Now I want to continue my discussion of the continuity and energy equations. And today we'll discuss the energy equations, the basic theory behind it. And the energy equation is more commonly known as the Bernoulli equation. And the Bernoulli equation related the variation of pressure, velocity and elevation in a flowing fluid. The basic equation which is an equation for consolation of mechanical energy for steady flow, in other words nothing is changing with time, and assuming no energy losses or additions is this. P over Gamma + V squared over 2 G + Z is equal to a constant along any straight line. In this equation the terms are P over Gamma is known as the pressure head. V squared over 2 G is known as the velocity head which is related to the kinetic energy of the flow. And Z is called the potential head which is related to the potential energy of the flow. So, the sum of these three terms is equal to a constant, in other wards, the mechanical energy is conserved. If we apply this equation between any two points on a stream line, for example, this stream line here, say between points 1 and point 2, then the equation becomes p1 over gamma, etc, is equal to p2 over gamma, etc. Or, here the two points could be here and here. This is the Bernoulli equation, and also, we have a little special terminology here. This streamline here that comes up to this point where the velocity is 0, any point in a flow where the velocity is 0 is called a stagnation point and the particular streamline that leads up to the stagnation point is called the stagnation streamline. This is the relevant section from the reference handbook, and for reasons which aren't clear, they refer to this equation here as the field equation. But, the more common name for it is the Bernoulli equation where the terms are as defined here. They also write the equation in another form here, which is equivalent, but the more common way for writing the equation in the secondary form is given here. P1 plus one half, over v1 squared plus gamma z1, is equal to a constant. So either of these two equations are acceptable. They're both equivalent, just written in slightly different ways. We can illustrate this principle by applying it to a simple syringe. Let's suppose that we have some fluid in the syringe for example water. And we apply a force f to the plunger here so we increase the pressure inside the syringe. As a result of which it issues from the orifice here. At 2, at a high velocity which is some time on the y side here, and then falls back. So, I can apply Bernoulli equation between this different points. First of all, I'll apply it from some point 1 within the syringe here to the nozzle at 2 so the equation is here. But then I can start to simplify it. If the cross-sectional area here in the syringe is very large, then the velocity V1 is negligible. At the here at station two, the air jet is in contact to the atmosphere therefore the pressure there is atmospheric pressure which is 0 gauge. So p2 goes out and if I neglect the small change in between 1 and 2, z1 and z2 are the same so they cancel out. So, we left with a very simple equation that p1 over gamma is equal to v2 squared over 2g. Similarly, I can apply bernoulli equation up to the top of the fountain here, at station three. So, again I apply the bernoulli equation from station one here, to station two at the top, or equivalently I could apply it from 2 to 3, it's the same thing. So the same approximations, v1 here is negligible. The pressure at the top here, p3, is 0, because it is atmospheric pressure. The velocity of the terminal rise height is 0. And if I measure elevations from the height of the nozzle tip, Z1 is 0. So I find, P1 over gamma is equal to Z3. So, combining those two equations, we have P1 over gamma is equal to V2 squared over 2g, is equal to Z3. In other words we have a conversion of P one over gamma here. The pressure head is converted into V2 squared over 2G the velocity head at the nozzle tip which is ultimately converted to potential energy, Z3 at the top. So this illustrates that the Bernoulli equation is simply an equation for conservation of energy. In the reference handbook, they refer to the energy equation as the Bernoulli equation with an additional term here, HF, where HF is an energy or a head loss. Which in this case is typically due to friction in the pipeline. And that's what they refer to as the energy equation. If we apply this equation to a simple situation, like a converging pipe here, so in this case I have a converging pipe. And let's neglect the head losses so the h f term goes out here. If I assume that the pipe is horizontal, z1 and z2 are the same, so those terms go out. Where 1 is the upstream station, 2 is the downstream station. So we have P1 of the gamma plus V1 squared over 2g is equal to P2 over gamma plus V2 squared over 2g. In this case, the pipe is converging. In other words, the flow is accelerating, so V2 is greater than V1, so to satisfy this equation here, the pressure must be decreasing. In other words, P2, the downstream pressure, must be less than P1 the upstream pressure. And the heights that the water or liquid rises to in these static pressure taps is equal to from hydrostatics P1 over gamma or P2 over gamma here. So the liquid level in the static pressure top taps. In the case of a converging pipe is dropping like this, the pressure is decreasing. If we apply that now to a more complex situation where here I have a converging pipe and the elevation is also increasing, again I apply Bernoulli equation to the upstream station here, which is one, and the downstream station two. In this example, we also show the pipers converging. Therefore the pressure is dropping. The liquid level here in the static pressure taps is dropping. If I draw a line through here I get a line which looks like that. Each of these terms in the Bernouli equation has dimensions of length. Some of those terms, p over gamma plus v squared over 2g plus c, is equal to a length which is the height of the total head or the total energy of the flow. And the height that that elevation reaches is called the energy grade line, or the EGL, which is this line right here, is the energy grade line, EGL. And similarly, this quantity, P over gamma plus Z is called the piezometric or the piezometric head and the height that that rises to is called the hydraulic grade line, the HGL. We can see also that the difference or the distance between these two lines. So here's the HGL. The difference between those two lines is the local velocity head, v squared over 2g. In the case where we have no energy losses or gains in the system, the total head or the elevation of the energy grade line is constant. In other words the energy grade line is horizontal. The hydraulic grade line though, this line here can either rise or fall depending on whether the pipe is converging or diverging. In the next section, we'll do some examples of application of the Bernoulli equation.
