'Development of the theory of critical flow through nozzles: an international achievement'

Business card stapled to top left corner:

Flow Measurement
Fluid Systems

Henry S. Hillbrath P.E.

P.O. Box 80271
Seattle, Wash. 98108

Tel. (206) 763-9233




Cover page:

DEVELOPMENT OF THE THEORY OF CRITICAL FLOW THROUGH NOZZLES
AN INTERNATIONAL ACHIEVEMENT

HENRY S. HILLBRATH, P. E.

PRESENTED TO:

WORKING GROUP ON CRITICAL FLOW
INTERNATIONAL ORGANIZATION FOR STANDARIZATION
ISO/TC30/SC2/WG5

AMSTERDAM, HOLLAND
OCT. 6, 1975




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The history of compressible fluid theory (and of Science itself) may be
considered to begin with the Greeks. The origins of pneumatic technology is
even older and many devices, such as the siphon and bellows, were known to
the Egyptians. However, the Greeks were the first to leave a written record
of their scientific speculations.

Pneumatics was an important part of the Greek science. Indeed, one early
philosopher, Anaximenes, believed that air was the elemental substance from
which the universe is made. Before 400 B. C., Empedocles and Anaxagoras were
giving what we would consider to be arguments from experimental evidence that
air is a material body.

Ctesibius, a contemporary of Archimedes, attained an international fame
for his construction of a variety of elaborate pneumatic and hydraulic machines
in about 200 B. C. These included water clocks, keyboard organs, animated
displays with sound effects, and fire pumps.

About 100 A. D., Hero of Alexandria wrote a sort of handbook which summarized
much of the science and technology of the era including pneumatics. The Aleoplie
is well know (although it is often not realized that even though Hero may have
considered it only a toy, which I doubt, many later engineers applied it to
practical service) but it is merely the tip of the iceberg of a whole catalog
of ideas which has directly stimulated innovators for nearly 19 centuries.
For example, all the ideas for the Savery and Newcomen steam engines, as well
as both the impulse and reaction turbines, are sprinkled through the work. Some
archeological discoveries indicate that rather than the literature being in
advance of the actual construction of devices, as scholars have been prone to
assume, that some of the actual devices constructed were actually far more complex
that the simple examples in the few surviving documents.

Although we now lump all these early philosiphers together as “Greek”,
Anaximenes and Anaxagoras lived in Turkey, Empedocles and Archimedes in
Sicily, and Ctesibius and Hero in Egypt.

Hero’s influence was very great, as his works survived and most of them
became available when the demand was right for them at the beginning of the
scientific revolution. Galileo, Boyle and Newton all began with Hero as a text.
The investigation of the properties of a vacuum (which was begun by the Greeks
and described in some detail by Hero) played an important part in the scientific
revolution.

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Beginning with Galileo and Torricelli in Italy, the study of vacuum was carried
on by Guericke in Germany, Pascal in France, and Boyle and Hooke in England.
By the late 1600’s, it reached the level of a socially popular fad.

By the 1680’s, the understanding of vacuum and fluid behavior had progressed
sufficiently that Dennis Papin, an associate of Boyle who might be said to be the
“Father of Compressible Fluid Dynamics,” could meaningfully address the question:
”How fast does air rush into a vacuum?” Papin was seeking to analyze the performance
of a vacuum gun he had built. He assumed, as many were to do after him, that the
same velocity would result as for an incompressible fluid. Papin was a Frenchman,
received his early training from a Dutchman (Huygens), and working in England, Italy,
and Germany.

By 1820, the static properties of compressible fluids, in particular the
variation of density with pressure and temperature, had been adequately estab-
lished, the flow of incompressible fluid had been extensively investigated by
the Bernoullis, Euler, and LeGrange, among others, the operation of many com-
mercially important devices (steam engines, in particular) depended on the flow
of compressible fluids and (at least in France) schools had been established which
were transmitting the mathematical and physical basis from which compressible flow
theory could be derived to relatively large numbers of students.

The most important contributors to the study of compressible fluid properties
were Boyle (in England) and Guy-Lussac (in France). The Bernoullis were Flemish,
but moved to Switzerland. Daniel, whose fluid mechanics works are best know, was
born in Groningen, lived most of his life in Basel, and wrote his most important
works in St. Petersburg. Euler was Swiss, and did most of his work in Russia and
Germany. LaGrange was born in Italy of French parents and spent most of his active
period in Germany. The steam engine was primary a British development.

Although the scientific and mathematical basis was now available, compressible
flow theory was in fact unchanged from Papin’s time. In 1828 Navier, in France,
published an analysis of which agree with the modern viewpoint. The phenomena of
critical flow is implicit in Navier’s equations, but he apparently did not observe
this.

In the mid 1830’s, de Pambour, in testing steam locomotives, observed results
which were suggestive of the phenomena. These results were discussed by Gaspard

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de Coriolis. Apparently at his suggestion, the subject was taken up by two of
Coriolis’ younger associates, A. J. C. Barre de Saint Venant and Laurent Wantzel.
In 1838 they published a classic paper in which they gave the “modern” equation
for the adiabatic flow of a compressible fluid through a nozzle. The correctly
argued that both their analyses and those of Navier implied the existence
of the critical flow regime. They also presented suppo9rting experimental evidence
that critical flow did occur. Although their analyses and experiments were, in
the modern view, correct, they were not convincing to anyone.

In 1843, Saint Venant and Wantzel became embroiled on controversy on their results
with J. V. Poncelet, who was said to be the leading mechanical engineer of the
period. Poncelet began with the assumption that Papin was right and strongly
attacked the concept of critical flow. He argued that in passing through the
mozzle that the fluid did not have time to relax and that its density remains
constant. After a brief debate, Poncelet apparently “pulled rank.” The result
was an impasse and little development occurred in this field in France for another
40 years.

Between 1843 and 1873, the subject was investigated by a number of workers with
varying success. These included William Froude, who held as other were to do
later that a compressible fluid could not flow into a vacuum at all! Eli Blake,
a Connecticut Yankee from New Haven and a nephew of Eli Whitney, gave a good
intuitive argument for and a reasonably satisfactory experimental confirmation
of critical flow before 1850. Joule and Thomson, in Britain, correctly deduced
the nozzle equation son thermodynamic principles, but also thought (as Fraude had),
that the flow rate was reduced at pressures below the critical value.

Julius Weisbach, in Germany, obtained the modern equation and gave some experi-
mental support at inlet pressures up to about two atmospheres, but did not deduce
the pheonomena of critical flow. Robert Napier, a Scott, ran tests with steam
from which he deduced the phenomena and gave his results in the form of empirical
equations. J. M. Rankine, also a Scott, gave a very satisfactory analysis of
Napier’s results.

The work of all of these individuals was flawed by the lack of knowledge of the
other work of individuals in the field. In particular, they were not aware of
Saint Venant and Wantzel and none of them advanced very much beyond their findings,
if any. (Equally, Saint Venant, although active until his death in 1886, never
seems to have reentered this particular fray after his bout with Poncelet. Wantzel
died in 1848).

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Just as they were unaware of their predecessors (with one exception),
none of these workers had any “progeny” – that is, they had no associate workers
to carry on with their work. The exception was Weisbach. One of his students,
Gustav Zeuner, played a vital (but now all but forgotten) role in the transmission
and development of the theory of critical flow; we can all trace a short and direct
line of descent from him to our own experience. At various times, Zeuner was a
professor at Freiburg (Weisbach’s school), Swiss Federal and Dresden. He
established a laboratory at Swiss Federal which was similar to Weisbach’s but
which could operate at higher pressures. He wrote a number of papers which
pulled together the work of Navier, Saint Venant and Wantzel with that of
Weisbach, Napier and Rankine. Zeuner was succeeded at Swuss Federal by
Albert Fliegner (a Swiss) and Aurel Strodola (who was born in Hungary). Their
work, together with De Laval’s (a Swede) application of the critical flow nozzle
to steam turbines, directly influenced Ludwig Prandtl at the University of
Gottingen in Germany.

The influence of Stodola (who probably was responsible for the term
“critical flow”) and Prandtl (through their publications and students) extends
to every text written in the 20th century – and this influence is flet by every
student who used them. In the United States, this can be seen very clearly.
For example, in the work of Joseph Keenan and Ascher Shapiro at Massachusetts
Institute of Technology and Theodor van Karman (originally Hungarian and a
student of Prandtl) at the California Institute of Technology.