Force generated by impingement of a jet upon a body

Fig.H-17
The body in the case shown in Fig.H-17 is a perpendicular flat plate - so large that the jet is divided into the two resultant flows which leave the control volume in direction perpendicular to the original motion. With pressure (almost) equal everywhere the pressure force term is dropped from the equation. Also dropped is the gravity force since its orientation angle (defined in Fig. H-2 ) here equals /2. What remains in the force balance equation
is just = = -
In principle this is a two-dimensional problem (note the two axes and in Fig.H-17) - however, for the task of determining the acting force (drag) only the components on the horizontal direction are relevant. In the difference , the subtracted term is therefore zero, because in the exit cross section there is no flow into the horizontal direction . The resultant expression
=
Fig.H-18
may be re-written in the form of the formula for the aerodynamic drag as
= with value of the drag coefficient = 2.
This is not the case if the body, upon which the jet impinges, is smaller so that fluid leaves the control volume at with some nonzero momentum component in the direction - as shown in Fig.H-18. In this case the generated force is, of course, smaller - as may be seen from the values of the drag coefficient approaching the above value 2.0 only gradually with increasing target body diameter. Since the change in fluid velocity (caused by friction) is usually small, the decrease in may be expressed approximately as dependent upon the exit angle :
= 2 (1 + cos ), because the horizontal velocity component is = cos( - )
Fig.H-19
Jet action on a moving body
- for the case of the flat plate body the derivation is in Fig.H-19. There are two possibilities of the derivation - either with steady control volume (case A) or with the control volume moving, attached to the plate. In the both cases the result obtained is the same. Note that the generated force depends upons the velocity ratio: its magnitude is zero if this ratio / equals 1.0 (because this means that the plate moves as fast as the jet, which then never reaches it). This or similar situation is encountered in hydraulic machines, such as the Pelton turbine Fig.H-20, where jet from a nozzle impinges upon buckets on the circumference of a wheel. The faster the wheel rotates, the lower is the force generated by the jet. It would seem advantageous to operate the turbine at low rotation speed - but this would generate low output power (which is porportional to wheel speed). This gives rise to an interesting classical problem of optimum wheel speed. In the case of Fig.H-19 the corresponding problem is the question of maximum achievable power transferred to the moving plate. It the plate is stationary, = 0, the force is maximum, but the transferred power, equal to = is, of course, zero. When is increased, again = 0 is reached when there is = . Somewhere between these two extreme situations the power attains a maximum. Determining this maximum is
Fig.H-20 Pelton turbine
performed in Fig.H-19 in the relative co-ordinates, from the condition that derivative at an extreme is zero. One of the extremes is a minimum. Note that insertion of the optimum
= / 3 into the expression for leads to maximum efficiency lower than 30 % . Note that instead of flat plates, the buckets of the Pelton wheel (Fig.H-20) have concave shape. This increases the magnitude of the generated force to nearly twice the flat-plate value ( near to 0 means
= 2 (1 + cos ) near to 4 ). The theoretical maximum efficiency (again attained for = / 3 ) is thus increased to 59.26 %. Pelton type turbines exhibit very low magnitude of the speed coefficient from 0.03 to 0.3 rad (compare with Fig.F-20) so that they are suitable for very high heads and small flows, where other turbine types would require for good efficiency choosing too high rotation speeds.


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This is page Nr. H07 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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