Fig.B-3 Hydrostatic efects in gases.
The only condition required for applying of the simple integration as described above is that the otherwise arbitrary course of the axis , in it role of the integration path, passes everywhere through the same fluid (if there are several different fluids, integration may be divided into corresponding segments with different ). The possibility of curvature of the axis is exploited in the case shown in Fig.B-4 where both positions as well as are on liquid surfaces. The task to be solved there is a reversal of the previous tasks: values of pressure at the surfaces are here known and it is now the height difference if the liquid columns, which are displaced from their original equilibrium position by a given acting pressure difference , which is be calculated. Using the same integration as in Fig.B-2 it is, on the other hand, possible also in the U-tube configuration shown in Fig.B-4 to determine pressure at , if there are known column heights and the starting value of pressure in . This means that providing the arms of the U-tube with suitable scales will convert this device into a manometer.

In fact, this is a well known and widely used principle of pressure measurement. The devices operation of which is based upon this principle are called liquid-filled manometers . In spite of being the simplest and cheapest instruments for pressure measurement, the may be - in suitable arrangement - very reliable and also very precise. For use in aerodynamic laboratories, special variants were developed with very good operational properties (so that they are gradually displaced from use only recently due to recent requirements of computer data logging, which requires the manometer outputs available in the form of electric signals). Basic forms of such manometers are shown schematically in Fig.B-5. The necessity to read the position of liquid surface usually requires the arms being made of glass. This makes them easily damageable and this - together with relatively large dimensions and necessary maintenance (liquid evaporates and must be replaced) means that they are typical laboratory instruments, not suitable for industrial use. With water as the liquid in the U-tube it is possible to measure in the basic arrangement A in Fig.B-5 pressure within the range from 10 Pa to 20 kPa (which corresponds to from 1 mm to 2 m). More than one order of magnitude higher pressure differences may be measured with mercury (height 2m, which is about the maximum dimension
Fig. B-4 Connection between heights of liquid columns and acting
pressure difference in a U-tube.
of a practical instrument, then corresponds to = 270 kPa). On the other hand, for more precise measurement od small pressure differences the suitable liquid has to be larger specific volume than water. In this direction, however, the possibilities are limited: practically it is only alcohol, which us used. It makes possible to obtain at the same acting pressure difference displaced column height 1,25-times larger than with water. Accuracy of liquid-filled manometers is limited on one hand by the precise knowledge of specific volume (which varies with impurities in the liquid, evaporation of a component in case od mixture; in the case of alcohol, which is hygroscopic, also by absorption of water from atmosphere), on the other hand also by capillary elevation. Its influence should be cancelled in the arrangement A by its acting in the both arms, in practice this is not so since capillary elevation is sensitive to even very small differences in pipe diameters (this is why precise, callibrated glass pipes are used) and contamination of walls, which is never exactly the same in both arms.
Numerous variants were developed, e.g. the Prandtl manometer (arrangement B in Fig.B-5) removes the necessity to read on two scales




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This is page Nr. B02 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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