MACHINERY.] HYDROMECHANICS 535 Ui (4). v,-i = t cosec 6 = V uf + If the pump is raising less or more than its proper quantity, 6 will not satisfy the last condition, and there is then some loss of head in shock. At the outer circumference of the wheel or outlet surface, v rn u n cosec <j>
Variation of Pressure in the Pump Disk. Precisely as in the case of turbines, it can be shown that the variation of pressure between the inlet and outlet surfaces of the pump is Inserting the values of v r0 , v rt in (4) and (5), we get for normal conditions of working h -h _ V o 3 ~ V 2 7/ Q 2 cosec 2( fr 20 20 V ft 2 u n z cosec " 20 20" (6). Hydraulic Efficiency of the Pump. Neglecting disk friction, journal friction, and leakage, the efficiency of the pump can be found in the same way as that of turbines ( 172). Let M be the moment of the couple rotating the pump, and a its angular velocity; w , r the tangential velocity of the water and radius at the outlet surface; ., r, the same quantities at the inlet surface. Q being the dis charge per second, the change of angular momentum per second is Hence M = ^
In normal working, w; = 0. Also, multiplyingby theangular velocity, the work done per second is But the useful work done in pumping is GQH. -efficiency is GQH 0H 0H
- 7 = -Ty -- = -- = - TT
Ma W r a W Therefore the 189. Case 1. Centrifugal Pump with no Whirlpool Chamber. When no special provision is made to utilize the energy of motion of the water leaving the wheel, and the pump discharges directly into a chamber in which the water is flowing to the discharge pipe, nearly the whole of the energy of the water leaving the disk is wasted. The water leaves the disk with the more or less considerable velocity v , and impinges on a mass flowing to the discharge pipe at the much slower velocity v t . The radial component of v is almost necessarily wasted. From the tangential component there is a gain of pressure 20 20 which will be small, if r, is small compared with w . Its greatest w 2 value, if t = i?0 , is | , which will always be a small part of the whole head. Suppose this neglected. The whole variation of pressure u - 2 in the pump disk then balances the lift and the head necessary to give the initial velocity of flow in the eye of the wheel. " COSCC 2 (|> U;~ TT Y 2 20 20 or V = /( 2 0H + V cosecV) and the efficiency of the pump is, from (7), . . . (8) Vo 2 - ;t 2 cosec ft
- 2V (V -w cot<)
For 2V* which is necessarily less than J. That is, half the work expended in driving the pump is wasted. By recurving the vanes, a plan intro duced by Mr Appold, the efficiency is increased, because the velocity v of discharge from the pump is diminished. If <f> is very small, and then = cosec cf> = cot (f> ; V 2 + M O cosec <ft which may approach the value 1, as <j> tends towards 0. Equation (8) shows that cosec < cannot be greater then V . Putting we get the following numerical values of the efficiency and the circumferential velocity of the pump : <*> 90 45 30 20 10 n 0-47 0-56 0-65 073 0-84 1-06 1-12 1-24 1-75 <f> cannot practically be made less than 20; and, allowing for the fric- tional losses neglected, the efficiency of a pump in which < = 20 is found to be about 60. 190. Case 2. Pump with a Whirlpool Chamber, as in fig. 203. Pro fessor James Thomson first suggested that the energy of the water after leaving the pump disk might be utilized, if a space were left in which a free vortex could be formed. In such a free vortex the velocity varies inversely as the radius. The gain of pressure in the vortex chamber is, putting r , r w for the radii to the outlet surface of wheel and to outside of free vortex, 2 / r 2 2 / ^L f i_Io l!a. f 1-P) 20 V r.V 20 V ) if The lift is then, adding this to the lift in the last case, H = ) V 2 - 2 cosec 2 4> + u 2 (l - & 2 ) | . A(] But V 2 - cot H j (2 - 2 )V 2 -2&V w cot <f> - F 2 cosec 2 <?i j (10). Putting this in the expression for the efficiency, we find a con siderable increase of efficiency. Thus with = 90 and &=4> 1 = nearly; <p a small angle and k= 4 > J = 1 nearly. With this arrangement of pump, theiefore, the angle at the outer ends of the vanes is of comparatively little importance. A moderate angle of 30 or 40 may very well be adopted. The following numerical values of the velocity of the circumference of the pump have been obtained by taking &= , and =0 25N/20H . 90 45 30 20 r , 762V20H 842 ,, 911 1-023 The quantity of water to be pumped by a centrifugal pump neces sarily varies, and an adjustment for different quantities of water can not easily be introduced. Hence it is that the average efficiency of pumps of this kind is in practice less than the efficiencies given above. The advantage of a vortex chamber is also generally neglected. The velocity in the supply and discharge pipes is also often made greater than is consistent with a high degree of efficiency. Velocities of 6 or 7 feet per second in the discharge and suction pipes, when the lift is small, cause a very sensible waste of energy ; 3 to 6 feet would be much better. Centrifugal pumps of very large size have been constructed. Messrs Easton and Anderson have made pumps lor the North Sea Canal in Holland which deliver each 670 tons of water per minute on a lift of 5 feet. The pump disks are 8 feet diameter. Messrs J. and H. Gwynne constructed some yiumps for draining the Ferrarese Marshes, which together deliver 2000 tons per minute. A pump made under Professor J. Thomson s direction for drainage works in Barbados had a pump disk 16 feet in diameter and a whirl pool chamber 32 feet in diameter. The efficiency of centrifugal pumps when delivering less or more than the normal quantity of water is discussed in a paper in the Proc. Inst. of Civil Engineers*
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