I. Impulse Turbines.II. Reaction Turbines.(Wheel passages not filled, and discharging above(Wheel passages filled, discharging above or belowthe tail water.)the tail water or into a suction-pipe.(a) Complete admission. (Rare.)Always with complete admission.(b) Partial admission. (Usual.)Axial flow, outward flow, inward flow, or mixed flow.Simple turbines; twin turbines; compound turbines.Fig. 183.§ 183.The Simple Reaction Wheel.—It has been shown, in § 162, that, when water issues from a vessel, there is a reaction on the vessel tending to cause motion in a direction opposite to that of the jet. This principle was applied in a rotating water motor at a very early period, and the Scotch turbine, at one time much used, differs in no essential respect from the older form of reaction wheel.The old reaction wheel consisted of a vertical pipe balanced on a vertical axis, and supplied with water (fig. 183). From the bottom of the vertical pipe two or more hollow horizontal arms extended, at the ends of which were orifices from which the water was discharged. The reaction of the jets caused the rotation of the machine.Let H be the available fall measured from the level of the water in the vertical pipe to the centres cf the orifices, r the radius from the axis of rotation to the centres of the orifices, v the velocity of discharge through the jets, α the angular velocity ofthe machine. When the machine is at rest the water issues from the orifices with the velocity √ (2gH) (friction being neglected). But when the machine rotates the water in the arms rotates also, and is in the condition of a forced vortex, all the particles having the same angular velocity. Consequently the pressure in the arms at the orifices is H + α2r2/2g ft. of water, and the velocity of discharge through the orifices is v = √ (2gH + α2r2). If the total area of the orifices is ω, the quantity discharged from the wheel per second isQ = ωv = ω √ (2gH + α2r2).While the water passes through the orifices with the velocity v, the orifices are moving in the opposite direction with the velocity αr. The absolute velocity of the water is thereforev − αr = √ (2gH + α2r2) − αr.The momentum generated per second is (GQ/g)(v − αr), which is numerically equal to the force driving the motor at the radius r. The work done by the water in rotating the wheel is therefore(GQ/g) (v − αr) αr foot-pounds per sec.The work expended by the water fall is GQH foot-pounds per second. Consequently the efficiency of the motor isη =(v − αr) αr={√ (2gH + α2r2) − αr} αr.gHgHLet√ (2gH + α2r2) = αr +gH−g2H2...αr2α3r3thenη = 1 − gH / 2αr + ...which increases towards the limit 1 as αr increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the wheel has an infinitely great velocity of rotation. But this condition is impracticable to realize, and even, at practicable but high velocities of rotation, the friction would considerably reduce the efficiency. Experiment seems to show that the best efficiency is reached when αr = √ (2gH). Then the efficiency apart from friction isη = {√ (2α2r2) − αr} αr / gH= 0.414 α2r2/ gH = 0.828,about 17% of the energy of the fall being carried away by the water discharged. The actual efficiency realized appears to be about 60%, so that about 21% of the energy of the fall is lost in friction, in addition to the energy carried away by the water.§ 184.General Statement of Hydrodynamical Principles necessary for the Theory of Turbines.(a) When water flows through any pipe-shaped passage, such as the passage between the vanes of a turbine wheel, the relation between the changes of pressure and velocity is given by Bernoulli’s theorem (§ 29). Suppose that, at a section A of such a passage, h1is the pressure measured in feet of water, v1the velocity, and z1the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by h2, v2, z2. Thenh1− h2= (v22− v12) / 2g + z2− z1.(1)If the flow is horizontal, z2= z1; andh1− h2= (v22− v12) / 2g. (la)(b) When there is an abrupt change of section of the passage, or an abrupt change of section of the stream due to a contraction, then, in applying Bernoulli’s equation allowance must be made for the loss of head in shock (§ 36). Let v1, v2be the velocities before and after the abrupt change, then a stream of velocity v1impinges on a stream at a velocity v2, and the relative velocity is v1− v2. The head lost is (v1− v2)2/2g. Then equation (1a) becomesh2− h1= (v12− v22) / 2g − (v1− v2)2/ 2g = v2(v1− v2) / g(2)Fig. 184.To diminish as much as possible the loss of energy from irregular eddying motions, the change of section in the turbine passages must be very gradual, and the curvature without discontinuity.(c)Equality of Angular Impulse and Change of Angular Momentum.—Suppose that a couple, the moment of which is M, acts on a body of weight W for t seconds, during which it moves from A1to A2(fig. 184). Let v1be the velocity of the body at A1, v2its velocity at A2, and let p1, p2be the perpendiculars from C on v1and v2. Then Mt is termed the angular impulse of the couple, and the quantity(W/g) (v2p2− v1p1)is the change of angular momentum relatively to C. Then, from the equality of angular impulse and change of angular momentumMt = (W/g) (v2p2− v1p1),or, if the change of momentum is estimated for one second,M = (W/g) (v2p2− v1p1).Let r1, r2be the radii drawn from C to A1, A2, and let w1, w2be the components of v1, v2, perpendicular to these radii, making angles β and α with v1, v2. Thenv1= w1sec β; v2= w2sec αp1= r1cos β; p2= r2cos α,∴ M = (W/g) (w2r2− w1r1),(3)where the moment of the couple is expressed in terms of the radii drawn to the positions of the body at the beginning and end of a second, and the tangential components of its velocity at those points.Now the water flowing through a turbine enters at the admission surface and leaves at the discharge surface of the wheel, with its angular momentum relatively to the axis of the wheel changed. It therefore exerts a couple −M tending to rotate the wheel, equal and opposite to the couple M which the wheel exerts on the water. Let Q cub. ft. enter and leave the wheel per second, and let w1, w2be the tangential components of the velocity of the water at the receiving and discharging surfaces of the wheel, r1, r2the radii of those surfaces. By the principle above,−M = (GQ/g) (w2r2− w1r1).(4)If α is the angular velocity of the wheel, the work done by the water on the wheel isT = Ma = (GQ/g) (w1r1− w2r2) α foot-pounds per second.(5)§ 185.Total and Available Fall.—Let Htbe the total difference of level from the head-water to the tail-water surface. Of this total head a portion is expended in overcoming the resistances of the head race, tail race, supply pipe, or other channel conveying the water. Let ɧpbe that loss of head, which varies with the local conditions in which the turbine is placed. ThenH = Ht− ɧpis the available head for working the turbine, and on this the calculations for the turbine should be based. In some cases it is necessary to place the turbine above the tail-water level, and there is then a fall ɧ from the centre of the outlet surface of the turbine to the tail-water level which is wasted, but which is properly one of the losses belonging to the turbine itself. In that case the velocities of the water in the turbine should be calculated for a head H − ɧ, but the efficiency of the turbine for the head H.§ 186.Gross Efficiency and Hydraulic Efficiency of a Turbine.—Let Tdbe the useful work done by the turbine, in foot-pounds per second, Ttthe work expended in friction of the turbine shaft, gearing, &c., a quantity which varies with the local conditions in which the turbine is placed. Then the effective work done by the water in the turbine isT = Td+ Tt.The gross efficiency of the whole arrangement of turbine, races, and transmissive machinery isηt= Td/ CQHt.(6)And the hydraulic efficiency of the turbine alone isη = T / GQH.(7)It is this last efficiency only with which the theory of turbines is concerned.From equations (5) and (7) we getηGQH = (GQ/g) (w1r1− w2r2) α;η = (w1r1− w2r2) α/gH.(8)This is the fundamental equation in the theory of turbines. In general,7w1and w2, the tangential components of the water’s motion on entering and leaving the wheel, are completely independent. That the efficiency may be as great as possible, it is obviously necessary that w2= 0. In that caseη = w1r1α / gH.(9)αr1is the circumferential velocity of the wheel at the inlet surface. Calling this V1, the equation becomesη = w1V1/ gH.(9a)This remarkably simple equation is the fundamental equation in the theory of turbines. It was first given by Reiche (Turbinenbaues, 1877).
§ 183.The Simple Reaction Wheel.—It has been shown, in § 162, that, when water issues from a vessel, there is a reaction on the vessel tending to cause motion in a direction opposite to that of the jet. This principle was applied in a rotating water motor at a very early period, and the Scotch turbine, at one time much used, differs in no essential respect from the older form of reaction wheel.
The old reaction wheel consisted of a vertical pipe balanced on a vertical axis, and supplied with water (fig. 183). From the bottom of the vertical pipe two or more hollow horizontal arms extended, at the ends of which were orifices from which the water was discharged. The reaction of the jets caused the rotation of the machine.
Let H be the available fall measured from the level of the water in the vertical pipe to the centres cf the orifices, r the radius from the axis of rotation to the centres of the orifices, v the velocity of discharge through the jets, α the angular velocity ofthe machine. When the machine is at rest the water issues from the orifices with the velocity √ (2gH) (friction being neglected). But when the machine rotates the water in the arms rotates also, and is in the condition of a forced vortex, all the particles having the same angular velocity. Consequently the pressure in the arms at the orifices is H + α2r2/2g ft. of water, and the velocity of discharge through the orifices is v = √ (2gH + α2r2). If the total area of the orifices is ω, the quantity discharged from the wheel per second is
Q = ωv = ω √ (2gH + α2r2).
While the water passes through the orifices with the velocity v, the orifices are moving in the opposite direction with the velocity αr. The absolute velocity of the water is therefore
v − αr = √ (2gH + α2r2) − αr.
The momentum generated per second is (GQ/g)(v − αr), which is numerically equal to the force driving the motor at the radius r. The work done by the water in rotating the wheel is therefore
(GQ/g) (v − αr) αr foot-pounds per sec.
The work expended by the water fall is GQH foot-pounds per second. Consequently the efficiency of the motor is
Let
then
η = 1 − gH / 2αr + ...
which increases towards the limit 1 as αr increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the wheel has an infinitely great velocity of rotation. But this condition is impracticable to realize, and even, at practicable but high velocities of rotation, the friction would considerably reduce the efficiency. Experiment seems to show that the best efficiency is reached when αr = √ (2gH). Then the efficiency apart from friction is
η = {√ (2α2r2) − αr} αr / gH= 0.414 α2r2/ gH = 0.828,
about 17% of the energy of the fall being carried away by the water discharged. The actual efficiency realized appears to be about 60%, so that about 21% of the energy of the fall is lost in friction, in addition to the energy carried away by the water.
§ 184.General Statement of Hydrodynamical Principles necessary for the Theory of Turbines.
(a) When water flows through any pipe-shaped passage, such as the passage between the vanes of a turbine wheel, the relation between the changes of pressure and velocity is given by Bernoulli’s theorem (§ 29). Suppose that, at a section A of such a passage, h1is the pressure measured in feet of water, v1the velocity, and z1the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by h2, v2, z2. Then
h1− h2= (v22− v12) / 2g + z2− z1.
(1)
If the flow is horizontal, z2= z1; and
h1− h2= (v22− v12) / 2g. (la)
(b) When there is an abrupt change of section of the passage, or an abrupt change of section of the stream due to a contraction, then, in applying Bernoulli’s equation allowance must be made for the loss of head in shock (§ 36). Let v1, v2be the velocities before and after the abrupt change, then a stream of velocity v1impinges on a stream at a velocity v2, and the relative velocity is v1− v2. The head lost is (v1− v2)2/2g. Then equation (1a) becomes
h2− h1= (v12− v22) / 2g − (v1− v2)2/ 2g = v2(v1− v2) / g
(2)
To diminish as much as possible the loss of energy from irregular eddying motions, the change of section in the turbine passages must be very gradual, and the curvature without discontinuity.
(c)Equality of Angular Impulse and Change of Angular Momentum.—Suppose that a couple, the moment of which is M, acts on a body of weight W for t seconds, during which it moves from A1to A2(fig. 184). Let v1be the velocity of the body at A1, v2its velocity at A2, and let p1, p2be the perpendiculars from C on v1and v2. Then Mt is termed the angular impulse of the couple, and the quantity
(W/g) (v2p2− v1p1)
is the change of angular momentum relatively to C. Then, from the equality of angular impulse and change of angular momentum
Mt = (W/g) (v2p2− v1p1),
or, if the change of momentum is estimated for one second,
M = (W/g) (v2p2− v1p1).
Let r1, r2be the radii drawn from C to A1, A2, and let w1, w2be the components of v1, v2, perpendicular to these radii, making angles β and α with v1, v2. Then
v1= w1sec β; v2= w2sec α
p1= r1cos β; p2= r2cos α,
∴ M = (W/g) (w2r2− w1r1),
(3)
where the moment of the couple is expressed in terms of the radii drawn to the positions of the body at the beginning and end of a second, and the tangential components of its velocity at those points.
Now the water flowing through a turbine enters at the admission surface and leaves at the discharge surface of the wheel, with its angular momentum relatively to the axis of the wheel changed. It therefore exerts a couple −M tending to rotate the wheel, equal and opposite to the couple M which the wheel exerts on the water. Let Q cub. ft. enter and leave the wheel per second, and let w1, w2be the tangential components of the velocity of the water at the receiving and discharging surfaces of the wheel, r1, r2the radii of those surfaces. By the principle above,
−M = (GQ/g) (w2r2− w1r1).
(4)
If α is the angular velocity of the wheel, the work done by the water on the wheel is
T = Ma = (GQ/g) (w1r1− w2r2) α foot-pounds per second.
(5)
§ 185.Total and Available Fall.—Let Htbe the total difference of level from the head-water to the tail-water surface. Of this total head a portion is expended in overcoming the resistances of the head race, tail race, supply pipe, or other channel conveying the water. Let ɧpbe that loss of head, which varies with the local conditions in which the turbine is placed. Then
H = Ht− ɧp
is the available head for working the turbine, and on this the calculations for the turbine should be based. In some cases it is necessary to place the turbine above the tail-water level, and there is then a fall ɧ from the centre of the outlet surface of the turbine to the tail-water level which is wasted, but which is properly one of the losses belonging to the turbine itself. In that case the velocities of the water in the turbine should be calculated for a head H − ɧ, but the efficiency of the turbine for the head H.
§ 186.Gross Efficiency and Hydraulic Efficiency of a Turbine.—Let Tdbe the useful work done by the turbine, in foot-pounds per second, Ttthe work expended in friction of the turbine shaft, gearing, &c., a quantity which varies with the local conditions in which the turbine is placed. Then the effective work done by the water in the turbine is
T = Td+ Tt.
The gross efficiency of the whole arrangement of turbine, races, and transmissive machinery is
ηt= Td/ CQHt.
(6)
And the hydraulic efficiency of the turbine alone is
η = T / GQH.
(7)
It is this last efficiency only with which the theory of turbines is concerned.
From equations (5) and (7) we get
ηGQH = (GQ/g) (w1r1− w2r2) α;
η = (w1r1− w2r2) α/gH.
(8)
This is the fundamental equation in the theory of turbines. In general,7w1and w2, the tangential components of the water’s motion on entering and leaving the wheel, are completely independent. That the efficiency may be as great as possible, it is obviously necessary that w2= 0. In that case
η = w1r1α / gH.
(9)
αr1is the circumferential velocity of the wheel at the inlet surface. Calling this V1, the equation becomes
η = w1V1/ gH.
(9a)
This remarkably simple equation is the fundamental equation in the theory of turbines. It was first given by Reiche (Turbinenbaues, 1877).
§ 187.General Description of a Reaction Turbine.—Professor James Thomson’s inward flow or vortex turbine has been selected as the type of reaction turbines. It is one of the best in normal conditions of working, and the mode of regulation introduced is decidedly superior to that in most reaction turbines. Figs. 185 and 186 are external views of the turbine case; figs. 187 and 188 are the corresponding sections; fig. 189 is the turbine wheel. The example chosen for illustration has suction pipes, which permit the turbine to be placed above the tail-water level. The water enters the turbine by cast-iron supply pipes at A, and is discharged through two suction pipes S, S. The wateron entering the case distributes itself through a rectangular supply chamber SC, from which it finds its way equally to the four guide-blade passages G, G, G, G. In these passages it acquires a velocity about equal to that due to half the fall, and is directed into the wheel at an angle of about 10° or 12° with the tangent to its circumference. The wheel W receives the water in equal proportions from each guide-blade passage. It consists of a centre plate p (fig. 189) keyed on the shaft aa, which passes through stuffing boxes on the suction pipes. On each side of the centre plate are the curved wheel vanes, on which the pressure of the water acts, and the vanes are bounded on each side by dished or conical cover plates c, c. Joint-rings j, j on the coverplates make a sufficiently water-tight joint with the casing, to prevent leakage from the guide-blade chamber into the suction pipes. The pressure near the joint rings is not very great, probably not one-fourth the total head. The wheel vanes receive the water without shock, and deliver it into central spaces, from which it flows on either side to the suction pipes. The mode of regulating the power of the turbine is very simple. The guide-blades are pivoted to the case at their inner ends, and they are connected by a link-work, so that they all open and close simultaneously and equally. In this way the area of opening through the guide-blades is altered without materially altering the angle or the other conditions of the delivery into the wheel. The guide-blade gear may be variously arranged. In this example four spindles, passing through the case, are linked to the guide-blades inside the case, and connected together by the links l, l, l on the outside of the case. A worm wheel on one of the spindles is rotated by a worm d, the motion being thus slow enough to adjust the guide-blades very exactly. These turbines are made by Messrs Gilkes & Co. of Kendal.
Fig. 190 shows another arrangement of a similar turbine, with some adjuncts not shown in the other drawings. In this case the turbine rotates horizontally, and the turbine case is placed entirely below the tail water. The water is supplied to the turbine by a vertical pipe, over which is a wooden pentrough, containing a strainer, which prevents sticks and other solid bodies getting into the turbine. The turbine rests on three foundation stones, and, the pivot for the vertical shaft being under water, there is a screw and lever arrangement for adjusting it as it wears. The vertical shaft gives motion to the machinery driven by a pair of bevel wheels. On the right are the worm and wheel for working the guide-blade gear.Fig. 191.§ 188.Hydraulic Power at Niagara.—The largest development of hydraulic power is that at Niagara. The Niagara Falls Power Company have constructed two power houses on the United States side, the first with 10 turbines of 5000 h.p. each, and the second with 10 turbines of 5500 h.p. The effective fall is 136 to 140 ft. In the first power house the turbines are twin outward flow reaction turbines with vertical shafts running at 250 revs. per minute and driving the dynamos direct. In the second power house the turbines are inward flow turbines with draft tubes or suction pipes. Fig. 191 shows a section of one of these turbines. There is a balancing piston keyed on the shaft, to the under side of which the pressure due to the fall is admitted, so that the weight of turbine, vertical shaft and part of the dynamo is water borne. About 70,000 h.p. is daily distributed electrically from these two power houses. The Canadian Niagara Power Company are erecting a power house to contain eleven units of 10,250 h.p. each, the turbines being twin inward flow reaction turbines. The Electrical Development Company of Ontario are erecting a power house to contain 11 units of 12,500 h.p. each. The Ontario Power Company are carrying out another scheme for developing 200,000 h.p. by twin inward flow turbines of 12,000 h.p. each. Lastly the Niagara Falls Power and Manufacturing Company on the United States side have a station giving 35,000 h.p. and are constructing another to furnish 100,000 h.p. The mean flow of the Niagara river is about 222,000 cub. ft. per second with a fall of 160 ft. The works in progress if completed will utilize 650,000 h.p. and require 48,000 cub. ft. per second or 211⁄2% of the mean flow of the river (Unwin, “The Niagara Falls Power Stations,”Proc. Inst. Mech. Eng., 1906).Fig. 192.§ 189.Different Forms of Turbine Wheel.—The wheel of a turbine or part of the machine on which the water acts is an annular space, furnished with curved vanes dividing it into passages exactly or roughly rectangular in cross section. For radial flow turbines the wheel may have the form A or B, fig. 192, A being most usual withinward, and B with outward flow turbines. In A the wheel vanes are fixed on each side of a centre plate keyed on the turbine shaft. The vanes are limited by slightly-coned annular cover plates. In B the vanes are fixed on one side of a disk, keyed on the shaft, and limited by a cover plate parallel to the disk. Parallel flow or axial flow turbines have the wheel as in C. The vanes are limited by two concentric cylinders.Theory of Reaction Turbines.Fig. 193.§ 190.Velocity of Whirl and Velocity of Flow.—Let acb (fig. 193) be the path of the particles of water in a turbine wheel. That path will be in a plane normal to the axis of rotation in radial flow turbines, and on a cylindrical surface in axial flow turbines. At any point c of the path the water will have some velocity v, in the direction of a tangent to the path. That velocity may be resolved into two components, a whirling velocity w in the direction of the wheel’s rotation at the point c, and a component u at right angles to this, radial in radial flow, and parallel to the axis in axial flow turbines. This second component is termed the velocity of flow. Let vo, wo, uobe the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and vi, wi, uithe same quantities at the inlet surface of the wheel. Let α and β be the angles which the water’s direction of motion makes with the direction of motion of the wheel at those surfaces. Thenwo= vocos β; uo= vosin βwi= vicos α; ui= visin α.(10)The velocities of flow are easily ascertained independently from the dimensions of the wheel. The velocities of flow at the inlet and outlet surfaces of the wheel are normal to those surfaces. Let Ωo, Ωibe the areas of the outlet and inlet surfaces of the wheel, and Q the volume of water passing through the wheel per second; thenv0= Q/Ωo; vi= Q/Ωi.(11)Using the notation in fig. 191, we have, for an inward flow turbine (neglecting the space occupied by the vanes),Ωo= 2πr0d0; Ωi= 2πridi.(12a)Similarly, for an outward flow turbine,Ωo= 2πrod; Ωi= 2πrid;(12b)and, for an axial flow turbine,Ωo= Ωi= π (r22− r12).(12c)Fig. 194.Relative and Common Velocity of the Water and Wheel.—There is another way of resolving the velocity of the water. Let V be the velocity of the wheel at the point c, fig. 194. Then the velocity of the water may be resolved into a component V, which the water has in common with the wheel, and a component vr, which is the velocity of the water relatively to the wheel.Velocity of Flow.—It is obvious that the frictional losses of head in the wheel passages will increase as the velocity of flow is greater, that is, the smaller the wheel is made. But if the wheel works under water, the skin friction of the wheel cover increases as the diameter of the wheel is made greater, and in any case the weight of the wheel and consequently the journal friction increase as the wheel is made larger. It is therefore desirable to choose, for the velocity of flow, as large a value as is consistent with the condition that the frictional losses in the wheel passages are a small fraction of the total head.The values most commonly assumed in practice are these:—In axial flow turbines,uo= ui= 0.15 to 0.2 √(2gH);In outward flow turbines,ui= 0.25 √2g (H − ɧ),uo= 0.21 to 0.17 √2g (H − ɧ);In inward flow turbines,uo= ui= 0.125 √(2gH).§ 191.Speed of the Wheel.—The best speed of the wheel depends partly on the frictional losses, which the ordinary theory of turbines disregards. It is best, therefore, to assume for Voand Vivalues which experiment has shown to be most advantageous.In axial flow turbines, the circumferential velocities at the mean radius of the wheel may be takenVo= Vi= 0.6 √2gHto 0.66 √2gH.In a radial outward flow turbine,Vi= 0.56 √2g(H − ɧ)Vo= Viro/ ri,where ro, riare the radii of the outlet and inlet surfaces.In a radial inward flow turbine,Vi= 0.66 √2gH,Vo= Viro/ ri.If the wheel were stationary and the water flowed through it, the water would follow paths parallel to the wheel vane curves, at least when the vanes were so close that irregular motion was prevented. Similarly, when the wheel is in motion, the water follows paths relatively to the wheel, which are curves parallel to the wheel vanes. Hence the relative component, vr, of the water’s motion at c is tangential to a wheel vane curve drawn through the point c. Let vo, Vo, vrobe the velocity of the water and its common and relative components at the outlet surface of the wheel, and vi, Vi, vribe the same quantities at the inlet surface; and let θ and φ be the angles the wheel vanes make with the inlet and outlet surfaces; thenvo2= √ (vro2+ Vo2− 2Vovrocos φ)vi= √ (vri2+ Vo2− 2Vivricos θ),(13)equations which may be used to determine φ and θ.Fig. 195.§ 192.Condition determining the Angle of the Vanes at the Outlet Surface of the Wheel.—It has been shown that, when the water leaves the wheel, it should have no tangential velocity, if the efficiency is to be as great as possible; that is, wo= 0. Hence, from (10), cos β = 0, β = 90°, Uo= Vo, and the direction of the water’s motion is normal to the outlet surface of the wheel, radial in radial flow, and axial in axial flow turbines.Drawing voor uoradial or axial as the case may be, and Votangential to the direction of motion, vrocan be found by the parallelogram of velocities. From fig. 195,tan φ = vo/ Vo= uo/ Vo;(14)but φ is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when the velocity of flow uoand velocity of the wheel Voare known. When φ is thus determined,vro= Uocosec φ = Vo√ (1 + uo2/ Vo2).(14a)Correction of the Angle φ to allow for Thickness of Vanes.—In determining φ, it is most convenient to calculate its value approximately at first, from a value of uoobtained by neglecting the thickness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards corrected to allow for the vane thickness.Letφ′ = tan−1(uo/ Vo) = tan−1(Q / ΩoVo)be the first or approximate value of φ, and let t be the thickness, and n the number of wheel vanes which reach the outlet surface of the wheel. As the vanes cut the outlet surface approximately at the angle φ′, their width measured on that surface is t cosec φ′. Hence the space occupied by the vanes on the outlet surface isFor A, fig. 192, ntdocosec φB, fig. 192, ntd cosec φC, fig. 192, nt (r2− r1) cosec φ.(15)Call this area occupied by the vanes ω. Then the true value of the clear discharging outlet of the wheel is Ωo− ω, and the true value of uois Q/(Ωo− ω). The corrected value of the angle of the vanes will beφ = tan [Q / Vo(Ωo− ω) ].(16)§ 193.Head producing Velocity with which the Water enters the Wheel.—Consider the variation of pressure in a wheel passage, which satisfies the condition that the sections change so gradually that there is no loss of head in shock. When the flow is in a horizontal plane, there is no work done by gravity on the water passing through the wheel. In the case of an axial flow turbine, in which the flow is vertical, the fall d between the inlet and outlet surfaces should be taken into account.Let Vi, Vobe the velocities of the wheel at the inlet and outlet surfaces,vi, vothe velocities of the water,ui, uothe velocities of flow,vri, vrothe relative velocities,hi, hothe pressures, measured in feet of water,ri, rothe radii of the wheel,α the angular velocity of the wheel.At any point in the path of a portion of water, at radius r, the velocity v of the water may be resolved into a component V = αr equal to the velocity at that point of the wheel, and a relative component vr. Hence the motion of the water may be considered to consist of two parts:—(a) a motion identical with that in a forced vortex of constant angular velocity α; (b) a flow along curves parallel to the wheel vane curves. Taking the latter first, and using Bernoulli’s theorem, the change of pressure due to flow through the wheel passages is given by the equationh′i+ vri2/ 2g = h′o+ vro2/ 2g;h′i− h′o= (vro2− vri2) / 2g.The variation of pressure due to rotation in a forced vortex ish″i− h″o= (Vi2− Vo2) / 2g.Consequently the whole difference of pressure at the inlet and outlet surfaces of the wheel ishi− ho= h′i+ h″i− h′o− h″o= (Vi2− Vo2) / 2g + (vro2− vri2) / 2g.(17)Case1.Axial Flow Turbines.—Vi= Vo; and the first term on the right, in equation 17, disappears. Adding, however, the work of gravity due to a fall of d ft. in passing through the wheel,hi− ho= (vro2− vri2) / 2g − d.17aCase2.Outward Flow Turbines.—The inlet radius is less than the outlet radius, and (Vi2− Vo2)/2g is negative. The centrifugal head diminishes the pressure at the inlet surface, and increases the velocity with which the water enters the wheel. This somewhat increases the frictional loss of head. Further, if the wheel varies in velocity from variations in the useful work done, the quantity (Vi2− Vo2)/2g increases when the turbine speed increases, and vice versa. Consequently the flow into the turbine increases when the speed increases, and diminishes when the speed diminishes, and this again augments the variation of speed. The action of the centrifugal head in an outward flow turbine is therefore prejudicial to steadiness of motion. For this reason ro: riis made small, generally about 5 : 4. Even then a governor is sometimes required to regulate the speed of the turbine.Case3.Inward Flow Turbines.—The inlet radius is greater than the outlet radius, and the centrifugal head diminishes the velocity of flow into the turbine. This tends to diminish the frictional losses, but it has a more important influence in securing steadiness of motion. Any increase of speed diminishes the flow into the turbine, and vice versa. Hence the variation of speed is less than the variation of resistance overcome. In the so-called centre vent wheels in America, the ratio ri: rois about 5 : 4, and then the influence of the centrifugal head is not very important. Professor James Thomson first pointed out the advantage of a much greater difference of radii. By making ri: ro= 2 : 1, the centrifugal head balances about half the head in the supply chamber. Then the velocity through the guide-blades does not exceed the velocity due to half the fall, and the action of the centrifugal head in securing steadiness of speed is considerable.Since the total head producing flow through the turbine is H − ɧ, of this hi− hois expended in overcoming the pressure in the wheel, the velocity of flow into the wheel isvi= cv√ {2g (H − ɧ − (Vi2− Vo2/ 2g + (vro2− vri2) / 2g) ],(18)where cvmay be taken 0.96.From (14a),vro= Vo√ (1 + uo2/ Vo2).It will be shown immediately thatvri= uicosec θ;or, as this is only a small term, and θ is on the average 90°, we may take, for the present purpose, vri= uinearly.Inserting these values, and remembering that for an axial flow turbine Vi= Vo, ɧ = 0, and the fall d in the wheel is to be added,vi= cv√ {2g(H −Vi2(1 +uo2)+ui2− d) }.2gVo22gFor an outward flow turbine,vi= cv√ [2g{H − ɧ −Vi2(1 +uo2)+ui2} ].2gVi22gFor an inward flow turbine,vi= cv√ [2g{H −Vi2(1 +uo2)+ui2} ].2gVi22g§ 194.Angle which the Guide-Blades make with the Circumference of the Wheel.—At the moment the water enters the wheel, the radial component of the velocity is ui, and the velocity is vi. Hence, if γ is the angle between the guide-blades and a tangent to the wheelγ = sin−1(ui/vi).This angle can, if necessary, be corrected to allow for the thickness of the guide-blades.Fig. 196.§ 195.Condition determining the Angle of the Vanes at the Inlet Surface of the Wheel.—The single condition necessary to be satisfied at the inlet surface of the wheel is that the water should enter the wheel without shock. This condition is satisfied if the direction of relative motion of the water and wheel is parallel to the first element of the wheel vanes.Let A (fig. 196) be a point on the inlet surface of the wheel, and let virepresent in magnitude and direction the velocity of the water entering the wheel, and Vithe velocity of the wheel. Completing the parallelogram, vriis the direction of relative motion. Hence the angle between vriand Viis the angle θ which the vanes should make with the inlet surface of the wheel.§ 196.Example of the Method of designing a Turbine. Professor James Thomson’s Inward Flow Turbine.—Let H = the available fall after deducting loss of head in pipes and channels from the gross fall;Q = the supply of water in cubic feet per second; andη = the efficiency of the turbine.The work done per second is ηGQH, and the horse-power of the turbine is h.p. = ηGQH/550. If η is taken at 0.75, an allowance will be made for the frictional losses in the turbine, the leakage and the friction of the turbine shaft. Then h.p. = 0.085QH.The velocity of flow through the turbine (uncorrected for the space occupied by the vanes and guide-blades) may be takenui= ui= 0.125 √2gH,in which case about1⁄64th of the energy of the fall is carried away by the water discharged.The areas of the outlet and inlet surface of the wheel are then2πrodo= 2πridi= Q / 0.125 √ (2gH).If we take ro, so that the axial velocity of discharge from the central orifices of the wheel is equal to uo, we getro= 0.3984 √ (Q/√H),do= ro.If, to obtain considerable steadying action of the centrifugal head, ri= 2ro, then di=1⁄2do.Speed of the Wheel.—Let Vi= 0.66 √2gH, or the speed due to half the fall nearly. Then the number of rotations of the turbine per second isN = Vi/ 2πri= 1.0579 √ (H √ H/Q);alsoVo= Viro/ ri= 0.33 √2gH.Angle of Vanes with Outlet Surface.Tan φ = uo/ Vo= 0.125 / 0.33 = .3788;φ = 21º nearly.If this value is revised for the vane thickness it will ordinarily become about 25º.Velocity with which the Water enters the Wheel.—The head producing the velocity isH − (Vi2/ 2g) (1 + uo2/ Vi2) + ui2/ 2g= H {1 − .4356 (1 + 0.0358) + .0156}= 0.5646H.Then the velocity isVi= .96 √2g (.5646H)= 0.721 √2gH.Angle of Guide-Blades.Sin γ = ui/ vi= 0.125 / 0.721 = 0.173;γ = 10° nearly.Tangential Velocity of Water entering Wheel.wi= vicos γ = 0.7101 √2gH.Angle of Vanes at Inlet Surface.Cot θ = (wi− Vi) / ui= (.7101 − .66) / .125 = .4008;θ = 68° nearly.Hydraulic Efficiency of Wheel.η = wiVi/ gH = .7101 × .66 × 2= 0.9373.This, however, neglects the friction of wheel covers and leakage. The efficiency from experiment has been found to be 0.75 to 0.80.
Fig. 190 shows another arrangement of a similar turbine, with some adjuncts not shown in the other drawings. In this case the turbine rotates horizontally, and the turbine case is placed entirely below the tail water. The water is supplied to the turbine by a vertical pipe, over which is a wooden pentrough, containing a strainer, which prevents sticks and other solid bodies getting into the turbine. The turbine rests on three foundation stones, and, the pivot for the vertical shaft being under water, there is a screw and lever arrangement for adjusting it as it wears. The vertical shaft gives motion to the machinery driven by a pair of bevel wheels. On the right are the worm and wheel for working the guide-blade gear.
§ 188.Hydraulic Power at Niagara.—The largest development of hydraulic power is that at Niagara. The Niagara Falls Power Company have constructed two power houses on the United States side, the first with 10 turbines of 5000 h.p. each, and the second with 10 turbines of 5500 h.p. The effective fall is 136 to 140 ft. In the first power house the turbines are twin outward flow reaction turbines with vertical shafts running at 250 revs. per minute and driving the dynamos direct. In the second power house the turbines are inward flow turbines with draft tubes or suction pipes. Fig. 191 shows a section of one of these turbines. There is a balancing piston keyed on the shaft, to the under side of which the pressure due to the fall is admitted, so that the weight of turbine, vertical shaft and part of the dynamo is water borne. About 70,000 h.p. is daily distributed electrically from these two power houses. The Canadian Niagara Power Company are erecting a power house to contain eleven units of 10,250 h.p. each, the turbines being twin inward flow reaction turbines. The Electrical Development Company of Ontario are erecting a power house to contain 11 units of 12,500 h.p. each. The Ontario Power Company are carrying out another scheme for developing 200,000 h.p. by twin inward flow turbines of 12,000 h.p. each. Lastly the Niagara Falls Power and Manufacturing Company on the United States side have a station giving 35,000 h.p. and are constructing another to furnish 100,000 h.p. The mean flow of the Niagara river is about 222,000 cub. ft. per second with a fall of 160 ft. The works in progress if completed will utilize 650,000 h.p. and require 48,000 cub. ft. per second or 211⁄2% of the mean flow of the river (Unwin, “The Niagara Falls Power Stations,”Proc. Inst. Mech. Eng., 1906).
§ 189.Different Forms of Turbine Wheel.—The wheel of a turbine or part of the machine on which the water acts is an annular space, furnished with curved vanes dividing it into passages exactly or roughly rectangular in cross section. For radial flow turbines the wheel may have the form A or B, fig. 192, A being most usual withinward, and B with outward flow turbines. In A the wheel vanes are fixed on each side of a centre plate keyed on the turbine shaft. The vanes are limited by slightly-coned annular cover plates. In B the vanes are fixed on one side of a disk, keyed on the shaft, and limited by a cover plate parallel to the disk. Parallel flow or axial flow turbines have the wheel as in C. The vanes are limited by two concentric cylinders.
Theory of Reaction Turbines.
§ 190.Velocity of Whirl and Velocity of Flow.—Let acb (fig. 193) be the path of the particles of water in a turbine wheel. That path will be in a plane normal to the axis of rotation in radial flow turbines, and on a cylindrical surface in axial flow turbines. At any point c of the path the water will have some velocity v, in the direction of a tangent to the path. That velocity may be resolved into two components, a whirling velocity w in the direction of the wheel’s rotation at the point c, and a component u at right angles to this, radial in radial flow, and parallel to the axis in axial flow turbines. This second component is termed the velocity of flow. Let vo, wo, uobe the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and vi, wi, uithe same quantities at the inlet surface of the wheel. Let α and β be the angles which the water’s direction of motion makes with the direction of motion of the wheel at those surfaces. Then
wo= vocos β; uo= vosin βwi= vicos α; ui= visin α.
(10)
The velocities of flow are easily ascertained independently from the dimensions of the wheel. The velocities of flow at the inlet and outlet surfaces of the wheel are normal to those surfaces. Let Ωo, Ωibe the areas of the outlet and inlet surfaces of the wheel, and Q the volume of water passing through the wheel per second; then
v0= Q/Ωo; vi= Q/Ωi.
(11)
Using the notation in fig. 191, we have, for an inward flow turbine (neglecting the space occupied by the vanes),
Ωo= 2πr0d0; Ωi= 2πridi.
(12a)
Similarly, for an outward flow turbine,
Ωo= 2πrod; Ωi= 2πrid;
(12b)
and, for an axial flow turbine,
Ωo= Ωi= π (r22− r12).
(12c)
Relative and Common Velocity of the Water and Wheel.—There is another way of resolving the velocity of the water. Let V be the velocity of the wheel at the point c, fig. 194. Then the velocity of the water may be resolved into a component V, which the water has in common with the wheel, and a component vr, which is the velocity of the water relatively to the wheel.
Velocity of Flow.—It is obvious that the frictional losses of head in the wheel passages will increase as the velocity of flow is greater, that is, the smaller the wheel is made. But if the wheel works under water, the skin friction of the wheel cover increases as the diameter of the wheel is made greater, and in any case the weight of the wheel and consequently the journal friction increase as the wheel is made larger. It is therefore desirable to choose, for the velocity of flow, as large a value as is consistent with the condition that the frictional losses in the wheel passages are a small fraction of the total head.
The values most commonly assumed in practice are these:—
§ 191.Speed of the Wheel.—The best speed of the wheel depends partly on the frictional losses, which the ordinary theory of turbines disregards. It is best, therefore, to assume for Voand Vivalues which experiment has shown to be most advantageous.
In axial flow turbines, the circumferential velocities at the mean radius of the wheel may be taken
Vo= Vi= 0.6 √2gHto 0.66 √2gH.
In a radial outward flow turbine,
Vi= 0.56 √2g(H − ɧ)
Vo= Viro/ ri,
where ro, riare the radii of the outlet and inlet surfaces.
In a radial inward flow turbine,
Vi= 0.66 √2gH,
Vo= Viro/ ri.
If the wheel were stationary and the water flowed through it, the water would follow paths parallel to the wheel vane curves, at least when the vanes were so close that irregular motion was prevented. Similarly, when the wheel is in motion, the water follows paths relatively to the wheel, which are curves parallel to the wheel vanes. Hence the relative component, vr, of the water’s motion at c is tangential to a wheel vane curve drawn through the point c. Let vo, Vo, vrobe the velocity of the water and its common and relative components at the outlet surface of the wheel, and vi, Vi, vribe the same quantities at the inlet surface; and let θ and φ be the angles the wheel vanes make with the inlet and outlet surfaces; then
vo2= √ (vro2+ Vo2− 2Vovrocos φ)vi= √ (vri2+ Vo2− 2Vivricos θ),
(13)
equations which may be used to determine φ and θ.
§ 192.Condition determining the Angle of the Vanes at the Outlet Surface of the Wheel.—It has been shown that, when the water leaves the wheel, it should have no tangential velocity, if the efficiency is to be as great as possible; that is, wo= 0. Hence, from (10), cos β = 0, β = 90°, Uo= Vo, and the direction of the water’s motion is normal to the outlet surface of the wheel, radial in radial flow, and axial in axial flow turbines.
Drawing voor uoradial or axial as the case may be, and Votangential to the direction of motion, vrocan be found by the parallelogram of velocities. From fig. 195,
tan φ = vo/ Vo= uo/ Vo;
(14)
but φ is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when the velocity of flow uoand velocity of the wheel Voare known. When φ is thus determined,
vro= Uocosec φ = Vo√ (1 + uo2/ Vo2).
(14a)
Correction of the Angle φ to allow for Thickness of Vanes.—In determining φ, it is most convenient to calculate its value approximately at first, from a value of uoobtained by neglecting the thickness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards corrected to allow for the vane thickness.
Let
φ′ = tan−1(uo/ Vo) = tan−1(Q / ΩoVo)
be the first or approximate value of φ, and let t be the thickness, and n the number of wheel vanes which reach the outlet surface of the wheel. As the vanes cut the outlet surface approximately at the angle φ′, their width measured on that surface is t cosec φ′. Hence the space occupied by the vanes on the outlet surface is
For A, fig. 192, ntdocosec φB, fig. 192, ntd cosec φC, fig. 192, nt (r2− r1) cosec φ.
For A, fig. 192, ntdocosec φ
B, fig. 192, ntd cosec φ
C, fig. 192, nt (r2− r1) cosec φ.
(15)
Call this area occupied by the vanes ω. Then the true value of the clear discharging outlet of the wheel is Ωo− ω, and the true value of uois Q/(Ωo− ω). The corrected value of the angle of the vanes will be
φ = tan [Q / Vo(Ωo− ω) ].
(16)
§ 193.Head producing Velocity with which the Water enters the Wheel.—Consider the variation of pressure in a wheel passage, which satisfies the condition that the sections change so gradually that there is no loss of head in shock. When the flow is in a horizontal plane, there is no work done by gravity on the water passing through the wheel. In the case of an axial flow turbine, in which the flow is vertical, the fall d between the inlet and outlet surfaces should be taken into account.
Let Vi, Vobe the velocities of the wheel at the inlet and outlet surfaces,vi, vothe velocities of the water,ui, uothe velocities of flow,vri, vrothe relative velocities,hi, hothe pressures, measured in feet of water,ri, rothe radii of the wheel,α the angular velocity of the wheel.
Let Vi, Vobe the velocities of the wheel at the inlet and outlet surfaces,
vi, vothe velocities of the water,
ui, uothe velocities of flow,
vri, vrothe relative velocities,
hi, hothe pressures, measured in feet of water,
ri, rothe radii of the wheel,
α the angular velocity of the wheel.
At any point in the path of a portion of water, at radius r, the velocity v of the water may be resolved into a component V = αr equal to the velocity at that point of the wheel, and a relative component vr. Hence the motion of the water may be considered to consist of two parts:—(a) a motion identical with that in a forced vortex of constant angular velocity α; (b) a flow along curves parallel to the wheel vane curves. Taking the latter first, and using Bernoulli’s theorem, the change of pressure due to flow through the wheel passages is given by the equation
h′i+ vri2/ 2g = h′o+ vro2/ 2g;h′i− h′o= (vro2− vri2) / 2g.
The variation of pressure due to rotation in a forced vortex is
h″i− h″o= (Vi2− Vo2) / 2g.
Consequently the whole difference of pressure at the inlet and outlet surfaces of the wheel is
hi− ho= h′i+ h″i− h′o− h″o= (Vi2− Vo2) / 2g + (vro2− vri2) / 2g.
(17)
Case1.Axial Flow Turbines.—Vi= Vo; and the first term on the right, in equation 17, disappears. Adding, however, the work of gravity due to a fall of d ft. in passing through the wheel,
hi− ho= (vro2− vri2) / 2g − d.
17a
Case2.Outward Flow Turbines.—The inlet radius is less than the outlet radius, and (Vi2− Vo2)/2g is negative. The centrifugal head diminishes the pressure at the inlet surface, and increases the velocity with which the water enters the wheel. This somewhat increases the frictional loss of head. Further, if the wheel varies in velocity from variations in the useful work done, the quantity (Vi2− Vo2)/2g increases when the turbine speed increases, and vice versa. Consequently the flow into the turbine increases when the speed increases, and diminishes when the speed diminishes, and this again augments the variation of speed. The action of the centrifugal head in an outward flow turbine is therefore prejudicial to steadiness of motion. For this reason ro: riis made small, generally about 5 : 4. Even then a governor is sometimes required to regulate the speed of the turbine.
Case3.Inward Flow Turbines.—The inlet radius is greater than the outlet radius, and the centrifugal head diminishes the velocity of flow into the turbine. This tends to diminish the frictional losses, but it has a more important influence in securing steadiness of motion. Any increase of speed diminishes the flow into the turbine, and vice versa. Hence the variation of speed is less than the variation of resistance overcome. In the so-called centre vent wheels in America, the ratio ri: rois about 5 : 4, and then the influence of the centrifugal head is not very important. Professor James Thomson first pointed out the advantage of a much greater difference of radii. By making ri: ro= 2 : 1, the centrifugal head balances about half the head in the supply chamber. Then the velocity through the guide-blades does not exceed the velocity due to half the fall, and the action of the centrifugal head in securing steadiness of speed is considerable.
Since the total head producing flow through the turbine is H − ɧ, of this hi− hois expended in overcoming the pressure in the wheel, the velocity of flow into the wheel is
vi= cv√ {2g (H − ɧ − (Vi2− Vo2/ 2g + (vro2− vri2) / 2g) ],
(18)
where cvmay be taken 0.96.
From (14a),
vro= Vo√ (1 + uo2/ Vo2).
It will be shown immediately that
vri= uicosec θ;
or, as this is only a small term, and θ is on the average 90°, we may take, for the present purpose, vri= uinearly.
Inserting these values, and remembering that for an axial flow turbine Vi= Vo, ɧ = 0, and the fall d in the wheel is to be added,
For an outward flow turbine,
For an inward flow turbine,
§ 194.Angle which the Guide-Blades make with the Circumference of the Wheel.—At the moment the water enters the wheel, the radial component of the velocity is ui, and the velocity is vi. Hence, if γ is the angle between the guide-blades and a tangent to the wheel
γ = sin−1(ui/vi).
This angle can, if necessary, be corrected to allow for the thickness of the guide-blades.
§ 195.Condition determining the Angle of the Vanes at the Inlet Surface of the Wheel.—The single condition necessary to be satisfied at the inlet surface of the wheel is that the water should enter the wheel without shock. This condition is satisfied if the direction of relative motion of the water and wheel is parallel to the first element of the wheel vanes.
Let A (fig. 196) be a point on the inlet surface of the wheel, and let virepresent in magnitude and direction the velocity of the water entering the wheel, and Vithe velocity of the wheel. Completing the parallelogram, vriis the direction of relative motion. Hence the angle between vriand Viis the angle θ which the vanes should make with the inlet surface of the wheel.
§ 196.Example of the Method of designing a Turbine. Professor James Thomson’s Inward Flow Turbine.—
Let H = the available fall after deducting loss of head in pipes and channels from the gross fall;Q = the supply of water in cubic feet per second; andη = the efficiency of the turbine.
Let H = the available fall after deducting loss of head in pipes and channels from the gross fall;
Q = the supply of water in cubic feet per second; and
η = the efficiency of the turbine.
The work done per second is ηGQH, and the horse-power of the turbine is h.p. = ηGQH/550. If η is taken at 0.75, an allowance will be made for the frictional losses in the turbine, the leakage and the friction of the turbine shaft. Then h.p. = 0.085QH.
The velocity of flow through the turbine (uncorrected for the space occupied by the vanes and guide-blades) may be taken
ui= ui= 0.125 √2gH,
in which case about1⁄64th of the energy of the fall is carried away by the water discharged.
The areas of the outlet and inlet surface of the wheel are then
2πrodo= 2πridi= Q / 0.125 √ (2gH).
If we take ro, so that the axial velocity of discharge from the central orifices of the wheel is equal to uo, we get
ro= 0.3984 √ (Q/√H),do= ro.
ro= 0.3984 √ (Q/√H),
do= ro.
If, to obtain considerable steadying action of the centrifugal head, ri= 2ro, then di=1⁄2do.
Speed of the Wheel.—Let Vi= 0.66 √2gH, or the speed due to half the fall nearly. Then the number of rotations of the turbine per second is
N = Vi/ 2πri= 1.0579 √ (H √ H/Q);
also
Vo= Viro/ ri= 0.33 √2gH.
Angle of Vanes with Outlet Surface.
Tan φ = uo/ Vo= 0.125 / 0.33 = .3788;
φ = 21º nearly.
If this value is revised for the vane thickness it will ordinarily become about 25º.
Velocity with which the Water enters the Wheel.—The head producing the velocity is
H − (Vi2/ 2g) (1 + uo2/ Vi2) + ui2/ 2g= H {1 − .4356 (1 + 0.0358) + .0156}= 0.5646H.
H − (Vi2/ 2g) (1 + uo2/ Vi2) + ui2/ 2g
= H {1 − .4356 (1 + 0.0358) + .0156}
= 0.5646H.
Then the velocity is
Vi= .96 √2g (.5646H)= 0.721 √2gH.
Angle of Guide-Blades.
Sin γ = ui/ vi= 0.125 / 0.721 = 0.173;
γ = 10° nearly.
Tangential Velocity of Water entering Wheel.
wi= vicos γ = 0.7101 √2gH.
Angle of Vanes at Inlet Surface.
Cot θ = (wi− Vi) / ui= (.7101 − .66) / .125 = .4008;
θ = 68° nearly.
Hydraulic Efficiency of Wheel.
η = wiVi/ gH = .7101 × .66 × 2= 0.9373.
η = wiVi/ gH = .7101 × .66 × 2
= 0.9373.
This, however, neglects the friction of wheel covers and leakage. The efficiency from experiment has been found to be 0.75 to 0.80.
Impulse and Partial Admission Turbines.
§ 197. The principal defect of most turbines with complete admission is the imperfection of the arrangements for working with less than the normal supply. With many forms of reaction turbine the efficiency is considerably reduced when the regulatingsluices are partially closed, but it is exactly when the supply of water is deficient that it is most important to get out of it the greatest possible amount of work. The imperfection of the regulating arrangements is therefore, from the practical point of view, a serious defect. All turbine makers have sought by various methods to improve the regulating mechanism. B. Fourneyron, by dividing his wheel by horizontal diaphragms, virtually obtained three or more separate radial flow turbines, which could be successively set in action at their full power, but the arrangement is not altogether successful, because of the spreading of the water in the space between the wheel and guide-blades. Fontaine similarly employed two concentric axial flow turbines formed in the same casing. One was worked at full power, the other regulated. By this arrangement the loss of efficiency due to the action of the regulating sluice affected only half the water power. Many makers have adopted the expedient of erecting two or three separate turbines on the same waterfall. Then one or more could be put out of action and the others worked at full power. All these methods are rather palliatives than remedies. The movable guide-blades of Professor James Thomson meet the difficulty directly, but they are not applicable to every form of turbine.
C. Callon, in 1840, patented an arrangement of sluices for axial or outward flow turbines, which were to be closed successively as the water supply diminished. By preference the sluices were closed by pairs, two diametrically opposite sluices forming a pair. The water was thus admitted to opposite but equal arcs of the wheel, and the forces driving the turbine were symmetrically placed. As soon as this arrangement was adopted, a modification of the mode of action of the water in the turbine became necessary. If the turbine wheel passages remain full of water during the whole rotation, the water contained in each passage must be put into motion each time it passes an open portion of the sluice, and stopped each time it passes a closed portion of the sluice. It is thus put into motion and stopped twice in each rotation. This gives rise to violent eddying motions and great loss of energy in shock. To prevent this, the turbine wheel with partial admission must be placed above the tail water, and the wheel passages be allowed to clear themselves of water, while passing from one open portion of the sluices to the next.
But if the wheel passages are free of water when they arrive at the open guide passages, then there can be no pressure other than atmospheric pressure in the clearance space between guides and wheel. The water must issue from the sluices with the whole velocity due to the head; received on the curved vanes of the wheel, the jets must be gradually deviated and discharged with a small final velocity only, precisely in the same way as when a single jet strikes a curved vane in the free air. Turbines of this kind are therefore termed turbines of free deviation. There is no variation of pressure in the jet during the whole time of its action on the wheel, and the whole energy of the jet is imparted to the wheel, simply by the impulse due to its gradual change of momentum. It is clear that the water may be admitted in exactly the same way to any fraction of the circumference at pleasure, without altering the efficiency of the wheel. The diameter of the wheel may be made as large as convenient, and the water admitted to a small fraction of the circumference only. Then the number of revolutions is independent of the water velocity, and may be kept down to a manageable value.