I) CONSERVATION OF MASS
Already about one and a half century before Lavoisier's (1743-1794) general formulation of the mass conservation law, in hydromechanics a particular case valid for fluid flow was recognised and used. It was called CASTELLI THEOREM (Benedetto Castelli, 1577-1644, pupil of Galileo Galiei, in "Della misura dell'acque correnti", 1628) and this name remains in use even now. In its simplest form it may be expressed as:

- which means that the mass flow rate [kg/s] does not change. Note that in the present case of one-dimensional steady flow, the only possible change would be a change along the co-ordinate . The meaning of the equation therefore threfore is that if we move along the axis (e.g. along the axis of a pipe (Fig. A-16) and investigate en route the mass flowrates in all the individual cross sections , the magnitude in all of them will be equal.
Flow rate measurement:
measuring flow rate is the most often performed measurement in industrial applications, because fluids are moved around in almost all technological processes and we need to know how much fluid is moved (especially so if fluids are mixed to form a mixture with prescribed composition - or simply whenever transported fluid has to be paid for). The ubiquitous nature of this measurement has led to development of a large number of methods, which form the subject
of extensive literature. Most methods used in practice are indirect nature - which means that the flowrate sensors based upon them must be first calibrated. Usually, the calibration is peformed by comparison with one of the two direct methods:
Liquid flow: is measured by weighing the outflowing liquid collected during a time period. It follows from the definition

that mass flow rate is evaluated from the measured according to the formula
Gas flow: the small weight in a much more heavy vessel would be difficult to measure accurately. It is, therefore, experient to measure the volume flow rate by storing the captured outflowing gas under the movable, water-sealed bell according to Fig.A-19. From the definition follows that the mass flow rate is evaluated as
Fig.A-20
Connection with fluid velocity: In the assumed one-dimensional flow we work with fluid velocity (= specific momentum). In the present one-dimensional approach it is the mean value in a given cross sectional area (the symbol "" is from German "die Flache"). During an elementary time interval dt the volume passing through this area equals dV and the volume flow rate therefore equals ... this leads to the important formula at the bottom of Fig.A-20. Because in this textbook we do not consider compressibility, (and, of course, also 1/v = const) so that the Castelli theorem on the top of this page may be written as
d() = 0
- product of cross-sectional area and velocity is constant (increasing cross section leads to decrease of velocity, which is the original formulation of the theorem by Castelli).


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This is page Nr. A09 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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