk, and a gallon of nearly every other

liquid.

The actual values are 8.34 lb./gal. for water and 8.6 lb./gal. for

whole milk.

Gasoline is a lightweight, weighing in at just 6.1 lb./gal. Pure

octane has a density of 5.84 lb./gal, while

ethanol has a density of 6.58 lb./gal.

Another rule of thumb that I learned from a

metallurgy professor is that the

density of everything is five. That's because the average density of the

Earth is 5.5, since it's a composite of

crustal rock (density about 2.5), and the density of

iron (about 7.87) comprising its

core.

I wrote about other

scientific rules of thumb in an

earlier article (Rules of Thumb, September 5, 2008). These include the

Dulong and Petit Law, that the

specific heat of a

material is always three times the

gas constant (R) divided by the

molar mass, and the

Wiedemann-Franz Law, that the

ratio of the

thermal conductivity to the

electrical conductivity of a

metal is

proportional to the

temperature.

It's not surprising that there's a

web site,

rulesofthumb.org, devoted to this topic. This site has more than five thousand such rules, just a small fraction of which involve

science and

technology. One of my favorites there is

Rule 1966, "The shorter the life of an

elementary particle, the more it costs to make." Also entertaining is

Rule 1686, "For

mathematics professors, each

published math paper is worth $10,000 in

salary."

Once you have such basic data, it's time for

calculation. As I wrote in

"Estimation" (December 21, 2011),

Nobel physics laureate, Enrico Fermi was a master at estimating nearly anything, including the number of piano tuners in Chicago (in the days when many people still had mechanical pianos in their

homes). His

back-of-the-envelope calculations would infer the desired quantity by a very long chain of estimated values using simple models. There was a book, "Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin," published on this topic in 2008.[1]

All this is preface to an important

chemist's rule of thumb that the

rate of most

reactions doubles every ten

degrees Celsius. This rule is a consequence of the

Arrhenius equation, which expresses the

reaction rate **k** in terms of an

activation energy **E**_{a} and

absolute temperature **T**; viz.,

Where

**R** is the

gas constant. This could be called the "chemist's" version of the equation, especially when the gas constant is expressed in

calories as 1.9872 cal/K/mol. The "

physicist's" version of the equation, shown below, uses the

Boltzmann constant **k**_{B}, instead. The units for the activation energy are different between these equations.

Activation energies vary widely, but a reasonable value is 12.5

kcal. Plugging this into the Arrhenius equation gives a rate change of 1.96 when going from 300

K to 310 K. This large temperature dependence of reaction rate allows us to speed

exothermic reactions by

heating, or slow them by

cooling. Changing the reaction rate also changes the

equilibrium mixture of

products and

reactants.

Scientists from the

Department of Materials Science and Engineering,

Stanford University (Stanford, California), the

Department of Mechanical Engineering,

Massachusetts Institute of Technology (Cambridge, Massachusetts), and the

Stanford Institute for Materials and Energy Sciences,

SLAC National Accelerator Laboratory (Menlo Park, California) have demonstrated that such

thermochemical effects can be used for

thermal energy-harvesting at small differential temperatures. Their thermogalvanic cell is described in a recent issue of

Nature Communications.[2-3]

The operation of such a thermogalvanic cell for energy-harvesting is shown in the figure. An

electrochemical cell is heated so that its

voltage becomes lower. It is then

charged at a higher temperature, using a low voltage. After the cell is cooled, its voltage becomes higher, and it's

discharged at this lower temperature at the high voltage. The

heat energy is harvested from the voltage difference between the high and low temperature states.

The thermogalvanic cell is proposed for

generating electricity through energy harvesting of the

waste heat available from many processes. This heat is generally available only at a temperature just a hundred degrees

Celsius higher than the

Environment.[3] Energy -harvesting of such heat sources is presently done by low

efficiency thermoelectric devices.[2] MIT's

Gang Chen, an

author of the study, explains that the concept goes back to the 1950s, but

materials at that time could not make a reasonably effective device.[3]

The electrochemical system is based on a

copper hexacyanoferrate cathode and a Cu/Cu2+

anode.[2] The device showed an

energy conversion efficiency of 5.7% in cycling between 10°C and 60°C.[2-3] Says

Yuan Yang, an author of the study and a

postdoctoral associate in MIT's

Mechanical Engineering Department, "One-third of all energy consumption in the

United States ends up as low-grade heat."[3]

This method of thermal energy conversion has several present problems, the first of which is that it doesn't operate against a

temperature gradient as thermoelectrics do; rather, the entire device needs to be heated and cooled in a cycle. Another problem, shared with all

battery systems, is its low

power density. Then there's the speed of charging and discharging.[3]

Funding for this research came from the

US Department of Energy, the

US Air Force, and the

National Research Foundation of Korea.[3]

### References:

- Lawrence Weinstein and John A. Adam, "Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, Princeton University Press, April 21, 2008, (via Amazon).
- Seok Woo Lee, Yuan Yang, Hyun-Wook Lee, Hadi Ghasemi, Daniel Kraemer, Gang Chen and Yi Cui, "An electrochemical system for efficiently harvesting low-grade heat energy," Nature Communications, vol. 5, article no. 3942 (May 21, 2014), doi:10.1038/ncomms4942.
- David L. Chandler, "A new way to harness waste heat," MIT Press Release, May 21, 2014.