Fourier
Is this French
mathematician the true father
of modern engineering?
Concepts
that engineers use every day—as fundamental as the homogeneity
of equations and the heat transfer coefficient—were pioneered by
a French thinker who died in 1830. His name was Joseph Fourier. He is
better known for his career in mathematics, but his contributions to engineering
science are so important that a case can be made for calling him the father
of modern engineering.
Fourier's contributions to engineering science, many of which were presented
in his 1822 book, The Analytical Theory of Heat, include the original
view of dimensional homogeneity. The heat transfer science it presented
has been handed down to us virtually unchanged, and has served as a model
for other branches of engineering.
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Joseph Fourier
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The book also presented his concept of "flux" (that is, a flow of something
per unit area and time), his view of homogeneity, and his original methods
for solving engineering problems, all of which are used today in many
branches of engineering and science.
In short, this treatise by Fourier presented the groundwork, as well as
some of the finish work, for modern engineering. Fourier conceived the
view that scientifically rigorous equations must be dimensionally identical—that
is, each term in an equation must have the same dimension. For example,
if the left side of an equation is pounds per cubic foot, the dimension
of the right side must also be pounds per cubic foot. If the left side
is measured in pounds per cubic foot and the right side feet per second,
the equation is irrational.
Fourier's view of homogeneity is now considered almost self-evident, but
in the early 19th century, it was revolutionary. It required the multiplication
and division of dimensions—mathematical operations that had been
deemed irrational for more than 2,000 years.
Fourier was born in Auxerre, France, in 1768. He trained for the priesthood,
but spent much of his life teaching mathematics at French universities,
principally the École Polytechnique. He was active in the French
Revolution, and was twice imprisoned for his political activism. He acted
as Napoleon's scientific adviser and sometime administrator in the Egyptian
campaign, from 1798 to 1801. From 1804 to 1807, he was prefect of Grenoble,
a post that he reluctantly accepted because the appointment was made by
Napoleon. He was elected to the Académie des Sciences in 1817,
and served as its secretary.
Before Fourier's time, Newton and his successors generally expressed laws
in the form of proportional expressions. For example, Hooke's law says
that stress is proportional to strain. Newton's second law holds that
the change of motion is proportional to the impressed force.
But Fourier was not satisfied with proportional expressions. He wanted
to arrive at laws in the form of equations that could be used quantitatively
to describe and predict natural behavior, specifically of heat transfer.
To do it, Fourier had to create a new kind of parameter.
During his years at Grenoble, he conducted heat transfer experiments.
In the manner of his predecessors, he could have considered his work finished
when he observed that convective heat flux is proportional to temperature
difference and that conductive heat flux is proportional to temperature
gradient. Neither expression will yield a homogeneous equation.
The proportional expression for convective heat transfer is qconv
= aDT, where a
is a pure number generally referred to as the constant of proportionality.
Fourier would not accept that as a law because q and DT
have different dimensions: The left side is energy flow per unit time
and area, and the right side is temperature.
He solved the dilemma by stating that a is
a parameter with the same dimension as q/DT,
which makes the equation homogeneous. Rather than retain a generic name
and symbol for the new parameter, he called it "heat transfer coefficient"
and gave it the symbol h. The end result is Fourier's law of convective
heat transfer, qconv = h
DT. (American heat transfer texts
call this equation "Newton's law of cooling," but it should be attributed
to Fourier.)
By a similar process, Fourier arrived at the law of conductive heat transfer,
qcond = k dT/dx where
the constant of proportionality has been assigned the name "thermal conductivity,"
the symbol k, and the same dimension as q/(dT/dx).
Fourier's view of homogeneity makes it necessary to create parameters
such as resistances and coefficients because without them, engineering
phenomena cannot be described by homogeneous equations. Engineering phenomena
are cause-and-effect processes: electromotive force causes electric current;
temperature difference causes heat flux; stress causes strain.
Because cause and effect generally have different dimensions, a third
parameter is necessary to obtain a homogeneous equation.
Ohm's law underwent a transformation from its original form to make it
homogeneous. Georg Ohm published his treatise, The Galvanic Circuit
Investigated Mathematically, in 1827. He originally expressed his
famous law as: "The force of the current in a galvanic circuit is directly
as the sum of all the tensions, and inversely as the entire reduced length
of the circuit." Reduced length is the equivalent length of a copper wire
of a standard diameter.
As an equation, it was I = E/L, which does not conform to Fourier's
view of homogeneity. To render it homogeneous, a parameter was later assigned
the dimension "ohm" (a synonym for volts per ampere), and it is now called
"electrical resistance." The homogeneous form of the equation is E
= IR.
Hooke's law, that "stress is proportional to strain," also was transformed
into a homogeneous equation in the manner pioneered by Fourier. It was
stated that the proportionality constant between stress and strain was
a parameter. The parameter was assigned the same dimension as stress,
since that would make the equation homogeneous. This parameter is now
called "material modulus." The homogeneous equation based on Hooke's law
is called "Young's law."
Fourier's contemporaries forestalled the general publication of his work
for 15 years while they claimed to find fault with it. For example, they
strongly objected to his concept of flux, a concept that now seems so
simple and straightforward as to border on the obvious.
They ultimately accepted his revolutionary view of homogeneity, solely
because he was able to solve many practical and theoretical problems that
had never been solved. He attributed his success to the homogeneity in
his equations.
Eugene F. Adiutori is the author of The New Heat
Transfer, which was published in the 1970s in English and Russian. His
article, "Origins of the Heat Transfer Coefficient," appeared
in Mechanical Engineering magazine in August 1990.
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