Editing Pelton wheel (section)

== Turbine physics and derivation ==
{{originalresearch-section|date=December 2024}}
=== Energy and initial jet velocity ===

In the ideal ([[friction]]less) case, all of the hydraulic [[potential energy]] (''E''<sub>''p''</sub>&nbsp;= ''mgh'') is converted into [[kinetic energy]] (''E''<sub>''k''</sub>&nbsp;= ''mv''<sup>2</sup>/2) (see [[Bernoulli's principle]]).  Equating these two equations and solving for the initial jet velocity (''V''<sub>''i''</sub>) indicates that the theoretical (maximum) jet velocity is ''V''<sub>''i''</sub>&nbsp;= {{radic|2''gh''}}.  For simplicity, assume that all of the velocity vectors are parallel to each other.  Defining the velocity of the wheel runner as: (''u''), then as the jet approaches the runner, the initial jet velocity relative to the runner is: (''V''<sub>''i''</sub>&nbsp;−&nbsp;''u'').<ref name = jcalvert/>
The initial velocity of jet is ''V''<sub>''i''</sub>.

=== Final jet velocity ===

Assuming that the jet velocity is higher than the runner velocity, if the water is not to become backed-up in runner, then due to conservation of mass, the mass entering the runner must equal the mass leaving the runner.  The fluid is assumed to be incompressible (an accurate assumption for most liquids).  Also, it is assumed that the cross-sectional area of the jet is constant. The fluid impacts the runner, slowing down and transferring momentum from the jet to the wheel. Assuming no losses, the fluid leaving the runner's velocity is reduced by the velocity difference between the jet and the runner. The jet ''[[speed]]'' remains constant relative to the runner.  So as the jet recedes from the runner, the jet velocity relative to the runner is: − (''V''<sub>''i''</sub>&nbsp;− ''u'')&nbsp;= −''V''<sub>''i''</sub>&nbsp;+ ''u''. Note the minus sign indicates the reduction in stream velocity from impacting the runner. With the assumption that jet velocity is higher than runner velocity, the result is a "rebound" resulting in the jet flowing away from the runner. In the standard reference frame (relative to the earth), the final velocity is then: ''V''<sub>''f''</sub>&nbsp;= (−''V''<sub>''i''</sub>&nbsp;+ u)&nbsp;+ ''u''&nbsp;= −''V''<sub>''i''</sub>&nbsp;+ 2''u''.

The value ''V''<sub>''f''</sub> is bounded by two cases: a stationary runner and a runner moving at the velocity of the stream, ''V''<sub>''i''</sub>. For the stationary runner case, the stream velocity is -Vf, indicating the fluid fully reversed direction. In this case the force on the wheel is the highest (due to the largest possible velocity change), but power delivered is zero, since there is no movement. For the case where the runner is moving at the speed of the stream, the velocity of the wheel is the highest, but power delivered is also zero, since there is no torque or force imparted on the wheel (due to no change is stream velocity). 
<!-- Note: reference is believed to have two sign errors, that are believed to be corrected here.  The two sign errors cancel each other, so the end result is the same.-->

=== Optimal wheel speed ===

The ideal runner speed will cause all of the kinetic energy in the jet to be transferred to the wheel.  In this case the final jet velocity must be zero.  If −''V''<sub>''i''</sub>&nbsp;+ 2''u''&nbsp;= 0, then the optimal runner speed will be ''u''&nbsp;= ''V''<sub>''i''</sub> /2, or half the initial jet velocity.

=== Torque ===

By [[Newton's laws of motion|Newton's second and third laws]], the force ''F'' imposed by the jet on the runner is equal but opposite to the rate of momentum change of the fluid, so
: ''F'' = −''m''(''V''<sub>f</sub> − ''V''<sub>i</sub>)/''t'' = −''ρQ''[(−''V''<sub>i</sub> + 2''u'') − ''V''<sub>i</sub>] = −''ρQ''(−2''V''<sub>i</sub> + 2''u'') = 2''ρQ''(''V''<sub>i</sub> − ''u''),
where ''ρ'' is the density, and ''Q'' is the volume rate of flow of fluid.  If ''D'' is the wheel diameter, the torque on the runner is.
: ''T'' = ''F''(''D''/2) = ''ρQD''(''V''<sub>i</sub> − ''u'').
The torque is maximal when the runner is stopped (i.e. when ''u''&nbsp;= 0, ''T''&nbsp;= ''ρQDV''<sub>i</sub>).  When the speed of the runner is equal to the initial jet velocity, the torque is zero (i.e., when ''u''&nbsp;= ''V''<sub>i</sub>, then ''T''&nbsp;=&nbsp;0).  On a plot of torque versus runner speed, the torque curve is a straight line between these two points: (0, ''pQDV''<sub>i</sub>) and (''V''<sub>i</sub>, 0).<ref name = jcalvert/>
Nozzle efficiency is the ratio of the jet power to the waterpower at the base of the nozzle.

=== Power ===

The power ''P''&nbsp;= ''Fu''&nbsp;= ''Tω'', where ''ω'' is the angular velocity of the wheel.  Substituting for ''F'', we have ''P''&nbsp;= 2''ρQ''(''V''<sub>''i''</sub>&nbsp;− ''u'')''u''.  To find the runner speed at maximum power, take the derivative of ''P'' with respect to ''u'' and set it equal to zero, [''dP''/''du''&nbsp;= 2''ρQ''(''V''<sub>''i''</sub>&nbsp;− 2''u'')].  Maximum power occurs when ''u''&nbsp;= ''V''<sub>''i''</sub> /2.  ''P''<sub>max</sub> = ''ρQV''<sub>''i''</sub><sup>2</sup>/2.  Substituting the initial jet power ''V''<sub>''i''</sub>&nbsp;= {{radic|2''gh''}}, this simplifies to ''P''<sub>max</sub>&nbsp;= ''ρghQ''.  This quantity exactly equals the kinetic power of the jet, so in this ideal case, the efficiency is 100%, since all the energy in the jet is converted to shaft output.<ref name = jcalvert/>

=== Efficiency ===

A wheel power divided by the initial jet power, is the turbine efficiency, ''η''&nbsp;= 4''u''(''V''<sub>''i''</sub>&nbsp;− ''u'')/''V''<sub>''i''</sub><sup>2</sup>. It is zero for ''u''&nbsp;=&nbsp;0 and for ''u''&nbsp;=&nbsp;''V''<sub>''i''</sub>.  As the equations indicate, when a real Pelton wheel is working close to maximum efficiency, the fluid flows off the wheel with very little residual velocity.<ref name = jcalvert/>  In theory, the [[Efficient energy use|energy efficiency]] varies only with the efficiency of the nozzle and wheel, and does not vary with hydraulic head.<ref>[http://people.rit.edu/rfaite/courses/tflab/Cussons/pelton/pelton.htm Pelton Wheel Water Turbine], Ron Amberger's Pages</ref>
The term "efficiency" can refer to: Hydraulic, Mechanical, Volumetric, Wheel, or overall efficiency.