Continuing our discussion of hydraulics and hydrologic systems, in this segment, I'm going to discuss flow in pipes. And these are the overall topics we'll look at in flow in pipes, and in this segment, we'll be looking at the be, Bernoulli equation. With energy losses and gains, computation of friction headloss, the Moody Chart, and equations for non-circular ducts or conduits. So, as a starting point, let's consider a flow of fluid like water or air, or anything, in a pipeline, and let's suppose the water flows between these two stations, one and two. The basic Bernoulli equation that applies to this pipeline, if there are no energy losses or gains, is quite familiar. It is given here, which relates the variation of pressure, velocity, and elevation in the pipe. But now, let's suppose that we have energy losses and gains between these two stations, as shown here. Where, in this diagram, the energy losses and gains are firstly, h p is the mechanical energy added to the flow. The most common way of doing that of course, would be by means of a pump, h t is mechanical energy extracted from the flow for example by a turbine. H f is the head loss or energy loss due to friction in the pipe. And finally, h m are so called minor losses, for example due to valves, elbows, etc and other fittings in a hydraulic system. So, in that case, our modified Bernoulli equation, or the extended Bernoulli equation, is shown here. Which is similar to the first version, except with the addition of all of these terms, the addition, or, addition or extraction of energy from the flow. And here is the corresponding section from the f e reference manual, where what I am calling h m here for minor losses, they are calling h f, or head loss for fittings. So, let's first of all do a simple example on flow in a pipeline with no energy losses or gains. So, we have this converging pipeline here with the gain in elevation and given that the flow rate is 0.05 cubic meters per second. Upstream and downstream elevations are one meter and five meters and the upstream pipe diameter is 0.2 meters, and the downstream pipe diameter is 0.1 meter. We're given, that the upstream pressure, in other words, at station one, is a hundred and twenty kilopascals. The downstream pressure head, in other words, at this sta, station here is most nearly, which of these alternatives? So, our starting point is the Bernoulli equation with no energy losses or gains is given here. And in this case, we're asked to calculate the downstream pressure and downstream our label station two, upstream is one. So, rearranging the downstream pressure head p two over gamma is given by this expression here. So, to evaluate this first of all we need to compute the velocities as volume flow rate divided by cross-sectional area. So, v one is 1.59 meters per second similarly, the two downstream is q over a two is equal to 6.37 meters per second. So, plugging in all the numbers, we have p two over gamma is p one over gamma p one is a hundred and twenty kilopascals. And that should be a hundred and twenty there, times ten cubed divided by ninety eight hundred the specific weight. And continuing on here with the elevations, and separating out all those numbers, the different heads are given there. So, the answer is p two, the pressure head of p two over gamma, the pressure head at station two is 4.26 meters, the closest answer is b, 4.3 meters. It wasn't asked, but if you want to continue with this and calculate the actual pressure at station two is equal to specific weight gamma times 4.26. So, is equal to ninety eight hundred times 4.26 or 41.7 kilopascals. Now also, it's instructive to compute all of these head terms in this equation. So, at station one here, we have the elevation z one is one meter then the first term here is p one over gamma is the pressure head. And that is the height to which the liquid column in the static pressure tube rises. So, that distance is 10.2 meters, this distance, the distance from the top of the pitot tube, to the pitot static tube, is the local velocity head, v squared over 2 g or 0.13 meters. And the height that this rises to, is by definition, the energy grade line. Similarly, at station two here, this elevation is five meters, this distance is the pressure head, p two over gamma which is this term, which is 2.07. And this distance, the difference between the distance between the two liquid levels is the velocity head downstream v two squared over two g. Which is, I'm sorry, that is 2.07 meters and this term, the pressure head, is 0.13 meters. If you add all of these terms up, you'll find that this elevation is constant. In other words, the height that these levels rise to is the energy grade line which is, by definition, constant or horizontal because there are no energy losses or gains in the system. And the energy gain grade line passes through the fluid level at the top of the pitot tube. Similarly, this height here, the distance between these two lines is the velocity head, v squared over two g. And the height that that level rises to is the hydraulic grade line, which is dropping in this case. So, compute all these heights here and just convince yourselves that indeed the height of the energy grade line is constant, which it must be from Bernoulli equation. Now, let's look at some situations where we have head losses due to friction in the pipe. So, our Bernoulli equation in this case becomes this term where h f is the head loss due to friction. And the basic equation for computing headloss is the so called Darcy-Weisbach equation h f is equal to f l over d v squared over two g. Where f in that equation, is some function of the Reynolds number and the roughness of the pipe, and f is known as the friction factor. And in this equation r e is the Reynolds number, epsilon is a height of the roughness element or a measure of the height of the roughness element in the pipe material. And the ratio epsilon over d, the height of the roughness elements divided by the pipe diameter is called the relative roughness, relative roughness. And here is the corresponding section from the, from the manual which gives that equation. And generally speaking, friction factor is a function of Reynolds number and relative roughness. And that relationship is given by the Moody chart, which is this diagram shown here. Typical of that found in any fluid mechanics textbook and one of the most famous and important on charts in fluid mechanics. So in this chart here, we have the Reynolds number on the horizontal axis, f, the friction factor, on the vertical axis. And then, each line here is a line of constant relative roughness, epsilon over d. Now, the, whether or not the flow is laminar or turbulent depends on the Reynolds number. If the Reynolds number is less than about 2,000, the flow is laminar. If the Reynolds number is between about 2,000 and 4,000, the flow is in a transition between laminar and turbulent flow. And if the Reynolds number is greater than 4,000, the flow is turbulent. So, for laminar flows in this region here, the friction factor depends only on the Reynolds number. And in that case, we actually have an analytical solution which I'll, I'll show later. In the turbulent region here to the right, the friction factor depends on the Reynold, Reynolds Number and the relative roughness. We can also apply that equation to non-circular conduits or ducts. For example, the rectangular shape here and in this case, I'll define a as being the cross-section area of the conduit. And the quantity which I'll call p the wetted perimeter is the length of the perimeter, which is in contact with the fluid. So in this case, if the pipe or conduit is flowing full, it's the total length of the perimeter there. And then we define a quantity called the hydraulic diameter which is four times the cross sectional area divided by the wetted perimeter. And then we can use our regular equations for a round pipe. In other words, the head loss is f l over d h v squared over two g, where we simply replace the diameter of the round pipe by the equivalent hydrolic diameter. So, the Reynolds number is now based on the hydraulic diameter and the relative roughness is also based on the hydraulic diameter. And the hydraulic diameter appears in the, in the Dossawise-Speck equation. So we can just, to a reasonable approximation. Approximate this as a round pipe with an equivalent hydraulic diameter and use the same equations. So, for this simple case for a rectangular pipe here, the cross sectional area is the width b times the height a. And the wetted perimeter is the total length here, which is two times a plus b. So, the hydraulic diameter is two a b over a plus b for that case. Now, generally speaking for all flows, the friction factor depends on the Reynolds number and the relative roughness. And we have the Dar, Darcy-Weisbach equation for laminar flow. However, there is an exact solution to the equations of motion, which tells us that the friction factor is equal to 64 divided by the Reynolds number. Which only applies for laminar flow, and this particular flow is called the Hagen-Poiseuille flow, or more commonly just Poiseuille flow. And, if we equate this head loss to the pressure drop in the pipe, that leads to this equation combining with, these two equations together. That the volume flow rate, is pi d to the fourth times the pressure drop, divided by 128 mu times l, where mu's the viscosity and l is the pipe length. So, the volume flow rate you see is very sensitive to the pipe diameter, or conversely the pressure drop in a pipe of length l, is given by this equation as a function of flow rate. And this is the corresponding equation for laminar flow, the same equation given in the handbook. So, those are the basic equations and basic theorem for predicting head losses and pressure drops. In pipe flow, and in the next section we'll do some examples to illustrate that.
