The next topic in Hydraulics and Hydrologic systems is analysis of Pumps and Turbines. In other words, devices for adding or extracting energy from flows. And in this section we will cover the basic equations. Characteristic curves of pumps, matching pumps to systems hea, heads, and then show some examples of pump and turbines. So firstly, schematically a pump might look something like this, where we have an inflow station here, a flow rate going through here, and the inlet station I'll call station 1, outlet station, station two. And if I apply the new lead equation between those two stations, our general equation is given here. But in this case, we can neglect the head loss due to friction, hf. We can neglect the head loss, due to a turbine, ht, and neglect the head losses. The minor head losses so the relationship is P1 over gamma plus V1 squared is equal to P2 over gamma plus V2 squared over 2g where I'm neglecting the small elevation difference here Z2 minus Z1. Or rearranging hp, the head ended by the pump, is P2 over gamma, plus V2 squared over 2G, et cetera. And, I can show this relationship graphically, because the sum of these two terms here, P2 over gamma, plus V2 squared over 2G is the elevation of the energy grade line upstream, EGL1. And the elevation of the energy grade line downstream here, EGL2, is here, so graphically this increase in the energy grade line across the pump is the head added by the pump hp. Now, other equations that are important is that the power, which is either added to the water by a pump or extracted from the water by a turbine, is given by this equation by a pump. The power added to the water is gamma Qhp. Where gamma is the specific weight of the water, Q is the volume flow rate, hp is the pump head or the pump, the head added. Similarly, for a turbine, the corresponding equation for the power extracted from the water is gamma Qht, where ht is the head extracted by the turbine. Another important equation is the efficiency, which we can define for a pump and turbine, by eta is equal to the power out divided by the power in. So, for a pump, this would be equal to eta is equal to gamma Qhp, the power out. In other words, the power which is extracted from the water, divided by w., the power which is required to drive the pump. In other words, supplied by the shaft. Similarly that equation applied to a turbine is the efficiency is equal to w., divided by gamma Qht where w., is the use for power, which is extracted by, by the turbine. In other words to generate electrical power. And gamma Qht is the power extracted from the water. So in that equation, w., is either the power which is supplied to the pump, or the power extracted from the turbine, the useful power. A note about units, power in both those equations is gamma Q times h, so in metric units. The fundamental units of specific weight on Newtons per cubic meter, flow rate is cubic meters per second, head is in meters, so putting that together, the fundamental units where power would be Newton-meter per second. However, a Newton-meter per second is a watt. And a 1,000 watts is a kilowatt. In British units or USCS units, the fundamental unit of specific weight is pounds-force per cubic feet. Flow rate is cubic feet per second. Head is in feet. Therefore the fundamental units of power are foot-pounds per second. However we more commonly use the horsepower. And a horsepower is 550 foot-pounds per second. Therefore the power is gamma Qhp divided by 550 horsepower. Now, the most common type of, pump that we'll, there are different types of pumps, but the most common one is the centrifugal pump which is shown here. And if I think about the characteristics of the pump, here is my diagram here. So let's suppose that I have a volume flow rate Q going through here, and the head added by this pump is hp. As I vary the flow rate and the head, I get curves which are called characteristic curves, which are shown here. Imagine I have a valve on here and I open and close the valve and therefore vary the flow rate and the head. That as they vary, I get graphs which look like this. So this is for example, the flow rate Q and this curve here is the head hp. And in particular if I completely close the valve so that there's no flow rate through here, I still have a head, I'm still increasing the pressure. And the head at zero flow rate is called the shut off head. Similarly, the power w., is given by this expression here and if I compute the ratio of those two, the power to the delivered to the water to the pumping power. I get the efficiency eta, which has a curve which looks something like this. And at some point here we have a maximum in the efficiency which is obviously the point where we have designed the pump to operate. Where that occurs is called sometimes the BEP. The best efficiency point and it occurs at the normal or the design flow rate. So, these curves for a pump are called the characteristic curves of the pump, and here is the corresponding section from the reference manual. That covers the basic ideas and equations of pumps and turbines, and in the next segment we'll look at some examples.