|
 |
 Photo collection by Th.Schiöler Applied mathematics concerning water-lifting 5: More theoretical efficiencies |
| More theoretical efficiencies |
 |
| About the theory of the watermills by Parent in 1704 Unfortunately, the paper by Parent is badly organized but the result is beautiful. He calculate that 4/27 of the power effet naturelle from the river is received of the waterwheel. Already in the 18th century they were aware that this theoretical figure was too little, but the outcome was that The French Academy were interested in the efficiency of the water wheel. A prize dissertation was arranged. We will now turn to the figure by Parent but using an other method than he did. If the water wheel is blocked then the paddle will be affected with a very big force. We assume that the wheel has only one paddle. If the wheel turns freely, the velocity of the rim will be equal to the velocity of the river. In both cases, the supplied power is equal to zero. This is because one of the two factors: force or velocity is zero. In the instance, the force is great but the velocity is zero. In the freely turning instance, we have the other way round. Between these two zeroes we must have a maximum.  This water-lifting wheel is a physical model of a real wheel. The diameter is equal to the height to which the water should be lifted. The buckets are kept close together and the total volume is V. The frequency f is equal to revolution per second. It could be 0.1 run per second. The pressure (force) on the paddle is deduced from the dynamic pressure. This pressure was for the first time used by Newton but not exactly correct. Experience with waterwheels shows another result.  Nomenclature
From the relative velocity – witch is the difference between the velocities of the river and the wheel – we get the force. The force on the paddle wheel is then:  The power is equal to the force multiplied with the velocity of the rim.   Although the wheel turns the problem is not dynamic but static. The generating torque should then be equal to the torque, which is made by the weight of the water in the buckets. We need the ratio of the velocities and the two torques.   The two torques are now equal to:  where:   where:   where:   introducing the dynamic pressure head:  we get:   The last fraction between in- and output is the efficiency. The powers PB and Pf are the results of : pressure head • the flow • the density • the acceleration of gravity. Using the nomenclature, you will get the result in watt, which is the dimension of power. The left hand side is identical with the efficiency, which again is the function of the velocity ratio.   If we use the velocity ratio – for which we have the maximum power – we get:  Shortly after it was obviously, that the figure was too small. However, never mind the mathematic was beautiful. From the formula, we can see how the efficiency varies according to the velocity-ratio. The formula can be rewritten as an equation of the third degree.  There is a misprint in the equation ……correct it. If we differentiate this equation, we get an equation of second degree.    To the first root, we have the maximum value. The second root gives the minimum. The value (2/pi) R is the distance to the centre of gravity. We will control this value. The centre of gravity of a wheel is placed in the centre of the rim. However, where is it placed for half a wheel?    |
