Introduction and History

Abstract

Differential-pressure flowmeters (called differential-pressure meters) remain the most common type of meter for large and medium flowrates: their demise has often been predicted, and yet large numbers of them are still both used and bought. The reasons for their continued popularity are considered: their advantages and disadvantages are listed. The theory (Bernoulli’s Theorem) for incompressible and compressible flow is considered, and both the basic equation for flow through differential-pressure meters and the expansibility-factor equation for Venturi tubes are derived. An introduction to Reynolds number and velocity profile, pipe roughness and pressure loss is provided. The essential requirements for differential-pressure flow measurement are described and discussed, and the location of material in other chapters required to meet these requirements is given. The history of flow measurement by Venturi tubes and orifice plates is described, with particular reference to Clemens Herschel in the main text and an Appendix on Sextus Julius Frontinus.

Appendix 1.A: Sextus Julius Frontinus

Probably the first person to describe flow measurement in closed pipes using a method whose descendant is that codified in modern differential-pressure standards was Sextus Julius Frontinus. He was born about 35 AD and was Provincial Governor of Britain about 76 AD, where he subdued the Silures (a tribe in Wales) and constructed the Via Julia. After his return to Rome he wrote Strategemata (his work on the art of war) about 90 AD. He then became Water Commissioner in Rome: he might have regarded this as a sinecure, but instead he was an outstanding public servant and wrote a treatise De Aquis (Frontinus c 97–98) around 97–98 AD on the Aqueducts of Rome (quotations below from De Aquis are from the Loeb edition with a translation by Charles E Bennett, a revision of that by Clemens Herschel). He describes how the aqueduct system of ancient Rome (depicted in Figs. 1.A.1 and 1.A.2) operated, and places particular emphasis on the flow measurement aspects. His book is a remarkable achievement for one who came to flow measurement so late in life.

Fig. 1.A.1

Aqua Claudia in the Parco degli Acquedotti: August 2008 taken by Agricmarketing (talk) Wikipedia under Parco degli Acquedotti

Fig. 1.A.2

Porta Maggiore, Rome, formed by a supporting arch of the Aqua Claudia and Rome’s highest elevated aqueduct Aqua Anio Novus, integrated into Aurelian Wall as a gate 271 AD. From Wikipedia under Roman aqueduct: it was taken by Diana

Standardized pipes

The system contained standardized pipes for offtakes and intakes from and to the main aqueducts. Frontinus described the permissible pipe sizes showing how they were constructed: the basic unit was the quinaria (or 5-pipe), a pipe with a diameter of 5 quarters of a digit, where a digit is 1/16 Roman foot (the Roman foot was equal to 0.967 imperial foot; the Roman foot could alternatively be divided into 12 inches). The senaria (or 6-pipe) had a diameter of 6 quarters of a digit and so up to the vicenaria (or 20-pipe) with a diameter of 20 quarters of a digit (i.e. 5 digits). Above that size the system was different and based on area not diameter: the vicenum quinum (or 25-pipe) had an area of 25 square digits, the tricenaria (30-pipe) of 30 square digits and so up to the centenum vicenum (120-pipe). If the second definition based on area had been used for the vicenaria (or 20-pipe) then the difference from the definition based on diameter would have been very small (less than 2 per cent on area). So there was little problem at the changeover between systems.

Frontinus discovered that in four pipe sizes (duodenaria (12-pipe), vicenaria (20-pipe), centenaria (100-pipe) and centenum vicenum (120-pipe)) the common usage was different from the systematic method described above. Although for the first of these the error was small, for the second, which was commonly used for delivery, the common usage was 20 % smaller than it should have been, the third and fourth were 13 and 68 % larger, respectively, than they should have been. He sums this up as follows:

In all there are 25 ajutages.^{1 , 2} They all conform to their computed and recorded capacities, barring these four which the water-men have altered. But everything embraced under the head of mensuration ought to be fixed, unchanged, and constant.³ For only so will any special computation accord with general principles. Just as a sextarius,⁴ for example, has a regular ratio to a cyathus,⁵ and similarly a modius ⁶ to both a cyathus and sextarius, so also the multiplication of the quinariae in case of the larger ajutages must follow a regular progression. However, when less is found in the delivery ajutages and more in the receiving ajutages, it is obvious that there is not error, but fraud.

Frontinus was a faithful public servant, honest and conscientious, basing his conclusions on personal examination. For clarity he listed in detail the permissible 25 pipe sizes (those not actually in use are here in italics): 5, 6, 7, 8, 10, 12, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 and 120. The problem with his system was that he assumed the flowrate would be proportional to the area alone. He made no direct allowance for velocity.

Meter installation

Frontinus knew that meter installation (in terms of the angle of the meter to the main flow) does make a difference:

⁷But the position of the calix is also a factor. Placed at right angles and level, it maintains the normal quantity. Set against the current of the water, and sloping downward, it will take in more. If it slopes to one side, so that the water flows by, and if it is inclined with the current, that is, is less favourably placed for taking in water,⁸ it will receive the water slowly and in scant quantity. The calix, now, is a bronze ajutage, inserted⁹ into a conduit or reservoir, and to it the service pipes are attached. Its length ought not to be less than 12 digits, while its orifice¹⁰ ought to have such capacity as is specified.

Calix is defined in the Oxford Latin Dictionary as ‘a short length of brass pipe of a specified diameter installed between a water main or reservoir and a private supply pipe to regulate the amount of water drawn off by the latter’.

Frontinus knew a little about the effect of pressure, but not enough to quantify the effect:

In setting ajutages^{11 , 12} also, care must be taken to set them on the level, and not place the one higher and the other lower down. The lower one will take in more; the higher one will suck in less, because the current of water is drawn in by the lower one.

Frontinus knew about the effect of unmatched upstream pipework:

In some of the reservoirs, though their ajutages^{13 , 14} were stamped in conformity with their lawful measurements, pipes of a greater diameter [than the ajutages] were at once attached to them. As a consequence, the water not being held together for the lawful distance,¹⁵ and being on the contrary forced through the short restricted distance, easily filled the adjoining larger pipes. Care should therefore be taken, as often as an ajutage¹⁶ is stamped, to stamp also the adjoining pipe over the length which we stated was prescribed by the resolution of the Senate.

Accreditation

Frontinus not only put forward a standard, but also an accreditation system based on standards and diligence:

The deputy must call in the levellers^{17 , 18} and provide that the calix is stamped as conforming to the deeded quantity, and must study¹⁹ the size of the ajutages²⁰ we have enumerated above, as well as have knowledge of their location, lest it rest with the caprice of the levellers to approve a calix of sometimes greater, or sometimes smaller, interior area, according as they interest themselves in the parties. Neither must the deputy permit the free option of connecting directly to the ajutages²¹ any sort of lead pipe, but there must rather be attached for a length of fifty feet one of the same interior area as that which the ajutage²² has been certified to have, as has been ordained by a vote of the Senate which follows.

For then and then only can the overseer be held to his full responsibility, when he understands that none but stamped pipes must be set in place.²³

Frontinus’s Legacy

So Frontinus produced a standard for flow measurement, took account of installation effects, and aimed for consistency and fairness. He even achieved pleasure in such worthwhile engineering as the aqueducts:

With such an array of indispensable structures carrying so many waters,²⁴ compare, if you will, the idle Pyramids or the useless, though famous, works of the Greeks!

As is stated in the Loeb edition of Frontinus’s works ‘it is the personality of the writer that one loves to contemplate, his sturdy honesty, his conscientious devotion to the duties of his office, his patient attention to details, his loyal attachment to the sovereign whom he delighted to serve, his willing labours in behalf of the people whose convenience, comfort and safety he aimed to promote’.