William John Macquorn Rankine



Born July 5, 1820 – Died December 24, 1872(aged 67)




William John Macquorn Rankine was a Scottish engineer and physicist.

He was a founding contributor, with Rudolf Clausius and William Thomson (1st Baron Kelvin), to the science of thermodynamics.
Rankine developed a complete theory of the steam engine and indeed of all heat engines.

His manuals of engineering science and practice were used for many decades after their publication in the 1850s and 1860s.
He published several hundred papers and notes on science and engineering topics, from 1840 onwards, and his interests were extremely varied, including, in his youth, botany, music theory and number theory, and, in his mature years, most major branches of science, mathematics and engineering.

He was an enthusiastic amateur singer, pianist and cellist who composed his own humorous songs. He was born in Edinburgh and died in Glasgow, a bachelor.

Thermodynamics

Work

Undaunted, he returned to his youthful fascination with the mechanics of the heat engine.
Though his theory of circulating streams of elastic vortices whose volumes spontaneously adapted to their environment sounds fanciful to scientists formed on a modern account, by 1849, he had succeeded in finding the relationship between saturated vapour pressure and temperature.

The following year, he used his theory to establish relationships between the temperature, pressure and density of gases, and expressions for the latent heat of evaporation of a liquid. He accurately predicted the surprising fact that the apparent specific heat of saturated steam would be negative.

Emboldened by his success, he set out to calculate the efficiency of heat engines and used his theory as a basis to deduce the principle, that the maximum efficiency of a heat engine is a function only of the two temperatures between which it operates.

Though a similar result had already been derived by Rudolf Clausius and William Thomson, 1st Baron Kelvin, Rankine claimed that his result rested upon his hypothesis of molecular vortices alone, rather than upon Carnot's theory or some other additional assumption.
The work marked the first step on Rankine's journey to develop a more complete theory of heat.

Rankine later recast the results of his molecular theories in terms of a macroscopic account of energy and its transformations.
He defined and distinguished between actual energy which was lost in dynamic processes and potential energy by which it was replaced.

He assumed the sum of the two energies to be constant, an idea already, although surely not for very long,
familiar in the law of conservation of energy.

From 1854, he made wide use of his thermodynamic function which he later realised was identical to the entropy of Clausius.

By 1855, Rankine had formulated a science of energetics which gave an account of dynamics in terms of energy and its transformations rather than force and motion. The theory was very influential in the 1890s.

Energetics offered Rankine an alternative, and rather more mainstream, approach, to his science and, from the mid 1850s, he made rather less use of his molecular vortices.
Yet he still claimed that Maxwell's work on electromagnetics was effectively an extension of his model. And, in 1864, he contended that the microscopic theories of heat proposed by Clausius and James Clerk Maxwell, based on linear atomic motion, were inadequate.

It was only in 1869 that Rankine admitted the success of these rival theories. By that time, his own model of the atom had become almost identical with that of Thomson.

As was his constant aim, especially as a teacher of engineering, he used his own theories to develop a number of practical results and to elucidate their physical principles including:

The Rankine-Hugoniot equation for propagation of shock waves,
governs the behaviour of shock waves normal to the oncoming flow.
It is named after physicists Rankine and the French engineer Pierre Henri Hugoniot;

The Rankine cycle, an analysis of an ideal heat-engine with a condensor.
Like other thermodynamic cycles, the maximum efficiency of the Rankine cycle is given by calculating the maximum efficiency of the Carnot cycle;

Properties of steam, gases and vapours.

The history of rotordynamics is replete with the interplay of theory and practice.
Rankine first performed an analysis of a spinning shaft in 1869, but his model was not adequate and he predicted that supercritical speeds could not be attained.

Rankine's Momentum theory

The momentum theory or Disk actuator theory is a theory describing a mathematical model of an ideal propeller or helicopter rotor, by W.J.M. Rankine (1865), Alfred George Greenhill (1888) and R.E. Froude (1889).

The rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation.
The basic state of an helicopter is hovering. This disc creates a flow around the rotor.

Under certain mathematical premises of the fluid, there can be extracted a mathematical connection between power, radius of the rotor, torque and induced velocity. Friction is not included.

The power required to produce a given thrust is:

momentum formula

Where:

  • T is the thrust
  • ρ is the density of air (or other medium)
  • A is the area of the rotor disc