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DISSIPATION OF HEAD IN
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DISSIPATION OF See also:HEAD IN See also:SHOCK § 36. Relation of Pressure and Velocity in a Stream in Steady See also:Motion when the Changes of See also:Section of the Stream are Abrupt.—When a stream changes section abruptly, rotating eddies are formed which dissipate See also:energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the See also:work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. Suppose fig. 38 to represent a stream having such an abrupt See also:change of section. Let AB, CD be normal sections at points where See also:ordinary stream See also:line motion has not been disturbed and where it has been re-established. Let w, p, v be the See also:area of section. pressure and velocity at AB, and PI, vi corresponding quantities at CD. Then if no work were expended internally, and assuming the stream See also:horizontal, we should have p/G+v2/2g = p,/G+v,2/2g• (I) But if work is expended in producing irregular eddying motion, the head at the section CD will be diminished. Suppose the See also:mass See also:ABCD comes in a See also:short See also:time t to A'B'C'D'. The resultant force parallel to the See also:axis of the stream is pw+Ps(cei—w)—piwi, where po is put for the unknown pressure on the See also:annular space between AB and EF. The impulse of that force is Ipw+po(wi—w)—piwi}t. The horizontal change of momentum in the same time is the difference of the momenta of rI c ,CDC'D' and See also:ABA'B', - . because the amount of momentum be- t / tween A'B' and CD remains unchanged if the motion is steady. The See also:volume -~ — of ABA'B' or CDC'D', being the inflow and outflow in the time , ) Qt=wvt=wivit, and the momentum of t these masses is The change of momentum is therefore (G/g)Qt(vi—v). Equating this to the impulse, t pw +po (-i—w)—piwi } t = (G/g)Qt(vi-v). Assume that po = p, the pressure at AB extending unchanged through the portions of fluid in contact with AE, BF which See also:lie out of the path of the stream. Then (since Q=wivi) (p-pi) = (G/g)vi (vi-v) ; p/G—pi/G =v1(vi—v)/g; (2) p/G+v2/2g =pi/G+v12/2g+(v-vi)2/2g• (3) This differs from the expression (I), § 29, obtained for cases where no sensible See also:internal work is done, by the last See also:term on the right. That is, (v—v1)"--/2g has to be added to the See also:total head at CD, which is piiG+v12/2g, to make it equal to the total head at AB, or (v—vi)2/2g is the head lost in shock at the abrupt change of section. But v—vi is the relative velocity of the two parts of the stream. Hence, when an abrupt change of section occurs, the head due to the relative velocity is lost in shock, or (v—vi)2/2g See also:foot-pounds of energy is wasted for each See also:pound of fluid. Experiment verifies this result, so that the See also:assumption that po = p appears to be admissible. If there is no shock, p1 /G = p/G + (v2—vi2) /-2g• It there is shock, pi/G = p/G-vi (vi--v)/g• Hence the pressure head at CD in the second See also:case is less than in the former by the quantity (t!—vi)2/2g, or, putting wive=wv, by the quantity (v2/2g) (1—w/wi)2. (4) V. THEORY OF THE See also:DISCHARGE FROM ORIFICES AND MOUTHPIECES 37. Minimum Coefficient of Contraction. Re-entrant Mouth- piece of See also:Borda.—In one See also:special case the coefficient of contraction can be determined theoretically, and, as it is the case where the convergence of the streams approaching the orifice takes See also:place through the greatest possible See also:angle, the co-efficient thus deter-See also:mined is the minimum coefficient. Let fig. 39 represent a See also:vessel with See also:vertical sides, 00 being the See also:free See also:water See also:surface, at which the pressure is pa. Suppose the liquid issues by a horizontal See also:mouthpiece, which is re-entrant and of the greatest length which permits the See also:jet to See also:spring clear from the inner end of the orifice, without adher- See also:ing to its sides. With such an orifice the velocity near the points CD is negligible, and the pressure at those points may be taken equal to the hydro- static pressure due to the See also:depth from the free surface. Let 12 be the area of the mouthpiece AB, w that of the contracted jet aa Suppose that in a short time t, the mass OOaa comes to the position 0'O' a'a'; the impulse of the horizontal See also:external forces acting on the mass during that time is equal to the horizontal change of momentum. The pressure on the See also:side OC of the mass will be balanced by the pressure on the opposite side OE, and so for all other portions of the vertical surfaces of the mass, excepting the portion EF opposite the mouthpiece and the surface AaaB of the jet. On EF the pressure is simply the hydrostatic pressure due to the depth, that is, (pa+Gh)St. On the surface and section AaaB of the jet, the horizontal resultant of the pressure is equal to the atmospheric pressure pa acting on the vertical See also:projection AB of the jet; that is, the resultant pressure is —pat2. Hence the resultant horizontal force for the whole mass OOaa is (pa+Gh)S2-pa[2=See also:Ghat. Its impulse in the time t is Ghst t. Since the motion is steady there is no change of momentum between O'O' and aa. The change of horizontal momentum is, therefore, the difference of the horizontal momentum lost in the space 000'0' and gained in the space See also:aaa'a'. In the former space there is no horizontal momentum. The volume of the space aaa'a' is wvt; the mass of liquid in that space is (G/g)wvt; its momentum is (G/g)wv2t. Equating impulse to momentum gained, Ght2t= (G/g)wv2t; .'.w/it = gh/v2. v2=2gh, and w/S2=es; w/It ==eal a result confirmed by experiment with mouthpieces of this See also:kind. A similar theoretical investigation is not possible for orifices in See also:plane surfaces, because the velocity along the sides of the vessel in the neighbourhood of the orifice is not so small that it can be neglected. The resultant horizontal pressure is therefore greater than Ght , and the contraction is less. The experimental values of the coefficient of discharge for a re-entrant mouthpiece are 0.5149 (Borda), 0.5547 (Bidone), 0.5324 (Weisbach), values which differ little from the theoretical value, 0.5, given above. § 38. Velocity of Filaments issuing in a Jet.—A jet is composed of fluid filaments or elementary streams, which start into motion at some point in the interior of the vessel from which the fluid is discharged, and gradually acquire the velocity of the jet. Let Mm, fig. 40 be such a filament, the point M being taken where the velocity is in-sensibly small, and m at the most contracted section of the jet, where the filaments have be- come parallel and exercise See also:uniform mutual pressure. Take the free surface AB for datum line, and let pi, vi, hi, be the pressure, velocity and depth below datum at M ; p, v, h, the corresponding quantities at m. Then § 29, eq. (3a), vi2/2g + pi /G—hi = v2/2g+p/G-h. (I) But at M, since the velocity is insensible, the pressure is the hydro-static pressure due to the depth; that is, vi = o, pi = P +Ghi. At m, p=pa, the atmospheric pressure See also:round the jet. Hence, inserting these values, or That is, neglecting the viscosity of the fluid, the velocity of filaments at the contracted section of the jet is simply the velocity due to the difference of level of the free surface in the See also:reservoir and the orifice. Additional information and CommentsThere are no comments yet for this article.
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