 Okay, so in the previous example calculated the pumping energy and power requirements for pumping water up 10 meters. In most cases in a solar array there's horizontal piping, not vertical piping. You're piping the fluid through the collector across a big horizontal surface area. So that still requires energy because of the friction in the pipe. So how do you translate that to an equivalent head loss essentially? And we do this with the Darcy Weisbach equation which says here Darcy equation which says that the head loss due to friction is equal to the Darcy friction factor which can be looked up based on your various fluid parameters such as the Reynolds number, whether your flow is turbulent or laminar, things like that, times the length of your pipe. So if you're pumping it a thousand meters that would go there and then the internal diameter of your pipe. Oh, back to the Darcy friction factor that also has to do with the roughness of the pipe. So if you have a very smooth pipe then you would get a lower friction factor times the average velocity of the fluid squared divided by two times Earth's acceleration on Earth, 9.81 meters per second squared. So you can see already that the head over here is a h sub f, the thing we're calculating, is affected by the square of velocity which shows that as speed increases in your fluid you're going to have much more, you're going to require much, you're going to have a much higher head loss which from the previous calculation shows you'd have much higher energy. So if we have a Darcy friction factor of say 0.2 and that's unitless and we want to say do a pipe that's a thousand meters long with an internal diameter of about an inch, 0.03 meters. We're going to do it for two different velocities here so let's say that the average velocity for the first round is three meters per second and we know that G is 9.81 meters per second squared. When we plug all these values in, 0.02 times meters, motor of 0.03 meters, velocity is three meters per second squared, 2 times 9.81 meters per second squared. So you can see some of these units cancel out, meters, meters, second squared, second squared. We have meters squared, meters in the denominator so we're going to end up with meters because there's two meters up top one below so in the end this is equivalent to meters, to units of meters. Once we crunch the numbers, 0.03 times 3 squared divided by 2 divided by 9.81, we end up with 306 meters of head loss so that's what this says is that one pipe that's a thousand meters long and one inch in diameter at three meters per second takes the same, has the same essentially energetic requirements as pumping fluid up, straight up with no friction losses but straight up 306 meters vertically on earth. If we're on the moon it would take less energy just to say side note because the moon has one-sixth the gravitational constant so it does matter these types of things though I doubt you'll be installing a solar collector on the moon for now but just a little side note there all these little details do matter and so let's see how this changes if instead of three meters per second average velocity we have say half a meter per second average velocity so up here this three would change to be a 0.5 and we would run that calculation again and we would get let's see here 0.2 times thousand times point five squared instead of three squared by 2 and 9.81 I forgot to divide by 0.3 there we go that's a better number there we get about 8.5 meters for the second version of this calculation and so you can see that by reducing the flow speed 0.5 to 0.5 we have a much less head loss or much less pumping energy required and so you know the implications of this are really that you can save a lot of energy by pumping slower at the same time in a solar thermal energy collection application that means your fluid would get hotter a lot faster and so you know this this it could be a good thing under certain circumstances and others it could be bad because your fluid could be overheating if it's going too slow through your collector and so this becomes an optimization problem where you have to balance all these different things happening at once the fluid flow as well as the rate of energy being absorbed as well as what those maximum temperature thresholds are for good operation of your system without damaging the fluid or any other components as well and so that's that's sort of a fine line that has to be walked to ensure that your system's system is working correctly so hopefully that gives you a little bit of an insight into that on a more technical level and thanks for listening