Escape from the Jail of Shape; Dimensionality and Engineering Science

  • Edwin T. LaytonJr.
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 144)


In V. Sackville-West’s Poem, “The Land” the poet deals with craftsmen. The noem reads in part:

All craftsmen share a knowledge. They have held

Reality down fluttering to a bench;

Cut wood to their own purposes; compelled

The growth of pattern with the patient shuttle,

Drained acres to a trench.

Control is theirs. They have ignored the subtle

Release of spirit from the jail of shape.

They have been concerned with prison, not escape.1


Heat Transfer Dimensionless Parameter Engineering Science Dimensional Analysis Fluid Resistance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    V. Sackville-West, ‘The Land — Summer’, in Collected Poems (London: Webb and Bower, 1989), p. 69. Reprinted courtesy of Curtis Brown and John Farquharson. I would like to thank Professor Rutherford Arris for calling my attention to this poem and for his inspiration and help over many years.Google Scholar
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    On the diameter of tubes in relation to their mechanical properties, see Charles MacGregor, ‘Mechanical Properties of Materials’, in Theodore Baumeister et al. (eds.), Marks’ Standard Handbook for Mechanical Engineers (New York: McGraw-Hill, 1978), Sec. 5, pp. 49–50. For a cylindrical tube the circumferential or “hoop” stress, S, is equal to S = pr/t where p is the internal pressure, r the radius and t the plate thickness.Google Scholar
  3. 3.
    On “specific speed” see W.G. Whippen, ‘Hydraulic Turbines’, in Baumeister, Marks’ Standard Handbook for Mechanical Engineers, Sec. 9, pp. 137–138. The discovery of specific speed will be discussed below. One of the most important keys to rational design of water wheels was to avoid “angle of incidence” losses due to shock at entrance. To eliminate shock it is necessary to have “tangent entry” of the water, which, in turn, required that the designer knew the speed of the rotor.Google Scholar
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    J.J. Cornish, ‘Aeronautics’, in Baumeister, Marks’ Standard Handbook for Mechanical Engineers, Sec. 11, pp. 60–64, 66–71. In the most obvious case, in laminar flow the boundary layer flows as if made up of many thin layers moving smoothly parallel to one another; in a turbulent boundary layer the flow is no longer parallel; there are irregular motions (such as eddies) normal to the surface of the layer.Google Scholar
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    There is a modern tendency to restrict the meaning of “dimensional analysis” to the contents of the Vaschy—Buckingham π theorem, but a much broader definition is used here. I have been much influenced by Stephen J. Kline, Similitude and Approximation Theory (New York: McGraw-Hill, 1965), who uses “dimensional analysis” in a broad sense. He analyzes four distinct types of dimensional analysis (pp. 262–215). These are (1) the π theorem (discussed below), (2) the use of governing equations, that is, normalizing the governing differential equations to make them dimensionally homogeneous. This method was pioneered by Lord Rayleigh; it is favored by physicists, and (3) the “method of similitude”, (which is largely Kline’s own creation). To this one should add the method of “configurational analysis” which is built upon Kline’s work, but which emphasizes the use of group theory for the analysis of dimensional issues. (For configurational analysis, see H.A. Becker, Dimensionless Parameters (New York: John Wiley, 1976), Vol. 3, pp. 10–11. To these I add historical methods of analyzing dimensions such as that of Fourier, who invented the first formalism of dimensional analysis. In addition, I follow Kline in including the derivation of model laws within dimensional analysis (pp. 2–7). Model laws are often called rules or principles of similitude (or similarity). Other works dealing with dimensional analysis that I found particularly helpful include Percy W. Bridgman, Dimensional Analysis (Cambridge, Mass: Harvard University Press, 1921); W.J. Duncan, Physical Similarity and Dimensional Analysis (London: Edward Arnold, 1953); H.E. Huntley, Dimensional Analysis (London: McDonald, 1953); Henry L. Langhaar, Dimensional Analysis and Theory of Models (New York: John Wiley, 1951); and L.I. Sedov, Similarity and Dimensional Methods in Mechanics (New York: Academic Press, 1959). On modelling theory I found Langhaar especially useful. Several of these contain useful bibliographies, notably Becker and Kline. Kline has comments on major contributions to the literature which can be found in the text . On the history of dimensional analysis I am deeply in the debt of Enzo O. Macagno, ‘Historico-Critical Review of Dimensional Analysis’, J. Franklin Inst. 232(1971), pp. 393–394. Macagno’s study is a critical and informative history of dimensional analysis; Macagno follows the more common practice of restricting the meaning of “dimensional analysis” to the Vaschy—Buckingham theorem and closely related matters.Google Scholar
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    The traditional phrase “Research and Development” reflects the now outdated “applied science” theory of technology, which has now been replaced by the “interactive” model. There is evidence suggesting that it is more common for research to follow the appearance of a novel engineering (or design) idea. For example, one of the largest and best of the innovations studies was done by the well-known economists Sumner Myers and D. G. Marquis, who found that successful innovations in the industries studied did not originate with research in 95 percent of the cases, but came about in implementing a technological idea. (See Sumner Myers and D.G. Marquis, Successful Industrial Innovations (Washington D. C.: The National Science Foundation, NSF 69–17, 1969) especially Table 19, p. 46.)Google Scholar
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    On the interactive model see Barry Barnes, ‘The Science—Technology Relationship: A Model and a Query’, Social Studies of Science 12 (February, 1982), pp. 166–171 and Edwin T. Layton, Jr., ‘Mirror Image Twins: The Communities of Science and Technology in 19thCentury America’, Technology and Culture 12 (October, 1971), pp. 562–580. Studies of technology as knowledge include Edwin T. Layton, Jr., ‘Technology as Knowledge’, Technology and Culture 15 (January, 1974), p. 41; Walter G. Vincenti, What Engineers Know and How they Know It, Analytical Studies from Aeronautical History (Baltimore: Johns Hopkins, 1990).CrossRefGoogle Scholar
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    Newton, Principia, Book I, section 12 deals with spherical bodies, section 13 extends the analysis to non-spherical bodies, pp. 193–225. See particularly Proposition 88, Theorem 45, “If the attractive forces of the equal particles of any body be as the distance of the places from the particles, the force of the whole body will tend to its center of gravity; and will be the same with the force of a globe, consisting of similar and equal matter, and having its center in the center of gravity.” (I, p. 216).Google Scholar
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    Newton’s ontology also included his “geometrization of space”, and other things. These issues have long been debated. Though now somewhat outdated, historically important works in understanding Newton’s ontology (or metaphysics) included Edwin A. Burtt, The Metaphysical Foundations of Modern Science: A Historical and Critical Essay (New York: Harcourt Brace, 1925) and Alexandre Koyré, Etudes Galiléennes (Paris: Hermann, 1939) and the same author’s, Metaphysics and Measurements: Essays in Scientific Revolution (London: Clapham and Hall, 1968).Google Scholar
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    Newton, Principia, I, pp. 328–336. Newton’s awareness of the limits, and possibilities, of his ontology is remarkable and evident in these propositions of fluid resistance. Newton did deal with some of the effects of the mutual interactions of the particles of a fluid, in Proposition 33, pp. 328–329, where he shows that the centripetal and centrifugal forces by which the particles of the system act on each other, can be reduced to simple considerations of the force and matter as with a simple system by using the similarity principles developed in Proposition 32. Thus Newton refuted Descartes, while not solving the problems of Fluid Mechanics with which modern engineers must deal. The fact that Newton’s physics was associated with a particular ontology does not mean that it was invalid except with that ontology. Thus it is possible (with later analytical tools) to expand Newton’s F= MA to get the Navier—Stokes equation, and by considering the ratios of two such expressions for two different systems to derive the key dimensional concepts of dynamic and kinematic similarity. This is done, for example, by Max Jakob, Heat Transfer, 2 vols. (New York: John Wiley, 1949, 1955), Vol. I, pp. 16–19, 429–430. It would be a-historical to expect Newton to have done this; the fact that it can be done is a tribute to the foundations which Newton provided for physical science. The Navier—Stokes theorem will be discussed below.Google Scholar
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    Froude’s relationship to Ferdinand Reech and the story of the law of similarity and the dimensionless parameter named after Froude is complicated and will not be discussed in this paper. As Rouse and Ince note, Froude did not originate the similarity rule (it was discovered by Reech), and Froude never used the “Froude Number” . On the other hand he did anticipate boundary layer analysis. See Rouse and Ince, History of Hydraulics, pp. 154–155, 182–187.Google Scholar
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    John Smeaton, ‘An Experimental Inquiry Concerning the Natural Powers of Water and Wind to Turn Mills, and Other Machines, Depending upon Circular Motion’, Phil. Trans. Royal Soc. 51(1, 1759–1760), pp. 100–101. Smeaton’s role in the experimental tradition in engineering is discussed in Terry S. Reynolds, Stronger than a Hundred Men, pp. 218–233.Google Scholar
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    Quoted in Edwin T. Layton, Jr., ‘American Ideologies of Science and Engineering’, Technology and Culture 17 (October, 1976), p. 693.Google Scholar
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    Francis, Lowell Hydraulic Experiments, p. 52; the experimental derivation of the Francis weir formula is on pp. 69–135.Google Scholar
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    Francis, Lowell Hydraulic Experiments, p. 52 and Table 4, pp. 53–54. The properties which Francis measured turned out to be dependent upon dimension, as will be noted below in discussing specific speed.Google Scholar
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    One can infer from Newton’s Proposition 32 basic geometric, time, and force similarity conditions, where 1 and 2 stand for similar systems, and S, T and F are any lengths, time intervals, and forces occurring in the similar systems, the relations S1/S2 = a, T1/T2 = b, and T1/T2 = c, where a, b, and c are factors of proportionality. A common alternative for this system of units is to consider length, time and force instead of mass.Google Scholar
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    There is an interesting discussion of graphical statics and the link between this science and design in David Billington, Robert Maillart’s Bridges. The Art of Engineering (Princeton: Princeton University Press, 1979), pp. 5–7Google Scholar
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    As noted above, I am much indebted to Kline and follow his very broad definition of dimensional analysis. He categorizes dimensional analysis as a part of “fractional analysis” (e.g., analysis which provides useful but not complete information) and sees the latter as a fundamental methodology in engineering. See Kline, Similitude and Approximation Theory, pp. v–vii, 2–7, 87–219.Google Scholar
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    Ibid. , 2. This is true for heat transfer where rates (that is time) are critical; but the later science of thermodynamics can be considered in terms of equilibrium, without concern for the rates at which the transfer of heat takes place.Google Scholar
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    Macagno, ‘Historico-Critical Review of Dimensional Analysis’, pp. 393–394. The further development of heat transfer added a good deal of complexity to Fourier’s theory. It would be surprising if the first investigator in a new science discovered everything of relevance and importance. Fourier’s theory was primarily useful in conduction; though he made beginnings in convective and radiative heat transfer, these latter theories were less satisfactory. Modern heat transfer uses the concept of thermal diffusivity in the study of heat conduction, and especially the Fourier number, a dimensionless parameter needed to correlate information in unsteady heat conduction. (E.R.G. Eckert and Robert M. Drake, Heat and Mass Transfer (New York: McGraw-Hill, 1959), p. 77).Google Scholar
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    Fourier, Theory, p. 127. See also pp. 126–130. The variables Fourier admitted were x (length), t (time), v (temperature), c (heat capacity), and two heat transfer coefficients (in Fourier’s nomenclature), “specific conductibility”, K, and “surface conductibility”, h (p. 130). See also Jakob, Heat Transfer, Vol. I, pp. 3–5. ‘Google Scholar
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    Macagno, Historico-Critical Review of Dimensional Analysis , pp. 391–402. See also H.E. Huntley, Dimensional Analysis (London: McDonald, 1952), contains a brief history (pp. 33–44). For another historical study see Alton C. Chick, ‘The Principle of Similitude’, in John R. Freeman (ed.), Hydraulic Laboratory Practice (New York: the American Society of Mechanical Engineers, 1929), pp. 796–797.Google Scholar
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    Lord Rayleigh (John William Strutt), ‘The Principle of Similitude’, Nature 95 (1915), pp. 66–68, 644 is a very remarkable virtuoso performance. Kline has a remarkably clear and insightful chapter in which he analyzes the comparatives advantages and drawbacks of Rayleigh’s method with that of Buckingham and others. (See Kline, Similitude and Approximation Theory, pp. 262–215.) For a succinct statement, see also Murdock, ‘Mechanics of Fluids’ in Baumeister, Marks’ Standard Handbook for Mechanical Engineers, Sec. 3, p. 50.CrossRefGoogle Scholar
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    Aimé Vaschy, ‘Sur les lois de similitude en physique’, p. 345. Vaschy’s statement is very compressed; his statement of his theorem takes less than a page, and he moves on to illustrations and applications at the bottom of the first page. At the outset of the theorem he states that , “The most general law of similitude in mechanics and physics results in the following theorem.” He then states that a function of n physical quantities (a1, a 2 ...,an) can be reduced to another of (n—p) parameters of the form f(x1, x2..., xn( p) in which the parameters x1, x2 xn p are monomial (”single term”) functions of a1, a2,.... Vaschy does not state that this is the most general or fundamental form for physical equations, only that it is possible. Vaschy, in his first statement, makes no statement suggesting that the x parameters are dimensionless. Later on in the paper, however, he does say that these terms are independent of the units in which these terms are expressed. This is today a definition of a dimensionless parameter, but perhaps in 1892 these useful entities were not so well known and an explicit statement might have helped some readers. In any case Vaschy and his fellow French telegraph engineers do not appear to have made wide use of the theorem. One problem appears to have been Vaschy’s extreme brevity. 70 Buckingham, ‘On Physically Similar Systems’, J. Wash. Acad. Sci. 4, pp. 345–350 (Sec. 2, pp. 345–346). As Langhaar has shown, the mere fact of dimensional homogeneity leads directly to the π theorem. (Langhaar, Dimensional Analysis, pp. 55–58). In some early works the π terms were called variables, but the term parameter is more correct. The dimensionless parameters are composed of terms which, in the non-dimensional form, are correctly called variables. But the process of dimensional analysis does more than just group together “variables” into groups in which the units of dimension cancel out. ” In his paper ‘Model Experiments and the Forms of Empirical Equations’, p. 265, Buckingham presented a short version of his theorem which is strikingly similar to Vaschy’s earlier statement of the theorem.Google Scholar
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Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • Edwin T. LaytonJr.
    • 1
  1. 1.University of MinnesotaUSA

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