Handbook of hydraulic resistance pdf

Handbook of Hydraulic Resistance [Fourth 4th ed. Handbook of sport psychology [Fourth edition. Oxford handbook of psychiatry [Fourth ed. Handbook of Statistical Genomics: Two Volume Set, Fourth Edition A timely update of a highly popular handbook on statistical genomics This new, two-volume edition of a classic text pro 78 24MB Read more.

Hydraulic Machines 2, 28MB Read more. Annular section: kan -AaA, where kz is determined from curve c of diagram ;. Re 20' 3. OWN e. Dh - T;lo 4P, perimeter 1. II k'an - k, V. Apoal till 1I! A, fljA '4 sp'. Re ' Annular section:. Circular and rectangular sections: Dh "o go perimeter.

Aan k3k,. Friction coefficient Diagram D and D. C resistancu. Friction coefficientI Diagram 2' D, is in meters; v is taken from I ,b. Nominal internal diameterdnom. Friction coefficient DiagrLm A where X - Re-- is determined from the curve k-t Re. A Ainoill :. URI a,. Ab-t f'W0,4 1. Dh i W; 1 -- perim eter 1. C ircular cross section: 6 - Y10H I. V I is taken according to ,b. II I 2CH l. It 4,j - 78o SPAl' :. The entry of a stream into a straight pipe or channel of constant cross section Figure depends on two characteristics: the relative thickness Dh of the pipe-inlet wall, and the relative distance I from the pipe edge to the wall where it is mounted.

The coefficient of resistance t of the straight inlet stretch is maximum at a completely sharp edge h 0 and infinite distance of the pipe edge from the wall ok 0.

In this. Its minimum value is equal to 0. This case corresponds to a stream entrance into a conduit whose edge is at a Dh great distance from the wall.

When entering a straight conduit the stream flows past the inlet edge; if, however, this is insufficiently rounded, the stream separates near the entrance Figure This stream separation, and the resulting formation of eddies, are the main cause of pressure losses at the inlet.

The stream separation from the pipeF walls leads to a decrease of the 0 jet cross section. The coefficient of jet contraction a. The thickening, cutting, or rounding of the inlet wall, and the nearness of the conduit edge to the wall in which the pipe is mounted, all lead to a smoother motion of the stream about the inlet edge and to a smaller zone of stream separation with a smaller inlet resistance.

Flow at the inlet to a straight pipe from an unlimited space. The greatest decrease of resistance is obtained for a stream entrance through a smooth bellmouthwhose section forms an arc of a curve circle, lemniscate, etc. Figure ,a. Thus, in the case of a circular intake with relative radius of curvature h0. A relatively low resistance is also created by a stream entrance through inlets shaped as truncated cones Figure ,b,c or as contracting stretches with transitions from rectangular to circular or from circular to rectangular Figure ,d.

The resistance coefficient of such transition pieces is a function of both the convergence angle. The optimum value of a for a relatively wide range of 0.

At such angles, e. Plan of smooth inlet stretches: a-belln'buth Whose section forms arc of a circle; b and c-bellmouths shaped like truncated cones; d-transition pieces.

When an inlet stretch is mounted in the end wall under an angle 8 Figure. The resistance coefficient of smooth intakes mounted flush with the wall is determined in the.

Inlet stretch with screen before the entrance. The phenomenon observed in inlet stretches in which the stream suddenly contracts, i. The only difference. Sudden contraction:. In the case of ,thei inlet edge of a narrow channel mounted flush with the end wall of a wider channel, Figure ,a , 'the resistance coefficient can vary within the limits 0 -, C The resistance coefficient of an inlet with a sudden contraction at Reynolds..

The resistance of a contracting stretch can be decreased considerably if the transition from the wide section to the narrow one is accomplished smoothly by means of a rectilinear or curvilinear adapter Figure The contraction losses decrease with the increase of the transition smoothness. In the case of a perfectly smooth contraction of the section, where the convergence angle is very small or the length of the contracting stretch sufficiently large, and where this stretch has a very smooth curvi- linear generatrix, the stream'does not separate from the wall, and the pressure losses reduce to friction losses.

Adapters a - rectilinear; b - curvilinear. The resistance coefficient of a rectilinear transition section Figure ,a can be approximately determined by the formula:. The resistance coefficient of a smooth curvilinear adapter Figure ,b is determined either as the friction coefficient of a bell-mouth orifice given in diagram , or as the friction coefficient of a rectilinear adapter with the same length and contraction ratio, from the data of diagrams to The resistance coefficient of inlet sections is also a function of their location and method of mounting in the wall [of the vessel or container into which they discharge.

A low resistance coefficient can be achieved by installing an annular rib or ledge before the inlet stretch enclosing the opening Figure If the. The eddies formed in the region of separation contribute to the smooth flow of the stream into the main inlet stretch of the pipe without separation.

Entrance through an annular belhinouth. The optimum dimensions of the widened stretch in which a bell-mouth ledge is placed, must closely correspond to the dimensions of the eddy region: a at a point upstream from the most contracted section of the stream jet at the inlet into a straight pipe with sharp edges; b to a pipe mounted flush with the wall.

The values of.. The pressurellosses in the case of a lateral entrance of a stream through the fir-st orifice in a constant-,section collecting pipe are much larger than in the case of a straight entrance. Entrance through side orifices is frequently used in rectangular-section ventilating shafts. In order to prevent the penetration of sediments, louvers are mounted in the orifices. The resistance coefficient of such shafts is a function not only of the relative area of the orifices, but of their relative location as well.

The resistance coefficients of intake shafts with differently disposed lateral orifices are given in diagrams and The values of are given for both orifices with and without fixed louvers. The resistance of intake shafts with straight entrance and canopies cf.

In the case of normal ventilating shafts of circular section, in which the relative thickness of the inlet edges lies within the limits 0. An increase, of this distance would require building too large a canopy hood in view of the possibility of rain or snow entering the shaft. Stream entrance in a straight stretch: 1 -- through an orifice; b -through a perforated plate; For cross section.

The recommended shaft design is the one with conical inlet stretch. T orifice; ,' t or 1 I is. When a perforated plate is installed at the stream entrance, the total resistance coefficient can be approximately determined as sum of the resistance coefficients of the plate and the inlet:. To each cross-section coefficient there is an optimum value of the relative depth. The selection of the louver with the optimum value of - is recommended. In the case of standard grids with fixed louvers, the inlet edges of the slats are cut along the vertical Figure 3- 10,a.

From the point of view of the resistance it is more expedient, however, to use louvers with inlet edges cut along the horizontal Figure ,b. Type of diagram Source No. Straight intake shafts of rectangular section; side Nosova and Experimental data orifices with fixed louvers and without them Tarasov Rectangular-section intake shafts with a bend: The same The same.

AH -a C2. Screen is determined from the curves -. Diagram 3- 5 i i V. Dh - k;. Values of C approximately C 42 4' 4 4 O Z 4: -. Inlet-section in the end wall -0 perimeter. Inlet edge rounded t -. Inlet edge moved forward relative to the end wall. Inlet edge sharp or. Inlet edge rounded , where C'is determined from the curvesC t on diagram curves a and c.

Inlet edge beveled wee i ', Dh VPt, "91"' where C' is determined from the curve C b', on diagram ; v is taken from ,b. AM ii. Re2 10 1 22 20 30 40 50 5"0 10s 2" 4"10 3 5" 3 Resistance coefficient.

Inlet conditions Diagram- Ti Entrance with end wall on 0. Entrance with end walfs'on 0. Entrance with end walli on four sides of the conduit 0.

Resistance coefficient C Inlet conditions. Entrance withdeflector at two sides of 0. Entrance to a conduit mounted on a 0. Entrance to a conduit mounted between 0. Entrance to a conduit in an angle.

Entrance to a conduit clamped between 0. Diagram Ns II. One orifice. Two orifices C Z without with 8b -A 0 louvers louvers - ;.

Values of C A ing No. Flat grid 4 Ior perimeter of the orifices b. OI Diagram i. Resistance coefficient Grid 0ID characteristic b. Ivor 4A 0a 4 a. N dtxr art. Converging bell- mouth orifice I C, -as above. For ,7- -. The sudden enlargement of the cross section of a conduit is the cause of sorcalled 'shock" losses'.

The resistance coefficient of a shock, with uniform velocity distribution. When a stream suddenly expands, a jet is formed in the expanded section. The shock losses at sudden expansion are due to the formation of eddies in stretch 1,. Actually, the velocity distribution in the stretch before a sudden expansion is generally nonuniform Figure This has a strong effect on the actual pressure losses, and considerably increases them above the values given by ;.

Nonuniform velocity distribution before a sudden expansion. In order to calculate the resistance coefficient of a shock in a stream with nonuniform velocity distribution, it is necessary to use a general formula for the shock, which allows. ZAH 2M The accuracy of this formula is the higher, the nearer N and M are to unity. Using the last expression, the following approximate formula is obtained for the resistance coeffici'ents:.

If the velocity distribution over a section is known, the coefficients M and N can be easily determined. If the velocity distribution is unknown, it must be determ,ined empirically. The 'values of M and N can then be determined by graphic integrat-ion from the velocity profiles obtained.

In practice a velocity profile approximating a rectangle is obtained for m as low as 8'to 10 Figure Such a value of m can be assumed for lengthy. N- m The velocity profile in lengthy straight stretches of conduits, with a distance from the inlet larger than IODh and laminar flow, is parabolic Figure :. Figure When a nonuniform velocity field is established in a conduit of constant cross section n 1 , the equalization of stream velocity is accompanied by irreversible pressure.

These losses are taken into account only where they have been neglected in determining the local resistance of fittings or obstructions in the straight stretch. Sinusoidal velocity profile behind grids and guide vanes. For the definition of the concept of "main" zone of a free jet, cf.

Section XI. Nonsymmetric velocity distribution behind an elbow or a diffuser with a di- vergence angle causing stream division. The magnitudes -- d i are functions of the relative length of the free jet. The resistance of a stretch with a sudden enlargement canbe reduced by installing baffles cf. In the-general case, the passage of a stream from one, volume into another through a hole in a' wall is accompanied bythe phenomena illustrated inFigure The stream passes from channel 1, located before the partition A with orifice of diameter D.

The two channels can have cross sections of arbitrary dimensions, provtided they are not smaller than the cross section of the, orifice of passage. Thee passage of the stream through the orifice is accompanied by the bending of the trajectories of the particles, the inertial forces causing them to continue.

The basic data to be used in the installation of such baffles are given in paragraph Straight section of the ejector mixing chamber. Velocity distribution in the main zone of the free jet after its entrance into the mixing chamber of the ejector: '. General case of stream flow from one volume into another through an orifice:.

This leads to a decrease of the jet section from its initial area F, to an area F. Starting with section c-c, the trajectories of the moving particles are straightened, and the normal phenomenon of sudden jet expansion takes places farther on.

The thickening Figure ,b , beveling Figure , c , or rounding Figure ,d of the orifice edges dead to the weakening of the jet-contraction effects in the orifice, i. Since it is this velocity which basically determines the shock losses at the discharge from the orifice, the total resistance of the passage through the orifice is decreased. In the case of beveled or rounded orifice edges, Cfr is assumed to be equal to zero. The subscript "o" corresponds here to the subscript "or", and the subscript "1" to the subscript "o" in Section XI.

Open test section of a wind tunnel. SH Thick- edged orifice. At low cross-section coefficients Lo of the restrictor, large velocities are obtained in its orifice even at relatively low stream velocities in the pipe. The resistance coefficient of a restrictor, taking the influence of compression into account, can be determined by the formula.

As with entry into a straight channel, a sharp decrease of orifice resistance is achieved by installing an annular rib or ledge at the orifice inlet Figure 4- Entrance to an orifice through an annular rib a or ledge b. When the stream passes through a smooth belImouth orifice set into a wall cf. The resistance coefficient of such a stretch can be determined by the formula.

When the stream passes through apertures in a wall fitted with various flaps, the resistance is higher than in the absence of flaps, since they cause a complex flow ,pattern. Here the resistance coefficient becomes a function of the angle of opening of Ifl the flaps a and their relative length I-f The open test section of a wind tunnel Figure can likewise be considered as a stretch with suddenenlargement. Ejection dissipation of energy is the main cause of losses in the open test section of a wind tunnel.

Another cause of losses is that part of the free jet is cut off by the diffuser. The kinetic energy of this portion of the jet is lost for the wind tunnel and, there- fore, represents a part of the resistance of the throat.

AH O 45It. Section IV Sudden expansion of a stream with uniform velocity distribution Diagram p. Cm4 2 curve 1 on graph a. With ba ffles. I I 'rg UOm F,. Values of C. Au" II 0. Sudden expansion after stretches with parabolic velocity SectionIV distribution. Exponential velocity profile 6.

M and N are determined from graph b of diagrams and The magnitudes C. M, and N are determined from the graph of S 9. SIDI, 0. W02 I0 I, is determined from. DhA L. N-, II IV a so0? ON ZO. Schematic diagram Resistance coefficient c- Orifice with rounded F F. C 7 3 Values of C I FIda. A is determined from diagrams to as a. The values of C are completely determined. Thick-walled orifice deep orifice 1.

V'is determined from the curve V-' I on graph b. Orifice with beveled -A'. Orifice edges beveled or rounded, cf. Z2 N1 41 ',1. Intake flap, single, top-hinged Section IV Diagram 4.

S ectio n Diagram IV Single flap, center-hinged. S 46 26 Double flaps, both top-hinged Section 4- Diagram IV F,; 'G 7- AH2" F. A diffuser is a gradually widening passage to make the transition from a narrow conduit to a wide one and the transformation of the kinetic energy of the stream into pressure energy, with minimum pressure losses.

In such a divergent pipe the intensity of turbulence is greater than in a straight pipe, and the local friction resistances are also greater. The increase in the pipe section is accompanied by a drop in the mean stream velocity. Therefore, the total resistance coefficient of the diffuser, expressed in terms of the velocity in the initial section, is less for divergence angles below a certain. The separation of the boundary layer from the walls Figure 5- 1 is due to a positive pressure gradient existing in the diffuser as a result of the velocity drop which accompanies -the ijincrease in cross section' Bernoulli equation.

The beginning, of stream separation. The velocities over the cross section in narrow-angle diffusers with nonseparating boundary layers are distributed symmetrically about the longitudinal axis Figure With the beginning of stream separation from one of the walls, the pressure increase through the diffuser is interrupted or reduced, and as a result there will be no stream separation from the opposite wall.

Consequently, the distribution of velocities over the diffuser section will be asymmetric Figures and Velocity profiles in plane diffusers with different divergence angles. In a perfectly symmetrical wide-angle diffuser the separation occurs alternately at one or the other side of the diffuser Figure , leading to strong fluctuations of the whole stream. The resistance coefficient of a diffuser is a function of several parameters: 1 the divergence angle a; 2 the area ratio n, -; the cross-section shape; 4 the.

Ik 1, k 2,. In practice an arbitrary method of loss separation is used. This is due to the scarcity of available data on the dependence of diffuser resistance on these parameters, and particularly on the Reynolds number.

With this method, the total resistance of the -". It is convenient to express the expansion losses by the coefficient of shock bf. In the case of uniform velocity distribution at the inlet k, -. In the case of a pyramidal diffuser with uneqlial divergence angles in the two planes, a is the larger angle.

The following value of k, is used for conical and plane diffusers:. Currently there is no theoretical analysis of the corresponding situation for wide-angle diffusers, which is by far tfie most interesting case. The coefficient of local resistance of expansion is expressed through the coefficient of shock as follows:. Cexp - g. The friction coefficient of a plane diffuser with inlet-section sides a, and b,, where b, is constant along the diffuser, is calculated by:.

The resistance coefficients of diffusers where the rectangular section changes to circular or vice versa, can be determined from the data for pyramidal diffusers with equivalent divergence angles. The equivalent angle ae is determined on the basis of the following formulaz :.

For the derivation cf. The magnitude of IL actually varies along the diffuser, but in practice it is assumed to be constant. In the case of a nonuniform velocity distribution at the inlet section], i. At the inlet, the dependence of the diffuser resistance coefficient on the state of the boundary layer, the velocity distribution is complex. In narrow-angle diffusers a nonuniform symmetric velocity profile with a maximum at the-center and mininrnum at the walls leads to a decrease of'the total resistance, since the frictional stress at the walls is decreased.

At the same time this velocity profile increases the possibility of stream s eparationiand displaces the point of separation toward the initial section of the diffuser, so that with the subsequent increase of the divergence angle the resistance will increase compared with resistance at a uniform velocity distribution. In a nonuniform; velocity profile, with lower velocities at the center and higher ones at the walls. The dependence of the coefficient k, on the divergence angle for a symmetric velocity profile at the inlet has.

The dependence! In the case. Since the smooth increase of a pipe section with narrow divergence angles leads to a decrease in the pressure losses compared with those in a pipe of constant section, and at wide divergence angles to the increase of these losses, there must obviously exist an optimum divergence angle at which minimum losses are obtained. This angle can be calculated for the case of a straight diffuser of circular section by:.

For a diffuser of i-- rectangular section, a. The flow conditions of a stream in short wide-angled diffusers can be considerably improved by preventing stream separation or reducing the formation of eddies. Different methods for improving the work of short diffusers:. When the boundary layer is sucked Figure ,a , thepart of the stream'separated from the wall again adheres to the surface, and as a result the zone of separation is displaced downstream, the flow becomes smoother, and the resistance decreases.

The blowing away of the boundary layer Figure ,b leads to an increase of stream velocity near the walls. As a result, the separation zone is also displaced downstream. Guide vanes or baffles Figure ,c deflect a part of the high-velocity stream core toward the boundary zone of separation.

The latter is reduced or even completely eliminated as a result. The effect of guide vanes is greatest at wide divergence. It is necessary to permit the stream to expand in the peripheral channels just as in the main channel. The dividing walls divide a wide-angle diffuser into several narrow-ahgle diffusers Figure ,d. The dividing walls are more efficient with the increase of the total divergence angle of the diffuser. At relatively narrow divergence angles the dividing walls can e',en increase the diffuser resistance, since ihey increase the total friction'surface.

The' following procedure is used when selecting and installing dividing wallis in wide- angle diffusers: a The number z of dividing walls is selected from Table as a function' of the divergence angre. The length of the protruding parts must not be smaller than 0. The variation of the pressure gradient along the diffuser is smoother in a diffuser with curved walls Figure ,e , in which the rate of increaseof the cross-section area is lower in the initial section than in the end section.

As a result, the main cause of stream Feparation is weakened, and the main source of losses is attenuated. A diffuser in which the pressure gradient remains constant along the channel! At lower divergence angles, e. The use of curved diffusers is therefore expedient at wide divergence angles only. Y, In a multistage diffuser Figure ,f , in which a sudden expansion takes place after a smooth variation of cross-section area, the main shock losses occur at relatively low velocities. As a result, the losses in the diffuser are reduced by a factor of two to.

The frictional losses in very wide-angled diffusers arc quite small. It is not necessary, therefore, to separate these losses from the total losses with curved diffusers which correspond to wide-angle straight diffusers.

To each area ratio n and each relative length L or -d of the multistage diffuser, there corresponds an optimum divergence angle aopt at which the total coefficient of resistance is minimum cf. The use of multistage diffusers with optimum expansion angles is recommended. The limiting divergence angle ali 1of the smooth part of the multistage-diffuser, i. The lines of optimum values of y- are represented in graphs a of. If the diffuser is installed behind a fan, it becomes necessary to allow for the flow pattern at the fan exit, which is quite different from that at an inlet to an isolated diffuser placed behind a straight stretch of constant section.

As a rule, the velocity profile behind a centrifugal fan is nonsymmetric, due to a certain stream deflection in the direction of fan rotation. The stream deflection in the direction of fan rotation makes it possible to use unusually wide divergence angles behind centrifugal fans and diffusers. W ,nax 1. When the space available for the diffuser behind the centrifugal fan is restricted, use can be made of a multistage diffuser, which is much shorter than a straight diffuser at equal resistance.

The optimum divergence angle of the diffuser, at which a minimum coefficient is obtained, can be calculated from the corresponding curves of diagram 5- The resistance coefficient of an annular diffuser formed by installing a conical diffuser behind an axial fan or compressor with converging back fairing cf. The resis-tance coefficient of an annular diffuser formed by installing a conic diffuser behind an, axial fan with diverging back fairing can be approximated by the formula.

Axial turbines use radial-annular diffusers in which the increase of area is mainly due to the. The, area-expansion ratio of radial-annular diffusers can be determined from the relation. The resistance. The magnitude of the resistance. The values of the resistance coefficients of radial-annular diffusers of an operating at c0, 0.

The axial-radial-annular diffuser is somewhat better from the aerodynamic point of view. Here, a radial bend follows a short annular diffuser Figure , b. In this diffuser the radial turn is achieved at lower stream velocities, and the pressure losses are, accordingly, somewhat lower. At the same time the axial dimensions are much larger than those of a radial-annular diffuser.

When installed behind an operating turbo-compressor, at c. The existence of a uniformly distributed resistance behind the diffuser promotes orderly streamline flow in the diffuser itself and in the channel behind it. This some- what reduces the losses in the diffuser itself. The total losses in the diffuser, however, remain roughly the same. Specifically for curved diffusers, and for straight diffusers of divergence angles from 40 to , these losses remain equal to the sum of the losses taken separately for the diffuser and the grid, screen, etc.

Diagram description Source No. Diffusers of arbitrary shape located at the discharge of long Section V stretches with nonuniform but symmetric velocity profile Diagram 5- 1 A. Values of kh. Initial zone -- free jet. Uniform velocity distribution at the diffuser inlet:.

I II II. D, a on diagrams to ; v is taken from ,b; A is taken from Table Nonuniform velocity distribution at the diffuser inlet:. Values of Cfr ROx P. Thexp vl e. At k0. Cg2XJ r. I," -! Plane diffuser in a line Section V Diagram Texp is determined approximately from the curve t.

Jb, At 0. F; on o graphs IT 0. FJF, a, Values of Cfr at a. Dh-f4F; I6- perimeter -kC,. Circular or pyramidal - '0- yod formula applicable in the range 0. Plane diffuser a, 1.

Circular or pyramidal diffuser. I- 1- No. Plane diffuser. Values of Cmin t'dI 1. Resistance coefficient C- Guiding device Schematic diagram. With dividing walls Number of dividing wallsz ra a'. With bafflesC Ad. Az, - - a-. It is characterized by for- mation of a laminar boundary layer at the walls even at large Reynolds numbers greatly ex- ceeding the critical Reynolds number.

This boundary layer downstream of the inlet becomes thicker and at some distance from it Xt at the "transition" point becomes turbulent Figure Figure 1. Flow separation and formation of eddies in a diffuser. Farther along, this turbulent layer fills up the whole cross section of the pipe, with the transverse velocity distribution asymptotically approaching that for a stabilized turbulent flow.

The relative distance Xr from the transition point to the inlet depends on the Reynolds 59 dium is deceIerated on enlargement of the flow area or accelerated on contraction of the number and can be approximated from a formula suggested by Filippov: flow area. The dependence of Xt on Re is given in Figure 1. A liquid or gas is in equilibrium if, for each arbitrarily isolated portion of it, the result of The thickness of the boundary layer at a given distance from the initial section of a all the forces acting on this portion is equal to zero.

The flow rate of a fluid liquid or gas is defined as the mass or volume of fluid passing referenee plane the corresponding geometrie heights, Figure 1. It is for this reason that, for example, the pressure exerted on the walls of avessei filied where w is the flow velocity at the given point across the pipe channel in mls.

The trans verse distribution of the velocities in the pipe is hardly ever uniform. To simplify the solution of practical problems use is made of a fictitious me an velocity of the flow:. The volumetrie flow rate and, eonsequently, the velocity of the gas flow depend on the fluid temperature, pressure, and humidity. Scheme of the flow and its basic parameters for two sections of a channeL in Pa. In the ease of a dry gas at apressure of Fa F].

On the basis of Equations 1. For a dry gas at apressure of In the general case, the continuity equation can be written for any distribution of velocities in two pipe sections and Figure 1.

Aecording to the law of eonservation of energy for the medium moving through a pipe Fa F] ehannel , the energy of the liquid gas flow passing through seetion per time unh see Figure 1. Having divided Equation 1. In most praetical eases, the statie pressure P in straight-line flow is eonstant aeross the mternal and external, that is, meehanieal over the segment eonsidered, flow, even when the veloeity distribution is greatly nonuniform. Therefore, in plaee of Equation 0.

By relating the energy of the flow to the volumetrie flow rate for example, to Qo we obtain the Bernoulli generalized equation in the form. In a number of cases, when performing approximate calculations, the process can be considered isentropic. For this process the polytropie exponent n in Equations 1. In some cases the state of the flow is changed, following an isotherm constant tem- perature. There the pressure is proportional to the gas density. All the terms of Equation 1.

In the case of a polytropic process, the gas parameters change according to Then, the polytropic exponent for air passing through a converging wye becomes n pS - p1- pn ' "'" 1. Formulas 1. Table The basie similarity groups of gas flows are the Maeh number or the reduced veloeity 1. The Maeh number is The flow veloeity equal to the loeal speed of sound and ealled the eritieal velocity is or, taking into aecount Equations 1. The speed of sound in a stagnated medium is.

For air On the basis of Equations 1. The reduced velocity is Taking into account the relation analogous to relation 1. The rate of mass flow is expressed in terms of the functions q Xc and Y Ac : 0. Expanding Equation 1. The gas dynamic functions 1. This table also eontains the funetions that eharaeterize the mass flux For a jet of an ineompressible fluid the total pressure is.

The quantity reciproeal to Y A c eharaeterizes the change in the statie momentum in the isen- s: -- PI - up po po -- PI [ po lIk - 1 "'" 2? Moreover, Table 1. The subscripts 0 and 1 relate to sections and of the given flow, respectively. For an incompressible liquid, to which gas at small flow velocities practically up to w '" rnJs can also be referred, Vo VI. Then, on the basis of Equation 1. On the basis of Equation 1. When the densities of the flowing medium, p, and of the surrounding atmosphere, Pa, sections and is are equal and the pipes flow channels are horizontal, then the elevation pressure net driving head is zero.

Then Equation 1. The excess elevation pressure net driving head is produced by the fluid, which tends to descend or rise depending on the medium lighter or heavier in which the fluid is located. This pressure can be positive or negative depending upon whether it promotes or hinders the 1. The mo- 1. Therefore, the total energy thermal energy inclusive 1. By solving Equation 1.

The temperature, on the other hand, does not change at constant velocity. Ptot - ps. Then sor, fan, etc. All of these phenomena contribute to the exchange of momentum between the moving fluid particles i. The loeal pressure los ses also include the dynamie pressure losses oceuring during liquid 1.

The phenomenon of flow separation and eddy formation is assoeiated with the differ- ence of veloeities over the cross seetion of the flow and with a positive pressure gradient For the ease of uniform distribution of statie pressure and density over the seetion, but along the flow.

The latter develops when the flow velocity is retarded for example, in an which are variable along the flow, the resistance eoefficient based on Equation 1. The difference in veloeities over the cross seetion of a negative pres- sure gradient e.

The total pressure los ses in any eomplex element of the pipeline are inseparable. The value of S depends on the velocity, and eonsequently on the flow cross section.

In value is eommensurable with L1Ploc. The fluid resistance coefficient is defined as the ratio of the total energy power lost over the given segment - to the kinetic energy power in the seetion taken for sinee example, or which is the same the ratio of the total pressure lost over the same seg- ment to the dynamic pressure in the seetion taken, so that on the basis of Equations 1. A and 1. The prineiple of superposition of losses ean be realized by two methods: 1 by sum- 1.

The overall fluid resistanee of any network element is to a eertain velocity and then expressing the total resistanee of the system through its total eoefficient of resistanee. POV - ':lOV rate. The coefficients ' The prineiple of superposition of los ses is used not only for ealculation of aseparate in the seetion Fi of the same seetion element. Generally, the eoeffieient Si ineludes also the element of the pipe ehannel , but also in the hydraulic ealeulation of the entire network.

This eorreetion for the mutual effeet of adjaeent elements of the network. Only the quantity PtfPa is unknown. In the majority of cases, the process can be regarded to be isentropic. Then, the expo- or nent n in Equation 1. For locking devkes n"" 1. For T-joints and other analogous shaped elements, when n "" 1 and the pressure is propor- tIOnal to the gas density [see Equation 1.

The loss of specific energy over any ith segment in a network can be defined through the 1. From this, the equation analogous to Equation 1. The internal flows can be induced by the influence of Archimedean forces on a fluid and 1. The total resistance to the motion of a Newtonian fluid can be considered as a sum of external magnetic fieId; eIectromagnetic forces that originate during the interaction of an elec- resistance forces: tric Iayer at the phase interfaces with the extern al eIectric and magnetic fields.

Viscous forces that hinder the irrotational laminar motion of fluid. Those opposing the change in the momentum of the system when secondary fluid flows around its axis. In heterogeneous nonuniform systems, the phases of which have substantially different 3. Scheme of intemal eddy motion of fluid and of the effect of extemal forces on it. Equation 1. It follows from Equation 1. When a fluid flows in a bent pipe, the system experiences the centrifugal inertia forces.

In this case, t.. In this case, the resistance coefficient is. For this case, the values of t.. During the flow of conducting fluids in pipes or channels in the trans verse magnetic Equation 1.

During the pipe flow of different oi1s the coefficient A. The velocity Weon of jet discharge for an incompressible fluid passing from the exit section for the channel flow A. The forces that induce intern al flows in a heterogenous system due to the relative mo- tion of phases depend on both the difference of the densitities of the fluid and dispersed par- 1. Then 2. In the general case of the flow from vessel A into vesse1 B see Figure 1.

In the ease of the liquid flow from vessel A into vessel B of large volume, that is, at Fo «F2. Diseharge from a submerged orifice. When the flow diseharges from vessel A through a nozzle in the bottom of the vessel, its veloeity is. Discharge from avesseI through an orifice in the bottom or wall. Bill E - o. Dependence of the discharge coefficient on the Reynolds number Re for dis charge from outer cylindrical nozzles: 55 1 dis charge coefficient for orifices in a thin wall; the length of the nozzle;.

The coefficient of discharge, Il, through nozzle in the bottom or in the wall of avesseI 0. Dependence of the discharge coefficient on Re number small Re's for discharge from neglected. The values of Il for orifices and nozzles of certain shapes Figure 1. I aGO 0. Dependence of the discharge coefficient on the relative thickness of the wall for the inner Figure 1. Discharge from avesseI through nozzles. Discharge of a Compressible Gas 1. When agas vapor, air issues at high pressure into the atmosphere, a significant change occurs in its volume.

Therefore, it is necessary to take into account the compressibility of the gas. In this case, the mass discharge is independent of the extern al where Psuc is the excess pressure in the suction volume, Pinj the excess pressure in the injec- pressure Po and is controlled by the pressure PI in the vessel, increasing with its rise.

To set a liquid or gas medium at the ends of a given piping system in motion, it is nec- Pinj , we have essary to create a difference of the total pressure means of a pressure-boosting device pump, 2 fan, flue-gas fan, compressor. Under normal operating conditions of the supercharger, Ptot is positive, that is,. At the same time both the static and the dynamic pressure downstream of the supercharger can be smaller than upstream of it.

In a specific case of equal cross-sectional areas of the suction and injection orifices,. A supereharger in the system. Psys redueed to the same volu- metrie flow rate. Example 1. Forced Ventilation System 8. Usually the volumetrie flow rate of the medium displaeed is a speeified quantity, while the pressure ereated by a supereharger is ealculated from Equations 1. To determine whether a given supereharger meets the required predietions of Qop and 2. Then if the flow rate of the medium 4.

Material from whieh the duets are made: sheet steel oil coated , roughness f.. In the case of high-head superchargers, the density of the medium being displaeed is related to the mean pressure on the rotor. Then Psup in Equation 1. Pinj - 0. Pinj are the pressure losses in the injection seetion of the system, in Pa, and f.. Psys are 11 tot 11 tot the total pressure losses in the whole system, in Pa.

The rated power on the supereharger shaft is. Scheme of calculation of the ventilation system network. S25 1. OIS 0. Installation for the Scrubbing of Sintering Gases The sehematie of the installation is shown in Figure 1. Given are: 3 1. Internal eoating of the gas mains eomparatively long : sheet steel, roughness taken to be the same as for seamless eorroded steel pipes after several years of service , Ll z 1.

Gas c1eaning, done in a wet scrubber, rate of spraying A z 0. Scheme of calculation of the installation for scrubbing sintering gases: Ca plan of an in- stallation; Cb side view. S00 "" 14 40 1. Exit from - 1. In the present ease, the gas temperature in the system varies due to eooling; therefore, as done in Example 1. Q ::l 4. Type of element Diagram and basic dimensions of the element Parameters Type of element Diagram and basic dimensions of the element Parameters 1 Circular open.

The ea1culation of the tunnel resistance is given in Table 1. Abramovich, G. Altshul, A. Mochan, S. The volumetrie air flow rate through the working seetion nozzle is 5. Branover, G. Nauk Latv. SSR, Sero Fiz. Nauk, no. Burdukov, A. Burdum, G. Standartov Press, Moscow, The power on the fan shaft at a fan effieieney lhot Z 0. Vakina, V. Vitkov, G. Lyatkher, V. Vulis, L. Makarov, A. Hartman, U. Malkov, M. GeIler, Z. L, Skobeltsyn, Yu. VUZ, Neft Gaz, no. Mergertroid, V.

Genin, L. Mikheev, M. Guizha, E. Monin, A. Gil, B. Nevelson, M. Petukhov, B. Guinevskiy, A. Pisarevskiy, V. VUZ, Mashinostroenie, , Grabovsky, A. Prandtl, L. Gubarev, N. Rikhter, G. Gukhman, A. Sedov, L. Deich, M. Skobeltsyn, Yu. VUZ, Heft Gaz, no. Elovskikh, Yu. Zelkin, G. Solodkin, E. Kiselev, P. Handbook of Chemistry, Vol.

Idelchik, I. Vargaftik, N. Moscow, , p. Idelchik, 1. Stepanov, P. Reclaimants, Kolos Press, Moscow, , p. Stochek, N.

VUz, Energetika, no. Kiselyov, P. Tananayev, A. Komlev, A. Blum, E. Fabrikant, N. Levin, V. Filippov, G. Frenkel, V. Loitsyanskiy, L. Khozhainov, A. Jen, P. Shiller, L. Schlichting, H. Shcherbinin, E. Shchukin, V. Elterrnan, V. Yuriev, B. Barach, A. Benedict, P. Boussinesq, 1. Pur Appl. Forst, T. Aero- naut. Iversen, H. ASME, vol. Jackson, R. A13, nos. Kolodzie, P. Maa Yer. Murakarni, M. D88, 2. Wielogorski, J.

The pressure losses along a straight tube conduit of constant cross section linear or fric- tion los ses are ca1culated from the Darcy-Weisbach equation:. The use of the hydraulic equivalent diameter Dh as the characteristic length in resis- tance Equations 2. Therefore, when Dh is used as a characteristic dimension, the resistance law for tubes of different cross sections remains about the same.

However, even in turbulent flow individual geometries have different friction resis- tance coefficients. The hydraulic resistance of a tube channel with a stabilized laminar flow cannot be calculated through the use of Dh.