Monson looks at a pelten wheel turbine in section 12.8. The basic idea with a pelten wheel is to have a jet that's formed by lethics and water out of a nozzle. So it comes out of a V nozzle with a cross-sectional area of A nozzle. And that impacts on blades that are sitting on this wheel and the wheel is turning. Now if we look from above in this direction looking down on those blades we'll see that the jet's coming in this way and then it's turning through almost exactly 180 degrees. Just a little bit less than 180 degrees to allow the water to escape out to the sides here instead of just getting in the way of the next bucket when it comes to a stop. Now this nozzle impacts on multiple blades sometimes all at the same time. So keep in mind that this all of this mass flow is eventually hitting one of these blades at approximately this radius r as this wheel goes around at this angular velocity omega. So the velocity of the buckets moving away in the x direction is equal to omega times r. At least if we approximate this curvature as being more or less a straight line and it's not a bad approximation. So we've got the fluid coming in at location 1 going out at location 2 and we've chosen a control volume here around our bucket that's moving at the bucket velocity whatever that bucket velocity happens to be so that this blade is stationary within our analysis volume. Because it's stationary we can apply Bernoulli. There'll be no losses because there's no work done on the blade. We're going to neglect friction and gravity as usual. So by Bernoulli V1 coming in at atmospheric pressure is going to be equal to V2 going out again at atmospheric pressure and neglecting gravity. And it's going to be equal to the nozzle velocity however fast this was going minus the blade velocity Vb however quickly the blade is moving away from the nozzle so we wind up with the relative velocity there and by continuity conservation of mass we know that whatever's coming in here is going out there as long as the flow is steady. So m dot 1 is going to be equal to m dot 2 is going to be equal to density times the velocity coming in at 1 that's the relative velocity here times the area at 1 that's this area here and it's the same as the area of the nozzle up there. The geometry remains the same. So that'll be equal to rho V nozzle minus V blade times the area of the nozzle. The force in the x direction and there's just one this is the force that's going to be acting on the on the fluid. It's going to be if that's the positive x direction it's going to be m dot u out minus m dot u in. M dot in both cases is the same rho V1A1 the outlet velocity in the extra component direction is actually going this way so it's negative and it's negative V2 which is equal to V1 and then minus rho V1A1 that's the mass flow in times the x component of velocity coming in that's V1 again it's going to be the relative velocity. So we wind up with negative 2 two terms negative there and negative there times density times the area of the nozzle that's A1 times the velocity of the nozzle minus the velocity of the blade. That's V2 and V1 and it's squared because there's two of them in each location. So this is the force acting on a single blade but keep in mind that all of the mass flow coming from the nozzle hits one blade or another. And in that case instead of having the m dot be rho V1A1 it's going to be rho VnA1 and we'll wind up with the sum of the forces in the extra action equal to negative 2 rho V nozzle A nozzle that's the total mass flow impacting all of the blades together times Vn minus Vb that's the velocity change associated with the relative velocity that's happening on each blade. So that's the force on multiple blades all taken together and that's the force acting on the water of course the force acting on the blade is equal and opposites so it'll be positive to rho VnA1 times the difference Vn minus Vb acting in the positive x direction on the blade. And if we look at how this works out we can draw a graph for varying blade velocities. If we plot the velocity of the blade in this direction then it's increasing up to the point where we get to the velocity of the nozzle and then the whole thing just doesn't work. So we can move from having the blades locked, the rotor locked up to freewheeling and this will be the increasing velocity of the blade. Now the force is going to be largest when the blades are locked when this Vb is equal to 0 and it's going to decrease to 0 when the whole thing is freewheeling. So we'll have the other side here a linear decrease like that of the force in the extra action. Now the power is going to be equal to the force acting on the blades, that's the force in the x direction, times the velocity of those blades. And if we plug that in we're going to get the force we've already got and then we'll wind up with the additional Vb term in there. So we'll wind up with 2 rho Vn Vb An times Vn minus Vb. So we plot that, it's going to have a maximum in the middle here and it's going to go to 0 at either side. So that's the power, it's going to be a maximum when the blade velocity is equal to half of the nozzle velocity. We're going to take out the largest amount of kinetic energy from that nozzle jet that we possibly can. So this simple analysis for a pelten wheel ignores the fact that this angle isn't quite exactly 180 degrees. It ignores the friction and it ignores the fact that the geometry is a little more complicated than we can capture with this Vb equal to omega R here. But what we will see is that the actual power output from a real pelten wheel doesn't look terribly different and we can get very close to the performance that we predict from this very simple analysis. So this whole integral momentum analysis will give us a good approximation of what we can expect in terms of the forces, power, and behavior of systems where we're dramatically changing the velocity of a large stream of water.